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Electrocutaneous stimulation via bipolar current pulses : models and experiments Bütikofer, Rudolf 1977

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ELECTROCUTANEOUS STIMULATION VIA BIPOLAR CURRENT PULSES: MODELS AND EXPERIMENTS by Rudolf Butikofer Diploma, Swiss Federal I n s t i t u t e of Technology (ETH-Z), Zurich, Switzerland, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © Rudolf Bufcifcofer, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at 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 , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i 1 a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g . o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f Electrical Engineering The 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 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date May 1977 ABSTRACT Mathematical models have been used to study the e f f e c t s of various e l e c t r i c a l s t i m u l i on nerve. The a p p l i c a b i l i t y of the findings to electrocutaneous stimulation i n man has been investigated experimentally. With the model of a nerve membrane the influence of v a r i a t i o n s i n the stimulus parameters have been in v e s t i g a t e d . This was done f or multiple b i p o l a r pulses by considering selected cases and for s i n g l e b i p o l a r pulses with a systematic i n v e s t i g a t i o n . The main findings were i ) that the threshold charge f o r a s i n g l e b i p o l a r pulse s changes only s l i g h t l y f o r d i f f e r e n t pulsewidths; i i ) that the threshold charge monotonically de-creases with pulsewidth and threshold charge also decreases monotonically with increasing delay of the symmetric negative pulse; i i i ) that threshold amplitude f o r multiple b i p o l a r pulses was only s l i g h t l y lower than the amplitude f o r a s i n g l e b i p o l a r pulse. The influence of d i f f e r e n t components involved i n cutaneous stimulation, such as skin, electrode, and neuroanatomy, have been examined. Corresponding models f or the passive components involved were selecte d . From these models the following l i m i t i n g conditions f o r the stimulus were derived: i ) the stimulus has to be current regulated; i i ) i t must be b i p o l a r (no net charge t r a n s f e r ) ; and i i i ) the electrode voltage must remain below the s k i n break-down voltage. The aspect of the conversion of stimulus' energy i n t o heat i n the skin has been examined i n d e t a i l . A review of mathematical models of the active nerve membrane i s presented and the a p p l i c a b i l i t y of a nerve model to the stimulation of per i p h e r a l nerve f i b r e s i n man i s discussed. Numerical methods were used i i to solve the model's d i f f e r e n t i a l equations. The e f f e c t s on the s o l u t i o n of d i f f e r e n t i n t e g r a t i o n methods and of d i f f e r e n t i n t e g r a t i o n step sizes has been assessed. Experiments with electrocutaneous stimulation have been performed using s i n g l e b i p o l a r current s t i m u l i . The duration of a pulse was les s than 100 microseconds. For the experiments, an e l e c t r i c a l l y i s o l a t e d stimulator has been designed and b u i l t . I t operated under the co n t r o l of a PDP-12 computer. The sensations produced were s l i g h t l y suprathreshold and pain-l e s s . The thenar region of the hand was stimulated using a concentric electrode. The r e s u l t s of the experiments supported the t h e o r e t i c a l pre-d i c t i o n s and indic a t e d the p o s s i b i l i t y of using models to in v e s t i g a t e the optimization of stimulus parameters wit h i n the range tested. The close correspondence between the experimental r e s u l t s and the nerve model calcu-l a t i o n s seems to provide some evidence f o r the hypothesis that i n electrocutaneous stimulation the nerve f i b r e s are stimulated d i r e c t l y . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF FIGURES " v LIST OF TABLES v i i ACKNOWLEDGEMENTS v i i i INTRODUCTION 1 Part I; THE ASPECTS OF ELECTRICAL STIMULATION 3 Chapter 1: Sensory stimulation and f u n c t i o n a l s t i m u l a t i o n 3 2: Neural organization and sensation: a review 6 3: The ski n i n t e r f a c e 16 4: Characterization of the stimulating wave form 22 5: Heat production by the stimulus 29 6: Electrode properties 42 7: On transmembrane currents i n a myelinated axon 45 placed i n a current f i e l d Summary I 53 Part I I : INVESTIGATION OF SINGLE NERVE FIBRE MODELS 56 Chapter 8: Theory of the nerve membrane 56 9: Four models f or nerve membranes 62 10: Evaluation of a model 78 11: Axon models and propagation of nerve impulses 86 12: The time constants i n membrane models 91 Summary II 100 Part I I I : CALCULATIONS AND EXPERIMENTS 103 Chapter 13: Dynamic threshold and the i n d i r e c t e f f e c t .103 of b i p o l a r stimulation 14: Investigations with selected current pulses 110 15: Systematic i n v e s t i g a t i o n of threshold s t i m u l i 126 16: Experiments with electrocutaneous stimulation 143 Summary I I I 154 CONCLUSIONS AND RECOMMENDATIONS 156 Appendix A: Integration methods f or nerve membrane models A 1 B: Threshold search program B 1 C: Current c o n t r o l l e d stimulator C 1 D: Fortran Program D 1 E: On n o n l i n e a r i t i e s i n the FH model and on E l performance of s i m p l i f i e d models F: Threshold and stimulus timing i n the F 1 l i n e a r V, m model References R 1 i v LIST OF FIGURES FIGURE PAGE 2--1 Schematic myelinated f i b r e 9 2--2 E l e c t r i c a l model of myelinated f i b r e 9 2--3 Cross section of skin 11 3--1 Skin model 18 3-•2 P o t e n t i a l d i s t r i b u t i o n 20 5--1 C i r c u i t f o r heat c a l c u l a t i o n s 29 5--2 Heat i n function of time 31 5--3 Energy with and without capacitor 32 5-•4 Heat i n RC model of skin 37 5-•5 Heat i n complete model of skin 39 6-•1 Concentric electrodes 44 7-•1 Model f or myelinated nerve i n current f i e l d 47 7-•2 Current f i e l d of concentric electrode 48 8-•1 Basic membrane c i r c u i t 60 9--1 Voltage r e l a t i o n s f o r an action p o t e n t i a l 63 9-•2 Depolarizing currents 63 9-•3 Space clamped membrane 64 9-•4 Hodgkin-Huxley (HH) membrane model 65 9-•5 Frankenhaeuser-Huxley (FH) model 70 9-•6 Membrane p o t e n t i a l and currents i n the FH model 72 9-•7 Dodge's membrane model 74 9-•8 Model of Purkinje f i b r e 76 11-•1 Cable structure of unmyelinated axon 86 11-•2 E l e c t r i c a l model f or myelinated axon 88 12-•1 Time constants f o r the HH model 92 12-•2 Time constants f o r the FH model 95 12-•3 FH model with conductances 97 12-•4 Time constant of the membrane voltage 99 13--1 Membrane action p o t e n t i a l 104 14-•1 Membrane action p o t e n t i a l s , several s t i m u l i 112 14--2 Membrane action p o t e n t i a l s , several s t i m u l i 114 14-•3 Threshold amplitude vs. number of pulses 117 14-•4 Membrane action p o t e n t i a l s , s e v e r a l s t i m u l i 120 14-•5 Membrane action p o t e n t i a l s , continuous s t i m u l i 123 15-•1 Variables of stimulus 126 15-•2 Threshold amplitudes (normalized) vs. spacing 130 15-•3 Threshold charge vs. pulsewidth 133 15-•4 Threshold charge vs. spacing 134 15-•5 Two threshold s t i m u l i 135 15-•6 Threshold charge vs. time tp + t ^ 136 15-•7 Rate constants of the FH model 140 15-•8 Threshold charge f o r long pulse duration 142 16-•1 Voltage and current of electrode 147 16-•2 as 16-1, with saturation 147 16-•3 Charges f o r equivalent s t i m u l i 151 A-•1 Mean square errors vs. step s i z e A 12 A-•2 Time maximum amplitude vs. step s i z e A 14 A-•3 Maximum amplitude vs. step s i z e A 14 A-•4 Po t e n t i a l s computed with two step s i z e s A 15 V FIGURE PAGE B- l Stimulus response curve B 1 C-l Structure of stimualtor and i s o l a t i o n C 2 C-2 Analog i s o l a t i o n C 3 C-3 Current output C 5 C-4 Low power current source C 8 C-5 Stimulator, p o s i t i v e pulses C 11 C-6 Stimulator, negative pulses C 12 C-7 Power supply C 13 C-8 Test c i r c u i t C 14 E-1 Membrane voltages, FH model and V, m model E 4 E-2 as E-1, other stimulus E 5 E-3 as E-1, other stimulus E 6 E-4 I-V r e l a t i o n f o r sodium E 12 E-5 I-V r e l a t i o n f o r potassium E 13 E-6 Linear V, m model E 16 E-7 Membrane voltages, FH model and l i n . V, m model E 17 E-8 as E-7, other stimulus E 18 E-9 as E-7, other sitmulus E 19 E-10 as E-7, other stimulus E 21 E - l l Response of a passive RC c i r c u i t E 23 F - l Response of the l i n . V, m model; V, t and V, m plane F 4 F-2 as F - l , b i p o l a r stimulus F 5 F-3 E f f e c t of b i p o l a r d i r a c stimulus F 8 F-4 V, m curves F 9 F-5 P l o t of Table F-2 F 10 v i LIST OF TABLES TABLE PAGE 2-1 Data for myelinated frog f i b r e 8 12-1 Time constants of the HH model 93 12-2 Time constants of the FH model 96 14-1 Adaptation of stimulus parameters to the model 111 14-2 Threshold amplitudes f o r several b i p o l a r s t i m u l i 115 14-3 Threshold charges 119 15-1 Threshold currents vs. pulsewidth and spacing 128 15-2 Normalized currents vs. pulsewidth and spacing 129 15-3 Threshold charge vs. pulsewidth and spacing 131 15-4 Peak voltages vs. pulsewidth and spacing 138 15-5 Depolarization voltages vs. pulsewidth and spacing 139 16-1 Normalized charges from experiment 149 16-2 Threshold charges f o r the FH model 149 A-1 D i f f e r e n t i n t e g r a t i o n step s i z e s A 11 E-1 Lin e a r i z e d conductances E 11 F - l Threshold values (V, m) i n the l i n . V, m model F 3 F-2 Pulse i n t e r v a l s f o r threshold de p o l a r i z a t i o n F 10 F-3 Accuracy of the l i n e a r V, m model F 12 v i i ACKNOWLEDGEMENTS I would l i k e to express my appreciation to Dr. Peter D. Lawrence for h i s invaluable suggestions and support throughout the research and the preparation of t h i s documentation. Thanks are also due to Mark Bunce for providing the software to in t e r f a c e the PDP - 12 computer with the stimulator and for h i s assistance during the preparation of the experiments, and to Mary - E l l e n Flanagan f or her very e f f i c i e n t typing of the th e s i s . F i n a l l y , ' I am indebted to the Canada Council f o r having enabled with i t s f i n a n c i a l support my stay i n Canada and also the research reported here. v i i i " I f you please - draw me a sheep..." ... But then I remembered how my studies had been concentrated on geography, h i s t o r y , arithmetic and grammar, and I t o l d the l i t t l e chap (a l i t t l e c r o s s l y , too) that I did not know how to draw. He answered me: "That doesn't matter. Draw me a sheep..." For Antoine de Saint-Exupery's " L i t t l e Prince" i x INTRODUCTION The motivation to carry out the present study was to provide means for more r e l i a b l e , pain-free, and graded stimulation of p e r i p h e r a l nerve f i b r e s . This stimulation i s p r i m a r i l y used i n aids for disabled persons. I t has been claimed that b r i e f , b i p o l a r pulses can be used to achieve that type of stimulation. The mechanisms of its precise function are unclear at t h i s time. The method used i s to i n v e s t i g a t e with appropriate models the conditions imposed on s t i m u l a t i o n by a l l of the major components involved. These are: electrode, s k i n , t i s s u e , neuroanatomy, and nerve f i b r e s . This approach distinguishes the present work from an empirical inves t i g a t i o n . The s p e c i f i c goal of the thesis i s to i n v e s t i g a t e the e f f e c t of stimulus parameter v a r i a t i o n s on p e r i p h e r a l nerve f i b r e s using a dynamic model of myelinated nerve with p a r t i c u l a r a p p l i c a t i o n to the optimization of parameters for electrocutaneous stimulation. The achieve-ment of t h i s objective i n i t s f u l l e s t sense requires consideration of a large number of i n t e r r e l a t e d t o p i c s . Part I deals with the passive e l e c t r i c a l properties of s k i n , nerve, and the e l e c t r i c a l i n t e r f a c e . Since one i s stimulating the s k i n , i t s anatomy, receptors and neural organization must be considered ( Chapter 2). A s i m p l i f i e d e l e c t r i c a l model of skin has been selected (Chapter 3). Stimulus parameters u t i l i z e d by other researchers have been reported and the range of stimulus parameters narrowed to short duration current pulses (Chapter 4). Each stimulus produces heating i n t i s s u e . One aspect of optimizing parameters i s to minimize the amount of heating (Chapter 5). The e f f e c t s of electrode materials and geometry have been considered i n Chapter 6,and the influence of myelinated nerve geometry and o r i e n t a t i o n i n the current f i e l d discussed i n Chapter 7, since both of these factors are important i n the stimulus-response coupling. Part II i s concerned with the s e l e c t i o n of a mathematical • model for a c t i v e nerve phenomena (Chapters 8 to 10), with the e f f e c t s of propagation of the nerve impulse (Chapter 11), and with the time constants of the model (Chapter 12). The l a t t e r have to be known f o r the numerical computations with the model and also f or possible model s i m p l i f i c a t i o n s . Part I I I deals with the r e s u l t s of the i n v e s t i g a t i o n s using the Frankenhaeuser-Huxley nerve model and also with the experimental v e r i f i c a t i o n of those r e s u l t s . Several concepts of threshold are discussed i n Chapter 13, since subsequent chapters deal with threshold s t i m u l i . Sample c a l c u l a t i o n s on the e f f e c t s of d i f f e r e n t s t i m u l i (Chapter 14) lead to the s e l e c t i o n of a s i n g l e b i p o l a r pulse to be investigated systematically i n Chapter 15. The experiments, designed to t e s t the p r e d i c t i o n s , are described i n Chapter 16. In the Appendix the methods used i n c a l c u l a t i o n s and experiments are described (Appendix A, B, C, D). Also, an extension of the work not relevant to the main body of the thesis i s presented i n Appendix E and F, where s i m p l i f i e d nerve models are i n v e s t i g a t e d . The major conclusions and recommendations are described on the pages preceeding the Appendix. The main findings of each part are reviewed i n the summaries at the end of each part. 3 PART I: THE ASPECTS OF ELECTRICAL STIMULATION Chapter 1: SENSORY STIMULATION AND FUNCTIONAL STIMULATION 1. Types of e l e c t r i c a l s timulation 2. Sensory cutaneous stimulation 1. Types of e l e c t r i c a l stimulation The following work i s focussed on sensory and f u n c t i o n a l stimu-l a t i o n and has been motivated by an i n t e r e s t i n aids f o r disabled people and medical a p p l i c a t i o n s . Types of stimulation used primarly i n research i n v o l v i n g animals, such as brain stimulation and e l e c t r i c a l l y induced pain for behaviour conditioning are not investigated. Sensory stimulation i s mainly used to transmit information, be i t feedback concerning a p o s i t i o n or concerning an exerted pressure of a part of a prosthesis, or be i t information f o r the b l i n d or deaf concern-ing t h e i r environment. Of growing c l i n i c a l importance i s e l e c t r i c a l s timulation to r e l i e v e the chronic sensation of pain (Linzer and Long, 1976). Functional stimulation i s used to produce muscle contractions e s p e c i a l l y i n cases where the c o n t r o l from the Central Nervous System i s interrupted. One important a p p l i c a t i o n , for example, i s the c o n t r o l of the bladder i n patients with s p i n a l l e s i o n s . I f the stimulation acts upon a nerve trunk there i s l i t t l e d i f f e r e n c e between the mechanisms f o r sensory or f u n c t i o n a l stimulation. Buchthal and Rosenfalck (1966) investigated sensory and motor f i b r e s i n the arm. They found "that the d i f f e r e n c e s i n threshold between motor and sensory f i b e r s are due rather to the p o s i t i o n of the stimulating e l e c -trode i n r e l a t i o n to the d i f f e r e n t f a s c i c l e s of the nerve than to d i f f -erences i n f i b r e s i z e " [1]. Evidence for the p o s s i b i l i t y of stimulating [1] p. 70, 71. 4 sensory nerve f a s c i c l e s i s also given by E l t a h i r (1965)• By p l a c i n g a stimulating skin electrode accurately above a nerve trunk at the w r i s t , he was able to produce l o c a l l y defined sensations i n the hand. The mechanisms involved i n pain suppression by e l e c t r i c a l stim-u l a t i o n are unclear. The main d i f f i c u l t y i s that pain i s only a l i n g -u i s t i c l a b l e , used for a large v a r i e t y of manifestations and although pain i s quite r e a l f o r the subject experiencing i t , i t i s not e a s i l y r e l a t e d to a s p e c i f i c neural a c t i v i t y . I t i s unclear what structures of the ner-vous system are stimulated and further i t i s unclear i f pain suppression i s due to nervous i n h i b i t i o n , a c t i v a t i o n , or blocking. However, evidence i s i n c r e a s i n g that pain suppression may be achieved by a c t i v a t i n g large A-type nerve f i b r e s , which support the pain gating theory of Melzack and Wall (1965). This theory w i l l be discussed i n Chapter 2. I f t h i s were true, pain suppression would be c l o s e l y r e l a t e d to sensory s t i m u l a t i o n . In the c l i n i c a l a p p l i c a t i o n there i s no doubt that t h i s method of pain suppression by e l e c t r i c a l stimulation works. I t y i e l d s good to e x c e l l e n t r e s u l t s i f a number of conditions are met. A large l i t e r a t u r e concerning t h i s technique i s developing. Linzer and Long (1976) monitored the par-ameters of the e l e c t r i c stimulus and the electrode l o c a t i o n s i n 23 p a t i e n t s , and they provide s t a t i s t i c a l data. Further references are, to mention only a few, K i r s c h et a l . (1975) and Burton and Maurer (1974). In the normal case of sensory cutaneous stimulation, the stim-ulus acts rather upon the dermal neural network than on nerve trunks. 2. Sensory cutaneous stimulation In t h i s s ection the properties of sensory stimulation produced by electrodes attached to the s k i n are examined i n d e t a i l . Often t h i s s t i m u l a t i o n i s r e f e r r e d to by the name "transcutaneous s t i m u l a t i o n " or 5 "electro-cutaneous s t i m u l a t i o n " (ECS). I t i s of s p e c i a l i n t e r e s t because the a p p l i c a t i o n of the electrodes i s noninvasive, because of the poten-t i a l use i n many information transmitting devices, and because of the ease with which experiments can be performed. Primarily we would l i k e to obtain the following properties i n the sensation: large dynamic range - f i n e gradation of s t i m u l a t i o n accuracy and consistency - minor or no adaptation to the stimulus. To achieve an optimum, the following c r i t i c a l parameters have to be set properly or to be. accounted f o r : the i n t e n s i t y , the duration, and the waveform of the s i n g l e pulse; the r e p e t i t i o n rate of that pulse, and the t o t a l duration of a t r a i n of pulses. Also of importance i s the area of the electrode as w e l l as the body locus. F i n a l l y , the state of adaptation, the e f f e c t s of other s t i m u l i i n s p a t i a l or temporal proximity, and the motivational state of the observer have to be observed, ( a f t e r Rollmann, 1973) . Note that here the measure of stimulation i s a sensation and not a p h y s i c a l l y measurable manifestation such as nerve f i r i n g r a t e . Despite the two a d d i t i o n a l d i f f i c u l t i e s encountered with cutan-eous sti m u l a t i o n , s k i n - i n t e r f a c e and sensation,we have chosen to i n v e s t i g a t e t h i s form of stimulation because of i t s importance and advantages i n a p p l i c a t i o n s . In the next chapter the neural organization i s reviewed i n order to define more p r e c i s e l y what should be stimulated and what p o t e n t i a l problems w i l l be encountered. 6 Chapter 2: NEURAL ORGANIZATION AND SENSATION: A REVIEW 1. Neural organization, d e f i n i t i o n s 2. Histology of skin and receptors 3. What i s stimulated: receptor or nerve 4. Sensation and pain. 1. Neural organization, d e f i n i t i o n s The i n v e s t i g a t i o n s are r e s t r i c t e d to the pe r i p h e r a l nervous system. A s i n g l e neural unit has three parts: the receptive part that generates graded e l e c t r i c a l p o t e n t i a l s (generator p o t e n t i a l s ) i n response to s t i m u l i of appropriate modality. - the axon, the transmission l i n e , which may be over one meter long and which conducts information a c t i v e l y by transmitting and by continuously reshaping the e l e c t r i c impulse, which i s c a l l e d the action p o t e n t i a l (A.P.) or spike. The A.P. o r i g i n a t e s at the jun c t i o n of the axon with the receptive part. The synaptic knobs, terminating on another neuron. Further, the neuron has a nucleus which may be located i n d i f f e r e n t places, depending on the type of neuron. Our in v e s t i g a t i o n s are r e s t r i c t e d to the nerve structures i n the periphery, i . e . to receptors and to axons. In t h e i r passage to the pe r i p h e r a l nerve, f i b r e s are c o l l e c t e d into bundles or f u n i c u l i . Associated with them are other tissues and structures to form nerve trunks. The d e f i n i t i o n of a nerve i s : a bundle or a group of bundles of nerve f i b r e s outside the c e n t r a l nervous system. A f i b r e i s defined as: a neuron or the axonal portion of a neuron. The reader i s r e f e r r e d to Sunderland (1968) [1] or to any standard textbook on phy-siology f o r more d e t a i l s . [1] Part I, chapter 1 and 2 7 C l a s s i f i c a t i o n P e r i p h e r a l nerve f i b r e s are c l a s s i f i e d into type A - f i b r e s , which are myelinated, and unmyelinated type C - f i b r e s . A - f i b r e s : These f i b r e s are covered by segments of an i n s u l a t i n g myelin sheath, which leaves the nerve membrane uncovered only at the nodes of Ranvier. The transmission of the ac t i o n p o t e n t i a l i s achieved by e x c i t a t i o n only at the nodes, which r e s u l t s i n a high propagation v e l o c i t y . The distance between two nodes i s proportional to the diameter and so i s the propagation speed. According to t h e i r diameter, the A - f i b r e s are divided i n sub-groups: diameter A-a 12-22 um A-g 5-12 um A-fi 2-5 um While A-a f i b r e s are re l a t e d to the Golgi tendon organ, A-3 f i b r e s are r e l a t e d to touch, pressure, and v i b r a t o r y receptors. A-6 f i b r e s are re-la t e d to free nerve endings and pain-temperature (from Mountcastle, 1974) [1]. The diameter of A - f i b r e s may change randomly along the f i b r e , f o r example i n one nerve f i b r e from 6.5 to 16 um (Sunderland,1968) [2]. Example: E l e c t r i c a l c h a r a c t e r i s t i c s of a Frog's myelinated f i b r e Many investigation's by several researchers have determined the e l e c t r i c a l c h a r a c t e r i s t i c s of myelinated f i b r e s . The following table from Hodgkin (1964) summarizes these r e s u l t s . A myelinated f i b r e i s shown schematically i n F i g . 2-1, and the equivalent e l e c t r i c c i r c u i t i s shown i n F i g . 2-2. The model w i l l be d i s -cussed i n Part I I . The number of d i s c r e t e segments to model the sheath [1] p. 75, [2] p. 9. 2 Fibre diameter (D) 14 u Thickness of myelin 2 u Distance between nodes (L) 2 mm 2 Area of nodal membrane (assumed) 22 u Resistance per unit length of axis c y l i n d e r (r^) 140 megohm/cm S p e c i f i c resistance of axoplasm 110 ohm cm Capacity per unit length of myelin sheath (C g) 10-16 pF/cm Capacity per unit area of myelin sheath 0.0025-0.005 uF/cm D i e l e c t r i c constant of myelin sheath 5-10 2 Resistance x un i t area of myelin sheath 0.1-0.16 megohm cm S p e c i f i c resistance of myelin sheath 500-800 megohm cm Capacity of node 0.6-1.5 pF 2 Capacity per unit area of nodal membrane (C^) 2-7 uF/cm Resistance of r e s t i n g node 40-80 megohm 2 Resistance x unit area of nodal membrane (R^) 10-20 ohm cm at r e s t i n g state Action p o t e n t i a l 116 mV , Resting p o t e n t i a l 71 mV 2 Peak inward current density 20 mA/cm Conduction v e l o c i t y 23 m/sec Table 2-1 Data for myelinated frog f i b r e ( a f t e r Hodgkin, 1964) 9 var i e s from 1 to 15. The myelin sheath i s composed of up to one hundred layers of membrane-like structures wound around the axon. Myelin Sheath Node of Ranvier Area 22 i i F i g . 2-1 Schematic myelinated f i b r e c _ 2/iF/crn —v— node AA/V A A / V W W . •AAA Q=10pF/cm -AAA-4l~0 outside R=15&cm inside j i — , » ^rr140MD-/cm myelin sheath node F i g . 2-2 E l e c t r i c a l model of myelinated f i b r e C-fibres These are small, unmyelinated and thus slowly conducting nerve f i b r e s . Their diameters range from 0.1-1.3 um and t h e i r f u n c t i o n i s r e -l a t e d to pain, temperature, and mechanoreceptors. Unmyelinated and 10 f i n e l y myelinated f i b r e s predominate i n the p e r i p h e r a l sensory system. In a cutaneous sensory nerve they may make up three-quarters of the f i b r e s . However, the sensation of touch and pressure i s mainly mediated by A-3 f i b r e s (see Section 4) and therefore the i n t e r e s t w i l l be focussed upon stimulation of these myelinated f i b r e s . Photographic References: Micrographic pictures documenting the growth of the sheath i n a young rat's s c i a t i c nerve are published by Shanes (1961) [1], In the same book there are also pictures of a cross s e c t i o n of a s c i a t i c nerve f i b r e i n a youngmouse [2]. Micrographic pi c t u r e s of cutaneous f i b r e s i n a cat are published by Gasser (1958), who compares structure and physiology of nerve f i b r e s . An exc e l l e n t d e s c r i p t i o n of per i p h e r a l nerve f i b r e s i l l u s -trated by many micrographic pictures i s given by Sunderland (1968). F i n a l -l y , Winkelmann (1960) presents large photographic data about nerve endings i n s k i n . 2. Histology of skin and receptors The main components of skin and cutaneous receptors are shown i n F i g . 2-3. A considerable v a r i e t y of nerve terminations i s found i n the skin l a y e r s . For a d e t a i l e d d e s c r i p t i o n see for example, Geldard (1972) [3]. Though the three main layers of the skin, epidermis, dermis, and subcutaneous tissue are found i n a l l types of skin, there are considerable differences i n skin structure depending on the body locus. Ducts of sweat glands and h a i r f o l l i c l e s may contribute most to a nonuniform d i s t r i b u -t i o n of an e l e c t r i c current across these s k i n layers when cutaneous e l -ectrodes are used. [1] p. 67 [2] p. 87 [3] p. 273 Moissner's corpuscle Nerve ending Subcutaneous Pacinian Duct of Ruf f in i around hair fat corpuscle sweat gland ending Fig. JM). Composite diagram of .the skin in cross section. The chief (aycrs, epidermis, dermis, ;md subcutaneous tissue, arc shown, as arc also a hair foilic'e, the smooth muscle which erects the hair, ami several kinds of nerve endings. In the epidermis are to be found tactile discs and free nerve endings; in the dermis are Meissner corpuscles, Krause end bulbs, Ruflini endings, and (around the base of (he hair) free terminations. The subcutaneous tissue is chiefly fatty and vas-cular but contains Pacinian corpuscles, the largest of the specialized endings. From Gardner (216) after W'oollard, WcddcIJ, and Harpman (652). By permis-sion of AV. B. Saunders Company, Philadelphia. F i g . 2-3 C r o s s s e c t i o n o f s k i n ( f r o m G e l d a r d 1972) . 12 Receptive f i e l d of a f i b r e (Dermatome) A s i n g l e nerve f i b r e branches i n t o many endings. The areas i n -nervated by one nerve f i b r e overlap areas innervated by other f i b r e s . For example, an i n v e s t i g a t i o n of the d i s t r i b u t i o n of t a c t i l e endings i n the s k i n of frog was made by Adrian et a l . (1931). The area served by a 2 s i n g l e nerve i s several cm i n areas of low s e n s i t i v i t y such as the back, and i t i s small i n areas of high s e n s i t i v i t y and high s p a t i a l r e s o l u t i o n , such as the hand. Figures f o r monkey are: the l a r g e s t f i e l d i n the upper 2 2 arm i s 2.8 cm and the smallest f i e l d i n a d i g i t i s 0.09 cm . 3. What i s stimulated: receptor or nerve? Despite several attempts published i n l i t e r a t u r e to solve t h i s problem, there i s no c l e a r answer. Of course, a preliminary condition to be met i s that the e l e c t r i c a l stimulus i s r e s t r i c t e d i n such a way that no thermal or chemical changes occur i n the s k i n , to make sure that receptors w i l l not react to such secondary events. E s p e c i a l l y , a p a i n f u l sensation might be caused by a chemical intermediary a c t i v a t i n g receptors. Evidence comes from the f a c t that the pain remains for a prolonged time a f t e r cessation of stimulation and from the f a c t that pain i s proportional to the charge applied. From Faraday's f i r s t law i t follows that the weight of a substance l i b e r a t e d from solutions by e l e c t r o l y s i s i s p r o p o r t i o n a l to the charge, which might propose a chemical mediation of pain. Also i n the case of a d i e l e c t r i c breakdown of the s k i n , causing small burned out current paths across the skin, i t may be a mere case of receptors reacting to such s t i m u l i . More serious attempts were made by t r y i n g to separate s p e c i f i c reactions of receptors and nerves. Vernon (1953) t r i e d to solve the controversy by studying the i n t e r a c t i o n s between simultaneous v i b r a t o r y 13 and e l e c t r i c a l stimulation of the f i n g e r - t i p . His experiments were not conclusive, but at l e a s t , they provided evidence against the p o s s i b i l i t y that the e l e c t r i c a l current would cause a mechanical v i b r a t i o n of the skin. In other experiments, however, i t also has been shown that the skin can be vibrated by the e l e c t r o s t a t i c forces of an e l e c t r i c f i e l d . Investigations of the e f f e c t of skin temperature upon threshold for e l e c t r i c a l stimulus by Hawkes (1962) showed that r a i s i n g the skin temperature up to 45°C had no e f f e c t on the threshold f o r electro-cutaneous stimulation, but i t increased the threshold f o r mechanical stimulation, which suggests that the current a f f e c t s the nerves d i r e c t l y . Summing up his experience, Rollmann (1973) states: "My own research ... as we l l as that of several others ... suggests that an e l e c t r i c a l stimulus applied to the skin surface also acts d i r e c t l y on the underlying p e r i p h e r a l nerves to i n i t i a t e an ac t i o n p o t e n t i a l , bypassing the receptors which respond when adequate modes of stimu l a t i o n are employed" [1], F i n a l l y , the ex-periment of E l t a h i r (1965) stimulating a nerve trunk at the wri s t and producing a sensation i n the nerve's receptive f i e l d i n the hand c l e a r l y demonstrates the p o s s i b i l i t y of d i r e c t neural stimulation. , To summarize, i t i s evident that e l e c t r i c a l s timulation acts d i r e c t l y upon nerves, but i t remains uncertain to what degree stimulated receptors also contribute to the nervous a c t i v i t y . 4. Sensation and pain Sensation i s based on the q u a l i t y of a stimulus (pressure, v i b r a t i o n , temperature, pain, etc.) as w e l l as on the quantity ( i n t e n s i t y ) of a stimulus. I t i s d i f f i c u l t to measure or to describe the q u a l i t a t i v e dimensions of a stimulus because the transfer function r e l a t i n g a sensa-[1] P. 39 14 ti o n to the perip h e r a l nerve a c t i v i t y i s nonlinear and dependent on many emotional and environmental v a r i a b l e s . However, the apparent correspon-dence between subjective i n t e n s i t y and impulse frequency i s s u r p r i s i n g . The p h i l o s o p h i c a l controversy concerning the coding of the neuronal i n f o r -mation was c a r r i e d on over many decades. One theory, based on a mechani-c a l conception, postulated highly s p e c i a l i z e d , s p e c i f i c receptors, each connected with a s p e c i f i c part of the c e n t r a l nervous system, whereas the other theory postulated few broadly tuned receptors and the coding of i n -formation i n f i r i n g patterns. Melzack and Wall (1965) give an excel l e n t review of these two theories and corresponding p h y s i o l o g i c a l evidence. They were able to combine these two theories into the gate theory, which i s supported by increasing evidence. The essence of the theory i s that there are two neural networks, the fa s t conducting one composed of A-3 f i b r e s and the slowly conducting one composed of C f i b r e s and A-6 f i b r e s , each of which codes the information by f i r i n g patterns. The mutual i n t e r a c t i o n determines i f a sensation i s p a i n f u l or not. For d e t a i l s see Melzack and Wall (1965) or physiology textbooks. S i m p l i f i e d , the p r e d i c t i o n of the theory i s , that a sensation i s p a i n f u l whenever the a c t i v i t y of the A-6 and C f i b r e s i s greater than that of the A-3 f i b r e s . Support f o r the theory comes, among others, from the experiments of H a l l i n and Torebjb'rk (1973) i n which they were able to measure and d i s t i n g u i s h the responses of A and C f i b r e s and were able to associate the a c t i v i t y of fa s t conducting A f i b r e s with v i b r a t o r y sensa-tions and to associate the a c t i v i t y of slowly conducting A-6 and C f i b r e s with the sensation of p r i c k i n g or pain. Experiments by Torebjb'rk and H a l l i n (1973), examining perceptual changes when one group of nerves was blocked, confirmed the f i n d i n g s . C o l l i n s et a l . (1960 and 1966) made 15 s i m i l a r experiments with d e t a i l e d observations. When only A-B f i b r e s were ac t i v a t e d , a l l experiences were reported by the subjects as nonpain-f u l . When, however, a stronger stimulus activated also the A-6 f i b r e s , a sharp change i n the sensation evoked occurred immediately, the sensation being sharp and p r i c k i n g l y p a i n f u l . When conduction was blocked i n the A f i b r e s and a much stronger stimulus was used to a c t i v a t e C f i b r e s , a sensation of unbearable pain was experienced. These findings are we l l proven and accepted i n the l i t e r a t u r e . Fortunately, the stimulating threshold increases with a decrease i n f i b r e diameter,, such that the A-3 f i b r e s respond f i r s t . However, as shown by Sassen and Zimmermann (1973) and also by Brown and Hamann (1972) the conduction i n large myelinated f i b r e s can be blocked by a high con-tinuous current which keeps the f i b r e depolarized and thus prevents these f i b r e s from recovering, while the smaller f i b r e s remain unaffected. To summarize, a stimulation i s not p a i n f u l provided: a) only the large myelinated f i b r e s are activa t e d b) these large myelinated f i b r e s are not blocked by continuous overstimulation. Therfore, two conditions f o r a pain free e l e c t r i c stimulus are: a) the i n t e n s i t y must be c o n t r o l l a b l e i n small gradation steps b) the stimulus must be pulsed (no DC-current). The next questions to answer are how the ski n l y i n g between the electrode and neural structures can be modelled i n order to under-stand the e f f e c t s of various s t i m u l i upon i t . 