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Modelling neural transdution Dean, Douglas Philip 1984

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MODELLING NEURAL TRANSDUCTION  by  DOUGLAS PHILIP DEAN  B.A.SC, Engineering Physics, The University of B r i t i s h Columbia, 1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l Engineering)  We accept this thesis as conforming fce~t.he required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1984  (c)  Douglas P h i l i p Dean, 1984  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  agree t h a t for  that  Library  s h a l l make  for reference  and  study.  I  f o r extensive copying of  h i s or  be  her  g r a n t e d by  shall  the  not  be  of  this  •g.uecm^OcC  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada  V6T  Date  1Y3  13L-/2J  this  & J 6 3 > ) U ^ a £ ^ »-JCq>  Columbia  thesis my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  further  head o f  representatives.  copying or p u b l i c a t i o n  f i n a n c i a l gain  University  the  s c h o l a r l y p u r p o s e s may  understood  the  the  I agree that  permission by  f u l f i l m e n t of  advanced degree a t  Columbia,  department or for  in partial  written  ABSTRACT  Neural transduction i s the process by which neurons convert externally applied  electrical  energy  into  s t a b l e , propagating voltage pulses.  some stimulus waveform with particular  parameters  Given  such as duration, phase  delay, e t c . , there i s a minimum stimulus amplitude  required  i n order for  transduction of the waveform to result i n an active neural response.  The  minimum amplitude f o r e x c i t a t i o n , the threshold amplitude, i s a strong function of many variables including stimulus waveshape, frequency and duration. This study reveals some d e t a i l s of the threshold c h a r a c t e r i s t i c s of the Frankenhaeuser-Huxley  (FH) model of myelinated nerve.  These threshold trans-  duction c h a r a c t e r i s t i c s are studied with the aid of phase-space a n a l y s i s , and are  used to produce a model of neural excitation which i s c l i n i c a l l y applic-  able  to human nerve.  The f u l l  FH system  of equations i s used  to predict  threshold behaviour for in-vivo human nerve, and the predictions are shown to be  i n good agreement with c l i n i c a l l y  concludes by examining  obtained threshold  data.  The study  some of the additions to the FH model which would be  necessary to model the accumulation of extra-nodal potassium ions.  ii  TABLE OF CONTENTS  Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF FIGURES  vi  ACKNOWLEDGEMENT  viii  PREFACE 1.  ix  INTRODUCTION  1  1.1  Modelling Neural Transduction  2  1.2  D e f i n i t i o n of Terms  4  1.3  Problem Definition  5  1.4  The Basis for FES as a Successful Mode of Treatment  7  1.5  Nerve -vs- Muscle Stimulation i n FES  8  1.6  Factors Influencing Stimulation  9  1.6.1  External Factors:  Stimulus Parameters,  Stimulus Design  1.7 2.  9  1.6.2  Electrode Tissue Interface  11  1.6.3  Internal Factors:  Membrane Transduction  12  1.6.4  Internal Factors:  Motor Unit Responses  12  Choosing a Computational Model of Nerve  13  PHASE-SPACE ANALYSIS OF THE FRANKENHAEUSER-HUXLEY MODEL  17  2.1  Introduction  17  2.2  Numerical Solution of the FH System  19  2.3  C l a s s i f i c a t i o n of the FH System  21  2.4  Phase Space Analysis of the FH System  22  2.4.1  Choice of Phase Space - A Reduced System  23  2.4.2  Isoclines and C r i t i c a l Points i n the Vm Phase Plane  2.4.3  Threshold  2.4.4  Phase Plane Variations with n, p, h  30  2.4.4.1  Potassium Activation Variable n  31  2.4.4.2  Unspecific Current Activation Variable p  .  Characteristics of the FH System  iii  25 28  . .  31  Page  2.5  2.4.4.3  Sodium Inactivation Variable h  31  2.4.4.4  Sodium and Potassium Permeability  35  Determining Threshold Stimulus Parameters  36  2.5.1  42  The Separatrix Definition of Threshold  2.6  Discussion  47  2.7  Summary  50  3. OPTIMIZATION OF NEURAL STIMULI BASED UPON A VARIABLE THRESHOLD POTENTIAL  51  3.1  Introduction  51  3.2  The Classical Strength-Duration Curve  53  3.2.1 3.3  Limitations of the RC Strength-Duration Curve  . . . .  55  The Threshold Potential Function  57  3.3.1  Monophasic Stimuli  58  3.3.2  Biphasic Stimuli  58  3.4  Threshold Current and Charge  60  3.5  Stimulation with Minimum Power  61  3.6  Stimulation with Minimum Charge  66  3.7  Clinical Determination of Threshold Parameters  68  3.8  Discussion  71  3.9  Conclusions  74  4. SIMULATION OF HUMAN MEDIAN NERVE THRESHOLD RESPONSE  75  4.1  Introduction  75  4.2  Enhancement and Extension of the Model  77  4.2.1  Temperature  78  4.2.2  Passive Tissues and Adjacent Nodes  81  4.3  Experimental Design  86  4.4. Experiments  89  4.4.1  Percutaneous Stimulation  91  4.4.1.1  Selection of Model  91  4.4.1.2  Electrophysiological Techniques  91  4.4.1.3  Stimulus Waveforms  93  iv  Page  4.4.2  5.  Reference Selection  93  4.4.1.5  Procedure  94  4.4.1.6  Comparison with the Model  95  Transcutaneous Stimulation  98  4.4.2.1  Selection of Model  98  4.4.2.2  Methods and Procedures  100  4.4.2.3  Stimulation  100  4.4.2.4  Comparison with the Model  101  4.5  Discussion  103  4.6  Conclusions  107  SIMULATION OF NODAL POTASSIUM ACCUMULATION  109  5.1  109  5.2  5.3  6.  4.4.1.4  Introduction Models of K  +  Accumulation  110  5.2.1  The Three Compartment Model  5.2.2  Diffusion i n an Unstirred Layer Model  Ill  Choice of a IT*" Accumulation Model f o r the FH System  Ill  +  DUSL Accumulation on the FH System  110  5.4  The Affects of K  5.5  The Influence of K+ Accumulation on Multiple Pulse Thresholds  117  5.6  Discussion  121  SUMMARY  . . . .  112  123  REFERENCES  128  APPENDIX I  134  v  LIST OF FIGURES  Page 1.1  Elements of Neural Stimulation  16  2.1  Standard Response of the FH Model  20  2.2  Quasi-Threshold Configuration of the FH Phase Plane  24  2.3  Resting State Isoclines of the Vm Phase Plane  27  2.4  Resting State Phase Plane Detail  29  2.5  Affects of Changing  'n' on the Vm Phase Plane  32  2.6  Affects of Changing  'p' on the Vm Phase Plane  33  2.7  Affects of Changing  'h' on the Vm Phase Plane  34  2.8  Vm Phase Plane at Peak P „ and Peak P., Na K Stimulus-Response Curves for 120 us Monophasic Square Stimuli at  37  20°C and 15°C  39  2.9 2.10  Subthreshold Response to 5 us Biphasic Square Stimulus 0.0 us Phase Delay  41  2.11  Threshold Amplitudes of Biphasic Stimuli Determined by the 44  2.12  Separatrix D e f i n i t i o n Comparison of Threshold Calculation Times for Separatrix and Level Definitions  46  2.13  Stimulus-Response  Curves for 120 us Monophasic Square Stimuli at  2.5°C, 20°C, and 40°C  3.1  49  RC Model Strength-Duration Curves Compared to FH Model Threshold Amplitudes  3.2  56  Threshold Potentials for Biphasic Square Stimuli from the FH Model  3.3  3.4  59  RC-Variable Threshold Potential Strength-Duration Curves Compared to FH Model Threshold Amplitudes  62  Pulse Widths Minimizing the Normalized Damage Function  69  vi  Page 4.1  Equivalent C i r c u i t of the Three-Node FH Model  82  4.2  Distortion of Injected Current Waveforms  4.3  DISA 15E07 Stimulus Waveform  85  4.4  Near Nerve Recordings  90  4.5  Assumed Geometry for Percutaneously Applied Stimuli  92  4.6  Model and C l i n i c a l Thresholds f o r Percutaneous Stimuli  96  4.7  Assumed Geometry for Transcutaneoujsly Applied Stimuli  99  4.8  Model and C l i n i c a l Thresholds f o r Transcutaneously Applied  . . .  Stimuli 4.9  84  102  C l i n i c a l Thresholds f o r Percutaneously and Transcutaneously Applied Stimuli  106  5.1  Potassium Current Flow i n Response to Standard Stimulus  5.2  FH Variables Affects by Potassium Ion Accumulation  5.3  Phase Plane Changes due to Potassium Ion Accumulation  116  5.4  Phase Plane Detail  118  5.5  Threshold Delay Times  120  vii  . . . .  114  .  115  ACKNOWLEDGEMENTS  I wish to thank Peter Lawrence for his assistance and guidance during the course of this study. have been f i n i s h e d .  Without h i s encouragement this thesis would not  I must also thank Theresa Barber and Dave MacDonald for  the part they played i n the completion of the work described on these pages. In addition, Paul Kendrick and  Steve Mennie spurred my creative urges with  observations and comments from a non-technical point of view — was  more important than they r e a l i z e .  Miss Heart, who  A special acknowledgement i s due to  r e a l l y started a l l of this i n 1962.  F i n a l l y , many thanks to  B i l l March for advice, stern warnings, and much much more.  viii  their input  PREFACE  This thesis i s presented as four self-contained chapters. t i o n of each chapter beginning  with  a  i s progressively less mathematical  analysis  The presenta-  theoretical and more of  the  properties  clinical, of the  Frankenhaeuser-Huxley system of equations, and leading to the presentation of c l i n i c a l data demonstrating the a p p l i c a b i l i t y of model results to human nerve stimulation. With the exception of the fourth chapter and the Introduction, a l l chapters have either appeared as papers i n the IEEE Transactions on Biomedical Engineering, or are awaiting p u b l i c a t i o n , or have been submitted for publicat i o n i n that j o u r n a l .  Consequently, each chapter contains i t s own introduc-  t i o n , summary, conclusions and discussion, and may be read without reference to preceeding or following chapters.  ix  1  1.  INTRODUCTION  The motivation f o r this work has been a perceived need to understand more about the threshold c h a r a c t e r i s t i c s of myelinated human peripheral nerve fibres.  Understanding  the nature of neural threshold i s an important goal  since many s i g n i f i c a n t implicitly parameter  involve  applications  making  settings  of neural control  decisions  regarding  which are ultimately  predicted  (as outlined below)  stimulus  waveshapes  and  upon aspects of neural  threshold. It i s d i f f i c u l t system. able.  to perform experiments on the human peripheral nervous  The system i s d e l i c a t e , important, and to a large extent nonregenerHence the primary source of investigation i n this work was not in-vivo  or i n - v i t r o human nerve, but rather a computational model implemented on a d i g i t a l computer.  By using such an 'electronic' nerve, i t was possible to  perform some threshold 'experiments', or experimental analogues, which would not have been possible to perform on a human subject.  Experiments with the  human nervous system were undertaken only to obtain data which could be used to provide an i n d i c a t i o n of the relevance of the computational model to the system being simulated. No experiments were performed on human nerve f o r the purpose of gathering physiological data which would be used to a l t e r model parameters. The question of model a p p l i c a b i l i t y to human nerve i s , of course, c r i t i cal  since  results  no matter  how convenient  are irrelevant  i f one wishes  the computational model may be, them  to provide insights  its  into the  behaviour of human nerve, but one finds that model dynamics have no resem-  2  blance to those of human peripheral nerve. model which has the greatest probable  The question of choosing a neural  applicability  to the human case i s  addressed at the end of this introduction. A transducer i s a device which converts the form of some applied signal into an a l t e r n a t e , usually e l e c t r i c , s i g n a l . signal  i s converted  into  the output  The process by which the input  signal i s known as transduction. The  neurons which compose the peripheral nervous system i n man behave l i k e transducers when subjected to externally applied e l e c t r i c a l energy;  however, the  neuron i s a rather unusual transducer of external e l e c t r i c a l signals i n that i t operates on an ' a l l or none' p r o t o c o l .  If the input signal (the external-  l y applied e l e c t r i c a l energy) meets certain c r i t e r i a , the neuron-transducer produces an output signal which has a c h a r a c t e r i s t i c amplitude and shape, and which w i l l travel from the s i t e of transduction at a c h a r a c t e r i s t i c v e l o c i t y . This s t a b l e , propagating voltage pulse i s known as the action p o t e n t i a l . If the input signal does not meet the threshold c r i t e r i a , no stable propagating action potential w i l l be produced.  1.1  Modelling Neural Transduction Regardless of the type of external neural stimulation being applied, the  properties of the neural membrane which f a c i l i t a t e action potential generat i o n are of c r i t i c a l importance.  The purpose of this work i s to analyse the  threshold transductive properties of the myelinated axon with the a i d of the Frankenhaeuser-Huxley model [1], and to discuss some of the r e s u l t i n g cal implications.  clini-  3  Within bundles  or  the peripheral nervous system, neurons (nerves) are arranged i n fasiculi  which  are  surrounded  by  Thousands of neurons, or neural elements, may bundle. are  tissue  sheaths.  be present i n a single neural  The neuron i s composed of a nerve c e l l body and i t s processes which  called  impulses  connective  dendrites  to other  dendritic  process  neurons.  The  and  axons.  The  axonal  process  neurons or to muscle fibres and serves  reader  as  a  point  i s referred  of  serves  gland  termination  to  conduct  c e l l s , while  for  axons  of  the  other  to an introductory textbook on the human  nervous system such as that by Barr and Kiernan [2] for further d e t a i l s . A block diagram, F i g . 1.1, ment are related  shows how  the properties of the neural ele-  to other factors influencing external neural stimulation.  In t h i s diagram, I  r e f e r s to the e x t e r n a l l y applied current, i n units of t ll  c u r r e n t d e n s i t y , which serves as an input signal to the n th i s the transduced response of the n  neuron to 1^.  neuron;  Although S  r  and  is strictly  a voltage s i g n a l , i t i s considered here as a dimensionless binary signal with the two binary states representing the a l l or none condition of transmission. A stimulus current I Q at some potential V lus  electrodes.  Assuming  the  use  q  i s applied through a set of stimu-  of a constant  current  stimulator, the  impedance of the electrode-tissue interface (ETI) does not a f f e c t the stimulus  c u r r e n t , and  surrounding  so a c u r r e n t of I  tissues.  The  action of  passes through the e l e c t r o d e s i n t o this  current on  the  ETI  can  produce  stimulation by-products which can, i n turn, influence both ETI impedance and the properties of the tissues near the electrodes. passes through passive tissues surrounding  As the stimulus current  the neural elements, some frac-  4  t i o n , T-n» i s d i v e r t e d to enter each of n myelinated neural elements through a node of R a n v i e r . sary  threshold  If the current I  requirements, then  entering the  the neural  element meets neces-  element w i l l  generate an active response - the action p o t e n t i a l .  be induced to  This active response,  S^, w i l l propagate from the s i t e of a c t i v a t i o n with a c h a r a c t e r i s t i c v e l o c i t y and  shape and w i l l be processed i n the central nervous system i f the neural  element was responsible for sensory input, or S r may r e s u l t i n a c t i v a t i o n of a motor unit i f the neural element was a moto-neuron.  1.2  D e f i n i t i o n of Terms E l e c t r i c a l stimulation may be applied to the body either for therapeutic  or diagnostic purposes as pointed out by Roberts e t . a l . [3]. The application of e l e c t r i c a l energy to human neurological tissues f o r the purpose of evoking active  neural  modulation.  responses  may  The therapeutic  be c a l l e d goals  applied  neural  of applied neural  control  or neuro-  c o n t r o l , the means of  applying and d e l i v e r i n g the e l e c t r i c a l energy to the body, and the targetted t i s s u e s , provide  convenient means of c l a s s i f y i n g the various neuromodulation  scenarios. The tinct  two most general classes of neuromodulation are created by two d i s -  classes of therapeutic  enhance  weak  neurological  application - one where the intention i s to  signals  or generate  non-existent  neurological  signals - the other where the purpose i s to mask or supress undesired activity.  neural  5  In cases where disease or trauma have damaged c r i t i c a l neural pathways, the r e s u l t i n g sensory or motor d e f i c i t s may neural  control.  Such control or  be somewhat a l l e v i a t e d by applied  information  signal enhancement has  been  c a l l e d neuroaugmentation and functional e l e c t r i c a l stimulation. Undesired neural a c t i v i t y with few exceptions means perception of pain, either chronic or acute.  In these cases the object of neuromodulation i s to  eliminate  of  the  perception  pain.  The  mechanism of  pain  suppression  by  neural stimulation i s not well understood, but i t i s thought that the release of endogenous opiates may  be  involved.  Further s u b - c l a s s i f i c a t i o n s of therapies i n the context of the above two classes of neuromodulation can be made on the basis of the method of applying the e l e c t r i c a l stimulus. cally  implanted  cortical  to  In spinal cord stimulation, electrodes are surgi-  stimulate  stimulation  inserted electrodes.  refers  the to  dorsal  column  stimulation  of  of the  the  spine,  brain  via  similarly surgically  Percutaneous stimulation, on the other hand, involves  the use of needle-like electrodes inserted through the s k i n , without s u r g i c a l intervention, f o r the purpose of stimulating peripheral nerves, while transcutaneous stimulation stimulates nerves through unbroken s k i n .  1.3  Problem D e f i n i t i o n Considering  lation  only functional e l e c t r i c a l stimulation (FES), and not stimu-  for pain r e l i e f , the goal of neuromodulation may  generation  be  said to be  the  of functional responses i n 'paralysed' end organs which are indis-  tinguishable  from the  responses of i d e n t i c a l  'unparalysed' organs, through  a p p l i c a t i o n of externally generated control s i g n a l s .  6  Human nerves can be of  externally  induced to transmit control signals by application  generated  stimuli.  