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Averaged equations for electrical potential and ion transport in brain tissue Mah, Christopher Derrick 1989

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A V E R A G E D EQUATIONS FOR ELECTRICAL POTENTIAL AND ION TRANSPORT IN BRAIN TISSUE by C H R I S T O P H E R D E R R I C K M A H A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Depar tment of Mathemat ics Institute of A p p l i e d Mathemat ics W e accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A u g u s t 1989 ® Chris topher Derr i ck M a h , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MATHEMATICS The University of British Columbia Vancouver, Canada Date SEPTEMBER 28th 1989 DE-6 (2/88) ABSTRACT The accurate modelling of bulk passive properties of neural tissue is essential to the modelling -of macroscopic phenomena in the brain such as spreading depression and epilepsy. Properties which characterise the passive and active flows of ions or electric current through tissue are referred to as transport properties. Such properties associated with passive flows include bulk conductivity, bulk diffusion coefficient, and those associated with electrically mediated ionic flux which is called 'spatial buffer flux'. While models for such transport properties of cortical tissue have been published, each of these models contained different assumptions about the structure of the tissue. Recent data on potassium transport through neural tissue are important for the construction of a unified model (i.e., based on a consistent set of assumptions) because they provide measurements of the amount of bulk electric current passing through cell membranes. In this thesis the Nernst-Planck equation is used as the governing equation for ion transport and electric potential, with specification of the jump conditions at the cell membrane. An asymptotic expansion and averaging procedure is -described which reduces the computation of bulk properties to a calculation for a single cell. The idea of transport numbers (a proportionality constant between ion transport and electric field vectors) in electrolytes is introduced and it is shown that this idea applies to bulk tissue. Estimates of the coefficients in the averaged equations are computed numerically for different geometries and a range of microscopic parameter values including cell size, membrane conductance, intracellular conduct! vitj-, extracellular space fractional volume. An important finding is that theoretical transcellular current, i.e., the bulk current flow through disconnected cells, is significant and relatively ii insensitive to several of these parameters, in particular cell size and membrane conductance. The role of electrotonic parameters (the parameters involving electrical constants) in the tissue model is discussed and a formal analogy between transcellular current and electrostatic polarization is introduced as an aid to physical understanding of the transport properties of arrays of disconnected (physically separated) cells. Asymptotic analyses of the electrotonic parameters are performed in order to supplement the numerical solutions with qualitative results, and it is shown how to incorporate asymptotic assumptions about these parameters into an asymptotic model. The properties of steady solutions to the averaged equations are discussed and it is shown that some coefficients of the equations cannot be estimated in a steady experiment. It is argued that the general model proposed here is simpler and more appropriate than cable theory for bulk tissue. For example, it is concluded that specialized transfer cells are unnecessary 'to -explain transcellular flux and spatial buffering, that disconnected cells cannot be neglected, and that cells of differing sizes may contribute .significantly to transcellular flux. Since transcellular flux is significant and insensitive to geometry and intracellular conductivity in our model, our results imply that spatial buffering occurs very generally. This model is chosen to include most measurable quantities such as extracellular potential and extracellular concentration, and to be mathematically simple. Since it is shown that the bulk parameters of the model are relative^ insensitive to many of the microscopic parameters of the tissue, the resulting governing equations should be applicable to many physiological iii situations. iv T A B L E O F C O N T E N T S Table of Contents v L i s t of Tables v i i L i s t of F igures ix Abstrac t ii Acknowledgement x i i I. General Introduction 1 1.0. Objectives 1 2.0. Physiology and Physics Develop in Para l le l 5 3.0. Neurons and G l i a 7 4.0. B u l k Tissue Properties 13 4.1. Electr ical Properties 13 4.2. Ion Transport Due to Dif fusion and Spat ia l Buf fer ing 18 5.0. N e u r a l Model l ing 25 5.1. Spreading Depression 30 6.0. Outline of Thesis 34 Glossary 36 II . The Model Equations 40 1.0. Introduction •. 40 1.1. Transport Equations in Electrolyte Solution 40 1.2. L imitat ions of the Nernst -Planck Equations 43 2.0. The Non-dimensional Model Equations 47 2.1. Scal ing of the Transport Equations 47 2.2. Asymptot ic Assumptions 50 2.3. Model Equations 55 2.4. J u m p Conditions and Boundary Conditions 58 2.5. Continuity and Smoothness Conventions 62 2.6. The Mathemat ica l A p p r o a c h 65 III. Asymptot ic Expans ion 66 1.0. Introduction 66 1.1. Expans ion 67 2.0. A v e r a g i n g 71 2.1. Introduction 71 2.2. A v e r a g i n g Procedure 74 Appendix I I I .A. Operator Expansions 82 Appendix III .B. Second Order Perturbations of the Dependent Variables 85 Appendix III .C. General A v e r a g i n g 88 I V . Canonical Problems and the Computat ion of Bulk Properties 93 1.0. Introduction 93 1.1. Transport N u m b e r s in Electrolyte 93 1.2. Discontinuous Ionic F l u x in B u l k Tissue 95 v 1.3. L o c a l Transport N u m b e r in B u l k Tissue 97 2.0. F i n a l Simplif ication of the B u l k Equat ion 99 3.0. Computat ion of Coefficient Est imates 101 3.1. Introduction 101 3.2. General Properties of Solutions 103 4.0. N u m e r i c a l Methods 119 4.1. Choice of Numer ica l Method 119 4.2. N u m e r i c a l A l g o r i t h m 123 5.0. Biological Parameter Selection 131 6.0. Parameter Studies 134 6.1. Computat ion of Coefficients f rom Canonical Problems 134 6.2. B u l k Conductivity and F l u x Proport ional to Electr ic F i e l d 136 6.3. Effect of Intracellular Conductivity on B u l k Conduct ivi ty 141 6.4. Tissue Structure 144 6.5. Ionic F l u x T e r m s Proportional to Concentration Gradient 155 6.6. Ionic F l u x Terms Proportional to N e r n s t Potential Gradient ... 161 Appendix I V . A . Consistency Checks on the Coupled Solution 164 Appendix I V . B . Uncoupled B u l k Conduct ivi ty 173 V . The Role of Electrotonic Parameters in Tissue Models 186 1.0. Interpretation of the Model 186 1.1. Introduction 186 1.2. F o r m a l Analogy Between Intracel lular F l u x and Electrostatic Polar izat ion 187 2.0. Asymptot ics in the Electrotonic Parameters 189 2.1. Correspondence w i t h Canonical Problem 189 2.2. Electrotonically Short Case and Transmembrane Transpor t 191 2.3. Electrotonically L o n g Case and Transmembrane Transport 196 2.4. Choice of Scal ing 199 3.0. Neurons , G l i a and Electr ical Scale 202 V I . S u m m a r y and Biological Conclusions 205 1.0. The Asymptot ic Approach 205 2.0. Biological Conclusions 207 2.1. Introduction 207 2.2. Properties of the Averaged Steady Equations 208 2.3. Magni tude of Spat ia l Buf fer ing 211 3.0. Implications For a Model of Tissue Structure 213 3.1 . Transfer Cells A r e Unnecessary 213 3.2. Disconnected Cells Cannot Be Neglected 215 3.3. Tier-structure M a y Be Important 216 4.0. Compar ison W i t h E m p i r i c a l Properties of B u l k Tissue 218 4.1. Introduction 218 4.2. Significance of Transcel lular C u r r e n t 218 4.3. Scale Effects 219 5.0. L imi ta t ions 220 5.1. Transport N u m b e r Simplif icat ion 220 5.2. Tortuosity and Geometry Assumpt ions 221 6.0. S u m m a r y 222 v i Appendix VI.A. Recent Literature 223 References 225 vii List of Tables Table II.2.1. Extracellular Ionic Concentrations 48 Table II.2.2. Characteristic Dimensional Parameters 56 Table II.2.3. Dimensionless Parameters 56 Table II.2.4. Dimensionless Variables 57 Table II.2.5. Corresponding Sign Conventions at Membrane 59 Table IV.4.1. Unit Cell Lengths 134 Table IV.6.1. E , , D,, and D 2 by Cell Length 140 Table IV.6.2.A. E , Coefficient Versus Intracellular Conductivity 144 Table IV.6.2.B. F , Coefficient Versus Intracellular Conductivity 144 Table IV.6.2.C. F 2 Coefficient Versus Intracellular Conductivity 146 Table IV.6.3. Fluxes due to 0 and C Perturbations 152 Table IV.6.4. Coefficients in (III.2.17) for Two-Tier Model 152 Table IV.6.5.A. D , and Array Geometry: One-Tier Model 153 Table. IV.6.5.B. D 2 and Array Geometry: One-Tier Model 153 Table IV.6.5.C. D , and Array Geometry: Two-Tier Model 154 Table IV.6.5.D. D 2 and Array Geometry: Two-Tier Model 154 Table A . l . A . Maxima of X and the Average a 0 M w { t (1+ V v -,)} 168 •p w o w p J_ Table A. l .B . Maxima of X p and the Average a 0 M w { t K t a ( l + V wX p l)} 169 Table A.2.A. Maxima of X y and the Average a 0 M w { t a (t^ + V wX y l)} 170 Table A.2.B. Maxima of xy and the Average a o M ^ t j ^ (t^  +VwXyl)} 170 Table A.2.C. Maxima of Ky and the Average aoM^tj^t^ t|3VwKvl} 1 7 0 Table A.3.A. The Average v^OoM^Jt V X -,} m u W o w cl Table A.3.B. The Average ^ ^ { M ^ t ^ t ^ x ^ } } • 171 Table A.3.C. The Average ^ { M w * (d C; v „ K . i + W 1 7 2 W 10 W C1 viii Table A.3.D. Maxima of x 172 c Table A.3.E. Maxima of K 172 c Table VI.2.1.A. One-Tier Coefficients in Equation (VI.2.4) 212 Table VI.2.1.B. Two-Tier Coefficients in Equation (VI.2.4) 212 Table VI.2.2. Lower Bound for C Coefficient 214 uu ix List of Figures Figure 1-4.1. Spat ial Buffer ing 23 F i g u r e II-2.1. Tissue Models 53 Figure T V - 3 . 1 . A r r a y Geometries 105 F i g u r e IV-3 .2 .a Xp^ <t> Perturbat ion Proport ional to eV<j>0 107 F igure IV-3 .2 .b X c : # Perturbat ion Proport ional to e V C 0 : 108 F igure IV-3.2.C X y : <t> Perturbat ion Proport ional to e V V 0 109 F igure IV-3 .2 .d KQ: C Perturbat ion Proport ional to e V C 0 110 F igure IV-3.2.e K ^ : C Perturbat ion Proport ional to eV</>0 I l l F igure IV-3 .2 . f K Y : C Perturbat ion Proport ional to e V V 0 112 F igure I V - 3 . 2 . g Xp^ <P Perturbat ion Proport ional to eV<j>0 113 F igure I V - 3 . 2 . h x^ 4> Perturbat ion Proport ional to e V C 0 114 F igure I V - 3 . 2 . i x y : <f> Perturbat ion Proport ional to e W 0 115 F igure IV-3.2.J K^: C Perturbat ion Proport ional to e V C 0 116 F i g u r e IV-3 .2 .k K^: C Perturbation Proport ional to e V 0 o 117 F igure IV-3.2.1 KV: C Perturbat ion Proport ional to e W 0 118 F igure T V - 4 . 1 . Conductivity Distr ibut ion in the U n i t Cel l 125 F igure I V - 4 . 2 . Locat ion of Points in Difference Formulas 128 F igure I V - 6 . 1 . Ce l l Size and Bulk Conduct ivi ty . 139 F igure I V - 6 . 2 . Intracellular Conductivity and B u l k Conductivity 143 F igure I V - 6 . 3 . A Two Dimensional Two-Tier M o d e l 147 F igure IV-6 .4 . A Two-Tier Conductance Study 150 2 F igure I V - 6 . 5 . Terms Proportional to V C 158 2 F igure IV-6 .6 . Terms Proportional to V C and Tier Structure 160 F i g u r e I V - 6 . 7 . E 2 Coefficient and Model Structure 163 x Figure IV-6 .8 . E 2 Coefficient and Conductance Frac t ion 166 Figure I V - B . l . Cel l Size and Bulk Conductivity 175 Figure I V - B . 2 . Intracellular Conductivity and Bulk C o n d u c t i v e 177 Figure I V - B . 3 . A Two-Tier Conductance Study 180 F igure I V - B . 4 . Extrace l lu lar Space Fract ion and Bulk Conduct ivi ty 183 Figure I V - B . 5 . A r r a y Geometry and B u l k Conductivity 185 xi A C K N O W L E D G E M E N T I thank my supervisor, Dr. Robert M. Miura, -for his constant support and •encouragement, for his careful reading of my thesis, and for suggesting the general topic of spatial buffering. I thank the members of my supervisory committee, Drs. Ernest Puil, Uri Ascher, and Peter Vaughan, for their advice during the critical early stages of this work and for their comments on drafts of the thesis. I am grateful to Dr. Vaughan for serving ably as my supervisor during Dr. Miura's absence. Finally, I must thank numerous friends at the University of British Columbia for moral and material support of every possible variety. xn ' When the mind wills to recall something, this volition causes the little [pineal] gland, by inclining successively to different sides, to impel the animal spirits toward different parts of the brain, until they come upon that part where the traces are left of the thing which it wishes to remember; for these traces are nothing else than the circumstance that the pores of the brain through which the spirits have already taken their course on presentation of the object, have thereby acquired a greater facility than the rest to be opened again the same way by the spirits which come to them; so that these spirits coming upon the pores enter therein more readily than into the others'. R. Descartes (1664) Passions of the Soul. Part I, Article 42 xiii ' The concept of the bra in cell microenvironment rests on a triadic relationship, as yet very incompletely defined, between neuron, gl ia , and the encompassing extracellular space. G l i a remain the least categorized element and for this reason are sometimes included as constitutents of the microenvironment and sometimes not .... A m b i g u i t y of definition is desirable in our present ignorance because too much rigor would stifle imaginative conception. The bra in cell microenvironment does indeed resemble that of a social environment, such as a c i ty, where both structure and space play constantly v a r y i n g roles in the total ambience.' : C . Nicholson, (1980) Dynamics of the bra in cell microenvironment. Neuroscience Research P r o g r a m Bul le t in , V o l . 1 8 , no. 2, p l 8 5 . x iv ' The postulate that the transfer cells form a syncyt ium is not strictly necessarj' . Independent transfer cells wi th processes that overlap wi th their neighbours by distances much greater than their electrical space constants would behave in essentially the same w a y . ' A . G a r d n e r - M e d w i n (1983 b) A n a l y s i s of potassium dynamics in m a m m a l i a n brain tissue. Journa l of Physiology, Vo l .335 , p397. xv I. GENERAL INTRODUCTION HO. OBJECTIVES Studies of the brain from a var ie ty of points of view, e.g., experimental , theoretical, microscopic, macroscopic, vertebrate, invertebrate, functional , s t ructural , chemical , electrical, etc. have enormously increased our understanding of m a n y different bra in phenomena . ! C u r r e n t research on excitable cells seeks to understand the sub-microscopic processes controlling the permeation (passage) of ions through the membrane which imbue the membrane w i t h so called active  properties. The behavior of single ion conducting channels is being studied in experimental electrophysiologic molecular biology, and by theoretical means. F r o m a more general point of view these studies are valuable for at least the following reasons. F i r s t l y , the study of the bra in ul t imately m a y i l luminate the nature of thought and behavior, as Descartes realized (see page xi) . Secondly, the medical treatment of pathological conditions such as epilepsy requires a basic scientific understanding of brain function. Since the operation of the bra in depends upon phj 'sical and chemical mechanisms, the physics and chemistry of the nervous system are relevant to thought and behavior, to medicine, and to the principles of bra in function. The present work is intended to advance the understanding of the physics of bra in function. t F o r a general introduction to neurophysiology, the reader is referred to a reference such as K a n d e l & Schwartz , (1983). Whi le it is not possible to provide a detailed introduction to neurophysiology here, m a n y terms which m a y not be fami l iar to every reader w i l l be defined. Despite Nicholson's remark (see page iii) , which referred to nomenclature rather than mathematical definitions, definitions are essential to construction of models. Biological terms which are s tandard, but which m a y not be fami l iar to mathematicians such as anion, membrane, etc. are underlined and defined i n the glossary which appears at the end of this introductory Chapter . 1 I. General Introduction / 2 In this thesis a mathematical model of electrical potential and ionic concentrations in m a m m a l i a n nervous tissue is derived, as much as possible, from 'first principles. The result ing equations are analysed, solved numerical l j ' , and the physiological implications of the analysis are discussed. O u r -ultimate a i m is to produce a single model for the macroscopic electrical and ionic properties of neural tissue, so that results on tissue conductivity, ion transport , cell swell ing and electrical potential can be correctly incorporated wi th in the same model. Here , we derive and solve (approximately) an equation for the membrane potentials of a collection of electrically discontinuous potassium permeable cells of differing physical and electrotonic lengths, subjected to an electric field, and spatial ly v a r y i n g potassium concentration. M a n y of the interesting aspects of bra in function, the most important example of which is behavior, occur at a macroscopic scale, involving the joint act ivity of m a n y cells. Unders tanding the physical processes under lying membrane potential changes in individual cells is generally recognized to be inadequate, in itself, to account for learning, memory , and the computing capabilities of the bra in (Kandel & Schwartz , 1983, . . p l l . 2 3 ; Lashle j r ,1950) . For this reason .it is desirable to study the properties of large numbers of cells in bulk. The cells which are thought to be important i n most present theories of bra in function are neurons (Kandel & Schwartz , 1983; Hebb, 1955,1958). While the active properties of neurons are seen as the p r i m a r y mediators of bra in function, m a n y significant manifestations of active properties such as neuronal f i r ing rates are sensitive to the rest ing - t ransmembrane potential. The resting transmembrane potential is determined to a great extent by the prevai l ing electrical potential gradients w i t h i n the tissue and the extracellular concentration I. General Introduction / 3 of potassium. Thus , an essential pre l iminary to the study of active properties i n •bulk tissue is an understanding of the factors affecting the electrical potential gradients and extracellular potassium concentration. The present work is a theoretical study ,of these factors. The work w i l l be directly relevant to several types of experiment; namely those which involve electrical b ra in st imulat ion, measure ionic concentrations i n tissue, measure effective bulk physical properties of tissue, such as impedance or diffusion coefficient, or involve electrically mediated potassium transport (which is discussed in detail below). F r o m an understanding of the factors influencing these parameters , (indirect) inferences m a y be d r a w n as to the electrical and ionic 'microenvironment ' or ambient conditions experienced by neurons (Nicholson, 1980) in vivo and hence about processes of direct physiological interest. A l o n g wi th the modelling of ion homeostasis, the determination of bulk current-voltage relations is an essential step i n developing accurate continuum models of gross phenomena in the nervous system, such as spreading cortical depression (Tuckwell & M i u r a , 1978) and epilepsy (Prince, 1978). The use of mathematical models in modelling .microscopic properties is wel l established (Hodgkin & H u x l e y , 1952 d). In the present chapter the classical (microscopic) model of the neuron is described and we indicate how basic physics as embodied i n the cable equation (described below) has influenced theory and experiment. The relat ively new field of macroscopic neural modelling is briefly described, w i t h its connections to neuroanatomy and the bra in cell microenvironment. If successful, the theory of tissue properties presented here w i l l play a role in macroscopic modell ing analogous to the role of cable theory in discussions of the neuron. I. General Introduction / 4 The general mathematical problem is to determine the bulk average flow of an ion which flows according to the Nernst-Planck -equations in an inhomogeneous medium containing periodically placed inclusions. Inside these inclusions the ionic concentrations and electrical conductivity are different from the surrounding medium, and the inclusions are surrounded by barriers (membranes) across which jump conditions are satisfied, relating the sizes of the discontinuities in electrical potential and ionic concentrations to the flux across the barrier. The solution of this mathematical problem will be applicable to the flow through tissue of permeating ions. Related abstract mathematical problems are discussed in Bensoussan et aL, (1978). There, it is shown how to construct formal multiple scale solutions to these related problems and the convergence of these expansions is proved under various assumptions. The application of such techniques to determining the average properties of inhomogeneous media is called homogenization. The application of the technique to the Nernst-Planck equations, and to the type of inhomogeneous medium described here is new, however. Exposition of the details of this reduction, solution, and interpretation of the mathematical problem and its application to ionic homeostasis form the substance of this thesis. There are some novel physical features in our derivation. The final averaged equations for the membrane potential are non-linear. Each biological cell generates current through the extracellular space because of variations in the ionic Nernst potentials along its length. When the flux lines of these (or other) current sources pass through the membranes of adjacent cells, ions must enter/leave the extracellular space, thus complicating the description of the ionic concentration profiles. I. General Introduction / 5 2.0. PHYSIOLOGY AND PHYSICS DEVELOP IN PARALLEL It will clarify the objectives to state our view of the relationship between neuroplrysiological phenomena and mathematical models of them, and to briefly review the history of this relationship. Rene Descartes (1545-1650) the French philosopher, was one of the first writers to formulate a neurophysiological model different from those of the Greeks. While his model was not mathematical, the model sketched in the introductory quotation (see page xi) is similar to modern neurophysiological models because of its dependence on physical mechanisms. Later, physiological models became more detailed in order to accommodate the measurements of a more mature physics and improved instrumentation technology. Once techniques had been invented to measure electrical phenomena, theoretical electrical mechanisms replaced the hydraulic physiological mechanisms postulated by Descartes, and theories began to be tested. In his memoir of 1791, Luigi Galvani described the response of a frog nerve-muscle preparation to electrical stimulation from a sparking machine, atmospheric electricity, and a bi-metallic arc. The galvanometer was invented by Ampere and Babinet (1822); and the 'action current' of muscle and nervet was discovered by du Bois-Reymond in the 1840's (e.g., du Bois-Reymond (1848,1849)). Theoretical developments in physics also influenced the development of neurophysiology. Maxwell's Treatise on Electricity and Magnetism appeared in 1873, and the Nernst-Planck equations for diffusion of charged particles in an electric field were formulated about 1890. The availability of these equations was t In modern terminology, the 'action potential' or 'nerve impulse'. I. General Introduction / 6 a factor in the development of several theories of ionically mediated bioelectric phenomena by Nerns t (1899), Cremer (1906,1909), and Bernstein (1912) in the early part of this century. The roles of ions and of electricity in physiology are inseparable. The dissociation of electrolytes in water into charged species called ' ions' was demonstrated by A r r h e n i u s in 1883. Theories of bioelectric phenomena prevalant since the late nineteenth century depend on the potassium ion, ( K + ) , and other ions such as sodium, ( N a + ), and chloride, (Cl ) (Biedermann, 1895; Os twald , 1890; D o n n a n , 1911). The apparently disparate topics of tissue electrical properties and ion transport are int imately intertwined from the physiological point of v iew. In the post-World W a r II period, H o d g k i n and H u x l e y (1952 a-d) used a mathemat ica l model to describe the action potential as a regenerative change in ionic permeabil i ty of nerve cell membrane and this model is accepted today in most essential features. Their theoretical work was confirmed by detailed measurements using the (then) new technique of the voltage clamp, and clever experimental protocols. T h u s , since the eighteenth century, mathematica l theory in physiology has advanced i n tandem w i t h physical theory and measurement techniques. Recently, the invention of ion-selective micro-electrodes has made possible the in vivo recording of variations i n extracellular ionic concentrations wi th in nervous tissue (Zeuthen, 1981). To complement this development, a more detailed and rigorous mathematical theory of ion transport in inhomogeneous media would be valuable . Fo l lowing A r i s (1978) a mathematical model m a y be defined as: 'any I. General Introduction / " 7 [mathematically] complete and consistent set of mathematical equations which is thought to correspond to some other entity, its prototype. ' B y enforcing mathematical consistency among the relations of the model it is possible to summarize empirical relations economically, discover sources -of inconsistency, and to formulate new testable hypotheses. A mathematical model applicable to neurophysiology w i l l serve these purposes for neurophysiologists. Therefore, an applicable model must be tractable, and logical completeness does not entirely determine its value. However , it is necessary to accommodate relevant physical measurements which are .presently possible. 3.0. NEURONS AND GLIA Biological cells consist of cytoplasm and cell organelles surrounded by a l ipid membrane . t Anatomists of the eighteenth century believed that the bra in was glandular and considered nerves to be ducts which conveyed the secretions of the bra in to the periphery. The foundations of modern neuroanatomy were laid by Santiago R a m o n 3' Cajal and Camil lo Golgi in the nineteenth centur}' (Cajal, 1892; Golgi , 1906) who (with others) developed the histological techniques such -as ! the silver impregnation method (which allowed the visual izat ion of an individual nerve cell in a tissue slice containing m a n y cells) and the conceptual foundation which led to the modern view of the neuron as the p r i m a r y active element in bra in function. Neurons are excitable cells, and they are able to t ransmi t information t The topics of these sections are technical, but treated in standard modern textbooks (e.g., K a n d e l & Schwartz , 1983; Jack et ah , 1975). M a t e r i a l to be used here w i l l be given in self-contained form, but w i t h relat ively little commentary . I. General Introduction / 8 down the long process of the neuron, called the axon, by means of propagating action potentials. The central nervous system of humans consists of some 10 ^ neurons of which 10 are in the cerebral cortex. Neurons are surrounded by satellite cells, g l ia in the bra in , and other morphologically distinct cells in the peripheral  nervous system. Centra l gl ia are classified by morphology and location into astrocytes, oligodendrocytes, microgl ia , and ependymal cells and outnumber central neurons by about nine to one. The functions of gl ia have not been completely elucidated and have been the subject of increased speculation and experiment (Varon & Somjen, 1979; W a l z & H e r t z , 1983). G l i a contribute a third to a hal f of the total intracellular volume, and are found in close association with both blood vessels and neurons. Because astrocytes typical ly possess m a n y radiat ing processes they contribute a substantial fraction of the large membrane surface area separating the intracel lular and extracellular spaces in bra in tissue (Hertz, 1982). t Oligodendrocytes are physiologically important because they form the n ^ e l i n which coats the axons of central neurons; however, data on the differences between the physical properties of astrocytes and ^oligodendrocytes are only recently becoming available (Hertz, 1982; Pevzner, 1982; K e t t e n m a n , et a l . , 1984 b), and data about other types of gl ia are sparse. The functions of g l ia other than astrocytes and oligodendrocytes have not been established. Because of this lack of data we w i l l not differentiate between types of gl ia in this thesis. Ions do not easily cross a l ipid membrane; however, neural membrane contains pores which selectively permit the passage of certain ions such as K + and N a + , and C l . A pore which selectively passes potassium ions is called a 1 , 2 t g T h e surface volume ratio i n m a m m a l i a n bra in has been estimated at 5n per H of tissue (Horstmann & Meves , 1959). I. General Introduction / 9 potassium channel and a pore which selectively passes sodium ions is called a sodium channel. The abundance and properties of different membrane channels determine the permeability of the membrane to each ion. The states of these channels, and hence, the membrane permeabilities, may depend on the electrical potential difference across the membrane, as discussed in Section 5.0. In addition, the states of some membrane channels are governed by chemical factors released by neurons at synapses during synaptic transmission. The passage of cations (positively charged ions) out of the cell, or anions (negatively charged ions) into the cell constitutes an outward electric current. For neurons and glia the intracellular potassium concentration, [ K + ]., exceeds the extracellular potassium concentration, [ K + ] q , (where the subscripts i and o refer to intracellular and extracellular concentrations, respectively), and vice versa for + + [Na ]. and [Na ] . Because the concentrations of the ions inside and outside the 1 o cell are different, they tend to move across the membrane, carrying electrical charge with them. The resulting transmembrane electrical potential (inside potential minus outside potential) opposes the chemical gradient due to the concentration differences. A resting transmembrane electrical potential, V , is attained when the net transmembrane current remains zero. If a membrane is permeable to only one ion, the transmembrane potential Vj associated with zero net transmembrane current is given by the Nernst equation: (3.1) V. = EL ln( C° / C1. ) i z.F i i where I. Genera l Introduction / 10 C° is the extracellular concentration and C 1 . is the intracel lular concentration 1 1 of the ion and z. is its ionic valence, t I W h e n the membrane is permeable -to several ions, the transmembrane potential is m o r e difficult to calculate. In .this case, the physics are approximately governed by the Nernst -P lanck equations discussed in Chapter II, however, the internal parameters of the membrane and the correct physical model for the membrane are not certain (Plonsey, 1969; M c G i l l i v r a y & H a r e , 1969). Such difficulties cannot be resolved by theoretical analyses alone. F o r this reason, we assume (Hodgkin & H u x l e y , 1952 a-d; K a n d e l & Schwartz , 1983) that the rest ing transmembrane potential satisfies the Goldman-Hodgkin-Katz equation (Goldman, 1943; Hodgkin & K a t z , 1949): (3.2) V R = ^ l n ( P i [ N a + ] Q + P 2 [ K + ] q + P , ^ ] . } P , [ N a + ]. + P 2 [ K + ] . + P 3[C1 ] Q where R is the gas constant (joule K ^mol "S, F is F a r a d a y ' s constant, T is the absolute temperature, and the subscripted quantities P . , i = 1,2.3 are the permeabilities of the membrane to N a + , K + , and C l , respectively. This equation is empir ical ly correct for the squid axon, (Hodgkin & K a t z , 1949) which is the classical physiological model for nerve membrane, and m a y be derived heurist ical ly f rom the Nernst -Planck equations assuming that the electric field wi th in the membrane is constant and separate ionic fluxes are uncoupled. W h e n the permeabil i ty to sodium, P 1 ? and to chloride, P 3 , are zero, equation (3.2) t Equations are numbered consecutively w i t h i n Sections. Thus , the first equation of Section 3 of Chapter I is equation (1.3.1). Subsection numbers do not appear in equation numbers and the Chapter prefix is only used to refer to equations f rom other Chapters . I. General Introduction / l l reduces to the N e r n s t equation (3.1) for potassium.. This assumption about the sodium and chloride permeabilities is supported by most reports on glial cells (Varon & Somjen, 1979), and (3.1) approximates the rest ing potential for neuronal .membrane, presumably because P 2 is relat ively large. Al though the net transmembrane current at rest is zero, the membrane m a y admit net fluxes of K + , Na~*" , and CI . A steady concentration of these ions in the cytoplasm is maintained by the sodium-potassium pump; (Glynn & K a r l i s h , 1975), a complicated assembly of protein subunits in the membrane which exchanges external K"*" ions for internal Na"*" at the expense of metabolic energy. Transport of ions which utilizes metabolic energy is called active  transport , (Kandel & Schwartz , 1983). The distribution of anions between the cytoplasm and extracellular region m a y be determined by the distribution of cations. The net charge of the cell interior relative to the extracellular region is l imited by the smal l capacitance of the membrane, so that the total electrical charges of intracel lular .anions and cations .are equal. In the invertebrate nervous system, the distribution of ions in extracellular space is influenced by the presence of negative charges on molecules in the intercellular clefts. Because diffusion in the m a m m a l i a n bra in is unaffected by the charge of the diffusing ion, (Nicholson & Phi l l ips , 1981) and in view of other evidence (Gardner-Medwin, 1983a) this effect is not considered here. The distribution of specific anions, especially CI and H C O g , m a y be modified by an anion transport system (Kimelberg & Bourke , 1982) in some cases. We w i l l not model active transport of ions in this thesis, however. Unless otherwise stated i t is s imply assumed that the active transport f lux is chosen to cancel the net transmembrane fluxes of K + and N a + over the entire cell at the resting potential. " I. Genera l Introduction / 12 It is assumed that neural membrane is of two types. F i r s t , gl ial membrane, which has a fixed permeabil i ty to K + (Pape & K a t z m a n , 1972; Somjen & Trachtenberg, 1979) and exhibits a membrane potential given by (3.1), and second, neuronal membrane which ;has voltage- and time-dependent permeabilities to K + , N a + , and C l w i t h resting potential given by (3.2). The separate transmembrane ionic fluxes I. are assumed to follow the equations: (3.3) I i = S i( v ~ VP where I. is the -outwardly directed f lux of the i^1 ion, V is the transmembrane potential, V. is the N e r n s t potential for the i ^ ion, and g. is the ionic conductance of the membrane for the i^1 ion (mS). E a c h of I., g., V , and V . is , i n general, a function of space and time. F o r the case of gl ia l membrane i t is assumed that the g. are fixed, and equal to zero for al l ions except K + . F o r neuronal membrane, the g. w i l l var}' in a manner to be described. These assumptions about neural membrane are simplifications of the real membrane properties of .central ^neurons which are not completely known (Cri l l & Schwindt, 1986). F o r example, dendritic membrane has a significant permeabil i ty 2 + to the calc ium ion C a at some membrane potentials (Kandel & Schwartz , 1983). This permeabil i ty is not expected to have a large influence on potassium concentrations, which are the focus of this work. Techniques which permit voltage clamp experiments on isolated microscopic patches of membrane have led to the recent discovery of m a n y channels wi th v a r y i n g properties, (e.g. Sonnhof, 1987) and the detailed characterisation of those already known (Aldrich et aL , 1983). The attempts to understand bra in function have involved m a n y different I. Genera l Introduction / 13 types of experiment. The under ly ing purpose of such experiments is to understand physiological -processes i n vivo . However , for such understanding to be made precise in a mathematical model, it is necessary to obtain measurements of cell and tissue characteristics under controlled (or known) physical conditions, e.g., f rom in vitro experimentation. 