16 Chapter 3: THE SKIN INTERFACE 1. Physical changes to be avoided 2. E l e c t r i c model of skin 3. Investigation of model properties 4. Electrode areas and body p o t e n t i a l 1. P h y s i c a l changes to be avoided I f the stimulus causes a p h y s i c a l change i n the s k i n , i t i s l i k e l y that ( i ) receptors w i l l respond and ( i i ) the sensation w i l l be p a i n f u l . The following three e f f e c t s have to be avoided: the d i e l e c t r i c breakdown of the skin r e s u l t i n g i n a few high density current channels across the s k i n . e l e c t r o l y t i c processes, eventually producing gas. a large production of heat i n the s k i n resistance. High density current channels across the s k i n may be formed by high voltages causing a breakdown, or by inhomogeneiti.es of the s k i n , for instance by conductivity through sweat pores, or by inhomogensLties of the electrode-skin contact. Saunders (1973) shows that with a concentric electrode and with current stimulation the electrode voltage saturates at 250 V. The corresponding sensation i s p a i n f u l . Therefore voltages across the s k i n i n excess of about 150 V should be avoided. (The voltage of 250 V, mentioned above, i s d i s t r i b u t e d between two s k i n crossings, see Section 3-4). Also, to avoid excessive currents i n the case of a breakdown, the stimulator has to be current regulated. Mason and Mackay (1976) also investigated the sensations of s t i n g i n g pain and found t h i s pain r e l a t e d to the "thermal damage to the corneal layer of the s k i n . The high-energy d e n s i t i e s required to create t h i s damage often occur, even at moderate stimulation currents, because 17 of the highly nonhomoger.eousnature of the skin-electrode i n t e r f a c e " . They used long b i p o l a r pulses (pulsewidth t = 1 ns, separation t ^ = 2 ms) i n t h e i r i n v e s t i g a t i o n s . To avoid e l e c t r o l y t i c processes, two necessary conditions are ( i ) that the average charge i s zero, which means that only b i p o l a r pulses can be used, and ( i i ) that the charge of a pulse of one p o l a r i t y i s l i m -i t e d such that electrochemical processes are l i m i t e d to double layer charg-ing and surface processes, excluding the production of reaction products leaving the electrode surface (Brummer and Turner, 1977). F i n a l l y , the heat generation w i l l be discussed i n chapter 5. 2. E l e c t r i c a l model of the skin The skin shows a complex e l e c t r i c a l behaviour: - The resistance depends on the s k i n preparation (dry, wet, sanded skin) (Burton 1974). The resistance depends on the psychological conditions of the test subject. - The impedance changes slowly when a stimulus i s applied: warm-up e f f e c t (Saunders, 1973). The resistance and the capacitance depend on the current applied (Gibson, 1968). The impedance depends on the body locus (Johnson, 1975) . Further, there i s a i n t e r n a l skin p o t e n t i a l of 15 ... 40 mV. For stimu-l a t i o n , however, t h i s can be neglected. I t i s not the o b j e c t i v e of t h i s t hesis to i n v e s t i g a t e the properties of the s k i n . Edelberg (1972) gives an ex-c e l l e n t review, r e f e r r i n g to 115 further references. See also Burton (1974), Saunders (1973), Stephens (1963), and Gibson (1968). To summarize, an approximately equivalent c i r c u i t f o r stimu-l a t i o n has the form of an RC network (Edelberg, 1972), shown i n F i g . 3-1. -2 -2 With 0.02 uFcm < C < 0.04 yFcm 2 2 100 fi cm < R < 500 fi cm s 2 2 32 k fi cm < R < 710 k fi cm inside of skin layers R i c outside o r F i g . 3-1 Skin model Rg i s p a r t i a l l y a t t r i b u t e d to the inner t i s s u e resistance. Although not stated e x p l i c i t l y i n Edelberg's d e s c r i p t i o n , t h i s c i r c u i t seems to i n -clude the electrode-skin i n t e r f a c e . 3. Investigation of model properties Several basic properties follow from the RC structure of the equivalent skin c i r c u i t . For s i m p l i c i t y i t i s assumed that no current spreads out to adjacent skin areas. Thus, the ski n area considered i s the same as the electrode area. The time constant T = RC i s independent of the electrode area. 2 -2 The minimum value i s : T = 32 k fi cm -0.02 uFcm = 0.64 ms. Gibson m 2 (1968) reports the same values for a current step of about 0.1 mA/cm , however, i f the amplitude increases to 2.5 mA/cm , the measured resistance R decreases to about 16 k era' and the measured minimal time constant i s as low as 0.07 ms. If a voltage step V q i s applied, a current spike r e s u l t s . I t i s a decreasing exponential function, the time constant i n t h i s case depending on the small R g i n ser i e s with C: with R_ << R I Because the s e r i e s resistance R i s small t h i s time constant i s extremelv s } 2 2 short: CRg % 0.02 yF/cm * 100 Q cm = 2 ys. Such current spikes r e s u l t i n poor c o n t r o l over stimulation and can lead to p a i n f u l sensations. Conversely, i f a current step i s applied, the skin voltage i s : Note, that the voltage only r i s e s with the long time constant T = RC > 0.07 ms. C l e a r l y , to avoid these current spikes and to make the stimulus accurately c o n t r o l l a b l e , i t has to be current c o n t r o l l e d . This conclu-sion i s supported by nearly a l l i n v e s t i g a t i o n s reported i n the past ten years. The maximum allowable charge which can be transmitted by a cur-rent pulse through the capacitor across the skin can be ca l c u l a t e d . An approximation f o r short pulses i s to only consider the capacitor (R = <*>, Rg = 0). Then, the charge i s : Q = V gC. Assuming the maximum voltage across the skin, V to be ±100 V and taking C = 0.02 uFcm , we get: s —2 |Q| <^  2 ucoul - cm 4. Electrode areas and body p o t e n t i a l Two electrodes with the areas A^ and A^ are needed f o r stimu-l a t i o n . Assuming that the electrodes make i d e a l contacts with the skin, and assuming that the sk i n i s homogeneous, we get the c i r c u i t of F i g . 3-2. Electrode I v, t ) r skin under electrode I \ P o r e ntial of core of body ? skin under 21 electrode I Electrode 1 F i g . 3-2 P o t e n t i a l d i s t r i b u t i o n Denoting with Z q the skin impedance per unit area [Q, cm" 2], F i g . 3-2, we write Z = — 1 A i 7 = — 2 A2 and therefore the voltages are: thus If electrode 2 i s the i n d i f f e r e n t electrode with >> A^ then % 0 and the p o t e n t i a l of the core of the body (for ex. of a foot, i f the arm i s stimulated) i s approximately the same as that of terminal B of the electrode i n F i g . 3-2. There i s no shock hazard even i f point B i s con-nected with ground. I f , however, both electrodes have the same area, then = V^. To prevent shock hazards, i . e . to connect point C with ground without any e f f e c t s , i t i s necessary that neither terminal A nor B be con-nected to ground. To summarize, i t i s necessary that the stimulator be i s o l a t e d from any ground connection to prevent shock hazards. A proper d e s c r i p t i o n of the stimulus parameters i s given i n the next chapter. Throughout the thesis these notations w i l l be used. Also, the spectrum of s t i m u l i used by researchers i s surveyed. Chapter 4: CHARACTERIZATION OF THE STIMULATING WAVEFORM 1 . Variables of the waveform 2. S t i m u l i used i n l i t e r a t u r e 3. Discussion 1 . Variables of the waveform In general, an e l e c t r i c a l stimulus has the following p r o p e r t i e s : the amplitude of e i t h e r the voltage or the current can be chosen as parameters (but not both). Correspondingly, the stimulator i s designed to supply a defined current or voltage waveform, the stimulus may occur continuously or be gated, the stimulus may be monopolar or b i p o l a r . the stimulus s t a r t s with e i t h e r the negative or the p o s i t i v e amplitude. - the amplitude i s a function of time, normally sine wave or rec-tangular pulses. Thus i t has: a r e p e t i t i o n frequency T/2 a charge Q = / i dt of the half-wave o - a pulse width ( i f i t i s a pulse) - the stimulus may be gated into bursts i n which case i t has a burst r e p e t i t i o n frequency. Conventional p h y s i o l o g i c a l c h a r a c t e r i z a t i o n only deals with amplitude and time: Rheobase i s defined as the amplitude of a stimulus of threshold strength which j u s t causes a stimulation but only a f t e r a long time. Chronaxie i s defined as the stimulus duration of threshold strength when the stimulus has twice the amplitude of Rheobase. 23 2. St i m u l i used i n l i t e r a t u r e A waveform i s defined by the following notation: a) Sine wave + A -A b) Pulse + A f r - j ^ M -A. r t T = f = % = t = r f = r A = period 1/T: frequency duration of burst burst r e p e t i t i o n i n t e r v a l l / t r burst r e p e t i t i o n frequency amplitude i II n JU u III II 11 T P+ P-duration of p o s i t i v e pulse duration of negative pulse i n t e r v a l between end of p o s i t i v e and beginning of negative pulse (spacing) delay of negative pulse The following table i s a survey of stimulus waveforms used by the several researchers. ECS stands f o r electro-cutaneous stimulation with human subjects. a) current c o n t r o l l e d s t i m u l i AUTHOR OBJECT WAVEFORM Saunders (1973) ECS, Abdomen bi p o l a r pulses t = t = 0 .. . 20 us p+ p-t n = 25 ys -n-u ru^.. T = 100 ys, f = 10 kHz f = 250 Hz r Sundstrom (1974) ECS, Forearm as Saunders, but t = 20 us P parameter: // of pulses per burst, amplitude Hardt (1974) ECS, Ventral forearm, thenar, index f i n g e r as Saunders parameter: amplitude, t P Linz e r and Long (1976) ECS f o r pain r e l i e f , several body locati o n s mostly used pulse form: monopolar pulses t = 50 ... P mn ]i« n n .._ f = 14 ... 60 Hz Pr i o r (1972) ECS, d i f f e r e n t body locati o n s many d i f f e r e n t pulses monopolar pulses are best t = 1 ys ... 5000 ys f = low frequency 25 b) voltage c o n t r o l l e d s t i m u l i AUTHOR OBJECT PULSEFORM E l t a h i r (1965) ECS, forearm, hand b i p o l a r pulse: t = t = p+ p-80 ys, t^ = 0 (no spacing) -TJ parameter: # of pulses per burst, f , amplitude Schwarz and Volkmer (1965) Frog, s c i a t i c nerve, s i n g l e node b i p o l a r pulse: t = t p+ p-60 ys, t^ = 0 (no spacing) parameter: # of pulses f o r one A.P., f = 270 ... 7500 Hz, p o l a r i t y Bromm and Girndt (1967) Frog, s c i a t i c nerve, s i n g l e node, 14°C a) b i p o l a r pulse t , = t = P+ p-25 ... 250 ys, t± = 0, -TJ T r -b) gated and phase locked sine wave , -Vl^I/ l / l r - f = 5 kHz Bromm (1966) Frog, s c i a t i c nerve, multiple f i b r e s , 15°C comparison between monopolar pulse and sine wave of same burst duration f = 5 kHz Bromm and L u l l i e s (1966) Frog, s c i a t i c nerve, multiple f i b r e s , 7°C ... 20°C sine wave, gated and phase locked, f = 20 Hz ... 50 kHz — J | 26 AUTHOR OBJECT PULSEFORM Schwarz (1966) ECS, musculus sine wave, parameters: o r b i c u l a r i s o r i s frequency and // of periods (i n face) Wyss (1967a) Frog, s c i a t i c sine wave, gated, f = 20 kHz nerve, multiple parameters: burst duration, fibres, 18-20°C amplitude Wyss (1967b) Frog, s c i a t i c sine wave, gated f = 2 ... nerve, multiple 5 kHz f i b r e s , 20°C Rabbit, vagus nerve, multiple sine wave, gated f = 18 kHz f i b r e s , 38°C 3. Discussion Frequency of stimulus: The highest f i r i n g frequency of periph-e r a l f i b r e s i n man i s below 500 Hz. Using frequencies above t h i s value r e s u l t s i n the summation of the e f f e c t s of each period u n t i l eventually an A.P. i s generated. This important e f f e c t , c a l l e d medium frequency stimulation or sometimes c a l l e d quantal stimulation (Saunders) w i l l be considered i n d e t a i l i n subsequent chapters. These medium frequency waves are gated and the r e p e t i t i o n rate of the bursts i s below 500 Hz. Note that DC-stimulation i s not used. Pulse p o l a r i t y : S t r i c t l y monopolar pulses must be rejected because they cause an e l e c t r o l y t i c r eation. I f , however, the pulse i s biased such that the average charge i s zero, they may be used e f f e c t i v e l y . In t h i s case the pulse i s , s t r i c t l y speaking, an asymmetric b i p o l a r pulse: A • *• A A ft//// t Current or voltage regulated? It w i l l be shown conclusively in the next chapters that for ECS applications a current regulation is re-quired. That this notion.is a recent one is seen from the publications: in the 1960's experiments were performed with voltage stimuli, in the 1970's with current stimuli. Sine wave or rectangular pulses? As i t will be shown later, charge is an important parameter in stimulation. If the frequency of a sine wave is changed, the charge of a half cycle is changed/too. If, however, a waveform is used consisting of short pulses at the begining of each half period, the frequency f may be varied without affecting the amount of charge, provided f remains below f^ = ^ ^ : m P 28 The sine wave contains the s i n g l e frequency f. Besides the same frequency f, the pulse frequency spectrum contains also many higher frequency com-ponents. In the next chapter i t w i l l be seen that the ca p a c i t i v e property of the skin f a c i l i t a t e s transmission of high frequencies, an advantage f o r pulses. Further, pulses o f f e r the p o s s i b i l i t y of varying the p o s i t i o n of the negative pulse within the period T, which introduces a new and us e f u l con-t r o l v a r i a b l e . Because of these advantages further i n v e s t i g a t i o n s are r e s t r i c t e d to rectangular pulses. Since our aim i s to stimulate nerve f i b r e s d i r e c t l y , we have to make sure that the a c t i v a t i o n of receptors v i a thermal e f f e c t s w i l l not i n t e r f e r e with that goal. This i s investigated i n the following chapter. 29 Chapter 5: HEAT PRODUCTION BY THE STIMULUS 1. Description of the problem 2. Heat versus time 3. Bipolar stimulus: heat versus pulse waveform 4. Discussion More than 99% of the stimulating energy acts not on the nerve but i s converted into heat i n the various t i s s u e l a y e r s : a current con-t r o l l e d stimulus requires a voltage swing of up to 100 V to produce a voltage change i n the nerve membrane of several tens of m i l l i v o l t s . The s e r i e s resistance R g of the ski n model (Fig. 3-2) i s small compared with R. Therefore we w i l l l i m i t most of our in v e s t i g a t i o n s to the simple RC p a r a l l e l c i r c u i t of F i g . 5-1. The complete c i r c u i t i s discussed i n Section 4. This c i r c u i t i s such a basic model that i t applies not only to the skin but also to the electrode i n t e r f a c e (see Chapter 6) and also to the sheath of connective t i s s u e , the perineurium,,surrounding a nerve bundle. As i t w i l l be seen i n Part I I , the most important parameter for nerve stimulation i s the charge. The in v e s t i g a t i o n s focus on the question of how to transmit a charge across the skin and how to l e t t h i s charge act for a "long time" upon the nerve membrane before i t i s removed 1. Description of the problem F i g . 5-1 C i r c u i t f o r heat c a l c u l a t i o n s by the charge-equalizing negative stimulus, while minimizing the amount of heat produced. 2. Heat versus time The energy E which i s converted i n t o heat depends on the current through the resistance R and the time: t « ry E(t) = f I (t) R dt, i n Watt-sec-cnf . R o The current through the r e s i s t o r , I , depends only on the voltage V„ R C across the capacitor. h = V R Assuming that a constant current pulse i s applied with amplitude I, we get with T = RC: V c = I R (1 - e " t / T ) and t V 2 t f ' kR E(t) = / (-£) R dt = I2 R / (1 - e " t / T ) dt o E(t) = I 2 • R (t - | T + 2 T e " t / T - \ e " 2 t / T ) v Energy * The time dependent expression "Energy *" i s cal c u l a t e d i n F i g . 5-2. The normalized time i s t/T. For comparison, the l i n e a r function Energy'* = t i s also p l o t t e d . This represents the heat produced i n R i f there were no capacitor. The di f f e r e n c e between the two curves thus repre-sent the energy stored i n the capacitor. The comparison of the two fun-ctions at d i f f e r e n t times i s as follows: 31 a ' TIME (NORMRLIZED) F i g . 5-2 Heat production i n function of time f o r a constant current applied to the RC network. Abscissa: normalized time = — = r~ T RC Eft) Ordinate: Energy* = ^ ' I R t = Energy* Energy1* 0.1 T 0.3% T 16% 2 T 38% 3 T - • 53% 9 • 100 T • • 100% The interpretation of these figures is that at the beginning of a stimulus most of the current is conducted as displacement current through the capacitor and only gradually the current through the resistor increases, giving rise to heat production. Of course, i f the current pulse ends, the energy stored in the capacitor is also gradually converted into heat as C is discharged by R. Fig. 5-3 shows the qualitative curve. E n e - p V only R Stimulus I r t Fig. 5-3 Energy with and without capacitor However, i f at the end of the positive pulse the energy stored in the capacitor is removed by a following negative pulse, the maximum of the energy will be less than the final value in Fig. 5-3. This i s one reason f o r the use of short, b i p o l a r current pulses Saunders' (1973) explanation for the use of short b i p o l a r pulses em-ploys s i m i l a r but q u a l i t a t i v e arguments. Hardy et a l . (1952) determined -1 -2 the value of 250 m c a l - s -cm for the burning pain threshold from radiant heat. In e l e c t r i c a l u n i t s , t h i s value i s : 250 m c a l s _ 1cm~ 2*4.184 = 1.05 Watt-cm" 2 Note that t h i s refers to heating power, not to energy, and thus describe t h e ' a b i l i t y of the s k i n to conduct heat away from the point of occurence. Gibson (1968) [1] c a l c u l a t e d that during a normal stimulation the average -2 power i s l e s s than 0.1 Watt-cm . However, i f some current paths are -2 formed, the f i g u r e of 1 Watt-cm may w e l l be exceeded l o c a l l y . The r e l a t i o n between heat production and waveform parameters i s i n v e s t i g a t e d i n the next se c t i o n . 3. Bipolar stimulus: heat versus pulse waveform The pulse waveform i s defined by the notation: Amplitude -fl - I The current within an i n t e r v a l i s a piecewise constant function. What i s the t o t a l energy converted i n t o heat i n the r e s i s t o r R as a function of t p , and t^? The i n v e s t i g a t i o n s are made under the [1] p. 243 s i m p l i f y i n g r e s t r i c t i o n s : a) the p o s i t i v e and the negative pulse have the same amplitude and duration. b) the charge Q = I t of a pulse i s kept constant f o r d i f f e r e n t pulse widths, thus the amplitude i s determined: I = Q/t c) only the energy of one b i p o l a r pulse i s considered on the i n -t e r v a l t = 0 ... =» (no pulse t r a i n s ) . General s o l u t i o n for constant current: The energy i s determined by the current I through R. I depends on the capacitor's voltage V . Li E(t) - /" I R 2 R dx ; I - ^ o with V = V„(t = 0) and T = RC o C V_(t) = RI • (1 -e" t / T) + V e " t / T = RI + ( V - R I ) e " t / T C o o 2 E(t) = / • R dx = | / ( R 2 I 2 + 2RI (V - RI) e " T / T + o R o ( V - R I ) 2 e " 2 x / T ) dx o the s o l u t i o n i s : 2 E(t) = R I 2 (t - | T + 2T e " t / T - | e " 2 t / T ) + V q IT (1 - e " t / T ) -2 (1 _ e " 2 t / T ) 2R Tota l energy: The current i s piecewise constant. The energy can be deter-mined f o r each of the i n t e r v a l s 1 to 4 and the t o t a l energy i s : E = E± + E 2 + E 3 + E 4 The voltages at the beginning of an i n t e r v a l are: 35 > t Abbreviations; - t /T (1 - e ? ) -2t /T (1 - e P ) -2t /T (1 - e 1 ) - t /T Q P = a = b = c ~t±/T Then: -2t /T P '01 f 0 y i e l d s : V Q 2 = R I a V Q 3 = R I a e '04 = R I a e d - R I a = RIa (ed - 1) The use of the general s o l u t i o n with appropriate v a r i a b l e s I n t e r v a l 1: Amplitude = t = V E n = R I 2 (t - 4 T + 2T 1 P 2 + I [0, t p ] V01 = 0 d - j • f) I n t e r v a l 2: Amplitude = 0 t = [0, t^] V = V o v02 2 T 2 E 2 = R I • -±- • a c I n t e r v a l 3: Amplitude = - I t - [0, t p ] V o = V03 2 2 7 T 7 7 E 3 = - R I • a * e • T • a + R I .« a e • b Interval, 4: Amplitude = 0 t = [0, »] V = V o v04 2 T 2 7 E 4 = R I • a (ed - 1 ) Z Thus, the t o t a l energy E as a function of pulsewidth t and spacing t ^ i s , where a, b, c, d, e, f are functions of t, and t : 1 P E(t , t.) = R I 2 [2(t - 4 T + 2Td - \ f) - T a 3e + P 1 P 2 2 T / 2 i 2, 2 ^ 2, , 1 N 2 . •j (a c + a be + a (ed -1) ) 2 = R I ' funct (t , t., , T) P 1 This function with the values R = 1, T = RC = 1 and I = 1/t i s p l o t t e d P i n F i g . 5-4 with the pulsewidth t as a v a r i a b l e and the spacing t^ as a parameter. 37 F i g . 5-4 Heat produced i n RC model of s k i n by a b i p o l a r pulse. The charge of the pulse i s constant. Parameter: t ^ = spacing between p o s i t i v e and negative pulse. Abscissa: pulse width divided by T = RC. 4. Discussion As seen e a r l i e r , the time constant T of the skin v a r i e s con-siderably during an experiment between 0.64 ms and 0.07 ms. Because the r e s u l t s depend heavily upon T ;the discussion can be only q u a l i t a t i v e . We see: 1. The spacing t ^ between the pulses has not much influence for long pulsewidths (t > 3 T). • 2. I f t^ < 0.8 T, the heat does not increase for short pulses; i t even may decrease. These findings a r e > s l i g h t l y modified i f the s e r i e s resistance R g i s not neglected. The heating energy produced by a pulse i n R g i s : E = I 2 ' R * t s s p I f , as above, the charge Q i s kept constant, I = then E g increases for P decreasing t ^ . The t o t a l energy of a b i p o l a r pulse i s : E f c ^ = R I 2 f u n c t ( t , t . , T) + R I 2 • t -2 tot p' 1' s p Example: Taking f o r REdelberg's (1972) low value of 32 k fi cm2 and for 2 R g the maximum value of 500 fi cm , we get i n t h i s worst case f o r the normalized R s _ 500 fi ^ R s n " 32000 fi ^ 0 , 0 1 6 In F i g . 5-5 the r e s u l t s are shown and may be compared with F i g . 5-4. The i n t e r e s t i n g r e s u l t i s that the heat production i s independent of the pulse width for spacings t ^ shorter than 0.6 T and pulse width t longer than 0.25 T. In absolute values these boundaries for skin are: n 1 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3. PULSEWIDTH TP (NORMALIZED) Fig.-5-5 Heat produced in the skin model of Fig. 4-2 with R =0.016 R. Rest as in Fig. 5.4. T = 640 ys T = 70 ys t < 0.6 T t > 0.25 T P 384 ys 160 ys 42 ys 17 1/2 Thus, the worst case conditions to make the production of heat independent of the stimulus, are: spacing t ^ < 42 ys pulsewidth t > 160 ys. P According to Gibson (1968), the time constant T decreases with current i n t e n s i t y , which in d i c a t e s that t may e f f e c t i v e l y be shorter than 160 ys. His measurements also i n d i c a t e a reduction i n the skin resistance R with increasing current d e n s i t i e s . Since the t o t a l heat i s proportional to R, and since i n our c a l c u l a t i o n s R was set a r b i t r a r i l y to unity (R = 1 ft), a decrease i n R means i n r e a l i t y an o v e r a l l decrease i n heat production, which also favours the use of short s t i m u l i with high current d e n s i t i e s . However, as the mechanisms of changes i n the ski n resistance are not c l e a r l y understood, i t might w e l l be that Gibson's r e s u l t s are also i n -fluenced by other factors than the current density, and i t also might we l l be that the skin reacts d i f f e r e n t l y to pulses of several tens of micro seconds than to pulses of several m i l l i s e c o n d s . Further i n v e s t i -gations, which exceed the scope of t h i s t h e s i s , are necessary to deter-mine the time constant and the resistance of the skin . In general, these c a l c u l a t i o n s show that i t i s favourable to use a stimulus with a long pulsewidth and a short spacing. This contra-d i c t s completely the requirements f o r nerve stimulation discussed i n Chapter 15. Thus the optimum i s obtained when the long spacing t ^ = 0.6 T i s combined with the short pulse width t = 0.25 T. P Note, that these r e s t r i c t i o n s do not apply to electrodes im-planted under the skin (subcutaneous or percutaneous s t i m u l a t i o n ) . The current d i s t r i b u t i o n i n the tissue and hence the a c t i v a t i o n of nerve f i b r e s depends heavily on the electrode geometry. In order to transmit a current from the stimulator output to the t i s s u e , the e f f e c t s of electrode materials and of the electrode-skin i n t e r f a c e have to be investigated. These problems are b r i e f l y discussed i n the next chapter. 42 Chapter 6: ELECTRODE PROPERTIES 1. Electrode skin i n t e r f a c e 2. Materials 3. Geometry 1. Electrode skin i n t e r f a c e Since the skin i s e i t h e r moistened or there i s some sweat, the i n t e r f a c e to consider i s the e l e c t r o d e - e l e c t r o l y t e i n t e r f a c e . This junc-t i o n has the three properties of a p o t e n t i a l d i f f e r e n c e , a r e s i s t a n c e , and a capacitance, which i s of i o n i c o r i g i n . A l l properties depend on the type of metal and on the e l e c t r o l y t e . For d e t a i l s see Cosentino (1973), Geddes (1972), or Bergveld (1976). A l i n e a r approximation f o r the electrode i s a p a r a l l e l RC c i r c u i t . When the stimulus i s a current, then t h i s current passes through the electrode, no matter what the e l e c -trode's c i r c u i t looks l i k e . Therefore we need not go i n t o further d e t a i l . For stimulation we do not have to worry about the electrode p o t e n t i a l of about 100 mV or l e s s . However, i f a current i s passed through the e l e c -trode a p o l a r i z a t i o n i s b u i l t up. This e f f e c t i s described by Bergveld (1976) where he shows that "to ensure a minimal molecular gas generation or other objectionable i r r e v e r s i b l e reactions", the voltage developed between the electrodes by a current pulse must never reach a constant value. To c o n t r o l t h i s p o l a r i z a t i o n e f f e c t i s one further reason to use current c o n t r o l l e d stimulation. Bergveld's findings support the s t a t e -ment of Saunders' (1973) that one should use short b i p o l a r pulses. His findings were that sensation tends to be p a i n f u l i f the electrode voltage of a current stimulus reaches a constant l e v e l . 2. Materials D i f f e r e n t materials have d i f f e r e n t p o l a r i z a t i o n p o t e n t i a l s , an important factor for recordings. For stimulation, however, these polar-43 i z a t i o n p o t e n t i a l s have no influence, which allows the use of several d i f f e r e n t m a terials. Of importance may be the p o l a r i z a t i o n r e s i s t a n c e , which also depends on the materials used. E s p e c i a l l y the two metals s i l v e r and s t a i n l e s s s t e e l have proven s a t i s f a c t o r y i n experiments. Of course, materials forming nonconductive compounds when i n contact with the s k i n , such as brass or aluminium, cannot be used. With metal electrodes, p a r t i c u l a r l y on dry s k i n , the electrode's pressure exerted on the skin i s an important parameter. Also of import-ance i s the time the electrode remains on the s k i n . A low resistance w i l l eventually be< formed from sweat trapped between electrode and skin (Mason and Mackay, 1976). A metal electrode on dry s k i n may have the further disadvantage of nonhomogeneous contact, which r e s u l t s i n a few high density current channels. Other materials used are a v a r i e t y of s a l i n e pastes and, for large electrodes, conductive rubber (Brennen 1976) or epiductive e l e c -trodes, c o n s i s t i n g of a conductive paint (Burton and Maurer, 1974, 1976). 3. Geometry For electrocutaneous stimulation concentric electrodes have proven superior to other types(Tursky et a l . 1965). The main advantage i s the precise d e l i m i t a t i o n of the area of stimulation. A disadvantage i n m u l t i p l e electrode a p p l i -cations i s that there i s no common ground (see Chapter 3-4). This com-p l i c a t e s the stimulator design considerably. The electrode's design i s shown i n F i g . 6-1. Version a) has the disadvantage that a current shunting pathway i s formed by the sweat captured below the i s o l a t i o n . Version b) allows the skin to remain dry. A l l concentric electrodes can only be used e f f i c i e n t l y on dry s k i n . Any electrode p a s t e w i l l form a s h o r t c i r c u i t i n g path between the center e l e c -trode and the r i n g electrode. lot or b) F i g . 6-1 Concentric electrodes Experiments have proven that a center electrode with a diameter of l e s s than 3 mm w i l l produce pain (Hardt, 1974; Saunders, 1973). The electrodes create a current f i e l d i n the tis s u e containing the nerve f i b r e s to be stimulated. The geometrical aspects of t h i s f i e l d d i s t r i b u t i o n are of prime importance, since they determine the n o n l i n e a r i t y of the r e l a t i o n between nerve f i b r e diameter and the required stimulus i n t e n s i t y to a c t i v a t e the f i b r e . A few general statements are deduced i n the next chapter. Many problems remain unsolved. 45 Chapter 7: ON TRANSMEMBRANE CURRENTS IN A MYELINATED  AXON PLACED IN A CURRENT FIELD 1. R e s t r i c t i o n s 2. Myelinated nerve model i n a current f i e l d 3. The influence of the current f i e l d 4. Threshold - diameter r e l a t i o n 5. A d d i t i o n a l factors 1. ' R e s t r i c t i o n s Stimulation of an axon i s mediated by the transmembrane current at the nodes. We w i l l t r y to get a rough idea of what t h i s nodal current looks l i k e by examining several i d e a l i z e d s i t u a t i o n s . The l i t e r a t u r e about t h i s problem i s sparse. The current f i e l d d i s t r i b u t i o n i n the t i s s u e i s probably quite inhomogenous due to l o c a l v a r i a t i o n s i n the tissue (glands, small muscles, h a i r s , varying thickness of the epidermis; see F i g . 2-3). Thus, c a l c u l a t i o n s which assume a homo-genous tis s u e surrounding the nerve f i b r e are only approximations. Of mathematical i n t e r e s t only are the two i n v e s t i g a t i o n s of Klee and Plonsey (1974, 1976), c a l c u l a t i n g stimulation of round c e l l s i n a homogenous f i e l d . Experimental i n v e s t i g a t i o n s on f i e l d d i s t r i b u t i o n s i n a homogenous medium can be made with the e l e c t r o l y t i c tank method. Gold and Schuder (1975) published the example of two electrodes placed on the surface of a homogenous medium. This method would allow the study of the i n t e r a c t i o n of a model nerve with the surrounding f i e l d . Our i n v e s t i g a t i o n s are l i m i t e d to i d e a l i z e d s i t u a t i o n s . 2. Myelinated nerve model i n a current f i e l d The following assumptions are made: the current f i e l d i s par-a l l e l , homogenous and undisturbed by the presence of the nerve. Per unit length dx, the voltage d i f f e r e n c e i s the s c a l a r product -*• -»-dV = j • p dx J e with j denoting the current density vector [A cm ] and denoting the r e s i s t i v i t y [fi cm] of the medium. Thus,' the voltage between two nodes L AV = / j p e dx o with L denoting the internodal length. Parameters and data f o r myelinated f i b r e : L = int e r n o d a l length D = outside diameter of myelin d = diameter of node G = t o t a l nodal membrane conductance ( i n r e s t i n g state) m G. = internodal conductance of axoplasm 1 V = membrane voltage m 2 = 2uF/cm membrane capacity 1 = 2.5 ym nodal gap width (Dodge and Frankenhaeuser, 1959) = 110 fi cm r e s i s t i v i t y of axoplasm (Stampfli, 1952) rjj 300 fi cm r e s i s t i v i t y of external medium 2 &m = 30.4 m mho/cm membrane conductance f o r r e s t i n g state per uni t area (Frankenhaeuser and Huxley, 1964) d = 0.7 D Mountcastle (1972) L = 100 D Dodge and Frankenhaeuser (1959) G. £ 3 G Dodge and Frankenhaeuser (1959) l m With the further approximation that the myelin i s an i d e a l i n s u l a t o r , th equivalent c i r c u i t of the f i b r e i s that of F i g . 7-1. The equivalent circuit of the surrounding medium is omitted. Only AV is important, which is generated by the external current. A V node n node n + l Fig. 7-1 Model for myelinated nerve in current field 3. The influence of the current field The mathematical abstraction is examined of a straight nerve fibre of infinite length placed in a parallel current field. Disregard-ing the effects at both ends where small transmembrane currents occur, we see that for the major part of the fibre there will be no transmem-brane currents, because the current flowing along the inside of the axon has (except at both ends), such an amplitude that i t generates across the G^ 's exactly the same voltage difference between two nodes as the AV generated by the external current field. The same is true for any orien-tation of this nerve fibre in the current field, with the modification that the amplitude of the internal current now depends on the orientation as does also the external voltage difference between two nodes. A neces-sary condition for the occurrence of a transmembrane current (ie: for stimulation) i s , therefore, that either the current field is not homogen-ous (not parallel) or that the nerve fibre is not straight. I t i s conceivable, that a concentric electrode, producing the non p a r a l l e l current f i e l d of F i g . 7-2, could cause an optimum of trans-membrane currents to flow, because of (i ) the inhomogenous f i e l d and be-cause of ( i i ) the summing e f f e c t of the " l e f t " and the " r i g h t " current f i e l d seen by a nerve placed below the electrode. + F i g . 7-2 Current f i e l d of concentric electrode The s i t u a t i o n of a s t r a i g h t nerve i n the divergent current f i e l d of a s i n g l e electrode was ca l c u l a t e d by McNeal (1976). His p u b l i c a t i o n i n s p i r e d many ideas expressed i n this chapter. I f , due to a n o n p a r a l l e l f i e l d , there i s a voltage d i f f e r e n c e between the outside of two nodes which d i f f e r s from that of the i n s i d e , a transmembrane current w i l l be flowing which may t r i g g e r an ac t i o n poten-t i a l . I t i s important to note t h i s two stage a c t i o n : the e x t e r n a l l y applied constant current causes a constant internodal voltage d i f f e r e n c e and only t h i s voltage controls the membrane currents. As shown by McNeal (1976), the time course of the membrane current d i f f e r s considerably from the stimulating pulse, mainly because of the internodal voltage acting upon the RC c i r c u i t of the membrane. The fa c t that the membrane current i s greatest at the beginning of a pulse i s another good reason to use short pulses. 49 -> t stimulus current internodal voltage membrane current These findings show that experiments with f i b r e preparations using voltage sources correspond to skin stimulations using current sources (see Chapter 4-2). 4. Threshold - diameter r e l a t i o n We w i l l show that f i b r e s with large diameters are more l i k e l y to be stimulated f i r s t . The nodal gap 1 i s independent of f i b r e diameter (Dodge and Frankenhaeuser, 1959). Thus, both conductances are proportion-a l to the nodal axon diameter d: m and with L g m IT Id 100 D = TT d 100 0.7 d = C d; C = const. G i 4 p. L 4 C • p. " l i I t w i l l be seen i n Part II that the current density across the nodal membrane t r i g g e r s an ac t i o n p o t e n t i a l . If the voltage d i f f e r e n c e AV between two nodes i s assumed to be constant f o r any f i b r e , and i f i t i s assumed that a current through the axon only flows between these two nodes, the coarse approximation for the s t a t i c current density i n the membrane i s (see F i g . 