F o r t u i t o u s l y , such control  signals  are  i d e n t i c a l to those which are naturally generated, and hence the organs under control of  the  a r t i f i c i a l l y stimulted  nerves w i l l  manner to the a r t i f i c i a l control s i g n a l s . able  to  restore  lost  motor  or  respond i n an i d e n t i c a l  In p r i n c i p l e then, one  perceptual  function  nerves c o n t r o l l i n g the organs i n question.  by  should be  stimulation  of  the  In p r a c t i c e , such a restoration  of lost function embodies a large number of c r i t i c a l factors from stimulator design  and  electrode  positioning,  neural membranes being  gives an  the  transductive  study the c r i t e r i a which must be met  of  the  by the stimulus  C  p a s s i n g i n t o the n ^ neural element so that the neural membrane  active response.  How  the r e s u l t i n g signals are processed, or  they activate motor end units are not studied. passive  properties  stimulated.  In t h i s t h e s i s , we current I  to  tissues  are  addressed  only  insofar  how  The properties of surrounding as  i s required  correspondence of c l i n i c a l data to model p r e d i c t i o n s .  to examine  the  It i s not the purpose  of t h i s work to model i n d e t a i l passive t i s s u e s , electrode-tissue  interfaces  or geometries of stimulating electrodes.  Rather, the threshold properties of  neural elements i n response to external  s t i m u l i and  c l i n i c a l implications are The FES c r i t e r i a may 1.  Nerves  must  stimulator  some of the r e s u l t i n g  investigated. be stated as follows: be  stimulated  by  pulses  which meets established  generated  safety  from  a  requirements;  7  and  which  produces  comfortable,  effective  trains  of  electrical stimuli. 2.  The  s t i m u l i must be  applied  through  electrodes  which  minimize discomfort or damage to surrounding tissues which have a geometry which i s optimal for the  and  effect  desired. 3.  The  stimulus  etc.)  parameters  must simultaneously  selected activate  (frequency, the  and be perceived as comfortable by the 4.  The  induced  control  target  duration, neurons,  patient.  signals must have the  temporal properties to produce smooth regular  spatial  and  contraction  of the muscle which i s innervated by the neural elements being stimulated;  or i n the case of sensory organs, that  an appropriate sensory response i s generated. In p r a c t i c e , 4 i s extremely d i f f i c u l t no  noticeable  difference  to achieve to the extent that there i s  between a naturally generated muscle  contraction,  and one generated by techniques of functional electrostimulation.  1.4  The Basis for FES The  the  as a Successful Mode of Treatment  p r i n c i p l e of neural control signal i n d i s t i n g u i s h a b i l i t y i s one  fundamental axioms of functional electrostimulation.  of  It i s well known  that the fundamental unit of neural communication, the action p o t e n t i a l , can be The  generated by  subjecting  an  in-vivo nerve to a changing e l e c t r i c  field.  physiological mechanisms of the neural membrane which are activated  by  these ' a r t i f i c i a l ' means to produce a stable propagating action potential are  8  Identical  to the mechanisms  conduction.  Away from  the immediate v i c i n i t y  stimulus there i s therefore generated  and a r t i f i c i a l l y  under control stimulus  of 'natural' action  action  of the applied  electrical  no distinguishable difference between naturally generated  of the neural  induced  potential generation and  action p o t e n t i a l s .  element being stimulated  potential i n exactly  Hence, any organ  will  respond  to the  the same manner as i t would  respond to a similar naturally generated action p o t e n t i a l .  1.5  Nerve -vs- Muscle Stimulation i n FES When applying  functional  effects  FES to patients, i t i s possible to produce the desired by d i r e c t  stimulation  of nerve, or by stimulation of  s k e l e t a l muscle, as stated by Mortimer [4]. by many d i f f e r e n t research  Both approaches have been used  groups i n the past, but d i r e c t nerve stimulation  has proven to have several advantages compared to muscle stimulation: a)  Lower stimulus amplitudes are required  - this  prolongs  stimulator battery l i f e and reduces possible stimulation side e f f e c t s . b)  Electrodes  can be located  electrode  displacement  i n areas which do not move -  (or breaking  i n the case of  implants) i s common when the electrodes must be located on the body of the muscle which i s being c)  stimulated.  One muscle function can be controlled by a single nerve.  The chief problem of d i r e c t nerve stimulation - that of fatigue at the neuromuscular junction - occurring  only i n the case of implanted stimulus elec-  trodes, has been l a r g e l y eliminated by the technique of 'karusellstimulation'  9  developed i n Austria  by  Holle  et a l . [ 5 ] .  With t h i s  technique, a  inhomogeneous e l e c t r i c a l f i e l d i s generated by implanted electrodes ing  the  nerve to be  eliminates  stimulated.  The  rapid v a r i a t i o n of  neuro-muscular junction fatigue by inducing  the  cyclic  surround-  cyclic  field  a more equitable  dis-  t r i b u t i o n of control signal conduction throughout the target nerve.  1.6  Factors Influencing Factors of  gories  [4].  Stimulation  interest i n FES  may  These c l a s s i f i c a t i o n s  be  roughly organized into three  are  defined  body - mainly those concerned with stimulator waveform, stimulation parameters); interface such as electrode  by  factors external  design and  function  factors influencing the  impedance and  of  the  afferent  neural  membrane, and  stimulation by-products;  sensory neurons.  shown i n F i g . 1.  'central  These factors and  processing'  to  (stimulus  of  and  signals  from  their Inter-relationships  are  Further c l a s s i f i c a t i o n s are usually made based upon  the  stimulation, or peripheral nervous system (PNS) of  fac-  proper-  nature of the neural elements being stimulated - central nervous system  class  the  electrode-tissue  tors wholly i n t e r n a l - motor unit responses to s t i m u l i , transductive ties  cate-  a p p l i c a t i o n s , there  are  further  stimulation.  (CNS)  Within the  s u b - c l a s s i f i c a t i o n s according  PNS to  stimulation of either efferent or afferent pathways.  1.6.1  External Factors:  Stimulus Parameters, Stimulator Design  It i s accepted by most researchers that charge per stimulus i s a c r u c i a l factor  in  stimulation,  current i s necessary.  thus  implying  that  complete control  over  stimulus  This further implies that a constant current stimula-  10  tor  i s preferable to a constant  current  output  is  a  strong  function  (Butikofer and Lawrence [ 6 ] ) . current v a r i e t y .  p o t e n t i a l stimulator since i n the of  tissue  and  electrode  latter,  impedance  Most FES stimulators are thus of the constant  To avoid dangerous increases i n current stimulator driving  potential which can occur  i n the case of an increasing electrode impedance,  an over-voltage detection c i r c u i t should be incorporated into the stimulator design.  These two  considerations, constant  voltage detection, are recognized  current  stimulation and  by nearly a l l researchers  over-  to be the  two  is s t i l l  an  most fundamental requirements of any FES stimulator design. Although  stimulus  waveform  and  parameter  investigation  active area of research, c e r t a i n parameter ranges and stimulus waveforms are agreed upon as most e f f e c t i v e .  In general, p u l s a t i l e biphasic stimuli are  preferred, with stimulus durations ranging from 20 [is to 300 taneously stimuli  applied  (Vossius  s t i m u l i , and [7]).  1 |is  Pulsatile  t o  300  rather  us for transcu-  us for percutaneously  applied  than sinusoidal s t i m u l i are  pre-  ferred since with sinusoidal stimulation charge delivered per stimulus phase i s dependent upon stimulus stimulation.  frequency.  Stimuli should  be  This i s not the case with  biphasic and  charge balanced  pulsatile  i n order  to  avoid perturbing electro-chemical e q u i l i b r i a at the electrode-tissue i n t e r face (McHardy et a l . [ 8 ] , Pudenz et a l . [ 9 ] ) . Recent studies have shown that d i f f e r e n t stimulus waveforms may respect during  to  minimal  be arranged i n a heirarchy of e f f i c a c y with  stimulation amplitudes  stimulation (Janko and  Trontelj  and  [10]).  minimal noxious  sensations  Although there i s a general  concensus regarding the form of the most favourable stimulus waveshape, there  11  are  still  many aspects  of stimulus  parameter selection which are not  well  understood and which require further research.  1.6.2  Electrode Tissue Interface Electrode-tissue interface (ETI) considerations have important implica-  tions for stimulator design. interface  impedance, and  The p r i n c i p a l factors of concern relate to the  to the dissemination  of electro-chemical reaction  products to tissues i n contact with the stimulating electrodes. If reasons  the of  electrode diminishing  current density may associated  painful  impedance electrode  during  stimulation becomes too  contact,  local  regions  high  of high  stimulus  occur resulting i n tissue damage due to ohmic heating sensations  (Burton  and  Maurer  [11], Mason and  for  and  MacKay  [12]). Charge imbalanced stimuli can perturb the thermodynamic electro-chemical e q u i l i b r i a of the ETI r e s u l t i n g i n corrosion of the stimulus electrode. same phenomena may sively  long  The  also occur with charge balanced stimuli which have exces-  phase durations.  Transcutaneously,  t h i s equilibrium imbalance  may manifest i t s e l f as l o c a l reddening and i r r i t a t i o n of the skin [12], while in  the  case of implanted  electrodes toxic electro-chemical by-products  may  induce l o c a l tissue breakdown (Pudenz et a l . [13]). A d d i t i o n a l l y , the d i f f e r e n t current dispersion influences of various ETI configurations can s i g n i f i c a n t l y a l t e r the appropriate magnitudes of stimulus current waveforms, even changing the neural response patterns as functions of stimulation parameters.  This factor must be considered when selecting an FES  stimulator for a p a r t i c u l a r a p p l i c a t i o n .  12  1.6.3  Internal Factors;  Membrane Transduction  The threshold properties of a neural element are of prime importance i n stimulus parameter s e l e c t i o n , stimulus waveform selection and electrode positioning.  Ideally one would l i k e to induce neural a c t i v i t y with the minimum  possible stimulus amplitude i n order  to minimize both ETI problems and to  minimize current drain on the batteries powering the e l e c t r i c a l stimulator. Based  upon  threshold  modelling stimulus  studies amplitude  and  experimental  loci  as  verification  functions  of  ([14]-[16]),  stimulus  waveform  parameters are known and can be used to select stimulus parameter ranges to optimize  stimulus  efficacy  (Gorman and Mortimer  [17]).  In a d d i t i o n , some  recent work has detailed the form which an optimal stimulus should take [18].  1.6.4  Internal Factors;  Motor Unit Responses  Afferent nerve stimulation, achieved by a c t i v a t i o n of skin receptors and the  largest  diameter  A-p  fibres  of the afferent nerve  bundles,  spinal r e f l e x arcs and may be used to generate complex movements.  triggers  It i s well  known that afferent f i b r e FES has b e n e f i c i a l secondary e f f e c t s on the a c t i vated motor u n i t s , decreasing muscle atrophy, increasing both muscle tone and blood  flow, as well as decreasing  i n a c t i v i t y osteoporosis i n bone [2], [7]  and [19]. Direct stimulation of efferent nerve f i b r e s produces useful muscle contractions,  and has been  demonstrated  (Vodovnik et a l . [19]), with [10]),  and with  standing  applied FES signals  gait  assists  with  the electronic  peroneal  assists  i n hemiplegics  (Stanic et a l .  f o r paraplegics using  [21], and implanted  brace  transcutaneously  electrodes (Sthor et a l . [22]). In  13  addition i t has been shown that stimulation of efferent nerve fibres improves volitional  control  (Gracanin [23]) and reduces s p a s t i c i t y  (Rebersek et a l .  [24], Bowman and Bajd [25]). A combination of afferent  and  p l e g i c s , using efferent FES during afferent  stimulation  efferent FES has been applied to hemithe gait  swing  phase with simultaneous  of the non-moving leg to produce  a flexion  response  [21].  1.7  Choosing A Computational Model of Nerve A large number of computational models of nerve have been presented i n  the l i t e r a t u r e .  Cole,  [26], provides a comprehensive  important neural models.  review of the most  In general, a model of nerve which accounts for the  observed threshold c h a r a c t e r i s t i c s must consist of a capacitance i n p a r a l l e l with a time-varying non-linear resistance. unmyelinated nerve was  The Hodgkin-Huxley  [2 7] model of  the f i r s t model to separate the membrane resistance +  1  into d i s t i n c t components - one for each ionic species, Na , K"" and C l ~ . Perhaps Hodgkin-Huxley Huxley  the most well known models of nerve are the afore-mentioned (HH) model of squid unmyelinated axon;  (FH) model of myelinated nerve  from  the Frankenhaeuser-  the frog Xenopus leavis [1];  Dodge's model of Rana pipiens myelinated nerve [28];  and Noble's model of  mammalian Purkinje f i b r e [29]. Noble's model i s unsuited to investigations of peripheral nerve since i t emulates the dynamics of the pacemaker nerve f i b r e s i n mammalian heart whose behaviour i s s t r i k i n g l y d i f f e r e n t from peripheral nerves.  Dodge's model i s  not scaled to unit area and i s , i n f a c t , a model of a single node of the many  14  which were investigated ("node 7"), and hence i s unsuitable for our application.  Although the HH model i s valid, strictly speaking, only for the giant  axon of squid at 2.5°C, and the FH model i s valid only for the nerve fibres of  Xenopus laevis, a more optimistic view may  generic model of unmyelinated  be that the HH model i s a  nerve, and the FH model one of myelinated  nerve. Such a hypothesis cannot be proven, but many examples abound in the literature which seem to indicate that the HH and FH models emulate many of the mechanisms of excitation in species other than those which supplied the data bases for the original models. For example, Fitzhugh [30] showed using dimensional analysis that results from one nerve model can be transformed to apply to another fibre of similar structure having different model constants. The experimentally verified linear diameter-velocity relation of myelinated fibres was derived from the FH equations by Goldman and Albus [31], while Buchthal and Rosenfalck [32] determined that the relation between temperature and conduction velocity in human sensory nerve was similar to that in frog nerve and in the vagus and saphenus nerves of cat. Although encouraging, the above examples by no means prove that such models can be used to produce results which predict human nerve dynamics. Since we are interested in the large diameter myelinated fibres of the human peripheral nervous system, the FH equations were chosen as the basis of the computational model in this study.  In chapter 2, the dynamics of the FH  model are studied using phase-space analysis.  The analysis of chapter 2  leads to a precise definition of threshold which i s useful in determining threshold  amplitude  loci  of  various  stimuli  as  functions  of stimulus  parameters.  Chapter 3 makes use of the threshold d e f i n i t i o n i n chapter 2 to  suggest an algorithm for selecting 'optimal' stimulus waveforms and parameter settings.  In chapter 4, we w i l l present c l i n i c a l results from human subjects  which compare favourably to analogue FH model computations. will  provide  some  Frankenhaeuser-Huxley  justification model  as  a  f o r the means  of  properties of myelinated human peripheral nerve.  decision emulating  to the  These results employ  the  transductive  16  CENTRAL PROCESSING (SENSORYI  CONSTANT  ELECTRODETlSSUE  CURRENT  TISSUE  STMULATOR  MTERFACE  NEURAL ELEMENT  X  I I I  JL.  MOTOR  STMULATON  UNIT  BVPROOUCTS  ACTIVATION  Figure 1.1 Elements of Neural Stimulation  17  2.  PHASE SPACE ANALYSIS OF THE FRANKENHAEUSER-HUXLEY MODEL  2.1  INTRODUCTION  The Frankenhaeuser-Huxley (FH) equations are a system of f i v e non-linear ordinary d i f f e r e n t i a l equations which describe the space-clamped response of a myelinated Ranvier.  toad  (Xenopus laevis) nerve to current  s t i m u l i at a node of  The equations were empirically derived by Frankenhaeuser, Dodge and  Moore over a number of years using the voltage clamp technique i n a series of experiments [33] - [42]. of  equations  are  The r e s u l t s of the experiments and the derived set  summarized and  discussed by  Frankenhaeuser and  Huxley i n  [1], and are repeated i n Appendix I. The to  equations  give an expression for the f i r s t derivative with respect  time of transmembrane potential as a sum  sodium, potassium,  an  unspecific component  of i o n i c current components of (primarily  i o n s ) , a stimulus current and.a leakage current. T h e +  (K , Na  +  and  unspecific) are expressed  carried  by  sodium  primary ionic currents  as a product of time-varying perme-  a b i l i t y of the membrane to the ionic species i n question, and a non-linear Goldman f a c t o r .  The Goldman f a c t o r , which describes the observed r e c t i f y i n g  properties of a semi-permeable membrane, can be derived from the assumption of  a constant  [33],  f i e l d membrane.  It was  observed by Dodge and  Frankenhaeuser  [34], that peak i n i t i a l current densities caused by step depolariza-  tions of the membrane departed s i g n i f i c a n t l y from a l i n e a r form i n the region of  the largest d e p o l a r i z a t i o n s .  In  [34] they proposed that the i o n i c cur-  rents would best be described by the constant f i e l d Goldman equation [43]. Introduction  of  the  delayed  unspecific ionic  current  component, and  description of the i o n i c currents using Goldman factors are the fundamental  18  differences between the FH system and the well known Hodgkin-Huxley equations for unmyelinated nerve [2 7].  