4.0. BULK TISSUE PROPERTIES 4.1. Electrical Properties The f irst systematic study of the effect of electrical bra in st imulation was made by F r i t s c h & H i t z i g in 1870, on the motor cortex of the dog. The f irst recording of electrical act ivity from the bra in was reported by Richard Caton in 1875. The recent experimental l i terature on electrical currents in the bra in is too vast to summarize here, though ear ly references m a y be found in Braz ier (1961). In this section we discuss the models which have been used to predict the passive electrical rproperties of cells and tissue. The extensive use of electrophysiological methods as an investigative tool is an important reason to establish the bulk current-voltage relations of -cerebral tissue. F o r example, the bulk current-voltage characteristics of cerebral tissue must be k n o w n in order to compute the distribution of current injected dur ing st imulation experiments (Ranck, 1975), to compute the distribution of current sources in the cerebral cortex (Nicholson, 1973), or to interpret the impedance characteristics of neural tissue (Ranck, 1964). W h e n neurons are st imulated electrically w i t h sufficiently smal l voltages, action potentials are not generated. Under these circumstances, the intracel lular , extracel lular , and transmembrane electric potentials are described by the classical I. Genera l Introduction / 14 M a x w e l l governing equations for electricity in a conducting and/or dielectric medium. N e u r a l s ignall ing which depends on such phenomena is called electrotonic transmission. Electrotonic transmission in neurons or other cells depends on cell geometry, membrane properties, and electrical properties of the extracellular and intracel lular media . The use of the cable equation (described below) as a model for electrotonic transmission in the neuron is well-established (Jack et ah , 1975). Generalizations of the cable equation have been discussed for branching structures, (Rai l , 1959, 1969) bundles (Clark & Plonsey, 1970, a, b), and syncyt ia (Jack et a l , 1975). The cable model and its generalizations are useful and are the motivation of much experimental work on the microscopic electrical properties of neural tissue (Pellionisz & L l i n a s , 1977; Johnston, 1980; Stafstrom et a L , 1984; Turner & Schwartzkroin , 1984). Because the nervous system is complex both microscopically and macroscopically, modelling of bulk tissue properties has not "been attempted often (Ranck, 1964; H a v s t a d , 1976). The lack of theory for the bulk electrical properties of tissue is part icular ly noticeable. For example, the f i rs t modern review of the distribution of s t imulat ing electrical currents is due to J . B . Ranck (1975). Genera l results on the bulk electrical properties of neural tissue (apart f rom ion transport) are useful in themselves (cf. Nicholson, 1973) because of their application to such experiments. Cerebral tissue is an inhomogeneous medium of considerable complexity. In part icular , conductivity varies w i t h the direction and (more subtly) w i t h the length scale on which it is being measured (Ranck & Bement. 1965; Nicholson & Phi l l ips , 1981). Th is is because tissue elements which are sufficiently extended I. General Introduction / 15 relative to the distance between electrodes w i l l alter impedance measurements in a complicated manner related to this distance. Several treatments of the conductivity of cellular tissue have .previously been published, (Ranck,1963; Nicholson, 1973; Nicholson & F r e e m a n , 1975; H a v s t a d , 1976; Eisenberg et ah , 1979; G a r d n e r - M e d w i n , 1983; ). These treatments have modelled the effects described above in various w a y s . The two m a i n assumptions used in electrophysiological models for the electrical potential in bulk tissue are the syncyt ia l assumption (Jack et ah , 1975) and the assumption of infinite membrane resistance (Nicholson, 1973). It is also possible to discuss effective bulk conductivity from an empirical point of v iew, so that no specific assumption about current through cells is made (Nicholson & F r e e m a n , 1975). A s discussed later in this chapter, however, current through cells is an important part of the present work. In electrophysiology, the term syncyt ium refers to a collection of distinct biological cells whose intracel lular regions are electrically continuous with each other, possibly due to gap junctions. The syncyt ia l assumption is known to be val id for invertebrate glia (Varon & Somjen, 1979). Equations describing the electrical potential in syncyt ia l tissue were discussed by Jack et ak, (1975), Eisenberg et a L , (1979), M a t h i a s et ak, (1979), and Peskoff (1979). Syncyt ia l tissues have been modelled by three-dimensional versions of cable theory assuming the extracellular and intracel lular spaces are extensively interdigitated. The existence of a glial syncyt ium in m a m m a l i a n cortical tissue is st i l l controversial . While intercell junctions have been observed (Varon & Somjen, 1979) and there is electrophysiological evidence of coupling between some gl ia l I. Genera l Introduction / 16 cells (Somjen, 1984; Schoffeniels et aL_, 1978), studies of bulk properties do not suggest that there is extensive coupling in m a m m a l i a n tissue (Hounsgaard & Nicholson, 1983; G a r d n e r - M e d w i n , 1983 a, b). In view of the complex nature of cerebral tissue, it is desirable to investigate the statement that isolated cells can behave like a syncyt ium and to investigate alternative models for bulk tissue. The assumption of infinite membrane resistance implies that extracel lularly generated currents cannot pass through neurons and gl ia , which therefore form opaque obstacles to current f low. The problem of determining the electrical potential in tissue under this assumption is formal ly equivalent to the problem of determining the steady concentration profile of a non-permeating ion in neural tissue. Both the electrical potential and steady concentration satisfy Laplace 's equation in the extracellular space and no f lux conditions at the cell membranes. I f the classical equations of electricity govern the extracellular potential and membrane resistance is infinite, then according to porous media theory, (Gray & Lee, 1977) the bulk electrical potential m a y be described by an equation i n which the effective conductivity is a tensor o (Nicholson, 1973) and the extracel lular electrical potential <j> satisfies (4.1) I ff 114 = i P=I P a y where a , p = 1,2,3, are the constant components of a, i is a current source density, (possibty due to the currents generated by action potentials) and the potential 0 is an average, defined in some appropriate w a y . Equation (4.1) is shown later to be val id under our more general assumptions when extracellular I. General Introduction / 17 ionic concentrations are constant, and the averaging procedure is specified .precisely. The explicit dependencies of a on geometry, the conductivities of the intracel lular and extracellular media , and membrane resistance have not been calculated. The assumption of infinite membrane resistance has been useful in the interpretation of field potentials (Nicholson & F r e e m a n , 1975) because the fraction of extracellular current which passes through cells is s m a l l in m a n y experimental paradigms. However , evaluation of the anisotropy of conductivity measurements in m a m m a l i a n cortex (Gardner-Medwin, 1980) suggests that electric current passes through cells in sufficient quantity to appreciably influence the bulk resistance of neural tissue. Most important ly , because electrically mediated potassium transport occurs chiefly by means of current flow through cells, it would be inappropriate to employ the assumption of infinite membrane resistance in the computation of ion transport properties. For example, in Gardner -Medwin ' s current passing experiment, the smal l fraction of imposed current which passed through cells apparently accounted for a significant potassium flux. The formulation of more general systematic physical models for tissue electrical potential might help to resolve difficulties in the interpretation of electrophysiological data in bulk tissue (cf. Somjen & Trachtenberg, 1979). F o r example, i t is not k n o w n whether high or low membrane resistance of the cells (neurons or glia) might favor electrically mediated ion transport in tissue which is not syncyt ia l . I. General Introduction / 18 4.2. Ion Transport Due to Diffusion and Spatial Buffering In this thesis the term " transport" usual ly refers to the f lux of some conserved quanti ty , such as net electric charge or an ionic species, in response to a gradient of intensity (e.g., potential , concentration) of that quantity (cf. Batchelor, 1974). Whi le electrophysiology dates f rom the mid-nineteenth century (du Bois -Reymond, 1848; Helmhol tz , 1850,a,b) the concomitant measurement of ionic effects has become possible only recently. Exper iments wi th squid axon to measure transmembrane ionic fluxes us ing radioactive tracers date f rom the 1950's (Hodgkin & Keynes , 1957), while useful ion selective microelectrodes became available only in the 1960's (Zeuthen, 1981; Nicholson, 1980). Thus , some of the electrophysiological l iterature does not explicit ly treat ions and it is indeed possible to obtain theoretical predictions about electrical properties of tissue without including ionic effects. Recently, it has become clear that wi th in .the m a m m a l i a n central nervous system, physiological states such as spreading depression (Leao, 1944; Grafs te in , 1956) and seizure activity ( Moody et ah , 1974; F isher et a l , 1976; -Futamachi et ah , 1979) can give rise to spatial variat ions in extracellular [ K + ] . These spatial variat ions can v a r y in their characteristic spatial wavelengths f rom several m m to . 5 m m (approximately hal f the length of a Purkinje cell arborization). Thus , these variat ions have spat ia l wavelengths which are long compared to most cortical cells. Potass ium release occurs during nervous act ivity due to K"*~ efflux f rom neurons dur ing the repolarization phase of action potentials. Spat ial gradients in depth presumably develop because K~^ release is p r i m a r i l y f rom the cell bodies I. General ^Introduction / 19 of neurons, which are concentrated in part icular layers of the cortex (Futamachi et a l , 1974; Moody et a l , 1974; Sypert & W a r d , 1974). Spat ia l gradients paral le l to the surface of the cortex may occur because of an advancing wave of spreading depression, -which is a lways accompanied by drastic changes in extracellular ion concentrations. The potassium released under the conditions described above is cleared f r o m bulk tissue by several mechanisms, including active transport , diffusion, and spatial buffering. The diffusion of ions i n neural tissue is different f rom diffusion in a medium without cells because ions do not move freely across cell membranes. Solution of the diffusion equation for non-permeating ions is complicated by the tortuous geometry of the m e d i u m . In the physiological l i terature (Nicholson & Phi l l ips , 1981) diffusion of non-permeating ions has been described by the equation: (4.2) D , V 2 C + •£ = 3C A a "St 'where D is the diffusion coefficient, a is the extracellular space volume fraction, Q is a source density and C is an averaged local extracellular concentration. The constant X is a dimensionless geometrical factor called the tortuosity factor. Equat ion (4.2) is not the most general equation for diffusion in an inhomogeneous medium, because it has been assumed that diffusion is isotropic. A s s u m i n g that the diffusion equation holds extracel lular ly for a non-permeating ion, the theory of porous media (Lehner, 1979; G r a y & Lee, 1977) shows that equation (4.2) is a correct description of steady diffusion in a geometrically complicated m e d i u m , provided that the average is taken in an I. General Introduction / 20 appropriate w a y . The averaging procedure is precisely specified, and equation (4.2) can be derived using .this procedure i f there are no electrical forces and the ions are non-permeating. The explicit relation between X and the geometry and parameters of the medium has not been determined in a physiological context. The constants a and X have been determined empir ica l ly , however, for non-permeating ions in bra in tissue (Nicholson & Phi l l ips , 1981). The factor X is approximately 1.5, and a about .2, in m a m m a l i a n cerebellum and cerebral cortex. Because potassium ions K + cross cell membranes under rest ing conditions, equation (4.2) is not appropriate for accurate computation of potassium spatial transport. The approach taken i n this work w i l l lead to the derivation of a more appropriate governing equation. The gradients of [K + ] Q described above give rise to electrical currents which cause an electrically mediated transfer of K + f rom regions of high [K + ] 0 to regions of low [ K + ] q , a transport mechanism called spatial buffering. Electr ical ly mediated spatial transport of potassium was f i rs t described by Orkand et aL , (1966) and is usual ly attributed to gl ia l cells. In detail , the mechanism may be explained as follows. Gl ia l and resting nerve membranes are predominantly permeable to K + . Thus , extracellular [K + ] is the p r i m a r y factor determining the local transmembrane potential (cf. (3.1)). W h e n the transmembrane potential varies along the length of an electrically continuous elongated cell , the longitudinal voltage gradient causes current flow through the cells and extracellular space. A displacement of the membrane potential V toward zero is called a depolarization and a displacement of the membrane potential toward more negative potentials is called a hyperpolarizat ion. Higher [K~^] corresponds to I. Genera l Introduction / 21 depolarized transmembrane potentials, and lower ^ + ] 0 ^° hyperpolarized transmembrane potentials relative to the rest ing value. W h e n a single cell is exposed to a spatial gradient of [K + ] o the local N e r n s t potentials (3.1) become different at different points of the cell. Since a return ,path for the current exists v i a the intracel lular and extracel lular media, a closed current loop is formed which passes inward through the cell membrane at one point, goes through the intracel lular space, passes outward through the membrane at another point, and f inal ly completes the loop through the extracellular medium (see figure I-4.1).t A s a result of this current f low, the true transmembrane potentials differ sl ightly from the Nerns t potentials; and these differences w i l l be precisely calculated in subsequent chapters. In addition to the currents described, extracellular current m a y be imposed on tissue due to the mass f i r ing (production of action potentials) of m a n y neurons (Nicholson & F r e e m a n , 1975) or by an experimental current generator, and these currents m a y result i n t ransmembrane currents which enter cells at one location and leave at other remote locations. W h e n electric current flows in a l iquid medium containing ions (electrolyte) the •flow of charge is due to the movement of ions through the medium. Charge -1 -2 -1 -2 f lux or current is defined by (mol sec cm or amp sec cm ) (4.3) 1 = J C - J A where the f lux of cations, JQ , and of anions, J ^ , are the s u m , respectively, of the signed fluxes of each cation and anion in the solution. The symbol 3 w i l l be used to denote a f lux which consists of the weighted sum of several ionic fluxes, t The f irst figure of Section 4 of Chapter I is numbered 1-4.1, and so on. I I. General Introduction / 22 Figure 4.1. The spatial relationships between extracellular K concentration, depcjilarization/hyperpolarization of biological cell membranes, and the flux of K , Na , and Cl ions results in the transport of K from regions of high to low concentration. ELEVATED [K+! LONERED [K+] E L a u r e 4 .1 I. General Introduction / 24 while "j w i l l be emploj'ed to denote a specific ionic f lux. I f electric current in an electrolytic medium consisted entirely of the f lux of potassium, then by conservation of current, no concentration changes of potassium would occur along the current path. However , only about 1.2% of extracellular currents consist of potassium flux (Gardner-Medwin, 1983). Extrace l lu lar currents consist main ly of sodium and chloride fluxes, while transmembrane and intracel lular currents consist main ly of potassium f lux. Hence, in the current loop described above, a net efflux of potassium occurs f rom the extracellular space at one location and a net influx occurs at others, w i t h the result that local + is altered. Other sources of extracellular current such as those due to an experimental source maj^ also modify [K +] Q. Such spatial transport of potassium (or other ions) by electric current w i l l be referred to as electrically mediated spatial transport. Reference to F igure 4.1 shows that when current loops result f rom spat ia l gradients of [ K + ] o , the effect of electrically mediated spatial transport is to reduce ['K^ ~] where it is high and to increase [ K + ] Q where i t is low. Therefore, this 1 phenomenon is called spatial buffering. To estimate the magnitude of these effects, A . G a r d n e r - M e d w i n (1983 a, b; G a r d n e r - M e d w i n & Nicholson, 1983) performed experiments i n which electric current was passed through bra in tissue and the consequent movement of K + ions was measured. Current was passed perpendicular to the cortical surface through f luid contained in a circular cup placed on the cortical surface. The potassium contents of the cup was monitored during current passage. It was verif ied by means of intracellular recording and application to the cortex of tetrodotoxin ( T T X ) , a pharmacological agent which suppresses the generation of I. General Introduction / 25 action potentials by acting upon sodium channels, that the observed effects were independent of any effect on action potentials of current passage. For some experiments, the extracellular potassium concentration, [K + ] q , was monitored at various depths in the cortex 'beneath the cup. Finally, the results were interpreted according to a theoretical model. The results were consistent with the conclusion that electrically mediated potassium (spatial buffer) flux is about five times the diffusive flux for potassium distributions varying over distances much greater than 200 nm in the rat brain. The theoretical model was necessary because spatial buffering and diffusion both result from spatial gradients of [ K + ] q , so that the results of such an experiment are confounded, and theoretical assumptions must be made in order to attribute a definite fraction of the potassium flux to either diffusion or electrically mediated flux. The theoretical model for the tissue electrical potentials used by Gardner-Medwin to interpret data was the cable model. It was assumed that intracellular current flowed through a coupled cell population in the cortex, called 'transfer cells', and that intracellular current consisted of potassium flux. However, the cell .population which was supposed to be the substrate of the cable equation was not identified. Thus, it was necessary to estimate the parameters of this putative population from the data. This procedure, while useful, does not indicate the relationship between the observations and independent measurements of the microscopic parameters of cell populations. 5.0. N E U R A L M O D E L L I N G In this section we summarize Hodgkin and Huxley's model of the action potential. The relation between models for active and for electrotonic properties of nerve is illustrated for the case of Hodgkin and Huxley's model. This is expected I. G e n e r a l , Introduction / 26 to be instructive in evaluat ing the present work , though it is wel l -known to the biological reader. While the Hodgkin-Huxley equations were original ly developed for the squid giant axon, they represent the classical physiological model of the action potential , which is assumed (Cri l l ,& Schwindt, 1983, 1986) to -apply to the neurons in m a m m a l i a n central nervous system in some possibly generalized form. F o r example, dendritic membrane exhibits action potentials based on the voltage dependent calcium permeabil i ty (Kandel & Schwartz , 1983). B y means of a series of experiments Hodgkin and H u x l e y , (1952a-d)t were able to formulate quantitative descriptions of the quantities g. in equation (3.2) for the squid axon. It was found that the transmembrane current could be satisfactorily described as the sum of the capacitive current, 1^,, the potassium current Ij^, the sodium current, I j ^ a , and the current carried by chloride and other ions, Ij, called the leakage current. Thus , the total t ransmembrane current _2 I (uraA c m ) is given by: (5.1) I = I c + I R + I N a + I r The ionic conductances of (3.2) satisfy a set of ordinary differential equations w i t h coefficients depending on the membrane potential V . A n action potential is a rapid (lasting ca. 1 msec) regenerative depolarization of the neuronal membrane as a result of a complicated set of changes i n the membrane conductances. It is characterized by a rapidly r is ing phase in w h i c h the membrane sodium conductance increases dramat ica l ly , and a t F o r an elementary account, see A i d l e y , 1978, or m a n y other standard texts. This model contains explicit hypotheses on the non-linear voltage and time dependent 'active' properties of neurons. I. General Introduction / 27 somewhat less rapid return to equilibrium in which the sodium conductance decreases and potassium conductance increases. In general, excitable membranes contain voltage dependent sodium channels (Aidley,1978). From the experimentally determined formulas describing the ionic conductances, it was possible to reconstruct the action potential, assuming that Vj^, V j ^ a > and Vj are constant. When the membrane potential V is constant in space, the associated action potential is called 'space clamped' and does not depend on the electrotonic properties of the axon or cell. To calculate a propagating action potential, a model for the spatial dependence of the potential is needed. For this purpose, Hodgkin and Huxley employed the core conductor model of Hermann (1879), also known as the cable theory t It is assumed that the axon is an elongated cylinder of uniform cross section aligned with the Z axis, and only the axial coordinate is considered. Since the axon is a three dimensional object, radial current flow must occur, so that these assumptions are artificial in some respects. The model equations of cable theory are: (5.2) m (5.3) 1° = - I 1 t t f . The original cable equation of classical physics is due to Kelvin. I. General Introduction / 28 (5.4) M = - r , 1° oz t (5.5) | / = - r 2 l [ where 1° is the axia l current flow in the external medium, I 1 , the axia l current flow in the internal medium, i is the transmembrane current per unit length, 0° is the potential in the external medium, <j>1 is the potential in the internal medium, r^ is the resistance of the external medium per unit length, and is the resistance of the internal medium per unit length, A standard argument us ing (5.2) - (5.5) yields: (5.6) i = H*. where V = 6l-d>°. For a fibre of radius a, i = 27ral and r„ = RV7ra^ m 2 where R 1 is the specific resist ivi ty of the cytoplasm (Q cm). The relation between r^ and R ° the resist ivity of the external medium is complicated because the three-dimensional current flow is not ax ia l . Because R ° is smal l however, Hodgkin and H u x l e y assumed R ° = 0, and thus obtained: (5.7) I - ™ = a ^ 27ra 2R1 I. General Introduction / 29 The val idi ty of the assumptions (5.2) - (5.5) has been investigated experimental ly by Hodgkin and Rushton (1946), and Lorente de N o (1947),t and mathemat ica l ly by Clark and Plonsey (1966). Tt is found that these assumptions are approximately correct for isolated preparations . In addition, the use of the cable model for the intracellular and transmembrane potentials (but not the extracel lular potential) can be mathemat ica l ly justified by a three-dimensional analysis of the electrical potential (Clark & Plonsey, 1966). A steady propagating action potential must have the form V = V(z-c?t) , where 6 is the conduction velocity. The propagating action potential , which satisfies an equation obtained by equating (5.1) and (5.7) has a propagation speed (or wave speed) t? 0 which is determined from the equations by a numerica l iteration procedure. The propagation speed and the dependence of the action potential on z - dt depend upon the simplification (5.7) and the values of a, the fibre diameter, and Cjyj- the membrane capacity. Despite the necessity to estimate these parameters , and the various s impl i fy ing assumptions, the predicted propagation speed (18.8m/s) of the action potential (Hodgkin & H u x l e y , 1952d) closely matched the observed speed (21.2m/s). The use of cable theory was an essential step of this calculation, though the cable model for electrotonic properties is s imple. The role of cable theory in the model of the action potential shows that (microscopic) electrotonic properties of neurons play an essential role in their physiology and it m a y be anticipated that bulk passive properties w i l l be important in the formulation of an}' macroscopic model. t I a m indebted to D r . E . P u i l for pointing out this fundamental ear ly reference. I. General Introduction / '30 5.1. Spreading Depression Spreading depression (SD) was discovered by the physiologist Le'ao (1944) dur ing studies of experimental epilepsy in the cerebral cortices of rabbits. S D is characterised by a marked and prolonged -depression of the spontaneous electrical act ivity of the brain . Subsequently, S D has been observed in m a n y brain structures and species (Bures et ah , 1974). In addition to the electrical changes, later studies revealed large changes in the normal ionic equi l ibr ium of the bra in . K r a i g and Nicholson (1978) measured changes in the extracellular concentrations of several physiological ions during S D in the cerebellum, namely , an elevation in the concentration of potassium from 4 m M to 40-50 m M , a fal l in the concentration of sodium from 80 m M to 60 m M , a fal l in the concentration of calc ium from 1.2 m M to .12 m M and a fal l in the concentration of chloride f rom 85 m M to 50 m M . Spreading depression has been observed in cerebellum, olfactory bulb, hippocampus, and in vitro in the slice preparation (Somjen & A i k e n , 1984). This section is intended to indicate what factors must be included in any model of S D which accounts for its observable effects. Nei ther a review of the S D literature nor an evaluation of competing hypotheses is intended. A satisfactory model must include the factors which interact to produce the p r i m a r y observable effect, but it need not include epiphenomena which accompany the m a i n phenomenon. The factors producing the m a i n phenomenon w i l l be referred to as mediat ing factors. L i k e the nerve impulse, the experimental phenomenon exhibits a wavelike character but with a (much slower) wave speed of 1-9 m m / m i n . Also it exhibits recovery since the electrical act ivity and ionic concentrations of the tissue return I. General Introduction / 31 to their original levels after several minutes. A detailed comparison of the S D wave and the nerve impulse ma}' be found in M i u r a (1981). These changes m a y be initiated by a var iety of s t imul i , including electrical s t imulat ion, mechanical s t imulat ion, and the local application of K C 1 . The pronounced and robust nature of S D affords a unique opportunity to study the interaction of electrical and ionic mechanisms w i t h i n neural tissue at a macroscopic level. Several hypotheses have been advanced as to the mediat ing factors of S D . D u r i n g the 1950's it was known that extracellular potassium concentrations increased greatly during S D (the magnitude of the increase was later measured to be 40 - 50 m M , in some preparations (Nicholson & K r a i g . 1981)). Grafste in (1956) hypothesized that spreading depression was due to the spread of extracellular potassium, which depolarized neurons and caused further potassium release by means of action potentials. A t that t ime, the role of other released factors such as glutamate, which is an important intracellular anion, (Puil , 1981) was not understood. V a n H a r r e v e l d (1978) postulated that the principal chemical can be potassium, or the neurotransmitter L-glutamate , result ing in two distinct mechanisms for S D . However , the phenomenon does not depend crit ical ly on the generation of (sodium) action potentials. The S D wave is not antagonized by treatment of the cortex w i t h tetrodotoxin (TTX) (Sugaya et ah , 1975). T u c k w e l l and M i u r a (TM) (1978) formulated a simplified mathematical model for S D , t which accounts for the essential features of the phenomenon t O u r calculations are not intended exclusively to extend the S D model. A detailed evaluation and extension of this model from a physiological point of view, would require experimental work , which is beyond the scope of this thesis. T u c k w e l l and M i u r a ' s work is cited here as a paradigm for macroscopic modelling of neurophysiological phenomena. I. General Introduction / 32 (depression of spontaneous bra in act ivi ty , wavelike propagation, ionic concentration changes, and recovery) by incorporating several physiological mechanisms, wel l -known through microscopic studies, into a set of equations. Tuckwel l and M i u r a hypothesized that conductance changes in post-synaptic membrane rather than action potentials are responsible for the release of potassium by neurons dur ing S D . The increase in extracel lular potassium leads to depolarization of neuronal membrane according to equations (3.1) and (3.2). This depolarization leads to the entry of calcium into pre-synaptic terminals which causes release of neurotransmitter (Katz , 1969; Krn jev ic , 1974). The neurotransmitter (likely L-glutamate) acts at post-synaptic (and pre-synaptic) receptor sites, changing the conductivities of post-sj'-naptic neurons and leading, to further potass ium release. It is not necessary to specify the identity of the neurotransmitter wi th in the mathematica l model, and it is possible that other neurotransmitters , such as acetylcholine might play the same or a s imi lar role. According -to this model, propagation of the S D wave depends on the spatial transport of potassium from the site of the init ial s t imulus to remote regions of the extracellular space. The wave speed predicted by this model is about l m m / m i n , which is at the lower end of the observed wave speeds of l - 9 m m / m i n in m a m m a l i a n cortex. This discrepancy is smal l , given the s implic i ty of the model, but because m a n y of the other parameters of the model were obtained by f i t t ing the data for the observed wave , the model wave speed might be expected to depend p r i m a r i l y on the diffusion coefficient of potass ium, which is accurately known (Horvath , 1985). Hence the discrepancy l ikely results f rom the omission of some mediat ing factor, and not from inaccuracy in the numerical parameters . I. Genera l Introduction / 33 Because potassium transport is the basic factor in the propagation of the S D wave in the T M model, refinements of this model might na tura l ly begin wi th a more complete account of potassium transport in the cortex. Whi le a tortuosity factor was used to account for the effect of geometry on the effective diffusion coefficient for K + the T M model contains no specific assumption (analogous to cable theory) concerning the effect of geometry of the extracel lular and intracellular spaces on the transmembrane potential . Therefore, the formulat ion of a model for this relationship is a second natura l refinement of the S D model. Several empirical correlates of S D are not presentty k n o w n to be mediat ing or epiphenomenal. The observations of Nicholson .and K r a i g , (1981) suggest that S D is accompanied by massive movement of an (unknown) anion f rom the intracel lular space. The mechanism for this putative anion movement is not k n o w n and it is possible that this ion m a y be one of several organic anions, including glutamate. It is found (Van "Harreveld & Khat tab , 1967) that S D is accompanied by a dramatic reduction of the extracellular space due (at least part ly) to swelling of the dendrites in the affected areas. It is l ikely that this swel l ing is osmotic and is caused by a net f lux of salt (anion and cation) into the intracel lular space. Since the osmolari ty of intracellular space must be preserved, water enters cells, causing an increase in intracel lular volume. A t the same t ime, the electrical resistance of the cortex rises dramat ica l ly during S D ( V a n H a r r e v e l d & Ochs, 1957; Ranck, 1964;). The role of these concomitant factors in S D has not yet been clarif ied, and some theory would be required to obtain the relation between cell swell ing, elevated tissue resistance, t ransmembrane potentials, and spat ia l buffering wi th in I. General Introduction / 34 the tissue itself. The calculations performed in the present work do not apply to the extreme conditions which prevai l during S D , however, thej ' are compatible w i t h the observed correlation between cell swell ing and the changes in tissue resistance. Elevations in the extracellular potassium concentration m a y produce slow field potentials by depolarizing the membranes of gl ial cells (Kuff ler , 1967; Somjen, 1975). A relation between extracellular potassium, gl ia , and slow field potentials is strongly suspected on experimental grounds. Extrace l lu lar potassium concentration m a y be involved in certain types of neural signall ing (Lebovitz, 1970) and a var ie ty of metabolic effects (Krnjevic & M o r r i s , 1981), even though it may not be the single most crit ical factor in every case (Somjen, 1984). A theory of potassium transport and the relation between extracellular, intracel lular , and transmembrane potentials is a basic prerequisite for accurate macroscopic modell ing of S D and other phenomena, e.g., epilepsy. (Prince, 1978; C r i l l & Schwindt, 1986; Traub et aL, 1985 a,b) in which ion concentrations and extracellular potentials in bulk tissue m a y play a mediat ing role. 6 . 0 . O U T L I N E O F T H E S I S In Section II. 1.It we introduce the Nernst -Planck equation, which is the governing equation for ion transport and electric potential, and specify the j u m p conditions at the cell membrane. In Section II. 2, appropriate scalings for the equations are chosen. This section also deals w i t h the structure of the tissue model and methods for incorporating the j u m p conditions into formal calculations. In Chapter III, an asymptotic expansion and averaging procedure is t Chapters are referred to by capital r o m a n numerals and sections by arabic numerals separated by a decimal point. Thus Section II. 1.1 refers to subsection 1 of section 1 of Chapter II. I. Genera l Introduction / 35 described which reduces the computation of bulk properties to a calculation for a single cell. In Section I V . 1, the idea of transport numbers in electrolytic media (Horvath , 1985) is introduced and it is shown that this idea applies to bulk tissue. Coefficient estimates in the averaged equations are computed numerical ly for a range of microscopic parameter values including cell size, membrane conductance, intracel lular conductivity, extracellular space and geometry, in Sections I V . 3 - I V . 6. A n important f inding is that theoretical transcellular current, the bulk current flow through disconnected cells, is significant and relatively insensitive to man}' of these parameters , depending p r i m a r i l y on cell size and membrane conductance. In Chapter V , the role of electrotonic parameters (the parameters involving electrical constants) in the tissue model is discussed. Section V . 1 . 