7-1) 'm AV • (2G + G.) m l TT Id [A/cm2] AV (2 gm + 4 C p. V l which i s independent of the diameter because both the conductances and the nodal area are proportional to d. Thus, a l l f i b r e s were stimulated equally, unless the external voltage AV would be d i f f e r e n t for d i f f e r e n t f i b r e s . This i n f a c t , comes about because the internodal length L i s proportional to the diameter L = cd; c = constant ->-Denoting by the vector dx the o r i e n t a t i o n of the axon, we get the voltage as a function of the f i b r e diameter and as a function of the geometric properties of the external current f i e l d j : node 2 AV = p e J j dx node 1 For a s t r a i g h t axon i n a p a r a l l e l current f i e l d : AV = cd p £ • |J| i n which case there would be a l i n e a r r e l a t i o n between f i b r e diameter and threshold with the large f i b r e s being the ones responding f i r s t . Note the t h e o r e t i c a l d i f f i c u l t i e s : previously we have shown that for a s t r a i g h t axon i n a p a r a l l e l current f i e l d no e x c i t a t i o n occurs; now, l i m -i t i n g the axonal current to two nodes (clamping the axon), we show that e x c i t a t i o n i s proportional to f i b r e diameter. Both cases are t h e o r e t i c a l abstractions which are only used to gain some i n s i g h t into the conditions governing e x c i t a t i o n . McNeal (1976) ca l c u l a t e d the threshold - diameter r e l a t i o n f o r the divergent current f i e l d of a s i n g l e electrode. His i n v e s t i g a t i o n s also take i n t o account several aspects neglected here (several nodes, membrane capacity, changes i n during the action p o t e n t i a l ) . In that case the r e l a t i o n i s nonlinear and has a higher s e l e c t i v i t y among small f i b r e s : f or the large diameters (d > 15 um) threshold current I i s approximately i n v e r s e l y proportional to the square root of f i b r e diameter ( I t a d x /^) and for the small diameter range (d < 5 u m) i t approaches -2 an inverse square r e l a t i o n s h i p (I a d ). This i s consistent with p h y s i o l o g i c a l experiments. In theory and i n laboratory experiments,three d i f f e r e n t types of stimulation are distinguished depending on how the current flows with-i n the nerve: - l o n g i t u d i n a l : a current enters the f i b r e at one node and leaves at another node; tr a n s v e r s a l : a current enters the f i b r e at one side of a node and leaves i t at the other side of the same node; point electrode: there i s a current source assumed at the centre of the f i b r e at one node. Whereas the l a t t e r i s a mathematical s i m p l i f i c a t i o n , l o n g i t u d i -n a l and t r a n s v e r s a l stimulations both e x i s t i n nerve stimulation. The normal case i s probably l o n g i t u d i n a l stimulation. Wyss (1967a) showed that stimulation i s possible (at l e a s t i n experiments with an excised nerve) using a transverse b i p o l a r current of medium frequency (5 ... 20 kHz) . In summary, i t i s seen that the geometrical arrangement i s an extremely important factor which influences the diameter - threshold r e l a t i o n . 5. A d d i t i o n a l factors So f a r the i m p l i c i t assumption was that the one dimensional c i r c u i t of F i g . 7-1 represents the three dimensional axon. As a con-sequence, only a stimulation mediated by a l o n g i t u d i n a l current from node to node was considered. Considering a nerve bundle, another complication a r i s e s due to the sheaths surrounding nerve bundles, causing inhomogenities i n the current f i e l d . However, as pointed out by Bromm (1966), i t s influence i s not of a prime concern, e s p e c i a l l y not for stimulation with symmetric b i p o l a r pulses where i t acts as a capacitor. SUMMARY I The i n v e s t i g a t i o n s of th i s f i r s t part concentrated upon the aspects of stimulating p e r i p h e r a l nerve f i b r e s by electro-cutaneous stim-u l a t i o n . Primarily of i n t e r e s t i s the sensory stimulation. However, most findings also apply to f u n c t i o n a l stimulation. The p h y s i o l o g i c a l condition to achieve a pain free sensation s i m i l a r to v i b r a t i o n or touch i s to stimulate s e l e c t i v e l y the large mye-l i n a t e d A-3 f i b r e s without stimulating A-6 f i b r e s or C f i b r e s which make up to 75% of the f i b r e s . The stimulus has to be pulsed to prevent a blocking of the large f i b r e s by continuous d e p o l a r i z a t i o n . Depending on properties of the stimulus there may be receptors a c t i v a t e d , but there i s evidence that the e l e c t r i c stimulus acts d i r e c t l y on the nerve f i b r e s . To apply a stimulus, two electrodes are needed. The core of the body has a voltage with respect to one electrode which i s determined by the t o t a l voltage developed between the two electrodes and by t h e i r areas. To allow the body to make contact with ground, none of the e l e c -trodes may be connected to ground, i . e . the stimulator must be i s o l a t e d to prevent shock hazards. To prevent p h y s i c a l or chemical changes which would produce a p a i n f u l sensation, the stimulus has to s a t i s f y the conditions: to avoid e l e c t r o l y t i c e f f e c t s , the stimulus must be b i p o l a r with a zero average charge. to prevent breakdowns of the skin, the voltage across the skin should remain below 150 V. to prevent an excessive production of heat i n the skin, the stim-ulus should be of short duration, such that the ca p a c i t i v e pro-p e r t i e s of the ski n can be used favourably. If the spacing between the end of the p o s i t i v e and the beginning of the negative pulse i s le s s than about 40 us and i f the pulsewidth i s longer than about 20 ys the heat production i s independent of the pulse width. For extremely short pulses with correspondingly high amplitudes, the heat produced i n the small seri e s resistance dominates and cancels the theo r e t i c improvement from the p a r a l l e l RC c i r c u i t . Rectangular current pulses are the waveform commonly used. Short pulses have the advantage over sine waves that the charge can be c o n t r o l l e d independently from the frequency. F i n a l l y , ' the following e f f e c t s require a current c o n t r o l l e d stimulation (as opposed to voltage c o n t r o l l e d stimulation used i n early s t i m u l a t i o n s ) : - with current stimulation, the electrode i n t e r f a c e i s of minor -importance: i t only has to pass the current. - p o l a r i z a t i o n e f f e c t s ( e l e c t r o l y t i c e f f e c t s ) can be c o n t r o l l e d . a current i s passed without degradation through the RC structure of the sk i n . - i n case of a breakdown of the skin , the current i s preset and cannot increase. a current c o n t r o l l e d f i e l d i s required i n the ti s s u e to produce between two nodes of a nerve f i b r e a voltage d i f f e r e n c e which w i l l a c t i v a t e the nerve. In the next part a mathematical model of nerve w i l l be selected to enable i n Part III the i n v e s t i g a t i o n of the nerve's a c t i v e response to d i f f e r e n t s t i m u l i . The model w i l l be selected based on the current status of mathematical nerve models (Chapters 8 to 1 0 ) . The e f f e c t of propagation on the action p o t e n t i a l i s considered i n Chapter 1 1 . F i n a l l y , the e f f e c t of the model's time constants on the numerical s o l u t i o n of the model equations i s investigated i n Chapter 1 2 . 56 PART I I : INVESTIGATION OF SINGLE NERVE FIBRE MODELS Chapter 8: THEORY OF THE NERVE MEMBRANE 1. Ionic theory of the action p o t e n t i a l 2. Membrane po t e n t i a l s 3. Membrane capacity 4. On modelling 1. Ionic theory of the action p o t e n t i a l There are diff e r e n c e s i n i o n i c concentrations between the in s i d e and the outside of the nerve membrane. For example, Frankenhaeuser and Huxley (1964) determined the i o n i c concentrations at 20°C f o r the r e s t i n g state f o r toad: outside membrane i n s i d e membrane Na + = 114.5 mM/1 Na + = 13.74 mM/1 K + = 2.4 K + = 120 These concentration differences are maintained by an active process i n the membrane. This process, which i s s t i l l not w e l l under-stood, i s normally described by the model of i o n i c pumps. The concent-r a t i o n differences create the e l e c t r i c r e s t i n g p o t e n t i a l of the membrane. The production of an action p o t e n t i a l (A.P.) i s characterized by the three cycles: a) When e x c i t a t i o n occurs, the permeability f o r Na + increases: Na + i n f l u x and the membrane p o t e n t i a l becomes more p o s i t i v e . b) The permeability for K + ions slowly increases: K + outflux. F i r s t , the p o t e n t i a l reaches a plateau, then i t i s reduced back to the -resting state. c) Both permeabilities return to t h e i r r e s t i n g values. Permeability to CI and other ions i s considered to remain constant. Although more recently there have been many i n v e s t i g a t i o n s examining the I | | j | j j | influence of polyvalent cations, such as Ca , Ni , Ba and La (Khodorov, 1974) , as c o n t r o l l i n g factors f o r the Na + permeability changes and as co n t r i b u t i o n d i r e c t l y to i o n i c currents, the basic concept of the creation of an A.P. as outlined above, i s s t i l l v a l i d , at l e a s t f o r a nerve functioning i n i t s normal environment. Independent i n v e s t i g a t i o n s ( e l e c t r i c a l , chemical, r a d i o a c t i v e tracing) a l l support the hypothesis that an action p o t e n t i a l i s i n i t i a t e d by sodium inflow and terminated by potassium outflow. 2. Membrane p o t e n t i a l s The t h e o r e t i c a l transmembrane voltage produced by a s i n g l e type of ion i s given by the Nernst equation. This equation i s derived from the e q u i l i b r i u m of the two currents, one caused by the concentration gradient, and the other caused by the e l e c t r i c p o t e n t i a l gradient. with: E = membrane p o t e n t i a l (inside with respect to outside) [volts] R = 8.314 jo u l e (gmole • °K) ^  gas constant T = 293.16 °K f o r 20.0°C;absolute temperature 4 -1 F = 9.649 • 10 coulomb (gmole) Faraday constant with 1 joule = 1 watt -sec = 1 v o l t - amp - sec 1 coulomb = 1 amp - sec The models discussed l a t e r are a l l based on the t h e o r e t i c a l model of the constant f i e l d membrane, which i s assumed to be planar, E - f- In Nernst Equation (O. (c) o , (C). = i o n i c concentrations outside, i n s i d e [Mol/cm ] uniform, and homogeneous. After having defined permeability c o e f f i c i e n t s P , which can only be determined experimentally, we can express the i o n i c current d e n s i t i e s across the membrane by the following equation, known as the Goldman equation: EF P F 2 E (C ) - (C ). • e R T _ _ _n t n o n 1  n ~ RT EF RT 1 - e _2 with I : current density of ion n [amp cm ] P : permeability of membrane f o r ion n [cm s n Descriptions of the theories and corresponding equations can be found i n a l l of the books: Adelman (1971), Plonsey (1969), Cole (1968), or Khodorov (1974). 3. Membrane capacitance The membrane capacitance i s constant wihin 2% during the occur-rence of an a c t i o n p o t e n t i a l . The capacitance seems to be a passive mem-brane c h a r a c t e r i s t i c . To c i t e Cole (1968) [1]: "A wide v a r i e t y of meas-urements on many d i f f e r e n t l i v i n g c e l l s have established the value of 2 1 uF/cm as a b i o l o g i c a l constant, with extreme values of about 0.5 and 2 2.0 uF/cm ". Further, Tasaki (1955) reports for frog nerve f i b r e that "the capacitance of the nodal membrane, j u s t as that of the myelin sheath i s not a l t e r e d by chemicals". Frankenhaeuser and Huxley (1964) found 2 that a capacitance of 2 uF/cm had to be used for modelling best the s c i a t i c nerve f i b r e of toad. In subsequent chapters we w i l l use that model. [1] p. 103 60 4. On modelling A large v a r i e t y of models have been put foreward i n the l i t e r -ature. The most basic one shown i n F i g . 8-1 i s that of a v a r i a b l e , non-l i n e a r r e s i s t o r i n p a r a l l e l with the capacitance, and with a voltage source for the r e s t i n g p o t e n t i a l V . T outside C inside F i g . 8-1 Basic membrane c i r c u i t The models d i f f e r i n the way R and V are represented. An e x c e l l e n t J m r r review of the development of models i s given by Cole (1968). Only the separation of R into three d i f f e r e n t branches for Na +, K +, and CI , put m r foreward i n the famous model by Hodgkin and Huxley (1952) , enabled the mathematical, q u a n t i t a t i v e d e s c r i p t i o n , which w i l l be discussed i n the next chapter. Because a model i s an abstraction, there are many models for the same r e a l i t y . Subsequently, more general mathematical models were developed. See, f o r instance, F i n k e l s t e i n and Mauro (1963), or FitzHugh (1961, 1969, 1973). Of a more v i t a l i n t e r e s t i s the question of errors introduced by a one dimensional model. S t r i c t l y speaking, t h i s model would only be accurate for an axon, i f the inner electrode were a l i n e i n the centre of the axon, and i f the outer electrode were a c y l i n d e r surrounding the axon. Clark and Plonsey (1966) inv e s t i g a t e d that problem. In most cases, the e r r o r introduced seems not to be s i g n i f i c a n t . In order to f i n d an appropriate model f or electrocutaneous st i m u l a t i o n , four models f or nerve membranes are reviewed i n the next chapter. These four were the only models found which describe accurately a s p e c i f i c nerve membrane by a mathematical model. Which of the models w i l l be used i s only decided a f t e r the discussion i n Chapter 1 0 . 62 Chapter 9: FOUR MODELS FOR NERVE MEMBRANES 1. Nomenclature 2. Unmyelinated axon: Hodgkin and Huxley's model for squid 3. Myelinated axon: Frankenhaeuser and Huxley's model for toad 4. Myelinated axon: Dodge's model for frog 5. Purkinje f i b r e : Noble's model 1. Nomenclature For a l l models, the following p o l a r i t i e s are used i n agreement with the l i t e r a t u r e since 1960: - current i s p o s i t i v e i f i t flows outwards. I t i s also c a l l e d a depo l a r i z i n g or cathodal current. Thus, an inward, anodal^hyper-p o l a r i z i n g current i s c a l l e d negative. - voltage i s measured as the p o t e n t i a l of the in s i d e minus that of the outside. - voltage reference: f o r convenience, the r e s t i n g p o t e n t i a l i s chosen as reference. A l l r e l a t i v e voltages (V) are expressed with respect to that r e s t i n g p o t e n t i a l (V = 0). They are r e -late d to the absolute p o t e n t i a l E by: V = E - E^ , E^ = r e s t i n g p o t e n t i a l - graphs: upward d e f l e c t i o n i s always p o s i t i v e . F i g . 9-1 summarizes the r e l a t i o n s for squid standard data. In the models there i s a separate branch for each i o n . Note that the d e s c r i p t i o n of the current's e f f e c t s only apply to the capaci-tance as shown i n F i g . 9-2: a depo l a r i z i n g e f f e c t , i . e . a p o s i t i v e I , can be caused either by a p o s i t i v e current I from outside, or by a neg-a t i v e , inflowing sodium current I (driven by concentration gradients). 63 absolute relative potential E i V t Fig. 9-1 Voltage relations for an action potential outside t o K inside ^membrane Fig. 9-2 Depolarizing currents In general, the measured data is scaled to a unit area of 1 cm' of membrane. The inexact determination of the nodal area in a myelinated axon is a major source of error in scaling data of a node. This makes, 2 2 for example, the calculated capacitance vary from 1 uF/cm up to 7 yF/cm . To reduce complexity, most investigations in subsequent chapters will be limited to the space-clamped membrane assuming that propagation along the axon will occur as soon as the membrane is activated at one place. In-vestigations of the membrane without propagation are referred to by names such as: • - membrane voltage - space clamped membrane current clamp. Fig. 9-3 illustrates the situation. For the membrane potential, i t is assumed that a segment of an axon is considered with a) equal voltage distribution and b) no propagating currents: i ^ = i ^ = 0. Thus, the externally applied stimulating current I is equal to the membrane current I. The errors resulting will be discussed in Chapter 11. M •A/W—fy—** H-A/W-M Fig. 9-3 Space clamped membrane, M = membrane model. 1 - i = 0, I = I . o i ' s 65 2. Unmyelinated axon: Hodgkin arid Huxley's model for squid (HH model) Using the special techinque known as the voltage clamp (Hodgkin, Huxley and Katz 1952), which consists of controlling the voltage of a piece of membrane by using internal and external electrodes while measur-ing the ionic currents, Hodgkin and Huxley (1952) were able to describe the membrane by the following, quantitative model. Measurements were made on the giant axon of squid. The normal temperature is 6.3°C. Cor-rection formulas apply for other temperatures. Fig. 9-4 shows the mem-brane model. The polarities are inverted with respect to the original model (Hodgkin and Huxley, 1952) to conform with the nomenclature out-lined above. outside 4 inside . Fig. 9-4 Hodgkin-Huxley (HH) membrane model The basic equations are: dV and with I = C • ^— C at 66 V 4 o / t [ I - ( INa + XK + \ ^ d t The i o n i c voltages are due to concentration d i f f e r e n c e s between the i n s i d e and outside. They are determined by the Nernst equation \ = ¥ ' l n TcTT - V k = N a> K k 1 where V, and E are i n the unit V o l t s f o r t h i s equation. The constants k r are defined i n Chapter 8. The t h i r d voltage, V^, with the index 1 standing f o r leakage, i s mainly due to CI . In the model i t i s calculated to be the sum of I„ + I,, + I, = 0 f o r V = 0. Standard values f o r 6.3°C Na K 1 are: 2 Capacitance: C = luF/cm Resting p o t e n t i a l : E^ = -60 mV Voltages: V„ = +115 mV Na V^ = -12 mV V x = +10.613 mV For squid the i o n i c currents are l i n e a r functions of the corresponding electrochemical p o t e n t i a l s . I = s (v - V ) Na s N a v Na' h = % ( v - v h - s ^ . - V where g ^ , g R , and g represents the sodium, potassium and leakage con-ductances r e s p e c t i v e l y . 2 The conductance g^ remains constant: g^ = 0.3 m mho/cm . The conductances g^ and g vary both with time and voltage according to the following expressions: % a = % a " with g^ a = 120 m mho/cm dm dt a (.1 - m) - 3 • m m m m m 0.1 V - 25 i ,25-V, 1 - exp ( - jg - ) 4 e x p ( - -^g) dh ,„ dt = a h ( 1 " h ) " 6 h ' h a h =  6 h = 0.07 e x p ( - ^ ) l J. /30-V^ and 8K = ^ ' % — 2 with = 36 m mho/cm dn dT = a n ( 1 " n ) ~ 6 n * n a - 0.01 V " 1 0 n /10-VN 0.125 exp(-where V i s expressed i n mV. 2 2 The current d e n s i t i e s are i n uA/cm , conductances i n m mho/cm , capaci-2 tance i n uF/cm , and time i n msec. Only the ex's and 3's are considered to be influenced by tempera-ture. For temperatures T d i f f e r e n t from 6.3°C, they must be m u l t i p l i e d by <f> corresponding to a Q^Q of 3: 68 V(t) 8Na • m' m = a (1 m - m) - 3m n = a n ( l - n) -h - a h ( l - h) -(T - 6.3)/10. _ * - Q 1 0 ; Q i o ~ 3 Experiments of Guttman and B a r n h i l l (1968) showed that the temperature c o e f f i c i e n t i s not the same for a l l a's and g's and that the model should be modified i n that area. However, they did not c a l c u l a t e new formulas to be used i n the model. To summarize, the following four d i f f e r e n t i a l equations have to be solved simultaneously f o r c a l c u l a t i n g the action p o t e n t i a l V ( t ) : - 3MV-V N a) - g K • n 4(V-V K) - g^V-V^ ] m n h Possible i n t e g r a t i o n methods and t h e i r errors are discussed i n Appendix A. 3. Myelinated axon: Frankenhaeuser and Huxley's model f o r toad (FH Model) Frankenhaeuser and Huxley (1964) published a nerve model f or the membrane of the A f r i c a n toad Xenopus Laevis. As the HH model, t h i s q u a n t i t a t i v e d e s c r i p t i o n i s also based on voltage clamp data. A s p e c i f i c experimental set-up, which takes advantage of the myelin sheath, makes i t easy to in v e s t i g a t e s i n g l e nodes. The method i s described i n many places, e.g. Tasaki (1953), H i l l e (1971), or as a r e -view: Adelman (1971). The equations f o r the sodium and the potassium current were published e a r l i e r by Frankenhaeuser (1960, 1963). The main di f f e r e n c e s between t h i s model and the HH model are: - The in t r o d u c t i o n of a fourth i o n i c current, the non-spe c i f i c , de-layed current I_. 69 - Experimental evidence indicated that the voltage-current charac-t e r i s t i c i s better described by the nonlinear r e l a t i o n s h i p derived from the constant f i e l d membrane theory than the l i n e a r conductance approxima-t i o n of Hodgkin and Huxley; pe r m e a b i l i t i e s instead of conductances are used except f o r the leakage current. - The leakage conductance g^ i s 100 times larger than f o r squid. - The current d e n s i t i e s are 5 to 10 times higher than i n squid. 2 3 2 4 - exponents are changed: m instead of m , and n instead of n . 2 - The capacitance i s 2uF/cm instead of luF/cm . Mcllroy (1975) comes to the conclusion a f t e r i n v e s t i g a t i n g the t h e o r e t i -c a l "instantaneous" current-voltage r e l a t i o n i n a membrane model, where the i o n i c conduction occurs through pores: " i t i s suggested that, pro-vided the mean distance between conducting pores i s greater than the mem-brane Debye length, the current-voltage r e l a t i o n s h i p w i l l be l i n e a r , as i n the case of the n a t u r a l squid giant axon. (...). The theory also suggests that i f the foregoing i n e q u a l i t y i s reversed, the c u r r e n t - v o l -tage r e l a t i o n s h i p w i l l be non-linear, the case of the frog node of Ranvier being c i t e d as an example". The schematic model i s depicted i n F i g . 9-5. The i o n i c current 2 d e n s i t i e s are i n amp/cm with E expressed i n v o l t s : with a: = — I V \ F ( C N a ) ° ~ ^ ' I XT- m h • a  'Na "Na "' " , a 1 - e T p 2 r ( V o - < V i • e° I K = P K n • aF — 1 - e _ (C._ ) - (C„ ). • e a  T — 2 Na o Na I I = P p • aF P P ! _ e * 70 i x = 8 l (v - v p The equation f o r the membrane voltage i s : The equations for m, n, h and p: d i — = 0^(1 - i ) - 3 • i , with i = m, n, h, p C 4 outside V inside F i g . 9-5 Schematic representation of the Frankenhaeuser-Huxley (FH) model With the voltage i n mV and the time i n ms, the rate constants (in ms 1) are f o r 20°C: a h = o.i (_ V - 10)/(1 - exp( V + 1 0 ) ) BH = 4.5/(1 + e x p ( 4 5 1 ^ V )) a m = 0.36 (V - 2 2 ) / ( l - e x p ( - ~ - - ) ) m 0.4 (13 - V ) / ( l - exp(V 2 Q1 3 ) ) a n = 0.02 (V - 3 5 ) / ( l - e x p ( 3 5 1 0 V ) ) B n = 0.05 (10 - V ) / ( l - exp( V ~ Q 1 0 ) ) a p = 0.006 (V - 4 0 ) / ( l - e x p ( 4 ° 1 ~ V ) ) B p = 0.09 (-25 - V ) / ( l - exp( V 2 Q 2 5 ) ) Frankenhaeuser and Moore (1963) have published i n d i v i d u a l Q-^Q's f ° r t n e c o r r e c t i o n of the rate constants f o r other temperatures than 20°C. A l l ((T - 20°)/10) a's and 3's must be m u l t i p l i e d by the f a c t o r <j> = Q-^ Q For temperatures between 2.5°C and 22°C the Q-^Q'S are: cv : 2.8 a : 1.8 a : 3.2 P„ : 1.3 h m n Na 8 U : 2.9 B : 1.7 B : 2.8 PT, : 1.2 h m n K There i s no temperature compensation f o r the small, delayed current I p . No Q-J^ Q'S are published f o r higher temperatures than 22°C. Noting that the v a r i a b l e m controls the a c t i v a t i n g system and both n and h co n t r o l the i n a c t i v a t i o n , we see that a c t i v a t i o n , which corresponds to the r i s i n g part of the act i o n p o t e n t i a l , i s l e s s affected by temperature than i n a c t i v a t i o n . To enable comparisons between d i f f e r e n t computa-ti o n s , the following standard data f o r the constants was defined (Frankenhaeuser and Huxley, 1964): ( CNa>o = 114.5 mM/1 E = r - 70 mV ( C N A = 13.74 mM/1 g l = 2 30.3 mmho/cm ( V o = 2.5 mM/1 V l = 0.026 mV ( V i = 120 mM/1 C = 2 uF/cm2 p Na _3 8.10 cm/s 3. ,  I 1 1 I n , °-0 0.333 0.667 1.0 1.333 1.667 TIME (MILLISEC) 2.0 0.0 0.333 T 0.667 1.0 1.333 TIME (MILLISEC) 1.667 2.0 F i g . 9-6 Membrane p o t e n t i a l and i o n i c currents i n the FH model. . 2 Monopolar stimulus of 1 mA/cm during [0, 0.12 ms]. Integration: P r e d i c t o r - c o r r e c t o r (Appendix A 2.5); step s i z e : 0.5 ys; standard data. (Recalculation 'of Frankenhaeuser and Huxley's (1964) F i g . l and 6) 73 P = 0.54 • 10 3cm/s P P R = 1.2 • 10"3cm/s The leak voltage i s adjusted to give zero t o t a l membrane current i n the r e s t i n g state. There i s no a n a l y t i c a l method for the determination of V^. Resting state conditions: a m m o a + B a t V=0 s i m i l a r expressions apply for n , h , p m m u u v ^ m = 0.0005, n = 0.0268, h = 0.8249, p = 0.0049 o o o o F i g . 9-6 shows the,membrane p o t e n t i a l and the i o n i c currents of the model. Important c h a r a c t e r i s t i c s to note are the early and sharp r i s e i n I„ at the s t a r t of an A.P. and that I has only a small peak value. Na P M o d i f i c a t i o n Bretag and Stampfli (1975) investigated d i f f e r e n c e s between sensory and motor nerve f i b r e s i n frog (Rana Esculenta). They found that the "standard data" model (Xenopus Laevis) of Frankenhaeuser and Huxley (1964) can be considered as a good model also of a sensory f i b r e action p o t e n t i a l from Rana Esculenta. For motor f i b r e s they s l i g h t l y modified the dynamics of the potassium system. The f a c t o r s A, B, C i n the two equations a and 3 are: n n a A(V - B) A(B - V) 1 - exp(- — - — ) 1 - exp( -—) Sensory (FH-model) Motor a 3 a g n n n n A(msec - 1) 0.02 0.05 0.012 0.05 B(mV) +35 +10 +15 +10 C(mV) 10 10 9 9 74 These modifications influence the shape and the duration of the ac t i o n p o t e n t i a l . However, they have v i r t u a l l y no influence on the onset of the action p o t e n t i a l , which depends p r i m a r i l y on the fa s t changes i n the sodium current. Therefore, t h i s model w i l l not be discussed f u r t h e r . 4. ' Myelinated axon: Dodge's model f or frog Dodge (1961, 1963) created a model for the nodal membrane of a s c i a t i c nerve f i b r e i n Rana Pipiens. He had helped to e s t a b l i s h the data base for the FH model (Dodge and Frankenhaeuser, 1959). For Rana pipiens he found: - For sodium, the I - V r e l a t i o n was nonlinear and i t had to be ex-pressed by the permeability as i n the FH model. - There was no need f o r a nonspecific current. However, the rate constant required a two-term expression. - Potassium current Iv and leakage current I 1 could be expressed l i n e a r l y as i n the HH model. Figure 9-7 shows the model. F i g . 9-7: Dodge's membrane model The equation for sodium i s : T - P 3, ( c , F 2E £ X ^ ( E - h ^ 7 ^ 1 Na Na™ 1 Na"o RT exp (EF/RT) - 1 which i s derived from the Goldman constant f i e l d equation. With the Nernst equation rearranged to: ( C N a } i = ^ 0 ' e X P ( " ENa ' ! f > the equation above i s i d e n t i c a l to the equation for 1^ used i n the FH model: 3 F 2 E CCNa)o ~ ( CNa>± ' exp(EF/RT) Na Na™ RT, 1 - exp(EF/RT) 3 2 with the only d i f f e r e n c e being that m i s used instead of m . The other equations are: f 4 ( I " ha ~ h " V I R = g K n 4 ( V - V K) i x = I x ( v - v x) 4r = a (1 - i ) - 6. • i , i = m, n, h at l I The major drawback of t h i s model i s that i t i s formulated only for one node investigated ("node 7"), and that i t i s not scaled to unit areas. Also, there i s no evaluation of temperature c o r r e c t i n g f a c t o r s . Therefore, we r e f r a i n from l i s t i n g the empirical formulas and constants These are summarized i n Adelman (1971) p. 242. 5. Purkinje f i b r e : Noble's model The only model that e x i s t s f or warm blooded animals i s that for the Purkinje f i b r e s of the mammalian heart. Noble (1962) does npt s p e c i f y what animal's heart he used. Noble and Tsien (1968) investigated the heart of sheep. However, the pacemaker p o t e n t i a l s are so d i f f e r e n t from normal action potentials, that this model cannot be used for our purposes. For completeness, a short description follows. Noble (1962) published the first model which was a modification of the HH-model. The main difference was the introduction of two potas-sium conductances. Further investigations led to modifications in the description of these two conductances (Noble and Tsien, 1968). An out-standing feature of Purkinje fibres is the large capacitance of about 12uF/cm . Therefore, McAllister (1968) made the further modification of introducing a resistance and capacitance in series, as i t was proposed by 2 Fozzard (1966), who measured a total capacitance of 9.4uF/cm where 2 2 7uF/cm were in series with a resistance of 300ftcm . Similar effects are also observed in skeletal muscles (Falk and Fatt, 1964). The equivalent circuit is shown in Fig. 9-8. 1/ outside j inside Fig. 9-8: Model of Purkinje fibre According to M c A l l i s t e r (1968), the mathematical expressions of Noble's (1962) model were not so w e l l founded. For instance, he was forced to introduce an empirical constant i n g„ to make the model corres-Na pond to measured data. The leakage conductance g^ and are modified as required f or d i f f e r e n t c a l c u l a t i o n s . The potassium current i s determined (a) by g ^ * which shows inward r e c t i f i c a t i o n and (b) by g j ^ ' which i s "proportional to a v a r i a b l e obeying f i r s t - o r d e r voltage-dependent k i n e t i c s of the Hodgkin-Huxley type, but with extremely long time constants". (Noble and Tsien, 1968). Concerning Purkinje f i b r e s , a large l i t e r a t u r e of i t s own i s developing. However, t h i s subject i s not of fur t h e r i n -ter e s t f o r t h i s t h e s i s . Having reviewed these models, we t r y to answer i n the next chapter the questions of which of these models we can use f o r our purpose and under what conditions. 78 Chapter 10 : EVALUATION OF A MODEL 1. The l i m i t s of the models 2. R e l i a b i l i t y of the data base 3. Choosing a model 4. Time Scale adaptation 1. The l i m i t s of the models The only two models to be discussed are those of Hodgkin-Huxley (HH) and of Frankenhaeuser-Huxley (FH). The model of the Purkinje f i b r e only was mentioned to show pos s i b l e v a r i a t i o n s of the models. Dodge's model, though an i n t e r e s t i n g combination of the models of HH and FH, cannot be used because i t i s not scaled to a u n i t area. The f a c t to keep i n mind while discussing models i s , that by d e f i n i t i o n , a model reproduces only c e r t a i n aspects of the r e a l object and therefore the use of the model i s r e s t r i c t e d a p r i o r i . Nevertheless, the degree of correspondence between measurements and model p r e d i c t i o n s i s impressive, i n p a r t i c u l a r because the model, based on the s t a t i c voltage clamp measurements, p r e d i c t s the dynamics of an a c t i o n p o t e n t i a l . HH-model: The HH model f i t s the experimental data f o r : - membrane p o t e n t i a l - propagated action p o t e n t i a l and v e l o c i t y - i o n i c charges (net f l u x of sodium and potassium per impulse) - absolute r e f r a c t o r y period - recovery of e x c i t a b i l i t y during the r e l a t i v e r e f r a c t o r y period - value of threshold to short current pulses - subthreshold responses - strength-duration r e l a t i o n of stimulus The HH model f i t s data only p a r t i a l l y f o r : - temperature dependence of accommodation and e x c i t a t i o n (Guttman and B a r n h i l l , 1968) The HH model y i e l d s wrong r e s u l t s for r e p e t i t i v e f i r i n g : - f o r step current, a r e a l axon generates not more than 4 spikes but the HH model eventually produces an i n f i n i t e p u l s e - t r a i n . - f o r ramp current, a r e a l axon generates a s i n g l e spike or none, depending on but the HH model y i e l d s r e p e t i t i v e pulses as for a step current when the amplitude i s i n the same range. The range of experimental conditions f o r which the membrane conductances were obtained i s : absolute p o t e n t i a l E = -80 mV. ... + 60 mV temperature T = 4°C. ... 23°C time t = lOus. ... 10 ms FH model The membrane p o t e n t i a l calculated by the model was found to be very s i m i l a r i n i t s parameters to the experimental spike. As with the HH model, the FH model y i e l d s i n c o r r e c t r e s u l t s f o r r e p e t i t i v e f i r i n g . (Bromm and Frankenhaeuser, 1972; Connor, 1975; Frankenhaeuser and Vallbo, 1965; Tasaki, 1950; Frankenhauser, 1965). 2. R e l i a b i l i t y of the data base Data was taken from d i f f e r e n t axons under varying conditions. Therefore, sets of standard data have been defined to enable comparison of c a l c u l a t i o n s . Throughout t h i s thesis only t h i s standard data w i l l be used. For the HH model i t i s defined and compared with measurements i n Table 3 reproduced here of the o r i g i n a l paper by Hodgkin and Huxley (1952) Val uo Experimental values , Constant chosen Moan Rimgo (1) (2) (3) (4) C„ (,<]'7cm») 1-0 O'Ol 0-8 to l-S I'N, (mV) -113 -100 -0.1 to -119 (mV) +12 +11 + 9 to + 14 V, (mV) -10-013* - 1 1 - 4 to - 22 fi...(m.mho/cm«)- 120 {*> gK (ra.mho/cmJ) 36 34 26 to 49 g, (m.mlio/cmJ) 0-3 0-26 0-13 to 0-50 * Exact value chosen to make the total ionio current zero at the resting potential (V — 0). The following three comments apply to the HH model: (i ) the model does not take into account the d i e l e c t r i c l o s s of the membrane, i . e . the capacitor i s assumed to be i d e a l . Hodgkin and Huxley (1952, p. 505) comment: "There i s no simple way of e s t i -mating the error introduced by t h i s approximation, but i t i s not thought to be large since the time course of the cap a c i t i v e surge was reasonably close to that calculated f or a per f e c t condenser", ( i i ) The equations governing the potassium conductance do not give as much delay i n the conductance r i s e on d e p o l a r i z a t i o n as was ob-served i n voltage clamps. ( i i i ) Threshold c a l c u l a t i o n s "must depend c r i t i c a l l y on such things as the leak conductance g^, whose value was not very w e l l determined 2) experimentally". Frankenhaeuser and Huxley (1964) discuss the standard data for the FH model: "The equations and the qua n t i t a t i v e data obtained i n the 1) HH (1952), p. 509. 2) HH (1952), p. 535. voltage clamp analysis must to a large extent be considered as approxi-mations. A f a i r amount of sc a t t e r appears e s p e c i a l l y between values from d i f f e r e n t f i b r e s " . In that paper as well as i n that of Frankenhaeuser and Vallbo (1965) the e f f e c t s of varying s i n g l e parameters of the model are investigated. Note that also i n the FH model the membrane capacitance i s assumed to be i d e a l , one reason being that "the time r e s o l u t i o n of the clamp technique i s not s u f f i c i e n t f o r r e l i a b l e measurements of I ". [1] Despite these reservations, both models are excel l e n t descriptions of the nerve membrane under the s p e c i f i e d conditions. Both models have been used by many other i n v e s t i g a t o r s . 