In the Hodgkin-Huxley  system, a l i n e a r r e l a t i o n  between membrane current and transmembrane potential was used i n conjunction with a time varying ionic conductivity. Numerical Lawrence  solutions  of the FH equations were used  [44] to calculate  by Butikofer and  minimum amplitudes of biphasic  square  s t i m u l i required to e l i c i t action potentials i n the FH system. paper, the authors demonstrated  current  In the same  the a b i l i t y of FH model solutions to predict  trends of experimental data corresponding to bulk stimulation of non-spaceclamped human nerves. previously  calculated  generated  i n the skin  In a l a t e r threshold during  paper  data  to  the course  [6] the same authors used determine of a  the  thermal  their energy  transcutaneously applied  threshold stimulus. The FH model was used by McNeal [16] to investigate current stimulation of a multi-node myelinated nerve.  In order to reduce the complexity of h i s  multi-node model, McNeal used a s i m p l i f i e d passive form of the FH equations and (<15  calculated  membrane  current  mV) where the assumption  and potential  f o r small  depolarizations  of passive RC response by the FH system i s  valid. Mortimer  [4] employed the model described by McNeal [16] to determine  the threshold dependence of myelinated nerve as a function of axon diameter and position with respect to stimulating electrode.  The threshold character-  i s t i c s were investigated for monophasic constant-current s t i m u l i . In [45] the FH equations were used by Clopton e t . a l . to produce tuning curves f o r single cycles of sinusoidal current s t i m u l i .  The model generated  19  tuning curves similar  to those obtained from the cochlea of guinea pigs i n  spite of a number of e s s e n t i a l differences between model and  experiment.  Applications of the FH system involve determining the nature of changes i n the model threshold as a function of certain parameters of i n t e r e s t .  In  the following sections, we present an investigation of the properties of the FH system using techniques of phase-space a n a l y s i s . of  Based upon the results  the phase-space a n a l y s i s , a d e f i n i t i o n of threshold  will  stimulus amplitude  be proposed, and w i l l be used to produce threshold amplitude l o c i for  various current s t i m u l i .  The  advantages of the phase-based d e f i n i t i o n of  threshold w i l l be shown to be i n terms of increased computational e f f i c i e n c y , and lack of ambiguity.  2.2  NUMERICAL SOLUTION OF THE FH SYSTEM  The FH equations were numerically solved by using the predictor-corrector  algorithm described by  Cooley  and  Dodge  [46].  The  integration  time  i n t e r v a l was set at 0.5 |j.s following the recommendations of Butikofer [47] to y i e l d an accurate numerical c a l c u l a t i o n with a reasonable computation  time.  A l l calculations were performed using a PDP-11/23 computer. Numerical  calculations  course of V, P ^ ,  were  checked  by  comparing  P R , P p ; m, n, p, h; 1^, I R , 1^,  to a 'standard' stimulus defined by Frankenhaeuser son with the standard responses i n [1] was  I  data  for the  time  ( F i g . 2.1) i n response  and Huxley [ 1 ] . Compari-  q u a l i t a t i v e as no numerical data  were published. However, the responses appeared to be the same with peak and minimum values occurring at the same times with similar magnitudes as far as could be determined from the graphic data.  20  F i g . 2.1 Standard Response of the FH Model 2  Computed response of the FH model to 'standard* 120 us duration 1.0 mA/cm monophaslc current stimulus, a) Transmembrane potential V. b) Ionic current components; sodium, I j j a ; potanssium, 1^; u n s p e c i f i c , Ipj leakage, I ^ . c) Membrane p e r m e a b i l i t i e s ; sodium, P N a ; p o t a s s i u m , P R ; u n s p e c i f i c , Pp. d) Sodium a c t i v a t i o n v a r i a b l e m, sodium i n a c t i v a t i o n v a r i a b l e , n; unspecific. a c t i v a t i o n v a r i a b l e p.  21  2.3  CLASSIFICATION OF THE FH SYSTEM  In [48], FitzHugh proposed three c l a s s i f i c a t i o n s for mathematical models of threshold nerve membrane behaviour. non  The discontinuous  threshold phenome-  (DTP) " . . . i n which d i f f e r e n t i a l equations with discontinuous  provide  functions  a d i s c o n t i n u i t y of response as a function of stimulus i n t e n s i t y at  threshold  ..." [48].  The singular-point threshold phenomenon (STP) was pro-  duced by a system of d i f f e r e n t i a l equations described by a n a l y t i c functions with a saddle point present i n the phase space. threshold. sponded  The t h i r d  to a system  c l a s s , the quasi described  threshold  Such a system has an exact phenomenon (QTP), corre-  by analytic functions with  neither  a dis-  continuous response, nor a sharp threshold. It was shown i n [48] and i n [49] that the Hodgkin-Huxley model of nerve was of the QTP c l a s s .  The quasi-threshold nature of the HH equations led to  obvious graded responses especially at higher temperatures as i l l u s t r a t e d i n [50] and [51].  In [51], Cole, Guttman and Bezanilla provided  experimental  confirmation of the t h e o r e t i c a l l y predicted graded responses with a set of experiments on squid giant axon at temperatures up to 35°C. The  Frankenhaeuser-Huxley system i s , l i k e the Hodgkin-Huxley system, of  the quasi-threshold v a r i e t y . The f i v e equations describing the FH system are of the form: dV = F1(m,h,n,p,V)  £  The  -  F  2  ( . . V ) .  §  - F ( h , V ) £ = F C„,V), * E - FjCp.V) 3  >  set of a l l possible c r i t i c a l  4  points of the five-dimensional  system i s  22  given by the use of  F± = 0, i = 2,3,4,5 in F^m.h.n.p.V) = 0. The  projection of this set of points into the two-dimensional  F i g . 2.2 shows that only one intersection (0,0) with ^ definition  of FitzHugh  [48],  V-m plane i n  = o exists.  the FH equations thus form  By the  a quasi-threshold  system.  2.4  PHASE SPACE ANALYSIS OF THE FH SYSTEM  The FH system of d i f f e r e n t i a l equations expresses the time derivatives of i t s f i v e p r i n c i p a l v a r i a b l e s , V, m, h, n and p (discussed i n d e t a i l i n a following  section) as functions  of one another.  Solutions to the system  would y i e l d expressions f o r each of the f i v e variables as functions of time, V=<()1(t), m=<t>2(t)  . . . p=<t>5(t).  Such a solution could be interpreted  as a  point tracing a curve i n the space formed by the f i v e v a r i a b l e s , with the <t>^(t) being a parametric representation of this curve.  The space formed by  the f i v e v a r i a b l e s , with time as a paramter, i s known as a phase space.  A  plane formed by two of the v a r i a b l e s , say V and m, would be the Vm phase plane.  A curve i n the phase space described parametrically by a solution of  the FH system i s usually referred to as an o r b i t , path or phase trajectory of the system.  23  A  phase  governed by the  trajectory  the  of  the  locations i n the  f i v e variables  syst em  will  exhibit  where the  influenced  by  points  i n the  system variables  Phase t r a j e c t o r i e s are also  phase space where i s o c l i n e s i n t e r s e c t .  These points of i n t e r s e c t i o n , or c r i t i c a l  points, may  belong to several d i f -  ferent c l a s s e s , with each class influencing system s t a b i l i t y and i n a d i f f e r e n t manner.  For more information  excitability  on phase spaces, t r a j e c t o r i e s ,  and c r i t i c a l points, the reader i s referred to [52] by Boyce and  2.4.1  is  Curves i n the phase space  time derivatives of the  are zero are known as n u l l c l i n e s or i s o c l i n e s . highly  which  phase space where the time derivatives of  forming the system are zero.  corresponding to points  behaviour  DiPrima.  Choice of Phase Space - A Reduced System A reduced system for the Hodgkin-Huxley (HH)  equations was  proposed by  FitzHugh [49] based upon the observation that i n i t i a l l y only two  of the four  HH variables changed s i g n i f i c a n t l y . at  their resting values and  the  The  remaining two  variables were fixed  r e s u l t i n g two-variable system was  studied  with the aid of the corresponding phase plane. Inspection  of responses for the FH variables V, m,  i n i t i a l l y , as i n the HH able change.  The  system, only the V and m variables undergo appreci-  standard responses i n F i g . 2.1  (transmembrane potential) and first  the  most active  potential has occurs as  the  n, p, h shows that  of  the  i l l u s t r a t e the fact that V  m (sodium current a c t i v a t i o n variable) are five  a r i s i n g phase and  FH  variables.  a f a l l i n g phase.  transmembrane potential i s driven  potential i n response to an increasing permeability  P h y s i c a l l y , the The  at  action  i n i t i a l r i s i n g phase  towards the  sodium Nernst  of the membrane to sodium  24  V (volts)  Figure 2 . 2 Quasi-Threshold Configuration of the FH Phase Plane  P r o j e c t i o n of set of FH c r i t i c a l p o i n t s onto the Vm phase p l a n e . Intersection of the locus with dm/dt-0 corresponds to a stable resting point, hence the system i s QTP.  25  ions.  The falling phase is due  potassium ions.  to an increased membrane permeability to  This simple physical argument also leads one to the conclu-  sion that i n i t i a l activity should be confined to the sodium system of any neural model. The m and V variables were thus chosen to form the FH phase plane. With the remaining three variables n, p, h at their resting values, the resulting phase plane was representative of the system for brief periods of time after stimulation.  For longer periods of time after stimulation, the same V-m  phase plane was  used, but with constant values of the variables n, p, h  chosen which were more representative of the state of the FH system at the time of interest.  2.4.2  Isoclines and C r i t i c a l Points in the V-m Phase Plane With the three variables n, p, h held constant at their resting values  n , p , h , the l o c i of points corresponding o o o calculated.  to F. =0 1  and F„ = 0 may be 2  From the FH equations with zero stimulus current;  ?  G  v or • z (-NA>HB<> - V ¥ >" Vo W > " 0  where GNfl(V) and G R (V)  2(  v  2(  8  *  (V  " V)  are the Goldman factors for sodium and potassium dV  respectively.  v  Solving the above equations for —  dm = 0 and —  = 0 gives two  equations for m in terms of V, dV  n  = 0 :  2 -SA(V - V m =  - VQ^CV) ^*w h Na o Na(V) G  M  \»Q\W 1)  26  dm = dT  °  m  *  =  m a (V) + B (V) m  Since 0 < m  (at  m  < 1 the negative root i n 1) i s ignored, and  root i s r e t a i n e d . positive.  „.  2 )  Since G „ (V) < 0 f o r V < 0.123 Na  only the positive 2  v o l t s , the value of m  The factor G^a(V) becomes zero for V approximately equal to 123  is mV  20°C), hence a v e r t i c a l asymptote of the V i s o c l i n e i s expected near that  value.  The  denominator of the m i s o c l i n e  term i s never zero, however, i t  a s y m p t o t i c a l l y approaches the numerator a (V) as V increases. As V decreasm e s , <xm(V) approaches z e r o , and  thus we  expect the m i s o c l i n e to be a ramp  shaped curve starting near zero increasing to some constant value less than 1 for  larger values of V. The  two  isoclines  are  plotted  in  Fig.  2.3  as  functions  of  the  independent variable V, by regularly increasing V over the i n t e r v a l 0 < v < 120.0  mV and calculating the corresponding m values from 1) and 2 ) .  With n,  p, h held constant at their i n i t i a l values, there are three intersections of the i s o c l i n e s , and hence three c r i t i c a l point (0.0 mV,  points i n the V-m  phase plane.  The  0.0005) i s a stable improper node, the point (17.0 mV, 0.055)  i s an saddle p o i n t , and the point (120.0 mV,  0.98)  i s another stable improper  node. The values of V and m corresponding  to the c r i t i c a l points were deter-  mined by l i n e a r i n t e r p o l a t i o n to l o c a t e the p o i n t s where m = V.  To more  accurately determine the location of the stable resting p o i n t , n, p, h were held constant at t h e i r resting values and the response to a n u l l stimulus was calculated.  The r e s u l t i n g values of m and V quickly approached those corres-  27  Figure 2.3  Resting State Isoclines of the Vm Pahse Plane  Resting state i s o c l i n e s dV/dt-0, dm/dt-0, of the reduced Vm phase plane.  28  ponding to the stable resting point - these values were (-0.16 mV, 0.00048). Similarly, for the stable point near 120 mV, the equations were solved with n, p, h constant at their resting values, starting with V=120 mV, m=0.98. The  computed response converged quickly to (119.3 mV,  saddle point near 17 mV, the sign of  0.994).  For the  was altered to transform the point  into a stable c r i t i c a l point and the computed response from an i n i t i a l value in  the neighbourhood of the point was observed to converge to (17.0  mV,  0.0544).  2.4.3  Threshold Characteristics of the FH System Although the complete FH system i s quasi-threshold in performance, for  certain  restricted  behaviour.  parameter  ranges i t exhibits  singular-point  threshold  The c r i t i c a l parameter ranges are illustrated in Figs. 2.5-2.8  and are discussed below. The saddle point at (17.0 mV,  0.0544) gives the FH system i t s sharp  threshold characteristics for certain parameter ranges. With one exception, a l l trajectories eventually diverge from the neighbourhood of the unstable saddle point. divergent  The single convergent trajectory separates two families of  trajectories  of which one group w i l l  veer towards  the stable  resting node while the other group w i l l head towards the stable excited node. These  two  distinct  trajectory  groups represent  threshold trajectories of the FH system. or  separatrix, may  supra-threshold  The single convergent trajectory,  be calculated by numerical reverse integration from a  starting value in the neighbourhood of the saddle point. resulting  from  and sub-  such a  calculation  i s shown in F i g . 2.4  The separatrix along  with a  29  Figure  2.4  K e a t i n g S t a t e Phase P l a n e D e t a i l  R e s t i n g s t a t e phase p l a n e near (dotted) divides sub-threshold phase plane t r a j e c t o r i e s .  the u n s t a b l e saddle p o i n t . The s e p a r a t r i x ( l e f t m o s t ) and s u p r a - t h r e s h o l d (rightmost)  30  representative trajectory from each of the supra-threshold and sub-threshold groups. The presence of this saddle point, and hence the STP behaviour of the FH system, i s transient for the f u l l set of variables and for any reduced system which includes the effects of recovery from the excited state.  2.4.4  Phase Plane Variations with n, p, h Although the presence of a stable excited node is indicated by  this  reduced phase analysis, calculated solutions of the f u l l FH equations show no such stable excited transient and  response.  excited  state of the  full  system is  the computed transmembrane potential returns to i t s original  stable resting value.  The reduced phase plane, however, represents only that  part of the five-dimensional space with the  The  phase space formed by the intersection of the  surface defined  by the i n i t i a l values of n, p, h.  Hence  variations in the other three variables n, p, h must be considered to understand the transient nature of the excited state. Assuming the FH standard response to be fairly typical, the following variations in the parameters n, p, h during the course of an action potential were noted: 0.0269 < n < 0.56 0.0049 < p < 0.02  0.188  < h < 0.8249  To observe the effects of changing n, p, h upon the phase plane, isoclines were re-calculated for values of each variable within the above ranges.  31  2.4.4.1 Potassium Activation Variable n As n increases from i t s i n i t i a l value of 0.0268, the magnitude of the potassium current component also increases.  In Fig. 2.5 are a series of  level curves of dV/dt = 0 for various values of n. The m isocline i s not altered by changes in n. The general effect of increasing n i s to raise the activation threshold through a translocation of the unstable saddle point to a position 'higher' on the m isocline, and to 'lower' the stable excited node.  2.4.4.2 Unspecific Current Activation Variable p As p increases from i t s i n i t i a l value of 0.0049 to i t s maximum near 0.188, the unspecific current component increases. A series of plots of the V isocline for various p values are shown in Fig. 2.6. Increasing p has the effect  of causing a coalescence of the stable  resting node and the unstable saddle point u n t i l , i f p i s large enough, the two  critical  points vanish.  remains, and every  In this case only the stable excited node  trajectory becomes supra-threshold.  The saddle point  vanishes for p > 0.13 (approximately).  2.4.4.3  Sodium Inactivation Variable h  The resting value of h i s the maximum value attained by h during the course of an action potential.  Decreasing h from i t s resting value decreases  the magnitude of the sodium current and, i f h i s small enough, turns the current o f f .  In F i g . 2.7 a series of V isoclines i s plotted i n the phase  plane for various h values.  32  Figure 2.5 A f f e c t s of Changing 'n' on the Vm Phase Plane The a f f e c t of changing potassium a c t i v a t i o n v a r i a b l e , n, on the phase p i The m i s o c l i n e i n unaffected.  33  Figure 2.6 A f f e c t s of Changing 'p' on the Vm Phase Plane  The a f f e c t of changing unspecific a c t i v a t i o n v a r i a b l e , p, on the phase plane. The m i s o c l i n e i s unaffected.  34  Figure  2.7  A f f e c t s of Changing * h ' on the Vm Phase P l a n e  The  affect  of  The m i s o l c i n e  changing is  sodium a c t i v a t i o n  unaffected.  variable,  h,  on the  phase  pi  35  As h decreases, no change i n the position of the resting node occurs. However, the excited node i s shifted to the l e f t along the m i s o c l i n e and the saddle point i s shifted to the right along the m i s o c l i n e .  These two nodes  ultimately coalesce at some value of h near h = 0.02, then both nodes vanish for smaller h values leaving the resting node as the only c r i t i c a l point i n the system.  