2 presents a formal analogy between transcellular current and electrostatic polarization as an aid to physical understanding of the transport properties of a r r a y s of disconnected cells. In Section "V.2, asymptotic analyses in the electrotonic parameters are performed in order to supplement the numerical solutions w i t h qualitative results, while in Section V . 3 it is shown how to build asj 'mptotic assumptions about electrotonic parameters into the model. In Chapter V I , we discuss biological implications of the analyses from the body of the thesis. The properties of steady solutions to the averaged equations are discussed i n Section V I . 2 and it is shown that some coefficients of the equations cannot be estimated in a steady experiment. It is argued that the general model proposed here is simpler and more appropriate than (syncytial) cable theory for bulk tissue. F o r example, Section V I . 3 . 1 concludes that specialized transfer cells are unnecessary to explain transcellular f lux and spatial I. General Introduction / 36 buffering, while Sections VI.3.2 - VI.3.3 conclude that disconnected cells cannot be neglected, and that tissue structure may be important. Section VI. 4 discusses the significance of current through cells and observed phenomena which may be affected by the length scales which are imposed by experimental observations. Limitations of our approach are discussed in Section VI. 5. Finalfy, Section VI. 6 briefly contrasts the present model with previous models. The present model is chosen to include measurable quantities such as extracellular potential and extracellular K"*" concentration and to be mathematically simple. Since it is shown that the bulk parameters of the model are relatively insensitive to many of the microscopic parameters of the tissue, the resulting governing equations should be applicable to many physiological situations. G L O S S A R Y Action potential , A rapid (lasting ca. 1 msec) regenerative depolarization of the neuronal membrane accompanied by a complicated set of changes in the membrane conductances. In the classical case, it is characterized by a rapidly rising -phase -in which the membrane sodium conductance increases dramatically, and a somewhat less rapid return to equilibrium in which sodium conductance decreases and potassium conductance increases. Active properties In Hodgkin-Huxley theory, properties associated with the voltage-sensitive membrane ionic channels. Active transport Net movement of molecules or ions, often between intracellular and extracellular spaces, which depends on metabolic energy. The use of the term "transport" in this phrase differs from the use of the term in the rest of the thesis. Anion A negatively charged ion. I. General Introduction / 37 Axon An extended tubular process of the neuron which projects to neighbouring neurons, sometimes over a considerable distance (lm or more). Capacitance The capability of storing electrical .energy by the separation of opposite electrical charges. Cation A positively charged ion. Central nervous S3',stem The higher portion of the nervous system, including the spinal cord, brain stem, cerebellum, basal ganglia, diencephalon and cerebral hemispheres. Cerebellum A large part • of the brain with motor functions, situated near the base of the brain and with a cellular architecture similar to cerebral cortex. Cerebral cortex The outer . layer of gray matter of the cerebral hemispheres, associated with higher perceptual, cognitive and motor functions and having a layered cellular architecture. Cytoplasm The jelly-like material surrounding the nucleus of a biological cell. Epilepsy A disorder of the nervous system which results when a large collection of neurons discharge in synchrony. Along with this discharge, stereotyped behaviors may occur, including convulsions. Excitable Capable of producing action potentials. Extracellular space The region exterior to biological cells within a tissue. Gap junctions Intercellular junctions at which cells are connected b)' transmembrane pores, permitting the exchange of intracellular molecules and electrical charge. Glia The passive interstitial cells of the central nervous system. Hippocampus A subcortical brain structure with a distinctive shape (the name is from the Greek for seahorse) and regular cellular architecture. It is associated with memory in man. Ion An electrically charged atom or group of atoms. Intracellular space The regions interior to biological cells within a tissue, taken together. In vitro A phrase used to characterize biological experiments made under artificial conditions, and not in a living animal. I. General Introduction / 38 In vivo A phrase used to characterize biological experiments or conditions in a living animal; as opposed to in vitro. Invertebrate The -biological phylum consisting of animals which lack backbones, such as the squid, leech, insects, etc.. Membrane A thin flexible sheet composed of lipid -molecules which forms the surface of biological cells or organelles, and which separates intracellular and extracellular regions. Metabolic Pertaining to the chemical processes occurring within a biological cell. Motor cortex A region of the cerebral cortex associated with motor function. Myelin Layers of lipid and protein substances composing a sheath around nerve fibers. Neuron The primary cell type of the nervous system, which is capable of producing action potentials. It consists of the nerve cell body and its processes, the axon, and other processes called dendrites. Olfactory bulb A region of the cerebral cortex associated with the sense of smell. Osmolarity The sum of the total concentrations of all solutes in a solution, so called because of the importance of this quantity for osmotic phenomena. Osmotic Pertaining to the process by which two different solutions which are mechanically separated (for example, by a membrane) tend to equalize their respective total solute concentrations by the flow of solvent from one to the other. Organelle A membrane-enclosed structure with a specific biochemical function within a biological cell. Passive A technical term referring to membrane properties which are not voltage dependent, as opposed to the active voltage- and time-dependent conductances of the Hodgkin-Huxley model. Peripheral nervous system The parts of the nervous system outside the central nervous system. Permeability The relative ease with which an ion passes through membrane pores. This may be measured by a mathematical coefficient. Preparation Part of an organism which is "prepared" in some way (in vivo or in vitro) which facilitates physiological study, e.g., of squid axon, frog muscle, etc.. I. General Introduction /*39 Processes Continuous anatomical extensions of a biological cell body, which may be tubular or sheetlike. Resting transmembrane potential The transmembrane potential across excitable membrane in the absence of stimulation, when action potentials are not occurring. Squid giant axon A physiological preparation of the axon of the squid Loligo , used because its large size facilitates experiments. Slice preparation A thin section of tissue, often from the cerebral cortex or hippocampus which preserves cellular architecture and function in vitro . Soma The cell body of a neuron, containing the nucleus of the cell. Synapse A specialized contact zone at which two neurons are closely apposed and where communication occurs by electrical or (typically) chemical transmission. Synaptic transmission The process by which the information of the action potential is transmitted from one neuron to another, typically through the release of a chemical factor which diffuses across the synapse. II. T H E MODEL EQUATIONS 1.0. INTRODUCTION In this chapter, the mathematical problem for determination of electrical and ion transport properties from a model of neural tissue containing two types of cell (neurons and glial cells) is specified precisely. The objectives of this mathematical formulation are to calculate the averaged electrical and ion transport properties of the tissue in a systematic way and to exhibit the dependence of macroscopic tissue properties on the the microscopic properties of cells. This provides a theoretical connection between studies of microscopic cellular properties, such as those described by Turner and Schwartzkroin (1984), and studies of macroscopic tissue properties such as those by Ranck (1963,1964) and Gardner-Medwin (1983 a, b). 1.1. Transport Equations in Electrolyte Solution We now sketch the physical chemistry thought to apply to --ion transport in solution (Horvath, 1985; Carnie & Torrie, 1984; Fuoss & Accasina, 1959) and justify the transport equations to be used in the remainder of this thesis, t This section is intended to justify the use of these equations here, rather than to derive them, since these derivations are described in elementary textbooks or in the voluminous literature of classical chemistry. The discussion of transport given in this section refers to electrolyte solutions such as those of the extracellular medium in neural tissue. In these basic electrochemical equations, averages are t As stated in Chapter I the term "transport" refers here to the flux of some conserved quantity such as net electric charge or an ionic species, in response to a gradient of intensity (e.g., potential, concentration) of that conserved quantity (cf. Batchelor, 1974). 40 "II . The Model Equations / 41 taken over length scales of 10 - 100 A 0 rather than 10 - 100 Mm as in the tissue equations 1.(4.1) - 1.(4.2). It is assumed that no convective transport occurs and the discussion is restricted to transport due only to diffusion and electrical forces. F r o m a theoretical point of v iew, the simplest transport process is diffusion. If there are neither electrical forces nor concentration gradients, then ionic and solvent molecules undergo brownian motions due to thermal energy, but have no average relative motion. In the presence of an ionic concentration gradient these random motions result in movement (diffusion) of ions from regions of high concentration to regions of low concentration. The result of the individual motions of the molecules and average f lux in aqueous solution (Robinson & Stokes, 1955) is precisely described by F ick ' s l a w : (1.1) " J D = - D . V C . , -3 -* where C is the ionic concentration (mol cm ), J.-Q is the ionic f lux vector in mol sec ^cm 2 , and D . is the diffusion coefficient (cm^ sec )^ for the ionic species. The description of ionic f lux in an electric field requires some assumptions, however. F o r reasons discussed below, it w i l l be assumed that the sum of the total concentration of cations mult ipl ied by their valence is equal to the sum of the total concentration of anions mult ipl ied by their valence in any non-zero volume. This assumption is referred to as 'electroneutrality' because it implies zero net charge wi th in any non-zero volume. Since variations in the D. ' s are unl ikely to be important under p l^s io logica l conditions of moderate variat ion in temperature and concentration, the diffusion constants D . w i l l be treated as II. The Model Equations / 42 empir ical constants depending on the identities of solvent and solute, and independent of concentration. U s i n g electroneutrality this assumption implies that Z z . D . V C j = 0, where z. is the valence of the I^1 ionic species, so that the net electric f lux due to diffusion is zero throughout the so lut ion . ! W h e n both electrical and concentration gradients exist, we assume that the total f lux "jj (mol sec ~ c m 2 ) of the i ^ 1 ion is given by the classical Nernst -Planck equation (1.2) " j . = - D . { V C . + z .C . L V.<j>}, J i il l i i R T The first term on the right-hand-side of (1.2) is a f lux proportional to concentration gradient and, therefore, w i l l be referred to as the diffusive transport term. The second term on the right-hand-side of (1.2) is proportional to an electric potential gradient and w i l l be referred to as the electrical t ransport te rm. U s i n g :.the above assumptions, summation of (1.2) gives the total diffusive mass flux vector: •+ 3 (1.3) J D = -Z DJVCJ, -* -1 -2 where the f lux (mol sec cm ) is a sum of diffusive fluxes. Mult ip l ica t ion of t Since F i ck ' s law is va l id for electrolytes, the coefficients D . , i = l , . . n , m u s t be consistent w i t h the possibility that (electrical current) z D V C + z D V C = 0 C C C 3. 3. 9. at the boundary for a solution composed of a single cation and anion, where C is the concentration of cations and C the concentration of anions, and z ancl z are, respectively, the valences of cation and anion. Because of electroneutrality, z V C + z V C = 0 throughout the solution. Combining the latter two equations implies either D = D , or V C = V C =0, so that the above result is true throughout the solution. II. The Model Equations / 43 -1 -2 (1.2) by the valence z. and summation gives the net current in mol sec cm (1.4) 1 = - F ^ V c ^ where a : = Zz. DjC. (mol sec cm ) will be referred to as mass conductivity where 'mass' refers to the dimensions of this quantity. It is shown below that this quantity, a, is proportional to the electrical conductivity of the solution. When the electrolyte considered consists only of uni-valent ions, Vo is just the total diffusive flux of ions. Thus, the form of the Nernst-Planck equations is simplified by assuming electroneutrality. Equation (1.4) asserts that under the stated assumptions, a spatially dependent conductivity K = K ® , (mS cm ) may be assigned to electrolyte solutions of given composition. The electric current ~ i may be written in the local form of Ohm's law as; (1.5) 1 = -KV<P, 2 where K :— F o/RT has the dimensions of conductivity. 1.2. Limitations of the Nernst-Planck Equations While the Nernst-Planck equations (1.2) are taken as the model equations, it is important to note that this entails some compromise. The Nernst-Planck equations (1.2) give each ionic flux independently of the others. For low concentrations or strong electrolytes, this prediction is born out empirical^ and is known as Kohlrausch's law of independent migration of ions. Equation (1.5) II. The Model Equations / 44 asserts that an electrolyte solution behaves ohmically and the effective conductivity is a l inear function of the concentrations of each ion. The ohmic behavior has considerable support while the status of the l inear i ty assumption is less certain. There is considerable experimental evidence (Horvath , 1985) that electrolyte solutions are ohmic (1 = K V $ ) ; however, at low concentrations ( < < . 1 M ) , electrolyte solutions exhibit conductivities given asymptotical ly by ; (1.6) K = C ( A 0 - b/C") where C is the ionic strength and AQ and b are constants independent of concentration. In addition to (1.6), there are a large number of other semi-empirical non-linear formulas applicable to part icular electrolyte types (Horvath , 1985). It is not clear which i f any of these formulas are applicable in the physiological context. A t physiological concentrations, the dependence of the conductivity of a strong electrolyte on concentration is approximately l inear. A s C —>0 in (1.6), K—> CA^, where K was defined in Section 1.1, so that AQ is the l imi t ing molar conductivity at infinite di lution. The quantity AQ m a y be approximately calculated from thermodynamic arguments. The theoretical value appears in the Nerns t -P lanck equations (1.2) and m a y be calculated using (1.4), (1.5), and (1.6). F o r example, for a solution of a single uni-univalent electrolyte (such as KC1): (1-7) A 0 = 2 F^D II. The Model ' "Equat ions / 45 where D is the diffusion coefficient for the electrolyte, and A 0 is computed using the definition of a w i t h two uni-valent ions. F o r self-consistency of the model, the assumed diffusion coefficients w i l l be the effective coefficients at physiological temperatures and concentrations, and solution conductivities w i l l be computed f rom these values. The result ing conductivities w i l l be less than the tabulated infinite dilution conductivities (Robinson & Stokes, 1955) but this procedure should produce more accurate approximations to the solution conductivity than the use of infinite dilution data , since it incorporates an empir ical correction for concentration. In principle, the individual ionic fluxes should be governed by non-linear formulas analogous to (1.6) and, due to non-linearity, each f lux must be a function of the local concentration of a l l the solute ions. However , in contrast to the experimental data leading to (1.6), these more detailed data are not available. In view of this s i tuation, i t is assumed in our model that the local ion fluxes are predicted -by the Nernst -P lanck equations. Because the conductivity of physiological solutions is determined largely by strong electrolytes, the non-linear correction which is neglected should be smal l . A theoretical account of the observed deviations of transport properties f rom those predicted by (1.2) is complicated. F i r s t , the Nernst -P lanck equations (1.2) do not specify the electric potential <j>. Since this equation describes the motions of charged particles (ions), a completely satisfactory model would determine <j> so that i t included the effect of the ionic charges. The correct resolution of this problem, using the principles of statistical mechanics is the subject of current research ( N . Patey, personal communication). The use of Poisson's equation to compute <f> is inappropriate because i t II. The Model Equations / 46 assumes that a smooth, stationary charge density is a va l id approximation to the collective effect of individual charges. Ionic charges of opposite sign rapidly cluster around one another, however, in an effect known as charge screening, so that the charge density is not smooth or stationary in time. In this thesis, we have accepted, w i t h others (Plonsey, 1969; Carnie & Torr ie , 1984), a classical treatment of this problem known as the Gouey-Chapman theory (Gouey, 1910; C h a p m a n , 1911). Gouey-Chapman theory consists of s tudying the properties of the Poisson-Boltzmann equation: 9 n 3 -z.q 0 /RT (1.8) V I = -% L z .C.e 1 e % 1 = 1 1 1 where q g is the charge of an electron, and is a dielectric permit t iv i ty (farads c m "*"). The Poisson-Boltzmann equation assumes the chemical solution has a bulk dielectric permit t iv i ty , e^, and identifies the -electrostatic potential wi th the corresponding thermodynamic potential. The reader is referred to Plonsey (1969) for mathematical details, and to Carnie and Torrie (1984) for the relationship of G o u e y - C h a p m a n theory to other theories. Phys ica l ly , the theory asserts that charge separation cannot occur over a large region because charge separation requires energies w h i c h are large compared to the available thermal energy. G o u e y - C h a p m a n theory predicts electroneutrality of the solution over any length scale larger than the Debye shielding distance. This distance is 9.6 A° units in .1 molar uni-univalent electrolyte solution (McGi l l ivray & H a r e , 1969; Plonsey, 1969). Thus , in this work the concentrations of anion and cation are taken as equal in any volume element, and no charge separation occurs in the bulk II. The M o d e l Equations / 47 solution, t 2.0. T H E N O N - D I M E N S I O N A L M O D E L E Q U A T I O N S 2.1 . S c a l i n g o f t h e T r a n s p o r t E q u a t i o n s In order to prepare the transport equations for asymptotic and numerical analyses, it is useful to choose non-dimensional variables w h i c h reflect the magnitudes of the physical quantities of interest. These choices are made both for reasons of numerical convenience, and in order to identify smal l non-dimensional parameters which m a y be used to construct asymptotic approximations to the ful l equations. We begin wi th a brief description of the electrical and ionic environment w i t h i n nervous tissue. In a common experimental preparation (mammal ian bra in slice) typica l extracellular concentrations of physiological ions are as shown in Table 2.1 (Ll inas & Sugimori , 1980). F o r comparison, typical concentrations of physiological ions in human cerebrospinal f luid (CSF) (Davson, 1976) are also 2 + 2 + shown. M a g n e s i u m , M g , and calc ium, C a , are also present in the bra in , in cerebrospinal f luid , and are included in experimental bathing solutions, at about 1 4 m M concentration, but they w i l l p lay no role in our transport equations 2 + because these ions contribute little to solution conductivity. The C a ion m a y play some role i n determining the rest ing membrane potentials of some neurons, however, this omission is not expected to quali tat ively affect the conclusions of the present work w i t h respect to potassium and electrical potential . Intracellular concentrations of ions are less certain but m a y be estimated t This assumption is not satisfied for ionic fluxes across biological membranes, and the treatment of f lux across membranes involves model equations different f rom those given here (Shultz, 1980; Plonsey, 1969). II. The Model Equations / 48 Table 2.1. Extrace l lu lar -Ionic Concentrations. Ion Slice C S F N a + 150 m M 147 m M K + 6.2 m M 2.86 m M C f 131 m M 113 m M H C 0 3 ~ 26 m M 23.3 m M ( m M = mill i -molar) as N a + (30 m M ) , K + (130 m M ) , C f (10 m M ) , and organic anions (150 m M ) (Llinas et ah, 1980), which consist mainry of glutamate and aspartate (Puil , 1981). Hence the intracel lular concentration of potassium, [ K + ] . , exceeds the extracellular concentration, [ K + ] o , by a factor of 20-40, and the extracellular concentration of sodium, [Na + ] o , exceeds the intracel lular concentration [Na + ]. by a factor of 5. Differences also occur for the other physiological ions. The •concentration n K + ] Q m a y v a r y considerably during abnormal physiological states, reaching 12 m M during epileptic act ivity (Prince, 1978) and 40-50 m M during spreading depression (Nicholson -& K r a i g , 1981). Spreading depression is also + - 2 + accompanied by large changes in the concentrations of N a , C l , and C a Tabulated conductivity data for electrolyte solutions give an estimate for the conductivity of the extracellular f luid of 20 m S c m ^ at 3 7 C ° which is in agreement w i t h the observed conductivity of cerebrospinal f luid (Nicholson, 1980). Correct values of the intracel lular conductivity are considerably less certain owing to the complicated morphology of neurons and the fact that axoplasm is not a simple electrolyte solution. Typica l measured values are between 1 to 4 times the res is t ivi ty predicted by the composition of the intracel lular medium (Barrett & II. The Model Equations /-'49 C r i l l , 1974; Carpenter et a l . , 1971, 1973; Schanne & R u i z P . -Ceret t i , 1978). U s i n g the concentrations cited above, this leads to an estimate for the internal resist ivi ty , r . , between 67-268 J2cm, while the resist ivity of somatic cytoplasm is even more variable (Schanne & R u i z P . -Ceret t i , 1978). Extrace l lular electrical potential gradients are usual ly between |V0| = 1 m V to 250 m V c m " 1 (Somjen, 1979). T y p i c a l length scales for cells in bra in tissue are L = 10 jum to 100 ( im. F o r example, dendrites m a y be 1 um in diameter, the cell body of a Purkinje cell m a y have a radius of 10 (im, astrocyte processes m a y extend 40 /um to 50 /um, while the complete arborization of a Purkinje cell may extend for 500 Mm (Hounsgaard & Nicholson, 1983). A t physiological temperatures and concentrations, diffusion coefficients for -5 -5 2 -1 KC1 and N a C l are approximately 1.7 x 10 and 1.3 x 10 c m sec , respectively, and at the physiological temperature, T = 37C, R T / F is 2 7 m V . Non-dimensional variables are chosen so that under typical conditions the magnitudes of relevant quantities are close to unity. A -convenient (though arbitrary) choice is to take the non-dimensional voltage gradient to be near unit}'. The -scaled spatial coordinate is defined as •x. = X j / L and the scaled voltage v!f> = F 0 / R T where vRT/F is the typical voltage variat ion over the length L and J" is a non-dimensional constant. Denoting a typica l ionic concentration by C , we obtain the non-dimensional concentrations; C. = C / C , which w i l l differ between I I ' intracel lular and extracellular environments. F o r the case of potassium, K + , the magnitudes of the terms in the Nernst -Planck equations m a j ' be roughly deduced as follows. E a c h nerve impulse + -9 -2 -1 releases K at approximately 2 x 10 m mole cm sec (Orkand, 1980), at a frequency between 1 H z and 100 H z . A t 100 H z the f lux associated w i t h this II. The Model Equations / 50 -5 -2 release rate corresponds to a current density of 2 x 10 amp cm . In contrast, an extracellular potential gradient of .1 V c m (Somjen, 1981) at an _5 effective conductivity t of .31mS leads to a current density of 3.1 x 10 amp -2 cm . Hence, the diffusive and electrical terms are of comparable magnitudes. Dimensionless variables are selected so that typical concentrations and voltage gradients w i l l be of order uni ty . Thus , i t is appropriate in the case of + -5 2 - 1 the potassium ion, K , to take L = 50 nm, D . = .85 x 10 cm sec , C = 5 m M , and *>RT/F = . 5 m V (that is . 5 m V 7 50Mm = 1 0 0 m V / l c m ) which implies v — .0185. Since v is smal l , the f lux associated w i t h a voltage gradient of order uni ty is smal l compared to the f lux associated wi th a concentration gradient of order unity. N o special use w i l l be made of this fact, but. it is necessary for later calculations that the electrical f lux be at most of the order of the diffusive f lux. It is important to note, however, that the magnitudes of the characteristic electric potential gradients on long and short length scales m a y differ because of fine structure in the tissue conductivity induced by the presence of cells. A s imi lar choice of scalings is appropriate for equations (1.1) - (1.5). It is convenient to scale a by d~ = Z D . C , while the variables x and <t> are scaled as before. 2.2. Asymptotic Assumptions In this work , we focus on results which are independent of detailed considerations of cell geometry and placement, because such results are more l ikely to be applicable to a var iety of different preparations. It is assumed for f The approximate effective par t ia l conductivity due to potassium i n the extracel lular medium. II. The M o d e l Equations / 51 convenience that the tissue contains a large number of periodically arranged cells. The periodic domains correspond to the smallest repeating subunits of the periodic structure of the tissue model, which w i l l be called crystallographic unit cells, t The assumption <of .periodicity and other assumptions made later about cell shape are convenient for computation. While the mathematical model employed could be used in two or three dimensions, it is convenient to perform the numerical calculations in two dimensions. It is expected that the properties of this abstract model, and properties of the real tissue, wi l l depend in s imilar w a y s on characteristic dimensionless parameters related to the extracellular space fractional volume, electrical properties of the intracellular and extracellular media, and cell size, since these model parameters can be matched to the real ones. It is assumed that there are two fine characteristic length scales; a fine length scale, which is characteristic of neuron size and a finest length, L 2 which is characteristic of glial cell size, (see Figure 2 .1 ) . Since i t . w i l l not be assumed that neuronal membrane has properties distinct f rom glia i n this model, •the 'neuronal ' cell "population could be any asymptotical ly larger cell population. This two-tier structure is chosen to model the structure of real tissue, which contains both coarse and fine structures of various kinds. F o r example, dendrites, and gl ial cell bodes and processes are expected to have finer spatial dimensions than neuronal cell bodies and axons (Peters et ah , 1976). The asymptotic expansion i n Chapter III w i l l be constructed using this assumption (referred to as a two-tier model). The calculation for a simple periodic a r r a y (one-tier model) forms a part of the two-tier calculation and this simpler model is considered i n t This term is borrowed from elementary chemistry. II. The Model Equations / 52 Figure 2.1. Periodic arrays of model cells can be arranged in different waj 's to match the properties of biological tissue. Possible shapes for the biological cell are shown, and assumptions about the spatial length scales L 0 , and are i l lustrated. The model tissue used later w i l l be in two dimensions (compare Figures IV-3 .1 and IV-6.3) . Tissue models ECS pothi incr*a>*d contlHvovt In length tltiu* >»••« <halch«d> FIGURE 2.1 II. The Model Equations / 54 many of the numerical calculations. While our results show that the two-tier model may be useful, further experimental data would be required to justify more detailed study of this model. Figure 2.1 illustrates the hypothesis that larger cells are aligned anisotropically, while finer cells are aligned isotropically. Such an alignment might be expected because neurons and glia have complicated branching structures, so that the finer structures tend to have less spatial organization. To completely justify this model however, it will be necessary to gather detailed quantitative data on the relative size, shapes, and positions of cells in neural tissue. In order to determine relationships between macroscopic and microscopic properties of tissue, we calculate the macroscopic properties of this tissue model over a coarse length scale, L = L 0 . The notation L 0 , L 1 ; and L 2 will denote asymptotic length scales which may correspond to various physical quantities. It 2 will always be assumed that L J / L Q = e and L 2 / L 0 = 0(e ) where 0 (») is the usual order notation. The extracellular and intracellular media are assumed to include onry the uni-valent ions, Na~*", K"*~, and CI . Neuronal and glial membrane are assumed to be permeable only to K + ions. This is correct for glial membrane and is approximately correct for resting neuronal membrane (Kandel & Schwartz, 1983, p40). The electric potential, 0, and concentration, C., are discontinuous across the cell membranes. The time-dependent equations for the electrical and ion transport properties of tissue are more complicated than those formulated here. Such equations would have included capacitive current and the time dependence of the membrane conductances. However, appreciable ion transport takes place only over times on the order of a second or longer. Thus, the time scale of interest (several II. The Model Equations / 55 seconds) is long compared to the time constant (a few milliseconds) for charging of the membrane capacity, or the time constant(s) for relaxation of the membrane conductances toward their steady state values. It m a y be shown by an elementary calculation that the charge or ion transport associated wi th charging of membrane capacity and relaxation of conductances is smaller (by a factor of about 1000) than the net charge and ion transport which can occur over several seconds. A characteristic t ime, t , of this order w i l l be chosen and c w i l l appear in the definitions of the non-dimensional variables. Thus , in computing the electrical and ion transport properties of neural tissue i t w i l l be assumed that the membrane charge and conductances have attained their steady state values. The extracellular ion concentrations w i l l be (slowly) t ime dependent, and the equations specifying this t ime dependence w i l l be given. Thus , a quasi-steady non-linear averaged equation for the evolution of the extracel lular potassium concentration is derived. The present analysis is different f rom previous analyses (see Chapter I) because it does not assume that the tissue is syncyt ia l , includes more than one type of cell , and explicit ly treats electrodiffusion of ions in the extracellular medium according to equation (1.2). 2.3. Model Equations Table 2.2 shows the dimensional parameters and Table 2.3, the dimensionless parameters to be used in the model equations. Table 2.4 gives the definitions of dimensionless variables used in the model equations. The tilde ~ denotes dimensionless variables in Tables 2.2 - 2.4, and w i l l be dropped in the text. Displayed equations are a lways given using the dimensionless variables of Table 2.4. II. The Model Equations / 56 Table 2.2. Characteristic Dimensional Parameters. Parameter Description L length L 0 measurement length L , fine tissue structure length L 2 finest tissue structure length C concentration (7 = ED.C mass conductivity t time c — — 2 S = a F / RT conductivity / cm Table 2.3. Dimensionless Parameters. Parameter Description v voltage gradient parameter 2— — T = L C I to time constant parameter 8.— D . / Z D ^ ionic diffusion parameters Table 2.4. Dimensionless Variables Description space coordinate time concentration Parameter x. = x . /L J J T = t/t c c. = c . / c 1 1 II. The M o d e l Equations / 5 7 4> 0 * i = F0 / * » R T = a / tr T 2 / -electric potential conductivity concentration source density - Z L ^ q . / a I ionic source density Q 2 = L L ^ z . q . / a i n i charge source density V V r = F V / v R T = F V /i>RT r t ransmembrane potential rest ing transmembrane potential ? i = L g i / S membrane conductance Pi = L p . / F a M active transport ionic f lux P i = Z P i active transport ionic f lux P 2 = E z . p . i i active transport charge f lux If "jj is the flux of the I" ionic species given by the Nernst -Planck equation and each ionic species is conserved, it is shown (by a derivation identical to that for the diffusion equation (Carrier & Pearson, 1976) wi th ~\-substituted for the diffusive flux) that: (2.1) - V •"].= 3C. + q. X J l TjT 1 M l where V x is the gradient operator w i t h respect to x and q. is a source density -3 (mol/sec cm ). The ionic source q. is zero in the intracellular and extracellular media unless ions are introduced experimental ly . Equat ion (2.1) and al l following equations hold in the interiors of the extracellular and intracel lular regions. The II. The Model Equations / 58 quantities "j-, C., q^ , are functions of the spatial coordinates x. Using the Nernst-Planck equations (1.2), the dimensionless form of (2.1) becomes: (2.2) where the variables and parameters are defined in Tables 2.2 - 2.4. Note that since 6. - D./ZD,, the sum Z0.C. is a. i l k i i Summing (2.2) over i gives: (2.3) V*o = TZ 9C. + Q, x i=l ' 5 t 1 2 — where L Z^ qV a. Multiplying (2.2) by z., the ionic valence, and summing over i gives: (2.4) '"V '(aV-0) = Q 2 2 — where the new quantity Q 2 is defined (Table 2.4) as Q 2 := Z. L z.q./ a. The time derivatives on the right-hand-side of (2.2) sum to zero by electroneutrality. 2.4. Jump Conditions and Boundary Conditions Sign conventions are illustrated in Table 2.5. The abbreviations ECS and ICS denote the extracellular and intracellular spaces, respectively. The normal n to the cell membrane is outward pointing. The signs of the ion fluxes, "j-, imply II. The Model Equations / 59 Table 2.5. Corresponding Sign Conventions at Membrane. Name ECS ICS Symbol Direction Normal Vector < n Membrane Potential + - V : ^ 1 - ^ 0 < 0 Current > I»n <0 Potential Gradient < V0 - n >0 Cation Flux > "l . -n <0 that cations flow in the same direction as electric current. Thus, the electric potential, <j>, is defined so that cation flux is in the direction of decreasing potential and diffusive flux (the first term of (1.2)) is in the direction of decreasing concentration. As a result of these conventions, the values of V and aV0*n have opposite sign. Using equation 1.(3.3) for the transmembrane current, and the definition of jj, the jump conditions for <j> and C. at cell membranes are: (2.5) -0.V C.-n - j>z.0.CV0«n = i»z.g.(V - V.) + p. I X I i i i ^ i & i r l l where p. is an active transport or 'pump' term and the transmembrane potential V is defined by V:= 01 - <j>° where and <j>° are the intracellular and extracellular potential, respectively, at adjacent points across the cell membrane. The electrical potential <j>, concentrations C., and their spatial derivatives, are discontinuous at the cell membranes. The quantity V r is the membrane resting II. The: Model Equations / 60 potential given by the Goldmann-Hodgkin-Katz formula (equation (1.2) of Chapter I) and the surface integral is evaluated over the membrane of a single cell. The dimensionless Nernst potentials, V., are given by: (2.6) vV. = J_ ln( C° / C1. ) 1 z. 1 1 l where C? denotes extracellular and C1. intracellular ionic concentrations, l l The jump conditions (2.5) hold intracellularly and extracellularly, and so (2.5) specifies two equations. Conditions of the same form hold at the cell membranes of both the smaller and the larger cell populations. In addition, boundary conditions are prescribed on the boundary of a region which is large compared to both cell lengths (order one in the L 0 scale). The pump term p. is chosen so that no net ionic flux occurs between intracellular and extracellular space. This is plrysiologically correct under normal conditions over many seconds. "It is assumed that: (2.7) / {vzgAY - V) + P}dS = 0 M 1 1 Summation of (2.5) produces, using electroneutrality: (2.8) - V a .n = Z *»z.g.