3. Choosing a model Which model can be used to examine sensory or f u n c t i o n a l stimu-l a t i o n i n man? S t r i c t l y speaking, the HH model i s only correct f o r the giant axon of squid and the FH model applies only f o r the amphibian Xenopus Laevis. In a more general way, the HH model i s used for unmyel-inated nerve f i b r e s (type C f i b r e s ) , and the FH model i s used for myel-inated ones (type A f i b r e s ) . FitzHugh (1973) showed with a dimensional analysis of nerve models that r e s u l t s f o r one f i b r e can be transformed to apply to another f i b r e of s i m i l a r structure which has d i f f e r e n t values for the constants. For myelinated f i b r e s , Goldman and Albus (1968) were able to c a l c u l a t e from the t h e o r e t i c a l FH model the l i n e a r diameter -v e l o c i t y r e l a t i o n , which was established by many experiments. Buchthal and Rosenfalck (1966) found that the influence of temperature upon the change i n conduction v e l o c i t y i n sensory nerve f i b r e s i n man was very s i m i l a r to that i n frog's nerve and also s i m i l a r to that i n cat's vagus and saphenus nerves. [2] [1] FH (1964), p. 302. [2] p. 55. These f a c t s c i t e d are no proof for the correctness of the use of a mathematical model applied to man, but they give evidence that the mechanisms of e x c i t a t i o n are s i m i l a r even between amphibian and mammals. The problem that the models give inaccurate r e s u l t s f o r r e p e t i t i v e f i r i n g i s avoided by r e s t r i c t i n g the stimulus to low r e p e t i t i o n frequencies; each pulse or p u l s e - t r a i n w i l l t r i g g e r only a s i n g l e a c t i o n p o t e n t i a l and the next stimulation only occurs when the membrane has again reached the r e s t i n g state conditions. As shown i n Part I, the aim i s to stimulate large myelinated A - 3 f i b r e s . Therefore, the Frankenhaeuser-Huxley (FH) model was chosen to be used i n the following i n v e s t i g a t i o n s . Occasionally a few c a l c u l a -tions were also made with the HH model. The methods used for the c a l c u l a t i o n s are described i n Appendix A. 4. Time scale adaptation The problem i s to f i n d a s c a l i n g f a c t o r such that c a l c u l a t i o n s with the FH model can be applied to man. Because of the large v a r i e t y of nerve f i b r e s and because of the many parameters involved, t h i s problem cannot be solved e a s i l y . The l i n k between membrane p o t e n t i a l and pro-pagation for the FH model was c a l c u l a t e d by Goldman and Albus (1968). Also, comparing the shape of the propagated a c t i o n p o t e n t i a l with the shape of a membrane p o t e n t i a l , we see that propagation has nearly no i n -fluence upon the shape. This can also be seen by making the rough assump-t i o n that propagation p r i m a r i l y means an increase i n capacitance f o r the nodal membrane. Frankenhaeuser and Huxley (1964, F i g . 7) showed that an 2 increase of the capacitance from 2 to 4uF/cm has only the e f f e c t of de-l a y i n g the A.P., but i t has nearly no e f f e c t upon the shape. The approach of the problem from the other side shows evidence of good agreement between the measurements on per i p h e r a l nerves i n man (Buchthal and Rosenfalck, 1966) and measurements on cat's vagus nerve ( P a i n t a l , 1966). Thus, the problem i s to compare P a i n t a l ' s measurements of propagated action p o t e n t i a l s i n cat with the r e s u l t s from the FH model for the nerve membrane. Now, what are the parameters we are intere s t e d i n : spike dura-t i o n , r i s e time and f a l l time of spike, propagation v e o l c i t y , or duration of the r e f r a c t o r y period? Of course, a l l parameters are functions of temperature and they are also functions of the f i b r e diameter. There i s no uniform treatment pos s i b l e . Comparisons between e x i s t i n g studies f or some of these parameters w i l l be made i n the next se c t i o n . V e l o c i t y Though there i s no d i r e c t , a n a l y t i c a l r e l a t i o n s h i p between mem-brane properties and v e l o c i t y f o r the myelinated n e r v e ^ i t i s worthwhile to i n v e s t i g a t e t h i s aspect. There i s a general agreement that the r e l a -t i o n between v e l o c i t y v and f i b r e diameter D i s l i n e a r . This has been shown for frog and for cat. Further, Goldman and Albus (1968) have shown that t h i s l i n e a r i t y also holds for the model, i f the internodal length and the nodal diameter are both proportional to the f i b r e ' s diameter. The pro p o r t i o n a l i t y f a c tors are temperature dependent. FH model: v = 1.125 D @ 20°C Goldman & Albus (1968) Xenopus Laevis measurement: v = 2.6 ... 2.0 D @ 23°C Hutchinson et a l . (1970) Frog measurement: v = 2.1 D @ 24°C Mountcastle (1972) Cat measurement: v = 5.5 D @ 37°C Waxmann & Bennet _3_ (1972) with v i n m/s, D i n pm and the f a c t o r i n (us) . The d i f f e r e n c e by a fa c t o r of 2 between the f i r s t two values i s s t r i k i n g . I t i n d i c a t e s that probably no exact co r r e c t i o n factor can be established. Goldman and Albus point out that the numerical value should not be expected to com-pare s t r i c t l y with a particular set of experiments because they used data from different fibres in their model and because the resistance r of the o f l u i d surrounding the axon has been set arbi t r a r i l y . The general expression for the temperature correction factor i s ( T _ T Win * = ?10 ' T ' To : T e m P e r a t u r e s i n ° c The Q^Q for velocities are Animal Q 1 Q Range Source cat 1.6 27 to 37°C Paintal (1966) man 1.5 25 to 36°C Buchthal and Rosenfalck(1966) Rana Pipiens 2.95 around 15°C Hardy (1973) Thus, reducing the cat measurements to 24°C with Q ^ =1.6 v - 5.5 • 1.6<24 " 3 7 ) / 1 ° • D - 2.98 D we get a value which does not agree with that for frog measurements (2.0 to 2.6)'. To match theoretically the FH model with the cat measure-ments, a Q^Q = 2.54 would be required. The interpretation of these results i s , that despite existing s i m i l a r i t i e s , a direct transformation i s not possible. It i s l i k e l y I | that differences in ionic concentrations, especially i n C , have an 3. important influence. Hardy (1973) reports similar contradictions for the velocity's temperature dependence in frog (Rana Pipiens) at 15°C, where he found a theoretical temperature coefficient (from Dodge's model) Q^Q = 1.5 but where he measured Q^Q = 2.95. With this rather unsatisfactory result we end this discussion, noting that the proper investigation of these relations yields material enough for another thesis. To enable further investigations, we define an arbitrary scaling factor based on the action potential duration. For most large A fibres in mammals at 37°C the duration of an action potential is 0.4 to 0.5 ms. In the FH model at the standard temperature (20°C) i t is 1.1 to 1.3 ms. Therefore, we define a general scaling factor for the time-axis: , 1.25ms „ c * = O^ mT" = 2'5 to be used in subsequent chapters. This corresponds to a calculated ^10 " x ' 7 -In order not to restrict the generality, a l l calculations will be made at the standard temperature of 20°C. If .necessary, the timing of the*stimulus will be scaled to 37°C by dividing the time scale by 2.5 to allow for comparisons with experiments. The complexity of calculations is reduced considerably i f only the space-clamped potential has to be calculated, as opposed to calculation of a propagated potential. In electrocutaneous stimulation, the action potential is propagated. The corresponding nerve models and the errors resulting from space-clamp are discussed in the next chapter. 86 Chapter 1 1 : AXON MODELS AND PROPAGATION  OF NERVE IMPULSES 1. Unmyelinated nerve 2. Myelinated nerve 3. Influence of the cable s t r u c t u r e upon space-clamp c a l c u l a t i o n s 1. Unmyelinated nerve A general representation f o r the nerve i s the cable s t r u c t u r e of F i g . 11-1. For the r e s t i n g state and f o r voltage deviations of l e s s than 10 mV, R^ can be taken as constant. For la r g e voltage deviations R^  m changes n o n l i n e a r l y , where the current I can be described by the HH model: R Na K 1 m •AA/V A A / V AAAA-AAAA-outside inside x axis F i g . 11-1. Cable s t r u c t u r e of unmyelinated axon. With r Q equal to the resis t a n c e of the external medium per u n i t length and r equal to the resistance of axoplasm per un i t length, propagation i n an i n f i n i t e axon i s described by the equation (Hodgkin and Huxley, 1952) _ J _ . i£v ! , c iZ + ! r + r . „ 2 m d t R o 1 9x m If the axon i s immersed i n a large volume of f l u i d , the value of r b o becomes so small that i t can be disregarded: r =0. The s p e c i a l case ° o of propagation at a constant v e l o c i t y 6 enables the s i m p l i f i c a t i o n to: A -6 " 3t 2 2 3 V _ jTV 1_ 2 ~ 2 ' ? 3x at %l Thus: - J L . a!v = c |i + + Q2 2 8t Na K 1 r. 8 3t i Hodgkin and Huxley (1952) solved this equation for f i x e d 6's. Cooley and Dodge (1966) describe the algorithm to solve the general case on a d i g i t a l computer. Owen et a l . (1976) describe new, more e f f i c i e n t algorithm to solve the same general case. L i e b e r s t e i n and Mahrous (1970) and L i e b e r s t e i n (1967) made a s u b s t a n t i a l change i n the axonal model, demonstrating that a l i n e induc-tance must be included i n any mathematical formulation of the e l e c t r i c a l properties of axons. The inductance r e s u l t s from the c i r c u l a r currents involved i n propagation. An extremely s i m p l i f i e d model where each mem-brane element has only an "on" and an " o f f " state i s used by Donati and Kunov (1976) to study v e l o c i t y v a r i a t i o n s . For more d e t a i l s see the l i t e r a t u r e . 2. Myelinated nerve The myelinated nerve can be represented by a purely passive cable for the length between nodes and by an active membrane c i r c u i t at the nodes. The action p o t e n t i a l i s propagated from node to node. This i s c a l l e d s a l t a t o r y conduction. Because the nerve i s nonuniform, there i s no closed mathematical d e s c r i p t i o n of propagation. FitzHugh (1962) solved the equations on a d i g i t a l computer for the c i r c u i t of F i g . 11-2. N stands f o r the act i v e nodal membrane. He used the HH equations. Goldman and Albus (1968) solved the same equations but using the more appropriate FH equations at the nodes. (The FH equations were only published i n 1964). T yVA- -AA -AA node spacing F i g . 11-2. E l e c t r i c a l model f or myelinated axon. These computations are quite time consuming, since the wave length of the propagating action p o t e n t i a l (duration * v e l o c i t y ) includes about 20 nodes (Dodge, 1963) [1] for each of which the nonlinear d i f f e r e n t i a l equations have to be solved simultaneously. McNeal (1976) made the s i m p l i f i c a t i o n of t r e a t i n g the nodal membranes as passive elements as long as the voltage at the corresponding nodes was below a c e r t a i n l e v e l , a measure which d r a s t i c a l l y saves computational time. [1] p. 87 89 3. Influence of the cable structure upon space-clamp c a l c u l a t i o n s There are two questions of i n t e r e s t concerning the r e l a t i o n between an i s o l a t e d membrane p o t e n t i a l and a propagating one: (i ) what i s the di f f e r e n c e i n threshold f o r the stimulus? and ( i i ) how i s the action p o t e n t i a l a f f e c t e d by propagation? To answer the f i r s t question we consider an axon of i n f i n i t e length, the axon being i n the r e s t i n g s t a t e . Stimulation occurs at one singl e point. For a short stimulus i t can be assumed that adjacent membrane segments i n i t i a l l y w i l l remain near t h e i r r e s t i n g state, such that they can be represented by the passive c i r c u i t of the membrane with a constant r e s i s t a n c e . Thus the axon represents a passive cable load, shunting a portion of the stimulating current. Cooley and Dodge (1966, F i g . 3) ca l c u l a t e d with the HH model strength-duration curves f o r both, the continuous (unmyelinated) axon and the space-clamped one. The d i f -ferences between the two normalized curves are s u r p r i s i n g l y small. This indicates that the axon has p r i m a r i l y an e f f e c t upon the amplitude of the required stimulus, which can be accounted for by a s c a l i n g f a c t o r , but that the axon does not degrade s i g n i f i c a n t l y the conditions p r e v a i l -ing i n the space-clamped membrane. For myelinated axons, i t can be s a f e l y assumed that the e f f e c t of the axon upon a l o c a l membrane ( i . e . node) i s even les s pronounced than i n the unmyelinated case, since the internodal part of an axon represents a l i g h t e r load than a continuous excitable membrane (see Fi g . 2-2 and Table 2-1). The exact c a l c u l a t i o n would require extensive computer c a l c u l a t i o n s . The second question i s more easy to answer. A f t e r an action p o t e n t i a l i s triggered, i t i s hardly a f f e c t e d anymore by stimulating or 90 propagating currents. Considering the active membrane as a source, we see that for Xenopus Laevis (FH-model) the source resistance drops to 2 2 ficm at the peak of the action p o t e n t i a l from the r e s t i n g value of 2 32 ficm , a reduction by a factor 16. This has the e f f e c t of reducing the membrane's time constant T = R^C by the same fa c t o r , which means that any disturbance i s n e u t r a l i z e d f a s t e r during an action p o t e n t i a l . I t i s e a s i l y seen that propagation has v i r t u a l l y no e f f e c t upon the shape of the action p o t e n t i a l by ( i ) comparing the cal c u l a t e d responses for propagation (Goldman and Albus, 1968) with space-clamped ones, by ( i i ) i n v e s t i g a t i n g the influence of an ongoing, b i p o l a r pulse upon the space-clamped p o t e n t i a l (see Chapter 14), and by ( i i i ) comparing experi-mental r e s u l t s (Tasaki, 1956; Bromm and Girndt, 1967). In order to set up proper parameters i n the methods of numerical in t e g r a t i o n s discussed i n Appendix A, i t i s necessary to have a knowledge of the system time constants. This forms the subject of the next chapter. 91 Chapter 12: THE TIME CONSTANTS IN MEMBRANE MODELS 1. Introduction 2. Time constants i n the HH model 3. Time constants of the rate v a r i a b l e s i n the FH model 4. FH model: Time constant of the membrane voltage and membrane conductance G 1. Introduction The time constants determine how r a p i d l y the membrane p o t e n t i a l w i l l respond to disturbances. For the numerical i n t e g r a t i o n s , described i n Appendix A, i t i s important to know the smallest possible time con-stant, which w i l l determine the maximal allowable step s i z e i n the i n t e -gration algorithm. Also, i t w i l l be seen that the sodium a c t i v a t i o n pro-cess ( v a r i a b l e m) responds at l e a s t ten times f a s t e r than the i n a c t i v a t i o n processes (variables h, n), an important feature enabling the construc-t i o n of s i m p l i f i e d models (see Appendix E). 2. Time constants i n the HH model Each of the four f i r s t order d i f f e r e n t i a l equations of the HH model has i t s own time constant. For the three rate constants m, n and h, the corresponding time constants are defined as functions of the membrane voltage V and temperature T: T i ( V ' T ) = 4>(T) • (a.(V) + p.(V)) W ± t h 1 = m ' n ' h' 1 1 _(T - 6.3°C)/10 9 = 3 T = temperature i n °C The three functions are shown i n F i g . 12-1 and l i s t e d i n Table 12-1 for T = 6.3°C ( 9 = 1 ) . 92 F i g . 12-1 Time constants of the rate constants i n the HH model as a function of membrane voltage. Temp-erature: 6.3 °C. Ordinate: l o g . scale. -30.0000 -25.0000 -20.0000 -15.0000 -10.0000 -5.0000 0.0 .5.0000 10.0000 15.0000 •20. 0000 25.0000 30. 0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70. 0000 75.0000 80.0000 85.0000 90.0000 95.0000 100.0000 105.0000 110.0000 115.0000 120.0000 0.04717 0.06221 0.08196 0. 10773 0.14123 0.18389 0.23677 0.299 14 0.36686 0.4 30 97 0.47904 0.50065 0.49352 0.46420 0.42296 0.37859 0.33644 0.29890 0.26655 0. 23908-0.21587 0.19624 0.17955 0.16528 0.15298 0.14231 0.13299 0.12478 0.11750 0.11101 0.10520 3. 16265 4.02586 5.07685 6.28232 7. 4964 3 8. 38969 8. 5160 1 7. 6702 3 6. 18582 4. 64 05 6 3. 3933 6 2.51512 1. 93 94 2 1.57574 1.35036 1.21219 1. 12798 1.07687 1.04596 1.02732 1.01613 1 . 0094 3 1.00 54 4 1.0030 8 1.00170 1.00090 1.00044 1.00018 1.00005 0.99998 0. 99995 5.28159 5.50198 5.67466 5.77584 5.78212 5.67717 5.45359 5.14135 4.75484 4.33457 3.91317 3.51451 3.15244 2.33236 2.55405 2.31417 2.10806 1.93084 1.77793 1.64548 1.52999 1 .42872 1.33936 1.26006 1.18927 1.12575 1.06846 1.01656 0.96932 0.92617 0.33660 Table 12-1 Values of the time constants (HH model) as a function of membrane voltage. Units: m i l l i v o l t s and m i l l i s e c . Note that f o r V < 15 mV x i s at le a s t ten times smaller than m or x^ which means that around the r e s t i n g p o t e n t i a l the a c t i v a t i n g sodium system reacts ten times f a s t e r than the i n h i b i t i n g mechanisms. The fourth time constant, x^ i s that of the membrane voltage. It i s defined by C C x V G GNa + GK + G l Thus, v a r i e s with the conductances between the two values: 2 r e s t i n g state (V = 0) = 2.5 i s ; G = 0.4m mho/cm peak of A.P. (V = 110 mV) T v(110) = 18 ys; G = 53.4 m mho/cm2 3. Time constants of the rate v a r i a b l e s i n the FH model Similar to the HH model, the time constants are cal c u l a t e d as a function of the membrane voltage V and temperature T: T i ( V ' T ) = 9.(T)(a.(V) + B.(V)) 5 i = m, n, h, p •-u A n (T-20)/l0 with $ ± = Q 1 Q i The only differences are (i ) that the i n d i v i d u a l Q ^ Q ' S l i s t e d i n Chapter 9-3 apply, ( i i ) that the standard temperature i s 20°, and ( i i i ) that there i s a fourth time constant, x . P The minimum values are: x = 28 ys at V = 120 mV m x = 490 ys at V = -30 mV n x, = 220 ys at V = 120 mV h x = 490 ys at V = -30 mV P F i g . 12-2 shows the functions and Table 12-2 the values. 1 r 1 1 1 1 -30.0 0.0 30.0 60.0 30.0 120.0 V ( M I L L I V O L T S ] F i g . 12-2 Time constants of the rate v a r i a b l e s i n the FH model as a function of membrane voltage V. Temperature 20°C. Ordinate: log. scale. 96 V = - 3 0 . O C O O - 2 5 . 0 0 0 0 - 2 0 . 0 0 0 0 - 1 5 . 0 0 0 0 - 1 0 . 0 0 0 0 - 5 . 0 0 0 0 0 . 0 5 . 0 0 0 0 1 0 . 0 0 0 0 1 5 . 0 0 0 0 2 0 . 0 0 0 0 2 5 . 0 0 0 0 3 0 . 0 0 0 0 3 5 . 0 0 00 4 0 . 0 0 0 0 4 5 . 0 0 0 0 5 0 . 0 0 0 0 5 5 . 0 0 0 0 6 0 . 0 0 0 0 6 5 . 0 0 0 0 7 0 . 0 0 0 0 7 5 . 0 0 0 0 8 0 . 0 0 0 0 8 5 . 0 0 0 0 9 0 . 0 0 0 0 9 5 . 0 0 0 0 1 0 0 . 0 0 0 0 1 0 5 , 0 0 0 0 1 1 0 . 0 0 0 0 1 1 5 . 0 0 0 0 1 2 0 . 0 0 0 0 - Li — 0 . 0 5 1 3 7 0 . 0 5 5 9 5 0 . 0 6 1 2 1 0 . 0 6 7 2 7 0 . 0 7 4 2 8 0 . 0 8 2 4 1 0 . 0 9 187 0 . 1 0 2 8 0 0 . 1 1 5 0 0 0 . 1 2 6 9 5 0 . 1 3 4 3 9 0 . 13 2 5 0 0 . 1 2 2 3 8 0 . 1 0 9 4 9 0 . 0 9 7 3 2 0 . 0 8 6 7 9 0 . 0 7 7 8 7 0 . 0 7 0 3 0 0 . 0 6 3 8 5 0 . 0 5 8 3 1 0 . 0 5 3 5 3 0 . 0 4 9 3 8 0 . 0 4 5 7 6 0 . 0 4 2 5 7 0 . 0 3 9 7 6 0 . 0 3 7 27 0 . 0 3 5 0 4 0 . 0 3 3 0 5 0 . 0 3 1 2 6 0 . 0 2 9 6 5 0 . 0 2 8 1 8 T H-0 . 4 8 1 5 9 0 . 6 1 0 4 1 0 . 8 0 6 7 . 1. 1 1 6 7 5 1 . 6 1 7 3 0 2 . 4 1 2 8 7 3 . 5 4 2 3 6 4 . 6 4 9 5 2 4 . 8 5 6 9 0 3 . 9 5 5 9 1 2 . 7 6 4 6 2 1 . 8 2 9 1 8 1 . 2 1 0 6 4 0 . 8 2 4 5 9 0 . 5 8 8 1 9 0 . 4 4 4 3 3 0 . 3 5 6 9 7 0 . 3 0 3 9 6 0 . 2 7 1 8 0 0 . 2 5 22 9 0 . 2 4 0 4 6 0 . 2 3 3 2 9 0 . 2 2 8 9 3 0 . 2 2 6 2 9 0 . 2'246 9 0 . 2 2 3 7 2 0 . 2 2 3 1 3 0 . 2 2 2 7 7 0 . 2 2 2 5 6 0 . 2 2 2 4 2 0 . 2 2 2 3 5 0 . 4 9 0 3 7 0 . 5 5 3 26 0 . 6 3 1 6 7 0 . 7 3 0 6 9 0 . 8 5 7 1 7 1 . 0 2 0 0 5 1 . 2 3 0 34 1 . 4 9 9 6 7 1 . 8 3 5 8 3 2 . 2 3 2 2 4 2 . 6 5 1 4 4 3 . 0 1 3 7 9 3 . 2 1 8 8 8 3 . 2 0 7 3 7 3 . 0 0 5 3 2 2 . 6 9 6 2 4 2 . 3 6 1 3 9 2 . 0 4 9 6 7 1 . 7 8 0 4 0 1 . 5 5 5 8 8 1 . 3 7 1 2 7 1 . 2 1 9 7 8 1 . 0 9 4 9 2 0 . 9 9 1 2 2 0 . 9 0 4 2 8 0 . 8 3 0 6 7 0 . 7 6 7 7 5 0 . 7 1 3 4 5 0 . 6 6 6 2 0 0 . 6 2 4 7 3 0 . 5 8 8 0 8 TP = 0 . 4 9 1 4 6 0 . 5 5 5 3 7 0 . 6 3 0 8 1 0 . 7 2 0 1 0 0 . 8 2 6 0 2 0 . 9 5 1 8 5 1 . 1 0 1 3 6 1. 2 7 8 7 3 1 . 4 8 8 2 1 1. 7 3 3 4 7 2 . 0 1 6 3 7 2 . 3 3 5 0 1 2 . 6 8 1 1 6 3 . 0 3 7 8 4 3 . - 3 7 8 6 2 3 . 6 7 0 9 1 3 . 8 8 3 7 0 3 . 9 9 7 0 8 4 . 0 0 8 3 0 3 . 9 3 0 7 0 3 . 7 8 7 1 8 3 . 6 0 2 4 1 3 . 3 9 7 5 6 3 . 1 8 8 1 8 2 . 9 8 4 3 4 2 . 7 9 1 8 0 2 . 6 1 3 3 3 2 . 4 4 9 7 6 2 . 3 0 0 8 3 2 . 1 6 5 6 8 2 . 0 4 3 1 7 Table 12-2 Values of the time constants (FH model) as a function of membrane voltage. Units: m i l l i v o l t s and m i l l i s e c . 97 4. FH model: Time constant of the membrane voltage T y and  membrane conductance G For the numerical integration methods and for simplifications of the model, the minimum value of T ^ has to be known. Because of the nonlinear I-V characteristic in the FH model, we have first to define instantaneous conductances. Calculation is based on the known ionic currents and on a model similar to the HH model. The model is shown in Fig. 12-3. f outside ? t Fig. 12-3 FH model with (instantaneous) conductances The voltages V^a and V^ are determined by the Nernst equation (see Chap-ter 8), and V^ is known. For Xenopus Laevis' standard data the values are: VXT = 123.56 mV Na V K = -27.79 mV V = 0.02506 mV With ionic currents determined by the FH model equations, the three con-ductances are defined: The leakage conductance i s known from the model g^ = g^ = const. Thus the t o t a l membrane conductance i s : G = % a + gK + gp + g l For standard data and for a monopolar stimulating pulse ( 1 = 1 mA/cm on [0, 0.12 ms]) the maximal conductance with the FH model i s : 2 G = 512.9 m mho/cm max occuring at t = 225 ys. NOTE: Because the membrane contains a c t i v e elements, the steady state I. d e f i n i t i o n G* = — , with 1^ denoting the sum of the i o n i c currents, i s not appropriate because i n th i s case G* includes an ac t i v e source, which makes G* negative during the r i s i n g part of an act i o n poten-t i a l and zero at i t s peak and at the r e s t i n g s t a t e . F i g . 12-4 shows the calculated time course of T = — during an V CJ action potential-(standard data, 1 mA/cm2 on [0, 0.12 ms], C = 2 uF/cm 2). The extremes are: maximum: = 65 ys at t = 0 minimum: T,, = 3.8 ys at t = 225 ys 1 1 1 — 1 1 1 0.0 0.333 0.667 1.0 1.333 1.667 2. TIME (MILLISECJ F i g . 12-4 Time constant of the membrane voltage t during an action potential.- Standard data, 2 stimulus: 1 mA/cm on (0, 0.12 ms) FH model. SUMMARY II In Part II several nerve membrane models have been examined. The The two most u s e f u l models are that of Hodgkin and Huxley (1952) (HH-model), which models the membrane of the unmyelinated giant axon of squid at 6.3°C, and that of Frankenhaeuser and Huxley (1964) (FH-model), which models the nodal membrane of a myelinated s c i a t i c nerve f i b r e of the toad Xenopus Laevis at 20°C. The models reproduce the electro-chemical events of changes i n the membrane's per m e a b i l i t i e s to sodium, potassium, and other l e s s impor-tant ions. I f the membrane i s stimulated, an i n i t i a l i n f l u x of sodium ions causes a membrane de p o l a r i z a t i o n , and the following outflux of po-tassium brings the act i o n p o t e n t i a l back to the r e s t i n g s t a t e . The FH model consits of f i v e nonlinear, f i r s t order d i f f e r e n t i a l equations (four f o r the HH model) which have to be solved simultaneously by a numerical i n t e g r a t i o n method. Several methods are discussed i n Appendix A. For both models, temperature c o r r e c t i o n factors are known. However, they are only defined i n the range 2°C to 20°C. Our i n t e r e s t i s focussed on the i n i t i a t i o n of ac t i o n p o t e n t i a l s i n large myelinated f i b r e s i n man. Therefore, the FH model was chosen for f u r t h e r i n v e s t i g a t i o n s . The question of the appropriateness of t h i s choice i s an extremely d i f f i c u l t one to answer, e s p e c i a l l y because there i s hardly any s p e c i f i c data a v a i l a b l e f o r nervous membrane i n man. Not expecting complete agreement between experiments i n man and predictions of the model, we have chosen the FH model as the best approximation a v a i l -101 able. To account for the d i f f e r e n c e i n temperature (37°C, 20°C), a time s c a l i n g f a c t o r was introduced, which i s based upon the duration of cor-responding action p o t e n t i a l s , and which makes the FH p o t e n t i a l l a s t 2.5 times longer than that i n man. This f a c t o r i s only important f o r the time s c a l i n g of the stimulus used i n experiments, since a l l c a l c u l a t i o n s are made at 20°C. The problem of the model's inaccurate modelling of repet-i t i v e f i r i n g i s i r r e l e v a n t i n our i n v e s t i g a t i o n s , because the i n t e r e s t i s focussed on i n i t i a t i o n of the f i r s t a c t i o n p o t e n t i a l , and because the stimulus i s gated i n t o bursts at a low r e p e t i t i o n rate (< 200 Hz) to pre-vent adaptaticnof the sensation. Thus i t i s expected that a f i b r e f i r e s only once during a stimulus burst. Models are also known to c a l c u l a t e the propagation of an ac t i o n p o t e n t i a l along the axon. Since, during the time an ac t i o n p o t e n t i a l de-velopes at one node, up to twenty other nodes along the axon are also activ a t e d by the propagating wave, the c a l c u l a t i o n of propagated poten-t i a l s i s at l e a s t 20 times more time consuming. Results published i n the l i t e r a t u r e i n d i c a t e that tte errors introduced by i n v e s t i g a t i n g only a s i n g l e node (space-clamp) are not s i g n i f i c a n t . Our i n v e s t i g a t i o n s are l i m i t e d to the model of the space-clamped membrane. The nerve membrane i s not an energy converter or transducer, but an active element. Thus the stimulus' energy i s not d i r e c t l y r e l a t e d to the generation of an actim p o t e n t i a l . The FH model membrane i s des-cribed by the f i v e s t a t e s : Voltage and four i o n i c currents. A l l that the stimulus does i s to cause changes i n the ( i n i t i a l ) states of these v a r i -ables, such that the ac t i v e process of the generation of an action poten-t i a l w i l l take place as soon as the threshold condition i s exceeded. 102 C l e a r l y , threshold not only depends on the membrane voltage, but also on the induced imbalance i n the i o n i c currents. In p a r t i c u l a r , i t i s pos-s i b l e with a b i p o l a r stimulus to reach threshold by i n f l u e n c i n g the i o n i c currents, while the membrane voltage i s near the r e s t i n g state at the end of that stimulus. Since the model i s nonlinear, threshold can only be determined numerically. The corresponding search algorithm i s described i n Appendix B. The time constants of the model's v a r i a b l e s have been ca l c u l a t e d . The shortest time constant = 3.8 ys (FH model, 20°C) determines an upper l i m i t f o r the step s i z e i n the numerical i n t e g r a t i o n methods (see Appendix A) . The f a c t that the time constants of the v a r i a b l e s n, h and p are at l e a s t ten times longer than that of V and m enables the construction of a s i m p l i f i e d model. This aspect i s inve s t i g a t e d i n Appendix F. In the next part, responses to several s t i m u l i of both, the FH model and sensory nerve f i b r e s i n man, w i l l be inve s t i g a t e d . PART I I I : CALCULATIONS AND EXPERIMENTS Chapter 13: DYNAMIC THRESHOLD AND THE INDIRECT EFFECTS OF BIPOLAR STIMULATION 1. Threshold 2. Temporal summation 3. B i p o l a r s t i m u l a t i o n 1. Threshold For short current pulses, the t r a d i t i o n a l view of threshold has to be modified. Often threshold i s r e f e r r e d to as a d e p o l a r i z a t i o n v o l -•tage necessary to t r i g g e r an ac t i o n p o t e n t i a l . F i g . 13-1 i l l u s t r a t e s that with short pulses the voltage d e f i n i t i o n i s not s u f f i c i e n t . At the end of the stimulus (50 ys i n F i g . 13-1) the voltage begins to drop u n t i l the dynamic changes i n the membrane have increased the inflowing sodium cur-rent so much that i t prevents the voltage from fur t h e r dropping and then, due to a further increase, the current w i l l r a i s e the voltage to the ac-t i o n p o t e n t i a l . C l e a r l y , the i n i t i a l voltage peak (at 50 ys i n F i g . 13-1) i s only r e l a t e d to threshold by the complex dynamics of the membrane. Threshold can be defined i n a general way as the l i m i t i n g bound for membrane properties beyond which an ac t i o n p o t e n t i a l w i l l occur. This d e f i n i t i o n imposes no r e s t r i c t i o n s on the dynamics of the stimulus nor does i t define a voltage. I t i s only based upon the f a c t of the occurrence or non-occurrence of an action p o t e n t i a l . No conditions are made when the action p o t e n t i a l has to occur. The following l e v e l d e f i n i t i o n i s used i n thi s thesis to determine s t i m u l i of threshold strength: i f the membrane voltage exceeded 80 mV, the stimulus was c l a s s i f i e d as suprathreshold. The search algorithm to determine a bracketing p a i r of stimulus amplitudes, one sub- and one suprathreshold, i s described i n Appendix B. — 1 — 0.2 —I 0.3 —I 0.4 TIME O.o 0.1 — I 1— 0.5 0.6 (M1LL1SEC) -1— 0.7 o.e 0.9 1.0 F i g . 13-1 Membrane action p o t e n t i a l triggered by a short, s l i g h t l y suprathreshold monopolar stimulus. Trace below time-axis i n d i c a t e s stimulus timing. FH-model, standard data, stimulating current: 2 1.462 mA/cm during 50 ys. 105. The membrane model consists of the four interacting variables V, m, n, h, for the HH model, arid the five interacting variables V, m, n, h, p for the FH model. Because also the dynamic changes in the ionic currents have an influence, a voltage definition for threshold is only possible i f the timing of the stimulus remains the same for a l l investi-gations. In the general case of the models, threshold is defined in the five (four) dimensional space V, m, n, h, (p) as the four (three) dimen-sional space separating supra- from sub-threshold conditions. Fortunately i t is possible to reduce the models to the two variables V, m without in-troducing too large an error, in which case threshold is a line separat-ing those V, m combinations leading to an action potential from those not leading to one, (see AppendixF). In the time domain, a sufficient condition for the occurrence of an action potential i s , that after cessation of the stimulus, the depolar-ization continues, which is expressed by Thus, threshold is defined by V = ^£ = 0 = £ (membrane currents) provided V > 0 after this event and provided that i t is not caused by another stimulus. It is impractical to use this condition as a criterion. 106 The important f a c t to note i s that an action p o t e n t i a l not only may be triggered by depolarizing the membrane but also by i n f l u e n c i n g the balance of i o n i c currents. A s t r i k i n g example i s the anodal break stimu-l a t i o n (Guttman and Hachmeister, 1972; FitzHugh 1976) where the a p p l i c a -t i o n of a negative membrane voltage (hyperpolarization) causes the i o n i c currents to change i n such a way, that a f t e r the stimulus' end the voltage not only returns to zero, but continues to grow and produces an action p o t e n t i a l . Another important a p p l i c a t i o n i s to b i p o l a r stimulation, where the voltage at the end of the stimulus i s near zero but where the i o n i c currents have changed such that an ac t i o n p o t e n t i a l occurs. The t h i r d example i s accommodation, defined as the slope of a l i n e a r l y i ncreasing d i stimulus: . A slowly i n c r e a s i n g stimulus w i l l hardly t r i g g e r an acti o n p o t e n t i a l , no matter what the stimulus' amplitude i s , because the i o n i c currents adapt continuously to compensate i t s e f f e c t (Frankenhaeuser and Vallbo, 1965). In t h i s work the conditions f o r a stimulus to reach threshold are only determined f o r a membrane which was previously i n the r e s t i n g s t a t e . During the r e f r a c t o r y period the V, m, n, h, p combination d i f f e r s from that of the r e s t i n g state and thus another stimulus strength would be required f o r th i s case to reach threshold conditions. 107 2. Temporal summation Because of the RC structure of the membrane (see F i g . 8-1), i t i s possible that the capacitor summates charges from s t i m u l i or from i o n i c currents f o r a time i n t e r v a l not much longer than the system's time constants. This enables t r i g g e r i n g of an ac t i o n p o t e n t i a l by a t r a i n of successive s t i m u l i , each of which i s of subthreshold strength. In prac-t i c e , the lower l i m i t f o r the r e p e t i t i o n frequency i s 2 to 5 kHz, depend-ing on the nerve's temperature. (Bromm, 1966; Bromm and Girndt, 1967; Schwarz and Volkmer, 1965). 3. B i p o l a r stimulation In a monopolar stimulation the stimulus produces a depolariza-t i o n of the membrane voltage, which i n turn influences the amplitude of the i o n i c currents, which w i l l increase the d e p o l a r i z a t i o n and produce an a c t i o n p o t e n t i a l . With a b i p o l a r stimulus the e f f e c t i s an i n d i r e c t one. The membrane i s depolarized only during the duration of the stimulus. During t h i s time a c t i v e changes i n the i o n i c currents are induced. At the stim-ulus' end, however, the voltage i s reduced again to a voltage near zero. The e f f e c t of the stimulus i s balanced out except f o r the modified amp-l i t u d e s of the i o n i c current. Only t h i s net i o n i c current w i l l generate an action p o t e n t i a l . This i n d i r e c t e f f e c t i s a demonstration that the generation of an ac t i o n p o t e n t i a l i s an act i v e process, which i s not re-late d to the stimulus' energy; the membrane i s not an energy converter (transducer). A l l the stimulus i s required to do i s to induce a supra-threshold condition, defined by an appropriate combination of the values V, m, n, h, p at the end of the stimulus. 108 The discussion may w e l l be extended to a continuous t r a i n of bi p o l a r pulses. The pulses occuring a f t e r threshold conditions have been metjhave l i t t l e i n fluence upon the f a c t that an action p o t e n t i a l i s gen-erated (see F i g . 14-5). Though, there i s one exception: i f , a f t e r thres-hold conditions have been met, a strong hyperpolarizing pulse occurs, i t may be possible that the state v a r i a b l e s are brought back to subthreshold values and no action p o t e n t i a l i s generated. This was investigated ex-perimentally by Tasaki (1956) and i n the model by Bromm and Girndt (1967). In Appendix F t h i s e f f e c t i s in v e s t i g a t e d i n d e t a i l . To summarize we note that threshold i s dependent not only on the d e p o l a r i z i n g voltage but also on the dynamics of the i o n i c currents and further, that the combination of r e c t i f y i n g e f f e c t s and temporal sum-mation may lead to an ac t i o n p o t e n t i a l even i f the i n d i v i d u a l stimulus i s of subthreshold strength and completely symmetric. NOTE: The term "threshold" i s used f o r two d i f f e r e n t manifestations. In measurements and i n computations with the model, threshold defines the i n -put to the membrane i . e . a s p e c i f i c stimulus amplitude f o r a given stimu-lus timing. In th i s chapter and i n Appendix E the word threshold i s used to describe a l l the c r i t i c a l combinations of the membrane parameters beyond which an action p o t e n t i a l occurs. The l a t t e r describes ( i n t e r n a l ) membrane prop e r t i e s ; i t i s i r r e l e v a n t i n which way (by what type of input) a s p e c i -f i c state i s reached. The second d e f i n i t i o n i s the key to the concept of b i p o l a r •emulation. 