2.4. A.A  Sodium and Potassium Permeability  From  F i g . l c ) and d) values  determined  at the instants  sodium and potassium.  of the variables  corresponding  n, p and h can be  to peak membrane permeability to  The phase plane i s shown i n F i g . 2.8 as i t appears f o r  maximum sodium and potassium permeability. At peak P^a the phase plane changes are not l a r g e . threshold i s observed  A s l i g h t increase i n  due to the small change i n saddle point l o c a t i o n .  At  dm the same time the stable excited node moves l e f t along  = 0.  At peak P^ the phase plane i s d r a s t i c a l l y a l t e r e d .  Both the stable  excited node and the saddle point have vanished, leaving the stable resting node as the only c r i t i c a l point i n the phase plane. thus  completely  un-excitable  in a  state  At peak P^ the system i s  corresponding  to the absolute  refractory period of the nerve, and w i l l remain so u n t i l the V i s o c l i n e again intersects  the m i s o c l i n e  to produce a stable  excited node.  When the V  dm i s o c l i n e does r e - i n t e r s e c t - 3 — = 0, the t h r e s h o l d w i l l i n i t i a l l y be quite dt high, since the saddle point w i l l be located high on the m i s o c l i n e , but w i l l decrease  as the saddle point approaches i t s resting  location i n the phase-  plane - this corresponds to the nerve's r e l a t i v e refractory period.  36  2.5  DETERMINING THRESHOLD STIMULUS PARAMETERS  Given the sharp threshold c h a r a c t e r i s t i c s of the FH system when a saddle point i s present determine  i n the V-m phase planes, i t i s of interest to be able to  the minimum  stimulus  amplitude  which  will  elicit  an action  potential. In [53] Frankenhaeuser computed a series of responses to short duration monophasic current stimuli " . . . to f i n d a threshold strength f o r regenerative activity". such  To do t h i s , a stimulus amplitude was changed i n successive runs  that  i t converged  to the l i m i t i n g  Frankenhaeuser i d e n t i f i e d  threshold  strength.  stimuli as supra-threshold  Apparently  by observing  the com-  puted action potential - i f the computed potentials did not show an active response  (by demonstrating the t y p i c a l  120 mV action potential) they were  c l a s s i f i e d as sub-threshold. The i n t u i t i v e meaning of threshold stimulus amplitude i s c l e a r , but more precise  definitions  are required  to automate  threshold  detection and to  remove descriptive ambiguities which can arise when applying the i n t u i t i v e threshold  amplitude d e f i n i t i o n  definitions inflection  of threshold definition.  i n [51].  stimulus While  FitzHugh  amplitude,  these  provided  the l e v e l  definitions  two a n a l y t i c a l  definition  were formulated  and the f o r the  Hodgkin-Huxley system, they are v a l i d f o r the Frankenhaeuser-Huxley equations as w e l l . The  l e v e l d e f i n i t i o n of threshold requires a c r i t i c a l value, V , of the  transmembrane potential V to be selected, and to be used i n conjunction with a  stimulus-response  stimulus-response  curve  curve  to define  i s a continuous  threshold  stimulus  amplitude.  The  curve of peak transmembrane poten-  37  Figure 2.8 Vm Phase Plane at Peak P N a and Peak P R The phase plane at the instant of maximum sodium permeability, PN_ (h - 0.76, p - 0 . 0 3 , n - 0 . 1 3 ) , and maximum potassium p e r m e a b i l i t y , P R ( h - 0.025), n - 0.555, p - 0.19).  38  t i a l , V , as a f u n c t i o n o f stimulus amplitude f o r some p a r t i c u l a r waveform.  A set of stimulus-response  square stimuli  curves  f o r FH standard  at two temperatures are shown i n F i g . 2.9.  stimulus  monophasic  According to the  l e v e l d e f i n i t i o n , the threshold stimulus amplitude i s the stimulus amplitude at w h i c h  the s t i m u l u s  response  curve  crosses the c r i t i c a l  level V . c 2  A r b i t r a r i l y choosing Vc=60 mV, t h i s gives a threshold of about 0.75 mA/cm at 2  20°C, and 0.84 mA/cm The  value  at 15°C f o r the FH standard  of threshold amplitude obtained  120 us monophasic pulse.  i n this manner i s only s l i g h t l y  a f f e c t e d by the c h o i c e of V c as long as the maximum slope of the stimulusresponse curve i s l a r g e . large.  For a DTP or STP system, the slope w i l l always be  For a OTP system the responses may be graded, and hence the slope  values may be more moderate. The  i n f l e c t i o n d e f i n i t i o n of threshold requires that one calculate the  slope of the stimulus-response stimulus  amplitude  curve  corresponding  at various stimulus amplitudes.  to maximum  defined to be the threshold amplitude.  stimulus-response  The  slope i s  Although requiring more computation  than the l e v e l d e f i n i t i o n , the i n f l e c t i o n d e f i n i t i o n  has the advantage of  giving a unique threshold amplitude even for stimulus-response  curves showing  a graded, rather than sharp, threshold response. For a given stimulus waveform, to determine the threshold amplitude, a series  of computations are required  stimulus response curve.  to obtain a number of points on the  If the l e v e l d e f i n i t i o n i s being used, computation  w i l l cease when the peak response exceeds the p r e v i o u s l y established V level.  If one i s using  the i n f l e c t i o n d e f i n i t i o n  of threshold then the  39  F i g u r e 2.9 Stimulus-Response Curves f o r 120 ws Monophasic Square S t i m u l i at 2 0 ° C , and 1 5 ° C .  Threshold V ^ 0 mV; £  amplitudes determined by the l e v e l d e f i n i t i o n o f 0.75 mA/cm @ 2 0 ° C , and 0.84 mA/cm @ 1 5 ° C . 2  2  threshold  with  40  threshold amplitude has been reached when the next stimulus amplitude results i n a lower stimulus-response slope. The i n f l e c t i o n d e f i n i t i o n was used by FitzHugh [ 5 0 ] to calculate threshold  amplitudes f o r anodal break e x c i t a t i o n i n the Hodgkin-Huxley  l e v e l d e f i n i t i o n was used by B u t i k o f e r and Lawrence produce  threshold  charge  curves f o r biphasic  d e f i n i t i o n s were adequate  square  [44]  system.  with  V =80 c  stimuli.  A  mV to  While  both  i n each of the a p p l i c a t i o n s , the d e f i n i t i o n s are  computationally i n e f f i c i e n t and, i n extreme cases, ambiguous. Ambiguities duration  can arise  biphasic  potentials  pulses.  of almost  when trying Such  arbitrary  to determine  pulses  thresholds  can generate  transient  magnitude while e l i c i t i n g  f o r short membrane  an o v e r a l l  sub-  threshold response and subsequently confusing the stimulus-response p i c t u r e . 2.10.  An example of such a response i s shown i n F i g . ties  f o r short  biphasic  stimuli  i t would  To avoid these ambigui-  be necessary  to exclude  from  consideration responses during the stimulus, and to note the maximum response upon conclusion of the stimulus. Computational definition  inefficiencies  result  of sub-threshold response.  due Thus  to the lack  of a  the threshold  positive  program  may  continue to compute a stimulus response when, i n f a c t , there i s no p o s s i b i l ity  of that  response generating an action p o t e n t i a l .  One normally would  select a time l i m i t and state that a response which was s t i l l not active at the  end of the time l i m i t was sub-threshold.  Shortening the allowed time  i n t e r v a l to improve computation e f f i e n c y , however, w i l l have the effect of a r t i f i c i a l l y i n f l a t i n g the threshold amplitude.  41  8 0)  >  0)  o  Q.  c 01  fg c  8 'o.oo  o'.io  oTaJ  oTao  Tiae (•i Ill-seconds)  0.40  Figure 2.10 Sub-theshold Response to 5 ps Biphasic Square Stimulus 0.0 ps Phase Delay 2  Subthreshold response to a 34 mA/cm 5 us wide biphasic square stimulus with 0.0 us phase delay. Note that threshold, according to the l e v e l d e f i n i t i o n , i s exceeded i n this example.  42  2.5.1  The Separatrix D e f i n i t i o n of Threshold  The phase plane and separatrix of F i g . 2.4 suggest an unambiguous and computationally e f f i c i e n t  threshold d e f i n i t i o n which may be applied as long  as the saddle point and associated separatrix exist i n the V-m phase plane. If,  at the completion  of a  stimulus  of duration  t the resulting  phase  trajectory has not escaped to the supra-threshold region of the phase plane as defined by the separatrix, then the stimulus i s sub-threshold. problem of threshold d e f i n i t i o n becomes one of determining  Thus the  on which side of  the separatrix a point l i e s which corresponds to the phase plane value of the trajectory  at stimulus  end.  In the case  of an exponentially  decaying  stimulus, some convenient d e f i n i t i o n of stimulus end must be employed. To  implement  this  threshold d e f i n i t i o n ,  a least-squares surface was  f i t t e d to a series of separatrices corresponding to V-m separatrix curves f o r various values of the sodium i n a c t i v a t i o n v a r i a b l e , h. This surface gave the sodium a c t i v a t i o n variable m as a function of V and h. t i o n t w i l l give r i s e  A stimulus of dura-  to a point ( V t , mt) i n the phase plane with a corre-  sponding h value of h^, when the stimulus has just ended.  If the separatrix  i s g i v e n by m =m (V,h), then a stimulus i s defined as being supra-threshold s s i f and only i f mt >  VVV  Hence, at stimulus end, the nature of the response can immediately be determined.  This  definition  of threshold  i s insensitive  to large  transient  fluctuations i n V since i t i s concerned only with the value of V, m, and h immediately  after the stimulus i s over.  It o f f e r s a s i g n i f i c a n t improvement  43  i n computational  e f f i c i e n c y since i t i s unnecessary to perform a protracted  computation to ensure that no delayed a c t i v i t y i s p o s s i b l e . The separatrix d e f i n i t i o n does not provide a d e f i n i t i o n of the threshold amplitude as do the l e v e l or i n f l e c t i o n d e f i n i t i o n s . means of determining  However i t provides a  threshold amplitude to an a r b i t r a r y degree of accuracy.  Let e>0 be the accuracy parameter; i f , m  m  (V  ,h  lt " s t t l )  then  the  stimulus  amplitude. nr  t  If i  t  and  m  < £  >  t  WV  a m p l i t u d e g i v i n g r i s e to ( V t » m^)  i s the t h r e s h o l d  - i (V^.li ) < 0 then the stimulus i s sub-threshold, i f s t t  - m (v^,h ) > e , s t ' t '  then  the stimulus i s supra-threshold.  In p r a c t i c e , e  would be bounded below by some f i n i t e non-zero value which would depend upon the degree of influence of the neglected n and  p.  This treatment of theshold separatrix as a series of l e v e l curves i n the V-m  plane, parameterized  by the i n a c t i v a t i o n variable h, ignores the effect  of the variables n and p upon threshold and determines threshold by using the full  sodium  system.  With  the  exception  of  long  duration  r e p e t i t i v e s t i m u l i , neglecting the influence of n and is  stimuli,  or  p upon the separatrix  an acceptable s i m p l i f i c a t i o n . Threshold amplitude l o c i for some t y p i c a l biphasic and monophasic square  pulses determined using  the  Although only r e s u l t s up that  threshold  determined consistently  to  to 100  amplitudes 1%  separatrix d e f i n i t i o n are  of  accuracy.  inaccurate  shown i n F i g .  (is duration are reported, i t was  pulses  up  to  Biphasic  results,  that  500  pulses  us  2.11.  observed  i n duration could  longer  I s , amplitudes  than reported  1  ms as  be  gave supra-  44  Figure 2 . 1 1 Threshold Amplitudes of Biphasic Stimuli Determined by the Separatrix D e f i n i t i o n  Threshold amplitude determined by separatrix d e f i n i t i o n as a function of biphasic current pulse width with phase separation as parameter. Infinite separation corresponds to monophasic pulse.  45  threshold actually generated no action p o t e n t i a l s , however i t was possible to determine to within 1% the threshold amplitude of a monophasic square pulse 1 ms i n duration. The  sources of error i n this method of threshold amplitude c a l c u l a t i o n  include; surface the  neglecting  and  p effects  to f i t the separatrix, and  stimulus.  introduced error  n  It does not  into  i s less  separatrix  1%.  A l l threshold  stimulus  stimuli were observed  amplitude was  by  i t can  amplitude  computing  with  the  approximate  calculate the be  inferred  values  obtained  error  that  the  by  the  the response of a f u l l  threshold amplitude.  response was  FH  A l l such  to generate action potentials and  reduced by 1% - a sub-threshold  In F i g . 2.12  an  neglecting separatrix changes following  amplitude, but  d e f i n i t i o n were tested particular  threshold, using  seem possible to d i r e c t l y  threshold  than  system to the threshold  the  on  i f the  noted.  the r a t i o of the time taken to compute threshold using the  l e v e l d e f i n i t i o n to the time taken using the separatrix d e f i n i t i o n i s plotted for the case of a monophasic square stimulus. observed faster  at  lower  pulse  durations,  for a 5 us wide pulse.  At  the 100  The greatest improvements are  separatrix method us pulse durations  being  over  60x  the improvement  factor i s a more modest 5x. Simulation  of  1  seconds on a PDP 11/23 PASCAL.  ms  of  membrane a c t i v i t y  requires  approximately  120  with an extended i n s t r u c t i o n set for a model coded i n  The parameters governing the numerical integration algorithm are set  to y i e l d highly accurate r e s u l t s , the integration i n t e r v a l being 0.5 a corrector tolerance of 10  -lf  .  us with  Increasing either of these parameters would  decrease the execution time at the expense of numerical accuracy.  46  8  Figure 2.12 Comparison of Threshold C a l c u l a t i o n Times for Separatrix and Level D e f i n i t i o n s  Ratio of time taken to calculate threshold amplitude of monophasic pulse using l e v e l d e f i n i t i o n , to time taken using separatrix d e f i n i t i o n , as a function of pulse width.  47  In  calculating  influenced  threshold  by the i n i t i a l l y  stimulus  provided  amplitudes,  supra-threshold  execution  time i s  amplitude,  and,  as  previously mentioned, by the simulation time l i m i t imposed during each cycle of  the threshold  threshold twice  iteration  process.  When using  a level  at 20°C, one must generally choose a l i m i t i n g  d e f i n i t i o n of  time no less  than  that of the stimulus duration unless the stimulus i s quite brief i n  which case a l i m i t i n g required.  Temperature  time more than will  twice  influence  the stimulus  the l i m i t i n g  duration may be  time  f o r any given  stimulus• Specifically,  f o r a monophasic  current  pulse  of 100 us duration, 2  2  starting with threshold bracketing amplitudes of 1.0 mA/cm and 0.0 cm , with a  limiting  time  of 750 us per i t e r a t i o n  the following execution  times  resulted: - separatrix d e f i n i t i o n : - l e v e l defintion:  93 s e c . 589 sec. 2  Both d e f i n i t i o n s gave a threshold amplitude of 0.8672 mA/cm .  2.6 A measure of the "sharpness"  DISCUSSION of the Frankenhaeuser-Huxley QTP adapted  from that of FitzHugh [48] i s given by: |V(I2;t2) - V ( I 1 ; t 2 ) | |V(I2;t1) - V d ^ t ^ l where I 2 and I± are current pulses of duration t j , respectively  producing  48  "all" and  and "none" responses, V(I;t) i s the computed response to I at time t , t 2 i s the time of maximum response  r a t i o , the more sharp i s the  to stimulus I j -  5  system based upon unpublished data of Cole.  hold  larger  this  QTP.  In [48] FitzHugh reported a sharpness of 9.8*10  system  The  f o r the  Hodgkin-Huxley  Similar calculations for the FH  show that for monophasic Dirac current pulses, s l i g h t l y supra-thresand  slightly  sub-threshold pulses y i e l d  transmembrane potentials  29.7871 mV and 29.7872 mV respectively - a difference of 10 nV.  of  The d i f f e r -  ence i n V f o r the two cases measured at the time corresponding to the peak response of the supra-threshold I 2 i s 103.3 7  bound of 1*10  mV.  This establishes a lower  for the sharpness measure of the FH system.  A similar sharp7  ness c a l c u l a t i o n for 100 us current pulses yields a value of 1.1*10 . Since the FH system exhibits a sharpness parameter two orders of magnitude greater than that of the HH system, one would expect to see less response gradation by the FH system. i n F i g . 2.13.  This i s , i n f a c t , the case as i l l u s t r a t e d  This figure shows stimulus response curves of the FH system  for a 120 us monophasic pulse at temperatures of 2.5°C, 20°C and 40°C. must be  noted  that  a  temperature  of  40°C i s a s i g n i f i c a n t  It  extrapolation  beyond the o r i g i n a l 2.5°C to 22°C data base f o r which the various Q^Q'S to the fact that the  W E R E  measured.  The rounding of the 40°C curve i s due  Q10  parameters  for the sodium i n a c t i v a t i o n system are larger than those of the  sodium a c t i v a t i o n system thus increasing the rate of accommodation at higher temperatures.  The curve exhibits no gradation, however, i n comparison with  the HH system at similar temperatures [50], [51].  It i s interesting to note  that Increasing temperature past 40°C eventually results i n a transformation  49  Figure 2.13 Stimulus-Response Curves for 120 us Monophasic Square Stimuli at 2.5° C, 2 0 ° C, and 40°C  The curves exhibit l i t t l e response gradation even at high temperatures. This i s a r e f l e c t i o n of the large 'sharpness' parameter associated with the FH model.  50  of the FH system to a singular-point phenomenon.  T h i s , however, occurs at a  temperature of 375°C and hence i s of no p r a c t i c a l value.  2.7  SUMMARY  The Frankenhaeuser-Huxley model of myelinated nerve has been shown to be a system of the quasi-threshold type for p h y s i c a l l y meaningful temperatures, with a sharpness parameter two Hodgkin-Huxley  system. The  stimulus-response  curves  orders of magnitude greater than that of the  sharpness are  not  of  this  system  observed,  even  i s such at  that  graded  relatively  high  temperatures. The e x c i t a b i l i t y of the FH system, investigated using phase plane analysis,  was  shown  to  depend  upon  the  location  of  associated separatrix i n a reduced phase plane. surface  to  variables  the V,  separatrix as  m and  h, i t was  a  function of  the  a  saddle  point  and  its  By f i t t i n g a least-squares initially  most active  FH  possible to automate a separatrix threshold  d e f i n i t i o n which resulted i n a s i g n i f i c a n t reduction of computation time when iterating  to  a  value  of  threshold  stimulus  amplitude.  