(V - V.) + P, x 1 & 1 r 1 and (2.8) - (2.9) hold at the cell membranes. Multiplication of (2.5) by z. and summation results in: II. The Model . Equations / 61 (2.9) x 3 X i=l * g . ( V V.) + P 2 • W h e n g. = 0 = p. for i such that sgn(z.) = const, the membrane is permeable to only cations or only anions and a useful s implif icat ion ensues. F o r 3 example i f the membrane is permeable only to cations; L {vg.(V - V . ) + z.p.} 3 1 = 1 = I {i>z.g.(V - V.) + p.} so that f rom (2.8) - (2.9). (2.10) V a-n = voV <*«n, x x at the membrane. This condition w i l l be applied at the membrane of model gl ial cells since they are permeable only to potassium, K + . t Because the quantities gj, P^, and q.^ contain the dimension of length, independent assumptions must be made about the asymptotic orders of the corresponding dimensionless quantities, just as assumptions must be made about the asymptotic orders of cell lengths. Different assumptions correspond to tissue models wi th different physical properties. It should be noted that the definitions of g . and p. imply that the dimensionless quantities m a y become large as L — > » . f The model equations remain ful ly coupled because of the time derivative occurring in (2.3). II. The Model Equations / 62 2.5. Continuity and Smoothness Conventions The dependent variables are differentiable everywhere, except possiblj ' at the cell membrane, where jumps may occur. It is assumed that the dependent variables <j>, and C . , containing transmembrane jumps take some bounded values inside the membrane, but only the transmembrane jumps w i l l enter the calculation. The ionic and charge fluxes are continuous, and given by (2.2) and (2.4) respectively, a w a y from the membrane. W h e n the electric potential and ion concentrations j u m p across the membrane, the derivatives of these quantities do not exist at the membrane. Thus , the intracel lular and extracellular solutions are coupled through the j u m p conditions (2.5) at the membrane. Whi le the above conditions completely specify the solutions, it is sti l l necessary to evaluate averages of the spat ia l derivatives of the electrical potential and ionic concentrations over the complete unit cell , including the membrane, dur ing the formal calculation of bulk properties. One s tra ightforward approach to this computation would be to replace the membrane by a thin region of low conductivity and to take l imits as the thickness of the region and its conductivity jointly tend to zero. To avoid such lengthy analytic arguments i n the course of the asymptotic calculation, however, it is useful to extend the interpretation of the spatial derivatives and to use an extension of the divergence theorem, discussed in this section. While s imi lar results could be obtained i n three dimensions, we only need the two-dimensional result in later chapters. The divergence theorem m a y be extended to functions wi th j u m p discontinuities along a piecewise smooth curve in two dimensions as follows. L e t F* be a vector field to which the divergence theorem applies on two disjoint open regions, and Rg, wi th boundaries, dR^ and 9R2, where R^ and are II. The Model Equations / 63 separated by a curve, M, which lies in BR^ and 9Rg. A jump discontinuity in F* occurs on M, and the divergence operator V* is interpreted so that (cf. Royden, 1968): (2.11) lim / V-FdA= / {F^-F^-n^ds, 6->0 M g M i V i where F ,^, r=l,2, are the values of ? in R r adjacent to M, n ^ is a unit normal to M pointing into R^ and M is traversed once. The right-hand-side may be computed as the limit as 6—>0 of a sequence of functions F*g where each F*g changes smoothly from to in the neighbourhood of M, Mg. Then: / V-FdA = f V-FdA + J V-F* dA + lim / V-FdA RUM intR, intR2 5->0 M, . (2.12) = j ? -n ds + • / ? , n . , • ds + / F - n ds 9R,-M M i V i 9R 2 -M - ] l 2 . i t -ds + J {F2 - F , } . n M ds M M M M / ? -n ds 9R-M where n is an outward pointing unit normal on 9R, U 9R2 - M, boundary arcs are traversed counterclockwise, and R = Rj U R^. This is the same formal result as the divergence theorem except that F* is discontinuous. To illustrate the extended divergence theorem, let S(x) be defined by: II . The Model Equations / '64 S 1 (x) for x i n E C S (2.13) S(x) = S ° ( x ) for x i n M S*(x) for x in I C S where S , r = 0,1,2, are continuous i n their respective domains, E C S denotes the extracellular space, I C S the intracel lular space, and M the separating membrane. For example, the function S(x) could be the conductivity function, o. L e t (J be continuous except for a jump across the membrane. The function 6 corresponds to a dependent variable <j> or C . . U s i n g the extended divergence theorem (2.12) on R = ( ICS)U(ECS)U(M) , we obtain (2.14) TV-SC* dA = J s5-n dS R 9 R - M which is zero when there are periodic boundary conditions on R - M . A similar calculation for the integral : (2.15) / S V - G " d A = / S V - G * d A + / S(5 2-S 1)-n dS R R - M M yields two terms. W h e n (a is the gradient of concentration or electric potential , the first integral of the right-hand-side of (2.15) represents the average f lux of the II. The Model Equations / 65 associated quantity and the second integral is proportional to the net transmembrane f lux. Thus , i f the associated quanti ty is conserved wi th in the cell, the second integral is zero. These results are used later to s impli fy expressions of the form of the left hand sides of (2.14) - (2.15), by removing the need to integrate over the membrane. 2.6. The Mathematical Approach The equations, non-dimensionalizations, and assumptions presented in this Chapter complete the specification of the mathematical model for bulk tissue properties. This model corresponds mathematical ly to a non-linear init ial-boundary-value problem w i t h coefficients rapidly v a r y i n g in space. In Chapter III i t w i l l be shown that computation of the macroscopic properties of this model can be reduced, us ing the method of multiple scales, to a sequence of numerical boundarj^-value problems on periodic domains (Keller, 1977; Bensoussan et a L , 1978). These problems are called 'cel l ' problems in Bensoussan et ah , and are called 'canonical ' problems here to avoid confusion. Whereas the solution of the original problem when e is smal l is an ill-conditioned and complicated computational problem (Traub et a L , 1985 a,b; B a b u s k a , 1976), the numerical solution of the canonical problem is s traightforward. The application of the method of mult iple scales to the computation of averaged properties of inhomogeneous media is called 'homogenization' (Babuska, 1976; Bensoussan et aL , 1978; Sanchez-Palencia, 1980). III. ASYMPTOTIC EXPANSION 1.0. INTRODUCTION The dependent variables </>, o, and G are functions of time and the spatial coordinates, x. In order to apply the method of multiple scales, additional scaled spatial coordinates are defined by, u := x / L 0 , v := x / L 1 ; w := x / L 2 . The conductances, gj, appearing in the j u m p conditions 11.(2.5) are each defined by scaling wi th respect to an arb i t rary length L in Table II.2.4. Because we are interested in the bulk properties of cell a r rays , the scaled conductance is equal to the entry in Table II.2.4 with L = L 0 , where L 0 is long compared to the cell length. A n additionally scaled value of g. is defined as follows. Since experimental data (Turner & Schwartzkro in , 1984) show that neuronal electrotonic length scales (length scales formed f rom the electrical parameters) and the cell length are of s imilar magnitude, the scaled value of g. w i t h L = L , is 0(1). Hence, we define a .new g. using L = L,-] and rewrite the scaled conductance as e "*g. where the new g. is 0(1). A s imi lar argument -is used to define rescaled pump fluxes. The .pump fluxes must have the same order as g. i n order to balance the transmembrane fluxes as described in Chapter II. Since the asymptotical ly larger population might not be neurons, other rescalings of g. are possible. However , the above assumption is the least-order assumption about membrane conductance which allows 0(1) f lux through this population. The role of electrotonic parameters in tissue models is discussed in more detail in Chapter V . In the asymptotical ly finer 'g l ia l ' population, g. is rescaled in the same w a y as above, and we rewrite the scaled conductance as e ^g. where the new 66 III. Asymptot ic Expans ion / 67 g. is 0(1) . Thus , electrotonic length scales have been assumed to be the same in the two populations, so that the membranes and intracel lular and extracellular media of the finer population are assumed to have the same properties as in the coarser population. This assumption is made because there is relat ively little data about membrane properties of very f i n e tissue structures. Because the leading-order equations up to 0(1) w i l l contain no time derivatives, the time dependence is suppressed. 1.1. Expansion W i t h the above definition of the scaled spatial coordinates, we have the formal correspondence: V > e " 2 V + e'1? + V X W V u 0 r v , -+ (1.1) : = Z e m D m = -2 It is assumed that -<j>, a, and C. m a y be expanded in the form: <j> = 00 00 oo Z e <j> , o =• Z e a , and C.= Z e C . Detai ls of the expansion of n = 0 n n = 0 n 1 n = 0 m operators and boundary conditions are given i n Appendix I I I .A . Collecting the equations f rom (A.4) - (A.6) for <j>0, a0, and C j we have: V »(a0V 0O) = 0, w u w ° (1.2) v f a 0 = 0, III. Asymptot ic Expans ion / : 6 8 V -(c?.V C. ) + vz.V '(B.C. V <t>0) = 0. W 1 W 10 1 W 1 10 w " The j u m p conditions in w are applied at glial membrane. Since the membrane is permeable onty to K + , we use the simplif ication (II.2.10) to obtain explicit ly; P o V w 0 o * n = 0, (1.3) V a 0 - n = 0, w u 6».V C. - n = 0. 1 w 10 These conditions together w i t h periodicity in w specify <t>0, a0, and CJQ, intracel lular ly and extracel lularly, though not the transmembrane jump. Because the equations (1.2) are potential equations and the j u m p conditions (1.3) are homogeneous N e u m a n n conditions, o 0 is a function of u and v alone. Thus (p0 is a function of u and v alone. U s i n g these observations it is concluded that C . q is a function of u and v alone. The j u m p across the membrane w i l l be determined later us ing (A.8) - (A. 10). Gather ing equations for <j>,, a 1 ; and C. ^ , in the expansions (A.4) - (A.6) produces: III. Asymptot i c Expans ion / 69 (1.4) A _ 3 c 0 + V ^ o , = 0, ? . A „ C . + V «(0 .V C ) + i>z.{d.A\(t>0 + V '(B.C. V 0,)} = 0, I -3 io w I w n i i -3 ° w I io w where A f = V - ( f 0 V ) + V « ( f 0 V ) + V - ( f , V ) , and f rom (A.8) - (A.10): -3 v ° w w u v w 1 w 3 " ^ o i V ^ r n + V v 0 o - n } = I _ ^ g i o ( ^ V 0 - f V . ) + P 2 , 3 (1.5) - V a , - n - V o0-n = Z z.g. ( y V 0 - v V . ) + P , , w 1 v u i = i 1 - 0 . { V C. - n + V C. -n} = z . ( l - 0 .C. / a 0 ) g - ( i » V 0 - i » V . ) + p. . l l W 11 V 10 J 1 1 10 o / & 1 0 ° 1 1 The integral of the transmembrane fluxes (1.5) over the cell membrane m u s t be zero, where the integral over 'g l ia l ' membrane is performed in the w •coordinate. Since the 0(1) solution of (1.2) - (1.3) implies that the r ight hand sides are functions of u and v alone, this implies that the r ight hand sides of (1.5) are identically zero. The definition of A ^ Q and the form of <j>0, a 0 , and C . i n w imply that -o 10 f A g annihilates each of the operands in (1.4). The equations (1.4) for a 1 ; and C in the intracel lular and extracellular media are thus identical to those I i for . 0 O , a 0 , and C. . Equations (1.4) reduce to potential equations because of the form of 0 O,O O, and C - o in w . However , the j u m p conditions depend on 0 O , a 0 , and III. Asymptot ic Expans ion / 70 C. which are arb i t rary functions of v. F o r example, because <j>0 is an arb i t rary function of v , each derivative V y 0 o i s an arb i t rary function in the solution for <t>i. It m a y be verified by direct substitution into (1.4) and (1.5) that if <j> ^, a 1 } and C. ^, respectively, have the form; <j>]=- J ' ^ o + $ i> ° i = V'^v°o + 6 1 , and, C- = / c . * V C + C . , where $ , , a 1 ; and C . are arb i t rary 1 ' ' 1 1 1 v io n ' Y 1 ' 1 ' 1 1 J functions of u , and x> 7? and K satisfy the vector equations: (1.6) V - V 77 = 0, V -(c?.V K . ) = 0, W 1 W 1 w i t h jump conditions at 'g l ia l ' membrane: V x * n = ~ n . w (1.7) V ^?-n = - n \ w ' V7 "* ~* -V K.' n = - n , w 1 then a solution is obtained. Because (1.4) reduce to potential equations, no other non-tr ivia l w dependence is possible, by applying the same argument that was used at order III. Asymptot i c Expans ion / 71 one to the difference of two solutions. It is only necessary to know the general form of <f>2, o2, and C in the computation of the u-dependence of 0 O, o 0 , and -C. . The computation which specifies (f>2, o2, and -C. is .given in Appendix III .B. It is shown that the intracel lular and extracellular equations for these quantities are potential equations and the solutions assume a simple form analogous to that of <j> ^ , a ^ , and C . ^. 2.0. AVERAGING 2.1. Introduction The sj^mbol M (•) denotes the average over the w unit cell; w (2.1) M W ( F ) = 1 , / F d A wv ; TW| J w - M where W denotes the w unit cell, dA — d w , d w 9 , and ( 2 . 2 ) |W| = f F dA . W - M The interpretation of averaging for our asymptotic solutions is now given, and the asymptotic size of the averaging region is specified precisely. This clarifies the relationship between the periodic average given by (2.1) - (2.2) and the average over a 'sufficiently large ' region in the unsealed spatial variable , often referred to in the physical l i terature (Garland & Tanner , 1978) without mathematica l definition. Suppose that a formal f lux vector, is given by the multiple scales III. Asymptot i c Expans ion / 72 procedure where x = ( x - ^ ^ ) are the space coordinates, 5^  = (X-^Xg) : = (x-j/e^/e ) , and J* is doubly periodic in ( X p X g ) and a smooth function of x. Then the spatial f lux vector is defined by "j (x ,e) := -3 (x,x/e) . It w i l l be shown that a smooth flux vector can be defined at each point XQ := (X-^Q, X2Q) by averaging "j(x,e) over a square region, R ^ , of side u centered at x^ , assuming that p. = for some p, 0 < p < 1 as e -> 0, and recalling that the cells have length 0(e). The result J^(^Q ) is independent of the choice of p, 0 < p < 1 . A property which holds locally w i l l hold -within an asj'mptotic region of this size. A l s o , it wi l l be shown that averages over R ^ are functions only of the f irst set of arguments of 3 (coarse variables), and m a y be evaluated b}' integrating over a single period of the second set of arguments (fine variables). The above statements follow from computing the average of "j over R ^ , a square area element of side p=e^, centered at x^ : (2.3) " j R ( e ' V = IfKl J *{*0' V e ) d A ' pi R where | R £ p | = e^. Then the average flux vector w i l l be the l imi t of (2.3) as e^>0, i.e., J A = l i m e e > Q 1R(e,xQ). To evaluate the l imi t it is useful to make the change of variables in the integrand: x~^ = e 1 (x-^ - X-^Q) and — E ^ (X2 - ^g), o r m vector form x=e \ x - XQ). It then follows that d A = dx-^dx2 = dx*^ dx ^ —dA~ and the regions of integration transform R^p > R f p - l where III. Asymptot ic Expans ion / 73 (2.4) R p = { ( x 1 ; x 2 ) : x 1 0 - e P / 2 < x-,^  < x 1 0 + e P / 2 ; x 2 Q - e P / 2 < x 2 < x 2 0 + e P / 2 } R p - i = { ( x l l X 9 ) : - e 9 ' 1 ^ < x- n < e p _ 1 / 2 ; - e ' 3 " 1 ^ < x 0 < eH!2}. e , p - l — \\x-i>x-2'' i ' 2 The vector m a y now be expressed as an integral over the fine variables ( X j , X 2 ) since as e ->0 : = * ™ A V e ' V = H™„ 7^P J ^ ( x (f e *' V e " ^ ) d A e ^ O " u e ^ O e ^ R e P (2.5) ' = lim 1 J ^ ( x n + 3(e P), x n / e - x ) d A " Hm 1 / J " j (x,X) d X 2 d X 2 T S _ j T , S — > » 4 S T - T - S where the last equality is obtained f rom the multi-periodicity of J (x , •) , setting S = T = e P ^/2, = _ x ^ j + x l ( / e ' a n d ^ 2 = ~*~2 x 2 ( / e " ^ o r ^ l x e a ^ (x10' X 2 Q ) , the periodicity of J implies that the average in (2.5) m a y be evaluated over a single period. Thus , for regions of appropriate asymptotic size, an average f lux vector is defined even though 3 m a y not be continuous, and J ^ ( x ) is a function of the coarse variables alone. III. Asymptot i c Expans ion / 74 2.2. Averaging Procedure In this section the specification of the average governing equation at coarse length scales is completed. The v-dependence of (p0, a 0 , ' C . , is deduced f rom a necessary condition for the existence of a bounded solution for </> 3 , o 3, and O , namely the Fredholm Al ternat ive applied to a constant function at _2 0(e ) in (A.4) - (A.6). This condition is equivalent to the physical observation that there can be no steady periodic solution to the potential equation unless (charge or concentration) is conserved, so that the periodic source density must have zero integral over each period. This leads to the conditions: y A - 2 a ° + A - 3 a i } d A W = 0 , (2.6) Sw{A°2(j>0 + A V l } d A W = 0 y i < A 2 C i o + A - 3 C i i + " W a C 2 * ° + A C 3 * i » d A W = 0 where A . f = V^^VJ + Vffo^) + V ( f ' V w ) + V ^ 7 ^ + + V w * ( f 2 V w ) , and the integrals are interpreted as described in Section II.2.5. Thus , us ing the extended divergence theorem, the integrals of V w applied to discontinuous w-periodic quantities are zero. To obtain the u-dependence of <pQ , o0 , and , the same necessary condition for the existence of bounded <pn > -o H > a n c l C j is applied to the 0(1) equations f rom the expansion of Appendix I I I .A . These equations are integrated over w and v to give: III. Asymptotic Expansion / 75 J V { A 0 o O + A l f f n +A 2 a 2 } d A y d A w = / ^ {Q, + rL 9 ^ 0 } d A v d A w (2-7) / r v i A u ^ > o + A a l 0 1 + A a 2 0 2 } d A v d A w = f v v Q 2 d A y d A w f 0.{A„C. + A . C . + A „ C . } d A „ d A w i l 0 io -1 n -2 i 2 J V W + tfzi°i ^ v { A o * o + A C l ^ + A C 2 ^ 2 } d A v d A w J {q. + T-SCjoldA^dA,,,, WV 1 tt* V W where WV is the unit cell in v space and A^, = V • (f0 V ) + V • (f i V ) + 0 u u u u 1 v V -(f,V ) + V -(f 2V ) + V «(f 2V ) + V -(f 3V ) + V «(f 3V ) + V 1 U U W W W W U V 3 W W 3 V V -(f«V ) . w w This averaging procedure, as applied to the computation of bulk properties of inhomogeneous materials is discussed b}' Bensoussan et ah, (1978) but has not previously been applied to any non-linear equation with jump conditions in the interior of the domain. However, the main objective here is the application of the analysis, rather than mathematical novelty. These equations provide an alternative to the cable model in computing the bulk properties of brain tissue and this treatment differs from those of Ranck (1964) and Havstad (1976) by the use of a systematic averaging procedure and the incorporation of ion transport into the model. The resulting averaged equations have ah extracellular and intracellular part. In order to apply the averaged equations it is necessary to determine the III. Asymptotic Expansion / 76 extracellular and intracellular parts. Thus we now investigate the specific form of the solutions to (2.6) (2.7). This part of the calculation appears to be new. It is assumed that a geometry has been chosen in which the averaged coefficients reduce to scalars. In this case, the .averaged equations (2.6) become eV^Oo = 0, (2.8) V y - ( a a o V v 0 o ) = 0, V «(0.e.V C. ) = 0, V 1 1 V 10 with periodic boundary conditions on the v unit cell (neuronal scale) and the jump conditions at neuronal membrane: - e V v a 0 « n = P, + g 2 0 { * ' V 0 - » » V k } (2.9) - ^ a a o V y 0 o - n = P 2 +g 2 0 {^V 0 - vVk} -e.C. V 0o «n - 0.e.V C. -n = p. + z.g. {vV0-vV,} I io v v o I I v io * i i 6 io 1 ° kJ where v^j^-= m ( C 2 0 / C 2 0 ) and the definitions of a, e, and e. are given in Appendix III.C with the more general calculation. The right-hand-sides of these equations are zero by an argument similar to that given for the right-hand-sides of equations (1.5). These equations have continuous intracellular and extracellular III. Asymptotic Expansion / 77 solutions <t>0, a0, C - o , where <j>l0 — <t>°o + Vj^ while OQ and are constants. After averaging in w (described in Appendix III.C) the 0(e) equations are: e V 2 ^ = 0, v 1 (2.10) V v - ( a a o V v 0 , ) = 0, V -(0.e.V C. ) + vz.V '(d.e.C. V 4>,) = 0, V 1 1 V 11 I V 1 1 1 0 V 1 with jump conditions at 'neuronal' membrane: -e{V vai-n + V^Q-n} = g20{vV, + ( C 2 1 - C ° i )}, r i p o ^ 2 0 ^ 2 0 (2.11) - a a 0 { V v f , - n + V ^ Q - n } = g20{uV, + ( c l i - C 2 i )}, p i p O ^ 2 0 ^ 2 0 -c?.e.{V C. -n + V C. -n} = z.(l-0.e.C. /aa0)g2o{»'V1 +( C 2 1 - C 2 i )}, 1 V V 11 U 10 1 1 1 10 U 1 — : p i p O ^ 2 0 ^ 2 0 and periodic boundary conditions on the unit cell. The periodic boundary conditions follow from the periodic structure of the tissue. These equations (2.11) hold intracellularly and extracellular^. The equations (2.10) - (2.11) are linear and the solutions may be obtained by the substitutions III. Asymptot ic Expans ion / 78 1 l p u T U i y u v *C U 10 's U u 1 (2.12) v<t>, = u( X p - V u 0 o + X v - V u V 0 ) + X c ' V u C i o + X g - V ^ o + C = vC ( K - V 0o + « - V V 0 ) + C. ( K «V C + K 'V a0 ) + C , . 11 10 p V U U 10 C u 10 s u 0 1 where X J X > X , X , 77, 7? , 77, 77, K , K , K , and K are to be A p j A y > /v c > A g , i p > ( y , i c , ( g j p , y , c > g determined. We write a 0 = a o " , #. = ej#. and drop the tildes in what follows. Subst i tut ing as described into (2.10) - (2.11), it is found that the variables to be determined in (2.12) satisfy the potential equation extracel lularly and intracel lular ly , w i t h jump conditions at the cell membrane given by : ~ - | V v V " = •g20{5^ 1 -^ l +4 n-K° 1}, (2.13) - o 0 { V V X p 1 - n + n , } = g 2 0 { X P 1 " 5$ , + ^ ~ }, - ^ i o ^ ^ i ^ = ^ o ( 1 " t f i C i o / a < » ) % " 3 S i + RPi"4i>' V 7 j y 1 - n = g z o i X ^ - X?M + 4" *vi}> III. Asymptot ic Expans ion / -79 (2.14) - f f o V / v 1 ' « = - O o { V v i ' i + n , } = g 2 0 { " X? M + 4 " " ^ i o ^ v ^ i ' 2 = g i o ( 1 " e i C i o / ( 7 o ) { X v 1 - ^ i + < n - < i ^ (2.15) - a 0 V v x s 1 - n = g 2 0 { 4 - & + 4 " ,^)> - • 7 V v ^ i - n - G 2 0 { 4 - « 1 + 4 - « L } (2.16) - a 0 V v X c 1 - " = 8 2 0 ( 4 - ^ 1 + 4 ~ = g i o ( 1 - 0 i C i o / a o ) ^ - ^ 1 + 4 - ^ i } where the subscript 1 denotes the first components of the quantities x , ??, and K. L a t e r i t is shown that the canonical problem for each component is the same and thus, that these coefficients reduce to scalars under appropriate geometrical III. Asymptotic Expansion / 80 assumptions. Since the variables 77 do not appear on the right-hand-side of (2.11), it is possible to solve (2.10) - (2.11) by first solving for x p ! Xy, X c > Xg, K p , K y , and K g because they are independent of the v solutions. In addition, as described in Chapter IV, it is assumed that V u a o = 0 . Hence the 77 solutions do not appear in the expressions for 0,, a,, and Cj 1 and need not be computed. The x and K solutions, which are calculated numerically in Chapter IV, will be substituted into the averaging conditions (2.7) to obtain the average governing equations which are the goal of this chapter. Substitution of the x and K solutions into (2.7) gives the equations: D , ^ + E , v V 0 + F ,v jc 0 = Q 2 (2.17) D a V ^ o + E 2 v V 0 + F 2 v jc 0 = q + r ^ o where CQ is the potassium ionic concentration and the coefficients are defined using (2.1) - (2.2) by D 1 = a 0 M v { t a ( l + V vx p l)}, E 1 = a 0 M v { t a ( t / 3 + V vx v l)}, (2.18) III. Asymptotic Expansion / 81 D , = a 0Mw{t t ,t (1+ V x -,) + V K A 2 o v LT£ a v A p l v p r ' E 2 = a 0 M v { t K t a ( ( t / 3 + V y x v l ) + V V K V 1 ) } , F 2 = / 1 { M V { ( e C 0 V v / c c l + 6) + ao^VV^}}, where t^ is unity intracellularly and zero otherwise, and t^= OQIO0Q. The periodic problems for the n, X, and K variables depend on o 0 , g2o> and Cj through (2.13) - (2.16) and thus are, in general, functions of v. Thus an infinite number of the canonical periodic problems must be solved in order to obtain the (variable) coefficient functions of (2.17). This is because the original mathematical problem was non-linear. It is shown in Chapter IV that it is appropriate to approximate (2.17) by a system with constant coefficients obtained from a single set of canonical problems in which V u a = 0. As it is impractical to solve numerically for the variable coefficients this is a useful simplification. The coefficients D 1 ; D 2, E 1 ; E 2, F 1 ; and F 2 are tabulated in Chapter TV using this assumption. The calculations of this chapter show that the coefficients of the average governing equation are different depending . on the length scale. This has implications in the interpretation of data from experiments on the bulk properties of brain tissue, and some specific comparisons of one-tier with two-tier models are made later. III. Asymptot ic Expans ion / 82 APPENDIX III.A. OPERATOR EXPANSIONS U s i n g the correspondence (1.1) it follows that: V 2 > e " 4 V 2 + e' 3(V - V + e"2(V -V + V - V + V2) X w v w u w w u v (A.l) + e'-'tV - V + V - V ) +V 2 U V V u u := I e A , — A n r m = -4 where mixed part ials such as V • V and V • V must be distinguished from u v v u each other in view of the discontinuity at the cell membrane. While the boundary-value problem is formulated using j u m p conditions at the membrane, i t is s t i l l necessary to interpret expressions such as ^'(aV^^) a n ^ ^w*( a^ u0) * n performing averages of derivatives over the unit cell R (intracellular and extracellular spaces and membrane). oo Thus i f f = 1 e n f , n = 0 n f V > e " 2 f 0 V + e ' V o V + f , V ) X w u V 1 w (A.2) + Z en(f V + f , -V + f ,„V ) _ n n u n + l v n + 2 w n = 0 := Z en$ n = -2 In general III. Asymptot ic Expans ion / 83 V -(f V ) >( e" 2V + e ^ V +V )-(Z e"!^ ) x x w v u _ 0 n n = -2 (A.3) := Z enA1 . n = -4 n So that the model equation (II.2.3) becomes: 0 oo oo 3 (A.4) Z Z e m n A a = Q, + TZ Z e 1 " 9 ^ m = -4 n = 0 m n n = 0 i = 1 <" and (II.2.4) becomes: 0 oo (A.5) uL Z e m n A a 0 = Q 2 . ™ _ , „ _ n n r n ^ z m — 4 n — u C f We define the operator t?.A := A , where f = 6-C The differential l m m i i equations (II.2.2) for C. become: 0 oo CO 0 0 Z v m + n „ . „ „ „ m t n „ . L , Z e t ? A C . + fz.Z Z e e?.A 6 m = _ 4 n = 0 1 m i n 4n = -4 n = 0 1 ^ n (A .6) °° 3 = Z Z e n 3 C . + q . . n = 0 i = 1 T h e j u m p conditions are now expanded i n the new variables. These conditions are to be applied at the asymptotic scales corresponding to the III. Asymptot ic Expans ion / 84 assumptions about cell sizes. The condition for 'g l ia l ' membrane is applied in the w variable and the condition for 'neuronal ' membrane is applied in the v variable . oo -co We define g. by expanding: -g.(L e n V )= Z e " g . , so that m 1 n = 0 n n = 0 i n g i 0 = g i ( V 0 } ' g i l = g ' i ( V 0 ) V l ' g i 2 = T { g " i V l + 2 g ' i V 2 } w h e r e t h e p r i m e ' oo denotes dg.(V)/dV and define h . by expanding pg.(V~V.)= Z e n h . so that: I in 1 1 n = 0 m h . = g . J y V . - i l n ( C ° / d . )}, lO 6iO L 0 z. io io •" i (A. 7) h i i = g i i ^ v o - f < 1 4 o » + ^ { v N i - mA - ^L)h * i d C° 10 10 etc. Then the jump conditions (II.2.5) become: 0 0 0 Z { e m + n 6 . I ) C. -2 + vz. em + ne:T>C4> -5} m = . 2 n = 0 1 m m 1 1 m n • Oo (A.8) = Z e n " 1 z . h . + e ' V _ n 1 i n 1 n = 0 at the 'g l ia l ' and 'neuronal ' membranes, where the scaling of p. was discussed in the introduction to this chapter. The equation (II.2.8) transforms to: •0 °° . , °= (A.9) - Z Z D ( en mo ) - n = Z e n z . h . + P , _ „ _ „ m n _ „ 1 in ' m = -2 n = 0 n = 0 and (II.2.9) to: III. Asymptotic Expansion / 85 (A.10) -vL Z L ^ ( e n + I % J - n = Z e n h.„ + P 2 _ o _ r . n n n _ n in m = -2 n = 0 n = 0 at the glial membrane. A P P E N D I X III.B. S E C O N D O R D E R P E R T U R B A T I O N S O F T H E D E P E N D E N T V A R I A B L E S To 0(e"2) (A.4) - (A.6) imply that </>2, a 2 , and satisfy: A a 2 0 o + A°3d>, + V w . ( a o V w 0 2 ) = 0, (B.l) A _ 2 a 0 + A 3 a, + V ^ a 2 = 0, 0 . A o . C . + e?.AQ.C. +V -(t?.V C. ) i -2i io l -3i 11 w l w 1 2 + vz.{6.ACo.(j>0 + e . A ^ . 0 ! + V .(e.C. V <t>2)} = 0, r i -2i U I -3i 1 w I io w " where A f„ = V -(f 0 V ) + V . ( f 0 V ) + V . ( f , V ) + V . ( f , V ) + -2 u " w w ° u w 1 v v 1 w V w • (f2 V ) , while the jump conditions at glial membrane become: Eh- = - g 2 o { ^ V 1 + ( C 2 i - C 2 1 ) } = ^ ^ ^ . g + V J o - S 1 1 . w V 1 c 1 r° ^ 2 0 L / 2 0 + ^a0{Vw<A2 .n + V v 0 ! «n + Vu</>o *"} > III. Asymptotic Expansion / 86 (B.2) -Z-zjh h = - g 2 o { ^ V 1 + ( _ ^ £ 2 - ^ 2 l ) } = V w a 2 - n + V v a ! .n + V u a 0 - n , i r i r o ^20 ^20 -h . 11 -g 2 O ( i -e .c i o /0 O ) { * 'V 1 + c 2 1 -^2 1 )} = c 1 , e.{v a r u io •n+V C. v n n+V C W 12 20 '20 f f The definition of A 9 and the form of <f>0, o0, and C imply that A 0 reduces to V "(fjV ) +' V *(f 0 V ) , when applied to the 0(1) terms in (B.l). f No similar reduction is possible for A ^ but the equations (B.l) - (B.2) may be simplified by substitution of the known forms for <j> ^ , C. ^ . By an explicit calculation, we obtain the equivalent equations for 0: - V ' ( a 0 V 02) = 2V X ' V (V <pQ) + V ff0'V x*V 7 7 ' V <j>0 w ° W ^ z W V V ^ u v ° w w ' V ° (B.3) + V o-o-VJ7-V 60 + Oo^Jo + V a 0 - V . v0o The equations for a are (cancelling ."0.) (B.4) - V 2 a 2 = 2V ^ - V (V 0O) + ^ a 0 , W W V V v while the equations for C. are: V 2 C + 2V K « V . ( V C. ) + ^ C . + vz.{V .(C. V 02) W 12 W V V 10 V 10 1 W 10 W ' III. Asymptot ic Expans ion / 87 (B.5) + 2 C V X - V (V <j>0) + V C. - V x - V ^ - V -0O 10 W V V " V 10 W W ' V u + V C. - V K - V C . + C. V ^ o + V C. - V f>0} = 0. V 1 0 W V ] 0 10 V u V 1 0 V u where x> and K are defined in (1.6) - (1.7). B y the solution to (1.4) - (1.5), the f irst term in the definition of h . ^ is zero, so that the jump condition for <f> i n equation (B . l ) has been simplif ied to: ~ g 2 0 { " V l " ( C ' i 1 C o - C ° 1 C ° » = v o i C v ^ i - n + V v 0 o * n ) + f a o ( V w 0 2 - n + V ^ ^ T - n + V^Q-n). A l s o , equations (1.6) - (1.7) imply that the terms proportional to a ^ vanish in the latter expression. Hence the expression reduces to: *>0 o{v w 0 2.n + (x-V v)-V vtf> 0-n + V^^n + V u 0 o*n} (B.6) = -g20{vV,+(.^-Q_)} . r i p O ^20 ^20 S i m i l a r l y : vo0{Vwo2>n + ( ^ - V V ) . V v a o - n + V y 0 , . n + V u 0 o -n* } = - g 2 0 { » ' V 1 +(_CJ_I-^2_|_)}, p i pO ^20 ^20 (B.7) #.{V C. - n + ( K - V )-V C - n + V C. - n + V C -n} 1 W 12 V V 10 V 11 U 10 = - g 2 o ( l - f l i C i o / o - 0 ) { i ' V 1 +( C 2 1 - C ° i )}. p i p o ^20 ^ 2 0 III. Asymptot ic Expans ion / 88 Since - 0 O , a 0 , and C. 10 do not depend on v, under the specific assumptions made here, the functions <j> ^ , o ^ , and C. i do not depend on v by (2.8) - (2.9). Thus , inspection of (B.3) - (B.7) shows that the intracel lular and extracellular equations for 0 2 , a 2 , and C are potential equations and the solutions assume simple forms analogous to those for 0 1 ; a 1 ; and C. ^ This observation simplifies the computation of the u-dependence of 0 O , a0, and C.Q. A P P E N D I X I I I . C . G E N E R A L A V E R A G I N G In this appendix, the averaging calculations are carried out in the general case. Without special assumptions about the geometry, the computed average coefficients do not reduce to scalars, even though the microscopic parameters , solution conductivity, and diffusion coefficients are constant. Because the objective of this calculation is only the general form of the average coefficients, i t is not necessary to separately consider the extracellular and intracel lular parts of the solutions. Substitution into (2.6) of the previously derived forms for 0 , , a 1 ; and C. yields the averaged equations at 'neuronal ' length scales: 11 2 3 { e i k 9 ^ o ; ( C . l ) III. Asymptotic Expansion / 89 Z[d{6e 9C,0} + vz 9{e.C i o a Ho}] = 0 where a j k= M w (6 j k +9 X k /9W.) , ^ejk= M w ( 6 j k + 3rjk/3W.) , e j k i = M, i r(6., + 3 K /9W.) , and 6 is the Kronecker delta, 8., = 1 when j = k and 0 W jk j jk J otherwise. It is shown in Chapter IV that for the geometries assumed here; a j k = a(u), e. k = e(u), e j k i=e.(u). The quantities <f>Q , o0, and C. satisfy jump conditions obtained by averaging fluxes of order e 1 at 'neuronal' membrane since (1.1) implies that aV <t> is associated with a flux of this order. These conditions are: v^ - I e 3o.on = g 2 0 {^V 0 - vNA + P, j,k J k !9^ k J k (C.2) -L vo0adjLon. = g 2 0 { ^ V 0 - y V , } + P 2 j,k J k 9vk J k - Z 0.e., . 9C: on. = g 2 O ( l -0 .C . /o0){vVQ-vV,} + p. i jki ^ ? k j 6 , ! 0 i io 0 / 1 ° kJ * i The v-dependence of <j>, , a , , and C. 1 is deduced from a necessary condition for the existence of a bounded solution for <j> 3 , a 3 , and C. , 13 namel}', the Fredholm Alternative applied at order e ^; yielding III. Asymptotic Expansion / 90 J " w ^ A . i a o + A . 2 a i + A - 3 a 2 } d A w •= 0 (C.3) / w { A A L0 o + A A 20 1 + A A 30 2 } d A w = 0 V i { A - i C i o + A - 2 C i < + ^ = ° w h e r e A ^ = V -(f 0 V ) + V -(f 0 V ) + V .(f,V ) + V . ( f , 7 ) + -1 U ° V V ° u u 1 w w 1 u V . ( f 2 V ) + V . ( f 2 V ) + V -(f 3 V ) . V ^ w w w v W J W Hence, these averages yield (corresponding to 2.10): Z 9 {e., 3a r j,k3vj J k ^ k (C.4) Z _9 {o 0 a..3li} = 0 j,k 9VJ J K ^ v k Z [9 {0.e., + vz. 9 {B.C. a., 3*i}] = 0 with jump conditions: - E e , { ^ l n + 8 a o n } = g 2 0 { „ V , +( C 2 1 - C 2 , ) } j,k J k 3vk J 9u k J j — ^20 ^20 III. Asymptotic Expansion / 91 (C.5) -vo0L a .,{30in . + 3>n} = g20{z>V , + ( C 2 i - C ° i )} j,k J K J ^ k J r i _o ^ 2 0 ^ 2 0 ^ 2 0 ^ 2 0 This is a system similar to (1.5) - (1.6) except that the coefficients now contain averages and the boundary conditions are applied at 'neuronal' membrane. The explicit form of the solutions when the coefficients are scalar is given in equations (2.10) - (2.15). Finally, to obtain the u-dependence of 0O , o0 , and C . q , the necessary condition, equation (2.7), for bounded <pn , On and C is applied to the w-averaged equation at 0(1) which yields, using the previously computed v dependence of $ 0 , o0, C.Q, 0 1 ; a 1 , and C. ^ : j,k 9 U J J K ouk i 9 t (C.6) L 3 {a05., 3*o} = Q 2 j ,k^uj. J k d u k L [ 3 { 0 e k 9 ^ o } + V-L 9{8C. *-k3>}] = r9C, 0 + q j,k5uj 1 J k l 3"k 1 ^ U J 1 1 0 J K ttxk W 1 where fi^, e"^, and ^ jki' a r e defined i " a manner analogous to a.^, e.^ , and e., ., by taking v-averages over v-unit cells of canonical problems with w III. Asymptotic Expansion / 92 averaged coefficients, in the same way that (C.l) were obtained by averaging over w unit cells. IV. CANONICAL PROBLEMS AND T H E COMPUTATION OF BULK PROPERTIES 1.0. INTRODUCTION The physical theory of ion transport is based on ideal assumptions which are useful for obtaining the qualitative features of such transport (Horvath, 1985). It is desirable to incorporate these idealizations into our model because most relevant experimental work on ion transport i n biology is described w i t h reference to this (transport number) theory (Horvath, 1985; G a r d n e r - M e d w i n , 1983; B a r r y & Hope, 1969 a, b). More generally, the physical applications of results f rom homogenization theory have been relat ively neglected, (I. Rubenstein, personal communication) and much of the theoretical work has onl j ' reproduced results already k n o w n to experimentalists or derivable through other techniques (Batchelor 1974; Lehner , 1979). It is expected that the incorporation of physical theory w i l l facilitate new physical insights as wel l as applications. Thus , in this chapter, we are concerned w i t h the relationship between the expansion procedure of Chapter III and the existing physica l theory of ion transport . It is shown that this transport number theorj ' applies to bulk tissue, and a further simplif ication of the canonical problems of Chapter III is made for consistency w i t h the physical theory. F i n a l l y , bulk properties of the tissue model are computed for a var ie ty of parameters . 