109 In both cases threshold i s measured by the response of the sys-tem's output. input • — — ^ membrane output states » stimulus Threshold \ membrane voltage property of stimulus property of membrane states In the next chapter the threshold conditions f o r various s t i m u l i are c a l c u l a t e d . The aim i s to obtain some i n d i c a t i o n s concerning the e f f e c t s of d i f f e r e n t s t i m u l i . Based on this , t h e scope of further i n v e s t i g a t i o n s w i l l be narrowed. Chapter 14 INVESTIGATIONS WITH SELECTED CURRENT PULSES 1. Computational methods 2. Threshold amplitude f o r multiple pulses 3. Pulse duration and charge 4. Continuous pulses 5. Conclusions With a s e r i e s of p i l o t c a l c u l a t i o n s we try to explore the e f f e c t s of d i f f e r e n t s t i m u l i i n the nodal membrane model. Because of the model's complexity, solutions are only obtained by c a l c u l a t i n g f or each case the model's response. A more systematic i n v e s t i g a t i o n f o r a s i n g l e , b i p o l a r stimulus i s presented i n Chapter 15. 1. Computational methods For a l l c a l c u l a t i o n s i n t h i s chapter the complete Frankenhaeuser Huxley (1964) model was used with the standard data. Temperature i s 20°C. The i n t e g r a t i o n method used i s the trapezoidal p r e d i c t o r - c o r r e c t o r method described i n Appendix A. Step s i z e f or a l l integrations i s 0.5 microsec. Threshold strength of a stimulus was determined by the l e v e l d e f i n i t i o n described i n Appendix B. Two bracketing values were determined, one above and one below threshold which d i f f e r e d not more than 1% from each other. For s e v e r a l cases the accuracy had to be increased to 0.1%. C a l c u l a t i o n s were made on an IBM 370/168 i n FORTRAN, using double p r e c i s i o n . 2. Threshold amplitude for multiple pulses The stimulus' threshold amplitude depends on several factors such as p o l a r i t y of the f i r s t pulse, the spacing, the r e p e t i t i o n frequency and pulse duration. I l l In t h e i r tests of cutaneous stimulation with human subjects, Saunders (1973), Hardt (1974), and Sundstrom (1974) used b i p o l a r pulses with the parameters of Table 14-1. With an average f a c t o r of 2.5 f o r temperature c o r r e c t i o n (see Chapter 10), the corresponding parameters for the FH model are shown i n the same table. Saunders 37°C FH model 20°C t p 20 ys 50 ys h 5 ys 12.5 ys T 100 ys 250 ys f = 1/T 10 kHz 4 kHz Table 14-1 Adaptation of stimulus parameters. For d e f i n i t i o n s see Chapter 3-2. In a l l examples the pulsewidth t p was kept constant (50 ys) as also was the r e p e t i t i o n frequency f (4 kHz). The spacing t^ may vary. F i g . 14-1, A shows the reaction to one b i p o l a r pulse with t ^ =0. In F i g . 14-1, B the same stimulus, but with decreased amplitude produces a temporal summation and gradual changes i n the i o n i c currents, such that an action p o t e n t i a l i s triggered only a f t e r the fourth pulse. The ampli-tude can be further reduced i f a spacing i s introduced ( t ^ = 12.5 ys): F i g . 14-1, C. The negative onset of a stimulus has a considerably d i f -ferent e f f e c t ( F i g . 14-1, D) and requires a high stimulus amplitude. A long delay of the negative pulse ( t ^ = 75 ys) may have the e f f e c t that the a c t i o n p o t e n t i a l may have developed already so f a r , that the negative pulse has no influence anymore: F i g . 14-2, A. In t h i s f i g u r e i t i s also seen that the remainder of the stimulus has only a min-or and transient e f f e c t on the action p o t e n t i a l . F i n a l l y , f o r comparison, F i g . 14-1 Membrane p o t e n t i a l of FH model reacting to d i f f e r e n t s t i m u l i . For a l l s t i m u l i : t = 50 us and T = 250 ys. A) A = 1.956 mA/cm2, t = 0; B) A = 1.668 mA/cm2, t± = 0. The trace below each graph indicates the timing of the rectangular pulses. L n I—1 -t> CTT F i g . 14-2, B shows the action p o t e n t i a l produced by four monopolar pulses The corresponding threshold amplitudes are l i s t e d i n Table 14-2 An i n t e r e s t i n g r e s u l t i s that i n the cases where the negative pulse i s l a t e (t^ >_ 75 ys) or where i t has a low amplitude (asymmetric case), an action p o t e n t i a l i s triggered e i t h e r a f t e r the t h i r d p o s i t i v e pulse or not at a l l . The increase i n accuracy of threshold determination to 0.1% or even 0.01% does not change that r e s u l t . t 1 (ys) 1 pulse 2 pulses 3 pulses 4 pulses 5 pulses 0 1.9562 1.7812 1.7109 1.6809 1.6687 0 - - - -2.550 -12.5 1.7687 1.6281 1.5812 • 1.5437 1.5375 12.5 - - - -2.1750 -75 - - - 1.4516 1 ) -150 - - 1.4570 2 ) -POS 1.4625 1.300 1.2375 1.2125 1.1937 ASYM 1.6625 1.6250 3) 1.6163 1.6172 4 ) 1.6163 5 ) NOTES: 1) accuracy 0.1%; peak of A.p. at 0.7 ms i . e . during 3rd pulse. 2) accuracy 0.1%; peak of A.P. at 0.5 ms i . e . during 2nd pulse. 3) accuracy 0.01%; peak of A.P. at 0.55 ms i . e . during 3rd pulse. 4) accuracy 0.1%; peak of A.P. at 0.53 ms i . e . during 3rd pulse. 5) accuracy 0.01%; peak of A.P. at 0.55 ms i . e . during 3rd pulse. Table 14-2 Threshold amplitudes for several b i p o l a r s t i m u l i . Units: mA/cm2. "POS" denotes only p o s i t i v e pulses, "ASYM" de-notes asymmetric pulse. t = 50 ys, T = 250 ys, standard data. P The waveforms of Table 14-2 are: f — H t 1 n T L.J POS n u T ASYM + H --fl-r In ASYM, the negative pulse extends from the end of the p o s i t i v e one t i l l the end of the period. I t i s only used to equalize the charge. This wave form acts b a s i c a l l y l i k e the monopolar pulse "POS", but i t s a t i s f i e s the requirement of zero average charge, which has to be s a t i s f i e d to pre-vent e l e c t r o l y t i c reactions (see Part I ) . Because i t s c o n t r o l over t r i g -gering an action p o t e n t i a l i s s i m i l a r to that of a monopolar pulse, i t i s not further i n v e s t i g a t e d . required i s shown i n F i g . 14-3. The two main r e s u l t s are: a) b i p o l a r stimulation requires higher amplitude than correspond-ing monopolar pulses; the higher the amplitude, the shorter the spacing time t.,. The r e l a t i o n between threshold amplitude and the number of pulse A/cm2) 2. Li t 1.6 l.tf 1.1 T f = ^ KHz 'rep 10°C ••9~~ _ *~ „ t, ~o " - - - - .-• *, - n s ^ 9 po s I. + **of pulses F i g . 14-3 Threshold amplitude versus number of pulses. FH model, Standard data, t = 50 us, T = 250 us, p o s i t i v e and negative pulse are equal i n amplitude and duration. 118 b) Only minute amplitude differences e x i s t between a stimulus t r i g g e r i n g an action p o t e n t i a l a f t e r three pulses and that t r i g g e r i n g one a f t e r f i v e pulses. 3. Pulse duration and charge What i s the r e l a t i o n between the pulse duration and the charge of a threshold stimulus? Charge i s defined as the charge contained i n a si n g l e rectangular pulse: Q = A * t ^ . This d e f i n i t i o n i s needed because to s a t i s f y the condition derived i n Part I the t o t a l charge of a b i p o l a r pulse i s zero, i . e . the charge of the p o s i t i v e pulse i s equal to that of the negative pulse. The equivalent c i r c u i t of the membrane i s b a s i c a l l y a p a r a l l e l RC c i r c u i t . I f i t were only a capacitor, the charge applied would cause the membrane p o t e n t i a l to change by: A * t AV = — = P-C C In the RC c i r c u i t , a f r a c t i o n of Q i s drained by R. Since the rate of change of m increases with i n c r e a s i n g V, m being the mediator of the sodium a c t i v a t i o n i n the FH model, one would expect that minimum charge would be required f o r a threshold stimulus with a pulse duration as short as p o s s i b l e . Three sample cases were ca l c u l a t e d where the stimulus pulses are converted i n t o d i r a c impulses, causing instantaneous displacements of the membrane voltage. In Table 14-3 threshold charges are compared between pulses with tp = 50 ys and dirac pulses (t = 0). In F i g . 14-4 the correspond-ing action p o t e n t i a l s are shown. These r e s u l t s i n d i c a t e c l e a r l y that the stimulus which requires minimum charge i s the shortest one f e a s i b l e . t = P 50 ys Dirac ( t p = 0) Q F i g . // Q F i g . # si n g l e p o s i t i v e pulse 73.125 - 60.0 14-4, A four p o s i t i v e pulses, period T = 250 us 60.625 14-2, B 50.624 14-4, B si n g l e b i p o l a r pulse, negative pulse s t a r t s at 50 ys 97.810 14-1, A 67.5 14-4, C" Table 14-3: Threshold charges. Q denotes the charge of one pulse [nanocoulomb/cm 2]. (t 25.312 1.6 3.0 J.4 TIME (M1LLISEC) 1.1 4.1 Fig 14-4 Membrane p o t e n t i a l of the FH model reacting to dirac s t i m u l i . 2 A) single p o s i t i v e pulse Q = 60 nanocoul/cm ; . B) 4 pos i t i v e pulses, T= 250 ps, Q of each = 50.624 nanocoul/ cm i—1 O 122 4. Continuous pulses A few sample c a l c u l a t i o n s were made on the e f f e c t of continuous pulses. This aspect i s not stressed as i t i s not important f o r our stimu-l a t i o n s , where the stimulus i s gated ( i . e . not continuous) to prevent ad-aptation. Nevertheless, i n t e r e s t i n g e f f e c t s are occurring. F i r s t , i t i s r e a d i l y seen that during the action p o t e n t i a l the influence of the stimu-lus i s markedly reduced. This i s due to the reduced membrane resistance during the a c t i o n p o t e n t i a l (see Chapter 12-4). Secondly, a r e p e t i t i v e f i r i n g may r e s u l t : F i g . 14-5, A. I t i s important to note that t h i s i s a completely new and d i f f e r e n t s i t u a t i o n . In 14-2 i t has been shown that the p o l a r i t y of the s t a r t i n g pulse i s important ( F i g . 14-1, C and D). The t r i g g e r i n g of the f i r s t a c t i o n p o t e n t i a l i s r e l a t e d to the transient res-ponse of the membrane c i r c u i t to the onset of the stimulus. A f t e r the f i r s t a c t i o n p o t e n t i a l , however, t h i s transient e f f e c t has disappeared and other mechanisms are required to generate another a c t i o n p o t e n t i a l . Two such mechanisms are: ( i ) the average voltage produced by a stimulus across a membrane which remains i n i t s r e s t i n g state depends on the stimu-l u s ' timing and i t i s not n e c e s s a r i l y zero (see Appendix E-6); ( i i ) the time constant of m i s voltage dependent which can r e s u l t i n a r e c t i f y i n g e f f e c t . In F i g . 14-5, B the stimulus i s that of Table 14-2, "ASYM". Here, no second p o t e n t i a l seems to develop. F i n a l l y , i n F i g . 14-5, C and D a subthreshold response follows the action p o t e n t i a l , but i t i s not strong enough to t r i g g e r another spike. I t i s known that the model y i e l d s wrong r e s u l t s for r e p e t i t i v e f i r i n g (see Chapter 10), which made us decide not to continue i n v e s t i g a t i o n s i n t h i s d i r e c t i o n . F i g . 14-5 Membrane potentials of the FII nodel under the influence of a continuous stimulus. For a l l graphs; t = 50 ys, T = 250 ys. A) A = 1.768 mA/cm2, t = 12.5 ys; 2 B) asymmetric pulse (see Table 14-2) A = 1.625 mA/cm The trace below each graph indicates the stimulus timing. D i s t o r t i o n of the rectangular pulse i s a drawing error. i — u> -<.0 -Z>.0 0.0 K.C *.D 50.0 W.O 100.0 a 73T 5. Conclusions Further i n v e s t i g a t i o n s w i l l be l i m i t e d to the e f f e c t of a si n g l e pulse upon the nodal membrane, which i s i n i t i a l l y i n the r e s t i n g state. An increase i n the number of pulses beyond three has only a minor e f f e c t , i f any. The l i m i t a t i o n to one pulse r e s u l t s from the p r a c t i c a l aspect of reducing the number of v a r i a b l e s . The continuous stimulation i s excluded a p r i o r i because the stimulus has to be gated to prevent adaptation. The fact to r e t a i n i s that ongoing stimulation does not s e r i o u s l y i n t e r f e r e with an ac t i o n p o t e n t i a l already under way. F i n a l l y , the reduction i n charge f o r short pulses encourages fur t h e r i n v e s t i g a t i o n with short pulses For these reasons, a systematic i n v e s t i g a t i o n f o r a si n g l e b i p o l a r pulse was undertaken. The r e s u l t s are presented i n the next chapter. 126 Chapter 15: SYSTEMATIC INVESTIGATION OF THRESHOLD STIMULI 1. Method 2. Results 3. Conclusions 1. Method The amplitude for a threshold stimulus was c a l c u l a t e d with the method described i n Appendix B. Two amplitude values were determined, one below and one above threshold. The values d i f f e r l e s s than 1% from each other. Except i n Table 15-5, the suprathreshold value i s used i n the tables and graphs. For the c a l c u l a t i o n s , the complete Frankenhaeuser-Huxley (1964) model was used with standard data and with temperature = 20°C. The i n t e g r a t i o n method used i s the f i r s t order pr e d i c t o r - corrector meth-od ( t r a p e z o i d a l rule) discussed i n Appendix A. The step s i z e f or i n t e -g ration i s , as usual,0.5 ysec. To reduce complexity, i n v e s t i g a t i o n s were l i m i t e d to a s i n g l e , symmetric, b i p o l a r pulse. The examples presented i n Chapter 14 suggest that the r e s u l t s w i l l not be s i g n i f i c a n t l y d i f f e r e n t f o r multiple pulses. The three v a r i a b l e s of the stimulus are: amplitude A, pulse-width tp, and spacing t^ between the end of the p o s i t i v e pulse and the begining of the negative pulse. Varying i n steps of 5 microsec the pulse-i l 4 -P F i g . 15-1 Variables of stimulus. 127 width between 5 and 75 microsec and the spacing between 0 and 75 microsec, 239 d i f f e r e n t combinations were ca l c u l a t e d . 2. Results  Amplitude of stimulus In Table 15-1 the amplitudes A f o r a s l i g h t l y suprathreshold stimulus are l i s t e d as functions of the pulsewidth t and spacing t ^ . The accuracy i s b e t t e r than 1%, i . e . a decrease of the amplitude by 1% of the l i s t e d value w i l l cause the stimulus to be subthreshold. The stimulus' amplitude i s not a good c r i t e r i o n to c l a s s i f y the effectiveness of a stim-ulus. A l l that can be said i s that A increases with decreasing pulse duration. The l i n e "POS" i n Table 15-1 shows the asymptotic threshold values when the spacing increases towards i n f i n i t y , i . e . when only the p o s i t i v e pulse has any e f f e c t . To i n v e s t i g a t e the influence of the negative pulse and the i n -fluence of the time when i t occurs (spacing - time t ^ ) , the amplitudes were normalized f o r a given pulsewidth by the corresponding amplitude for the monopolar pulse. The values are l i s t e d i n Table 15-2 and p l o t t e d i n F i g . 15-2. The s t r i k i n g r e s u l t i s that for pulses with t > 20 ys a large v a r i a t i o n of the spacing time has only a small e f f e c t upon the amplitude. Charge of a pulse Because of the large membrane capacitance, the charge d e l i v e r e d to the membrane i s more us e f u l to c l a s s i f y a pulse than i s the amplitude. Only the charge of the p o s i t i v e pulse i s considered: Q = A * t p . (The negative pulse removes the same amount of charge, such that at the end of the stimulus the t o t a l charge i s zero, a prime condition of b i p o l a r sti m u l a t i o n discussed i n Part I ) . In Table 15-3 the charges for threshold AMPLITUDES SPACISw PULSEWIDTH 0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 PCS 13.914 8.047 5.578 4 .219 3.406 2.662 2.461 2.168 1.950 1.766 1.618 1.499 1.395 1.313 24.265 10.761 6.790 4 .881 3.823 3.140 2.661 2.326 2.066 1.859 1.697 1.567 '1.452 1.351 1.271 19.526 9.332 6.047 4 .500 3.554 2.968 2.536 2.217 1.986 1.801 1.6U4 1.518 1.407 1.319 1.242 17.390 8.530 5.669 4 .218 3.388 2.829 2.437 2.148 1.924 1.744 1.605 1.483 1.385 1.299 1.222 16.168 8.064 5.359 4 .054 3.255 2.741 2.361 2.081 1.878 1.703 1.567 1.448 1.353 1.278 1.203 15.283 7.686 5.192 3.927 3.154 2.655 2.305 2.048 1.834 1.677 1.543 1.425 1.331 1.258 1.184 14.e06 7.445 5.029 3.804 3.080 2.614 2.269 2.000 1.806 1.651 1.519 1.403 1.321 1.239 1.175 14.343 7.271 4.911 3.745 3.032 2.552 2.234 1.969 1.778 1.625 1.495 1.392 1.300 1.229 1.157 14.111 7.157 4.835 3.686 2.984 2.533 2.199 1.954 1.764 1.612 1.483 1.381 1.290 1.219 1.143 13.890 7.046 4.759 3.629 2.938 2.493 2.165 1.923 1.736 1.587 1.472 1.371 1.280 1 .210 1.139 13.673 6.935 4.685 3.572 2.915 2.473 2.148 1.908 1.723 1.574 1.460 1.360 1.270 1.200 1.130 13.45? 6.827 4.648 3.544 2.892 2.435 2.131 1.893 1.709 1.569 1.449 1.349 1.260 1.191 1.130 13.354 6.774 4.612 3.516 2.869 2.416 2.114 1.878 1.696 1.557 1.438 1.339 1.250 1.182 1.121 13.250 6.721 4.576 3.489 2.847 2.397 2.098 1.864 1.683 1.545 1.426 1.328 1.250 1.180 1.112 13.146 6.668 4.540 3.462 2.825 2.397 2.081 1.849 1.683 1.533 1.415 1.328 1.241 1.171 1.112 13.044 6.616 4.505 3.435 2.803 2.378 2.081 1.849 1.669 1.533 1.415 1.318 1.240 1.171 1.104 12.269 6.230 4.244 3.250 2.S56 2.257 1.972 1.759 1.594 1.460 1.350 1.261 1.191 1.124 1.071 2 Table 15-1 Threshold current amplitudes A (mA/cm ) of the stimulating b i p o l a r pulse ( F i g . 15-1) as a function of pulsewidth t and spacing t (ysec). The l i n e "POS" denotes thres-hold amplitudes for a sing l e monopolar, p o s i t i v e pulse (for which t^ = °°). • NORMALIZED AMPLITUDES SFACING , ' 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0 PULSEWIDTH 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 , 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 1 0 . U 4 5 . 0 5 0 . 0 5 5 . U 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 1 . 9 7 8 1 . 5 9 2 1 . 4 1 7 1 . 3 1 8 1 . 2 4 6 1. 207 1 . 169 1. 150 1. 132 1 . 1 1 4 1 . 0 9 7 1. 088 1. 080 1 .072 1. 063 2. 233 1 . 7 2 7 1 . 4 9 8 1. 369 1. 294 1. 234 1. 195 1. 167 1 . 1 4 9 1. 131 1. 113 1 . 0 9 6 1. 087 1 . 0 7 9 1 . 0 7 0 1 . 0 6 2 1. 896 1 . 6 0 0 1 . 4 2 5 1 . 3 3 6 1 . 2 6 3 1 . 2 2 3 1 . 1 8 5 1 . 1 5 7 1 . 1 3 9 1. 121 1 . 1 0 4 1 . 0 9 5 1 .08 7 1 . 0 7 8 1 . 0 7 0 1 . 0 6 1 1 . 7 1 7 1 . 5 0 2 1 . 3 8 5 1 . 2 9 8 1 . 2 4 7 1. 208 1. 171 1 . 152 1. 134 1 . 1 1 7 1 . 0 9 9 1 .091 1 . 0 8 2 1 . 0 7 4 1 . 0 6 5 1 . 0 5 7 1. 589 1 . 4 4 0 1 . 3 3 9 1. 276 1 . 2 2 6 1. 188 1. 160 1. 142 1. 124 1. 106 1 . 0 9 8 1. 089 1 . 0 8 1 1 .072 1. 064 1 . 0 5 5 1 . 5 0 9 1 .391 1 . 3 1 5 1 . 2 5 3 1 . 2 1 4 1 . 1 7 6 1 . 1 5 8 1 .131 1 . 1 2 2 1 .104 1 . 0 9 6 1 . 0 7 9 1 . 0 7 0 1 . 0 6 2 1 . 0 6 2 1 . 0 5 4 1. 452 1. 349 1. 286 1. 236 1. 197 1. 169 1. 151 1. 133 1 . 1 1 5 1. 098 1. 089 1.081 1. 072 1.06 4 1. 056 1. 056 1. 399 1. 323 1 .261 1. 221 1. 183 1 . 1 6 5 1. 137 1. 119 1 .111 1. 093 1 . 0 8 5 1. 076 1. 068 1. 060 1 .051 1 .051 1. 360 1. 296 1 . 2 4 6 1. 207 1. 178 1. 151 1 . 1 3 3 1. 115 1 . 1 0 6 1. 089 1. 081 1 . 0 7 2 1. 064 1 . 0 5 5 1 . 0 5 5 1 . 0 4 7 1 . 3 3 6 1. 273 1 . 2 3 4 1 . 1 9 5 1. 167 1 . 1 4 9 1 . 1 3 1 1 . 1 1 3 1 . 1 0 4 1 . 0 8 7 1 . 0 7 9 1 . 0 7 5 1 . 0 6 7 1 . 0 5 8 1 . 0 5 0 1 . 0 5 0 1. 308 1. 257 1 . 2 1 7 1. 189 1. 161 1. 143 1. 125 1. 107 1. 099 1. 090 1. 082 1. 073 1. 065 1 . 0 5 7 1. 048 1. 048 1. 283 1 . 2 4 3 1 . 2 0 4 1. 175 1. 148 1. 130 1 . 1 1 2 1 . 1 0 4 1 . 0 9 5 1 . 0 8 6 1. 078 1 . 0 7 0 1 . 0 6 1 1 . 0 5 3 1 . 0 5 3 1 . 0 4 5 . 1 . 259 1. 220 1. 182 1 . 1 6 3 1. 136 1 . 1 1 8 1 . 1 1 0 1 . 0 9 2 1. 084 1. 075 1. 067 1 . 0 5 9 1. 050 1. 050 1 . 0 4 2 1 . 0 4 2 1 . 2 4 1 1. 202 1. 174 1. 156 1. 138 1 . 1 2 0 1. 103 1 . 0 9 4 1 . 0 8 5 1 . 0 7 7 1 . 0 6 8 1 . 0 6 0 1 . 0 5 2 1 . 0 5 0 1 . 0 4 2 1. 042 1. 226 1. 187 1. 159 1. 141 1. 124 1. 106 1 . 0 9 7 1. 080 1. 072 1. 063 1. 055 1 . 0 5 5 1. 047 1 . 0 3 9 1 . 0 3 9 1 . 0 3 1 Table 15-2 Current amplitudes of b i p o l a r pulse divided by amplitude of s i n g l e monopolar pulse of same pulsewidth. (Table 15-1, l i n e "POS"). 130 F i g . 15-2 Threshold amplitudes of b i p o l a r pulse normalized by the amplitude of the corresponding monopolar pulse. Parameter: pulsewidth. Each l i n e rep-resents a column of Table 15-2. CHA EG E PER HALF PULSE O.u 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . U 7 0 . 0 7 5 . 0 PCS PULSEWIDTH 10^0 1 5 . 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 8 9 . 5 9 0 . 2 8 8 . 1 8 8 . 8 1 3 9 . 1 1 2 0 . 7 1 1 1 . 6 1 0 5 . 5 1 0 2 . 2 1 0 0 . 2 9 8 . 4 9 7 . 6 9 7 . 5 9 7 . 1 9 7 . 1 9 7 . 4 9 7 . 6 9 8 . 4 1 2 1 . 3 1 0 7 . 6 1 0 1 . 8 9 7 . 6 9 5 . 6 9 4 . 2 9 3 . 1 9 3 . 1 9 3 . 0 9 2 . 9 9 3 . 3 9 4 . 0 9 4 . 4 9 4 . 6 9 5 . 4 9 7 . 6 9 3 . 3 9 0 . 7 9 0 . 0 8 8 . 9 8 9 . 1 8 8 . 8 8 8 . 7 8 9 . 4 9 0 . 0 9 0 . 4 9 1 . 1 9 1 . 5 9 2 . 4 9 3 . 1 8 7 . 0 8 5 . 3 8 5 . 0 8 4 . 4 8 4 . 7 8 4 . 9 8 5 . 3 8 5 . 9 8 6 . 6 8 7 . 2 8 8 . 3 8 9 . 0 9 0 . 0 9 0 . 9 9 1 . 7 8 0 . 8 8 0 . 6 8 0 . 4 8 1 . 1 8 1 . 4 8 2 . 2 8 2 . 6 8 3 . 2 8 4 . 5 8 5 . 2 8 6 . 2 8 6 . 9 8 7 . 9 7 6 . 4 7 6 . 9 7 7 . 9 7 8 . 5 7 8 . 8 7 9 . 7 8 0 . 7 8 1 . 9 8 2 . 6 8 3 . 8 8 4 . 9 8 5 . 5 8 6 . 5 7 4 . 0 7 4 . 5 7 5 . 4 7 6 . 1 7 7 . 0 7 8 . 4 7 9 . 4 8 0 . 0 8 1 . 3 8 2 . 5 8 3 . 5 8 4 . 2 8 5 . 9 7 1 . 7 7 2 . 7 7 3 . 7 7 4 . 9 7 5 . 8 7 6 . 6 7 8 . 2 7 8 . 8 8 0 . 0 8 1 . 2 8 2 . 2 8 3 . 5 7 0 . 6 7 1 . 6 7 2 . 5 7 3 . 7 7 4 . 6 7 6 . 0 7 7 . 0 7 8 . 1 7 9 . 4 8 0 . 6 8 1 . 6 8 2 . 9 8 3 . 9 6 9 . 5 7 0 . 5 7 1 . 4 7 2 . 6 7 3 . 4 7 4 . 8 7 5 . 8 7 6 . 9 7 8 . 1 7 9 . 3 8 1 . 0 8 2 . 2 8 3 . 2 8 4 . 7 8 5 . 4 6 8 . 4 6 9 . 4 7 0 . 3 7 1 . 4 7 2 . 9 7 4 . 2 7 5 . 2 7 6 . 3 7 7 . 5 7 8 . 7 8 0 . 3 8 1 . 6 8 2 . 6 8 4 . 0 8 4 . 7 6 7 . 3 6 8 . 3 6 9 . 7 7 0 . 9 7 2 . 3 7 3 . 0 7 4 . 6 7 5 . 7 7 6 . 9 7 8 . 5 7 9 . 7 8 1 . 0 8 1 . 9 6 6 . 8 6 7 . 7 6 9 . 2 7 0 . 3 7 1 . 7 7 2 . 5 7 4 . 0 7 5 . 1 7 6 . 3 7 7 . 8 7 9 . 1 8 0 . 3 6 6 . 2 6 7 . 2 6 8 . 6 6 9 . 8 7 1 . 2 7 1 . 9 7 3 . 4 7 4 . 6 7 5 . 7 7 7 . 2 7 8 . 5 7 9 . 7 8 1 . 3 8 2 . 6 8 3 . 4 6 5 . 7 6 6 . 7 6 8 . 1 6 9 . 2 7 0 . 6 7 1 . 9 7 2 . 8 7 4 . 0 7 5 . 7 7 6 . 6 7 7 . 8 7 9 . 7 8 0 . 6 8 2 . 0 8 3 . 4 6 5 . 2 6 6 . 2 6 7 . 6 6 8 . 7 7 0 . 1 7 1 . 3 7 2 . 8 7 4 . 0 7 5 . 1 7 6 . 6 7 7 . 8 7 9 . 1 8 0 . 6 8 2 . 0 8 2 . 8 8 6 . 7 8 8 . 1 8 4 . 5 8 6 . 0 8 6 . 8 8 5 . 4 8 6 . 1 8 3 . 4 8 4 . 7 8 1 . 3 8 2 . 7 8 4 . 1 6 1 . 3 6 2 . 3 6 3 . 7 6 5 . 0 6 6 . 4 6 7 . 7 6 9 . 0 7 0 . 4 7 1 . 7 7 3 . 0 7 4 . 3 7 5 . 7 7 7 . 4 7 8 . 6 8 0 . 3 2 Table 15-3 Charge per h a l f pulse (Q = A * t ), (nanocoul/cm ) as a function of pulsewidth t P p and spacing t . Rest l i k e i n Table 15-1. 132 s t i m u l i are l i s t e d as a function of the stimulus' pulsewidth t and P spacing t ^ . These values are pl o t t e d i n the figures 15-3 and 15-4. In Fi g . 15-3 each of the 16 l i n e s represents one row of Table 15-3, with spacing t ^ being the parameter, and i n F i g . 15-4 each of the 15 l i n e s represents one column of Table 15-3, with the pulsewidth being the parameter. As expected, a short p o s i t i v e pulse followed s h o r t l y afterwards by the negative one (t = 5 ys, t ^ = 5 ys) requires an extremely large charge to ac t i v a t e the membrane. This stimulus produces a membrane de-p o l a r i z a t i o n which i s immediately cancelled by the negative pulse ( F i g . 15-5, l e f t ) . The r e s u l t of Chapter 14 that an i n f i n i t e l y short pulse requires the l e a s t charge i s consistent with the present r e s u l t s , where the l e a s t charge i s required f o r a 5 microsec pulse with 75 microsec spacing. This pulse r a p i d l y produces a membrane de p o l a r i z a t i o n i n the f i r s t 5 ysec. During the following 75 ysec the sodium system w i l l be gradually activated, and only then the negative pulse occurs, reducing the membrane p o t e n t i a l again. This case i s i l l u s t r a t e d i n F i g . 15-5, r i g h t . In F i g . 15-6 the data of Table 15-3 i s plotted i n a t h i r d way, taking as v a r i a b l e not the spacing time t ^ but the t o t a l time from the beginning of the p o s i t i v e pulse u n t i l the beginning of the negative pulse, i . e . t + t^. To the r i g h t , the extension of the l i n e s would approach h o r i z o n t a l asymptotes. Again, the influence of t h i s time span (t + t..) upon threshold charge i s s u p r i s i n g l y small f o r time spans, say, greater than 30 microsec. Threshold charge depends more on the pulsewidth. 1 1 1 1 1 1 1 1 1 1 0.0 • 7.S 15.0 22.5 30.0 37.5 <C.O S2.5 60.0 67.5 73 PULSEWIDTH IMICROSEC) Fxg. 15-3 Bipolar stimulus. Threshold charge (Q = A • t ) P versus pulsewidth t with, spacing t± as parameter. Variables as defined i n F i g . 15-1. Each l i n e re-presents a row of Table 15-3. i 1 1 1 1 i 1 1 i 1 1 0.0 7.5 15.0 22.5 30.0 37.5 45.0 S2.S • 60.0 67.5 75 SPACING IMICROSEC) F i g . 15-4 As F i g . 15-3, except: charge versus spacing t^ with pulsewidth t as parameter. Each l i n e re-presents a column of Table 15-3. I I 1 1 1 1 1 1 1 1 0.0 15.0 30.0 . 45.0 60.0 75.0 90.0 105.0 120 0 13S 0 150 TIME TO NEG. PULSE IMICROSEC) F i g . 15-6 B i p o l a r stimulus. Threshold charge (Q = A • t ) P versus time (t + t..) with pulsewidth as parame-p 1 . r ter (5 to 75 ys). Data taken from Table 15-3. Membrane Voltages As a byproduct of the threshold determination, the peak membrane voltages were stored on a tape. In Table 15-4 the peak voltages are l i s t e d which were produced when an action p o t e n t i a l was triggered by the stimulus l i s t e d i n Table 15-1. The values vary le s s than 1 mV despite the wide v a r i a t i o n of the stimulus. F i n a l l y , Table 15-5 l i s t s the membrane voltages at the end of the de p o l a r i z i n g pulse. The pulse i s s l i g h t l y subthreshold, however, an increase i n amplitude by 1% would make i t suprathreshold. 3. Conclusions The f a c t that threshold charge remains approximately constant f o r d i f f e r e n t pulse widths, whereas the stimulus' amplitude v a r i e s considerably, has been reported f o r electrocutaneous s t i m u l a t i o n (Saunders, 1973 [1]; Hardt, 1974 [2]; Rollman, 1973 [3]). No such report has been found f o r the FH model. The i n t e r p r e t a t i o n of the e f f e c t of pulsewidth t ^ and spacing t ^ upon threshold charge follows e a s i l y from the FH model. The sequence of events f o r t r i g g e r i n g an ac t i o n p o t e n t i a l i s : f i r s t , the membrane i s depolarized by a voltage d i f f e r e n c e V^; then, the rate constant m i s i n -fluenced by V^, which leads to an increase of the sodium current; t h i s current then generates the action p o t e n t i a l . The other v a r i a b l e s of the model remain nearly constant i n t h i s i n i t i a l phase. Because the membrane behaves l i k e a p a r a l l e l RC c i r c u i t , the shorter the time used to apply a given charge to the c i r c u i t , the higher the d e p o l a r i z a t i o n voltage. Now, as shown i n F i g . 15-7, a increases with m V^ which means that the higher V^, the f a s t e r m changes. The combination [1] F i g . 7, p. 24; [2] F i g . 30, p. 91; [3] F i g . 8, p. 42. P E A K V O L T A G E S OF A . P . S P A C I N G , /S.O 1 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0 PULSEWIDTH 3 0 . 0 3 5 . 0 0 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 0 . 0 5 . 0 1 0 . J 1 5 . 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 P C S 1 1 4 . 6 1 1 1 4 . 5 7 1 1 4 . 6 3 1 1 4 . 6 5 1 1 4 . 2 9 1 1 4 . 6 0 1 1 4 . 1 9 1 1 4 . 6 4 1 1 4 . 6 9 1 1 4 . 6 7 1 1 4 . 5 4 1 1 4 . 6 6 1 1 4 . 6 9 1 1 4 . 7 0 1 1 4 . 6 8 1 1 4 . 3 5 1 1 4 . 4 1 1 1 4 . 5 3 1 1 4 . 5 2 1 1 4 . 6 4 1 1 4 . 3 6 1 1 4 . 3 5 1 1 4 . 4 6 1 1 4 . 6 5 1 1 4 . 6 7 1 1 4 . 6 1 1 1 4 . 1 3 1 1 4 . 5 1 1 1 4 . 5 7 1 1 4 . 5 5 1 1 4 . 4 6 1 1 4 . 5 8 1 1 4 . 6 4 1 1 4 . 3 3 1 1 4 . 6 5 1 1 4 . 4 3 1 1 4 . 6 5 1 1 4 . 5 5 1 1 4 . 4 9 1 1 4 . 6 2 1 1 4 . 6 0 1 1 4 . 4 1 1 1 4 . 6 1 1 1 4 . 6 7 1 1 4 . 6 8 1 1 4 . 6 7 1 1 4 . 6 2 1 1 4 . 6 4 1 1 4 . 4 9 1 1 4 . 6 4 1 1 4 . 4 8 1 1 4 . 6 2 1 1 4 . 6 5 1 1 4 . 3 1 1 1 4 . 6 3 1 1 4 . 6 7 1 1 4 . 6 3 1 1 4 . 4 4 1 1 4 . 6 0 1 1 4 . 6 4 1 1 4 . 6 4 1 1 4 . 6 1 1 1 4 . 5 3 1 1 4 . 5 0 1 1 4 . 5 7 1 1 4 . 4 7 1 1 4 . 6 0 1 1 4 . 5 4 1 1 4 . 3 7 1 1 4 . 3 2 1 1 4 . 5 8 1 1 4 , 5 9 1 1 4 . 4 5 1 1 4 . 6 3 1 1 4 . 6 9 1 1 4 . 7 0 1 1 4 . 7 0 1 1 4 . 6 7 1 14-. 61 114. 54 1 1 4 . 5 6 1 1 4 . 6 5 1 1 4 . 5 5 1 1 4 . 6 2 1 1 4 . 3 4 1 1 4 . 6 5 1 1 4 . 2 9 1 1 4 . 6 8 1 1 4 . 6 0 1 1 4 . 6 8 1 1 4 . 3 4 1 1 4 . 3 9 1 1 4 . 2 4 1 1 4 . 6 8 1 1 4 . 6 2 1 1 4 . 5 8 1 1 4 . 3 6 1 1 4 . 5 7 1 1 4 . 5 4 1 1 4 . 4 6 1 1 4 . 5 0 1 1 4 . 6 5 1 1 4 . 6 7 1 1 4 . 6 3 1 1 4 . 4 5 1 1 4 . 5 9 1 1 4 . 6 3 1 1 4 . 6 3 1 1 4 . 5 9 1 1 4 , 5 0 1 1 4 . 7 0 1 1 4 . 4 0 1 1 4 . 5 6 1 1 4 . 4 0 1 1 4 . 5 7 1 1 4 . 2 2 1 1 4 . 6 5 1 1 4 . 4 0 1 1 4 . 3 4 1 1 4 . 6 5 1 1 4 . 4 3 1 1 4 . 5 5 1 1 4 . 5 8 1 1 4 . 5 7 1 1 4 . 5 0 1 1 4 . 1 0 1 1 4 . 6 4 1 1 4 . 3 7 1 1 4 . 5 7 1 1 4 . 5 8 1 1 4 . 5 7 1 1 4 . 6 2 1 1 4 . 5 0 1 1 4 . 5 6 1 1 4 . 5 0 1 1 4 . 6 6 1 1 4 . 4 3 1 1 4 . 5 3 1 1 4 . 5 4 1 1 4 . 5 0 1 1 4 . 3 1 1 1 4 . 6 6 1 1 4 . 5 6 1 1 4 . 5 9 1 1 4 . 5 0 1 1 4 . 6 4 1 1 4 . 5 4 1 1 4 . 5 1 1 1 4 . 6 3 1 1 4 . 6 4 1 1 4 . 5 7 1 1 4 . 6 8 1 1 4 . 4 6 1 1 4 . 5 2 1 1 4 . 6 7 1 1 4 . 6 5 1 1 4 . 5 8 1 1 4 . 4 2 1 1 4 . 6 6 1 1 4 . 5 1 1 1 4 . 5 5 1 1 4 . 5 7 1 1 4 . 6 4 1 1 4 . 5 7 1 1 4 . 6 3 1 1 4 . 6 1 1 1 4 . 4 8 1 1 4 . 6 2 1 1 4 . 6 6 1 1 4 . 6 8 1 1 4 . 6 7 1 1 4 . 6 3 1 1 4 . 5 4 1 1 4 . 1 6 1 1 4 . 6 2 1 1 4 . 4 7 1 1 4 . 6 4 1 1 4 . 5 8 1 1 4 . 5 9 1 1 4 . 3 6 1 1 4 . 4 3 1 1 3 . 9 7 1 1 4 . 5 7 1 1 4 . 6 5 1 1 4 . 6 8 1 1 4 . 6 9 1 1 4 . 6 7 1 1 4 . 6 2 1 1 4 . 5 1 1 1 4 . 7 0 1 1 4 . 5 8 1 1 4 . 5 5 1 1 4 . 6 0 1 1 4 . 3 0 1 1 4 . 6 5 1 1 4 . 4 1 1 1 4 . 3 8 1 1 4 . 6 4 1 1 4 . 4 5 1 1 4 . 5 7 1 1 4 . 5 9 1 1 4 . 5 8 1 1 4 . 5 2 1 1 4 . 3 2 1 1 4 . 6 6 1 1 4 . 5 2 1 1 4 . 7 0 1 1 4 . 4 6 1 1 4 . 3 2 1 1 4 . 4 4 1 1 4 . 6 4 1 1 4 . 6 7 1 1 4 . 6 4 1 1 4 . 5 2 1 1 4 . 6 3 1 1 4 . 6 7 1 1 4 . 6 8 1 1 4 . 6 7 1 1 4 . 6 2 1 1 4 . 5 2 1 1 4 . 6 8 1 1 4 . 5 3 1 1 4 . 7 1 1 1 4 . 6 1 1 1 4 . 4 5 1 1 4 . 3 6 1 1 4 . 5 7 1 1 4 . 5 8 1 1 4 . 4 8 1 1 4 . 6 4 1 1 4 . 2 1 1 1 4 . 3 9 1 1 4 . 3 7 1 1 3 . 9 7 1 1 4 . 6 6 1 1 4 . 5 6 1 1 4 . 2 1 1 1 4 . 6 3 1 1 4 . 2 7 1 1 4 . 6 4 1 1 4 . 3 9 1 1 4 . 4 8 1 1 4 . 5 0 1 1 4 . 5 5 1 1 4 . 5 5 1 1 4 . 5 2 1 1 4 . 5 2 1 1 4 . 5 2 1 1 4 . 4 3 1 1 4 . 3 0 1 1 4 . 3 8 1 1 4 . 5 6 1 1 4 . 5 1 1 1 4 . 6 0 Table 15-4 Peak voltage of the action p o t e n t i a l (mV) triggered by the threshold stimulus. Rest l i k e i n Table 15-1. VOLTAGES AT END C? POSIT IVE PULSE SPACING/ 1 0 . 0 1 5 . 0 2 0 . 0 2 5 . 0 PULSEWIDTH 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 0 . 0 5 . 0 1 0 . 0 1 5 , 0 2 0 . 0 2 5 . 0 3 0 . 0 3 5 . 0 4 0 . 0 4 5 . 0 5 0 . 0 5 5 . 0 6 0 . 0 6 5 . 0 7 0 . 0 7 5 . 0 PCS 5 7 . 7 4 4 6 . 5 6 4 1 . 5 0 3 8 . 6 0 3 6 . 4 9 3 5 . 3 6 3 4 . 2 5 3 3 . 7 7 3 3 . 1 8 3 2 . 6 6 3 2 . 1 5 3 1 . 9 0 3 1 . 6 5 3 1 . 4 0 3 1 . 1 6 6 4 . 3 3 4 9 . 7 5 4 2 . 9 4 3 9 . 2 6 3 7 . 12 3 5 . 3 8 3 4 . 2 7 3 3 . 4 7 3 2 . 95 3 2 . 4 3 3 1 . 9 3 3 1 . 4 3 3 1 . 1 8 3 0 . 94 3 0 . 7 0 3 0 . 46 5 3 . 7 9 4 5 . 2 9 4 0 . 30 3 7 . 7 8 3 5 . 7 0 3 4 . 5 8 3 3 . 5 0 3 2 . 7 1 3 2 . 2 0 3 1 . 6 9 3 1 . 1 9 3 0 . 9 5 3 0 . 7 1 3 0 . 4 7 30. 23 2 9 . 9 9 4 8 . 15 4 1 . 9 9 3 8 . 6 6 3 6 . 20 3 4 . 7 8 3 3 . 6 8 3 2 . 6 1 3 2 . 1 0 3 1 . 6 0 3 1 . 1 0 3 0 . 6 1 3 0 . 3 7 3 0 . 13 2 S . 8 9 2 9 . 6 6 2 9 . 4 2 4 4 . 14 3 9 . 8 3 3 6 . 9 4 3 5 . 16 3 3 . 7 5 3 2 . 6 8 3 1 . 9 0 3 1 . 3 9 3 0 . 8 9 3 0 . 4 0 3 0 . 16 2 9 . 9 2 2 9 . 6 9 2 9 . 4 5 2 9 . 2 2 2 8 . 9 9 4 1 . 5 1 3 8 . 07 3 5 . 87 3 4 . 11 3 3 . 0 1 3 1 . 9 4 3 1 . 43 3 0 . 6 7 3 0 . 43 2 9 . 9 4 2 9 . 7 0 2 9 . 22 2 8 . 99 2 8 . 7 6 2 8 . 76 2 8 . 5 3 3 9 . 5 4 3 6 . 5 1 34 . 6 7 3 3 . 2 3 3 2 . 1 3 3 1 . 3 4 3 0 . 8 3 3 0 . 3 3 2 9 . 8 3 2 9 . 3 5 2 9 . 1 1 2 8 . 8 8 2 8 . 6 4 2 8 . 4 1 2 8 . 1 8 2 8 . 1 9 3 7 . 74 3 5 . 46 3 3 . 6 2 3 2 . 48 3 1 . 3 9 3 0 . 86 3 0 . 0 9 2 9 . 6 0 2 9 . 35 2 8 . 8 7 2 8 . 6 3 2 8 . 40 2 8 . 17 2 7 . 94 2 7 . 7 1 2 7 . 7 1 3 6 . 4 2 3 4 . 4 6 3 2 . 9 3 3 1 . 7 8 3 0 . 96 3 0 . 17 2 9 . 6 6 2 9 . 1 6 2 8 . 91 2 8 . 43 2 8 . 19 2 7 . 96 2 7 . 7 3 2 7 . 5 0 2 7 . 5 0 2 7 . 2 7 3 5 . 53 3 3 . 51 3 2 . 2 9 3 1 . 13 3 0 . 30 2 9 . 77 2 9 . 26 2 8 . 7 5 2 8 . 51 2 8 . 0 2 2 7 . 78 2 7 . 6 2 2 7 . 45 2 7 . 22 2 6 . 9 9 2 6 . 9 9 3 4 . 4 3 3 2 . 7 9 3 1 . 5 5 3 0 . 6 8 2 9 . 84 2 9 . 3 0 2 8 . 78 2 8 . 27 2 8 . 0 3 2 7 . 78 2 7 . 5 4 2 7 . 3 0 2 7 . 0 7 2 6 . 84 2 6 . 61 2 6 . 6 1 3 3 . 55 32 . 24 3 0 . 98 3 0 . 10 2 9 . 25 2 8 . 72 2 8 . 19 2 7 . 94 2 7 . 69 2 7 . 44 2 7 . 20 2 6 . 96 2 6 . 72 2 6 . 49 2 6 . 49 2 6 . 26 3 2 . 85 3 1 . 56 3 0 . 2 9 2 9 . 7 0 2 8 . 83 2 8 . 29 2 8 . 03 2 7 . 5 1 2 7 . 2 5 2 7 . 0 1 2 6 . 76 2 6 . 52 2 6 . 28 2 6 . 28 2 6 . 05 2 6 . 0 4 3 2 . 1 3 3 0 . 81 2 9 . 85 2 9 . 24 2 8 . 66 2 8 . 10 2 7 . 56 2 7 . 29 2 7 . 0 4 2 6 . 78 2 6 . 5 3 2 6 . 2 9 2 6 . 04 2 6 . 0 0 2 5 . 7 6 2 5 . 7 6 3 1 . 7 4 3 0 . 36 2 9 . 3 8 2 8 . 76 2 8 . 17 2 7 . 6 0 2 7 . 33 2 6 . 7 9 2 6 . 5 4 2 6 . 2 8 2 6 . 0 3 2 6 . 0 3 2 5 . 7 9 2 5 . 5 4 2 5 . 5 5 2 5 . 3 1 2 9 . 2 4 2 8 . 6 4 2 8 . 2 8 2 7 . 8 8 2 7 . 5 1 2 7 . 1 2 2 6 . 7 3 2 6 . 3 8 2 6 . 0 5 2 5 . 6 9 2 5 . 3 6 2 5 . 1 0 2 4 . 8 1 2 4 . 5 1 2 4 . 3 8 Table 15-5 Voltages (mV) at the end of the p o s i t i v e , depolarizing pulse. Rest l i k e i n Table 15-1. so 0 '50 WO mV Fig. 28. Rate constants a and 8 as functions of the membrane potential in the node of RanVier of Xeno-pus lacvis (a, b, c). Abscissa, change in membrane potential V, in mV; ordinate, rate constants.. Broken lines (in graph a) show variations in for different fibers (Frankenhaeuser, 19G0; Frankenhaeuser and Hux-ley. 19G4). F i g . 15-7 Rate constants of the FH model. 141 of the two e f f e c t s means that a short pulse i s more e f f i c i e n t . Thus, threshold charge for a short pulse i s lower than for a long l a s t i n g pulse ( F i g . 15-3). The e f f e c t of the negative pulse i s to reduce the d e p o l a r i z a t i o n and to make i t more d i f f i c u l t to t r i g g e r an action p o t e n t i a l . Hence, the shorter the time i s during which the d e p o l a r i z a t i o n i s large, the higher the threshold charge needed. This i s c l e a r l y seen i n F i g . 15-6. I t i s noteworthy i n F i g . 15-3 that for spacings t ^ > 30 ys the r e l a t i o n between threshold charge and pulsewidth t i s l i n e a r . To test P t h i s , two a d d i t i o n a l points have been c a l c u l a t e d f o r a spacing t ^ = 75 ys: t = 150 ys Q = 102.48 nC P t = 300 ys Q = 145.90 nC P (For t = 300 ys, the peak of the action p o t e n t i a l occurs already at 498 ys, i . e . during the negative stimulus). The curve f o r t ^ = 75 ys i s p l o t -ted i n F i g . 15-8. For long pulsewidths, the curve i s rather an exponential function than a l i n e a r one. Again, t h i s agrees with what would be expected from the RC-structure of the membrane and i t also agrees with the three measurements c i t e d above. > To summarize, the stimulus r e q u i r i n g the l e a s t threshold charge i s a high amplitude pulse as short as f e a s i b l e , followed only a f t e r "a long time" (e.g. 75 ys) by the symmetric negative pulse. In the next chapter, s i m i l a r trends w i l l be established for electrocutaneous stimula-t i o n . The experiments were designed to test how well the t h e o r e t i c a l findings apply to the r e a l s i t u a t i o n . 142 100 200 ' 300 (jus) F i g . 15-8: Threshold charge for long pulse duration. Bipolar pulse, t = 75 ps. 143 Chapter 16: EXPERIMENTS WITH ELECTROCUTANEOUS STIMULATION 1. Purpose 2. Method 3. Checks and comparison with l i t e r a t u r e 4. Results 5. Comparison with model and discussion 1. Purpose The purpose of the experiments was to achieve some i n d i c a t i o n as to whether the c a l c u l a t i o n s based on the Frankenhaeuser-Huxley (FH) model for toad bore some relevance to humans. Exact qu a n t i t a t i v e agreement was not expected due to the large number of a d d i t i o n a l factors to be considered such as electrode geometry, skin i n t e r f a c e , propagation of the action p o t e n t i a l , and neuroanatomy. Since the objective of t h i s thesis i s to shed some l i g h t on the parameters for electrocutaneous stimulation p r i m a r i l y , the experiments and equipment were designed for t h i s purpose. 