This  separatrix  d e f i n i t i o n of threshold i s unambiguous i n cases where the l e v e l and tion  threshold  determination ithm that can  definitions  fail.  Although not  of experimental thresholds be  d i r e c t l y applicable to  the  t h i s d e f i n i t i o n provides an algor-  used to extract computer model thresholds  with experimental data.  inflec-  for comparison  51  3.  OPTIMIZATION OF NEURAL STIMULI  BASED UPON A VARIABLE THRESHOLD POTENTIAL  3.1 The  problem of determining  INTRODUCTION the minimum stimulus amplitude required to  e l i c i t a nerve action potential - the threshold problem - i s one of considerable p r a c t i c a l s i g n i f i c a n c e .  With knowledge of the l o c i of minimum ampli-  tudes as functions of various stimulus parameters and waveform types, one can establish a heirarchy of stimuli ordered by r e l a t i v e comparisons of threshold amplitudes. constant  We refer here to 'amplitude' as the current parameter which i s a  i n the mathematical expression  Although stimulus  of time-varying  stimulus  current.  current may also be defined i n terms of peak or average  current, energy or charge - we propose to discuss threshold with respect to a parameter which may be controlled  directly  discussing  to parameters which would be calculated  threshold with  respect  at the simulator, rather  from, or functions o f , the stimulus amplitude s e t t i n g .  than  For any given stimu-  l u s , the threshold locus may be inspected to select a range of parameters minimizing stimulus amplitude or charge per phase of the stimulus while ensuring generation of a nerve action p o t e n t i a l .  still  Minimizing stimulus ampli-  tude minimizes an amplitude dependent tissue damage component due to ohmic heating  [11], while  stimulator's  minimizing  power source  ponent [55], [ 9 ] .  stimulus  charge maximizes the l i f e  of the  and minimizes a charge proportional damage com-  It i s variously reported i n the l i t e r a t u r e that the tem-  perature increases responsible f o r heating damage are large enough to cause l o c a l blackening of the skin [12], and small enough to be n e g l i g i b l e [56].  52  Such  threshold amplitude  elusive.  The familiar  threshold  amplitude  or charge  loci  strength duration  function  f o r the case  a r e , however, a n a l y t i c a l l y  (SD) curve  provides  a  of monophasic square  useable  constant-  current s t i m u l i , but few other deterministic expressions of threshold amplitude have been previously reported. The  emphasis of this  paper w i l l  expression of threshold amplitude constant current s t i m u l i .  be on developing a simple  f o r the case  analytic  of monophasic and biphasic  The l o c i of threshold amplitude (as a function of  stimulus duration at some constant phase separation) produced by the aforementioned  function,  Frankenhaeuser-Huxley  will  be  compared  amplitude  minimizes  will  tissue  numerical  (FH) model of myelinated  amplitude l o c i of the c l a s s i c a l SD curve. rent  to  also  be used  nerve  results  from  the  and to the threshold  The expression for threshold cur-  to describe the stimulus waveform which  damage due to both  thermal  power d i s s i p a t i o n  and charge  injection. In  [16], McNeal presented a model emulating the response of an eleven-  node myelinated f i b r e to constant current s t i m u l i .  The intention of McNeal's  work was to develop a modelling system where the effects on neural threshold of a r b i t r a r y electrode configurations and f i b r e geometries could be assessed. With  emphasis  passive  on the geometric  Frankenhaeuser-Huxley  aspects  systems  of neural threshold, McNeal used  at each of the eleven nodes.  The  membrane impedances were fixed at values corresponding to the quiescent state of the FH system and were not allowed to vary with transmembrane p o t e n t i a l . The  emphasis i n this  electrodes  and f i b r e s ,  paper i s away from the geometric  so well covered  influences of  by McNeal, and more towards the  53  implications that membrane n o n - l i n e a r i t i e s have with regard to threshold and strength-duration to current  relations.  Hence, our approach i s to calculate responses  stimuli of a single f u l l y active FH node which has a l l physical  parameters normalized with respect to unit surface area, ignoring the effects of electrode geometry. As pointed out by McNeal i n [16], this amounts to an assumption of a stimulus applied to a single node which i s v a l i d i n the case of stimulation with a microelectrode.  Experimental results i n [44] demon-  strate that single node FH model data agree closely with threshold data from transcutaneously extendable  applied  to stimulus  stimuli, scenarios  indicating other  than  that  the assumption  may be  that  of percutaneous  micro-  electrodes.  3.2  THE CLASSICAL STRENGTH-DURATION CURVE  The strength-duration  curve, an expression  of threshold amplitude f o r  constant current monophasic square waves, f i r s t derived by Lapicque i n 1907 [57], takes the form  • w *  W> T  1  -  e  "  T / T  >  (  1  )  where T i s the pulse d u r a t i o n , T the membrane time constant and I ^ the rheobase current. constants  i n this  As pointed equation  may  out by Noble and Stein be functions  [58], although the  of the stimulation  scenario  (electrode geometry, distance from the nerve e t c . ) , the form of the r e l a t i o n is  independent  applicable.  of stimulation  scenario  thus  making  this  relation  widely  54  The  derivation  of this  equation  i s predicated upon two assumptions:  1) the nerve response may be approximated by a passive RC network, and 2) the nerve threshold may be taken to be some constant AVc above resting p o t e n t i a l . The  assumption of passive RC response i s recognized  transmembrane potential effects  becomes too large  become s i g n i f i c a n t  to be i n v a l i d  i f the  (> 20 mV), or i f accommodative  as i n the case of long duration s t i m u l i .  constant threshold approximation  The  was proposed by Hodgkin and Rushton i n 1946  [59], based upon a core conducting cable model. Given a constant  current stimulus of amplitude I, assuming an RC mem-  brane, the membrane potential above resting potential becomes V(t) = I.R.(1 - e "  t/T  )  (2)  where R i s the membrane r e s i s t a n c e , C the membrane capacity and T = RC the membrane time constant  [60].-  For a constant  d u r a t i o n T, assuming a constant  current monophasic pulse of  t h r e s h o l d potential of AVC> the threshold  amplitude, I t n » i s given by I t h = AV c /{R.(l - e " The  T/T  )}  (3)  r h e o b a s i c c u r r e n t , I , , i s defined as l i m I , (vT ) , and thus I , = AV /R ' rh' „ th " rh c  with the r e s u l t i n g SD curve given by  \* - W* - > 1  <>  rI/T  4  Since t = RC may be calculated from known membrane constants only I noting  remains to be determined.  f o r R and C,  This would usually be done d i r e c t l y by  the threshold stimulus amplitude approached by long duration pulses  55  rather than by measurement of the constant p o t e n t i a l threshold AV^ and subsequent c a l c u l a t i o n of the rheobasic current from I , = AV /R. rh c Using a threshold amplitude locus numerically obtained from the FH model as  a  basis  for  comparison  (at  20°C,  with  1/R  =  0.0303  2  mho»cm ,  C = 2.0 uF/cm), i t can be seen that the hyperbolic strength-duration r e l a t i o n does not agree very well with low pulse widths ( F i g . 3.1).  the corresponding  FH threshold amplitudes at  Agreement i s good for pulse widths large with  respect to x, since the asymptotic I ^ value was chosen to match the observed threshold current for long duration pulses.  3.2.1  Limitations of the RC Strength-Duration Curve Apart  from  increasing  deviation  from  the  FH  model  results  with  decreasing pulse widths and the i n a b i l i t y to predict long duration thresholds due  to the  exclusion of accommodation influences, the RC  curve cannot be applied i n the case of biphasic s t i m u l i . constant may  strength-duration  This i s because the  threshold potential of the passive RC network modelling  the nerve  be exceeded only during the leading phase of a biphasic stimulus.  The  biphasic thresholds so determined w i l l be independent of phase separation, a s i t u a t i o n which i s known not to be true [4], [6] and  [15].  In the case of long duration s t i m u l i , the accommodation i n s e n s i t i v e network i s not a good neural model.  In the case of monophasic stimuli  RC and  biphasic stimuli with phase separation greater than about 20 us, the transmembrane p o t e n t i a l does not passive RC network.  greatly deviate  from the  form produced by  a  This can be determined by observing responses of the FH  model to the s t i m u l i i n question.  The source of the strength-duration inade-  56  Figure 3.1  Model Strength-Duration Curves Compared to FH Model Threshold Amplitudes  57  quacies f o r short pulses and biphasic pulses i s thus the assumption of constant potential threshold. In  [15], i t was shown that a threshold separatrix exists i n the phase  space of the FH system.  This separatrix, which divides sub-threshold phase  t r a j e c t o r i e s from supra-threshold t r a j e c t o r i e s was shown to be a strong function of the sodium system of the FH model.  In p a r t i c u l a r , F i g . 2.4 ( F i g . 4  of [15] demonstrates that the threshold separatrix i s not a constant function of V, thus implying either.  that  the threshold potential function i s not constant  If the assumption of constant threshold potential was c o r r e c t , then  the separatrix of F i g . 2.4 would appear as a plane i n FH phase space, i n t e r secting  the V-m phase plane  as a v e r t i c a l  line  independent of m, passing  through the FH unstable saddle point•  3.3 If  THE THRESHOLD POTENTIAL FUNCTION  the assumption of constant  potential threshold i s not adequate f o r  certain pulse width ranges and f o r certain types of stimulus waveforms, then what form would a variable potential threshold take?  The FH model of myeli-  nated nerve was used to provide some insight into this question f o r monophasic and biphasic s t i m u l i . Calculations with the model reveal that there i s a value of transmembrane potential which must be attained by stimulus end i n order to e l i c i t an action p o t e n t i a l .  However, the potential which must be exceeded i s not a  constant but instead appears to be a function of pulse width with short duration pulses requiring higher potentials than longer s t i m u l i .  58  3.3.1  Monophasic Stimuli The locus of these FH system threshold potentials for monophasic stimuli i s shown i n F i g . 3 . 2 .  of various durations calculating  threshold  stimulus  These values were obtained by  amplitudes and then recording the transmem-  brane potential attained by the FH system at the end of the stimulus. given  stimulus  duration  T, the threshold stimulus  such that a 0.1% reduction response.  i n stimulus  At any  amplitude was determined  amplitude produced a  sub-threshold  T h i s locus of V ^ i s c l e a r l y a function of stimulus duration T,  and  i s not c o n s t a n t .  As T increases V ^ approaches some asymptotic value,  and  as T->0, V ^ takes on some other  f i n i t e value.  These trends suggest a  functional r e l a t i o n of the form  V t h ( T ) = a.e" The  constant  bT  + c  (5)  'c' corresponds  to the constant  threshold  potential of the  c l a s s i c a l SD curve, 'a' i s related to the displacement of V required to bring the i n i t i a l phase point of the FH system to rest on the threshold separatrix [15], while  'b' determines  the rate  of decay from the maximum threshold  potential at T=0, to the asymptotic value ' c ' .  3.3.2  Biphasic Stimuli Also  shown i n F i g . 3.2 are the threshold potential l o c i  square s t i m u l i with various phase separations. with  f o r biphasic  For consistency of d e f i n i t i o n  the monophasic potential thresholds, the biphasic potential thresholds  were defined to be the transmembrane potentials attained by the FH system at the  end of the leading  phase of the stimulus.  This  i s consistent  with  59  Figure 3.2  Threshold Potentials for Biphasic Square Stimuli from the FH Model  Threshold potential increments (above resting potential) required for excitat i o n at various pulse widths, f o r monophasic and biphasic square constant current s t i m u l i .  60  consideration of the monophasic pulse as a biphasic stimulus with the t r a i l ing  pulse  infinitely  removed.  The biphasic threshold potential  l o c i for  pulses with phase separation no less than about 20 us, can be represented by functions of the same form as the monophasic case V t h ( T ) = a.e" The  bT  + c  (6)  parameters a, b, and c are constant  are numerically determined by curve  f o r any given pulse separation and  fitting.  In general, a l l three  para-  meters increase as phase separation decreases.  3.4  THRESHOLD CURRENT AND CHARGE  For monophasic stimuli and biphasic s t i m u l i with phase separations not less  than  about  20 us, a  potential  threshold  constants with one exponential term may be used. for  function  requiring  The resulting  three  expression  threshold current as a function of pulse width at a particular  phase  separation i s  V t h  using  (1 - e  the variable  membrane.  I e  T / t  )  rh*  ( 1  "  (1 - e  e  5  T / t  )  threshold i n conjunction with  The constant 'b' i s the exponent which appears i n the V ^ expres-  s i o n , w h i l e x i s the membrane time constant. to  the assumption of an RC  the previously  defined  rheobasic  current  The constant I , i s i d e n t i c a l rn of the c l a s s i c a l  strength-  duration curve and i s determined from lim I , (T) with  _ I  r  h  - c/R  th  (8)  61  where ' c' i s the asymptotic potential threshold constant. the  theta current, i s related  function  stimulus, and  to the threshold potential of a Dirac delta  i s determined  from  constants  expression for exponential potential threshold. o l d c u r r e n t pulse, some charge Q potential of AVQ = QQ/C. o o  The parameter 1^,  'a' and  'c' i n the  For an instantaneous thresh-  i s injected with a r e s u l t i n g transmembrane  The I . i s defined to be o  I 0 = AVQ/R  (9)  The threshold current predicted from this form of strength-duration curve i s shown i n F i g . 3.3 along with the threshold current values determined from the FH model. Qth  The corresponding threshold charge function can be determined from = Ith(T)*T  (10)  These calculated threshold charge values w i l l  over estimate  the actual FH  charges due to the 1 ^ ( 1 ) r e s u l t s being greater than the corresponding FH threshold  amplitudes.  As can be seen from F i g . 3.3, the RC model i n conjunction with the exponential threshold potential y i e l d s r e l i a b l e r e s u l t s only up to pulses of about 100 us i n duration, with phase separations greater than 20 us.  3.5  STIMULATION WITH MINIMUM POWER  One of the most interesting results derived from the constant threshold SD r e l a t i o n i s the minimum power waveform of Offner [61].  Using a passive RC  network with a constant potential threshold, Offner showed that f o r a mono-  62  8 *1 8 9-  r  =8  8  loo  oTio  oTao  oTso  . o.40  oTii  o'.eo  rr o'.ao o'. 40  Pulse Width (•illi-seconds)  O.HO  8  8  r 8  •bToo  Pulse Width (•!Ill-seconds)  o.to  Figure 3.3  RC-Varlable Threshold Potential Strength-Duration Curves Compared to FH Model Threshold Amplitudes Threshold current amplitudes calculated by the FH model compared to the threshold amplitudes determined from a passive RC model using i n conjunction with a variable threshold potential function.  63  phasic stimulus the waveform of duration T minimizing  generated heat while  s t i l l exciting the nerve has the form of a r i s i n g exponential current 2  ^ f n e r ^ with  R(e  (  T/x  V  A  t/x  -T/T, - e )  x - RC the membrane time c o n s t a n t , and AV the constant ' c  threshold  potential. Assuming a variable potential threshold and a damage component  propor-  t i o n a l to power, as did Offner, one can minimize T  2 H = k / I (t)dt o  k = constant,  (12)  for a stimulus of duration T, subject to the constraint that the threshold condition i s met within time T, while  the transmembrane potential  evolves  according to  V(t) -  e  -tlx t . s/T - g — • / I(s)e ds o  (13)  This i s accomplished by introducing a Lagrange m u l t i p l i e r , X , to minimize  T  2  / (kl (t) - £ e o  - T / T  e  t/T  I(t))dt  (14)  subject to the constraint that V(T) = V t h ( T ) = a » e " From the Euler-Lagrange  bT  + c  equation of v a r i a t i o n a l c a l c u l u s ,  (15)  64  with the m u l t i p l i e r \ being eliminated to y i e l d  R(e  - e  )  bT  Imm.v (t) ' = RI- e " « csch(T/x) \ ' / • e The  t/T  v + IOffner ... (T)'  (18)  expression for heat generated by a stimulus i s minimized for pulses of  infinite s i m pv l e  duration. RC  T h i s , however, i s beyond the recognized  m o d e l , and '  hence any v a l u e of I . ' min  obtained  l i m i t s of  the  J by e v a l u a t i n&g  lim Im i.n (T) should be cautiously i n t e r p r e t e d . T+The  assumption of a potential threshold with  minimum power waveform which has  the  same r i s i n g  Offner minimum, but with a larger i n i t i a l value. s  v a r i a b l e threshold minimum approaches ^offner * range of pulse width where the constant  three constants yields a exponential  form of  As pulse width T grows, the n c e  potential  ar  ^- 8  e  ^ corresponds to a  threshold assumption i s  more applicable ( u n t i l the effects of accommodation become s i g n i f i c a n t ) . largest differences i n the two waveforms occur for small T values. the i n i t i a l  I . min  The  'a', parameterized  constant  separation, corresponding  The mizing  As  The TO  value exceeds that of I,^,-,by a factor of (1 + a/AV ) . J Offner c  the monophasic case, to  of 1.7 and 2.3  the  17.2  by phase separation, ranges from 9.3 mV for mV  for a biphasic pulse with a 20  to i n i t i a l  us phase  current amplitude increases by factors  respectively.  