1.1. Transport Numbers in Electrolyte It is observed experimentally that when a steady electric current is passed through an electrolyte solution, the current I = 2"JZ."J ^ is approximately divided into constant fractions, t., depending only on the composition of the electrolyte 93 I V . Canonical Problems and the Computat ion of B u l k Properties / 94 (Horvath , 1985). These transport numbers, t., have been justif ied wi th in the thermodynamic l i terature and tabulated (Robinson & Stokes, 1955). The theory of -transport numbers assumes that (ideally) ionic fluxes are proportional to electric current w i t h a single constant of proportionality over space and that the electric current is specified by a l inear equation w i t h constant coefficients. If these assumptions are true, it is simple to predict the effect of current passing experiments on the concentrations of ions. The appropriateness of this idealization for a simple electrolyte solution is now discussed. While our model equations (II. 1.2) represent a simplif ication of a more complete thermodynamic treatment (see Chapter I), they st i l l give rise to non-linear governing equations. Thus , in general, the transport number is not constant under this model. If we assume that (II. 1.2) is correct and electroneutrality holds, then we have in the steady state away f rom sources: V 2 o = 0, (1.1) V*(ffV<A) = 0, - V . j . = V-(0 .VC. ) + i>z.V«(0.C.V0) = 0 J 1 1 1 0 1 1 1 ^ Thus , when a is k n o w n , the computation of the electrical potential (p is s tra ightforward. W h e n Cj are constant, this implies that V C j = 0, the diffusive term is zero i n (II. 1.2), and a — constant because of its definition (Section II. 1.1). In this case the ion transport vector has the simple form: IV. Canonical' Problems and the Computation of Bulk Properties / 95 (1-2) "Ji = vz.d.C.V<j> 111 •v t.z.aV0, 1 I where t. = 8.C. I a. Under these conditions, L.t. = l . The numbers t. are the i i i 11 I theoretical transport numbers measured by passing current through electrolyte solutions (Horvath, 1985).t Since Q. is not constant in general, this assumption concerning the transport properties of electrolyte solution is an idealization which is reasonable as long as variations in C. are not large. The diffusive portion of the ionic flux is assumed to remain unchanged. Because the intracellular regions have transport properties different from the extracellular medium, the definition of bulk tissue transport numbers is more complicated than the definition of transport numbers in electrolyte. 1.2. Discontinuous Ionic Flux in Bulk Tissue The idealization of constant transport number in electrolyte solution is useful and accepted in phj^ sical theory. Fortunately, no further assumptions are necessary in order to derive bulk transport properties for tissue which are similar to those for electrolyte solution. That is, potassium transport in bulk t It is important to note that these are not the only conditions under which "j. may be linearly related to electric current. For example, if a — constant and C.Vp is small: PV'(0.C.V0) — 0 holds, and the electric potential satisfies, and C. nearly satisfies, \lie potential equation. If, on the boundary of some region, the potential <j> and the concentration C satisfy boundary conditions of the form 90/9n= K 9C / 9n, then V C will be proportional to V<£ everywhere in the region and so "jj1— #.VC. + i>t!z.aV0 is proportional to I the electric current. A physical situation similar to this one, with more complicated boundary conditions, occurs in bulk tissue and is reflected in the mathematical form of averaged coefficients in Sections 1.3 and 6.6. I V . Canonical Problems and the Computat ion of B u l k Properties / 96 tissue is a l inear function of the electric current, and the electric current is specified by a constant coefficient potential equation. The physical basis for the transport number theory in bulk tissue is now described. In the extracellular medium, K + ions have a relat ively low tabulated transport number of about .012, while N a + has a transport number of about 0.4, and CI of about 0.6. In contrast, because gl ia l and resting neuronal membrane are p r i m a r i l y permeable to K + (Dietzel et a L , 1980) the transport number for K + across such membranes is close to uni ty . It is difficult to estimate the transport number for K"*~ inside glia or neurons because the intracel lular medium is different f rom standard electrolyte solutions, containing cell organelles and a var iety of organic molecules. However , + + since [K ]. is considerably higher than [K ] , the intracel lular transport number • + + of K must be closer to uni ty and the f lux of K m u s t be a larger fraction of the intracel lular electrical current than in the extracellular current. Therefore, i t w i l l be assumed that the intracellular transport number for K + is uni ty . This assumption about intracel lular transport number is not crucial (though it simplifies calculation), but since we w i l l assume simple geometry it seems appropriate to make simple assumptions about transport numbers in this section. The above idealization permits a conceptually simple description of electrically mediated spatial potassium transport . Because of these assumptions, the K"*~ f lux vector is near ly unaffected by the electric field i n the extracellular m e d i u m , but is proportional to the electric current vector i n the intracel lular medium. The situation can be visualized by imagining that the electrical streamlines (charge paths) passing through the inhomogeneous m e d i u m consisting of IV. Canonical Problems and the Computation of Bulk Properties / 97 intracellular and extracellular compartments are coloured. Some of these streamlines will pass through one or more cells. If current streamlines are coloured blue extracellularly and red where they pass through cell interiors then relatively more electrical potassium transport occurs along the red segments of streamlines, and relatively little along the blue segments because of the difference in intracellular and extracellular transport numbers.! If there are no initial concentration gradients, then the average of the red flux is the average K flux at the instant the electric field is turned on. In the steady state, the average K + flux is the spatial average of the red intracellular flux and the diffusive flux due to extracellular concentration gradients. 1.3. Local Transport Number in Bulk Tissue Formal expressions for the fluxes of electric charge I (current) and potassium ^j^ : = ^2 a r e obtained from the multiple scales procedure. To define a local average transport number it must first be shown that average electric current is determined by a linear equation with constant coefficients over a region which includes a large number of cells. This requirement concerning the equation for <f> follows from the form of the multiple scales expansion and the discussion of averaging in Chapter III. As was described earlier, 'local' means, over a length scale a which includes many cells but is small compared to the measurement scale, L 0 , for example, p = e ^ for p < 1. We denote such a region by t In the extracellular medium, K + moves primarily by diffusion (in this idealization). Because the extracellular medium contains high concentrations of other current carrying ions, Na and CI , there is no inconsistency in the specification of <f>. Diffusion does not cancel the electrical transport because, unlike the electric flux, it has no preferred direction in free space or in cortical tissue (Nicholson & Phillips, 1981). I V . Canonical Problems and the Computat ion of B u l k Properties / 98 R . M A linear relationship between bulk current and potassium flux follows f rom an appropriate interpretation of the averaged coefficients occurring in the expansion since, physical ly , the averaged coefficients are equivalent to averaged fluxes. The linear relationship of averaged current and average ionic f lux implies that the local transport number theory carries over to tissue. B u l k transport numbers describe any local l inear relation between an electrical and ionic f lux. It is simple to define them physica l ly . We denote the extracellular and intracel lular transport numbers by and t j^ respectively. There are at least two approximate ways to define bulk transport numbers which are useful. In general, the relation between " j -^ and I w i l l be a matr ix function instead of a simple proportionality. The f irst definition assumes that " j -^ is zero in the extracellular space and thus the average of "j j^ involves an integral only over the intracel lular space. (1.3) MW(1K) := ^ ^ I d A , where W denotes the unit cell . Equat ion (1.3) defines a local l inear operator applied to I . Since any such linear operator is associated w i t h a matr ix , this defines a m a t r i x of 'local transport numbers ' . The second definition assumes that a = constant, vV • ( C 2 V<j>) — 0 and no average change in concentration occurs over the unit cell . This case is a modification of the situation described in the footnote of Section 1.1. The average of is given approximately by: I V . Canonical Problems and the Computat ion of Bulk Properties / 99 (1.4) M (]K) := 1{ / t i l d A + 1 / tJt'ldA + v J aV^ dA}, W * iwl I C S K lwl E C S K E C S J where \pj satisfies membrane j u m p conditions of the form I • n = v o^7$j' n and periodic boundary conditions on the unit cell . The function xj/j corresponds to the flux due to concentration gradients set up by the j u m p conditions. Despite the periodic boundary conditions, the average of Vi/>j is not zero because of the presence of the biological cell. Because \pj depends l inearly on I through the j u m p condition, equation (1.4) defines a linear operator applied to "I and this defines a matr ix of 'local transport numbers ' . Thus , assuming the transport number theory is val id in electrolyte, it m a y be deduced that bulk tissue has transport properties which paral le l those of electrolyte solution, though the transport numbers are not, in general, scalars. 2.0. FINAL SIMPLIFICATION OF THE B U L K EQUATION Inspection of the governing equations 'for <j>o,o0, and -C. (III.2.8) -(111.2.9) shows that these quantities are constant extracellularly and constant intracel lularly over for a=e^. Thus , the coefficients of the equations for the dependence of 4>y,o~y, and C. 1 on the fine variables are constant both extracel lularly and intracel lular ly over R ^ . Hence, locally, to leading order, the transport number theory is va l id intracel lularly and extracel lularly over R ^ . It is now assumed, in addition, that the transport number theory is va l id over asymptotical ly larger regions R ^ , u=e®, consistent wi th the transport number theory of the physical l i terature. Mathemat ica l ly , this assumption means that we assume VOQ^O and that 0 . C . is replaced by t.a0 in the expression for I V . Canonical Problems and the Computat ion of B u l k Properties / 100 electrical ionic f lux. In the calculations which follow, a geometry has been selected in which matr ix coefficients reduce to scalars. It has been shown that a bulk transport number theory is va l id locally. The assumption that transport numbers are constant throughout the extracellular and intracel lular spaces, therefore, implies a bulk transport number theory which is va l id throughout the tissue. Whi le this assumption is ad hoc in a mathematica l sense, it is important to note that the assumption a — constant is a consistent assumption because a does not appear on the right-hand-sides of (III. 2.11). Also these assumptions are supported by experimental studies (Gardner-Medwin, 1983 a, b; H a v s t a d , 1976) which found no evidence that such non-linear effects quali tat ively affected bulk properties. The idealization that the (electrolyte) transport numbers in the extracellular space and intracel lular space are constant w i l l convert the variable-coefficient averaged equation into an equation w i t h constant coefficients. This is useful because the coefficients of the averaged equation have to be computed n u m e r i c a l ^ and the numerica l computation of the values of variable coefficients is impract ical here. A t the same time, the dependence of the membrane potential on K + concentration and the presence of intracel lular and extracel lular compartments w i l l make the averaged equations different f rom those which hold in electrolyte solution. Inspection of (III.2.11) shows that it is not consistent to assume that the concentration C is constant in computing <f>. However , some comparisons w i l l be made w i t h such a model because it is simple, and, as we show by direct computation, qualitative!}' correct in some respects. IV. Canonical Problems and the Computation of Bulk Properties / 101 3.0. COMPUTATION OF COEFFICIENT ESTIMATES 3.1. Introduction In order to apply the results from Chapter III and the present chapter to ion transport in tissue, it is necessary to solve the canonical microscopic boundary-value problems for the variables x > Y > X > X , ><• , K , K , and K , r- ~p> / v y , ^ c > /v.g> p> y ' c> g> to be determined in (III.2.12). These quantities satisfy Laplace's equation intracellularly and extracellularly, periodic boundary conditions on a fundamental domain W, and the jump conditions (III.2.13) - (III.2.16) which have the form: - f f o { V VX p i - n + n,} = g{xp1~ * p i + Kpi "pM' (3.1) - 0 C o V v K p = g(l - 0 C o / a o ) ( X 1p 1 - X ^ , + 4i~ Kp,}: "ffo {V VX v 1-n+-n 1} = gjx^-XVT + (3.2) - 0 C o V v / c v 1 -n = g U - G - C o / a o M ^ - ^ + K ^ - K S , } , - a 0 V v X c 1 - n = Si * c i X^ T + *^i}> (3.3) - 0 C o V -n = -e{C°0Vv%rn- + n,} = gU-ec0/a0){3k->$1 + 4 - '$ i} ' IV. Canonical Problems and the Computation of Bulk Properties / 102 where g: = g 2 at the -cell membrane. Solutions must be obtained for a range of values of membrane and intracellular conductivity values. Since it is impractical, numerically, to obtain many solutions of these boundary-value problems in three dimensions, solutions were computed in two dimensions. Although bulk conductivity and other properties should depend on whether the calculations are carried out in two or in three dimensions, the values we obtain are consistent with the experiments. Also, the discussion in Chapter V of transcellular conductance (average current flowing through cells per unit voltage drop) when membrane resistance is extremely high or low will suggest that the qualitative behavior of this conductance does not depend on the number of dimensions in which the computations are conducted. Therefore, a two-dimensional computation should suffice to estimate the order of magnitude of the coefficients in (III.2.17) and to determine the nature of their dependence on a number of important physiological parameters such as cell size, membrane conductance, intracellular conductivity, the extracellular space fraction, and the relative positions of the cells. For convenience in computation, cells are assumed to have straight boundaries. It is assumed that cells are square, although rectangular, or other rectilinear cell shapes would pose no special difficulties. Below, it is shown that this assumption reduces x and K to scalars. I V . Canonical Problems and the Computat ion of B u l k Properties / 103 3.2. G e n e r a l P r o p e r t i e s o f S o l u t i o n s The unit cells W emploj'ed in straight and staggered arrays are shown in F igure 3.1 surrounded by dashed lines. The jump conditions (3.1) - (3.3) in which n ^ appears are equivalent to a set of line sources at the membrane surfaces (indicated in F igure 3.1 by ± ) . Because of the source distributions and geometry, solutions are odd functions about the lines O and even functions about the lines E . Typica l solution surfaces for Xp> Xc> Xy> K C , K ^ , and K ^ , which satisfy Laplace 's equation, periodic boundary conditions and jump conditions (3.1) - (3.3), are shown in Figures 3.2a-f. The horizontal axes are in units of h = (1/72)L where L\— -elu^lLy, is the length of the unit cell, h is the numerical space step for this solution, and the vert ical axis is dimensionless. The numerica l values of the solution surface w i l l depend on the conductivity distribution intracel lular ly , extracel lularly, and at the membrane, and the intracel lular and extracellular transport numbers, t ^ and t^. It is seen that the solution surfaces are discontinuous across the membranes. This j u m p apparently occurs over a length (1/72)L in the pictures, because it is convenient to represent the membrane as a region of reduced conductivity between mesh points, t There is an apparent non-uniqueness in the solution of (III.2.10) - (III.2.11), because the proportional boundary conditions in (III.2.11) give rise to an arb i t rary constant in the j u m p across the membrane. In addition, when the intracel lular transport number , t ^ is uni ty , the K quantities are given by an arb i t rary constant intracel lular ly . This non-uniqueness is resolved by the hypothesis in Chapter II t In a centred finite difference scheme the difference between the finite difference solution at adjacent mesh points represents the true solution derivative (flux) ha l f -way between the mesh points. I V . Canonical Problems and the Computat ion of B u l k Properties / 104 Figure 3.1. Two-dimensional biological cells can be arranged in straight or staggered periodic a r rays . The crystal lographic uni t cells in each case (indicated by dashed lines) are different. P lus ( + ) and minus (-) signs correspond to the signs of the v i r t u a l sources associated w i t h the j u m p conditions 3.1 - 3.2 at the membrane. "1'0 5 F i g u r e 3.1a S t r a i g h t U n i t C e l l i _ .± ' 0 F i g u r e 3.1b + 1. .4. 1 0 Staggerec U n i t C e l l Array Geometries I V . Canonica l : Problems and the Computation o f . B u l k Properties / 106 F igure 3.2. Typica l ( L = 7 2 . 1 Mm) solution surfaces for canonical solutions X , X , X , K , K , and K are shown in F igure 3.2.a -3.2.f, while solution surfaces for v c p V staggered a r r a y s are shown in Figures 3.2.g 3.2.1. These solutions satisfy Laplace 's equation, periodic boundary conditions, and the j u m p conditions (3.1) -(3.2). E a c h solution was computed on a 72 x 72 grid w i t h an extracellular space fraction of 23%. These solutions represent, respectively, the 0 (e ) perturbations indicated in each figure title. A description of the m a i n features of these solutions is given in Section 3.2. Kp: Perturbation in cp oc s Vcp0 FIGURE 3.2.a L = 72.1/z Straight Array Perturbation in cp £ FIGURE 3.2.b L = 72.1/i, Straight Array FIGURE 3.2.C L = 72. tyx Straight Array M O K'- Perturbation in C °c £ VC G 0 FIGURE 3.2.d L = 72.1/x Straight Array K ' Perturbation in C £ Vcp. FIGURE 3.2.e L = 72.1/x Straight Array |1.12 Perturbation in C £ VV FIGURE 3.2.f L = 72.1/i. Straight Array j 113 X \ Perturbation in cp ^ £ VcpQ FIGURE 3.2.g. L = 72.1/i, Staggered Array Perturbation in cp £ wmmmmmm / FIGURE 3.2.h L = 72A/J, Staggered Array -1-15 FIGURE 3 . 2 . i L = 72.1/i. Staggered Array 116 L = 72.1/i. Staggered Array 117 h y Perturbation in C °< E VcpQ FIGURE 3.2.k L = 72.1/i, Staggered Array r i 8 Perturbation in C ° c s VV v 0 FIGURE 3.2.1 L = 72.1/x Staggered Array IV. Canonical Problems and the Computation of Bulk Properties / 119 that the pump fluxes force the net ionic flux and electric current to be zero over the cell membrane. Most of the unit cell is occupied by the biological cell, which is surrounded by a narrow extracellular gap. Solutions are zero along the two opposite ends of the unit cell which are axes of odd symmetry and have zero normal derivative along the sides of the unit cell which are axes of even symmetry. The solutions have two types of appearance; one type of solution is nearly a planar segment intracellularly, which is non-zero and tilted because of the presence of source terms, and the second type of solution is non-zero because of flux conditions at the membrane and exhibits considerably more curvature. Solution maxima ranged from approximately .15 to 50, depending on the parameters and type of solution. 4.0. N U M E R I C A L M E T H O D S 4 .1 . C h o i c e o f N u m e r i c a l M e t h o d The canonical microscopic problems specified by -(111.2.10) - (III.2.15), have been solved using finite differences. Details of the numerical algorithm are discussed later in this section. It is shown here that the features of the biological modelling problem make our choice of numerical method more suitable than other possible choices. It is not suggested that other methods could not be applied, but that finite differences is a simple and appropriate method in view of the special features of the modelling problem. Two special features of this biological problem are the low membrane conductance, and the relatively small extracellular space fraction necessary to I V . Canonical Problems and the Computation of B u l k Properties / 120 simulate the biological problem. F i r s t , i t is easily verif ied that when the membrane conductance g->0, the problems (III.2.10) - (III.2.15) have no unique solution and the intracel lular and extracellular solutions differ by an arbi t rary constant. Thus , it is expected that at g = 0 the coefficient m a t r i x associated w i t h any numerica l method w i l l be singular and for smal l g it w i l l be ill-conditioned and difficult to invert numerical ly . I f g were zero, an a r b i t r a r y constraint could be added to the equations to make the solution unique, but physiological values of g are s m a l l , so that some other approach is necessary. Second, i t is desirable to be able to determine the dependence of solutions on the extracellular space fraction independent of the shape of the extracellular space and to employ biologically relevant extracellular space fractions less than 0.2. These features make certain classical methods for solving the potential equation, those involving integral equations, and those involving separation of variables, difficult to apply. One approach to solving the potential equation in a n a r r a y of periodic inclusions assumes that a separation of variables solution is available for .the potential equation in the intracel lular and extracellular domains, t M c P h e d r a n & M c K e n z i e (1978) used this technique to compute the average conductivity of an infinite a r r a y of spheres w i t h conductivity k^ imbedded i n a medium of conductivity k^. Al though the biological problem here differs f r o m their problem, i n that cells are surrounded by membrane, and our equations are coupled, the same techniques are applicable. The technique consists of equating the values of the periodic separation of variables solutions centred at different cells to obtain an infinite system of l inear equations f o r the series coefficients. This infinite system is truncated at some large M , where M is the row dimension of the t This technique was anticipated by M a x w e l l (1878), exactly 100 years earlier. IV. Canonical Problems and the Computation of Bulk Properties / 121 coefficient matrix, and solved exactly. Also, the use of a separation of variables solution depends on the availability of such solutions in the form of classical special functions, for example, those available on elliptical regions. The use of elliptical regions in two dimensions has drawbacks, however, because the largest packing fraction (fractional area occupied by cells) which can be attained in an array which is not staggered is 7r/4 = 0.785. Since the usual extracellular space fraction is about 0.2, it becomes necessary to use unit cells with geometries close to this theoretical packing fraction. At the theoretical limit (7T/4) the extracellular space has zero width where the cells touch and the width of this gap controls bulk conductivitj'. For example, if the conductivhy of the inclusions is zero, bulk conductivity is zero when they touch; while if the inclusions are perfect conductors, bulk conductivity is infinite when they are in contact. Therefore, if cells are elliptical, it is difficult to study the effect of small extracellular space fractions. Because this can only -be achieved by rearranging the cells, the effects of rearrangement and extracellular space fraction changes cannot be separated. Unfortunately, most obvious cell shapes for which separation of variable solutions are available have similar drawbacks. Other methods for solving the potential equation involve integral equations. We denote typical solutions of the canonical problems by Since A\p = 0 intracellularly and extracellularly, the solution of such a boundary-value problem is determined by the normal derivative dip / 3n of the solution at the cell membrane, using a Green's function. Solution techniques for the potential equation based on Green's functions when the conductivity is piecewise constant have been described by Geselowitz (1967). Such an approach is possible for any cell shape IV. Canonical Problems and the Computation of Bulk Properties / 122 including the square cell shape assumed here. However, the near singularity of the numerical problem for small values of the membrane conductance, g, is a difficult}' for this method since the coefficient matrices which arise do not have any useful structure. For these reasons it seemed desirable to choose a rectilinear shape for the inclusions and to solve the canonical problems using a finite-difference method. The choice of a square or rectangular shape creates additional difficulties because such domains contain corners at which the spatial derivatives do not exist. Thus, numerical methods based on the existence of these derivatives suffer a loss of accuracy near the corners. For the potential equation in which the solution (but not its derivatives) is continuous at the corners, this is not a serious difficulty. Near a two-dimensional corner, with angle a radians, the solution \p of the potential equation (where 9\///3n but not i// is discontinuous at the corner) takes the form: (4.1) Re{A(z - z 0 ) 1 + a/27r} + f(z), where z is a complex variable, f(z) is analytic, Z Q is the location of the corner in the complex plane, and A is a real constant. This formula becomes asymptotically correct near the corner so that at mesh points adjacent to the corner the solution has the form f(z) + 0(h~" + a ' ' 2 7 r) as h—>0. The finite-difference formula produces an approximation to f(z) which is second order away from a boundary or jump locus. Finite difference methods are less accurate at boundaries (Mitchell, 1969), however. If a method with second-order accuracy in space is used, for example, accuracy near a boundary (or jump locus) is first IV. Canonical Problems and the Computation of Bulk Properties / 123 order. That is, the local truncation error of the finite-difference formula near the boundary is 0(h) where h is the space step. Hence, the accuracy of the finite-difference formula near the corner, which is 0(h^ + a ^ 2 7 r), is as good as the usual accuracy at boundary points. The magnitude of the global error may be estimated by variation of the mesh size. 4.2. Numerical Algorithm The finite difference approximation to the Laplacian: (4.2) 61 51 U.. = U. . , ,+U. • ,+U. , n . + U. . -4U.. W 1 W 2 IJ 1J+1 1J-1 1 + l j 1 - l j IJ was employed in the interior of each (intracellular and extracellular) domain. From a computational point of view it was convenient to model the jump conditions as discontinuities in the coefficients of the original potential equation. Thus, the region in the Figure 4.1 is assigned a conductivity °~,mYi ~(mS) with the property that as h—>0 the approximation to the transmembrane current may be written equivalently as (4.3) ( ^ ( V - t f V h = a m (^ - / ) where o is in mS cm \ In this way the same numerical scheme could be m J used throughout intracellular and extracellular domains and across the membrane. This is an advantage because this means that the stability properties of an iterative numerical scheme are preserved. The line sources at the membrane surfaces which are equivalent to the I V . Canonical Problems and the Computat ion of B u l k Properties / 124 Figure 4.1. In the finite difference approximation, the membrane is represented by a region M , of thickness h and conductivit j ' a ^ ( m S ) . Conductivities in the extracellular ana intracellular spaces are o° and o , respectively. The unit cell is bounded by the dashed lines. ECS o a ICS ConductlvL11) D i s t r i b u t i o n Ln U n i t C e l l F i g u r e 4.1 IV. Canonical Problems and the Computation of Bulk Properties / 126 jump conditions (3.1) - (3.3) are of magnitude la°_arrinl extracellularly and |°"!-°"mnl intracellularly. The line sources are discretized, and represented by point sources. The intensities of the individual point sources on the source loci depend on h. since the sum of the point sources equals the total intensity of the line source. The difference equations for the model system adjacent to the membrane are (4.4) U. . ,+U. . . , + U . , - + X r U. , , .-(3 + X r)U.. + F(X r - l ) + q.. = 0, . r=l,2, ij + 1 I-IJ l + l j y TJ where q„ is a source density. This difference formula is zero when (ij) are on membrane loci of the form pictured in Figure 4.2.A within the same tolerance, where F is a source strength, \*: = hg/o°, and X 2 : = hg/a x . Thus, X r is proportional to the mesh size h, reflecting the fact that while LL^ + ^ - U.^ goes to zero as h—>0 intracellularly and extracellularly, the transmembrane jump tends to a non-zero value. At inside corners (4.4) -was modified to (Figure 4.2.B) (4.5) U. . ,+U. , - . + X 2 (U. . . + U. . , 1)-2(l+X 2)U.. + F ( l - X 2 ) + q.. = 0, and at outside corners: (4.6) U. . , t +U. • ,+U. , , .+U. , -4U.. + F ( X ° - l ) + q.. = 0. y + i i j-i i + i j i-i j y y In some cases X was taken as small as .5 x 10 . I V . Canonical Problems and the Computat ion of B u l k Properties / 127 F igure 4.2. The locations of the points ( i - l j - 1 ) , ( i - l j ) , (iJ-1), ( i j ) , etc., in the difference formulas (IV.4.4) - (IV.4.6), is shown near the membrane locus, indicated by the solid lines (compare F igure 4.1). o o o o o lL, j - l ) IL,j) (i,j+n -e- -e- IL+l,j) o o o o o o o o o o A (L-l,j) (L-l,j+l) '1-2 8 fl -A-<<-,j) u , . +1) A (L , j -n A A L (l + l. A B L o c a t i o n of p o i n t s Ln d i f f e r e n c e formulas F i g u r e 4.2 IV. Canonical Problems and the Computation -of Bulk Properties / 129 While the boundary-value problem being solved is ill-conditioned for small g, the corresponding initial-value problem contains no such difficulty. This suggests that certain types of iterative methods will not be affected by this type of ill-conditioning. For this reason, a commercial iterative solver for the two-dimensional potential equation with variable coefficients (NAG library D03UAF) was employed. This solver performed a single iteration of the strongly implicit procedure described by Ames (1965). Making use of the symmetries, the minimum area over which a numerical solution had to be obtained was one quadrant of the unit cell, bounded by the lines O, E, and the dashed lines in Figure 3.1. Because the staggered unit cell array was more complicated, the use of symmetry was less effective in reducing the size of the numerical problem for a fixed extracellular space gap. Because of the symmetry in the problem solved here, appropriately chosen Dirichlet and Neumann conditions were equivalent to periodic boundary conditions. The symmetry was necessary in order to apply the present numerical method, since Dirichlet, Neumann, or Robin conditions were required as input to the solver. The numerical iteration was started with an initial guess for ( Xp , ). Coupling of the pairs of dependent variables (x ,K ), (X ,K ), (x ,K ), was p p C C V V acheived by using each variable to generate a source term in the iterations performed on the other. For example, k. iterations of the solver are performed toward the solution for + * in V 2 ^ 4 " 1 = 0, (4.7) T V . Canonical Problems and the Computation of B u l k Properties / 130 - e C 0 V v * c * + 1 . n = g ( l - 0 C o / a o ) { x p r-Xpr + K £ + 1 > L - K * + 1 > 0 } , followed by 1. iterations of the solver toward the solution f o r X p ^ ^ i n v 2 x p n + 1 = 0, (4.8) - a 0 { V v 3 ^ + 1 .3 + n,) = g i X p ^ ^ - X p ^ ^ + K p ^ + ^ - ^ n + M ^ where (k.,1.) are a sequence of pairs of integers. A cyclic sequence for (k.,1.), i = l , . . . , 6 ; given by (20,1),(1,2),(1,2),(2,1),(2,1),(1,20) was found by t r i a l to be effective, and produced much faster convergence than (k.,L) = (1,1), i = l , . . . , N . However , iterations in the above sequence were terminated at a preset error tolerance and the overall solution algori thm terminated when the residuals in the finite difference equations -5 -5 were < 10 , and the solutions had not changed by more than 10 . In addition, i t was necessary to use large values of an internal D 0 3 U A F parameter A P A R A M , to rel iably avoid failure (blowup) of the iteration procedure. We took A P A R A M = 1 0 in the X iteration and A P A R A M =100 in the K solutions. Solutions at neighbouring values of X 1 were used to start the iteration for new values, a procedure called continuation. The results of v a r y i n g mesh size are described in Appendix I V . A , but the results from the finest grid used (72 x 72) w i l l be reported. W i t h continuation, some 100-300 iterations were required to obtain the accuracy described. It was convenient to implement this procedure on the Float ing I V . Canonical Problems and the Computat ion of B u l k Properties / 131 Point Systems processor model 164 ( F P S - 1 6 4 / M A X ) attached to the general purpose A m d a h l computer at the U n i v e r s i t y of B r i t i s h Columbia . , To assess the effect of the coupling between the concentration and potential equations, some solutions of an 'uncoupled' version of the Xp system, in w h i c h the K terms were omitted in the f irst equation of (3.1) are described in Appendix I V . B . Solving the uncoupled equations was less time consuming than solving the coupled equation w i t h j u m p conditions (3.1) - (3.3). However , there was some indication that the uncoupled equations were more difficult to solve, since continuation and parameter selection required greater care than the coupled case when the N A G routine was used. A finite difference method s imilar to that described for the coupled case was used to obtain the uncoupled solutions. 5.0. BIOLOGICAL P A R A M E T E R SELECTION Because the geometry selected for the cellular m a t r i x does not model the real geometry and because exact microscopic parameter values are not k n o w n i n some cases, i t is necessary to choose length scales and membrane conductances so that they approximate the characteristic values prevalent in the neural tissue and to explore the dependence of the solutions on these parameters in order to obtain qualitative results. Results which are not sensitive to the geometry and parameters chosen m a y be assumed to hold in vivo . A s w i l l be shown in Chapter V , for purposes of analysis , cell size and membrane conductance m a y be combined into a single parameter , analogous to the electrotonic length of cable theory. This fact was used in the choice of the order of the jump conditions in Chapter III. In what follows, it is assumed that 2 the cells under discussion have membrane resist ivi t j ' equal to 320 Ocm , which is H a v s t a d ' s (1976) estimate of gl ial membrane conductance. Deduced parameters IV. Canonical Problems and the Computation of Bulk Properties / 132 then depend on the characteristic size of the cells and their surface/volume ratio (discussion below). For other cells (neurons), membrane resistivity might be 3000 2 to 5000 Ocm , based on the resistivity of dendritic membrane. In this case the parameters employed in numerical studies correspond to cells which are larger by a factor equal to g / g, (or its square root, as we describe below) where g and g^ are glial and dendritic membrane conductances, respectively. In choosing biologically relevant parameters, we followed Gardner-Medwin (1983) in taking into account the surface/volume ratio for cells in the brain. This has been estimated at 5 um 1 (Horstmann & Meves, 1959) on the average, or 2 3 5 /um of membrane per Mm of tissue. The (two-dimensional) cells are square and, as discussed in Section 3.3, the choice of straight cell boundaries was made to facilitate the study of the effect of realistic extracellular space fraction values, as well as for ease in computation. The dimensional length of the unit cell is denoted by Z and the biological cell will have length fZ, where f is a fixed fraction close to unity. For an extracellular space fraction of a = 23%, the value of f is .875. Had these elements been extended normal to the W-plane, the 2 resulting square shafts would have had surface volume ratio AU 11 = 4f/Z. The parameter X1 can be written as X 1 =hg/ a ° = Zg/ 24 a ° and X 2 =hg/ a* = Zg/ 2 4 a 1 for a mesh which is 24 x 24. The value of g in two dimensions may be deduced from the assumption that these extended shafts have the membrane conductance observed experimentally. For each characteristic size Z, it was assumed that the effective membrane conductance was increased by a factor equal to that necessary to make the surface volume ratio of the unit cell equal to 5 Mm \ say Z=Z,. The formulas for X 1 when the surface-volume ratio is taken into account and when it is not are: I V . Canonical Problems and the Computat ion of B u l k Properties / 133 X 1 = (5.1) X 1 = J _ _ L L g _ i k = 24 F 4f 2 The correspondence between Z 0(/zm) and X 1 for g* = 320 flcm is given in Table 5.1. If the mesh were 48 x 48, the corresponding X 1 values would be half these values; or l0 twice as large for the same values of X 1 when the surface / volume correction is not applied. Our numerical studies of membrane conductance wi th a 24 x 24 mesh were carried out wi th X 1 values selected f rom those listed in Table 5.1 where Z 0 and I!, respectively, are the characteristic unit cell sizes associated w i t h corresponding assumptions about the surface / volume ratio. It is expected that characteristic sizes between 5 u m and 50 u m wil l be appropriate for gl ia while characteristic sizes between 50 u m and 500 Aim w i l l be appropriate for neurons or syncyt ia l glial elements of s imilar length. The intracellular conductivity is usual ly assumed to be 10 m S (Shelton, 1985; G a r d n e r - M e d w i n , 1983; H a v s t a d , 1976), but here some studies were carried out w i t h intracel lular conductivities of 4 m S and 2 m S to establish the sensit ivity of the results to the intracel lular conductivity, because this conductivity is uncertain. Knowledge of the sensit ivity of results is useful for comparisons w i t h the qualitative investigations of Chapter V . I V . Canonical Problems and the Computat ion of B u l k Properties / 134 Table 5.1. U n i t Ce l l Lengths. 