2. Method A group of f i v e naiive subjects were selected f o r the experiments. The task of each subject was to match the i n t e n s i t y of a test stimulus with that of a reference stimulus. Both s t i m u l i were alternated f or e i t h e r one or four seconds and using the same electrode. The four second i n t e r v a l was used for adjustments, the one second one to check the equivalence of the s t i m u l i . The test person chose the i n t e r v a l at w i l l . The experiment establishes an equivalence between two s l i g h t l y d i f f e r e n t s t i m u l i . In the FH model we searched for threshold stimulus 144 conditions. In Chapter 7 i t was demonstrated for perip h e r a l nerve f i b r e s that threshold i s a function of f i b e r diameter. Peripheral nerve f i b r e s have many d i f f e r e n t diameters and thus i t requires d i f f e r e n t stimulus intensities to activa t e d i f f e r e n t f i b r e s . Now, i f two d i f f e r e n t s t i m u l i produce exactly the same (suprathreshold) sensation f o r a s i n g l e stimulus pulse, we can conclude that the same number of nerve f i b r e s N have been stimulated, provided that also the q u a l i t y of the sensation i s the same. Thus, matching the test stimulus with the reference can be interpreted as e s t a b l i s h i n g the condition f o r stimulation of exactly N f i b r e s , which i s the threshold condition f o r the N f i b e r . The value of N i s determined by the reference stimulus and by the ski n q u a l i t y of the subject. With the concept of equivalence, the value of N does not need to be known. This concept of using the same electrode f o r the reference and the test stimulus has the following advantages: - The electrode-skin contact has no influence since i t i s the same for both s t i m u l i . The psychometric trans f e r function, r e l a t i n g the nerve a c t i v i t y to sensation, does not need to be known, since i t i s not important what the sensation i s but only that i t i s the same for both s t i m u l i . This function i s important f o r the (absolute) threshold determination, used by many i n v e s t i g a t o r s . - The d i s t r i b u t i o n of nerve f i b r e s i n the skin (number, function, place) has no influence since both s t i m u l i use the same electrode (electrode geometry) and ac t i v a t e the same structures. The reference stimulus was chosen a r b i t r a r i l y but i t was kept constant throughout a l l experiments. In successive presentations each of twenty 145 d i f f e r e n t test s t i m u l i had to be matched to the reference. The i n i t i a l amplitudes of the test pulses were preset randomly to produce a sensation to s t a r t with which was sometimes more, sometimes les s intense than the reference. The q u a l i t y of sensation remained the same with a l l test s t i m u l i since v a r i a t i o n s i n the stimulus parameters were r e l a t i v e l y small (from 10ps to 50ys). Implementation and procedures According to the recommendations of Hardt(1974), the following points have been observed: Electrode: s t a i n l e s s s t e e l , a i r gap, concentric, diameter of center electrode 3mm, r i n g electrode from 6 to 11 mm diameter. Areas: 2 2 center: 7.07 mm , r i n g : 66.8 mm . * M II mm - Pressure: A s p e c i a l electrode holder was designed here which pro-vides a constant force of the electrode of 100 grams against the s k i n . This was used since i t i s d i f f i c u l t f o r the subject to maintain a constant force against a f i x e d electrode. The thenar region of the hand yi e l d e d i n Hardt's experiments consistent, comfortable, d i s t i n c t sensations and was preferred by a l l subjects. We also used that region. To reduce external d i s t r a c t i o n s , the test person was l e f t alone i n a soundproof room during the experiment. 146 - The ski n was dry and unprepared. In a few cases small amounts of electrode paste were rubbed onto the ski n and removed before the experiment i n order to reduce the i n i t a l skin impedance. The stimulator used i s described i n Appendix C. According to the findings i n Part I i t was current regulated and i t was e l e c t r i c a l l y i s o l a t e d . The test procedure was under the con t r o l of a FORTRAN program on a PD.P-12 computer. The tes t subject c o n t r o l l e d the experiment by four switches: stop; slow/fast; decrease amplitude; increase amplitude. The subject could remove h i s hand from the electrode at any time. The stimulus was a s i n g l e , b i p o l a r pulse, s t a r t i n g with the p o s i t i v e pulse and repeated at a frequency of 100 Hz. The p o s i t i v e and the negative pulse were equal i n amplitude and duration. Pulsewidth t and spacing t ^ were defined as shown i n F i g 15-1. The increment (decre-ment) steps f o r the t e s t s t i m u l i were f or a l l waveforms 1% of the reference pulse's charge. The f a s t e s t change was one amplitude step per 0.5 seconds. 3. Checks and comparison with l i t e r a t u r e The current-voltage conditions observed by Hardt(1974) were exactly reproduced. F i g . 16-1 shows an example of the electrode voltage and current. The charge applied to the concentric electrode was 90 nano-coulomb to reach an electrode voltage of 100 V, which agrees with Hardt's f i n d i n g s . Agreement with c a l c u l a t i o n of the charge required to produce 100 V was not as good, but s t i l l reasonable. In Chapter 3 the required 2 charge f o r a ski n voltage of 100 V was calculated to be 2ucoul./cm . Since f o r our electrode the r i n g area i s much larger than that of the center, the ski n p o t e n t i a l below the center electrode i s approximately 9/10 of the p o t e n t i a l between the two electrodes (see Chapter 3-4). 147 0 V 0 rnA — r— t — i v -N 1114 I I I I I ' 1 1 I I I I I l l l I l l l i i i i 1 1 1 i I I i i I I i • f • t t 1 1 T T M t T I T 1111 J t t 1 t i t r M M 1 r T 1 M i l • A . . . i F i g . 16-1 Voltage and current of electrode Upper trace: voltage between the two electrodes: 50V/div. Lower trace: stimulating current: 5 mA/div. Time a x i s : 10 ps/div. 0 V I I I ! V 1 1 1 1 -4-J-t-t-V j M M X 1-4 1 1 i l l l i t i i 111! | M 1 1 lilt til l T T r r M M  \ r  "1 M t i ( II M M i t . F i g . 16-2 As F i g . 16-1. Saturation of stimulator at +140 V by a strong stimulus. Current scale changed to 10 mA/div. Thus, the calculated charge would be Q= 2ycoul v ' u ' > = 156 nanocoul 1 an 9 instead of the 90 nanocoul measured. However, the capacitance of s k i n and electrode varies considerably. Saunders (1973), for instance, reports 2 0.002uF/cm , i . e . a value ten times smaller than Edelberg's values used i n Chapter 3. F i g . 16-2 shows the case where the stimulator came in t o voltage saturation at +140 V and consequently the stimulating current dropped. To avoid these saturations, the experiments had to be l i m i t e d to s l i g h t l y suprathreshold s t i m u l i . P a i n f u l sensations could not be produced within the l i n e a r range of operation. Saturation was monitored continuously to ensure proper measurements. To summarize, our equipment yi e l d e d the same test r e s u l t s as previously published i n the l i t e r a t u r e . Therefore, we can be reasonably confident i n our measurements. 4. Results Each of f i v e subjects matched twenty d i f f e r e n t s t i m u l i to the a r b i t r a r i l y chosen reference stimulus with: t = 50 ys, t^= 20 ys, A= 2.35mA 2 and a charge of 117.6 nC, which corresponds to 1663.4 nC/cm . The r e p e t i t i o n frequency of these s i n g l e , b i p o l a r pulses i s 100 Hz. The charges per pulse which produce, with a d i f f e r e n t timing, the same sensation as the reference are l i s t e d i n Table 16-1. The values represent the mean of the f i v e subjects Q = \ I Q. , k = 5 and they are scaled by the f a c t o r (= 1/electrode area) to unit 0.07 cm area. 149 Spacing Pulsewidth t± ( M S ) 20 t P 30 (Ms) 40 50 10 1610 1665 1720 1809 20 1441 1530 1592 *) 1673 ; 30 1295 1446 1560 1630 40 1295 1376 1507 1618 50 1273 1343 1481 1579 Table 16-1 Normalized charges from the experiment i n nanocoulomb/cm for d i f f e r e n t s t i m u l i . *) reference (50,20) = 1663 nC/cm2. t and t are defined i n F i g . 15-1. Spacing Pulsewidth t 1 (Ms) t P (Ms) 50 75 100 125 25 83.34 88.59 94.14 100.10 50 78.51 84.84 90.92 96.97 75 76.21 82.73 89.16 95.31 100 74.89 81.68 88.09 93.75 125 74.11 80.98 87.50 93.75 Table 16-2 Threshold charges f o r the FH model, calculated with standard 2 data. Unit: nanocoulomb/cm . The times are 2.5 times those of Table 16-1. Integration step s i z e : 0.5 ys; accuracy <0.5%. 150 For each Q the sample standard deviation s, calculated by was determined. They are included i n F i g . 16-3. The average i s s = 3.1% . Hence, the probable error ( = 0.67 s ) i s 2%. Saunders (1973)[1] reports a probable error of 3.5% for a matching experiment with two electrodes. These low figures i n d i c a t e that the measurements are r e l a t i v e l y r e l i a b l e . The low fig u r e s r e s u l t from the f a c t that the subjects only had to make comparisons between two sensations, but they did not have to make absolute judgements, as i n the case of sensory threshold determination. 5. Comparison with model and discussion Is there any resemblence between the measurements and c a l c u l a t i o n s with the FH model? Applying a time s c a l i n g f a c t o r of 2.5 to t and t ^ to account f o r the d i f f e r e n t temperatures (20°, 37° C) (see Chapter 10-4), we cal c u l a t e d the corresponding threshold charges f o r the FH model. They are l i s t e d i n Table 16-2. If each charge from the measurements i s divided by the corresponding charge of the model, we get twenty d i f f e r e n t s c a l i n g f a c tors c_^  with a mean of c = 17.53 . Because the reference pulse's charge was set a r b i t r a r i l y , c i s also a r b i t r a r y . However, the standard deviation of a l l 20 c ^ ' s i - s only 4.1 %, which in d i c a t e s a close resemblence of the curves. The fac t o r c accounts p r i m a r i l y f o r losses i n the ti s s u e . The q u a l i t a t i v e s i m i l a r i t y of experiment and computations with the model i s shown i n F i g 16-3. A l l values of the model are m u l t i p l i e d by the s c a l i n g f a c t o r c = 17.53. Not only there i s a good agreement of the curve's shape and slope, but also of the curve's p o s i t i o n f o r d i f f e r e n t spacings t ^ . Complete qu a n t i t a t i v e agreement was not expected [1] P. 24 151 charge (nanocoulomb/cm1') 10 20 30 4 0 SO t H 1 1 1 1 - P 2 5 5 0 75 . 10 0 725 fryS; F i g . 16-3 Charges for equivalent s t i m u l i . S o l i d l i n e s : measurements. Dashed l i n e s : FH model with s c a l i n g f a c t o r 17.53 for charges. Short times:measurements,'long times: FH model. Bars i n d i c a t e the standard deviations of the measurements. Values from Table 16-1 and 16-2. 152 for several reasons: the temperature s c a l i n g f a c t o r may be d i f f e r e n t from 2.5; the electrode and skin i n t e r f a c e may have an influence which varies with the stimulus (e.g. frequency dependency); although the frequency spectrum of the test s t i m u l i did not vary considerably because of the small range of v a r i a t i o n s i n the timings 3 the frequency dependence of the current density d i s t r i b u t i o n i n the tiss u e might have a small influence; because only f i v e subjects were used f o r t h i s p i l o t experiment, the measurements are not highly r e l i a b l e ; the action p o t e n t i a l i s propagated i n the experiment, whereas i n the model i t i s a space-clamped p o t e n t i a l (see Chapter 11). The f a c t that the slope of the two curves t^=10 and t^=20 us i n F i g . 16-3 i s l e s s steep for the measurements than the c a l c u l a t i o n s , may i n d i c a t e that the s c a l i n g f a c t o r of 2.5 i s too large. F i g . 15-3 shows c l e a r l y f o r c a l c u l a t i o n s with the model that the same trend observed with the measurements occurs i n the model for shorter times than the ones used i n the comparison. Probably, the nerve temperature i s several degrees below 37°C. The s i m i l a r i t y between theory and experiment provides evidence that nerve f i b r e s are activ a t e d d i r e c t l y by the e l e c t r i c a l current. I f receptors were activa t e d by intermedial e f f e c t s of mechanical, thermal, or chemical nature, the measured curves would probably have other shapes, due to the a d d i t i o n a l transfer functions. To summarize, the p i l o t experiments i n d i c a t e that the Frankenhaeuser-Huxley (FH) model can be u s e f u l for t h e o r e t i c a l i n v e s t i -gations and for pr e d i c t i o n s concerning the e f f e c t s of current s t i m u l i 153 used i n electrocutaneous stimulation i n man. Large s e r i e s of experi-ments are necessary to define the exact conditions under which the model's r e s u l t can be applied to electrocutaneous stimulation. These experiments also have to include for the pulsewidth and for the spacing both, shorter and longer times than the ones used i n the present study. This would involve some redesign of the equipment to allow a larger voltage swing at the electrodes and higher current amplitudes for extremely short pulses. 154 SUMMARY I I I Threshold can be described e i t h e r i n terms of the stimulus par-ameters (input to the membrane) or by the va r i a b l e s of the membrane model. The f i r s t d e f i n i t i o n i s used i n experiments whereas the second enables us to i n v e s t i g a t e the e f f e c t s of b i p o l a r stimulation. A s u f f i c i e n t condition for threshold i s that a s p e c i f i e d v a r i a b l e combination be exceeded. The time h i s t o r y of how th i s combination was reached i s not important. A s e r i e s of sample c a l c u l a t i o n s has indic a t e d that the e f f e c t of multiple b i p o l a r pulses i s not very d i f f e r e n t from a s i n g l e b i p o l a r one. Further, an action p o t e n t i a l which i s triggered, f o r example, a f t e r the second pulse i s hardly a f f e c t e d i n i t s occurrence and i n the time of i t s maximum by a continuously ongoing pulse. A systematic i n v e s t i g a t i o n with the Frankenhaeuser-Huxley (FH) nerve model was c a r r i e d out i n Chapter 15. I t focusses on the influence of pulsewidths and spacing (between the end of the p o s i t i v e and the begin-ning of the negative pulse) upon threshold charge f o r a s i n g l e b i p o l a r pulse. The lowest charge i s required f o r a pulsewidth as short as f e a s i b l e and a spacing as long as s u i t a b l e . In the experiment with electrocutaneous sti m u l a t i o n the equi-valence between an a r b i t r a r i l y chosen reference stimulus and twenty d i f -ferent pulse timings was established. A l l pulses were b i p o l a r ones, re-peated at 100 Hz. The r e s u l t s i n d i c a t e a c l e a r correspondence between model c a l c u l a t i o n s and measurements i n man. The conclusions are (i ) that t h i s correspondence i s a strong evidence that nerve f i b r e s are d i r e c t l y stimulated (and not receptors) and ( i i ) that the FH model might be useful i n i n v e s t i g a t i n g the e f f e c t s of d i f f e r e n t s t i m u l i used i n sensory cutaneous stimulation i n man, under the condition where the stimulus i s near thresh' old. I f the model could also be used i n a more general sense remains to be demonstrated. CONCLUSIONS AND RECOMMENDATIONS 1. Conclusions 2. Recommendations f or electrocutaneous stimulation 3. Recommendations f o r further work 1. Conclusions Findings from i n v e s t i g a t i o n s with the complete Frankenhaeuser-Huxley model: A) The threshold charge f o r a si n g l e b i p o l a r pulse was found to change only s l i g h t l y f o r d i f f e r e n t pulsewidths. This has been observed experiment-a l l y by others (Saunders, 1973; Hardt, 1974). B) The minimum threshold charge was found for a s i n g l e b i p o l a r pulse with a pulsewidth as short as possible and a delay of the symmetric negative pulse as long as p o s s i b l e . C) The r e l a t i o n between threshold charge and pulsewidth t^ was approximat-e l y l i n e a r f o r t < 100 ys and f o r a delay of the negative pulse t ^ > 40 ys. For pulsewidth t > 100 ys threshold charge increases more than l i n e a r l y . In the l i t e r a t u r e t h i s has been reported f o r experi-ments . D) The e f f e c t s of b i p o l a r pulses cannot be inve s t i g a t e d with a simple (passive) RC model, because threshold conditions depend on s p e c i f i c combinations of the membrane voltage and of a l l i o n i c currents. Thus, threshold defined as a f i x e d voltage l e v e l i s i n s u f f i c i e n t . A b i p o l a r pulse induces v i a a temporary d e p o l a r i z a t i o n a change i n the sodium current p r i m a r i l y . With i t s negative pulse i t reduces the membrane voltage again to near the r e s t i n g state, such that i t i s the i o n i c currents' imbalance which leads to an action p o t e n t i a l and not the membrane de p o l a r i z a t i o n caused by the stimulus. E) A f t e r the action p o t e n t i a l has been triggered, i t i s r e l a t i v e l y insen-s i t i v e to a sustained stimulus; i t s occurence and the time when i t reaches i t s maximum remain nearly unaltered by an ongoing stimulus. F) The amplitude of a threshold stimulus c o n s i s t i n g of f i v e b i p o l a r pulses i s only about 16% lower than for a si n g l e b i p o l a r pulse. The energy used f o r the f i v e pulses i s several times that for the si n g l e thresh-ol d pulse. Therefore, i n v e s t i g a t i o n s i n this thesis were l i m i t e d to a s i n g l e , b i p o l a r pulse. A l l these findings were c a l c u l a t e d under the conditions: - the membrane i s i n i t i a l l y i n the r e s t i n g state the p o t e n t i a l i s not propagated (see Chapter 11) the constants used are the "standard data" of Frankenhaeuser and Huxley (1964) at 20°C the i n t e g r a t i o n method used f o r the equations i s a p r e d i c t o r - c o r r e c t o r algorithm with the Euler formula as pr e d i c t o r and the trape z o i d a l rule as corrector. The step s i z e i s 0.5 ys. a l l b i p o l a r s t i m u l i s t a r t with the p o s i t i v e pulse f i r s t ; p o s i t i v e and negative pulse are equal i n amplitude, duration, and charge. Findings from matching experiments with si n g l e b i p o l a r pulses A) The measurements agreed within 9.4% with the appropriately scaled model c a l c u l a t i o n s as to the required charges for d i f f e r e n t stimulus timings. Again, the lowest charge to produce a s p e c i f i c sensation i s required for a pulsewidth as short as possible and a delay of the negative pulse as long as po s s i b l e . The high degree of correspondence between the model and experiment indicates that the model can be u s e f u l f o r theor-e t i c a l i n v e s t i g a t i o n s and pr e d i c t i o n s concerning the e f f e c t s of current 158 s t i m u l i used i n electrocutaneous stimulation i n man. B) The s i m i l a r i t y of the r e s u l t s from the theory and experiment provides evidence that cutaneous nerve f i b r e s are activated d i r e c t l y by the e l e c t r i c a l current. I f receptors were activated, the measured curves would probably have other shapes. The measurements were made under the conditions: - pulses were repeated at 100 Hz i n t e r v a l s - the same concentric electrode was used for the test and the reference s t i m u l i , which were presented a l t e r n a t i v e l y . Normally the skin was dry (unprepared). the e l e c t r i c a l l y i s o l a t e d stimulator provided a constant current and had a voltage swing of ±140 V to f i t the model c a l c u l a t i o n s with the measurements, constant s c a l i n g f a c t ors were used. The reduced Frankenhaeuser-Huxley model It has been shown i n Appendix E and F that the complexity of the Frankenhaeuser-Huxley model for the nerve membrane can be reduced from f i v e v a r i a b l e s to the two v a r i a b l e s V and m. This reduced model can be used to i n v e s t i g a t e with considerably l e s s computational time e f f e c t s of s t i m u l i upon the membrane for membrane voltages below 60 mV. For s t i m u l i causing instantaneous voltage displacements (dirac current) t h i s model y i e l d e d charges for threshold s t i m u l i which deviated less that 1.5% from the r e s u l t s computed with the complete model. Heat production i n the skin I t follows from the e l e c t r i c a l model of the skin that the energy con-verted i n the skin i n t o heat i s minimal for long pulsewidths t (t > 0.25T) P P and for short spacings t., (t.,<0.6T). 159 Spacing denotes the time between the end of the p o s i t i v e and the begin-ning of the negative pulse where T denotes the time constant of the skin (70 ys < T < 640 ys). Under these conditions the capacitive properties of the skin can be used favourably. Conversely, minimum threshold charge for the nerve membrane requires a short t and a long t , . Thus, the optimum P 1 i s the short pulsewidth t = 0.25 T combined with the long spacing t.. = P 1 0.6 T. Note, that the factors 0.25 and 0.6 also depend on skin properties and that the values mentioned represent a worst case c a l c u l a t i o n . The l i n e a r , current - c o n t r o l l e d , i s o l a t e d stimulator An e l e c t r i c a l stimulator has been developed and was used i n the e x p e r i -mental work with the s p e c i f i c a t i o n s : - safety i s o l a t i o n voltage between pulse c o n t r o l inputs and output to electrode >_ 2.5 kV. - i s o l a t e d part powered by 12 V battery - current regulated output, i = ± 15 mA c max current amplitude accuracy 0.5% of i max r i s e time, f a l l time of pulses < 0.1 ys output i s c a p a c i t i v e l y coupled to prevent any net flow of charge - output voltage range of current regulation ± 140 V warning f l a g i s set i f output voltage exceeds l i n e a r range - slowly acting (f = 1 Hz), low current c i r c u i t readjusts average out-put charge to zero, i f i t deviates from there by any imbalance i n the b i p o l a r stimulus. - The output wave form i s l i m i t e d to rectangular p o s i t i v e and negative current pulses. 160 Importance of current regulated s t i m u l i Because of the following e f f e c t s , the stimulator has to be cur- rent-regulated and not voltage-regulated: the electrode-skin i n t e r f a c e does not a f f e c t the current stimulus; whatsoever i t s structure, i t has to transmit the current to the under-l y i n g t i s s u e . - i n the case of a d i e l e c t r i c breakdown of the s k i n the current i s pre-set and cannot increase - the stimulus can be b e t t e r c o n t r o l l e d because there are no current spikes such as occur with voltage stimulation; the stimulation depends on the current f i e l d i n the tissue containing the nerve, not on the electrode voltage. Further, the stimulus has to be b i p o l a r with equal charges i n the p o s i t i v e and the negative pulse to prevent any electrochemical e f f e c t s such as de-p o s i t i o n of electrode material i n the s k i n . To summarize, the main contributions of t h i s thesis are, i ) the demonstration that a mathematical nerve model can be used s u c c e s s f u l l y to c a l c u l a t e the e f f e c t s of stimulus parameter v a r i a t i o n s , and i i ) a general i n v e s t i g a t i o n of conditions for electrocutaneous stimulation. 2. Recommendations f o r electrocutaneous stimulation Based upon the findings of other researchers and of the present work the following points are recommended: concentric electrodes produce r e l i a b l e and l o c a l l y l i m i t e d sensations there should be an a i r gap between the center electrode and the r i n g to avoid the e f f e c t s of shunting due to sweat accumulation. 161 the geometry of the electrode has to be observed c a r e f u l l y , since i t i s an important parameter. Concentric electrodes with a center of l e s s than 3mm diameter tend to have a small dynamic range before pro-ducing pain (Saunders, 1973). the pressure of the electrode against the s k i n i s an important parameter which should be kept constant s t a i n l e s s s t e e l seems to be s a t i s f a c t o r y as electrode material - only b i p o l a r pulses with no net charge transfer should be used the stimulator has to provide constant current pulses the stimulator should be capable of a voltage swing of ± 250 V; the pulses should have r i s e and f a l l times of l e s s than 0.5 ys (measured with a t e s t load of 1 kfi) ; and the stimulator should be capable of 2 providing a charge of up to ± 2 y coulomb/cm per pulse (which makes, 2 e.g. for an electrode of 7.5 mm and a pulse width of minimal 5 ys a maximal current of ± 30 mA). - the stimulator should be c a p a c i t i v e l y coupled to prevent any net charge flow and also i t should be e l e c t r i c a l l y i s o l a t e d f or safety reasons pulses of short duration and with a long spacing require the lowest charge which r e s u l t s in the lowest energy consumption at the stimulator. 3. Recommendations f o r further work Despite the length of the present study, many questions remain to be answered. A) The work to follow immediately would be to extend the proof of the usefulness of the model. This includes: extension and continuation of the experiments, where more subjects 162 would have to be tested and where the timings of the pulses would have to be extended beyond the present scope (shorter and longer times) i n v e s t i g a t i o n s of the appropriatness of s c a l i n g f a c t o r s between the model and experiment, and, i f possible determination of a time s c a l i n g f a c t o r redesign of the stimulator to allow voltage swings of ± 250 V and max-imum output current pulses of about ± 30 ... 100 mA, where the maximum charge per pulse i s l i m i t e d to about 0.5 ... 1 u coulomb. B) Further i n v e s t i g a t i o n s of the accuracy and of the usefulness of the l i n e a r i z e d V, m model f o r threshold determinations e s p e c i a l l y f o r long l a s t i n g pulses and for multiple pulses would make a s p e c i a l t o p i c of i t s own. C) As a general t h e o r e t i c a l i n v e s t i g a t i o n , i t would be i n t e r e s t i n g to c a l -culate the e f f e c t s of the electrode geometry (variable areas of center and of r i n g , r a t i o of t h i s area, eventually also other electrodes than concentric ones) upon the threshold stimulus i n t e n s i t y - f i b e r diameter r e l a t i o n . This could be done with an i d e a l i z e d nerve i n an i d e a l i z e d medium of skin and tis s u e . Also, the e f f e c t s of frequency dependent current density d i s t r i b u t i o n i n such an i d e a l i z e d model would be of i n t e r e s t . Corresponding experiments could back up the theory. Model i n v e s t i g a t i o n s using an e l e c t r o l y t i c tank could be used as one possible method. D) The scope of the present i n v e s t i g a t i o n was l i m i t e d to a s i n g l e , sym-metric, b i p o l a r pulse. T h e o r e t i c a l and experimental i n v e s t i g a t i o n s of the e f f e c t s of varying other pulse parameters would be of great i n t e r e s t . The most i n t e r e s t i n g v a r i a t i o n s would be (i ) a si n g l e asym-163 metric pulse ( d i f f e r e n t durations for p o s i t i v e and negative pulse, but equal charge), ( i i ) negative s t a r t i n g s i n g l e pulses, ( i i i ) multiple pulses at d i f f e r e n t r e p e t i t i o n rates. E) Adaptation of the peresent r e s u l t s to f u n c t i o n a l stimulation and ex-perimental i n v e s t i g a t i o n of the usefulness for t h i s a p p l i c a t i o n of pre-d i c t i o n s which are based on the model. F) Further i n v e s t i g a t i o n s are also necessary concerning the maximum allow-able skin voltage which does not cause a d i e l e c t r i c breakdown and also how important, compared with the other e f f e c t s of s t i m u l a t i o n , the aspect i s of the energy conversion i n t o heat which takes place i n the ski n . 164 Appendix A: INTEGRATION METHODS FOR NERVE MEMBRANE MODELS  A 1 Introduction The problem i s to solve simultaneously the four f i r s t order d i f f e r e n t i a l equations of the Hodgkin-Huxley (HH) model and the f i v e f i r s t order d i f f e r e n t i a l equations of the Frankenhaeuser-Huxley(FH) model The equations are described i n Chapter 9-2 and 9-3. The i n i t i a l values of the equations are known. As Brady (1970) has shown for the HH model, the rate v a r i a b l e s m, n, h are bounded by zero and by 1, and V i s also bounded. Of s p e c i a l i n t e r e s t i s the s t a b i l i t y of an i n t e g r a t i o n method and i t s i n t e g r a t i o n e r r o r . Attempts to use analog or hybrid computers f a i l e d (FitzHugh, 1960) because of problems with voltage d r i f t s and ac-curacy. Only methods for d i g i t a l computers w i l l be discussed. A 2 Methods and Algorithms 1. Euler's Method and Algorithm 2. Runge-Kutta's Method 3. Predictor-Corrector methods, Adams' method 4. Adaptation of Adams' method to the equations 5. Predictor-Corrector algorithm. A l l numerical methods make use of a f i n i t e d i f f e r e n c e approximation. An important f a c t o r i s the proper choice of the step s i z e to minimize trun-cation or roundoff errors and to achieve s t a b i l i t y without increasing the computation time too much. Nomenclature The following notation i s used: y ( t ) : function to be calculated 165 A 2 V = y(0): known i n i t i a l value o dy , y = ~r- '• d e r i v a t i v e of y 3 dt 3 At: step s i z e y k = y(k'At) k = 0, 1, 2, ... y k + i = y ( ( k + 1 ) - A t > where k denotes the time of the ( l a s t ) known values and k+1 denotes the time f o r which the function i s to be evaluated. 1. Euler's Method and Algorithm The simplest approach to i n t e g r a t i o n i s to expand the function at time k in t o a Taylor s e r i e s : yk + i = y k + A t y k + < A t > 2 y k + ••• a-1* Since only the f i r s t d e r i v a t i v e i s known, Eq. (1-1) has to be truncated. The truncated form i s known as Euler's method. The step s i z e At has to 2 •» be chosen so small, that the truncation error ((At) y + ...) becomes small enough to be neglected. The algorithm i s : 1. With the known V , m^ , n^, h^, p^ determine the i o n i c currents I. and then l 2. With c a l c u l a t e the a's and 3's at time k and determine the four d e r i v a t i v e s *k = a y k ( 1 " y k } " 3yk " y k ; y " m ' n ' h ' P 3. The new values are: y k + l = y k + A t * y k ' y = V, m, n, h, p 4. Set y k + 1 = y k and go to 1. 166 A 3 The advantages are: simple implementation one step method: only the information at time k i s required, which makes i t s e l f - s t a r t i n g and allows for step s i z e changes whenever necessary. The disadvantages are: the step s i z e must be s u f f i c i e n t l y small such that the e r r o r by omitting the higher order terms (truncation error) remains small. - t h i s method i s at l e a s t inaccurate i f not unstable f o r integrations over long time i n t e r v a l s , because truncation errors are summed up i n each step. the required small step s i z e r e s u l t s i n greatly increased compu-t a t i o n a l time. This method i s used by P a l t i (1971) and by Schoepfle et a l . (1969). 2. Runge-Kutta Methods This i s a class of methods with d i f f e r e n t orders, d i f f e r i n g i n the number of points used f o r computation and thus d i f f e r i n g i n accuracy and number of required computations. The method normally r e f e r r e d to as Runge-Kutta i s the 4th order method, which ex i s t s at many computing centres as a l i b r a r y subroutine for FORTRAN programs. Therefore the algorithm i s not described here. A d e s c r i p t i o n and discussion of the RK method i s given, f o r example, i n Dorn and McCracken (1972). The advantages of RK are the s i n g l e step, s e l f s t a r t i n g properties and i t s easy implementation. The disadvantage i s that i t may be p a r t i a l l y unstable, i.e.. the computed r e s u l t may deviate d r a s t i c a l l y from the exact s o l u t i o n f or long i n t e g r a t i o n times because of the a d d i t i o n of roundoff and 167 A 4 truncation e r r o r s . This method i s used by Frankenhaeuser and Huxley (1964) and by Goldman and Albus (1968). 3. Predictor-Corrector Methods, Adams' Method These are multistep methods which c a l c u l a t e the new i n t e g r a t i o n value by one or several i t e r a t i o n s . Provided the step s i z e i s smaller than a given l i m i t , these i t e r a t i o n s converge towards a unique value. The b a s i c algorithm i s : 1. p r e d i c t y, , with method a) k+1 2. c a l c u l a t e values at time k+1, using the p r e d i c t i o n ^ + 1 ^ 3. use method b) to c a l c u l a t e the corrected value y . Method b) uses also the values at time k+1 c a l c u l a t e d i n step 2. 4. I f | v k + ] _ ^ ~ I > e w :*- t n e being an a r b i t r a r y error l i m i t , then set y, , .. ^ = y, , and go to 3. k+1 k+1 The index (0) or (1) denotes the number of c o r r e c t i o n s . There e x i s t s a large v a r i e t y of d i f f e r e n t equations used for method a) and method b), d i f f e r i n g i n the number of previous points k_^ ... k_^ used i n the equations and d i f f e r i n g i n weighting factors assigned to the points. The use of previous points generally increases accuracy, but i t also makes the i n t e g r a t i o n l e s s s t a b l e . In addition, i t has the disadvan-tage that at the beginning or at a change i n step s i z e another, s e l f s t a r t -ing method (e.g. Runge Kutta or Euler) has to be used to generate the re-quired values. The advantages of p r e d i c t o r - c o r r e c t o r methods are: The algorithm converges to a unique value i f the step s i z e i s below a given bound. With increasing number of steps, the error remains within bounds, 168 A 5 no matter how many steps are used. The method is stable. For y < 0 i t is absolute stable, i.e. the absolute error does not grow. For y > 0 the relative error does not grow. To reduce the broad spectrum, further discussion w i l l be limited to mostly commonly used class of predictor-corrector formulas known as Adams' methods. As a predictor the Adams-Bashforth equations employ foreward differences. For order q they are: q = ° yk+l " yk + A t \ q " 1 yk+l = yk + A t ( yk + \ ( yk " y k - l ) ; > q = 2 yk+l = yk + A t ( yk + I ( yk " y k - l ) } + 12 ( y k " 2 y k - l + yk-2» Henrici (1962, Chapter 5) gives a detailed discussion of this method and how to compute the given coefficients. Note that order zero is identical to Euler's method. The Adams-Basforth equations yield a predicted value y, ,, Hence, the f i r s t derivative y, ,, can be determined. An k+1 Jk+1 Adams-Moulton equation featuring backward differences can now be used for the iterative correction of y^+^' ^ o r different orders p, the equations are: (m+1) , * * (m) P = 0 yk+l " y k + A t yk+l . (m+1) . .^  Tl,' (m) . ' x n P " 1 yk+l " yk + A t [2 ( yk+l + y k ) ] 0 (m+1) . .1.