variable  Offner-like  threshold potential l o c i used to derive the power minis t i m u l i were produced with reference to biphasic square  65  waveforms rising  and hence may  stimuli.  not be applicable to the case  of exponentially  Calculation of threshold potential values  f o r the r i s i n g  exponential waveforms reveals that f o r short duration s t i m u l i the threshold potentials  are v i r t u a l l y  identical,  while  at longer  stimulus  s l i g h t differences i n threshold potentials were observed.  durations,  Quantitatively the  following differences were noted: - f o r 5us duration monophasic s t i m u l i : n square stimulus - V , = 29.45 mV th  i }  .„ _ AV = 0.01 mV  i }  , _ AV = 1.2 mV  n  i  exponential stimulus - V ^ = 29.44 mV - f o r 500 us duration monophasic s t i m u l i : square stimulus - V , = 20.68 mV th exponential stimulus - V ^ = 21.85 mV  - f o r 5 us duration 20 us phase separation biphasic s t i m u l i : i f  n square stimulus - V , = 38.74 mV th  .„ . AV = 0.02 mV n  T  exponential stimulus - V n = 38.72 mV - f o r 500 us duration 20 us phase separation biphasic s t i m u l i : M square stimulus - Vth , = 22.43 mV  l }  »TT= -i -i-? ™ AV 1.17 mV  exponential stimulus - V ^ = 23.60 mV At  low  threshold  stimulus  durations  potentials  for a l l phase  slightly  exceeded  separations,  those  the square  of the exponential  wave  stimuli,  while at long stimulus durations f o r a l l phase separations the square wave threshold  potentials were less than those of the corresponding  exponential  stimuli.  This result i s as expected since at short durations the exponential  'spike' causes the FH system to have a larger dV/dt at stimulus end than does a square wave; while at long stimulus widths the i n i t i a l low amplitude of the  66  exponential  stimuli  means  that  the sodium  increase, dm/dt, i s low thus implying  activation  variable  that a larger threshold  rate of  potential i s  required to excite the FH system [15]. The  slight  differences  between  threshold  potential  loci  f o r the two  types of s t i m u l i should not be s i g n i f i c a n t enough to cause concern over use of  the square  stimulus  potential  thresholds  to derive  the form  of the  exponential waves.  3.6  STIMULATION WITH MINIMUM CHARGE  In addition to the stimulation damage caused by heat d i s s i p a t i o n i n the t i s s u e , there i s a possible charge proportional damage component [9], which in  the case of a charge balanced  biphasic  stimulus would be i n terms of  charge per phase of the stimulus. Let  H be the heat related damage component, Q be the charge related  damage component and y be some dimensionless Define  D(T) to be a function  separation  which  expresses  of pulse  damage  parameter between 0 and 1.  duration  parameterized  due to stimulation  by phase  i n some normalized  dimensionless units as D(T) - Y H + (1 - Y)Q  (19)  with  H  2  /  I (t)dt  (20)  I(t)dt  (21)  o  T and  Q  / o  67  In this damage function the value of y would be set to r e f l e c t the r e l a t i v e degree of importance placed upon the two damage mechanisms. Minimizing  this new function subject to the same threshold constraint as  before gives V W  T  >  where  v t n  (T)  (-^-  (I) I  s  +  T  1  C l - e- ^))csCh(T/,)  •e ^  +  \  (22)  the previously defined threshold potential f u n c t i o n . Substi-  t u t i o n of t h i s  stimulus  waveform into  the damage function gives composite  damage as 2  D(T) = Y x A ( T ) c s c h ( T / T ) +  ( 1  (23)  with V  (T)  A(T) = ( - ^ —  e  T / 2 T  +  (LZJL)  sinh (T/2x) ).  (24)  The minimum of t h i s expression i s expected to occur at some f i n i t e non-zero value occurs  of T, parameterized as T+0  by y and phase separation, since minimum charge  [3], and minimum heating  occurs  as T-><» [61].  Calculation  reveals that the minimum of the damage function occurs when  2T The  = A(T)coth(T/x)  (25)  r e s u l t i n g expression i s awkward and i s best solved numerically.  selected  as 0.5  (equal  weight  With y  to heat and charge damage components) the  following damage minima were obtained for given phase separations: -  f o r a monophasic square pulse, minimum damage at a pulse width of about 400 us  68  -  f o r a biphasic square pulse, with 20 as phase separation, minimum damage at about 500 us pulse width.  The locus of damage minima as a function of y i s shown i n F i g . 3.4 f o r both v  of the above pulses.  Since the r e l a t i o n of the parameters determining  T  t  j1( )  i n terms of phase separation i s not e x p l i c i t , i t i s only possible to minimize D(T)  on a constant  separation isosurface, and numerical  comparisons would  have to be made between D(T) values to find a global minimum.  3.7  CLINICAL DETERMINATION OF THRESHOLD PARAMETERS  In order to use equation minimize for  (2 5) to optimize the stimulus pulse width to  stimulation damage, i t i s necessary to determine numerical  the threshold  parameters  parameters i n a c l i n i c a l  a, b and c  setting  A method  to evaluate  values these  f o r a biphasic stimulus with given phase  delay i s outlined below. The  expression  f o r threshold current calculated  from the RC-variable  threshold assumption i s given by:  T  It h (T ) =  —  —bT  _  -bT.  —— + — 1  (1 - e " ^ )  (1 - e  (26)  —.  )  C l i n i c a l l yJ , one can measure I . and I . d i r e c t l y . rh 9  It w i l l be shown that 'b'  may be r e a d i l y evaluated from knowledge of the chronaxie, T £ . The  rheobase  current, Irn»  can be e a s i l y obtained by n o t i n g the  t h r e s h o l d amplitude of long duration (but non-accommodative) pulses.  The 1^  parameters must be determined by noting the threshold amplitude of a short d u r a t i o n D i r a c d e l t a stimulus of width 6, denoted by 1^(6).  Ideally 5 i s  69  Figure  3.4  P u l s e Widths M i n i m i z i n g the Normalized Damage F u n c t i o n  P u l s e w i d t h v a l u e s m i n i m i z i n g the n o r m a l i z e d damage f u n c t i o n , by phase s e p a r a t i o n , as a f u n c t i o n of the damage parameter.  parameterized  70  zero, but p r a c t i c a l l y i t i s only necessary smaller than the membrane time constant 66 us).  significantly  x, (which for the FH model i s about  With a s u f f i c i e n t l y small 6 value, I 0 i s defined such that  6 • I . (6) I  =  ^ T  9 The  that i t be a value  Q (6) :  (27) '  V  T  i n t e r p r e t a t i o n o f I i s that i t represents the charge injected through 9  the membrane as capacitive displacement current i n time 6. i t i s related to the instantaneous  Mathematically,  potential increment required to displace  the phase point of the FH system to the 'base' of the threshold separatrix [15]. From the rheobase current one can determine the parameter 'c' using c = I , •R rh  (28)  The value of 'a' can be calculated from the rheobase and theta currents and the membrane resistance R: a  =  (I  I  9 ' rh  )  The decay constant  '  R  ( 2 9 )  'b' may be related to chronaxie by I  b =  i-  ln  —  9  I, rh  (30)  c  C l i n i c a l l y , the chronaxie amplitude  to twice  control u n t i l  Tc  rheobase, then  a setting  threshold amplitude.  T £ i s determined by f i r s t setting the stimulator  i s reached  by adjusting the stimulus  pulse  width  f o r which the chosen amplitude i s the  Substitution of the stimulator pulse width setting f o r  i n (3) w i l l allow c a l c u l a t i o n of 'b'. These variable threshold potential  71  parameters may now be used i n (25) to evaluate the optimal pulse width once the damage parameter y has been set to the desired value.  3.8 DISCUSSION In  the  preceeding  a n a l y s i s , i t has been assumed that the threshold  response of a myelinated nerve can be modelled by a passive RC network and a modified decaying exponential threshold potential function. Derived from the Frankenhaeuser-Huxley model of myelinated nerve, the potential function gives a threshold current locus which i s i n good agreement with the FH thresholds to a pulse width of about 100 us, and i n only f a i r  agreement with  pulses  longer than this to a reasonable l i m i t of about 500 us. Exhibiting the same functional form f o r monophasic and biphasic pulses, up to a phase separation of about 20 us, the threshold potential function contains three constants, a , b and c, which must be numerically determined f o r a given phase separation. The dependence of these parameters upon the phase separation of the stimulus i s notable, but the form of this dependence i s not known. Hence any results obtained using this threshold potential function are f o r some specified value of phase separation, and thus, i n the case of minimizing some stimulus damage function, the minimum separation value  can only be determined by r e l a t i v e  comparisons of the damage f u n c t i o n . For  equally weighted  contributions of power and charge damage compo-  nents, damage minima were found to l i e at pulse widths of about 400 us for monophasic pulses with  stimuli  (infinite  phase  20 us separation.  separation), and 500 us f o r biphasic  These pulse  separation settings represent  upper and lower l i m i t s for which the exponential threshold function i s a p p l i -  72  cable.  Any other  given separation between the monophasic and 20 us value  w i l l give a minimum between 400 us and 500 us pulse width., f o r a y of 0.5. This range of values i s just within the 500 us l i m i t beyond which the RCSelection of smaller y values which  variable threshold model i s inadequate.  increase the importance of the charge damage component, would give minimum pulse widths at lower values closer to the region of good agreement between the FH and RC-variable  threshold models.  For a y of 0.1, the monophasic  minimum occurs at 157 us, while the biphasic 20 us minimum i s near a pulse width of 234 us. optimal  pulse  Ay  widths  value of larger than 0.5 would begin to y i e l d past  the range  of a p p l i c a b i l i t y  threshold model, and thus such r e s u l t s  should  long  of the RC-variable  not be considered  accurate.  The optimum pulse width ranges as functions of y are shown i n F i g . 3.4. For biphasic stimuli with phase separations less than 20 us, Butikofer and  Lawrence [44] showed I n d i r e c t l y , using an FH model, that the minimum of  charge on a given threshold locus occurred at some non-zero pulse width setting.  As phase separation decreased  below 20 us, a maximum of threshold  charge was demonstrated by threshold Dirac delta function s t i m u l i i n contrast to the charge minima shown by Dirac pulses with phase separations of 20 us or greater.  Such trends lead one to expect minimum damage pulse width l o c i f o r  short separation biphasic pulses to closely resemble those of F i g . 3.4. However, the l o c i would have v e r t i c a l s h i f t s , so that as y+0 they would i n t e r sect the ordinate at a non-zero pulse width corresponding  to the location of  the charge minimum for the stimulus i n question. In  addition to minimizing  stimulus  damage  due  to charge  and heat  mechanisms, one may also elect to consider optimization of the stimulus with  73  respect to certain design features of the stimulator. such features i s the stimulator compliance.  One of a number of  A 'cost' function which mini-  mizes both damage and the voltage range of current stimulation i s C = Y H + (1 " Y)Q + Performing  ni(t)  (31)  the same minimization  procedure  subject  to the same threshold  constraints as before yields a 'minimum cost' stimulus waveform which s a t i s f i e s the d i f f e r e n t i a l equation  | i + kL(k2 - \e  t/T  )I = k3I  2  (32)  are c o n s t a n t s , x i s the membrane time c o n s t a n t , and \ i s a  where the  Lagrange m u l t i p l i e r .  This equation i s i n the form of a R i c c a t i equation and  has a solution t/T  l(t) =  expU-c . e e e  t/T  -k2t ) -e  rV- XT \ TUU  -j  The prospect of a global solution f o r this expression allowing determination of the m u l t i p l i e r X i s bleak, but asymptotic solutions f o r r e s t r i c t e d ranges of t may be p o s s i b l e . For pulses of very short duration equation  (33) may be simplified, to  give: \t I(t) ~  Sg-  ( l + (\ - k 2 ) t )  (34)  74  3.9 The  strength-duration curve  CONCLUSIONS derived from the assumption of passive RC  neural behaviour i n conjunction with a potential threshold function containing three constants, both produces a good match to the numerically determined FH  threshold amplitudes  neural threshold. monophasic  and result  i n a workable a n a l y t i c a l d e f i n i t i o n of  The r e s u l t i n g threshold d e f i n i t i o n i s applicable to both  and biphasic  pulses.  This  approach  to the threshold  problem  allows e x p l i c i t optimization of pulse widths with respect to a two component damage function which r e f l e c t s  the p o s s i b i l i t y of power and charge related  stimulation damage. The stimulus that minimizes the damage function exhibits the same r i s i n g exponential  form  as the minimum  waveforms are i d e n t i c a l  power waveform  i n the l i m i t  of Offner.  While  both  of i n f i n i t e pulse width, they demon-  strate greatest d i s s i m i l a r i t y f o r short duration s t i m u l i . exponential s t i m u l i are not the most convenient  In practice such  to generate;  but f o r pulses  of widths of up to one quarter the membrane time constant a constant current stimulus waveform i s a good approximation,  and beyond this width, up to a  l i m i t of about one membrane time constant, a l i n e a r l y increasing ramp-like stimulus may be used to approximate the r i s i n g exponential. Use  of the RC-variable  threshold model  allows  one to calculate  an  optimal pulse duration from (25) f o r either monophasic or biphasic waveforms once the threshold parameters a , b, and c have been determined.  A proposal  has been presented for determining the three parameters i n a c l i n i c a l setting that involves measuring rheobase current, theta current and chronaxie.  75  4.  SIMULATION OF HUMAN MEDIAN NERVE THRESHOLD RESPONSE  4.1 It  i s well  known  that  INTRODUCTION  a stable  propagating  action  potential can be  e l i c i t e d from an axon by e l e c t r i c a l stimulation i f the applied stimulus has an  amplitude which i s s u f f i c i e n t l y  large.  The minimum stimulus amplitude  necessary f o r generation of an action potential - the threshold amplitude i s a complicated function of many v a r i a b l e s .  These variables may be c l a s s i -  f i e d according to the nature of their influence upon threshold amplitude. Broadly  considered, there  are four  influencing factors may be separated; environmental.  classes  into which the threshold  f u n c t i o n a l , p h y s i c a l , geometrical, and  The functional class includes the general e f f e c t upon thresh-  old of stimulus waveshape, and the influence of stimulus parameters such as pulse width, phase separation  and frequency.  Temperature, l o c a l i o n i c con-  centrations and f i b r e diameter are examples of factors i n the physical category, while the geometrical grouping encompasses such variables as stimulus electrode configuration and the distance of the target axon from the s i t e of stimulation.  The environmental  class  includes  variables  due to adjacent  passive t i s s u e s , the presence of additional fibres i n the immediate v i c i n i t y of the target axon, and the presence of membranes acting as b a r r i e r s to l o c a l ionic  diffusion.  The boundaries  necessarily i n d i s t i n c t one  of d e f i n i t i o n  of these  since parameters of threshold  four  groups are  influence belonging to  group may have d i r e c t correspondence to influencing factors i n another  group.  76  If one attempts to simulate the threshold behaviour of an in-vivo axon, variables  from  a l l four  threshold  influencing groups must be  considered.  Several models of the nerve membrane have been proposed which simulate membrane dynamics  i n response  active membrane.  to  current  injection  at  an  Isolated patch  of  Such models, which include the Hodgkin-Huxley (HH) model of  unmyelinated nerve [27], the Frankenhaeuser-Huxley (FH) model of myelinated nerve [1], Dodge's myelinated nerve model [28] and others, are best suited to simulations this  seeking  type must be  to reveal functional threshold dependencies. embedded within  Models of  further models with broader horizons  simulate the influence of factors i n the p h y s i c a l , geometrical, and  to  environ-  mental threshold groups. In  [44], Butlkofer and  Lawrence used  an  FH model to obtain  loci  of  threshold amplitude and charge for a biphasic square stimulus, demonstrating the functional v a r i a t i o n of threshold with respect to the parameters of the biphasic stimulus.  To compare model results with threshold l o c i from human  subjects, the same authors devised a psycho-physical experimental protocol to account for environmental and geometrical threshold i n f l u e n c e s , and derived a uniform scale factor to deal with the effects of temperature. Using s i m p l i f i e d FH models to simulate neural a c t i v i t y at eleven sequential  nodes of  Ranvier,  McNeal  [16]  produced  a  threshold  amplitude  locus  r e f l e c t i n g functional threshold v a r i a t i o n s for a monophasic constant current stimulus. model  He also calculated threshold dependence upon f i b r e diameter.  proposed  by  f a c t o r s , accounting distance  McNeal also  allowed  determination  of  some  The  geometrical  for both the stimulus electrode geometry, and introducing  from stimulus  site  to target axon as a threshold parameter.  In  77  [4], Mortimer employed McNeal's model to determine the threshold dependence of myelinated nerve as a function of axon diameter and position with respect to the stimulating electrode. By extending the FH model to encompass more threshold f a c t o r s , and by appropriate design of an experimental protocol to allow f o r the effects of geometrical and environmental demonstrate the a b i l i t y  threshold  influences, this  of an enhanced FH model to predict experimentally  measured in-vivo thresholds from human median nerve. we w i l l  first  paper attempts to  In the following t e x t ,  address some of the general extensions to the FH model which  are required, and then discuss s p e c i f i c model enhancements and experimental protocol with respect to percutaneous and transcutaneous stimulation of human median nerve.  