1 0 3 X 1 l0(nm) 11 (Mm) 100 153600 328 50 76800 234 20 3072 146 10 1536 103 5 768 72.1 2 308 46.0 1 154 32.5 0.1 15.4 10.3 6.0. P A R A M E T E R S T U D I E S 6.1. C o m p u t a t i o n o f C o e f f i c i e n t s f r o m C a n o n i c a l P r o b l e m s Since the unit cells are unchanged by an interchange of the space coordinates, the canonical problems are the same for each coordinate direction. The computation of the bulk coefficients reduces to the evaluation of integrals of the form (6.1) f K,(w){K2 + dj_ }dA, W 3w. J where K . , j = l , 2 is proportional to o " ( w v , w 2 ) and \p, a typical canonical solution is independent of the spatial direction. T h u s , bulk coefficients reduce to scalars (cf. Section III.2.2). In addition, because of the s y m m e t r y discussed i n Section IV. Canonical Problems and the Computation of Bulk Properties / 135 3.2, the off-diagonal elements in the bulk coefficient formulas are zero under averaging. The integral (6.1) over the intracellular region may be evaluated directly since K , is constant intracellularly. It is convenient to avoid numerical integration of the derivative when the integral is taken over the extracellular region. For this purpose, the second term of (6.1) is integrated by parts in W. and the periodic boundary conditions are used to give As 3K, / 3 W , (the case of j = l) is a set of line sources, this reduces the calculation to a summation over the source locations. In computing coefficients from the finite difference solutions, it is possible to assume either that the membrane lies halfway between the adjacent mesh points on opposite sides of the membrane, or that the membrane contains the extreme extracellular and extreme intracellular mesh points. The first assumption necessitates an extrapolation to determine solution values at the membrane. This extrapolation cannot improve the asymptotic accuracj' (in h) of the coefficient estimates, so nothing is lost if (Figure 3.1 and 3.2) the available values at the extreme mesh points are used instead. Thus, the extracellular space fractions referred to below use the second assumption. (6.2) / i//9K l ClA. W "5w-J I V . Canonical Problems and the Computat ion of " B u l k Properties / 136 6.2. Bulk Conductivity and Flux Proportional to Electric Field The coefficients D , and D 2 of the average governing equations (III.2.17) contain the averages of the canonical solutions and X p which represent, physical ly , 0(e) fluxes proportional to the extracellular potential gradient, V0O-The fluxes proportional to ^wXp> where x p is the perturbation in <f>, and V^K^, where K p is the the perturbation in C , are discussed i n this section. U s i n g (III.2.17), the bulk conductivity m a y be identified w i t h the coefficient D , because DYV<l>0 is the bulk electrical current which is proportional to V 0 o - t The effective transcellular conductance is defined as the coefficient D 2 = a ^ M^r { t ^ t p ( l + V w X p - | _ ) + V K -.} and in the uncoupled model the transcellular conductance is defined w p i ' r by 0"^M^ r {t j^t^(l+ V w X p -^)} ' - The effective transcellular conductance is a weighted average of intracel lular and extracellular f lux w h i c h is predominantly intracel lular because t ^ < < t ^ and physical ly represents the ionic f lux proportional to <p0. The coefficient D 2 is compared to the transcel lular conductance of the uncoupled model in Appendix IV.B.. These quantities have s imilar numerical values because the intracel lular transport number — 1 and t ^ = 0. The flux associated w i t h "V K occurs in the definition of D 2 but not in the transcellular conductance, w p " This f lux m a y be interpreted physical ly as the diffusive ionic transport due to local concentration gradients caused by ionic f lux through cells. The average of V w / c p makes a positive contribution to D 2 of 9% at =32.5 n m and 5% at 1^=146 | im, compared to the total effective transcel lular conductance. Since X p j in the coupled and uncoupled models are distinct, however, V K does not w p determine the relative magnitudes of the conductances in the coupled and t A s noted in Chapter III, we do not dist inguish between the one- and two-tier models except in Section I V . 6 . 3 , so we write here: D 1 : = a ^ M ^ { t (1 + dx/dW))} where t ^ = a 0 / a ^ . I V . Canonical Problems and the Computat ion of B u l k Properties / 137 uncoupled models. Computed bulk conductivity versus cell sizes are shown in F igure 6 .1 .A, and corresponding effective bulk transcellular conductance estimates in Figure 6.1.B for straight arrays of square cells (72 x 72 point solutions w i t h an extracel lular space of 23%). These calculations assumed an extracellular conductivity of 20 m S and an intracel lular conductivity of 10 m S , which has been taken as typica l . The. bulk conductivity estimates range f rom 2.8 to 3.9 m S in the parameter range of biological interest (Z1 = 32.5 (im to ly = 146 ixm), and are an increasing function of the cell size (membrane conductance). F o r comparison, conductance estimates f rom an uncoupled calculation are presented i n the figures. W h e n Z 1 = 3 2 . 5 / i m , the coupled and uncoupled models agree to three significant figures, while at I, = 146 /xm, they differ in this coefficient by about 20%. These values are consistent w i t h observed conductivities (— 5 mS) for bulk cortex and cerebellum (Nicholson, 1980). A n extensive review of experimental data on electrical properties of biological tissue m a y be found in Schanne and R u i z P . -Ceret t i , (1978). A t low membrane conductances, L = 10.3 u m , bulk conductivity is 2.62 m S and transcellular conductance is approximately 2% of this . The bulk transcellular conductance ranges f rom 7% of bulk conductivity at ^ = 3 2 . 5 j i m , to 30% at Z 1 = 1 4 6 u m . This agrees closely w i t h the uncoupled model at 32.5 jum, but is some 13% less than the uncoupled model at 11=103 um. G a r d n e r - M e d w i n (1983) found that transcellular current, as measured by K + transport over a large (5 mm) region was about 6% of the total current. Hence, the present theory suggests that transcellular current is at least as large as was observed in that experiment, and supports the I V . Canonical Problems and the Computat ion of B u l k Properties / F igure 6.1. B u l k conductivity (A) and transcellular conductance (B) versus cell size (Z-|) are plotted for coupled and uncoupled models for a — 20 m S , and 10 m S . A logarithmic vertical scale is used in B . 1'3 9 >^ o D c o o O CD 8 T 6 + 4 + 0 0 • A A A U n c o u p l e d • C o u p l e d A A A A A 5 0 1 0 0 — I 1 1 5 0 2 0 0 C e l l S i z e (fi) 1 0 . 0 0 0 £ 1 . 0 0 0 o u D "D C o o u c o 0 . 1 0 0 + 0 . 0 1 0 + 0 . 0 0 5 0 A A A A A A A U n c o u p l e d • C o u p l e d B 5 0 1 0 0 1 5 0 2 0 0 Ce l l S i z e (/x) F i g u r e 6.1 Ce l l S i z e a n d Bu lk C o n d u c t i v i t y ' I V . Canonical Problems and the Computat ion of B u l k Properties / 140 Table 6.1. , P . , , and D 2 by_ Cel l Length . D 2 ( 1 0 mS) E 1 .024 .019 .040 .032 .069 .059 .096 .084 .122 .111 experimental demonstration that such currents are significant. It is useful to be able to deduce the values of coefficients other than D ^ and D 2 f rom fluxes proportional to V0 because these fluxes are easy to measure experimental ly (Gardner-Medwin, 1983a). The coefficient, E ^ = a 0 M ^ { t p ( t ^ + VwXv^)}> is formal ly distinct f rom the effective transcellular conductance, but it w i l l be shown by direct calculation that the effective transcellular conductance D 2 is a useful approximation to E i (and E 2 ) . The coefficients E ^ and E 2 p lay an important role in determining the magnitude of the spatial buffering effect. E x a m i n a t i o n of Table 6.1, ' shows that this approximation w i l l be in error by at most 4 to 18%. T y p i c a l values of P 1 ? D 2 , and are tabulated i n Table 6.1. The values of T>y correspond to F igure 6 .1 .A , and m a y be compared to uncoupled values displayed there, while P 2 and E , do not have analogues in the uncoupled model. It is seen that while D , does not v a r y greatly over the biological range of cell size/conductivity, (— 30%) the coefficients P 2 and E , v a r y by a factor of *1 (Mm) 32.5 46 72.1 103 146 D. , (10 mS) .280 .296 .327 .355 .386 I V . Canonica l "Problems and the Computat ion of B u l k Properties / 141 = 5. 6.3. E f f e c t o f I n t r a c e l l u l a r C o n d u c t i v i t y o n B u l k C o n d u c t i v i t y F igures 6 .2 .A and 6.2.B show the -bulk conductivity and effective transcel lular conductance when intracellular conductivity is 10 m S (Section 6.2), 4 m S , and 2 m S . While the influence of intracellular conductivity a 1 was relat ively slight f rom ly = 32.5 ;um up to Z , = 1 0 3 Mm, bulk transcel lular conductance was sensitive to the membrane conductance. Between Z, =32.5 Mm and Z ^ I O S Mm, the transcellular conductance changes by a m i n i m u m of 220% over each of the three choices of a" while the change in transcellular conductance for any X 1 f rom 10 m S to 2 m S is at most 27%. A t Z t = 146 Mm, a five-fold reduction in o 1 reduces transcellular current by 45%. These results demonstrate directly that for the parameter values discussed in Sections 6.2 - 6.3, the intracellular and extracellular electric potential are described qualitat ively by the asymptotic analysis (of Chapter V) for electrical space constants which are long compared to cell length. These asymptotic •solutions and hence, the numerical conclusions, do not depend strongly on geometry. In addition, it is biologically significant that these model cells which exhibit bulk transcellular currents f rom 7 to 30% of total current had electrical space constants which were long compared to the cell dimension, and thus were electrotonically unlike an electrical syncyt ium. In the one-tier model, coefficients other than D y and D 2 are also insensitive to the intracellular conductivity. The values of E y, F y, and F 2 for a" =10 , 4 and 2 m S for various membrane conductances are tabulated in Tables 6.2. The values of E 2 were close to those of E , . A s suggested above, E 2 is I V . Canonical Problems and the Computat ion of B u l k Properties / 142 F igure 6.2. B u l k conductivity (A) and transcel lular conductance (B) versus cell size (I,) are plotted for a = 10, 4, and 2 m S . A logarithmic vertical scale is used in B . 1%3 4 T > o •o C o o 3 + A CT'= 4 m S O a' = 2 m S O A O O A O 13 CD 0 50 TOO 150 2 0 0 Cell S ize (/u,) 1.500-r 0) u c •§ 1.000 ^ • C o . 2 0.500 a> o w c o O a = 10mS A cr '= 4mS O a'= 2mS O O A O o A O B 0.000 0 50 100 150 200 Cell Size O) Figure 6.2 Intracellular Conductivity and Bulk Conductivity I V . Canonical Problems and the Gomputation.,<of B u l k Properties / 144 closely approximated by the transcellular conductance for these cases and thus depends on the intracellular conductivity in a s imilar way. 6.4. Tissue Structure The derivation in Chapter III assumes that the tissue has a two-tier structure as discussed in Section II.2.2. Thus , i t is assumed that a periodic a r r a y of asymptotical ly larger cells is imbedded in a periodic a r r a y of asymptotical ly smaller cells. A one-tier model would assume that cells are surrounded by an extracellular medium consisting only of electrolyte w i t h conductivity = 20 m S , while a two-tier model assumes an extracellular medium containing another cell type, surrounded by extracellular electrolyte. The assumptions of such a model are i l lustrated schematically i n F igure 6.3. The cells have been assumed to be two-dimensional and square, as before, and the cells are not shown to scale. N u m e r i c a l solutions were obtained under the assumptions of the two-tier model. It is assumed that the areas N, in the unit cell V , and G , in the subcell W, have the same relative ICS .fraction, x, and that the total E C S fraction is a = 0.2. To obtain this E C S fraction, over the two-tier unit cell, the relative intracel lular space fraction, x, of each unit cell; V or the subcell W must satisfy x + x ( l - x) =0.8 since the total I C S in V is the space occupied by N plus the space occupied in its (relative) extracellular space by the areas G . Therefore the intracellular space fraction is x = 56%. F o r this reason, the solutions were obtained for a = 0.44, and assuming that the membrane conductance is zero at 0 (e ^), as already assumed in Chapter III. The bulk parameters for the extracellular space were then used in a further set of studies I V . Canonical Problems and the Computat ion of B u l k Properties / 145 Table 6 .2 .A. E 1 Coefficient Versus Intracellular Conduct ivi ty It 10 m S 4 m S 2 m S (Mm) 32.5 .019 .018 .017 46 .032 .031 .028 72.1 .059 .054 .046 103 .084 .074 .061 146 .111 .090 .075 Table 6.2.B. Fj_ Coefficient V e r s u s Intracellular Conduct ivi ty Z, 10 m S 4 m S 2 m S (Mm) 32.5 1.83 1.77 1.68 46 3.11 2.94 2.69 72.1 5.47 4.72 4:06 103 6.76 5.32 4.37 146 6.25 3.97 3.10 Table 6.2.C. F 2 Coefficient Versus Intracellular Conduct ivi ty Z, 10 m S 4 m S 2 m S (Mm) 32.5 4.55 4.49 4.40 46 5.93 5.76 5.51 I V . Canonical Problems and the Computat ion of B u l k Properties / 146 72.1 103 146 8.42 9.99 9.93 7.79 8.76 7.99 7.15 7.82 7.16 F igure 6.3. A two-dimensional two-tier model is schematically i l lustrated. The asymptotical ly larger population (N) is surrounded by an asymptotical ly smaller population (G). U n i t cells are bounded by dashed lines. The spatial coordinate i n the larger unit cell is V and in the smaller unit cell, W . The diagram is not to scale, since the ratio of the sizes of smaller to larger unit cells tends to zero in the mathematical model. 1*4 7 1 1 ICS L . IN) ICS. (GJ fl Two D i m e n s i o n a l T w o - t i e r Model F i g u r e 6 . 3 •, • I V . Canonical" Problems and the Computat ion o f B u l k Properties / 148 at a = 0.44. Thus , we took a o = 5 . 3 9 m S , in place of o"o=20 m S . A s described above, the total extracellular space remains the same. If bulk parameters depend sensitively on tier structure, these studies could produce very different results from the one-tier studies w i t h a = 0.2. The results for the uncoupled model, discussed in Appendix B , indicate that bulk conductivity is altered by a factor of 2 to 3 by differing tier structures (Figure B . 3 . A ) , but bulk transcellular conductance is not sensitive to tier structure. Conductivity estimates obtained for the coupled two-tier model are shown in F igure 6.4. These results indicate a strong coupling effect at a l l membrane conductance values, since they differ significantly f rom the uncoupled results. The computed values of bulk and transcellular conductance are also significantly different than those seen in the one-tier coupled model. Computed bulk conductivities are about half the one-tier results, ranging f r o m 1.56 m S to 1.82 m S . In addition, the transcellular conductance was consistently lower than the one-tier r e s u l t - b y a factor of 3 to 5. The transcellular conductance as a fraction of bulk conductivity varies f rom 7.5% at Z, =32.5 u m to 16% at Z 1 = 1 4 6 / i m (vs. 25% in the one-tier study). Thus , while the bulk .and transcellular conductances were not sensitive to the intracellular conductivity (Section 6.3), the extracellular conductivity has a significant effect on these quantities i n a coupled model. This effect is to decrease bulk and transcellular conductance, and to decrease transcel lular conductance as a fraction of bulk conductivity. The reasons 'for differences between one- and two-tier models are now discussed. The local, coupled, canonical problem for Xp has not been simplified by our use of the transport number theory. Thus , it is expected that for some parameter values, interactions between electrical potential and concentration w i l l I V . Canonical Problems and the Computat ion of B u l k Properties / 149 F igure 6.4. Bulk conductivities (A) and transcellular conductance (B) versus cell size (Z!) are plotted for one-tier and two-tier models. The one-tier model is indicated by filled squares and the two-tier model by open triangles. A logarithmic vert ical scale is used in B . T5 0 CD 5 T >s 4 > o D c o o 3 + 2 + 0 • cr = 10mS, CJ = 2 0 m S , a = .23 A a 1 = 10mS, u° = 5'.39mS, a = .44 A A A 50 A 100 A 150 200 Cell Size (fi) 1 . 0 0 0 -o c o u D X> C o CJ . -2 0.100 + O CO c o B 0.025 0 50 100 150 200 Cell Size (fi) Figure 6.4 A Two-Tier Conductance Study I V . Canonical Problems and the Computat ion of' B u l k Properties / ' 1 5 1 be significant. The average ionic fluxes associated wi th x p and K p are tabulated in Table -6.3. It is found that there is a large fractional change in ionic f lux associated w i t h the C perturbation (column 3), K^, over different membrane conductance values, and that this f lux m a j ' become negative at 7 1 = 146 jtxm. Thus , the coupling effect on transcellular conductance seen in the two-tier results m a y be interpreted physical ly as follows. Transcel lular conductance in an uncoupled model increases wi th the membrane conductance, because more electrical current proportional to V v $ 0 takes an intracel lular route. In a coupled model, inward transmembrane current results in a depletion of [K" 1"] near the outside of the membrane, and outward transmembrane current results in accumulation of [ K + ] near the outside of the membrane. This causes hyperpolarization of the membrane near the region of i n w a r d current and depolarization near the region of outward current (refer to F i g u r e 1-4.1 of Chapter I). The result ing intracellular current flows in the direction opposite to V v 0 o and tends to cancel it. This effect is accentuated as the extracellular conductivity becomes smaller relative to membrane conductance, because increased electrical K + transport through the membrane and decreased electrical K + transport in the extracellular medium causes more depletion (accumulation). The values of D , , D 2 , and E ^ for the two-tier model are tabulated in Table 6.4. It is seen that D 2 remains a reasonable approximation to E ^ Other coefficients are discussed in the next few sections. Studies wi th staggered a r r a y s were performed for o~°= 20.0 m S , a 1 = 10.0 m S , w i t h a — .2 for one- and two-tier models. There was little difference between the results for staggered arrays and straight arrays . The coefficients D , and D 2 are tabulated in Table 6 . 5 . A - C , for straight and staggered a r r a y s for I V . Canonical Problems . and the Computat ion of B u l k ^Properties / 152 Table 6.3. F luxes due to 0 and C Perturbations. * i M — f t ^ t (1 + V X J } M W { V K J 1 W l T C a w A p l J W L w p l J (Mm) 32.5 .0098 .0019 46.0 .0144 .0023 72.1 .0184 .0019 103.0 .0237 .0002 146.0 .0324 -.0038 Table 6.4. Coefficients in (III.2.17) for Two-Tier Model . Z, 0 , ( 1 0 mS) D 2 ( 1 0 mS) E , (Mm) 32.5 .156 .0117 .0079 46.0 .161 .0167 .0127 72.1 .166 .0203 .0164 103 .172 .0239 .0212 146 .182 .0286 .0289 the one-tier and two-tier model. The other coefficients followed the same pattern. This is i n contrast to the results for the uncoupled model (Appendix I V . B ) , in which the differences between staggered and straight arrays grew more pronounced as membrane conductance increased. Thus , the coupling between concentration and electrical potential , which is expected to become more significant w i t h higher membrane conductance, apparently reduced the effects of geometry. I V . Canonical Problems and the Computat ion of B u l k Properties / 153 Table 6 . 5 . A . D , and A r r a y Geometry: One-Tier Model . Z, Straight (mS) Staggered (mS) (Mm) 10.1 2.62 2.50 32.5 2.80 2.69 46.0 2.96 2.84 72.1 3.27 3.15 103.0 3.55 3.44 146.0 3.86 3.75 Table 6 .5 .B. D 2 and A r r a y Geometry: One-Tier Model . Z, S tra ight (mS) Staggered (mS) (Mm) 10.1 .056 .055 32.5 .242 .239 46.0 .397 .389 72.1 .693 .691 103.0 .955 .958 146.0 1.222 1.222 Table 6 .5 .C. Dj_ and A r r a y Geometry: Two-Tier Model Z, Straight (mS) Staggered (mS) (Mm) I V . Canonical Problems and the Computat ion of B u l k Properties / 154 32.5 46.0 72.1 103.0 146.0 1.56 1.61 1.66 1.72 1.82 1.43 1.48 1.53 1.60 1.70 Table 6 .5 .D. D 2 and A r r a y Geometry : Two-Tier Model / , Straight (mS) Staggered (mS) (Mm) 32.5 .117 .121 46.0 .167 .172 72.1 .202 .207 103.0 .239 .242 146.0 ..286 .290 The existence of tier-structure is an important theoretical possibility, because coupling between concentration and electrical potential becomes important in the two-tier model. A l s o , as discussed in Chapter V I , spatial buffering occurs by a different mechanism in this model. H o w e v e r , while the .two-tier model seems plausible, there is currently no experimental evidence to support this relat ively complicated assumption. Thus , the chief conclusion of this section is that the predictions of the one- and two-tier models are broadly s imi lar and the differences between straight and staggered arrays are not large. I V . Canonical Problems and the Computation of B u l k Properties / 155 6.5. Ionic Flux Terms Proportional to Concentration Gradient The coefficients specified by the and xc canonical problem are associated wi th the potassium concentration gradient. If the membrane conductance were zero, this canonical problem would s imply be the governing equation for diffusive f lux. Since the membrane is permeable to potass ium, however, its concentration at the membrane affects the N e r n s t potential to a l l orders. Thus , these coefficients do not depend exclusively on the diffusive properties of the tissue, but also reflect the interaction of the local Nernst potential w i t h 'unstirred layers ' (Schultz, 1980) at the membrane. These effects have not been included previously in a model of this kind and the coefficients ¥^=0^ ' ' " M ^ t V w x c j } and F , = ^ {M^MOC. V K - + $) + O o W t V x .}} do not correspond to £ 1 W L io w c l K a w c l " ^ any commonly employed physical quant i ty , such as conductivity or diffusion coefficient. F o r this reason, we do not discuss uncoupled estimates, or the behavior of the component averages of F , and F 2 . A s discussed in Appendix IV.A, the solutions for X £ and K £ converge slowly as mesh size tends to zero at larger values of the membrane conductance and our estimates of F , , and F 2 are l ikely less accurate than the estimates of D , and D 2 . A t smal l values of the membrane conductance, F 2 is proportional to the bulk diffusion coefficient, reduced by the extracellular space fraction and microscopic geometry of the medium. A t Z , = 10.3 n m , which corresponds to a high membrane resist ivi ty, bulk conductivity is 2.6 m S which is 2.6/20= .13 as a fraction of extracellular conductivity. O n this basis, it m a y be calculated (for 2 comparison) that F 2 , the coefficient of V C 0 i n (III.2.17) would be 2.64 wi th g=0. If g = 0, then F T = 0 in the model. In the one-tier models the effect of non-zero g is to increase F 2 by I V . Canonical Problems and the Computation of B u l k Properties / 156 factors between 1.7 and 3.8, above the value expected w i t h pure diffusion. In the one-tier model the coefficient F 2 is positive and comparable to F, i n magnitude. In the two-tier model, F , is nearly zero at I , = 103 u m and negative at Z i = 1 4 6 £im. In both models the ionic f lux associated w i t h the C perturbation, « c , is more stable to changes in membrane conductance, while the ionic f lux associated w i t h the <j> perturbation, xc> is more sensitive to such changes. The K - f lux varies from 2.9 to 4.55 in the one-tier model and f rom 7.22 to 8.81 in c the two-tier model, while the X -flux varies f rom 1.6 to 5.36 in the one-tier c model and from -.177 to 10.2 in the two-tier model. It is interesting that in the two-tier model, the variat ions in F , and F 2 are chiefly due to • x > the perturbation in the electrical potential <p. In our 2 models, F , and F 2 are the coefficients of V C in the bulk equation (III.2.17). 2 In the original physical equations for electrolyte solution, the coefficient of V C is s imply a diffusion coefficient. The values of F , and F 2 are shown in F igures 6 .5A and 6.5B, respectively, for various values of o\ the intracel lular conductivity, and a, the extracellular space fraction, in a one-tier model. In the one-tier model, F , and F 2 are increasing functions of membrane conductance up to Z 1 = 103 u m . However , both coefficients are relatively insensitive to the parameters over these ranges (i.e., v a r y by factors of about 2 to 3 when the independent parameters v a r y by factors of 5 to 20). A m a x i m u m in the values of F , and F 2 consistently occurs at 1 1 = 103 u m , though it is not pronounced. Electr ical ly mediated transcel lular ionic f lux associated with x is due to local c accumulation/depletion of potassium. Thus , the m a x i m u m m a y be due to the accumulation/depletion effect described in the last section. I V . Canonical Problems and the Computat ion of B u l k Properties / 157 F igure 6.5. Coefficients of V C (in III.2.17) versus cell size (1^) are plotted for different intracel lular conductivities (a =10 , 4, 2mS) and extracellular space fractions ( a = 0.16, 0.23, 0.37). For comparison, the value of F 2 corresponding to pure diffusion is indicated in B by a fil led square. 8.000 T 4.500 1.000 0 12 + •15 8 a = .23, a 1 = 10mS A a = V a' O a = O a = 4mS = 2mS 16 .37 O o 25 50 O A A a = 4mS V a' = 2mS O a = .16 O a = .37 75 6 V 100 o o A V a = .23, a' = 10mS I Pure Diffusion O O. 125 150 Cell Size (/i) O 7 + O O o o A V O B A — I 1 125 150 Cell Size (fj.) 0 25 . 50 75 100 Figure 6.5 Terms Proport ional to VC TV. Canonical Problems and the Computation of Bulk Properties / 159 Figure 6.6. Coefficients of V C (in III.2.17) versus cell size (I ^ ) are plotted for one- and two-tier models. Note that the coefficient F ^ is negative in A at ly = 1 4 6 Mm. 16-© 15 + • a = .23, CT - 10mS O Two-Tier 5 -5 + O O o - 1 5 + A O - 2 5 0 25 50 75 100 —I 1 125 150 Cell Size (fi) 15 5 + -5 + •15 + • • • a = .23, CT' = 10mS • Two-Tier • • -25 + H 1 125 150 Cell Size (/j,) 25 50 75 100 Figure 6.6 Terms Proportional to VC and Tier Structure I V . Canonical Problems and the Computat ion of B u l k Properties / 161 The coefficients F1 and F 2 for the two-tier model at various cell sizes are shown i n F igure 6.6. It is seen that F , and F 2 are not monotone functions of the membrane conductance in the two-tier model, w i t h F , is decreasing above If =46 /nm, becoming negative near ^ = 1 0 3 a m . This reversal of sign does not qual i tat ively affect the solutions to the bulk equation (III.2.17) since the effect of negative F , is balanced by other coefficients (see Chapter V I ) . The coefficient F 2 has a m i n i m u m near lf=10Z am. This cannot be s imply explained as an accumulation/depletion effect. Presumably , this m i n i m u m occurs because the solution surface for xc changes its shape at larger values of the membrane conductance (Z ,=146 a m ) . In other studies (not presented here), in which was lowered to .0074, this m i n i m u m did not occur. 6.6. Ionic Flux Terms Proportional to Nernst Potential Gradient The coefficients specified by the « v and X y canonical problems are associated w i t h the N e r n s t potential gradient. These coefficients are averages -of ionic and electric fluxes due to the ambient gradient in the N e r n s t potential determined by ln( C 2 o ' ' C 2 o > - These fluxes are different f rom the fluxes averaged in F , and F 2 because the F coefficients only reflect accumulation and depletion of K + near the cell . It is shown i n Chapter V I that the fluxes associated wi th the N e r n s t potential represent the most important contributions to spatial buffering at most cell sizes in the one-tier model. The values of E 2 are shown in F igure 6.7A for various values of a\ the intracel lular conductivity, and a, the extracellular space fraction. Values of E y are not shown because they are near ly identical to E 2 . The coefficients are relat ively insensitive to the parameters over these ranges (i.e., v a r y by factors of I V . Canonical Problems and the Computat ion of B u l k Properties / 162 Figure 6.7. In A , the coefficient of V V (in III.2.17) versus cell size (Z,) is plotted for different intracellular conductivities (a =10, 4, 2 mS) and extracellular space fractions (a = 0.16, 0.23, 0.37). The same coefficient is plotted against cell size for one- and two-tier models in B . Both figures use logarithmic vert ical scales. E 2 • a = .23, a 1 = lOmS i ; & 3 A a 1 = 4 m S V a ' = 2mS O O.IOoj O a = .16 ° 8 O a = .37 O 8 o o o A A 0.010 4 1 1 : 1 1 1 1 0 25 50 75 100 125 150 Cell S ize (ji) 0.100 + • a = .23, a ' = 1 OmS O Two-Tier 0.010 + O o o o o B 0.001 25 50 75 100 + -i 125 150 Cell Size (/J.) Figure 6.7 E2 Coefficient and Model Structure I V . Canonical Problems and the Computat ion of B u l k Properties / 164 about 5 when the independent parameters v a r y by factors of 5 to 20) and are an increasing function of cell size. The coefficient E 2 for the one-tier and two-tier models at various cell sizes is shown in Figure 6.7B. It is seen that this coefficient is considerably reduced in the two-tier model. The lower values of E 2 imply that spatial transport of potassium by means of f lux terms proportional to V 0 is significantly reduced in this model. The relationship between transcellular conductance and E 2 was maintained in both models, (with ratio = 1.1). M o r e precisely, the ratio of transcellular conductance to E 2 varies from approximately 1.2 (Z 1=32.5 Mm) to 1.0 ( ^ = 1 4 6 Mm), for both (one- and two-tier) cases. This is important because transcellular conductance can be measured more easily than E 2 , or E , . Relationships like the one between D 2 , E ^, and E 2 ( D 2 = E 1 = * E 2 ) are . useful because this m a y reduce the number of measurements which are necessary in practice. The ratio of effective transcellular conductance to bulk conductivity, D 2 / D 1 ; is the observed transport number in a current passing experiment (see Chapter VI ) . It is useful, to examine the relationships between D 2 / D ^ and E 2 since these might provide further reductions in labour (cf. G a r d n e r - M e d w i n , 1983b). A plot of E 2 versus D 2 / D , is presented in Figure 6.8, us ing log axes. It is seen that there is a linear trend in this plot, but that the data f rom different parameter • studies lie on different l ines. APPENDIX IV.A. CONSISTENCY CHECKS ON T H E COUPLED SOLUTION V a r i a t i o n of mesh size was complicated by the fact that both intracel lular and extracel lular spaces had to remain in the same proportion at the new mesh size. F o r example i f a solution was obtained on a 24 x 24 mesh w i t h an IV. Canonical Problems and the Computation of Bulk Properties / 165 Figure 6.8. The coefficient E 2 versus bulk transport number D 2 / D , is plotted for different cell sizes, values of a , and a, for one- and two-tier models. Logarithmic horizontal and vertical axes are used in A. In B, points from the two-tier model are plotted together with points from a one-tier model using linear axes. It is seen that relationships between E 2 and transport number were approximately linear within the studies identified by the legend symbols. Unless indicated otherwise by the legend, a = 0.23 and a = 10 mS. 16<6 1.000-T 0.100 0 . 0 1 0 -0.003 A a' = 4mS V a' = 2mS O a = .16 O a = .37 a - . 2 3 , CT' = 10mS Two—Tier ^ o 0.03 O o 2 V * o o o o 0.10 A 1.00 D 2 / D 1 0.100-r E 2 0.050 0.000 • Two—Tier. O a = .37 o o o o B o 0.00 0.10 0.20 Figure 6,8 E 2 Coeff ic ient and Conductance Fract ion I V . Canonical Problems and the Computat ion of B u l k Properties / 167 intracel lular dimension of (22-3)h = 19h and an extracellular dimension of 3h = (24 - (19 + 2))h, the new mesh had to preserve intracel lular proportions of 19/24 and extracellular proportions of 3/24 t so that the result ing numerical solutions would correspond to the same under lying cell shapes. F o r this reason, such variat ion of mesh size was l imited to the integer multiples of a smallest mesh size; 24 x 24 , 48 x 48, 72 x 72 and 96 x 96. The results suggested that a l l coefficient estimates except X c and were already accurate wi th in 5% at the 48 x 48 size. Whi le 72 x 72 results were obtained for most cases, it is clear, in retrospect, that the use of 48 x 48 solution results would have led to the same general conclusions as those d r a w n from the 72 x 72 results whenever both were available. In addition, 72 x 72 computations were necessary for extracellular space fractions < 0.2 in order to obtain a reasonable number of extracellular mesh points. Such studies did not compare the smaller and larger meshes, and were undertaken because the extracellular space ma}' be reduced under some physiological conditions. The solution properties for the pairs (Xp,Kp)> ( X y , K - y ) , are shown in Tables A . l and A . 2 , for a = .23. In the case of the coefficients X £ and K c > however, it was necessary to use 96 x 96 results for the largest values of the membrane conductance in order to be sure that the finite difference solutions converged as mesh size h ->0. A t the three largest values of membrane conductance the coefficient derived f rom the 72 x 72 solution appeared accurate wi th in 25, 10, and 5 %, respectively compared to the 96 x 96 results (column 3, Table A . 3 ) . More accurate solutions were not obtained because, in the absence of equally accurate t Th is is an extracellular space of 1 - (21/24) 2 = 23%. I V . Canonical Problems and the Computat ion of Bulk Properties / 168 experimental data , this did not seem to be useful . The properties of the solutions referred to in Tables A . l - A . 3. are representative of the other cases. Table A . l displays solution m a x i m a and selected averages of the canonical solutions X p and K ^ . These solutions appear in the terms of <p i and C. proportional to V 0 O in (III.2.12). The solution x 11 P occurs in the definitions of D , and D 2 and K occurs in the definition of D , in (III.2.18). Table A . l . A . M a x i m a of x and the P Average a 0 M ^ { t ^{1 + 1 0 3 X 1 48 _x 48 72 x_ 72 M a x Average M a x Average 20 4.28 .392 4.38 .386 10 4.45 .357 4.54 .355 5 4.61 .327 4.70 .327 2 4.80 .295 4.87 .296 1 4.88 .280 4.96 .280 Table A . 1 : B . M a x i m a of x and the P Average a 0 M w { t K t a ( l + V w * p l » 1 0 3 X 1 48 _x 48 11 JL II M a x Average M a x Average 20 4.28 .121 4.38 .116 10 4.45 .090 4.54 .088 5 4.61 .063 4.70 .063 2 4.80 .035 4.87 .036 1 4.88 .022 4.96 .022 IV. Canonical Problems and the Computation of Bulk .Properties /'169 Table A . l . C . Maxima o f and the Average a o ^ - y y { t j ^ t a v w K p ^ } . 