- (m) . • v 1 ,' (m) P = 2 yk+l " yk + A t [2 ( yk+l + V " 12 ( yk+l " 2 y k + y k - l ) ] 169 A 6 The equation of order p = 1 i s known as trapezoidal r u l e . For further discussion see H e n r i c i (1962). Higher order equations allow for l a r g e r step s i z e s f or the same error per step or, conversely,reduce the error f or the same step s i z e . Gear (1971 a, 1971 b) has elaborated a program with v a r i a b l e order to max-imize the step s i z e ^ r e s p e c t i v e l y to minimize the computation time. He assumes the er r o r of an optimal program to be proportional to 1/p, p de-noting the order. Low order methods f o r p r e d i c t i o n and c o r r e c t i o n were chosen because: ( i ) they tend to be more stable than higher order ones, ( i i ) they are s e l f s t a r t i n g , and ( i i i ) the step s i z e i s l i m i t e d anyway by short dur-ation pulses. The method consists of the Euler equation f o r p r e d i c t i o n (Adams-Bashforth, q = 0): y k + l < 0 ) = y k + A t y k and the t r a p e z o i d a l equation as corrector (Adams-Moulton, p = 1) y k + l = y k + T ( y k + y k + l } ' 1 = 0 , 1 , 2 , . . . This method was also used by Cooley and Dodge (1966). 4. Adaptation of Adams' Method to the equations  Convergence and step s i z e The i t e r a t i o n s converge only to a stable value i f the step s i z e i s smaller than an upper bound. The d i f f e r e n c e between two successive c o r r e c t i n g i t e r a t i o n s i s y _ y ( i ) = A t ( y ( i + D • ( i ) } y k + l y k + l 2 v y k + l y k + l ; 170 A 7 which must be smaller than 1 for convergence. According to Dorn and McCracken (1972), t h i s condition can be expressed At < with M = | ^ 1 M 1 9y For the membrane voltage V i n the HH model, t h i s condition becomes At < 2 . C , = 2 x„ — % a + g k + 8 l — ± For the FH model on the other hand the c a l c u l a t i o n i s complicated. How-ever, using the l i n e a r i z a t i o n described i n Appendix E which enables the d e f i n i t i o n of conductances, we a r r i v e at the same r e s u l t . For the rate v a r i a b l e s , the condition i s : 2 A t < a +B = 2 T x ' x = m, n, h, p x x Note that a l l x are temperature dependent. The various time constants are calc u l a t e d i n Chapter 12. The shortest time constants appear at the peak of the act i o n p o t e n t i a l . The l i m i t s f o r the step s i z e At are: HH - model, 6.3°C: At < 36 ys ( x y = 18 ys) FH - model, 20°C: At < 7.6 ys (x = 3.8 ys) Under the above condition the algorithm converges towards an unique value. This value i s not n e c e s s a r i l y i d e n t i c a l to the "true" value of the function. There i s no s p e c i f i c r u l e to choose a s p e c i f i c step s i z e . Dorn and McCracken (1972) mention, that experience i n d i c a t e s an optimum i f the step s i z e i s chosen such that there i s about one i t e r a t i o n f o r each step. In Section A3-2 we w i l l i n v e s t i g a t e the influence of the step s i z e upon the c a l c u l a t e d a c t i o n p o t e n t i a l . 171 A 8 C a l c u l a t i o n of the rate v a r i a b l e s The rate v a r i a b l e s are only functions of the voltage V and temperature. Therefore, a s p e c i a l equation can be derived, based d i r e c t l y on the trapezoidal r u l e . The equation i s derived for m, but i t applies also f o r n, h, and p. The indices m are omitted i n a and 8 . m m Knowing the predicted voltage ^ + 1 ^ ^ ' ay^+\ a n c * ^k+1 C 3 n ^ e determined. Now we assume the value has been predicted. Thus at time k+1 the d i f f e r e n t i a l equation i s : \+l - a k + l ( 1 ) - 6 k + l \ + l ( 1 ) The c o r r e c t i o n with the tr a p e z o i d a l r u l e gives: °wci) - (\+vr» <» Using f o r the unknown predicted value ^ ^ ^ ^ t n e corrected value ^ 4 . ^ ^ which i s the most accurate choice, equation (1) and (2) can be rearranged to the s i n g l e equation: \ + 2 ~ ( a k + l + m k ) \+i = . . At . — — 7 ( 3 ) 1 + T ( \ + l + < W for n, - n, ( 0 ) - m ( 1 ) f o r "V+l " "k+l " \ + l • Thus c a n ^ e c a l c u l a t e d d i r e c t l y . 5. P r e d i c t o r corrector algorithm The complete algorithm used for a l l c a l c u l a t i o n s i n t h i s thesis i s : 1. Determine with the model's equation. 2. P r e d i c t : V. . / ' = V, + At • V, k+1 k k 172 3. Determine m^ , n k > p>k> h k with the model's equations. 4. Calculate the new a's and 3's at time k+1 with V, ^ and k+1 (0) k+1 the model's equations. 5. Determine the new m, n, h, p (at time k+1) with equation (3). 6. Using the values of 5), equate with the model's equation V k + ^ . 7. correct: = V k + f (\ + • 8. Erro r t e s t : i f 1^ +1^  ~ Vk+i^°^I > e then use the corrected value V ^ as the new p r e d i c t i o n and repeat steps 3) to 8), els e set k = k+1 and go to 1. The program i n Appendix D i s an example of the implementation of th i s algorithm. A 3 Computations and Results 1. Method and Tests 2. Accuracy versus step s i z e 3. Comparison with l i t e r a t u r e and other methods 4. Conclusion 1. Method and Tests A l l computations were made i n FORTRAN on a IBM 370/168. Double P r e c i s i o n was necessary to avoid summation of roundoff e r r o r s . Reference s o l u t i o n : an "exact" reference s o l u t i o n had to be defined. T h e o r e t i c a l l y , the numerical s o l u t i o n approaches the exact s o l u t i o n as the step s i z e i s decreased towards zero. The shortest time constant i n the FH model i s 3.8 ys. Thus, a step s i z e of 0.05 ys i s s u f f i c i e n t l y small to be reasonably close to the exact s o l u t i o n . E s p e c i a l l y f o r th i s r e f -erence c a l c u l a t i o n s i n g l e p r e c i s i o n was not s u f f i c i e n t . 173 Tests: (i) Several figures and calculations published in the literature were reproduced to control the program. ( i i ) To compare responses cal-culated with different step sizes, care was taken that the stimulus timing was not distorted by the variable time discretization: for each case, the charge delivered to the membrane was the same within ±1 p.p.m. 2. Accuracy versus step size Choice of stimulus: The integration methods yield the largest errors for a stimulus of threshold strength. In this case the membrane potential has a long latency period before i t rises (see, for ex. Fig. A-1 or 13-1). 2 Therefore, the monopolar threshold stimulus of 1.462 mA/cm on [0, 50 us] was chosen as a test. For control, two calculations were made with the slightly stronger 2 stimulus of 1.50 mA/cm on [0, 50 ys]. The error between responses with different step sizes i s significantly smaller i n this case (Table A-1), which 2 indicates that the stimulus of 1.462 mA/cm on [0, 50 ys] represents a good test condition. Mean square error: To compare the results obtained with different step sizes, the mean square error e between the reference solution V (t) and the actual solution V(t) was determined over the time of 2 ms. Both voltages were sampled every 5 ys. k e 2 = £ I (V(t.) - V r ( t . ) ) 2 ; t. = i • 5 ys k - 1-22. . 400 5 ys CPU - time The computational time required for the integration algorithm 174 A 1: was measured. This CPU - time does not include input, output, or addi-t i o n a l computation. As shown i n Table A-1, the CPU time i s nearly propor-t i o n a l to the number of steps. Amplitude and Phase The maximum amplitude was determined by a simple search f o r the la r g e s t value which was stored together with the time of occurrence. A t 2 £ (mV) 2 V at max mV t max ms CPU sec Stim. mA/cm2 # of steps 0.05 us Reference 114.38 mv! 0.4222 ms 26.248 sec 40,000 0.1 0.198 • 10~ 3 114.38 1 0.42210 13.489 20,000 0.5 24.0 114.45 . 0.4015 3.071 4,000 1. 81.0 114.51 1 0.3820 1.888 1.46 2,000 5. 509.0 114.79 1 0.2900 0.971 400 10. - ; - - 200 0.1 ys 1. Reference 1.955 114.83 ! 114.86 l 0.27130 0.2650 13.39 1.8220 1.5 20,000 2,000 Table A-1 D i f f e r e n t i n t e g r a t i o n step s i z e s . Comparison of step s i z e s In Table A-1 the r e s u l t s are shown f or d i f f e r e n t i n t e g r a t i o n step s i z e s . A l l c a l c u l a t i o n s use the standard data of the FH model. The graph of the mean square e r r o r s , F i g . A-1, shows c l e a r l y the convergence 175 y^l h - i At F i g . A-1 Mean square error for d i f f e r e n t step s i z e s . 176 A 13 towards a unique s o l u t i o n f o r small At. Large step s i z e s tend to produce an early action p o t e n t i a l ( Fig. A-2). In th i s case, the i n a c t i v a t i o n v a r i a b l e s n and h had less time to grow, which accounts f o r the s l i g h t l y higher peak voltage with large step s i z e s ( F i g . A-3). In F i g . A-4 the reference response and that f o r At = 0.5 ys are shown. 3. Comparison with l i t e r a t u r e and other methods An evaluation of computation methods for the Frankenhauser-Huxley model has not been found i n l i t e r a t u r e . Evaluations e x i s t , how-ever, f o r the Hodgkin-Huxley model, which has the same mathematical stru c t u r e . Because the time constants are considerably d i f f e r e n t i n the two models, the step s i z e had to be evaluated s p e c i f i c a l l y f o r the FH model, whereas the evaluation of the method i s valuable for both models. Moore and Ramon (1974) made i n v e s t i g a t i o n s with the Hodgkin-Huxley model, covering e s p e c i a l l y the area of large step s i z e s (up to 100 ys), and examining only errors greater than 1%. They use a Runge Kutta 4th order method, Euler's Method, a modified Euler method, and a multistep Adams pr e d i c t o r corrector method of 4th order. Except f o r corrector i t e r a t i o n s used i n our trapezoidal method, our method corresponds to the modified Euler, sometimes also c a l l e d Heun's method ) as used by Moore and Ramon (1974). Our preliminary i n v e s t i g a t i o n s with the HH model confirmed the following of t h e i r findings:. the solutions with R.K(At = 5 ys) , Euler (At = 1 ys) or Adams (At = 1 ys) are a l l i d e n t i c a l to within 1% and near the exact s o l u t i o n . - The most s e n s i t i v e s i t u a t i o n to computational errors i s a s l i g h t l y suprathreshold stimulus. 177 i men (ms) 0.1*5 O.L,0 0.3S 4 0.30 0.2S 4 h tt F i g . A - 2 Time of maximum amplitude versus step s i z e . "max Hi*.SO 70 60 •• SO •• <r0 \ I-- t — s F i g . A-3 Maximum Amplitude versus step size, At 178 A 1 F i g . A-4 Action p o t e n t i a l s computed with two, step s i z e s . Stimulus: 1.462 mA/cm' on [0, 50 y s ] . 179 A 16 The amplitude of the A.P. i s quite i n s e n s i t i v e to methods or step s i z e s . The time when the peak, amplitude occurs varies considerably with method and/or step s i z e . Further, Moore and Ramon provide evidence that accurate c a l c u l a t i o n s during the latency period are more de c i s i v e than during the r i s i n g phase of the action p o t e n t i a l . They recommend that a v a r i a b l e step s i z e not be used during the c a l c u l a t i o n of an a c t i o n p o t e n t i a l . The published r e s u l t s are consistent with our own c a l c u l a t i o n s . For the HH model we found that f o r step s i z e v a r i a t i o n s from 0.5 ys to 10 ys, the Euler, R.K. and p r e d i c t o r - c o r r e c t o r methods a l l y i e l d accurate s o l u t i o n s . The v a r i a t i o n s i n the peak amplitude were less than 0.5% and the v a r i a t i o n s i n the times when the peak occurs were les s than 2%. If the step s i z e i s chosen f o r each method such that a l l three require the same CPU time (Euler: At = 1.25 ys; R.K. : At = 10 ys; Pred-corr: At = 2.5 ys), i t i s seen that the p r e d i c t o r - c o r r e c t o r method i s equivalent to R.K. and, as expected, s l i g h t l y superior to the simple Euler. 4. Conclusions Having found the qualitatively s i m i l a r behaviour between solutions for the FH and the HH model, we r e s t r i c t e d the d e t a i l e d i n v e s t i g a t i o n s f o r the FH model to the trap e z o i d a l p r e d i c t o r - c o r r e c t o r method described i n Section A 1-5. I t was chosen because of i t s properties of s t a b i l i t y , s t a r t without a u x i l i a r y routines, and speed (less CPU time than R.K.). A l l c a l c u l a t i o n s i n t h i s thesis were made with t h i s method. As a trade o f f between computational time and accuracy, a step s i z e of 0.5 ys was chosen f or a l l further i n v e s t i g a t i o n s . 180 B 1 Appendix B: THRESHOLD SEARCH PROGRAM A general method i s described to determine a stimulus of thres-hold strength f o r membrane models. In F i g . B - l the peak voltages of the membrane's responses to d i f f e r e n t stimulus i n t e n s i t i e s i s shown schemati-c a l l y . D e f i n i t i o n : Having determined two bracketing values f o r the stimulus, where I T i s subthreshold and I i s suprathreshold, and where the L ri XH * h d i f f e r e n c e — - < 1%, the upper bracketing value I i s defined to be I R - H the threshold stimulus. This d e f i n i t i o n , based only on the occurrence of an ac t i o n po--t e n t i a l , i s valuable f o r a l l types of stimulus waveforms, e s p e c i a l l y also for short pulses (see Chapter 13-1). llSmV \ 0 stirnuluS F i g . B - l Stimulus response curve 181 FitzHugh (1976) uses the same approach. However, he defines as threshold the i n f l e c t i o n point I i n the stimulus response curve ( F i g . r B - l ) . This requires considerably more computational e f f o r t s . For our purposes the accuracy of 1% i s s u f f i c i e n t i n most cases because there are many other factors also i n f l u e n c i n g threshold conditions. One exception i s discussed i n Chapter 14-2 (Table 14-2) where an accuracy of 0.01% was required. Search Algorithm The two i n i t i a l values I T n and I are required to s t a r t with. LU HU For each value the peak voltage of the membrane model's response i s determined. I f V exceeds 80 mV, i t i s c l a s s i f i e d as suprathreshold. P The l e v e l of 80 mV i s a r b i t r a r y . Because a subthreshold response may include a transient membrane voltage peak, caused by the stimulus, the l e v e l has to be f a i r l y high. 80 mV proved to be s a t i s f a c t o r y . Having determined that one of the bracketing values i s below ( I T O ) and the other one (I u„) i s above threshold, a new stimulus ampli-tude i s determined by taking the average of these values: *^ew = ho + ho 2 • The response to that stimulus i s c a l c u l a t e d and depending on whether i t was sub- or suprathreshold, the lower or the upper bracket value i s replaced by t h i s new value. The next stimulus i s the average of the new p a i r , and so on. C a l c u l a t i o n i s stopped i f the two values d i f f e r less than 1%. The advantage of t h i s method i s f a s t (geometric) convergence. 182 B R e a l i s a t i o n : G i v e n the s t i m u l u s , t he c a l c u l a t i o n o f t h e A . P . and d e t e r m i n a -t i o n o f t he peak v o l t a g e i s done i n t h e s u b r o u t i n e VMEM. VMEM i s t h e com-p l e t e p r o g r a m to s o l v e the F r a n k e n h a e u s e r H u x l e y e q u a t i o n s u s i n g the t r a -p e z o i d a l r u l e f o r i n t e g r a t i o n i n a p r e d i c t o r - c o r r e c t o r a l g o r i t h m . The peak v o l t a g e i s d e t e r m i n e d by a s i m p l e s e a r c h f o r t h e maximum. To s a v e t i m e , c o m p u t a t i o n i s s t o p p e d i f i n t he t i m e i n t e r v a l DELTA ( n o r m a l l y 0 . 5 ms) a f t e r t h e o c c u r r e n c e o f t he p e a k , no o t h e r peak o c c u r s . S a f e t y l i m i t s a r e p r o v i d e d t o l i m i t t he number o f l o o p s (LOOPM) and t h e t o t a l c o m p u t i n g t i m e (CPUM) . CPUM i s u s e d t o p r o c e e d t o t h e o u t -p u t i n c a s e o f an e r r o r , b e f o r e t h e s y s t e m ' s t i m e l i m i t a b o r t s t h e j o b . The p r o g r a m was t e s t e d t o p e r f o r m a l l b r a n c h i n g s p r o p e r l y and t h e r e s u l t s o f a t e s t s e r i e s were i d e n t i c a l t o p r e v i o u s c a l c u l a t i o n s . 183 B 4 Program TRN: Read: Stimulus timing i n i t i a l Bracket Amplitudes AL, AH LOOPM, CPUM Loop = 1 no no no no AL = A i no i CALL VMEM (AL, V , t ) i P P store AL, V , t _ L _ L p V > 80 mV P Loop^ = 2 CALL VMEM (AH, V , t ) i P P store AH, V , t j P P V < 80 mV P CPU > CPUM ^ I Loop = Loop + 1 A = (AL + AH)/2 I CALL VMEM (A, V , t ) | P P store A, V , t j P P — < ^ V p > 80 mV ^ > — yes P r i n t : AL not subthreshold yes • P r i n t : AH not suprathreshold • yes yes AH = A -<^ (AH - AL) /AH < ~0~ 01 < LOOP > LOOPM yes yes 3 output of a l l V 's, A's and P t 's 184 C 1 Appendix C: Current Controlled Stimulator 1. Design 2. Description of the c i r c u i t s 3. Performance 1. Design 1.1 Requirements and general structure The task was to b u i l d a stimulator with a current c o n t r o l l e d output, which works under computer co n t r o l and which has a p o t e n t i a l i s o l a t e d output. The requirements are: safety i s o l a t i o n voltage >_ 2.5 kV current regulated output, i = ±15 mA max - voltage range of current regulation ± 130 V current amplitude accuracy 0.5% r i s e time, f a l l time of pulses < 0.5 ys - output wave form l i m i t e d to rectangular pulses There are four inputs to c o n t r o l the stimulus: - one TTL input f o r timing of p o s i t i v e pulse i one TTL input f o r timing of negative pulse one analog input (0, +10 V) for p o s i t i v e current amplitude one analog input (0, +10 V) for negative current amplitude. Changes i n the amplitudes have to be possible within 0.5 ms. Therefore, the bandwidth f or the amplitude-channels i s chosen from DC to 3 kHz. The general structure i s shown i n F i g . C-1. To enable the f a s t pulse transmission and to maintain an accurate amplitude c o n t r o l across the s e c u r i t y i s o l a t i o n , the s i g n a l i s s p l i t up i n the pos./neg. amplitude 185 C 2 information and the pos./neg. timing s i g n a l s . + A 6 -A o f p o -P a 1.5 l<V isolator )2V Battery supply F i g . C-1 Structure of stimulator and i s o l a t i o n . 1.2 I s o l a t i o n There are several p o s s i b i l i t i e s to transmit the amplitude i n -formation across the i s o l a t i o n : a) D i g i t a l coupler: 8 b i t s e r i a l transmission and on the i s o l a t e d side 8 b i t b u f f e r and D/A converter. pro: - accuracy 1/4% 1 - l i n e a r i t y contra: - several l o g i c c i r c u i t s on i s o l a t e d side required. b) Transformer: modulation of the s i g n a l , transmission across the i s o -l a t i n g transformer, demodulation. contra: - modulator and demodulator required, l i n e a r i t y depends on implementation. c) Analog opto coupler with feedback. This method was chosen. 186 C 3 pro: l i n e a r i t y contra: s p e c i a l coupler required. 1.2.1 Analog opto coupler The photodiode's current produced by the incident l i g h t i s pro-p o r t i o n a l to the l i g h t over at l e a s t 4 decades of i n t e n s i t y . L i n e a r i z i n g the l i g h t output of the l i g h t emitting diode by a feedback loop, l i n e a r i t y of at l e a s t 0.5% should be possible f o r the o v e r a l l transmission. To eliminate the dark current of the receptor photodiode there i s no b i a s i n g of the diode. The basic c i r c u i t i s : F i g . C-2 Analog i s o l a t i o n An increase of the input voltage causes the output of 0P1 to increase. T r a n s i s t o r TI i s only used as a power output. The change of OPl's output 187 C 4 causes the current through the LED to increase and therefore to increase the LED's l i g h t emission. 0P2 acts as a current to voltage converter. Its output voltage i s proportional to the diode current, which, i n turn, i s s t r i c t l y p roportional to the i l l u m i n a t i o n . This voltage i s fed back to 0P1. Despite n o n l i n e a r i t i e s and temperature dependence of TI and the LED, the l i g h t i n t e n s i t y i s regulated to be s t r i c t l y p roportional to the input voltage. The output voltage ^ Q u ^ °f the receiver (0P3) on the i s o -l a t e d side i s proportional to the l i g h t i n t e n s i t y because of the diode's l i n e a r i t y . Hence V i s proportional to V. . J out r i n The capacitor C,. forms with R,_ a low pass f i l t e r ( f = f f c 3 kHz) to suppress high frequency noise. To avoid any o p t i c a l noise, the l i g h t transmission i s completely shielded against environmental l i g h t . 1.2.2 D i g i t a l pulse transmission The TTL-signal governing the stimulus timing i s transmitted over the i s o l a t i o n gap by a d i g i t a l opto coupler device (HP 5082-4360). The d i f f e r e n c e between the time delays f o r "on" and " o f f " i s l e s s than 60 ns, which ensures that the timing i s not s e r i o u s l y d i s t o r t e d . 1.3 Output stage Having transmitted the two analog and two d i g i t a l s ignals over the i s o l a t i o n , the next question i s how to r e a l i z e the stimulator, which provides the b i p o l a r current pulses for the electrode. 1.3.1 Structure of the stimulator The output current of the stimulator i s the sum of four current sources: for both p o l a r i t i e s there i s a) a pulse current source for stimu-l a t i o n and b) a low-amplitude, continuous current source for balancing the output over long time i n t e r v a l s . 188 C 5 Because of l i m i t e d power d i s s i p a t i o n of the high voltage trans-i s t o r s , the pulse current sources provide only a current during the pulse. In most other stimulators described i n l i t e r a t u r e the current sources work one against the other and the output i s the di f f e r e n c e of the currents (F i g . C-3). F i g . C-3 Current output In the present r e a l i z a t i o n , each current source i s an independent u n i t to enable accurate c o n t r o l . 1.3.2 Pulse current sources The following ideas are implemented i n the c i r c u i t (see F i g . C-5, C-6). - a constant current source i s amplitude regulated by an op. amp. feed-back loop. The feedback makes the current independent of n o n l i n e a r i t i e s or temperature e f f e c t s i n T3 or T4. the current monitored i s also the stimulating current, the t r a n s i s t o r s T4 and T5, T6 form a cascode c i r c u i t , decoupling the output voltage v a r i a t i o n s completely from the current s e t t i n g at T4. pulse switching: t r a n s i s t o r s T5, T6 act as a l t e r n a t i v e switches, guiding the current e i t h e r to the output or to an i n t e r n a l load. The advantage i s that the heat produced i n the high voltage t r a n s i s t o r i s reduced because of the short"on" cycles, and that the heat produced i n 189 C 6 T5 i s l i m i t e d , most of the power being d i s s i p a t e d i n i t s c o l l e c t o r resistance. A further advantage i s that transient noises i n f l u e n c i n g the analogue c i r c u i t s are markedly reduced because i n the major part of the ±150 V - c i r c u i t ( e s p e c i a l l y i n the supply l i n e s ) the current i s constant and not switched on and o f f with f a s t t r a n s i e n t s . 2. Description of the c i r c u i t 2.1 I s o l a t i o n The complete c i r c u i t i s shown i n F i g . C-5 and C-6. The trans-mission of the two analog s i g n a l s i s r e a l i z e d as described e a r l i e r . I t involves the a m p l i f i e r s 0P1, 0P2 and 0P3 for the p o s i t i v e pulses' ampli-tude and 0P1', 0P2', and 0P3' f o r the negative pulses' amplitude. The d i g i t a l s i g n a l s are transmitted by the integrated couplers 0P9 and 0P9'. 2.2 Analogue current s e t t i n g v The c i r c u i t consists of 0P4, T3, T4 and OP5 i n F i g . C-5 for the p o s i t i v e output current and i t consists of 0P4', T3', and 0P5' i n F i g . C-6 for the negative output current. Discussion i s r e s t r i c t e d to the c i r c u i t f o r p o s i t i v e current. The input d i v i d e r 15 k fi, 10 kfi(0P4) adjusts the voltage ampli-tude. 0P4 c o n t r o l l s the current of T3. For frequencies above 3 kHz feed-back i s achieved over the capacitor of 4.7 nF, for low frequencies feed-back i s achieved over 0P5. The c o l l e c t o r current of T3 controls T4 and thus the output (stimulating) current, which i s the c o l l e c t o r current of T4. The voltage drop caused by t h i s current flowing across the r e s i s -tor of 500 Q i s measured by the d i f f e r e n t i a l input c i r c u i t of 0P5. Because th i s voltage i s at a p o t e n t i a l of about +150 V, input attenuators to 0P5 190 C 7 are required, such that the input to the op. amp. remains below +10 V. The choice of the r e s t i n g p o t e n t i a l at -10 V enables a reduction of the attenuation required. TRI (22 k Q) i s adjusted for a high common mode r e j e c t i o n . The c i r c u i t of 0P5 also acts as a low pass f i l t e r with f = 6 kHz. The calculated input noise voltage for 0P5 i s l e s s than 30 uV rms. Therefore, there should be no problem with noise i n the feedback loop. Zero s e t t i n g Measurement of the current with 0P5 causes a current of 0.25 mA to flow through the r e s i s t o r s 500 0, and 560kft. To provide t h i s current, the input voltage has to be approximately +0.5 V f o r zero output current. The proper balancing of t h i s bias i s made on the primary side of the analog i s o l a t i o n ("offset" i n F i g . C-5). Comparison with other solutions The c i r c u i t described regulates the output current within 0.5% independently of t r a n s i s t o r n o n l i n e a r i t i e s or temperature influences. For an a p p l i c a t i o n where t h i s accuracy i s not required, the c i r c u i t of F i g . C-4 i s proposed. ' I i s exactly regulated by the op. amp. Then, R l I = — ( I - I - I ) - I out R 2 K o B l B2 ; B2 The output current depends on the base currents of TI and T2. The base currents depend on temperature and the operating point of the t r a n s i s t o r . Assuming R^ = 2 R^ and h ^ g >^  50 we get, with the approximation h " \ ' 1 / h f e = V 5 0 " 2 % ° f XC 191 F i g . C-4 Low power current source. 192 C 9 I = 2 (I - I J - 3I„. out v o B l B2 which means an e r r o r of about 6%. This c i r c u i t structure i s used, for example, i n the stimulator of Hardt (1974). 2.3 Switching of the current The accurately preset, continuous current of T4 flows normally over T5 and the 8.0 k ft r e s i s t o r . In the r e s t i n g state the output of the i n v e r t e r II i s low and thus T7 i s on, keeping the base p o t e n t i a l of T5 lower than that of T6. The output of II i s high during an output pulse, turning of T7 which causes the base p o t e n t i a l of T5 to be higher than that of T6 and therefore T6 c a r r i e s the t o t a l current. The c i r c u i t of T5 and T6 corresponds to that of a d i f f e r e n t i a l a m p l i f i e r . The capacitors from the TTL i n v e r t e r s to the bases of T5 and T6 increase the switching trans-i t i o n speed. I f the output voltage exceeds ±140 V during a pulse, which may be due to a high load (skin) resistance or due to a c a p a c i t i v e load, T6 becomes saturated, a f r a c t i o n of the current flows i n t o the base and causes such an increase i n the base p o t e n t i a l , that T5 s t a r t s to conduct and to take over that f r a c t i o n of the current unable to flow to the output. Thus, there i s no l a t c h up. The current i s switched back i n any case to T5 at the end of a pulse. Also, an open output i s allowed. The NAND - c i r c u i t of the two TTL o p t i c a l couplers (OP9, 0P9 1) eliminates the p o s s i b i l i t y that the p o s i t i v e and the negative current outputs are switched on simultaneously. 193 C 2.4 Output p o t e n t i a l monitor The output voltage i s monitored by the two comparators 0P7 and 0P8. I f the voltage i s higher than a maximum voltage Vm or lower than -Vm, i . e . i f the stimulator leaves the range of l i n e a r operation, one of the comparator outputs goes to the low s t a t e , turning on the LED of the o p t i c a l i s o l a t o r 0P10. This TTL feedback s i g n a l may be used to set a warning f l a g i n the computer. 2.5 Charge readjustment A slowly a c t i n g c i r c u i t (corner frequency f = 1 Hz) with a max-imum current of ±0.1 mA restores the output voltage to zero i f the average output voltage deviates from zero due to a s l i g h t asymmetry between the p o s i t i v e and the negative pulses. Assume the voltage i s negative. The output of 0P6 i s then pos-i t i v e and turns T9 on. The feedback c i r c u i t monitors the emitter current causing the c o l l e c t o r current of T9 to be pr o p o r t i o n a l to the input s i g n a l That current, i n turn, c o n t r o l l s the c o l l e c t o r current of T8 which w i l l cause the output voltage to become more p o s i t i v e . 2.6 Power supplies Hybrid DC to DC converters are used to provide the voltage ±15 V +150 V, and -150 V from a s i n g l e 12 V source. The +5 V supply i s derived from the +12 V. The current chopping at 14 kHz creates considerable cur-rent spikes, r e q u i r i n g large capacitors f o r f i l t e r i n g . The converters require a minimum load to prevent damage. The case and a l l the unused pins are connected to ground to increase s h i e l d i n g e f f e c t s . Fuses protect the converters from short c i r c u i t s . F i g . C-7 shows the c i r c u i t . 194 C 11 F i g . C - 5 S t i m u l a t o r , p o s i t i v e p u l s e s F i g . C-6 Stimulator, negative pulses o i—* 12. v/ +• — 9 9 Mm,* r fir TTT" C = 3 _ L _ HO*/ L r r '00/uF/<,oV — j > l?w/7 wo/ -*> 0.«/«r- LI 'h^ isoV -t ^ v -1" icomR + 0\/ \00„q A / — = £ -' i p t /rb i »« 3— "Cp i) I - l i t >l 3 — I1 '»3-l » 5 ) 7fZ 4 " 2 tOOxtf) too r c( ; • nv _ J I V1 e l * t M M * w k.'f* VO (Ti Si-F i g . C-7 Power Supply u> 197 C 14 3. Performance The c i r c u i t was b u i l t as described and i t s a t i s f i e s the required s p e c i f i c a t i o n s . A f t e r proper adjustment of the potentiometers, the over-a l l l i n e a r i t y i n c l u d i n g the i s o l a t i o n (analog input voltage, output cur-rent) i s better than 0.3%. The r e s i s t o r s of 0P5 and 0P5' had to be changed to 1% m e t a l l i c ones f o r better temperature s t a b i l i t y : s l i g h t r esistance changes due to temperature changes had a great influence upon the common mode r e j e c t i o n . With the test c i r c u i t of F i g . C-8, the switching times (in c l u d i n g i s o l a t i o n ) are: r i s e time f a l l time delay r i s e delay f a l l t < 20 ns r — t f <_ 20 ns t , = 110 ns dr t,f = 94 ns df To CUT with probe iOM£l/l^pF F i g . C-8 Test c i r c u i t The c i r c u i t worked s a t i s f a c t o r i l y during the experiments. 198 D 1 Appendix D: FORTRAN PROGRAM As an example, the FORTRAN program follows which c a l c u l a t e s the membrane voltage and the i o n i c current of the complete Frankenhaeuser-Huxley (1964) model. The i n t e g r a t i o n algorithm i s described i n Appendix A, (A2-5). The structure i s : data input k = 0 no t = k • At t > t, . -— f i n a l yes output re-i n t e g r a t i o n for one At k = k + 1 _ l 199 D 2 C » » • • • • N U M E R I C A L S O L U T I O N CF T H E F R A N K E N H A E U S E R - H U X L E Y EQ. FOB C A NODE CF X E N O F U S L A E V I S C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C * * * * * * N T F A 1 D . V S * * * * * * * * C c C F I R S T C P D E R P R E D I C T I O N C c C O K T C B E H 1 9 7 6 F . B t l T I K O F E R C C I N A , I K , I L , I P A R E T H E M E M E R A N E C U R R E N T S I N M I L L I A N P S . /C (1**2 C CUTWARD P O S I T I V C I S I S T H E S T I M U L U S WITH A M P L I T U D E S A ( 1 ) , A ( 2 ) , AND T I M I N G T ( 1 ) TO T ( 6 ) I N ( H A / C M * * 2 ) C T ( 7 ) = F I N A L T I M E , T ( 9 ) = S T E P S I Z E , T ( B ) = S A M P L I N G I N T E B V A L L F C B OUTPUT C A L L T I M E S ARE I N M T L L I S F C , A L L V O L T A G E S A B E I N M I L L I V O L T S C C M - MF .CERANE C A P A C I T Y IN M I C RO F A R A E / C M * * 2 C V M = M E M E R A N E P O T E N T I A L , V M = 0 FOR R E S T I N G S T A T E C P A R A M E T E R S A P S FOR 20 D E G R E E C E L S I U S D I M E N S I O N T I M (2) , V M ( 9 0 0 0 ) , T (9) , A (2 ) R E A L I N A . I K D O U B L E P R E C I S I O N V , V L , G L , C M , T 9 , X , R , I N A 2 , 1 K 2 , D V E , V N D O U B L E F P F C I S I O N 1 (2) , N (2) , H (2) , I S (2) , I L (2) , AN AX D O U B L E P R E C I S I O N A M , E M , A N , B N , A H , E H , D M , D N , D H , 0 V . V M E , I K 1 , I N A 1 r n U D L E P R E C I S I O N C B L E , D E X P , D A D S D O U B L E P R E C I S I O N ? E M P , F , P T . P N A , P K , P P , C N A O , C N A I , C K O , C K I , U , E R , U E , V H D O U B L E P R E C I S I O N P ( 2 ) , I F 1 , I P 2 , A F , B F , D P C C C O N S T A N T S : ( S T A N D A R D DATA) C V O L T A G E S : V = E - ER ; E R = - 7 0 M I L L I V O L T V = 0 . DO T E M P - 2 9 3 . 1 6 D 0 F = 9 6 . U 9 1 D 0 P= 0 . 0 0 8 3 1 U 3 2 D 0 P N A = 0 . 0 0 8 D 0 P P = 0 . 0 0 0 5 « D 0 • P K = 0 . 0 0 1 2 D O C N A O = 1 1 « . 5 D 0 C N A I = 1 3 . 7 I I D 0 CKC= 2 . 5 D 0 C K I = 1 2 0 . 0 D 0 E R = - 7 0 . O C O G L = 0 . 0 3 0 3 D 0 C M = 2 . D 0 C N O T E : TC A D J U S T V I N M I L L I V O L T S CM B E C O M E S C M * 0 . 0 0 1 C H = C M * 0 . 0 0 1 D 0 C A M A X = 0 . D 0 T1AX=0. C C P A R A M E T E R OF S T I M U L U S : R E A D ( 5 , 5 ) NO 5 FORMAT ( 1 5 ) R E A D ( 5 , 1 0 ) A ( 1 ) DO 2 0 1 = 1 , 9 2 0 R E A D ( 5 , 10) T ( I ) 1 0 FORMAT ( F 1 0 . 6) A ( 2 ) - A (1) * (T ( 2 ) - T (1) ) / (T ( U ) - T ( 3 ) ) « ( - 1 . ) 2 9 F O R M A T ( ' 0 ' , / / , 5 X , ' M E T H O D NTRA 1 D . V ' , 1 O X , • D A T A S E T NR = ' , I 5 , / / ) T9= D PL E (T (9) ) C K K - S T E P N R . : K = DUMMY • F C B 1 ( H I S T O R I C ) K=1 KK = 1 JR = 0 I D = 5 2 T A = 0 . KK K= 0 C H A = 0 . 200 D 3 C A L L T I M E ( O ) C C C A L C U L A T I O N S OF T H E I N I T I A L V A L U E S H (1) , N (1 ) , H (1) , ? { 1) C A L P H A (1 = A M , B E T A B = E M , E T C . A!1 = - 0 . 3 6 0 0 * 2 2 . D 0 / ( 1 . C 0 - C E X P ( 2 2 . 0 0 / 3 . DO) ) e« = 5 . 2 D C / ( 1 . E 0 - C E X P ( - 1 3 . D 0 / 2 0 . D 0 ) ) Al l = - 1. TO / (1. E O - D E X P (1 0 . E 0 / 6 . DO)) R H = » . 5 D O / ( 1 . C 0 * D E X P ( U . 5 C 0 ) ) AN = - 0 . 7 E 0 / ( 1 . E 0 - D E X P ( 3 . 5 E 0 ) ) PN = 0 . 5 E O / ( 1 . E O - D E X P ( - 1 . E O ) ) AP = - 0 . 2 U D 0 / ( 1 . D O - D E X P ( 1 . EO) ) BP = - 0 . 0 ° E 0 * 2 5 . D O / ( 1 . E O - D E X P ( 1 . 2 5 E 0 ) ) « ( 1 ) = A M / (AM*E,1) N (1) = A N / ( A S » P N ) P ( 1) = A F/ ( A P + BP) H (1) = A H / (AH + BH) W R I T E R , 9 1 ) 1(1) , N ( 1 ) , P ( 1 ) . H ( 1 ) 9 1 FORMAT ( ' 0 ' , « ( 5 X , E 1 2 . 5) ) C C R E S T I N G S T A T E C O N D I T I O N S R T = R * T E M P U - ( 0 . 0 0 1 D 0 * F * (V • ER) ) / P T U E = D E X P ( U ) F= ( F * U * ( C N A O - C N A I * U E ) ) / ( 1 . DO-OE) I N A 1 - P N A * M (1) * * (1) * H ( 1 ) *R I P 1 = F F * F (1) * P (1) « 3 I K 1 = ( P K * K ( 1 ) * N ( 1 ) * F * U * ( C K O - C r i * C F ) ) / ( 1 . DO-OE) V L = ( I N A 1 + I P 1 + I K 1 ) / G L C I T E R A T I C N S L O O P S T A R T S 100 C O N T I N U E C T 1 1 (K) = T (9 ) * F L O AT ( K K - 1 ) TP ( T I N (K) . G T . T (7) ) GOTO 2 5 0 C C D E F I N I T I O N OF THE S T I M U L U S T I = Tin (K) S=0. I F ( T I N (K) . G " . T (6 ) ) GOTO 1 0 9 10tt I F ( (T (5) - T I ) . G T . 1 . O E - U ) GOTO 1 0 5 T I = T I - T ( 5 ) GOTO 10U 1 0 5 I F ( T I . G E . T (1) ) S = A ( 1 ) I F ( T I . G E . 1 (2) ) S = 0 . I F ( T I . G E . T (3) ) S = A ( 2 ) I F ( T I . G E . T (U) ) S = 0. 1 0 9 I S ( 1 ) = 0 B L E (S) C H A = C H A » T (9) * S C C C A L C U L A T I O N CF THE MEMBRANE C U R R E N T S AND CF V L U - ( 0 . 0 0 1 D 0 * F * ( V + E R ) ) / R T UE = D F X P (U) R= ( F * U * ( C N A O - C * ! A I * U E ) ) / ( 1 . D 0 - U E ) I N A 1 = PNA*f1 (1) * M (1) *H(1) *R I P 1 = P P * P (1) * P (1) *R I K 1 = ( P K * N ( 1 ) * ! ! ( 1 ) * F * U * ( C K O - C K I * ! I E ) ) / (1 . D 0 - U 2 ) I L (K ) =GL * ( V - V L ) C C *** A L G C F I T H M FOP I N T E G R A T I O N S C DV = D E F I V A T . C F VH K 1 = K * 1 D V = ( I S ( 1 ) - I K 1 - I N A 1 - I P 1 - . I L ( K ) ) / C I 1 C V K E - E S T I M A T I O N OF NEW VM V E = T 9 * n V * V 1 9 0 C O N T I N U E C DER TV A T . OF M , N , H AT T I M E K : D M - A K * (1 . E O - M (K) ) - P fl * M ( K) DN = A N « ( 1 . D 0 - N (K) ) - E S * N ( K ) DH= AH* ( 1 . E O - H (K) ) - PH * H (K) D P - A F * ( 1 . E 0 - P ( K ) ) - P P * P ( K ) v 201 D 4 C MEW A L P H A S AND E ETA S WITH E S T I M A T E V.1E : © C I N C L U D I N G C C N T B C L T C - A V C I D D I V I S I O N BY ZERO VH = V . » E - 2 2 . DO I F ( D A B S (VH) . G T . 