4.2  ENHANCEMENT AND EXTENSION OF THE FH MODEL  The FH model i s a system of f i v e non-linear ordinary d i f f e r e n t i a l equations describing  the space-clamped  response  of a myelinated  toad (Xenopus  Laevis) nerve to current s t i m u l i injected at a single node of Ranvier. FH system class  The  has recently been shown to be a member of the 'quasi-threshold'  of excitable  systems  [15].  The model equations are summarized by  Frankenhaeuser  and Huxley i n [1], and by McNeal i n [16], and w i l l not be  repeated here.  Although derived from experimental observation of nerve from  the African toad, there i s some evidence that the model membrane emulates the membrane dynamics of human and other  nerve  fibres  over  certain  stimulus  78  parameter ranges [44], [45]. The enhancements and extensions discussed here are for the purposes of improving the applicability of the model to the human case, and for expanding upon the model range.  4.2.1  Temperature The effects of temperature are imparted to the FH model through a set of  temperature correction factors (applied to a l l of the model rate variables and ionic permeabilities) as well as through the temperature dependence of the Goldman factors  in expressions for sodium, potassium and  unspecific  currents. In [40], Frankenhaeuser and Moore published the results of experiments determining the behaviour of sodium and potassium rate variables and permeabilities  as  functions  of  temperature.  The  temperature  dependence  was  expressed as a set of multiplicative exponential functions of the form  • (I) - Q S " " 1 0  (1)  1 0  where T i s in degrees Celsius.  The temperature functions are dimensionless,  and have the effect of changing by factors of Q^Q for every ten degree temperature deviation relative to 20°C  In [40], Q1Q  values for the sodium and  potassium systems were presented, with subsequent haeuser  in  [53] of  Q,0  data for the unspecific  publication by Frankencurrent  system.  The  79  unspecific Q 1 Q  data were not experimentally determined, but were selected by  Frankenhaeuser using unstated c r i t e r i a . The  temperature dependence exhibited by the three Goldman factors for  sodium, potassium and unspecific currents, occurs as a product of a linear temperature component and an exponentially occurring temperature term.  With-  i n the Goldman f a c t o r s , temperature i s expressed i n Kelvin degrees and thus the r e l a t i v e change i n factor magnitude f o r a given temperature increment i s much less than the corresponding change undergone by the Celsius based  Q1Q  factors• The  first  s i t u a t i o n we  extension to the FH system arises because  wish to emulate  than the base  exists at a temperature s i g n i f i c a n t l y higher  20°C of the FH model.  fourth finger and at the w r i s t . near 30°C. mum  the experimental  Experimental stimulation was  at the  Both s i t e s were found to have temperatures  Such a temperture i s a s i g n i f i c a n t extrapolation beyond the maxi-  22.5°C of Frankenhaeuser's  nevertheless  used  o r i g i n a l data base, but the 33°C value was  for a l l predictive  model  calculations  i n view  of  the  i n a v a i l a b i l i t y of higher temperature data and the great experimental d i f f i culty i n obtaining i t . At  33°C, the normally constant -70  mV  (20°C) resting  potential  will  change appreciably due  to the r e s u l t i n g change i n thermodynamic e q u i l i b r i a  regulating  potential.  the  resting  Hence, the  resting  potential must  expressed as a function of temperature i n order to calculate the new value. ally  be  33°C  In order to do t h i s , one must introduce three constants, not o r i g i n -  given by Frankenhaeuser, which are related  to a chloride ion system.  These constants, external chloride ion concentration, i n t e r n a l chloride ion  80  concentration and membrane permeability to chloride ions, may be inferred from data quoted by Frankenhaeuser and from well known physiological constants.  The function using these constants which expresses resting potential  as a function of temperature arises from the assumption of a constant field membrane and gives:  V  ( T )=  R*T  r  F  P  l n n  T  + V >[ ]Q  +  T  ( p( >  (P(T>  +  P  ( T ) N a  )t  N a +  ]i  +  P  K  ( T )  K+  t  K +  ]i  +  +  P  P  T  C1  Cl< >[ ~li Cl  ( T )  t  C 1  ( 2 )  "]o  where R i s the gas constant, F i s Faraday's constant, T i s the absolute temperature, the Pj(t)  a r e  temperature dependent ionic permeabilities (with T  in degrees C e l s i u s ) , the [•] the  a r e  outside membrane ion concentrations, and  [*]^ are the inside ion concentrations.  chloride  With the exception of the  ion permeability and concentrations, a l l other factors are known  from measurements made by Frankenhaeuser.  If the three chloride constants  are not introduced, then i t i s not possible to reconcile the resting potent i a l calculated at 20°C using Frankenhaeuser's data, with the value of -70 mV measured at 20°C by Frankenhaeuser. The external  chloride ion concentration may be calculated as 110.5 mM  from the data given by Frankenhaeuser for his frog Ringer's solution [39]. The internal chloride  concentration can be calculated  Nernst potential at 20°C [62], and i s found to be 29.7 mM.  from the chloride The final unknown  - chloride ion permeability - i s calculated from the measurement by Franken7  haeuser of a 20°C resting potential of -70 mV as 1.2(10~ ) cm/s.  Using  81  equation (2) with T = 33°C, the resting potential i s found to be -75.14  mV.  Such a value can s i g n i f i c a n t l y change the threshold behaviour of the model from that at 20°C with V  = -70 mV. r  4.2.2  Passive Tissues and Adjacent Nodes In  order  to model r e a l i s t i c  c l i n i c a l situations where the stimulating  electrodes are f a r removed from the nodal membrane, one must consider other extensions to the FH model. Such extensions were f i r s t suggested by McNeal [16], when he proposed a model of myelinated  nerve which accounted  rounding passive t i s s u e s .  f o r a multi-node  f i b r e and sur-  Using a time invariant RC combination to model the  nodal membrane and assuming a r e s i s t i v e axonal core with myelin sheaths of i n f i n i t e resistance and zero capacitance, McNeal formed an eleven node f i b r e and  conceptually placed  this  fibre  into  an i n f i n i t e ,  isotropic  external  medium. He was able to calculate the responses of any of the eleven nodes to current stimuli by simultaneous  solution of eleven f i r s t order ordinary d i f -  f e r e n t i a l equations representing the evolving transmembrane potentials of the eleven nodes. We fibre,  implemented a substantially simpler version of McNeal's eleven node simulating the presence  infinite,  isotropic  medium  of only  ( F i g . 4.1).  two adjacent However,  passive nodes i n an  such  a  simplification  enabled us to r e t a i n the f u l l y active FH system of equations at the c e n t r a l , a c t i v e , node thus allowing the nodal impedance to vary during the course of a  82  Figure 4.1 Equivalent C i r c u i t of the Three-Node FH Model  Equivalent c i r c u i t f o r the three node f i b r e i n an i n f i n i t e , Isotropic medium. Only the central FH node i s f u l l y a c t i v e , the adjacent nodes are passive with impedance values taken from the quiescent state of the FH model.  83  current stimulus.  The  d i s t o r t i n g effect of such a f i b r e upon the applied  s t i m u l i i s i l l u s t r a t e d for two i n F i g . 4.2a  stimulus waveforms i n F i g . 4.2.  Illustrated  i s a current waveform entering the active FH node resulting from  stimulation with a biphasic square waveform.  The  e f f e c t s of the changing  active node impedance can be c l e a r l y seen i n F i g . 4.2a.  The leading positive  phase of the nodal stimulus shows a c h a r a c t e r i s t i c RC decay i n i t s amplitude. In the case i l l u s t r a t e d i n F i g . 4.2a  the leading stimulus phase i s s u f f i c i e n t  to cause an action potential with a concurrent ance.  decrease i n membrane imped-  The effects of t h i s reduced impedance on current entering the active  node can be seen during the t r a i l i n g phase of the stimulus i n F i g . 2a. current waveform i n F i g . 4.2b  The  i s that which enters the active node as a  r e s u l t of application of a ramped biphasic stimulus similar to the stimulus i l l u s t r a t e d i n F i g . 4.3. This model extension was approximation membrane was  of  stimulating electrodes  not adequate.  waveforms of F i g . 4.2, actual first  used to simulate c l i n i c a l situations where the i n direct  contact  with  the  nodal  From the tissue distorted transmembrane current  i t can  stimulating waveform can  be  seen that the applied stimulus and  differ  to demonstrate this f a c t , although  show none of the v a r i a b i l i t y due  greatly.  In  [16], McNeal was  the the  his distorted stimulus waveshapes  to changing membrane impedances because of  his constant impedance s i m p l i f i c a t i o n .  8  8  • •oo r  7!u TiM  7M  o'.ia.  (•illi-Mconds)  7!a  o.eo  Figure 4.2 D i s t o r t i o n of Injected Current Waveforms  The a f f e c t s of the three node model and the central node's non-constant impedance on the current waveform injected at the central node. a) Current crossing the central node of Ranvier, originating from a biphasic square current stimulus. Note the affects of changing membrane Impedance (time constant) on the two phases of the stimulus. b) Current crossing the central node of Ranvier, originating from a biphasic ramped current stimulus ( i l l u s t r a t e d i n F i g . 2 ) . The actual injected waveform i s s i g n i f i c a n t l y distorted from i t s form as i t leaves the stimulator.  85  8  o5o  oToJ  Tine  oToe  oTii  OTIS  (aim-seconds)  Figure  7.20  4.3  DISA 15E07 S t i m u l u s Waveform  A t y p i c a l s t i m u l u s waveform o r i g i n a t i n g from two DISA 15E07 c o n s t a n t c u r r e n t s t i m u l a t o r u n i t s w i t h i n t e r - p h a s e d e l a y p r o v i d e d by a DISA 15G01 d e l a y u n i t . The r i s e time i s approximately 0.8 us per m A / c m . 2  86  4.3 Due to several  EXPERIMENTAL DESIGN  essential differences between the neural model and the  in-vivo experimental s i t u a t i o n - differences which cannot be reconciled with simple model enhancements - judicious d e f i n i t i o n of the experimental protocol is  required.  Two  of the most  s i g n i f i c a n t differences  between model and  experiment are; the simulation of a space-clamped f i b r e response compared to an in-vivo saltatory response; and  an in-vivo stimulation of many nerve f i b r e s  subsequent measurement of a compound action potential (CAP) compared to  simulation of a single f i b r e response. The problem of comparing single f i b r e model responses to in-vivo bulk action potential recordings can be addressed by determining in-vivo olds  r e l a t i v e to an a r b i t r a r i l y  chosen set p o i n t .  thresh-  The CAP of some nerve  t r u n k , measured i n response to a stimulus of amplitude I  with some set of  pulse parameters p^, can be c o n s i d e r e d to be a l i n e a r superposition of N single f i b r e responses [63]:  y(l,P_,t) =  I g (t) i=l  where g^(t) i s the contributed number of active f i b r e s .  response of the i  (3)  t b  active f i b r e , and N i s the  Clearly N w i l l depend upon many factors from a l l  four threshold groups including the distance from stimulus electrode to f i b r e bundle, orientation of the bundle, diameter d i s t r i b u t i o n of f i b r e s within the  87  bundle and, of course, the stimulus amplitude and parameter s e t .  If the  location and placement of the stimulus electrodes i s not changed, and i f the f i b r e population remains the same, then any changes i n the CAP due stimulat i o n with amplitude I^-AI and the same parameter set p^, i s due to loss of the response contribution of the N  y ( l -AI,p ,t) =  f i b r e ( i f AI i s small enough) with:  I  N-1 g (t) < y ( l ,p , t ) i=l  (4)  Thus the stimulus amplitude I r i s defined to be the threshold amplitude of the N Pr.  tb  f i b r e i n the bundle i n response to the stimulus with parameter set  P r e c i s e l y which f i b r e of the t o t a l bundle population i s so close to  threshold response,  that some small amplitude decrement AI r e s u l t s i n a subthreshold i s determined  e l e c t r o d e s , the i n i t i a l  by  the  Initial  placement  of  the stimulating  amplitude I r , and the pulse parameter set p^. If  y ( l r > P r » t ) i s defined to be the reference CAP response, then the threshold of th — the N f i b r e f o r a t e s t stimulus with a d i f f e r e n t parameter set p^ may be obtained by adjusting the test stimulus amplitude I t so that:  (5)  y(It,Pt,t) = y(lr,pr,t)  The  r e s u l t i n g t h r e s h o l d amplitude at p t w i l l  be defined as the N  threshold r e l a t i v e to I at p , and w i l l be given as: r r  tb  fibre  88  W V  (6)  =  Inherent in this definition of relative threshold are the assumptions that any  two  equivalent CAPs correspond  to identical firing  identical orders of fibre recruitment. Janko and  Trontelj  in an  populations, with  These two factors were addressed by  excellent study  [10].  Using  microelectrodes  inserted into human ulnar, median and radial nerves, the authors  recorded  responses to various transcutaneously applied stimuli subsequently  showing  that when two stimuli were adjusted to the same neural response level, the same active population of fibres resulted with an identical sequence of recruitment. It has been shown by Cooley and Dodge [46]  that differences in threshold  loci between space-clamped and freely conducting axons can be accounted for, without significant error, by a uniform scale factor.  The influence of any  uniform scaling factor is eliminated by means of the definition of relative threshold which entails normalization of the threshold amplitude with respect to the set point threshold 1^. The above protocol definition results in the following sequence of steps which must be performed for either percutaneously or transcutaneously applied stimuli: 1.  Select a reference parameter set p r »  2.  Select a reference stimulus amplitude I .  3.  Stimulate with I at p ; record and retain the resulting set point CAP as the reference response.  4.  For various test parameter sets, pfc, determine the test amplitude Ifc required to match the recorded reference CAP and calculate I (p ) = I /I . th t t r  89  Three t y p i c a l near nerve recordings which i l l u s t r a t e this procedure are shown i n F i g . 4.4, form.  The  50  phase  p  us  along with a c h a r a c t e r i s t i c  DISA current stimulus wave-  central recording i s that obtained from a reference stimulus of duration  = (50,50,50)).  with  50  us  phase  separation  peak of 105.0  ing r e p r e s e n t s  setting,  The reference amplitude of Ir(50,50,50) = 25 mA i s chosen  using a protocol outlined i n the experimental r e s u l t i n g A-B  (50-50-50  that due  section of this paper.  mV i s retained for reference.  to a t e s t stimulus with p t  The upper record-  = (100,20,100).  measured A-p peak which results from application of this stimulus i s 101.3 i n amplitude, and i s lower than the 105.0  mV reference peak.  amplitude of the pulse must be increased to obtain an A-B  The  The mV  Hence the 19 mA  response equivalent  to that of the reference stimulus.  S i m i l a r l y , the lower recording shows the  response  25  Pt  to  a  test  stimulus  = (20,80,20), but  of  mA  at  a  pulse  parameter  set  of  i n t h i s case the t e s t response - and hence the test  amplitude - i s too large and must be reduced to y i e l d the correct r e l a t i v e threshold  amplitude.  4.4 Threshold  data  was  collected  EXPERIMENTS over  a  period  of  eight months i n six  separate t r i a l s involving either a percutaneous or transcutaneous stimulation scenario. sessions.  The  same healthy subject was  used for a l l of the  experimental  90  Near Nerve R e c o r d i n g  Near nerve r e c o r d i n g s demonstrating the concept o f a r e f e r e n c e s t i m u l u s (middle recording) and low and h i g h s t i m u l u s a m p l i t u d e s e t t i n g s a t pulse parameter s e t s o t h e r than t h a t o f the 50-50-50 r e f e r e n c e s e t . A typical b i p h a s i c ramp s t i m u l u s i s shown below the t h r e e CAP r e c o r d i n g s .  91  4.4.1  Percutaneous Stimulation  4.4.1.1  Selection of Model  With  appropriate  assumptions  concerning  stimulus  electrodes,  fibre  geometry and r e l a t i v e magnitude of c e r t a i n f i b r e impedances, a single FH node may  be used to emulate the response of median nerve f i b r e s to percutaneous  stimulation. We cross  first  assume that  s e c t i o n , that  the  fibres  of the median nerve are  circular  they are aligned so that a l l f i b r e axes are  in  parallel,  that they are arranged such that a transverse cross-section reveals a c i r c u l a r closest packing geometry ( F i g . 4.5), and diameter.  In  preferentially  a d d i t i o n , we oriented  assume that  transverse  to  the  that a l l f i b r e s are of the same  the  stimulus  fibre  current  axes with  vectors  are  no longitudinal  path component, and that the current vectors are contained within the bounds of a c y l i n d r i c a l element of unit cross-sectional area whose axis i s perpendicular to the axis of the fibres being stimulated. With  these  assumptions, a  single FH  node may  be  used  to model  the  percutaneous stimulation scenario.  4.4.1.2  E l e c t r o p h y s i o l o g i c a l Techniques  Using two DISA model 15E07 constant current stimulators and a DISA model 15G01  delay u n i t , biphasic s t i m u l i were generated and applied through needle  electrodes inserted at the w r i s t .  The stimulus electrodes were positioned as  close as possible on either side of the median nerve using anatomical land  92  Figure  4.5  Assumed Geometry f o r P e r c u t a n e o u s l y A p p l i e d  Stimuli  The assumed geometry, s t i m u l u s e l e c t r o d e o r i e n t a t i o n and s t i m u l u s p a t h f o r the case o f p e r c u t a n e o u s l y a p p l i e d s t i m u l i . The e q u i v a l e n t ( a s i n g l e a c t i v e FH node) i s a l s o shown.  