10V 48 _x 48 21 2L 11 Max Average Max Average 20 3.54 .0034 3.62 .0062 10 3.28 .0064 3.32 .0075 5 2.88 .0059 2.92 .0063 2 2.14 .0036 2.18 .0037 1 1.51 .0022 1.55 .0022 Table A. 2 displays solution maxima and selected averages of the canonical solutions X y and K^. These solutions appear in the terms of <j> ^ and C. 1 proportional to VV 0 in (III.2.12). The solution x y occurs in the definitions of E, and E 2 and occurs in the definition o f E 2 in (III.2.18). Table A.2.A. Maxima of x and the Average o 0 M w { t (t„ + V x -,)}• loV 48 x_ 48 72 x_ 72 Max Average Max Average 20 4.36 .116 4.36 .111 10 4.51 .086 4.60 .084 5 4.66 .059 4.74 .059 2 4.81 .032 4.90 .032 1 4.89 .019 5.00 .019 I V . Canonical Problems and the Computat ion of B u l k Properties / 170 Table A . 2 . B . M a x i m a of x V and the Average + V X n' w v l 1 0 3 X 1 48 _x 48 72 _x 72 M a x Average M a x Average 20 4.36 .105 4.36 .100 10 4.51 .077 4.60 .076 5 4.66 .053 4.74 .053 2 4.81 .029 4.90 .029 1 4.89 .017 5.00 .017 Table A . 2 . C . M a x i m a of K and the V Average o0M.^{tj^t^ p w v l J 1 0 3 X 1 48 x_ 48 11 JL 11 M a x Average M a x Average 20 3.36 .0032 3.42 .0055 10 •3.10 .0057 .3.16 .0066 5 2.72 .0052 2.77 .0055 2 2.02 .0032 2.07 .0033 1 1.41 .0019 1.47 .0019 Table A . 3 displays selected averages and solution m a x i m a of the canonical solutions x c and K c - These solutions appear in the terms of <f>, and C. I proportional to VC . q in (III.2.12). The solution x c occurs in the definitions of F : and F 2 and K c occurs i n the definition of F 2 in (111.2.18). Results from a 96 x 96-grid-solution are included since these solution pairs converged slowly as the • I V . Canonical Problems and the Computation of B u l k Properties / 171 mesh became finer. Table A . 3 . A . The Average / ^ o M ^ t V w x d } . 1 0 3 X 1 48 x_ 48 72 72 96 _x 96 20 1.81 4.74 6.25 10 4.84 6.12 6.76 5 4.69 5.21 5.47 2 2.95 3.11 1 1.77 1.83 Table A . 3 . B . The Average ^ O o i M ^ t ^ t ^ x ^ } } . 1 0 3 X 1 48 _x 48 72 x_ 72 96 _x_ 96 20 1.58 4.15 5.48 10 4,23 5.36 5.94 5 4.11 4.58 4.81 2 2.59 2.73 1 1.55 1.61 Table A . 3 . C . The Average f ' V ^ ^C[o^wK-cl + 1 0 3 X 1 48 _x 48 11 1.11 96 96 20 4.70 4.56 4.45 10 4.35 4.16 4.05 5 3.86 3.70 3.61 I V . Canonical Problems and the Computat ion of B u l k Properties / 172 2 3.28 3.20 1 2.99 Table A . 3 . D . M a x i m a of 2.94 X c • 1 0 3 X 1 48 _x_ 48 72 x. 72 96 _x 96 20 .219 .556 .721 10 .582 .714 .778 5 .564 .610 .631 2 .359 .367 1 .217 Table A . 3 . E . M a x i m a of .218 K . C 1 0 3 X 1 48 _x_ 48 72 x_ 72 96 _x_ 96 20 .756 1.93 2.52 10 3.40 4.20 4.59 5 5.93 6.45 6.71 2 8.62 8.89 1 9.93 10.10 The position of the m a x i m a in the coefficients of Table A . 3 at 10 X 1 = 1 0 for the 48 x 48 solution, m a y be inaccurate since it becomes less pronounced on larger grids, however, the m a x i m u m occurs consistently at this value of X 1 and is physica l ly reasonable, as discussed i n Section 6.3. I V . Canonical Problems and the Computation of B u l k Properties / ' 173 APPENDIX IV.B. UNCOUPLED BULK CONDUCTIVITY Results of the uncoupled calculations for straight arrays of square cells (48 x 48 point solutions wi th an extracellular space of 19.7%) are shown in Figure B . l . The bulk conductivity versus cell sizes and intracel lular conductivity are shown in Figure B . l . A , and the corresponding bulk transcel lular conductance estimates in F igure B . l . B . The bulk conductivity estimates range f rom 2 - 7 m S over a large range of the membrane conductance, Z ^ I O . 3 a to Z, =234 a, and are an increasing function of the cell size (membrane conductance). These values are consistent wi th conductivities observed for bulk cortex and cerebellum (Nicholson 1980). This agreement of the computed values wi th data occurs despite the fact that the model geometry is unrealistic. The bulk transcellular conductance was defined as: (B . l ) a / 3 : = M w { y i +3x/3W,)} and corresponds to the proportion of bulk conductivity due to current flow through cells. The factors affecting this quantity are investigated here. This bulk transcellular conductance ranges f rom 7% to 36% of bulk conductivity in the parameter range which is l ikely to correspond to glia (that is , cells w i t h characteristic size f rom 32.5 a -103 u). It is seen from Figure B . 2 that the influence of intracel lular conductivity a1 was relat ively slight, f rom smal l values of up to li=51u and that bulk transcellular conductance was determined by the membrane conductance. Between =32.5 a and Z -i = 103 a, w h i c h corresponds to a ten-fold change in X, the transcellular conductance changed by a factor of eight, while the effect of a I V . Canonical Problems and the Computat ion of B u l k Properties / 174 F igure B . l . B u l k conductivity (A) and transcel lular conductance (B) versus cell size are plotted for a = 10, 4 m S i n the uncoupled model. A logarithmic vertical scale is used in B . 17-5 10 8 i & \ V a = 10mS u - | A o-'= 4mS V c o ,_, o 4 | V D 00 V 0 -I 1 1 1 1 1 0 50 100 150 200 ' 250 Cell. Size (/Li) 10.000 -r .000 a> (j c o -t—' o D XI c o o ° 0.100 ~<L> o c a 0.010 A A A a i • a 10mS 4mS B A 0 50 100 150 200 250 Cell Size ((A) Figure B.1 Cell Size and Bulk Conductivity I V . Canonical Problems and the Computat ion of Bulk Properties / 176 F igure B . 2 . B u l k conductivity (A) and transcel lular conductance (B) versus cell size over a large range ( Z ^ O . l fx to —100 M) are plotted for a =10 , 4, 2 m S in the uncoupled model. Logar i thmic horizontal axes are used in A and B , and a logarithmic vert ical axis is used in B . 10 8 > O D X) c o o GQ 6 + 4 + 2 + 0 O a '= 10mS i • o-O a 4mS 2mS + 0.025 0.100 1.000 O o o • o 10.000 100.000 Cell Size (/x) 1 0.000 T CD c 1.000 a o • D c c3 0.100 + I 0.010 c o 0.001 O a A a = 10mS = 4mS O a = 2mS 0.025 0.100 1.000 O • O o B 10.000 100.000 Cell Size (/a) Figure B.2 Intracellular Conductivity and Bulk Conductivity I V . Canonical Problems and the Computat ion of B u l k Properties / 178 five-fold reduction in intracellular conductivity at Z, =32.5 u is 10% and at Z! = 103 a; 54%. It is biologically significant that these model cells which exhibited bulk intracel lular conductivities f rom 7 to 36% had electrical space constants which were long compared to the cell dimension, and thus were electrotonically unlike an electrical s y n c y t i u m . The derivation in Chapter III assumes that the tissue has a two-tier structure. Thus , it is assumed that a periodic a r r a y of asymptotical ly larger cells is imbedded in a periodic a r r a y of asymptot ical ly smaller cells. The assumptions of such a model were i l lustrated schematically in Figure 6.3 and are described in Section 6.4. Uncoupled conductivity estimates obtained in this w a y are shown in Figure B . 3 . If tier structure has no effect, the results (A) in F igure B .3 would be the same as the results for (O) except for the errors made in selecting the E C S fraction due to the discrete nature of the mesh and in selecting the extracellular -conductivitj ' . In fact the bulk conductivity estimates for a = 0.44 (two-tier model) are about 1.4 times the values obtained at a = 0.23 for a simple square cell. This is a significant but not large discrepancy, given the simplici ty of the model. O n the other hand, the agreement between the transcellular conductance values is remarkable . It is seen that these values are close to those for the one-tier studies, suggesting that the dependence on tier structure is not crit ical in the uncoupled case. In order to determine the effect of extracellular space (ECS) fraction on bulk parameters , numerical solutions were obtained for several values of the extracellular space fraction. F i r s t , the values of a = 0.2 and a = 0.23 were selected in order to establish the sensitivity of the solution to the E C S fraction near the IV.: Canonical Problems and the Computation of Bulk Properties / 179 Figure B.3. Bulk conductivities (A) and transcellular conductance (B) versus cell size (Z!) are plotted for one-tier and two-tier uncoupled models. The one-tier model is indicated by open circles and the two-tier model by open triangles. The filled squares are the results obtained at the finest length scale, which are used to determine a in the two-tier study. 180 10 • cr.'= 10mS, o-°= 2 0 m S , a = .44 A cr'= 1 0 m S , a °= 6 m S , a = .44 O CT'= 1 0 m S , cr°= 2 0 m S , a = .23 8 -— -> o ~o c o o 3 CQ 6 -4--2 O A O A O A O A A CD CJ C O -M CJ 13 "O-c o o _o "Q3 CJ cn c o 0 0 4 3 2 -50 100 o 150 2 0 0 Cell S ize Cu) cr = 1 0 m S , a = 2 0 m S , a = .44 A cr '= l O m S , CT°= 6 m S , a = .44 o O cr = 1 0 m S , a = 2 0 m S , a = .23 O A B 0 0 i 50 100 50 200 Cell S ize Gu) Figure B.3 A T w o - T i e r C o n d u c t a n c e Study I V . Canonical Problems and the Computat ion of B u l k Properties / . 181 normal physiological value. In addition solutions were obtained at a = 0.44, and a = 0.12. The bulk conductivity estimates for these E C S fractions are shown in F igure B .4 . The bulk conductivity is an increasing function of a while the transcellular conductance is a decreasing function of a. It is seen that the bulk conductivities are highly sensitive to the E C S fraction and increases by a factor between 2 and 6 between a = 0.12 and a = 0.44 for different characteristic cell lengths. Transcel lular conductance increases by about 50% at Z 1 = 73 n, as a changes from a = 0.44 to a = 0.12, however the effect of a change in E C S fraction f rom a = 0.2 to a = 0.12 is not large. Thus , for situations in which the E C S fraction a is lowered, it is expected (all else being equal) that bulk conductivity w i l l decrease and slightly more transcellular current w i l l f low. This is biologically significant in situations i n which the E C S fraction is decreased. Because simple geometrical assumptions have been made, i t is important to establish the sensitivity of the results to the geometrical arrangement of the cells. The results of studies undertaken wi th staggered arrays are shown in F igure B .5 along w i t h corresponding values for straight a r rays . It is seen that while staggered arrays exhibit significantly lower bulk conductivity values, transcellular conductance remains stable f rom one type of a r r a y to the other, across a large range of membrane and intracel lular conductivity values. In the biological parameter range it depends chiefly on characteristic size/membrane conductance. IV. Canonical Problems and the' Computation of Bulk Properties / 182 Figure B.4. Bulk conductivities (A) and transcellular conductance (B) versus cell size (I)) are plotted for a = 0.12, 0.2, 0.23, and 0.44 in the uncoupled model. A logarithmic vertical scale is used in B. u 3 TJ c o o 3 8 7 •5 5 4 + 3 + O A O A O A O 4-5 0 ' 1 0 0 O a = . 12 • a = . 2 0 A a = . 2 3 • a = .44 O a =. • a = A a =. A a = 186 1 2 2 0 2 3 4 4 O A 1 5 0 2 0 0 Cel l S ize (/x) 1 .000 + 0 . 1 0 0 + i i • i • 0 . 0 1 0 + 0 . 0 0 4 0 O 5 0 1 0 0 1 5 0 2 0 0 Cell Size (/J.) Figure B.4 Extracellular Space Fraction and Bulk Conductivity IV. Canonical Problems and the Computation of Bulk Properties / 184 Figure B.5. Bulk conductivities (A) and transcellular conductance (B) versus cell size (7.1) are plotted for each combination of a straight array, a staggered array, and o1—10 and 2 mS in the uncoupled model. A logarithmic vertical scale is used in B. 1-8 5 1 0 > o 3 C o o 3 CQ O Straight Array, a' = 10mS • Straight Array, a' = 2mS O Staggered Array, a' = 10mS A Staggered Array, a' = 2mS o • A • A 10. 100 Cell Size (fi) cu o c o o 3 TD C O o a _3 "cp o co c o 1.000 0.100 + 0.010 0.006 o B 10 100 Cell Size (fi) Figure B.5 Array Geometry and Bulk Conductivity V. T H E ROLE OF ELECTROTONIC PARAMETERS IN TISSUE MODELS 1.0. INTERPRETATION OF THE MODEL 1.1. Introduction The tissue model used here has been chosen to be simple and easy to solve, and involves a number of assumptions which may seem unrealistic, such as a square geometry, and asymptotic assumptions about lengths and membrane properties of cell populations. Further analysis is required to see which aspects of our artificial tissue model would be expected to reflect observations of real tissue. While numerical results are affected by these assumptions, dimensional analysis (Lin & Segel, 1965) predicts that the bulk physical properties of tissue depend on the dimensionless parameters formed from combinations of the characteristic physical parameters in the tissue, and we may hope to determine the nature of this dependence from the model. In fact the expansion procedure of Chapters II and III exploited the dependence of such properties on ratios of the characteristic cell lengths. However, the .dependence of bulk properties on dimensionless parameters associated with conductance properties of the cell has not been discussed. Dimensionless parameters formed by combining the characteristic conductance properties of the cell are called electrotonic parameters. An asymptotic analysis in these parameters gives information about the correspondence between the asymptotic model and real tissue, the dependence of transcellular current on geometry, and provides analytic confirmation of the main features of the numerical solutions. Comparison of the results of Sections IV.6.2 and IV.6.3 with Appendix 186 V . The Role of Electrotonic Parameters in Tissue Models / 187 I V . B demonstrate numerical ly that the averages of the coupled and uncoupled models yield averages which have s imilar dependence on electrotonic parameters. Thus , it is assumed that an analysis of the uncoupled case is sufficient to obtain the qualitative features of this dependence. The asymptotic analyses to be carried out in Sections 2.2 and 2.3 assume in addition, that the intracellular conductivity is less than the extracellular conductivity and that the membrane conductance is either large or s m a l l . The two-tier studies of Section IV .6 .4 do not satisfy these assumptions since the extracellular conductivity is less than the intracel lular conductivit.y. The analyses of this chapter do not apply to the two-tier case, however, as seen i n Appendix I V . B , the conclusions would remain correct i f there were no coupling between electrical potential and ionic concentration at 0 (e ) . Because this chapter focuses on the physical interpretation of the model, we give a physical interpretation of transcellular f lux before proceeding w i t h a perturbation analysis of the relationship between transcellular f lux and electrotonic parameters . In the l ight of this discussion, it is seen that the averaged parameters of Chapters II and III are not uniquely determined unless (as stated in Chapter II) specific assumptions are made concerning the asymptotic properties of electrotonic parameters. In Section 3.0, a conclusion is d r a w n concerning the relative contribution of cells of different sizes to transcellular current. 1.2. Formal Analogy Between Intracellular Flux and Electrostatic Polarization A s described in Section I V . 1.2, the electrical f lux component of the potassium flux vector (i = 2 in equation (IV. 1.2)) is discontinuous because of the V. The Role of Electrotonic Parameters in Tissue Models / 188 differences in extracellular and intracellular transport properties. Potassium disappears from the ECS at one location and reappears at another remote location with little time delay. This complicates the interpretation (though not the derivation) of differential equations for the average electrical flux of potassium. This problem of physical interpretation is not new, however, and occurs in a classical model of dielectric polarization (Landau & Lifshitz, 1960; Garland & Tanner, 1978). This analogy is demonstrated and it is suggested that the reader view bulk electrically mediated transport through inclusions as analogous to the existence of bulk polarization in an (inhomogeneous) dielectric medium. Let the flux strength of some quantity (such as potassium) at the cell membranes be f(w) = Vi// • n where \p is a potential function, n denotes the outward unit normal to the membrane and f(w) is doubly periodic. The quantity J" wf(w)dS will be referred to as the flux dipole moment of f. Although the flux M dipole moment of each cell is small, their sum may produce a bulk flux dipole moment which is significant. If the function f(w) were the density of electrical charge, the integral / wf(w)dS over a single cell would be the electrical dipole M moment of the charge distribution over a single cell. The bulk flux dipole moment is a vectorial quantity which reflects the intracellular flux through cells. If f represents transmembrane ionic flux, then the bulk flux dipole moment represents the intracellular ionic transport owing to the presence of the cells. Under certain circumstances it is equal to the average, of the ionic flux vector intracellularly. 2 If the intracellular potassium flux is equal to VuV, and V i^ = 0 then the bulk flux dipole moment associated with the flux distribution f(w) is numerically equal to the integral of the interior flux through all the cells. This follows by V . The Role of Electrotonic Parameters in Tissue Models / 189 applying the divergence theorem to the components of the vector V«(W.Vi//) where W . denotes the I^1 component of w: The intracellular f lux mentioned is a v i r t u a l f lux, rather than a real f lux 2 because V if/*0 at the membrane. However , the f lux dipole moment st i l l equals the integrated intracellular f lux in our model because the f lux dipole moment is additive over regions in space. Th is definition is p r i m a r i l y of theoretical interest (experimental measurement of bulk coefficients is discussed in Chapter VI) in providing a physical interpretation of the formal averaged coefficients. If spatial buffer capacity is defined as the average potassium flux associated w i t h a unit voltage gradient in the transmembrane potential, then the bulk flux dipole moment per unit voltage gradient is proportional to the spatial buffer capacit j ' of an arra3 r of cells. This m a j ' be verified by reference to the averaged equations (III.2.17) in which the coefficient of V"V corresponds to a bulk f lux dipole moment. 2.0. ASYMPTOTICS IN T H E ELECTROTONIC PARAMETERS 2.1. Correspondence with Canonical Problem In the calculations of Chapters II and III (see Table II.2.4) we defined g := L g / S where g:= g 2 (S cm )^ is the dimensional (two-dimensional) value of the membrane conductance to K + , -L is an appropriate length scale, and S depends on the extracellular conductivity and diffusion coefficients. The quantity (1.1) / V\//dA = I C S J ( V . ( w 1 V ^ ) , V . ( w 2 V i | / ) ) d A I C S 'V . The Role of Electrotonic Parameters in Tissue Models / 190 (g/S) \ which has the dimension of cm, is an electrotonic length scale arising from the electrical parameters only. In this section we let w = x / L , and ~g = L , g/ S (these do not correspond to definitions in Chapter III). This is done to make fluxes of the form g(0! - (j>°) and 90/9n of 0(1) rather than 0(e )^ in the analyses of this Section and Sections 2.2-2.3. The tilde in g is dropped in the calculations which follow. The canonical 0(e) problems (Chapter III) which determine average properties for conductivity, diffusion, and potassium transport consist of solving the potential equation with periodic boundary conditions and constant (intracellular and extracellular) conductivities. We now calculate the qualitative dependence of the solutions and the flux dipole moments on the electrotonic parameters. In the calculation which follows we consider the two-dimensional problems (2.1) V»(aV0) = 0, •where a is the electrolyte conductivity, <$> is the electrical potential and (2.1) holds extracellularly and intracellularly. The jump condition at the cell membrane is (2.2) o 9^ = g(^ - A which actually is two equations, with o30/9n evaluated intracellularly and extracellularly, n is an outward normal vector, g is membrane conductance, and is the transmembrane potential. Boundary conditions are given as <f> = A on V . The Role of Electrotonic Parameters in Tissue Models / 191 one edge of the unit cell (say x-^ = constant), <j> = B on the opposite edge, and 3 0 / 9 n =0 on the remaining edges of the crystallographic unit cell. The latter problem is equivalent to a typical canonical problem from (III.2.12) - (III.2.16) because the solution of (2.1) - (2.2) wi th the conditions described has the form <j> = A w ^  + X ^ where w t is a spat ia l coordinate and X 1 satisfies (2.3) V-(crVx1) = 0, w i t h periodic boundary conditions and j u m p conditions at the membrane given by (2.4) a fVx ^ n + n , } = g{x l i"X 1 0} where n , is the f irst component of n , the outward unit normal vector, and x 1 and x 2 i n x are analogous in form because the biological cells are assumed to be square. Note that x = constant does not satisfy the jump condition (2.4) which is non-homogeneous. By geometric s y m m e t r y , <j> must agree w i t h the previously stated conditions (2.1) - (2.2). Thus , because of the s y m m e t r y , the canonical problem for x m a y be replaced by a problem w i t h fixed rather than periodic boundary conditions and a fixed ambient voltage gradient. 2.2. Electrotonically Short Case and Transmembrane Transport W e define the new parameters y: = a1(a° + a1) * where a° and a1 are the extracellular and intracel lular conductivities, respectively, X: = a'g * (cm) where g is the membrane conductance, and £ : = /X where L t is the unit cell V . The Role of Electrotonic Parameters i n Tissue Models / 192 length. The asymptotic dependence of 0 on 7 and £ now -is calculated for the unit cell. In view of the equivalence described in Section 2.1, this calculation should give qualitative information on the effects of these parameters on bulk properties. Because the potential equation is unaffected by length scaling, and boundary and j u m p conditions w i l l be satisfied uni formly as the dimensionless conductivity 7—>0 and electrotonic length parameter £—>0 or °°, it is expected that the expansion i n 7 and £ w i l l be regular . Rewri t ing the jump conditions (2.2) using the definitions of 7, £ , and X and scaling w = x / L 1 , as in Chapter II, gives the equations: where (2.5i) arises f rom combining intracel lular and extracellular j u m p conditions. It is assumed that <p m a 3 ' be expanded in the form: where + and - correspond to £ —>0 and £ —><*>, respectively, and we w i l l wri te (J>Q for 0QQ - Since the boundary conditions on the unit cell have no £ or 7 dependence, 0° j = O o n the boundary of the unit cell when i or j >0. The leading-order behavior of <p w i l l be deduced f rom the dependence of the j u m p conditions on 7 and £. In this section we assume £—>0, i.e., the ratio of cell length to the electrical length scale X is smal l , and so the ' + ' sign is used in (2.6). (2.5) (2.6) <t> = 4>o + { \ i + 7 0 i o + 0 ( i ± 2 ) + 0 ( 7 ± 2 ) + V . The Role of Electrotonic Parameters i n Tissue Models / 193 U s i n g (2.6) in the jump conditions (2.5) yields: (2.7) 7 (30o + 73010 + £ 3 0 o i + •••) T=7""5n "cm 75n £(0o + 7 0 i o + £0qi + - - 00 - 7 0 ° o _ £0oV •••) (2.8) = 3^o + 73010 + £ 3 0 o i + .... 7m oh TTn Thus , to 0(1): (2.9) (i) 30o = 0, (ii) 30o = 0. 75n TJn Since (2.1) holds, the leading-order intracellular solution, 4>\ is constant and the extracellular solution, <j>°0 is determined and non-constant. Because V^<pOo=0 away from the membrane, the boundarj ' conditions 3 0 ° / 3n = 0 and 0 o = const = <pl0 would imply C6Q identically constant by a standard expansion procedure from the theory of par t ia l differential equations (Carrier & Pearson, 1978). Since 0o is not identically constant, 0o~ 0° * s n ° t identically constant on the membrane. To 0(7): (2.10) (i) 30o = 30^0 , (") 3 0 i o = 0. 7Tn "3n "cm The left side of (2.10i) vanishes by (2.9ii). Thus , by (2.10ii) 0 ^ 0 is constant V. The Role of Electrotonic Parameters in Tissue Models / 194 while <j>! o =0 because the boundary conditions have no y dependence on the unit cell. As in the case of 4>-^Q, <t>oi — 0 because the boundary conditions on the unit cell have no £ dependence. Since the transmembrane potential is 0(1), the transmembrane current to leading-order is i m = g(0o~ <Ao)- I n order to obtain a steady state solution at 0 (£ ) , the constant 0o must be chosen so that the average transmembrane current is zero. The flux dipole moment in this case may be estimated by assuming that (fy'o, the extracellular potential (which satisfies O4>°QI 9n = 0 at the membrane), is a linear function of the space variable. References to the discussion of (2.3) - (2.4) and the numerical solutions show that 4>°o w i l l have a voltage variation over the membrane which is greater than that which occurs if 4>°0 were linear. Since X is bounded, however, the voltage variation over the membrane locus will have the same asymptotic order as for the case of linear extracellular potential. The assumption of linearity has been used in models for conductivity of brain tissue and produced good agreement with data (e.g., Ranck, 1963). It has been assumed that the cell is square. The discussion of equations (2.5) - (2.11), with the assumption of linear extracellular potential implies that to leading order, the transmembrane current is a linear function of x^ '. The transmembrane current per unit extracellular voltage drop over the unit cell is To 0(|): (2.11) V . The Role of Electrotonic Parameters i n Tissue Models / 195 given by i = g x ^ / L , ( A m p / V c m ) where we use the dimensional value of g (S cm *). Thus , the f irst component of f lux dipole moment vector (1.1) per unit voltage gradient is: f L , / 2 f L , / 2 J X l i dl = 2 / x l g f l d X l + 2 J ! L i i f d x 2 M m - f L , / 2 L 1 - f L , / 2 2 2 (2.12) = - f - g f 3 ^ , (cmfi" 1 ) o where f is the cell length as a proportion of the unit cell length, dl is the differential line element on the membrane in two dimensions, and the second term comes f rom integrating over the ends of the cell. The second component of flux dipole moment is zero because the integrand is odd. Thus , the quanti ty (2.12) is proportional to the membrane conductance and the square of the cell length. This result could not be obtained f rom dimensional analysis alone since (as wi l l be shown in Section 2.3) the f lux dipole moment can have other forms. The dependence of the numerical solutions on g shows that this asymptotic case corresponds more nearly to the 'biological situation. By the previously demonstrated equivalence between f lux dipole moment and v i r t u a l intracellular f lux (as explained in Section 1.2), i t is seen that (2.12) is the (approximate) total transcellular current flow per unit voltage gradient per unit cell . It m a y be interpreted as a transcellular conductance. A s the units of a voltage gradient are V c m \ the product of the quanti ty (2.12) w i t h a voltage gradient has units of amperes. " V . The Role of Electrotonic Parameters i n Tissue Models / 196 2.3. Electrotonically Long Case and Transmembrane Transport In the electrotonically long case, i;"^—>0 and the ' - ' sign is selected in the expansion form (2.6). The boundary conditions (2.5) become, on substitution of the expansion (2.6): (2.13) 7 (3*o + TS&IO + I'13*oi + •••) T ^ l f r 9n lTn 11 i .1 . , .0 ,0 o - l . O * o + 7<Pio + £ * o i + ••• ~ * o ~ 7*10 ~ ? *01 ~ — (2.14) = ^(dj^o + 7 3 * i o + S ' ^ o i + ...)• on on ~b~n Thus to 0 (1) : (2.15) (i) 3 * o = 0, (ii) * o - * o = 0. "5n The extracel lular solution is the same as in the short case, and the intracel lular solution is determined from it v i a (2.15ii). Hence d<t>lJ 3 n * 0 , because * a = * o and * o is non-constant at the membrane by the same argument as that in Section 2.2 fol lowing equation (2.9). To 0 ( 7 ) : (2.16) (i) 3 * o = 3^.?o , (ii) * r o " * ? o = 0, on dn V . The Role of Electrotonic Parameters in Tissue Models / 197 while to CKS"1): (2.17) (i) 90^1 = 0, (ii) 0 O 1 - 0°, = 30o. o n "cm Since 0o = 0o> 01*0 = 0io> a n d 30V dn&O, the leading-order transmembrane potential is given by (2.17ii). The previous discussion (2.13) - (2.17) has established the leading-order form of the transmembrane current. The flux dipole moment vector is approximated using the same assumptions as before: 0Q is assumed linear and the cell is square. The transmembrane current per unit voltage gradient ( A m p / V cm) is i = gi ; ^(0oi~0oi ) = cr 1 ]^ ^ 30V 3n = OVL , on the membrane locus where X j = constant and is zero on the membrane locus where x 2 = constant, assuming that 0Q is l inear in the x^ direction and 0o = 0o a t t n e membrane. Hence, the f irst component of the f lux dipole moment vector per unit voltage gradient m a y be computed as: f L , / 2 fr j i~2 (2.18) J x , i dl = 2 f L k i £ d x 9 = a f L , , M 1 m -kyl2. 2 L , 2 ( c m P / 1 ) . The second component of the f lux dipole moment is zero by symmetry . These calculations give an approximation to the total intracellular current per uni t voltage gradient per unit cell when £ is large, i .e., physical cell length is large compared to the electrotonic length scale. This would be true for a s y n c y t i u m , or network of electrically connected cells. The asymptotic calculations were undertaken i n order to determine which V. The Role of Electrotonic Parameters in Tissue Models / 198 properties of the numerical solutions are likely to characterise real tissue. The bulk properties of tissue are expected to depend on both electrotonic parameters and shape-dependent geometric parameters which are 0(1). The asj^ mptotic analyses give the dependence of the flux dipole moment on the electrotonic parameters, ij and j, for extreme values of these parameters. These analyses do not indicate quantitatively how large or small such parameters must be in order for this description to be accurate. Also, because of the assumption that * ° is linear, this description does not take into account the effect of the square cell geometry on <6°. However, in the electrotonically short case, (which is biologically relevant) the intracellular solution is constant and hence independent of geometry.! Thus, in this case, the difference between the two-dimensional and three-dimensional results is a geometrical factor, independent of the solution for 0. Because of the assumption of linearity of <t>°, dependence of the flux dipole moment estimate on the geometry is due only to the fact that the membrane locus appears in the integral. Since the numerical solution indicates that the analysis of the short case is accurate for this geometry, the numerical results in two dimensions should be applicable to three dimensions in the manner described. In view of the complexity of the three-dimensional geometry of neural tissue, the selection of a square two-dimensional geometry is a drastic simplification. For example, neural tissue contains much fine structure and relatively complex shapes. Yet, in many respects, reasonable agreement with experiments was obtained. This likely occurs for the following reasons. First, geometry may be unimportant in the electrotonically short case, as suggested t A similar remark holds in the electrotonically long case. V. The Role of Electrotonic Parameters in Tissue Models / 199 above. Second, the qualitative features of the two-tier calculation of Chapter III show that there is no significant transcellular current when cell dimensions are sufficiently fine, if physical membrane properties of fine structures are the same as those of coarse structures. Other successful models have assumed (Ranck, 1963; Havstad, 1976) that because the tissue contains elements with random orientations, the electrical properties of neural tissue may be modelled by superposition of the properties of arrays of cylinders which are, perpendicular and parallel respectively, to the ambient gradients. This assumption will not be discussed in detail here, but we have followed these authors in using a geometrically simple model of the tissue. The length scales suggested for cells here (32 um - 72 um) are characteristic of the longitudinal dimensions of cell processes. Cylinders oriented perpendicular to ambient gradients would have small effective size and would not contribute much to bulk transcellular current. These calculations show that the parameter £, which depends on the cell length, determines the nature of the solution for the potential. Because the cell length formally depends on e, the asymptotic order of this ratio £ must be chosen in order to complete the formal asymptotic model. 2.4. Choice of Scal ing In the canonical problems of Chapter III, the dimensionless parameter g appears at different orders of e in the jump conditions. The reasoning used in selecting appropriate asymptotic assumptions for g is now discussed. A fundamental assumption of dimensional analysis asserts that any bulk coefficient P of an inhomogeneous medium will be some function of the V. The Role of Electrotonic Parameters in Tissue Models / 200 dimensionless parameters which characterize the medium multiplied by a dimensional constant K, (2.19) P = K V(e, a, p\ ... ) where V is to be determined from a canonical boundary-value-problem, and the selection of dimensionless parameters is not unique. An approximation for P is to be found by taking a limit as e —>0. In some possible selections of dimensionless parameters e , a ' , p", ... , the parameters a ' , ... may depend on e . For example, g may depend on e. Surprisingly, the dependence of g on e does not follow from the definition and must be chosen in the asymptotic model.t The definition "g = L 0 g / S is not an explicit function of e = L , / L 0 and it is not possible, in general, to pose the canonical boundary-value-problem for V in a form which does not involve g (dropping the tilde). The dependence of g on e may be chosen as follows. In the physical tissue, the parameters e and £ take definite values and an approximation to P evaluated at these parameters is desired. While the asymptotic approximation is formed from the limit e—->0, so that e is not fixed, it is reasonable to suppose that the best approximation to P is obtained by keeping the parameter(s) £, fixed at the values which characterise the tissue. This is equivalent to fixing L.g where L. is the (dimensionless) cell length. Using this principle, the order of g may be determined from the physical properties of the cells. t It has been previously observed that the homogenization. procedure (letting e —>0) may not produce a unique answer, without choices of the general type described here (cf. Babuska, 1976). V. The Role of Electrotonic Parameters in Tissue Models / 201 For example, we compare the case of small cells and moderate measurement length, for which 1^=0(6) and L 0 =O(l) , and the case of cells of moderate length and a large expanse of tissue, for which L T =0(1) and L 0 =0(e )^. Both situations result in L 1 / L 0 = 0 ( e ) . Suppose that the shapes and relative placement of the biological cells in crystallographic unit cells are geometrically similar. In order to keep £ fixed, g must be 0(e )^ in the first case and 0(1) in the second. These assumptions correspond to different dimensional values of g, i.e., different physical membrane properties. Such assumptions would lead to canonical boundary-value-problems which are identical, and thus lead to the same bulk coefficients. In contrast, keeping the dimensional g value fixed across the two cases, would lead to g appearing at 0(e) and 0(1) respectively, and produce different bulk coefficients for each case. Both sets of assumptions are mathematically consistent, but correspond to the different physical models described. When two electrotonic and two asymptotic length scales exist as described in Chapter II, this rule can still be used, though it will not be possible to fix all dimensionless parameters other than e. If there. are two cell lengths, and L , and two sets of membrane conductances, g and g , then the dimensionless n' g n membrane conductances are g = L_g /S and ~g =L g / S . If it is assumed g & S n n n that L , L , g , and g are such that L =0(e) , L =0(1) g =0(e *), and g n' 6 g ' 6 n g n 6 g g n = 0(l), then the electrotonic parameters £ of each population are fixed but the quantity L _g =0(e) is not fixed as e—>0. V . The Role of Electrotonic Parameters in Tissue Models / 202 3.0. Neurons, G l i a and Electrical Scale Neurons comprise some 50% of the tissue volume in the bra in , yet several recent models of the electrical properties of neural tissue have neglected current flow through neurons because of their high membrane resistance (Nicholson, 1973; G a r d n e r - M e d w i n , 1983).f Whi le the extent of connections between glia in vertebrate neura l tissue is not yet established (Gardner -Medwin , 1983), gl ia have been thought to contribute more significantly to electrical bulk tissue properties because of their low membrane resistance and the (possible) electrical connections between them. However , the analyses of this Chapter lead to the physiological conclusion that transcellular current through neurons should not be neglected. 