0 . 0 0 1 D 0 ) GOTO 1 1 A M = 1 . 0 8 D 0 GOTO 15 1 1 AM = 0 . 3 f i D O * ( V M f - 2 2 . D O ) / ( 1 . D 0 - D E X P ( ( - V I 1 E * 2 2 . D O ) / 3 . D 0 ) ) 1 5 V I I - V r E - 1 3 . DO I F ( D A B S ( V I I ) . G T . 0 . 0 0 1 D O ) GOTO 2 1 BM = 8 . D O GOTO 2 5 2 1 0.1 = 0 . U D O « ( * 1 3 . D O - V M E ) / ( 1 . D O - D E X P ( ( V M E - 1 3 . DO) / 2 0 . DO) ) 2 5 V H = V M F + 1 0 . D 0 I F ( D A B S (VH) . G T . 0 . 0 0 1 D O ) GOTO 3 1 A H = 0 . 6 D 0 GOTO 3 5 31 AH-= 0 . 1 D 0 * ( - 1 0 . D O - V M E ) / ( I . D O - D E X P ( (VM E * 1 0 . DO) / 6 . DO) ) 35 EH = U . 5 D 0 / ( 1 . D 0 * D E X P ( ( * U 5 . D 0 - V r . E ) / 1 0 . D 0 ) ) V H = V M E - 3 5 . D 0 I F ( D A B S (VH) . G T . 0 . 0 0 1 DO) GOTO HO A N = 0 . 2 D 0 GOTO U5 UO A N = 0 . 0 2 C 0 * ( V M E - 3 5 . D O ) / ( I . D O - D E X P ( ( * 3 5 . D O - V M E ) / 1 0 . D O ) ) ! »5 Vl l = v r E - 1 0 . DO I F ( D A B S ( V H ) . G T . 0 . 0 0 1 DO) GOTO 5 0 B N = 0 . 5 D 0 GOTO 5 5 5 0 EN = 0 . 0 5 D O « ( » 1 0 . D O - V M E ) / ( 1 . D O - D E X P ( ( V M E - 1 0 . D 0 J / 1 0 . D 0 ) ) 5 5 V H = V » E - « 0 . D O I F ( D A B S ( V H ) . G T . 0 . C 0 1 D O ) GOTO 6 0 A P = 0 . 0 6 D 0 GOTO 6 5 6 0 AP = 0 . 0 0 6 B 0 * ( V M S - U O . DO) / ( 1 . D O - D E X P ( ( • U O . D O - V M E ) / 1 0 . DO) ) 6 5 V l ! = V M E * 2 5 . D 0 I F ( D A B S (VII) . G T . 0 . 0 0 1 DO) GOTO 7 0 B P = 1 . 8 D 0 GOTO 7 5 7 0 B P = 0 . 0 9 B 0 * ( - 2 5 . D O - V M E ) / ( 1 . D O - D E X P ( ( V M E * 2 5 . D 0 ) / 2 0 . D 0 ) ) 7 5 C O N T I N U E v C C NEW M , N , H AND P : . F = 1 . D 0 * 0 . 5 D 0 * ( A M * E K ) * T 9 M ( K 1 ) = ( M ( K ) » 0 . 5 D 0 * T 9 * (AM + BM) )/B R= 1 . E 0 * - 0 . 5 D 0 * ( A N • B N ) *T9 N ( K 1 ) = ( N ( K ) » 0 . 5 D 0 » T 9 * ( A N * D N ) ) / R R= 1 . DO+ 0 . 5 D 0 * ( A I H P H ) *T9 H ( K 1 ) = ( R ( K ) » 0 . 5 D 0 * T 9 * ( A H » D H ) ) / 3 P= 1 . D O * C . 5 D 0 * T 9 * ( A P + E P ) P ( K 1 ) = ( F ( K ) • 0 . 5 D 0 » T 9 * ( A P * D P ) ) / f i C C E S T I M A T E D C U R R E N T S : V- ( 0 . 0 0 1 D 0 * ? * (VME + EP) ) / R T UF= D E X P ( U ) P = ( F * U * ( C N A O - C N A I * U E ) ) / ( 1 . D 0 - U E ) I N A 2 = P N A * " ( K 1 ) « M ( K 1 ) * H ( K 1 ) * R I P 2 = P P « F ( K 1 ) * P ( K 1 ) *R I K 2 = ( P K * N ( K 1 ) * N ( K 1 ) * F * U * ( C K O - C K I * U E ) ) / ( 1 . D 0 - U E ) I L ( K 1 ) = G1« ( V M E - V L ) C NEW E S T I M A T E D D E F I V A T . C F VK = DV E D V E = ( I S < 1 ) - I K 2 - I N A 2 - I P 2 - I I ( K 1 ) ) /CM C NEW V C L T A G E : V N = V * 0 . 5 D 0 * T 9 * ( D V * D V E ) C E R R C R T E S T X = D A E S ( V N - V M E ) I F ( X . L T . 1 . 0 D - U ) GOTO 2 0 0 VME = VN J R = J R * 1 GOTO 1 9 0 f 202 D 5 200 CONTINUE C KK= KK•1 H(K)=H(K1) N ( K) -N (K1) H (K) =n (K 1) P(K)=P(K1) C IP (AMAX.GT. VN) GOTO 210 AMAX=VN TBAX = T I « (K) ® 210 CONTINUE C C STORE SAMPLES EVEFY T(8) SECOND IF ( (TA-TIM (1) ) . GT. 1. OE-1) GOTO 350 IF (APS (S) . LT. 1. OE-U) SS = 0. IF (S.GT.1.OE-U) SS=300. IF<S.LT . -1 .0E -U) SS = - 3 0 0 . KKK =KKK+1 TA = TA *T (8) IF(KKK.GT.9000) KKK=900 V?! (KKK) =SNGL (V) * S S 350 CONTINUE V=VN GOTO 100 C 250 CONTINUE : Ou+pu* of clafa-203 E 1 Appendix E: ON NONLINEARITIES IN THE FH MODEL AND ON PERFORMANCE OF SIMPLIFIED MODELS 1. Introduction 2. The V, m system; (h, n, p = constant) 3. Membrane r e c t i f i c a t i o n and l i n e a r approximations 4. Li n e a r i z e d V, m system 5. Line a r i z e d V, m, h system 6. Further s i m p l i f i c a t i o n s 1. Introduction The main reason f o r s i m p l i f y i n g the Frankenhaeuser-Huxley (FH) model i s to allow q u a n t i t a t i v e predictions about how a stimulus w i l l act upon the space clamped membrane, and how i t t r i g g e r s an act i o n p o t e n t i a l . (A.P.)« The new model w i l l a lso be used to explain the e f f e c t of tempor-a l summation. Thus, the s i m p l i f i e d model has to reproduce the passive and the act i v e subthreshold responses as we l l as the onset of an A.P. up to approximately +60 mV. However, i t w i l l not reproduce the l a t e r phases of the A.P. Such a model was not found i n the l i t e r a t u r e . In the approximate FH model f o r the i o n i c currents to be des-cribed here, the two functions f ^ and f^ are introduced, such that: I ± = f 1 ± (V,t) * f 2 . (V) ; i = Na, K, p where f ^ (V,t) accounts f o r the dynamics of the i o n i c currents and where f^^(V) represents the nonlinear I-V r e l a t i o n , which only depends on the voltage. With the absolute p o t e n t i a l i n v o l t s , E = V + E = (V - 0.07) Vo l t s , the functions a r e : 204 I = f • f Na INa 2Na f l N a ^ = m ' h f m p *L_ [ CNa ]o - [ C N a ] i *** (EF/RT) 2 N a N a R T 1 - exp (EF/RT) I = f • f P IP 2p f l p (V.t) = P2 f m — EF^ [ CNa ]o - [ C N a ] . exp (EF/RT) 2p ( } p ' RT 1 - exp (EF/RT) = f • P /P 2Na p' Na I = f • f K IK 2K f l K ( V , t ) = ^ 2 [C ] - [C 1 exp (EF/RT) f ( v ) B p. . M _ . -A-2 K _ i 2 K 1 - exp (EF/RT) The constants and units are l i s t e d i n Chapter 8-2 and 9-3. F i r s t , the influence of s i m p l i f y i n g the f w i l l be i n v e s t i g a -ted, taking f o r h, n and p the constant r e s t i n g state value, such that the FH model i s reduced to the two v a r i a b l e V, m system. Then i n sec t i o n . E-3 the e f f e c t s of l i n e a r i z i n g the (membrane r e c t i f i c a t i o n ) are examined. In s e c t i o n E-4, a l i n e a r i z e d Vm model i s proposed and f i n a l l y i n section E-5 the influence of the parameter h i s investigated i n the l i n e a r V, m, h system. 2. The V, m system; (h, n, p) = constant FitzHugh (1960) reduced the HH model to the two v a r i a b l e system V, m, taking f o r n and h the fi x e d value of the r e s t i n g state. Since n and h change i n the complete model at a rate at l e a s t ten times slower than V or m (see F i g . 12-1), the error introduced by keeping these i n a c t i -205 vation v a r i a b l e s constant remains small during the onset of an A.P. For the study of the e f f e c t of short current pulses on t r i g -gering an A.P., the reduction of the Frankenhaeuser-Huxley model to a V, m model with n, h, and p taken constant can also be investigated. The e r r o r before and during the r i s i n g phase of the A.P. i s r e l a t i v e l y small because: (i) the n o n s p e c i f i c i o n i c current r e l a t e d to p (^p) changes slowly ( T (0) > 1 ms, F i g . 12-2), and the peak current I i n the com-plete model i s small; ( i i ) the potassium current I , r e l a t e d to n, changes slowly ( x (0) % K n 1 ms, F i g . 12-2); ( i i i ) the influence of h upon the sodium current i s small because h changes slowly ( T ^ (0) £ 5 ms, F i g . 12-2) and h i s l e s s e f f e c t i v e 2 2 than m during the A.P.'s r i s i n g phase i n L, = m h • f„„ . ° r Na 2Na In the reduced V, m system the i o n i c currents are: I N a - m2 (V,t) • h o • f ^ (V) h " % 2 ' f2K < V ) \ " Po' ' f2p <V> with h (V,t) = h (0,0) o n Q (V,t) = n (0,0) constant p (V,t) = p (0,0) O Only the a c t i v a t i n g m remains i n the o r i g i n a l form. The i n a c t i v a t i n g components n and h, which tend to bring the p o t e n t i a l back to the r e s t i n g state are blocked. I can be neglected. Because of the missing i n a c t i -v ation, the A.P. r i s e s e a r l i e r than i n the o r i g i n a l model, and i t reaches 206 fl: 1.625 •T= 0.000 — i — 0.6 — I — 0.9 1.1 0.0 o'.i n _ — i — 0.2 - 1 0.3 1 0.4 TIME — n 1 — B.i 0.6 I M J L U S E C ) — l — 0.7 F i g . E-1 Membrane Voltage computed with two models. Curve A: V, m system; Curve B: complete FH model. Asymmetric stimulus: t P+ 50 ys, t = 200 ys, t = 0, T = 250 ys, A+ = P- 2 1 1.625 mA/cm , A- chosen to compensate charge. Standard data. 207 i 1 1 1 1 1 1 1 1 1 0.0 0.2 0.4 O.E C.6 1.0 1.? 1.4 l.E l.B 2.0 TIME IM1LLJSEC) L ^ T M F i g . E-2 Membrane voltage computed with two models. As F i g . 15-1 except for stimulus: symmetric, b i p o l a r with t = 50 ys, t^ = 12.5 ys, T = 250 ys, A = -2.175 mA/cm2. 2 0 8 , 1— 0.0 0.2 0.4 i — 0.6 — I 1 1— 0.6 1.0 1.2 T I M E ( I U L L I S E C ) — I — 1.6 —1 1 i.a • 2.D F i g . E-3 Membrane Voltage computed with two models. ^ As F i g . E-2 except amplitude A = 1.544 mA/cm . 209 a second stable state near the sodium p o t e n t i a l V = 123.6 mV. r Na In the figures E-1 to E-3 the A.P. of the complete FH model i s compared with the reduced V, m system. The stimulating current i s chosen to represent conditions where the V, m model i s most l i k e l y to show deviations: the complete stimulus i s s l i g h t l y suprathreshold and long l a s t i n g . As expected, the error of the V, m system i s r e l a t i v e l y small for subthreshold responses - the V, m system i s a u s e f u l s i m p l i f i c a t i o n . A further discussion of the s i m i l a r V, m system f o r the HH model can be found i n FitzHugh (1960) and c a l c u l a t i o n s how m and h change i n response to a (50 + 50 us) b i p o l a r pulse are published i n Bromm and Frankenhaeuser (1968). 3. Membrane r e c t i f i c a t i o n and l i n e a r approximations In t h i s s ection the influence of l i n e a r i z i n g the current-voltage (I-V) r e l a t i o n s i s examined. For the i o n i c currents I„, , I T r, and I , the Na' K p I-V r e l a t i o n s are nonlinear and they have r e c t i f y i n g properties due to the differences of the i o n i c concentrations i n s i d e and outside the membrane. The I-V r e l a t i o n s are derived from the constant f i e l d membrane (Goldman eq.) and are described by the functions f„„ , f„T., f„ as defined i n the ^ J 2Na 2K 2p 2 previous section. For example, 1 ^ i s : I ^ & = m h • f£Na • The function f„ = f o x , • —^— behaves l i k e f_., and w i l l not be treated 2p 2Na — 2Na Na separately. In F i g . E-4 the function f o r sodium, f2Na ^ s s ^ o w n a n c * i n F i g . E-5 that f o r potassium, f 0„. ZK . 210 E Choice of l i n e a r forms for the I-V r e l a t i o n s f o j : In the Hodgkin-Huxley model (squid) the i o n i c currents are: I = m3 h • T • (V - V M ) Na BNa Na h = n 4 ' (V - V K) where the i o n i c p o t e n t i a l V„ and V„ are defined by the Nernst equation Na K . (see Chapter 8-2). The same l i n e a r form i s now adopted for the FH model. The l i n e a r i z e d functions w i l l be marked by a ': f ' = 2 ' (V - V ') 2Na % a v Na ' V = *K ( V " where g„ ', g ' , V„ V ' are constants. The l i n e a r functions must be TSIa K Na K equal to the o r i g i n a l ones at the r e s t i n g p o t e n t i a l i n order to keep V = 0 as a stable point. This means that at V = 0 the sum of the i o n i c currents has to be zero. Also, the l i n e a r i z a t i o n s should f i t ( i n a l e a s t squares sense) *2Na a n c * ^2K ^ e s t ^ n ^ & v i c i n i t y of the r e s t i n g p o t e n t i a l i n order to study threshold phenomena. These two conditions determine completely g„T ' and Vrr ' r e s p e c t i v e l y g ' and V ' . Note that r J &Na Na c J eK K these two constants are determined to s a t i s f y two mathematical conditions. They cannot be r e l a t e d d i r e c t l y to membrane properties. In contrast, i n the HH model for squid, the conductances g^ were determined experimentally. Least square curve f i t For s i m p l i c i t y the index Na and K w i l l be omitted where possible i n t h i s section because the algebra for both cases i s the same. We have: f2 (V): o r i g i n a l function of the FH model f2 ' = §' ( v _ v ± ' ) : l i n e a r i z e d f u n c t i o n , i = Na, K. 211 The condition f o r the r e s t i n g state f (0) = f ' (0) y i e l d s : thus: f 2 (0) = -g' V.' f 9 (0) v.' = 1 , 1 g VNa' = " f2Na ( 0 ) / % a ' V ' " f2K ( 0 ) V Thus, f 2 * can be wr i t t e n : = g' V + f 2 (0). The minimum mean square error E(g) between f 2 and f 2 ' i n the voltage range -30 mV to +60 mV ( i . e . around threshold) w i l l determine g. +60 +60 E(g) = / ( f 9 - V ) 2 dV = / (f (V) - g' V - f„(0)) 2 dV "2 - 2 ' J K 2 -30 -30 The minimum i s at dE d g = 0 = -2 / [ f 2 (V) - f 2 (0) - g' V] V dV solved f o r g', which i s a constant: +60 / V (f (V) - f (0)) dV -30 Z g = +60 / V dV -30 f ° r % a ' : % a ' = g ' a n d f2 = f2Na fo r g K ' : g K ' = g' and f 2 = f 2 K There i s no reason to extend the f i t above +60 mV, because the model i s confined to voltages i n the threshold and subthreshold region. Two programs INA* and IK* were wr i t t e n to c a l c u l a t e the value of g^ a' and V ( 8 K ' A N D A 1 S ° ' t h e f u n c t i o n s f 2 N a a n d f2Na' ( f2K a n d ^ K ^ were calculated i n steps of 1 mV between -30 mV and + 130 mV. In the 212 E 10 program, g' i s cal c u l a t e d by , i _ 91 I <f 2 (k) - f 2 (31)) (k - 31) k=l g = 91 I (k - 31) 2 k=l where the i n t e g r a l s are replaced by sums and where (k - 31) stands f o r V. As shown i n Section E - l the function f„ for the non s p e c i f i c 2p current I only d i f f e r s by the fa c t o r P /PXT from f„.7 : f_ = f_„ • P /~P . p J J p Na 2Na 2p 2Na p Na Therefore, with g ' = g.T ' • P /PXT ' °p &Na p Na f « = g ' (V - V ') 2p sp v Na ' Two point l i n e a r i z a t i o n For comparison, another l i n e a r function was also determined: f " = E " (V - V ) 2Na feNa K Na' f2K" " %" <V " V I I Besides the r e s t i n g state condition f 2 (0) = f 2 (0), i t s a t i s f i e s also f2Na" (VNa> = f2Na ( VNa ) r e s p e c t i v e l y f 2 R " (V R) = ( V R ) , which means that the l i n e a r i z e d function contains the i o n i c p o t e n t i a l s P e c : L -f i e d by the Nernst equation. Since two points have been selected on the s t r a i g h t l i n e l i n e a r i z a t i o n , g " i s thus uniquely determined. At the r e s t i n g state (V = 0): f2Na <°> = f2Na" <°> = % a " <° " VNa> I h u 8 ! % a " = " f2Na ( 0 ) / V N a g K " = - f 2 K ( o ) / v K e " = g " • p /P s p BNa p' Na 213 E 11 Results: The calculated values, based on standard data i n the o r i g i n a l model are l i s t e d i n Table E - l . current l e a s t square f i t V = g' (v - v.') two point f i t f" = g " ( V - V.) sodium potassium nonspecific leakage V,T ' = 91.165 mV Na 2 g.T ' = 2.845 mho/cm °Na V ' = -13.618 mV K g ' = 0.1259 mho/cm2 K V„ ' = 91.165 mV Na 2 g 1 = 0.1920 mho/cm V = +0.C V„ = +123.56 mV Na g X T " = 2.099 mho/cm2 °Na Vv = -27.79 mV g " = 0.0617 mho/cm2 K V X T = +123.56 mV Na g p " = 0.1417 mho/cm2 )251 mV g = 0.0303 mho/cm2 Table E - l Lin e a r i z e d conductances F i g . E-4 and E-5 show the three functions p l o t t e d versus voltage. Conclusion A l i n e a r i z a t i o n of the current-voltage r e l a t i o n around the re s t i n g p o t e n t i a l has been investigated. For sodium, the l i n e a r i z a t i o n with l e a s t square e r r o r , ^Na' ^ t s t* i e o r l g i n a l function f 2Na w e ^ » fc^e l a r g e s t error i n [-30, +60 mV] being 3% at -30 mV. For potassium, the two point approximation t " f i t s the o r i g i n a l function fnvr b e t t e r f o r 2K negative voltages, but for p o s i t i v e voltages the error (f„„ - f ' ' ) / f o i r d^K ZK 2K i s considerable, f o r example at +60 mV: 50%. The l e a s t square function f 2 j , ' also has considerable deviations from f 2^> D U t s t i l l i t i s preferred because of i t s better f i t for p o s i t i v e voltages. 214 E 12 F i g . E-4 I-V r e l a t i o n f o r sodium. INA* i s negative and -si 2 corre ponds to: INA* = I„ /m2 h. The un i t i s Na mA/cm f 2Na-^2Na': l l n e a r i z e d with l e a s t square f i t : f2Na' = 2 ' 8 ( V " 9 1 ' 2 ) ^2Na": i i n e a r i z e d with two point f i t : ^2Na: o r i g i n a l function of FH model f2Na" = 2 - 1 ( V " 1 2 3 ' 5 ) 215 F i g . E-5 I-V r e l a t i o n for potassium.IK* i s p o s i t i v e and corresponds to IK* = I /n 2. The unit i s mA/cm2 IN. f ^ K : o r i g i n a l function of FH model f ^ K ' : l i n e a r i z e d with l e a s t square f i t : f 2 K ' = 0.12 (V - 13.6) f ^ j , " : l i n e a r i z e d with two point f i t : f o v " = 0.06 (V - 27.8). 2 1 6 E 14 If f„ ' > f 0 , this means that the corresponding outward flowing 2K 2. potassium current i s larger than i n the o r i g i n a l model, which tends to bring V more ra p i d l y back to V = 0. In the previous s e c t i o n we have seen that taking n = n = const, r e s u l t s i n a I which i s too small, leading o K to an early A.P. Thus, with f ^ ' > f 2 K on [0, 40 mV] th i s e f f e c t i s s l i g h t l y reduced. The complete, l i n e a r i z e d model f o r the nodal membrane and f o r pot e n t i a l s below +60xmV i s : V = £ (I - I ' - I ' - I ' - I j C s Na K p 1 with I g : stimulating current 2 V - m h • % a ' ( V " W V = n* ' Y <v - Y> I ' = p 2 • g ' (V - V. ') p r °p Na I-L = g± (V - V 1) with g ', V ', g„', V ', g 1 as defined above and the other v a r i a b l e s JNa Na K K P as i n the o r i g i n a l FH model. For t h i s model no sample runs with diffe r e n t , s t i m u l i were made but only at the end of the next section, where i t i s combined with the V, m system. 4. Linea r i z e d V, m model With the l i n e a r i z a t i o n of the FH model and with the reduction to the V, m model, a very simple c i r c u i t can be derived f o r the membrane. If f o r h, n and p the constant values at the r e s t i n g p o t e n t i a l V = 0 are used, h Q = h (0), n Q = n (0), P Q = p (0), the currents 1^, Ip, and 1^ can be taken together to form a new current 1^, 217 E 15 The model i s : CV = I - I - I - I - I, = I - I - I. s Na K p 1 s Na 2 with I 0 = I T r + I + I, 2 K p 1 = n o 2 ' %' ( V " Y > + P o 2 * V ( V " V > + H ( V " V " V ( % 2 % ' + P o 2 8p' + 81> " ( n o 2 V K ' + V V 2 2 2 set: g = n g ' + P g ' + g n - const = 30.395 m mho/cm °o o °K o ° p 1 2 2 Tr o °K K o s p Na a l 1 ^ . m K Q 1 V = c = const. = -0.001593 mV o S o The values i n d i c a t e d are based on standard data for the FH model. Note that g Q % g^, i . e . the high leakage conductance, which i s c h a r a c t e r i s t i c for the nodal membrane, i s dominant. The current 1^ i s : i 2 = g c ( V - V Q ) . With g N a ( V , t ) = m2 h o g N a ' , the complete model, shown i n F i g . E-6, becomes: < v - 1 [ i s - ((v-v N a-) g N a ( v , t ) + i 2 ) ] m = a ( l - m ) - 3 * m m m The only remaining n o n l i n e a r i t y i s g ^ a ( V , t ) . This means that the a c t i v e subthreshold responses and the i n i t i a t i o n of an A.P. i s e x c l u s i v e l y con-t r o l l e d by the sodium current. Limits of the model This approximation i s valuable only for a c t i v e subthreshold responses and for the onset of an A.P. 218 E rc = cv So V c 6 Fig. E-6 Linear V, m model Performance of the model In the figures E-7 to E-9 three comparisons with the original model are shown. Considering the extreme simplifications in the linear model, the agreement is reasonably good. Of course differences increase for long-lasting stimuli and for only slightly suprathreshold stimuli, where the complete model yields long-latency periods before the potential increa-ses towards the A.P. » The model is simple enough without being too inaccurate to enable qualitative predictions on how the membrane potential will react upon a stimulus. However, controls and verifications should always be made with the complete model. 5. Linearized V, m, h model To test the influence of the variable h, the response of the V, m, h model was calculated. This model corresponds exactly to linearized V, m model with the only difference being that the variable h in the ex-219 E 17 F i g . E-7 A) l i n e a r i z e d V, m model, B) o r i g i n a l FH model. Stimulus and data as i n F i g . E-1. 220 E 18 F i g . E-8 A) l i n e a r i z e d V, m model, B) o r i g i n a l FH model. Stimulus and data as i n F i g . E-2. 221 E 19 F i g . E-9 A) l i n e a r i z e d V, m model, B) o r i g i n a l FH model. Stimulus and data as in' F i g . E-3. 222 pression for the sodium current I = m2 h g.T ' (V - V ') Na °Na Na i s not kept constant but corresponds to the h of the o r i g i n a l model. Again, sodium i s the only a c t i v e current. The c i r c u i t representation i s the same as that of F i g . E-6. F i g . E-10 shows the response to that stimulus where the biggest deviations occured with the other models. Despite the generation of a p o t e n t i a l which returns a f t e r a peak to the •resting state, also t h i s model i s only correct for the onset of the a c t i o n p o t e n t i a l . In the l i n e a r i z e d V, m model I was too large. In the l i n e a r -ized V, m,h model, the changes i n h tend to decrease I which r e s u l t s i n a decrease of the p o t e n t i a l deviations with respect to the complete model. The peak amplitude i n the V, m, h model i s only 88.5 mV because the l i n -e a r i z a t i o n y i e l d s f o r V„ ' the low value of: V„ ' = 91.2 mV. J Na Na 6. Further s i m p l i f i c a t i o n s To enable a better q u a l i t a t i v e understanding, several attempts have been made to reduce the l i n e a r two v a r i a b l e V, m model to a s i n g l e v a r i a b l e model. However, these attempts f a i l e d because the time constants of V and m are of the same order, which makes i t impossible to decouple the two v a r i a b l e s : a change i n one v a r i a b l e occuring i n a given time i n -t e r v a l causes the other v a r i a b l e to change i n the same time i n t e r v a l . One attempt was to divide the equivalent e l e c t r i c a l c i r c u i t ( F i g . E-6) i n t o a passive RC c i r c u i t and i n t o an a c t i v e component, rep-resenting the v a r i a b l e sodium current. This separation i s possible but i t y i e l d s no new aspects. Another attempt was to decouple V and m by assuming one of them to be constant during a given time i n t e r v a l . In t h i s case the d i f f e r e n t i a l 223 F i g . E-10 A) l i n e a r i z e d V, m, h model, B) o r i g i n a l FH model. Stimulus and data as i n F i g . E - 9 and E-3. 224 E equation for the remaining v a r i a b l e can be solved a n a l y t i c a l l y . For the time i n t e r v a l s of i n t e r e s t , the error introduced by taking one v a r i a b l e as constant i s so severe that i t prevents any conclusions from being drawn. One i n t e r e s t i n g aspect i s the e f f e c t of the negative pulse's p o s i t i o n within the period T upon the average voltage V*. This i s shown with the response of a passive RC c i r c u i t to a b i p o l a r stimulus c o n s i s t i n g of a p o s i t i v e charge i n j e c t i o n at the beginning and of a negative charge i n j e c t i o n a f t e r 50 ys. Both charges are equal. Repetition frequency i s 4 kHz. The resistance has the value of the r e s t i n g state and i t remains constant, i . e . m =m . o The average voltage T V* = -| / V(t) dt o i s not zero, despite the f a c t that the average charge provided by the stimulus i s zero. Note, that for f = 4 kHz there i s v i r t u a l l y no transient component i n the c i r c u i t ' s response. The p o s i t i v e V* would induce an i n -crease i n m, i . e . an a c t i v a t i o n of the sodium system, leading eventually to an action p o t e n t i a l . ^However, the i n t e r a c t i o n s are so complex that no general statements can be made. 225 Fig. E - l l Response of the passive RC circuit to a bipolar dirac stimulus. 226 F 1 Appendix F: THRESHOLD AND STIMULUS TIMING IN THE LINEAR V, m MODEL 1. Threshold 2. Bipolar stimulus • amplitude and timing 3. Discussion 1. Threshold With the V, m model i t i s possible to give an example for the rather abstract threshold d e f i n i t i o n of Chapter 13. The complete l i n e a r V, m model i s used as described i n Appendix E-4. The membrane system i s completely defined by the two v a r i a b l e s V and m. Threshold i s now defined as the l i n e , separating a l l V, m com-binations which w i l l lead to an action p o t e n t i a l i f there i s no further change produced by an external stimulus, from a l l the other V, m combin-ations which w i l l not lead to an a c t i o n p o t e n t i a l . Therefore, a s p e c i f i c approach becomes f e a s i b l e . I t i s possible to f i t an a n a l y t i c a l function to the threshold curve. Once t h i s threshold function m = f(V) i s known, i t i s s u f f i c i e n t to f i n d out on which side of the threshold l i n e the state (V, m) at the end of a stimulus l i e s , to determine whether or not an action p o t e n t i a l w i l l occur. I t i s not necessary to c a l c u l a t e the a c t i o n poten-t i a l . The extension of t h i s approach to the complete, f i v e v a r i a b l e model becomes too complex to be reasonable. R e s t r i c t i o n s : The assumptions are that p r i o r to a stimulus the membrane i s at the r e s t i n g s t a t e , that an action p o t e n t i a l w i l l develop ( i f ever) a f t e r the end of the stimulus, and that no second stimulus follows. Using the search algorithm described i n Appendix B, the threshold values of m were determined. These mark the boundary between those m leading to an action p o t e n t i a l and those which won't f o r a given i n i t i a l 227 F 2 voltage displacement. The values are l i s t e d i n Table F - l , rows "V" and "M". In the figures F-'l and F-2 the threshold condition i s p l o t t e d i n the V, m plane. Curve f i t : The threshold function mT = f(V) i s empirical. For easier use of t h i s r e l a t i o n i n subsequent c a l c u l a t i o n s a l i n e a r , polynomial function was determined which f i t s the o r i g i n a l data points. The c r i t e r i o n f o r the f i t was minimum square error . (U.B.C. computing Centre L i b r a r y Rou-tine LQF). The function i s : m = P Q + P 1 V + p 2 V 2 + p 3 V 3 + p 4 V 4 Depending on the order, the parameters are: P o P l P2 P 3 P 4 sum of square errors 8.5430-10"2 8.2925-10"2 8.3238-10"2 -1.0421-10" 3 -1.1217-10" 3 -1.2118-10" 3 -5.4401-10" 5 -4.8988-10" 5 -1.1761-10 - 5 0 -1.6502-10" 6 -5.1246-10" 6 0 0 2.8420-10"8 1.98-10"4 1.04"10~5 4.51-10"6 The fourth order f i t i s used. The values of m ca l c u l a t e d by t h i s formula are l i s t e d i n Table F - l , row "M f i t " . Examples: In F i g . F - l a supra- and a subthreshold response to a monopolar pulse are compared. Under the influence of the stimulus the state changes from the point of the r e s t i n g state (V = 0, m = 0.0005) to the point A (r e s p e c t i v e l y A'). With no further stimulus input, the system w i l l return to the r e s t i n g state from the point A 1 (subthreshold), but i t w i l l gener-ate an action p o t e n t i a l proceeding from point A (suprathreshold). 228 V = • -10.00000 -9.00000 -8.00000 -7.OOOOO -6.00000 -5.OOOOO -4.00000 -3.OOOOO -2.00000 -1.OOOOO 0.OOOOO 1.00000 2.00000 3.00000 4.ooooo 5.00000 6.00000 7.OOOOO 8.OOOOO 9.00000 10.00000 11.00000 12.OOOOO 13.00000 14.00000 15.OOOOO 16.OOOOO 17.OOOOO 18.00000 19.00000 20.00000 21.OOOOO 22.OOOOO 23.OOOOO 24.OOOOO 25.OOOOO 26.00000 27.00000 28.OOOOO 29.00000 30.OOOOO M = 0.09453 0.09297 0.09297 0. 09141 0.0 8 984 0.08828 0.08828 0.08672 0.08516 0.08516 0.08359 0.08203 0. 08047 0. 0789"! 0.07391 0.07695 0.07539 0.07383 0.07227 0.07070 0.06914 0.06758 0.06523 0.06367 0.06133 0.05898 0.05664 0.05430 0.05195 0.0 48 83 0.04570 0.04258 0.03837 0.03496 0.03105 0.02637 0.02168 0.01670 0.01103 0.00513 0.OOOOO M FIT = 0.09441 0.09338 0.09233 0.09125 0.09016 0.08905 0.08792 0.08678 0.08562 0.08444 0.08324 0.0R201 0.08076 0.07948 0.07816 0.07680 0.07540 0.07394 0.07241 0.07032 0.06915 0.06739 0.06553 0.06356 0.061U7 0.05925 0.05688 0.05435 0.05164 0.04375 0.04565 0.04233 0.03377 0.03496 0.03087 0.0264R 0.02179 0.01676 0.01137 0.00561 -0.00056 Table F - l Threshold values (V, m) i n the l i n e a r i z e d V, m model. The values M FIT are calculated by the fourth order polynominal. 229 F 4 fl= 1.462 F i g . F - l Response of the l i n e a r i z e d V, m model to supra-and subthreshold stimulus of 50,as duration. Standard data. ' Lower graph: Membrane voltage vs. time. The timing of the stimulus i s indic a t e d below graph. Upper graph: The same response i n the V, m plane. TH i s the cal c u l a t e d threshold condition. Stimulus' amplitudes: 1.462 mA/cm2 and 1.40 mA/cm2. 230 F i g . F-2 Response of the l i n e a r i z e d V, m model to a bi p o l a r (50 + 50 ys) stimulus. Amplitudes 1.956 mA/cm2 and 1.900 mA/cm2. Other con-d i t i o n s as i n F i g . F - l . 231 F 6 In F i g . F-2 the case of a b i p o l a r pulse (50 + 50 us) i s i l l u s -t rated. At the end of the p o s i t i v e pulse, which has a high amplitude, the system has reached at the point A ( r e s p e c t i v e l y A') a state from where an action p o t e n t i a l would be generated. Under the influence of the negative pulse, however, the state changes to the point B ( r e s p e c t i v e l y B'). B' l i e s i n the subthreshold region. Thus no action p o t e n t i a l i s generated i n t h i s case. As F i g . F - l and F-2 only serve as i l l u s t r a t i o n s , the ampli-tudes chosen are a r b i t r a r y and therefore r e l a t i v e l y f a r away from thres-hold conditions. These i n v e s t i g a t i o n s were not c a r r i e d further because they are p r i m a r i l y of mathematical i n t e r e s t . FitzHugh (1961) examined the s i m i l a r case f o r the HH model i n the V, m plane, as w e l l as the case of generalized, mathematical models. To summarize, we see that the occurrence of an a c t i o n p o t e n t i a l i s uniquely determined by the values of V and m at the end of the stimulus: the point (V, m) i n the V, m plane must l i e on the r i g h t side of threshold l i n e . No conditions are imposed on how t h i s point i s reached, i . e . no conditions are imposed on the stimulus (form or timing). This r e s u l t i s important for b i p o l a r stimulation. In the complete (FH) model there may be r e s t r i c t i o n s imposed for a long duration stimulus due to the i n t e r a c t i o n of the slowly varying v a r i a b l e s n, h, and p. 2. B i p o l a r stimulus; amplitude and timing What i s the influence of the amplitude and timing of a b i p o l a r pulse upon the v a r i a b l e s V and m? The seven parameters of a b i p o l a r stimulus are defined i n Chapter 3 . I t i s i m p r a c t i c a l to search i n t h i s seven dimensional space 232 F 7 for any conditions. Therefore, we are forced to introduce the following r e s t r i c t i o n s : only b i p o l a r pulses are considered which have equal p o s i t i v e and negative amplitudes. the average charge transfer i s zero. Thus, the p o s i t i v e and negative pulse have also equal duration. only short duration pulses are considered which produce an instantaneous voltage displacement: d i r a c impulse. In Chapter 14 i t i s shown that these are the optimum pulses, one s i n g l e b i p o l a r pulse i s considered. In Chapter 14 i t was demonstrated for the complete FH model that there i s only a small d i f f e r e n c e i n amplitude for t r i g g e r i n g an action poten-t i a l a f t e r the f i r s t or a f t e r the fourth pulse. Hence, the two v a r i a b l e s l e f t are the charge Q of the d i r a c stimulus producing a voltage displacement V x C the p o s i t i v e impulse and the negative impulse. = and the time i n t e r v a l t, between The r e s u l t i n g function i n the V, m plane, describing the changes i n V and m during t h i s stimulus i s shown schematically i n F i g . F-3. A f t e r the i n i t i a l displacement by V , m s t a r t s to increase while V decreases. A f t e r the time i n t e r v a l t.. , V i s displaced h o r i z o n t a l l y by -V of the negative X X 233 F V Fig. F-3 Effect of bipolar dirac stimulus dirac stimulus. Depending on the position of the point (V, m) with respect to the threshold function, the stimulus was suprathreshold or not. In Fig. F-3 the dashed line represents a l l possible points (V, m) for a l l stimuli with a given amplitude and different time intervals t^. In Fig. F-4 A) these calculated points are plotted. Each line 'represents a l l possible (V, m) points over a l l t^ for a given stimulus producing the voltage displacement V . V is the curve's parameter. We -X X see that must be greater than 30 mV to produce eventually an action potential. In Fig. F-4 B) the same curves are repeated.' In addition, points resulting from same time intervals t^ are connected to show the time course. The values are in microseconds. The time interval t^ necessary to reach the point of intersection with the threshold condition was determined. This relation between the step voltage V^ produced by the stimulus and the minimum duration of t^ to produce an action potential in the linearized V, m model is listed in Table F-2 and plotted in Fig. F-5. 234 F-4 A) V, m curves as functions of pulse interval time. Parameter voltage displacement produced by dirac stimulus. TH = thres-hold B) as A) with curves scaled every 10 microseconds. Explanation in text. 235 VOITAGI STEP EULSE INTERVAL (HI1LIVCLTS) (MICRCSIC) 26.000 1C00.000 28.000 1C00.000 30.000 . 1C00.000 32.000 69.000 3a.000 43.500 36.000 32.000 38.000 25.00C 40.000 20.500 42.000 17,50C 44.000 15.500 46.000 13.50 0 48.000 12.000 50.0CO 11.000 52.0C0 10.500 54.000 9.500 56.000 9.000 58.0C0 8.50C 60.000 8.000 62.000 7.500 64.000 7.000 Table F-2 Threshold condition f o r b i p o l a r d i r a c stimulus i n the l i n e a r V, m model. i 1 1 1 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 PULSE INTERVAL TIME IMICROSEC) F i g . F-5 P l o t of table F-2 236 F 3. Discussion The threshold charges for di r a c s t i m u l i compare favourably with the r e s u l t s from the complete FH model. In Table F-3 corresponding values are compared. Despite the d r a s t i c s i m p l i f i c a t i o n s i n the l i n e a r i z e d V,m model, the errors remain below 1.5%. This error includes the inaccuracy of threshold determination i n the FH model, which i s 1%. The error i s l i k e l y to increase f o r pulses other than d i r a c ones, e.g. where the time i s longer u n t i l threshold i s reached. The advantages of the l i n e a r i z e d V,m model are inexpensive (short) computations and s i m p l i c i t y . I t may be used i n cases, where the e f f e c t s of many d i f f e r e n t s t i m u l i on the corresponding threshold conditions have to be investigated. The p o s s i b i l i t y of depicting the va r i a b l e changes i n the V,m plane i s a further advantage i n i n v e s t i -gating d i f f e r e n t s t i m u l i , which may be us e f u l e s p e c i a l l y f o r b i p o l a r p u l s e t r a i n s . The e f f e c t of b i p o l a r pulses i s c l e a r l y demonstrated i n F i g . F-2. The p o s i t i v e pulse has to be suprathreshold and i t depends on the e f f e c t of the negative pulse i f an action p o t e n t i a l w i l l occur or not. The explanation of b i p o l a r pulses' e f f e c t s i s a major achievement of t h i s model. The i n i t i a l uncertanity of the model's accuracy prevented us from using i t for further i n v e s t i g a t i o n s made i n t h i s t h e s i s , a l l of which were c a r r i e d out with the complete model. 237 Spacing Threshold charge Error t 1 .(ys) nanocoul / cm % complete i l i n e a r i z e d FH model | V,m model 10.5 103.595 1 104.0 -0.39 15.5 88.125 | 88.0 0.14 20.5 80.935 1 i 80.0 1.17 25.0 i 76.875 | 76.0 1.15 32.0 1 72.810 , 72.0 1.13 43.5 I 68.905 1 1 68.0 1.33 69.0 i 64.685 1 1 1 64.0 1.07 Table F-3 Accuracy of the l i n e a r i z e d V,m model for d i r a c s t i m u l i . For the V,m model the charge was the independent v a r i a b l e . A l l c a l c u l a t i o n s with standard data and 0.5 ys i n t e g r a t i o n ' step s i z e . REFERENCES Adelman, W.J. , J r . (Ed.) (1971) Biophysics and physiology of excitable  membranes. 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