2 H a v s t a d (1976) suggested values of 3000 J2cm for the resist ivity of 2 neuronal membrane and 320 ficm for gl ial membrane, and 14 n m for the characteristic diameter of a neuronal segment and 1.8 p.m for a gl ial cell process. It is also expected that the cell lengths of the two populations w i l l be roughly in the same ratio, wi th gl ial processes extending some 20 n m - 50 /nm and neuronal processes some 100 nm - 5 0 0 nm, though w i t h much variat ion. Such values are approximate, but consistent, in general, w i t h values cited by others (Schanne & Ruiz-Ceret t i , 1978; Shelton, 1985; G a r d n e r - M e d w i n , 1983; Nicholson, 1973; La j tha , 1978). Thus , membrane resistivities and characteristic cell dimensions of neurons and gl ia differ by an order of magnitude, but m a i n t a i n roughly the same ratio. A s discussed below, the characteristic diameters, of cell segments determine (in part) the electrotonic length scales in three t These were not neglected by Ranck, (1963, 1964) and H a v s t a d (1976) in their studies of bra in impedance, but their studies dealt w i t h al ternating current (AC) for which capacitive rather than resistive properties of membrane are more significant. V . The Role of Electrotonic Parameters i n Tissue Models / 203 dimensions. The characteristic dimensions of the cells stated above suggest that neurons and glia are 'electrical scale models' of one another. The idea of an electrical scale model m a y be understood by reference to cable theory (Jack et a l . , 1975). A s stated in Chapter I, i t is commonly assumed that electrotonic t ransmiss ion in cyl indrical cells is described by the cable equation. The electrical space constant A (cm) is defined as (2.20) A 2 = R d / ( R + R.) , m o l where d is the diameter of the cell, R is membrane resist ivi ty and R and R. m o i are extracellular and intracel lular resistivities. If the space coordinate x : = x / L is normalized so that the length of the cell is uni ty , then the dimensionless ratio £ : = L / A , called the electrotonic length, appears in the exponential steady state solutions to the cable equation, thus determining the electrical properties of the cell. One cell is an 'electrical scale model ' of another cell i f i t has the same electrotonic length and geometrical shape as the other cell . This is because the solutions for the transmembrane potential differ only by a change of spat ia l scale. If two cells are geometrically s imi lar , their ratios of diameter to width are the same, and equation (2.20) and the definition of £ show that the membrane resistances of electrical-scale-model cells must be proportional to their length. This result in three dimensions is the same as the two-dimensional result described in Section 2.4. The discussion of Section 2.4 implies that two tissues composed of cells which are electrical scale models of one another in two dimensions have the V. The Role of Electrotonic "Parameters in Tissue Models / 204 same bulk electrical properties. According to the above discussion and the parameter values cited, if neuronal and glia geometry were the same (and if glia are not syncytial), neurons and glia would be three-dimensional electrical scale models of one another. Thus, it is expected that the leading-order bulk properties of a tissue composed entirely of neurons would be similar to the bulk properties of a tissue composed entirely of glia. Hence, the argument that neurons do not influence bulk DC tissue properties because of their high membrane resistance is incorrect. It is possible that factors other than membrane resistance, e.g., geometry, make the contribution of neurons to bulk tissue properties neglible. However, it seems unlikely that geometry alone could reduce the contribution of 50% of the tissue volume to a neglible amount. If glia are syncytial, then their contribution to tissue properties might be substantially larger than that of neurons; however, the analyses of this Chapter suggest that it is unlikely that this could reduce the contribution of neurons to a neglible amount since the effects of connections between glia (increasing average electrotonic length) are bounded. VI. SUMMARY AND BIOLOGICAL CONCLUSIONS 1.0. THE ASYMPTOTIC APPROACH Governing equations for averaged ion transport properties of a model of bra in tissue (III.2.17) have been derived using an asymptotic method that reduces the calculation of the averages to the solution of periodic boundary-value problems. A simple tissue model has been chosen for analysis and it has been argued that the properties of this model correspond to those in real tissue. S imple equations for the extracellular potential and potassium concentration are obtained for describing current passing and field potential experiments. While other approaches are possible (e.g., M c P h e d r a n & M c K e n z i e , 1978; M c K e n z i e et a l . , 1978; H a v s t a d , 1976; Lehner , 1979), the asymptotic approach has the fol lowing advantages: (i) It correctly specifies, in general, the canonical microscopic problem for a large number of disconnected (physically separated) cells. Other approaches, such as assuming a uniform potential gradient (Havstad, 1976) or _a_ priori sy mme try in the cells (McKenzie et ah , 1978), do not. -Examination of bulk conductivity versus extracellular space fraction (Figure IV-B.4) show that bulk conductivity increases by a factor between 11 and 2 as a changes from a = 0.12 to a = 0.44, for various cell sizes. Since the extracellular space fraction has changed by the same factor (0 .44/0 .12 = 3.66) for each cell size and the geometry has not changed, i t is seen that the effect of the adjacent cells on bulk f lux, precisely specified by our model, is significant. (ii) B u l k parameters are computed using only the microscopic parameters which appear in the model, and these need not be estimated from the bulk parameter data. 205 VI. Summary and Biological Conclusions / 206 (iii) The governing equation predicts all average bulk properties rather than simply a single one. This, with the above features (i-ii) means that bulk conductivity, current passing, and spatial buffering are all specified functions of the microscopic parameters. If all the experimental observations were consistent with these functions, the model would satisfy an exceptionally demanding criterion. (iv) The joint asymptotic analysis of dimensionless electrotonic and cell length parameters arises naturally in our approach and has not been appreciated in other approaches. This analysis (Chapter V) gives information about the sensitivity of the results to changes in membrane and intracellular conductivities. (v) Finally, a new and surprising result of this approach is that the systematically averaged equation for potassium concentration does not contain a 2 true diffusion term. The coefficients of the terms in (III.2.17) proportional to V C are different from those which would be obtained with a non-permeating ion and are a factor of 2 or 3 times larger than (it would be) if only -diffusion occurred. This is due to an 'unstirred layer' at the membrane which causes local variations in the transmembrane electrical potential. The term 'unstirred layer' refers to the concentration gradient which develops near a membrane due to electric current flow or other flux through the membrane. This fact could lead to experimentally observable results if the tissue has a two- or multi-tiered structure. Limitations of the present approach are discussed in Section 5.0. V I . S u m m a r y and Biological Conclusions / 207 2.0. B I O L O G I C A L C O N C L U S I O N S 2.1. I n t r o d u c t i o n In the model of spreading depression developed by Tuckwel l and M i u r a (1978), the advancement of the S D wave depends upon the electrical response of + + neural membrane to changes in [K ] . A l s o , i t has been speculated that [K ] q might (Prince, 1978), or might not (Somjen, 1984) p lay a significant role in epilepsy and other forms of bulk neuronal response (Leibovitz, 1977). F i n a l l y , variat ions i n the concentration [K- + ] o m a y have a wide var iety of physiological consequences under normal conditions (Krnjevic & M o r r i s , 1981). The role of [ K + ] q in these phenomena is not established, in part because of variat ions in physiological parameters between different preparations and var iat ion during the phenomena themselves. F o r example, the extracellular space and extracellular electrolyte composition are known to change during S D (Nicholson & K r a i g , 1981) and epilepsj ' (Prince, 1978). W h e n m a n y changes occur s imultaneously, it can be difficult to decide which factors are mediat ing and which are epiphenomenal. Therefore, i t is desirable to determine the existence and magnitude of potassium spatial transport mechanisms e.g., transcel lular current and spat ia l buffering, to describe the dependence of such transport on tissue parameters , as wel l as to estimate these parameters f rom experiments. In the following sections the implications of our model for these problems are discussed. O u r most important conclusion is that electrically mediated spatial t ransport does not require specialized cells and is re lat ively robust, i.e., this transport occurs in significant amounts w i t h physiologically relevant parameter values. Thus , this form of K + t ransport must occur and m u s t play a significant V I . S u m m a r y and Biological Conclusions / ,208 role in a wide variety of physiological situations. Such a conclusion is difficult to obtain f rom experimental studies. Some difficulties in incorporating ion transport theory into the bulk conductivity theory is described in Section 4. It is suggested that these difficulties can be resolved by experiment. 2.2. Properties of the Averaged Steady Equations In m a n y respects the properties of the averaged equations obtained here are s imi lar to those of the steady diffusion equation, i.e., Laplace 's equation and the ' spat ial buffering' equations based on the cable model ( S B C M ) used by G a r d n e r - M e d w i n (1983b). This is not surpris ing because these equations both describe conservative fluxes of electrical current and K + . The averaged equations of this thesis have a form s imi lar to the S B C M equations in the asymptotic l imi t of the electrical space constant for the cells going to zero. However , we emphasize that our equations cannot be 'derived' by the latter procedure. The S B C M equations contain bulk and transcellular conductances as empir ical , rather than derived, constants. In addition, the l imi t ing forms of the S B C M would contain a diffusion term, unlike the present -model. The values of the bulk coefficients in the present model are independent results of the model, unlike the coefficients of the S B C M . In this section we discuss properties of the averaged model equations in one dimension. To compare the results f rom our model w i t h experiments which have involved passing current across the cortical surface over relat ively long periods, i t is important to establish the steadj ' state properties of the solutions. Non-constant steady solutions to the model are obtained by prescribing C° = C ^ at some f ixed depth and C° = C at the surface of the cortex. Such boundary conditions V I . S u m m a r y and Biological Conclusions / 209 are chosen arb i t rar i ly , however. The most realistic steady solutions are those with finite non-zero [K + ] at inf ini ty since C° = [K + ] presumably tends to a o o constant deep w i t h i n the tissue. If C were governed by Laplace 's equation, it is obvious that the only such solution is C = constant. The same is true for our averaged equations (III.2.17) as shown below. The nature of the steady solutions for C m a y be deduced as follows. In one dimension, the governing equations (III.2.17) become, i f there are no sources: D,<2> + E , V + F , C = 0 , 1 u u 1 uu 1 uu (2.1) uu ' uu c uu t D2<6 + E 2 V + F 2 C = TC., where D . , E . , and F . , j = l , 2 V : = j ' " 1 l n ( C 0 / C 1 ) are as defined in Sections I V . 6 . 2 and I V . 6 . 6 and the subscript u denotes a par t ia l derivative wi th respect to the space variable u . In the steady state rC^. = 0, and the equations can be integrated directly. Integrating once, however, yields the first-order differential equations: (2.2) B.<j> + E . V + F . C = K . . J u j u j u j U s i n g the definition of V , e l iminat ing 3 * / 9 u f rom (2.2), and solving for C yields VI. Summary and Biological Conclusions / 210 (2.3) C u = HK,D2 ~ K . D Q C v(T)2F, - -DTF^C + E 2 - E , For steady solutions in which C is constant and non-zero, (2.3) requires D 2 K 1 - D 1 K 2 = 0 and C identically constant, as was stated above. The importance of this result is that the spatial derivatives of concentration and membrane potential C u = V u = 0 a n d hence the coefficients E. and F. do not appear in (2.2) and therefore, cannot be estimated from a steady state current passing experiment. However, as noted in Chapter IV, the coefficients, E , and E 2 , may be estimated within 10% to 15% (in any of the models employed here) from the value of D 2 , where D 2 is interpreted physically as the transcellular conductance. The significance of E 2 is described in the next Section 2.3. In the steady state with C=constant, the electrical flux is given by D 1 #u = K 1 and the ionic flux by D 2 0^ = K 2 . Since -both fluxes are proportional to the electric field, the ionic flux as a fraction of electric current in such an experiment is K 2 / K 1 = D 2 / D v Thus, D 2 / D , is the observed transport number in a current passing experiment. As shown in Chapter IV (Figure 6.8), the values of E 2 and D 2 / D , are not related. The value of the bulk conductivity D ! , however, with the observed transport number, will suffice to reliably determine E^ and E 2 . Since the present model predicts a reliable relationship between D , , D 2 and E , and E 2 , the governing equations could be tested by independent measurements of these quantities. While D ^ and D 2 could be obtained from steady state experiments, it would be necessary to measure membrane potentials V I . S u m m a r y and Biological Conclusions / 211 and K + transport during a time-dependent experiment to obtain E , and E 2 . If E , and E 2 are not as predicted, the model would have to be modified. 2.3. Magnitude of Spatial Buffering The relationship between the bulk coefficients and the magnitude of spatial buffering m a y be deduced by el iminat ing 0uu f rom (2.1) to obtain: (2.4) £v + fC TC„ u u u u t where £ = E 2 - ( D j / D ^ E , and £ = F 2 - ( D 2 / D 1 ) F 1 . The values of D 1 ; D 2 , E , , F , , and F 2 are given in Tables I V . 6 . 1 - I V . 6 . 5 , and E 2 , shown in F igure I V . 6 . 7 , is the same as E , to two significant figures. Mathemat ica l ly , we say that spat ia l buffering occurs when r C ^ is more negative (or positive) than i t would be i f diffusion alone were occurring for C y u ^ 0 (or > 0). The coefficient £ of V and £ of C in (2.4) are tabulated for (Case A) a = 20 m S , 0-uu UU 0 1 = 10 m S , a = 0.2, in Tables 2.1. A and for the two-tier study (Case B) in Table 2 . I .B . , where the values of £ and f in Table 2 .1 .A and 2.1.B differ by some 10% to 30% from the values of E 2 and F 2 . The m i n i m u m in £ in Tables 2.1.B and 2.2 corresponds to the m i n i m u m in F 2 discussed in Section I V . 6 . 5 . In order to compare spatial buffering to diffusion it is useful to compare the rates of decay of an ini t ia l concentration distribution, C = c ( l + asin(bu)) where 0 ^ a ^ 1, under (2.4) and under diffusion, respectively. To do this we 2 first obtain a bound for y V = C / C - (C / C ) in terms of C . For u u u u u u u u such that C < 0, we obtain: u u V I . S u m m a r y , and Biological Conclusions / 212 Table 2 . I . A . One-Tier Coefficients in Equation (VI.2.4). Z,(Mm) £ f 32.1 .0174 4.39 46.0 .0280 5.51 72.1 .0460 7.26 103. .0600 8.17 146. .0704 7.95 Table 2 . I .B . Two-Tier Coefficients in Equat ion (VI.2.4). /,(nm) £ t 32.1 .0072 11.39 46.0 .0109 12.88 72.1 .0131 12.18 103. .0154 8.66 146. .0184 20.80 „ V uu 2 - a b sin(bu) 1 + a sin(bu) (-(2.5) 2 < - a b sin(bu) 1 + a ab cos(bu) 1 + a sin(bu) 1 C . 1 + a uu Hence, concentration distributions wi th smal l amplitudes (a < < 1) lead to a V < v'^C . Hence, an approximate lower bound for the coefficient of C u u u u r r u u V I . S u m m a r y and Biological Conclusions / 213 in (2.4) is (t + v~l£). This quantity is tabulated in Table 2.2. It can be seen that the coefficient of C is not very sensitive to the uu model employed, and does not depend on I in a simple w a y . Since this coefficient would have been 2.64 w i t h pure diffusion, there is a consistent spatial buffer effect which is between 2 and 8 t imes the effect of diffusion. To obtain a spatial buffer effect which is 5 times that of diffusion, as deduced by G a r d n e r - M e d w i n (1983b), requires a coefficient of C u u of approximately 13, which is generally consistent w i t h Table 2.2. The £ V and F C terms both contribute significantly to spatial uu uu buffering in the one-tier case (Case A), but £ ^ u u contributes little in the two-tier case (Case B). This means that i n Case A, the spatial buffering effect m a y be estimated accurately f rom E . , while in Case B it is necessary to know all the bulk coefficients, including F . . The implications of this are discussed in Section 3.3. 3.0. I M P L I C A T I O N S F O R A M O D E L O F T I S S U E S T R U C T U R E 3.1. T r a n s f e r C e l l s A r e U n n e c e s s a r y In the S B C M it is postulated that a sparse network of electrically continuous ' transfer cells' is the substrate of the cable equations employed in that model. In this section, it is argued that the assumption of a gl ial syncyt ium is a complex assumption, and that it is unecessary. G a r d n e r - M e d w i n (1983b) states "the assumption of a s y n c y t i u m is not str ict ly necessary " and that an a r r a y of cells would behave in essentially the same w a y . In this thesis this statement has been tested by direct computation and found to be correct in m a n y respects. Differences between the present model and the S B C M are also V I . S u m m a r y and Biological Conclusions / 214 Table 2.2. L o w e r Bound for C Coefficient. — — u u Z, Case A Case B (Mm) 32.1 11.93 11.78 46.0 11.41 13.46 72.1 9.75 12.89 103. 7.03 9.49 146. 5.31 . 21.8 noted. The assumption is complex because the present results and the S B C M indicate that given reasonable assumptions about gl ial membrane, the bulk transport numbers would be m u c h larger than those observed i f a l l gl ia were electrically continuous. In addition, Hounsgaard and Nicholson (1983) have examined potassium transport experimental ly using ionophoretically applied K + and concluded that gl ia were not electrically continuous in vertebrate cerebellum. Therefore, i f the S B C M is used, it is necessary to postulate a sparse network of syncyt ia l t ransport cells w h i c h , because they are sparse, have sufficiently high internal resistance to account for observed bulk transport numbers . B u l k potassium current w i t h i n the tissue of between 7% and 30% of bulk electric current is consistent w i t h the present knowledge of parameters for a tissue consisting of disconnected cells (neurons or glia). Therefore, the assumption of syncyt ia l transfer cells is unnecessary, since the observed bulk transport numbers can be accounted for by current flows through non-syncyt ia l elements of the general cell population (neurons and gl ial cells). The magnitude of the current VI. Summary and Biological Conclusions / 215 depends mainly on the product of membrane conductance and cell length, and to some extent on intracellular resistance, extracellular space, and relative position of the cells with respect to each other. On the other hand, our analysis does not rule out the possibilhvy that specialized transfer cells exist. It may be possible to test for presence of transfer cells by performing current passage experiments in the presence of pharmacological agents which disrupt the putative coupling between glia (Tang et aL, 1985). The computed transport number range includes values close to experimental observations (0.06 over a 5 mm diameter region) as well as values considerably higher than those observed. If our model is correct, it is predicted that larger transport numbers will be observed with other preparations. In addition, higher transport numbers might be observed at finer length scales, since it is expected that the governing equations will be different on such scales. Differences between preparations also can occur because of tissue (tier) structure or differing surface / volume ratios of cells. 3.2. Disconnected Cells Cannot Be Neglected Many of the cells in neural tissue are not electrically continuous or syncytial and the present work was undertaken, in part, to assess the importance of such tissue components. The mathematical technique (homogenization) employed here derives for the first time the governing equation for electric current and ionic flux through closely apposed but disconnected cells. Although our model cells were electrically disconnected, they still exhibited transcellular currents comparable to those obtained from the SBCM. Therefore, even if a sparse syncytial network existed, the disconnected cells in the tissue VI. Summary and Biological Conclusions / 216 could not be neglected. Our model cells were different electrotonically from a syncj'tium because transcellular current was independent of the intracellular conductivity in the biological parameter range (Figure IV. 6.2). Instead, this current depends almost exclusively on the membrane conductance. As shown in Chapter V, a syncytium may be characterized electrotonically by the fact that transcellular current depends primarily on its intracellular conductivity. For high membrane conductance, the transcellular conductance of a coupled two-tier model, is considerably less than that of an uncoupled model. Most previous interpretations of such data have been based on uncoupled models, however. Somjen and Trachtenberg (1979) suggested that the relatively high K + conductance of glial membrane implies a bulk tissue conductivity much larger than the observed value. This suggestion seems to be based on the idea that if membrane conductance were high, then current would flow through the intracellular space, so that the bulk conductivity ought to be close to the intracellular conductivity. Our results show that this conclusion need not be correct in a coupled two-tier model. 3.3. Tier-structure May Be Important Because cells of different sizes coexist in neural tissue, we investigated the effect of cell populations with different asymptotic sizes. The results indicate that different governing equations will hold for passive transport at different length scales. Since transcellular flux depends almost entirely on cell size (or membrane conductance), different populations of cells of known size will characterize the tissue depending on the given length scale. Such effects have been modelled here with two cell populations, and it is supposed that the population composed of V I . S u m m a r y and Biological Conclusions / 217 smaller cells has the same membrane properties as the larger population. These simple assumptions have been used because little data exists to support more complicated assumptions and the results obtained. The results of Section 2.3 provide an experimental test of whether a given tissue has a two- or mult i - t ier structure. In a tissue accurately described by a one-tier model, the magnitude of spat ia l buffering observed can be predicted f rom D r , D 2 , E 1 ? and E 2 ; which m a y be deduced f r o m a current passing experiment. If the tissue is accurately described by the two-tier model, however, the prediction of spatial buffering derived using this method w i l l be smaller than the spatial buffering observed. Another k ind of tier structure w h i c h m a y exist, is a network of transfer cells surrounded by disconnected gl ia . If a sparse network of specialized transfer cells existed, i t is l ikely that surrounding glia would render it ineffective as a means of K + t ransport by ra is ing the effective extracellular transport number for K + . In Gardner -Medwin ' s S B C M (1983b) the parameter |3 m a y be interpreted as the ratio of the magnitude of spatial buffer f lux to diffusive flux and has the form .6= (t£) ^"R (R +R.) ^ where R. is the bulk transcellular -resistance and R ^ Tv o o i i o is the bulk extracellular resistance. Hence, the given expression indicates that relat ively smal l changes in extracel lular transport number t ^ can dramatica l ly alter the importance of spatial buffering i n a tissue corresponding to the S B C M model. For example, the conservative assumption that 7% of current is transcellular in the surrounding gl ia , reduces ^ by a factor of 0 . 0 7 / 0 . 0 1 2 = 5.8. The model which was used impl ic i t ly in the last paragraph was not investigated in this thesis. Therefore, further experimental and theoretical work are necessary to completely assess the possibility that s m a l l cells might reduce V I . S u m m a r y and Biological Conclusions / 218 the effectiveness of a transport network composed of larger or connected cells. 4.0. COMPARISON WITH EMPIRICAL PROPERTIES OF BULK TISSUE 4.1. Introduction Nicholson and Phi l l ips (1981) have investigated the properties of diffusive transport of non-permeating ions i n bra in tissue, and were able to describe such diffusion w i t h a simple isotropic model. The physiologically significant potassium ion, however, is not described by this model. In this section i t is shown that i t is necessary to use bulk governing equations and to consider diffusion, conductivity, and transport number data in order to obtain a complete model for potassium ion transport. 4.2. Significance of Transcellular Current If no bulk transcellular current flowed, bulk tissue would have few interesting electrical or ion transport properties. F o r this reason it is important to demonstrate that significant bulk transcellular current exists. A direct measurement of the bulk transcellular current is provided by the measurement of potassium transport through tissue i n the presence of an electrical current (Gardner -Medwin , 1983a). The fact that very few assumptions were made here in der iving a significant bulk transcel lular current makes our theoretical study an independent piece of evidence for the existence of significant transcellular f lux. The results reported here, however, show that the factors affecting transcellular f lux in a coupled model are complicated. Thus , the work presented here is not complete, because the results have not been compared to experiment. V I . S u m m a r y and Biological Conclusions / 219 4.3. Scale Effects The scale effects described in Sections V . 2 . 4 and V . 3 are likery to be useful for m a n y different situations because they are part icular ly simple. The predictions of the present model are s imi lar to those of the S B C M , as they should be in order to explain known observations. However , the present model also contains the possibility of increased, decreased, or anisotropic spatial buffer capacities at shorter or longer length scales. These possibilities have not been systematical ly investigated, though the existence of such phenomena is suggested by an isolated f inding of G a r d n e r - M e d w i n (1983a). H e found that the strength of field potentials due to superfusion of cortex wi th K + over an area of diameter 1 m m produces a smaller (30%) field potential than that over a 5 m m diameter area. According to the S B C M (Gardner -Medwin , 1983b), spatial buffer fluxes require potassium gradients to be extended for a longer distance than the electrical space constant of the 'transfer cells'. Thus , (it is argued) .at very short length scales spatial buffering is not significant compared to diffusion. This argument is correct for the S B C M model. However , we have shown that simple diffusion m a y not occur in bulk for the potassium ions since disconnected cells contribute to transcellular f lux even i f a transfer cell network also exists. Therefore this conclusion requires further investigation. The data on conductivity anisotropy appear to be inconsistent w i t h the f inding (Gardner-Medwin, 1983a) that only 6% of D C current passes through cells. If diffusion in the cortex is isotropic and the steady diffusion equation is the same as the equation for steady electric current, i t follows that anisotropy must be due to transcellular f lux. However , based on our results (or the S B C M ) , V I . S u m m a r y and Biological Conclusions / 220 it seems unl ike ly that transcellular flux could account for anisotropy of 3:1 to 5:1. Thus , i t seems l ikely that such discrepancies must arise f rom scale effects. These effects can be investigated by simultaneous measurement of anisotropic conductivity, diffusion, and spatial buffering at the same length scale. It might be convenient to perform such an experiment in a slice preparation. If the present model is correct, the conductivity, diffusion, and spatial buffering results w i l l be consistent when simultaneous measurements are performed at the same scale, but w i l l v a r y (together) as the scale of measurement is var ied. 5.0. LIMITATIONS 5.1. Transport Number Simplification The derivation of an averaged governing equation (III.2.17) for electrical current and ionic f lux begins w i t h a non-linear equation and results in a l inear equation w i t h constant coefficients. M o s t of the approximations employed can be justified asymptotical ly or in other, wel l defined senses (Bensoussan, L ions , & Papanicolaou, 1978). In addition, the convergence properties of s imilar mult iple scale expansions have been investigated numerical ly to some extent (Bourgat, 1977). The f ina l simplification of Chapter I V is not an asymptotic approximation of this type, however, and (until proven otherwise) is mathematical ly ad hoc . It is important to emphasize that existence of a transport number (with constant t ^ and ) is assumed throughout the experimental (Barry & Hope, 1969) and thermodynamic l iterature (Katchalsky & C u r r a n , 1965). In Chapter I V it is shown that no complications are introduced in extending this theorj ' to bulk tissue. E x p e r i m e n t a l evidence (Gardner-Medwin, 1983a; G a r d n e r - M e d w i n & V I . S u m m a r y and Biological Conclusions / 221 Nicholson, 1983; H a v s t a d , 1976; Nicholson, 1975) suggest that the non-linear effects neglected are not qual i tat ively significant under m a n y conditions of physiological interest. 5.2. Tortuosity and Geometry Assumptions It was not practical here to solve a canonical microscopic problem which exhibits the high tortuosity that is characteristic of rea l neural tissue. This shortcoming of the present work, together w i t h the fact that the coefficients were calculated in two dimensions, complicates the interpretation of our results. The l imited investigation here of the effect of geometry was undertaken in order to assess qualitatively the magnitude and direction of such effects. In this respect our study is s imi lar to the study by M c P h e d r a n and M c K e n z i e (1978) of the average conductivity of inhomogeneous media w i t h spherical inclusions arranged i n lattices of different types. M c P h e d r a n and M c K e n z i e ' s results are not directly applicable here because they studied spherical .inclusions without membranes. In general our results indicate that when membrane conductance is low in coupled or uncoupled models, there is little difference between the bulk conductivity of straight and staggered a r r a y s of cells. A t higher membrane conductances, the uncoupled model shows a reduction of global conductivity in the staggered a r r a y s because interaction between cells is more important at higher membrane conductances. For coupled models, however, the results for straight and staggered a r r a y s were near ly identical i n a l l of our studies because the effects of coupling between transmembrane potential and concentration dominate at higher membrane conductances. Thus , effects of geometry in the coupled model are V I . S u m m a r y and Biological Conclusions / 222 insignificant. Staggered arrays and straight ar rays conduct s imi lar proportions of transcel lular current in both coupled and uncoupled models. Thus , our results suggest that the unrealistic geometries used here wi l l not seriously affect conclusions about transcellular f lux. 6.0. SUMMARY Histor ica l ly , previous tissue models have been formulated in order to model a single bulk property such as impedance (Havstad, 1976), spat ia l buffer capacity (Gardner-Medwin, 1983a b), or diffusion properties (Nicholson & Phi l l ips , 1981). N o previous model has attempted to synthesize the modelling of all three properties. A n important reason to do this is that reliable observations from the experimental l i terature (isotropy of diffusion, anisotropy of conductivity, smal l bulk tissue transport number of K + ) cannot be easily reconciled w i t h previous models of cortical tissue structure. Electr ical models of tissue has been explored extensively; for example, Ranck (1963,1964), Eisenberg et ak, (1979), and Nicholson (1973) have given assumptions which have been useful experimental ly. U n l i k e the studies of K + transport , these impedance studies contained no direct measurements of transcel lular f lux. Previous models of cortical conductivity or impedance either have assumed that transcellular f lux is the same as it would be i n a uni form voltage gradient (Havstad 1976) (an assumption which is quali tat ively reasonable), or assumed that transcellular f lux is negligible (Nicholson, 1973), or not modelled transcel lular f lux at a l l (Nicholson & F r e e m a n , 1975). These previous studies also did not model unst irred layers at the membrane, which we have shown to be important under some conditions. APPENDIX VI.A. RECENT LITERATURE A s noted in the introduction, theoretical work on bulk tissue properties is difficult and has appeared infrequently. Recent work (1985-1989) on conductivity and spatial buffering has chiefty consisted of experimental work on microscopic preparations i n vitro and is therefore not directly relevant to the present work on bulk properties. Aspects of these studies are discussed below. The exceptions are Gardner -Medwin ' s (1986) theoretical exposition of the concept of ' spatial buffer capacity ' , and Dietzel and Heinemann's (1989) simultaneous experimental study of bulk current sources, spatial buffering, field potentials, and changes in extracellular space. Gardner -Medwin 's work is a simple and brief extension of the (1983b) theory. Dietzel and Heinemann found current sources and changes in extracellular space consistent w i t h our model (and the S B C M ) ; but also deduced that other, active, uptake processes played a significant role in the removal of potassium f rom the extracellular space. Exper imenta l studies of microscopic systems were of several types. Studies of isolated ret inal gl ial cells included studies of the distribution of potassium -conductance, ( B r e w & A t t w e l l , 1985; N e w m a n , 1986; Reichenbach & Eberhardt , 1988) or cell shape (Eberhardt & Reichenbach, 1987), and metabolic effects of K + (Coles, 1989). Other studies examined spatial buffering in intact re t ina • ( K a r w o s k i et ak, 1989a, b). These studies of ret inal cells are of independent interest, but give l imited information about the fundamental role of spatial buffering in m a m m a l i a n cortex and cerebellum. In addition, potassium channels of different kinds have been studied in ret inal glial cells (Newman, 1989) and cultured astrocytes (Gray & Ritchie, 1986; Sonnhof & Schachner, 1986; Sonnhof, 1987). These preparations are wel l suited to obtaining information about the 223 Appendix VI.A. Recent Literature / 224 membrane properties of glial cells and this work will likely lead to further refinement of the microscopic properties used in bulk tissue models. Such findings must be regarded as preliminary, however. In particular, data must be obtained regarding the generality of observed glial membrane properties across cell types and species before such properties are incorporated in general models. REFERENCES A i d l e y , D . J . , The Physiology of Excitable Cells , 2nd ed. , Cambridge U n i v e r s i t y Press , Cambridge, 1978. A d a m , G.(1973), The effect of potassium diffusion through the Schwann cell on potassium conductance of the squid axon. <L M e m b r . 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