current circuit  93  marks.  Electrode position was 'fine tuned' by stimulating with a low ampli-  tude monophasic pulse while  simultaneously adjusting the electrode position  u n t i l a maximal motor response to the stimulus was a t t a i n e d . Responses to the biphasic s t i m u l i , applied at a rate of 1 Hz, were recorded  using a near-nerve technique  median nerve at the upper arm.  [64] with electrodes placed near the  The CAP responses were amplified by a DISA  15C01 a m p l i f i e r , then passed to a DISA 15G01 signal averager  where sixteen  responses were averaged to reduce background noise.  4.4.1.3  Stimulus Waveforms  It was observed that the s t i m u l i generated by the DISA 15E07 units had non-zero r i s e / f a l l times which were proportional to stimulus amplitude. measured r i s e / f a l l  rate was approximately  density ( F i g . 4.3). to  0.8 us per milli-amp  The  of current  The i n t e r n a l c i r c u i t r y of the 15E07 u n i t s was modified  allow investigation of stimulus waveforms with pulse durations below the  normal ranging  100  us minimum.  With  this  modification, s t i m u l i  with  durations  from 20 us to 100 us and phase delay times of 20 us to 80 us were  investigated.  The biphasic pulses were generated by chaining two monophasic  pulses with the same duration, but opposite p o l a r i t y , through the DISA delay unit which was used to control phase separation.  4.4.1.4  Reference Selection  The parameter set f o r the reference stimulus and CAP was chosen to have a 50 us i n i t i a l duration, a 50 us phase separation, and a 50 us duration f i n a l reverse phase.  This 50-50-50 parameter set was selected because the  94  values corresponded to approximate mid-ranges of duration and separation for the parameter range under i n v e s t i g a t i o n .  The reference amplitude was chosen  to be the mean of sensory threshold amplitude and A-8 maximum amplitude. A-8  fibres (largest diameter, fastest conducting myelinated  to have the lowest threshold to stimulation [65].  The  f i b r e s ) are known  The A-8 maximum stimula-  t i o n amplitude i s the amplitude beyond which the CAP peak corresponding to this f i b r e population no longer increases. further  inactive  increasing  A-8 fibres  This indicates that there are no  i n the nerve trunk which can be activated by  the l e v e l of s t i m u l a t i o n .  Since  the A-8 group i s the fastest  conducting, i t s peak may be i d e n t i f i e d as the f i r s t detected  CAP.  occurring peak i n the  The reference 50-50-50 stimulus amplitude selected t y p i c a l l y  lay i n the range of 2.8 mA to 3.4 mA.  The reference CAP response to a t y p i -  c a l set point stimulus i s shown i n F i g . 4.4.  4.4.1.5 The  Procedure subject was allowed  to rest on a bed i n a darkened room, and the  stimulus and recording electrodes were then Inserted.  Sensory threshold and  the A-B maximal stimulus amplitudes were determined and from these values the 50-50-50  reference  amplitude  was c a l c u l a t e d .  A reference  CAP was  then  generated, recorded, and stored on one of the four channels of the DISA 15H01 display u n i t . parameter 100  sets  Relative threshold amplitudes were then calculated for twenty corresponding  us at four phase delay  to f i v e  pulse widths ranging  settings from 20 us to 80 us.  from 20 us to The threshold  points were determined by adjusting the test stimulus amplitude u n t i l the A-8 peak of the resulting CAP matched the 50-50-50 reference response.  As w e l l ,  95  stimulus amplitudes  required f o r a just noticeable increase and decrease of  the test CAP away from the threshold l e v e l were determined-  After every f i v e  test thresholds the reference CAP was re-evaluated to determine i f there had been a reference change due t o , f o r instance, movement of the stimulus or recording electrodes.  If there had been a reference change, the reference  CAP was redefined and threshold values r e l a t i v e to the new reference stimulus were determined f o r the previous f i v e test s e t t i n g s , whereupon the reference was again checked. would  typically  For the twenty parameter sets involved, t h i s procedure  take  from two to two and one-half  hours.  Throughout the  c l i n i c a l session, EMG a c t i v i t y was monitored by audio output.  In the event  that spurious contributions to the CAP were made by EMG s i g n a l s , that CAP was rejected and another was recorded.  If EMG a c t i v i t y persisted and It was not  possible to eliminate the a c t i v i t y by encouraging the subject to r e l a x , then the session was terminated.  4.4.1.6 The  Comparison with the Model relative  threshold amplitudes  shown i n F i g . 4.6.  exhibited by both data phase  .short  are favourable.  s e t s , highest  delay, and lowest  longest phase delay.  data are  In general, the comparisons between model and experi-  mental threshold amplitudes  shortest  of model and experimental  The same general  trends are  threshold at lowest pulse width with  threshold  at highest  pulse  width  with  The greatest discrepancies occur at low pulse width and  phase delay, where the model data  required threshold current.  consistently  under-estimates the  Allowing that the model thresholds are accurate  to within 1% of the true threshold value for each stimulus parameter s e t t i n g ,  96  8  20 us phase s e p a r a t i o n © 30 us phase s e p a r a t i o n <• 50 us phase s e p a r a t i o n  BO us phase s e p a r a t i o n -Model  data  -Experimental data  "VM  SJOO  4b.oo  ab.oo  Figure Model  and C l i n i c a l T h r e s h o l d s  Relative  amplitudes  compared  to  stimulation  tb.oo  Pulse Midth («icro-seconds)  of  percutaneously  experimental (solid  lines).  relative  iSo~ oo  Ik  00  4.6 f o r Percutaneous  derived  thresholds  model  Stimuli  data  obtained  (dashed  from  lines)  percutaneous  97  and that the error associated with each of the CAP amplitude readings i s 2% only the model and experimental  thresholds at the lowest phase separation  value of 20 us f a i l to lay within the bounds of experimental error. The modelling assumption that a l l of the stimulus current i s injected at a single node of Ranvier is highly simplified.  In practice this will cer-  tainly not be the case, some fraction of stimulus current w i l l be directly shunted to the cathode (not passing through the membrane) from the stimulus anode resulting stimuli.  in a more pronounced  effect  for the  shortest duration  This is demonstrated in Fig. 4.2 where comparisons between pure  injection stimulus waveforms and  tissue distorted waveforms are shown for  biphasic squares and biphasic ramped pulses. The effects of a time varying membrane impedance on trans-membrane current flow can also be seen in F i g . 4.2.  The non-constant membrane impedance  (and associated membrane time constant) has the effect of changing the fraction of stimultor current diverted through the nodal membrane. This can be seen in Fig. 4.2a where the current entering the active node, resulting from a biphasic square current stimulus, i s shown. The leading positive phase of the nodal stimulus shows a characteristic RC amplitude decay. As the node is excited, the resulting decrease in membrane impedance influences the fraction of stimulator current which flows into the node. This changing influence can be detected in the second stimulus phase where an obvious decrease in membrane time constant  can be observed.  The  same phenomenon for a ramped  biphasic square stimulus is shown in Fig. 4.2b. It can be seen from Fig. 4.2 that in the case of a stimulus with nonzero rise time, the effects of current shunting would be most significant for  98  the briefest pulses. the  predicted  It i s possible that the observed differences between  thresholds  and  the  measured  thresholds  are  due  to  the  assumption of complete stimulus i n j e c t i o n , the inadequacies of which are most pronounced for the shortest temporal  4.4.2  Transcutaneous  4.4.2.1 Due  parameters.  Stimulation  Selection of Model to the physical arrangement of the transcutaneous stimulating elec-  trodes, injected current w i l l pass from cathode to anode i n a d i r e c t i o n which i s p r e f e r e n t i a l l y oriented along the longitudinal axis of the median nerve. In doing so, a p a r t i c u l a r current path w i l l c e r t a i n l y traverse a large number of  excitable nodes -  this  i s i n contrast  to the percutaneous  current flows across the longitudinal axis of the median nerve.  case where Hence, for  the transcutaneous case, the presence of adjacent nodal membranes was deemed important, and such nearby nodes were included i n the model for transcutaneous s t i m u l i .  the nodal geometry, electrode orientation and stimulus current  path for t h i s case i s shown i n F i g . 4.7. A single active FH node with passive nodes on either side was  used to  simulate the median nerve response to the transcutaneously applied  stimuli.  The Only  e l e c t r i c a l network representation of t h i s model i s shown i n F i g . 4.1. two  nearby nodal membranes were simulated, and  unexcitable.  they were treated  Both of these s i m p l i f i c a t i o n s were made to reduce the com-  as  99  Figure 4.7 Assumed Geometry f o r Transcutaneously Applied Stimuli  The assumed geometry, stimulus electrode o r i e n t a t i o n and path f o r the case of transcutaneously applied s t i m u l i . stimulus current paths have been omitted. The equivalent f u l l y active FH node flanked by two passive nodes) Is c i r c u i t i s given i n more d e t a i l i n F i g . 4.1).  stimulus current For c l a r i t y , the c i r c u i t (a single also shown (this  100  plexity  and computation  represented  time  of the model.  The passive  nodes  were  by r e s i s t i v e and capacitive values from the resting state of an  FH node.  4.4.2.2  Methods and Procedures  Only the method of applying set of experiments.  the stimulus current was changed f o r this  A l l signal recording and reference selection was i d e n t i -  c a l to the percutaneous stimulation case. twenty parameter procedure  Data was collected f o r the same  sets as i n the percutaneous  of regularly checking  experiments, using  the reference  CAP during  the same  the threshold  measurements.  4.4.2.3 Two  Stimulation ring  electrodes with  2  surface areas of 4.8 cm , moistened with a  saline s o l u t i o n , were placed around the fourth f i n g e r .  Stimulus current was  applied through these electrodes, and twenty threshold points were measured relative  to a 50-50-50  reference  monitored with audio output.  setting.  As before, EMG a c t i v i t y was  The reference stimulus amplitude was chosen as  the mean of sensory threshold and A-8 maximal amplitudes, with t y p i c a l settings i n the range 14.8 mA to 15.4 mA, corresponding 2  to current densities of  2  3.1 mA/cm and 3.2 mA/cm , r e s p e c t i v e l y . The time taken to obtain the twenty threshold values  was generally t«o to two and one-half hours.  course of the experimental electrodes  was monitored  During the  session, a control set of stimulus and recording on the subject's  other  arm f o r the purposes of  detecting any changes i n threshold due to drying of the stimulus electrodes.  101  No such changes were observed during the length of time taken by an experimental session.  4.4.2.4  Comparison with the Model  The r e l a t i v e thresholds of model and experimental taneous case are shown i n F i g . 4.8.  data for the transcu-  As with the percutaneous case, the com-  parison of the two threshold data sets i s generally favourable.  However, for  a l l phase delay s e t t i n g s , the model data at long pulse durations i s consistently lower than the measured threshold amplitudes at 20 us and test  pulses  but with only the values  30 us being separated by more than experimental become shorter, correspondence  threshold values i s good.  of  the  error.  predicted and  This i s i n contrast to the general  As  the  measured  underestimation  of the experimental results for short pulses i n the percutaneous case. The low predicted threshold amplitudes  for long duration pulses (80  us  and 100 us duration) are possibly due to the highly s i m p l i f i e d network chosen to model the e f f e c t s of adjacent nodal membranes. are  increased  in  duration, the  adjacent  nodes  As the stimulating pulses begin  to  respond  to  the  presence of a stimulating current thus influencing the current d i s t r i b u t i o n near the active node.  The  simple network used accounts only f o r a single  passive node on either side of the active target node.  This s i m p l i f i c a t i o n  appears to become less adequate as the applied pulses become longer, and more nodes may  have to be simulated i n order to improve the accuracy of the model  predictions.  For pulses of durations up to 1 ms, McNeal [16] determined that  ten nodes, f i v e on either side of a target node, were necessary to obtain a highly  accurate  simulation of  fibre  activity.  It  should  be  noted  that  102  8  o  •J • c •  20 us phase s e p a r a t i o n © 30 us phase s e p a r a t i o n «> 50 us phase s e p a r a t i o n  D BO us phase s e p a r a t i o n -Model  data  -Experimental  ^00  80.00  data  46.00  86.00  86.00  Pulse Width (licro-seconds) Figure  i5o~  oo  Ik  4.8  Model and C l i n i c a l Thresholds f o r Transcutaneously Applied Stimuli  Relative amplitudes of transcutaneously derived model data (dashed lines) compared to experimental r e l a t i v e thresholds obtained from transcutaneous stimulation.  103  McNeal's simulations took place at a temperature of 20°C, while the data i n this  simulation correspond to a temperature of 33°C.  which adjacent environment  nodal a c t i v i t y becomes s i g n i f i c a n t w i l l  than  The pulse widths at be lower i n a 33°C  i n a 20°C environment due to an appreciable decrease i n  model time constants at the higher temperature.  4.5  DISCUSSION  With the exception of determining  sensory threshold, the subject had no  conscious input into the experimental data.  It was thus decided that a large  number of subjects was not required f o r this study.  Had the determination of  threshold amplitude involved a perceptual interpretation of sensation by the subject, then c l e a r l y a larger number of subjects would have been required for the study. lected,  Over the eight month period during which the data was c o l -  the r e l a t i v e  threshold  loci  were  found  to be quite  repeatable.  Although the absolute magnitude of the determined thresholds could, and d i d , vary from session to session, the thresholds r e l a t i v e to the 50-50-50 reference point exhibited l i t t l e v a r i a b i l i t y .  During a particular  experimental  session, v a r i a t i o n of the reference amplitude was not l a r g e , and more f r e quently  than not, there was no v a r i a t i o n at a l l .  reference threshold amplitude recorded s i o n , and 0.2 mA during  The largest ' d r i f t ' of  was 0.4 mA i n a transcutaneous ses-  a percutaneous session, representing  1.5% and 2%  amplitude v a r i a t i o n s , r e s p e c t i v e l y . These d r i f t s are l i k e l y due to s h i f t s i n  104  position of the recording electrodes, or i n the percutaneous case, the stimulus electrodes as well as the recording electrodes. In the case where assumption of a pure i n j e c t i o n waveform was not adequate, i t was found that the current waveform crossing the neural membrane was s i g n i f i c a n t l y electrodes.  distorted  from  i t s form as applied  through the stimulus  The d i s t o r t i o n proved to be most s i g n i f i c a n t f o r those stimuli  with ramped, non-zero, r i s e times due to the capacitive properties of the FH nodes. Use of a f u l l y active FH node allowed investigation of the influence of membrane n o n - l i n e a r i t i e s upon neural threshold.  In previous neural models  [16], passive FH nodes have been used to illuminate the effects of stimulus electrode and f i b r e  geometry upon neural threshold.  However, passive FH  nodes represented by time-invariant RC networks have the disadvantage that their threshold i s independent of the t r a i l i n g phase of any biphasic stimulus.  This  limits  the scope  of any waveform investigations  which  may be  undertaken with such a model. E a r l i e r applications of the FH model f o r threshold response prediction i n humans [44], have used less fundamental means to account f o r the influence of temperature upon threshold.  While global temperature scaling techniques  of the s t y l e employed i n [44] are adequate, they prevent detailed examination of the effects of temperature on the dynamics of the neural membrane. Only a subset of the f u l l geometric considerations addressed i n [16] has been implemented with t h i s FH modelling system, i n favour of greater attent i o n to the d e t a i l s of membrane non-linearity and time variance.  Although  emphasis has been placed upon the dynamics of the FH membrane, geometric  105  factors  are  still  important  as  results i n F i g . 4.6 and F i g . 4.8. different  clinical  stimulation  can  be  seen  when comparing  the  clinical  These figures represent two fundamentally geometries,  thresholds reveal the underlying geometric  and  the  clinically  recorded  influences i n both the rate of  change of r e l t i v e threshold amplitude with respect to pulse width (a lesser negative value i n the percutaneous case than i n the transcutaneous case), and the  rate  separation case) .  of change of r e l a t i v e (greater  i n the  threshold  amplitude with respect to phase  transcutaneous  case  than  i n the  percutaneous  A comparison between the two experimental r e l a t i v e threshold l o c i for  phase separations of 20 us and 80 us i s shown i n F i g . 4.9. The changes made to the FH model greatly enhanced Its a b i l i t y to simulate the experimental situations under i n v e s t i g a t i o n . ture Q 10 '  s  Although the tempera-  determined by Frankenhaeuser are, s t r i c t l y speaking, v a l i d only to  22.5°C, the extension of t h i s data base into temperature ranges more relevant to  human nerves was  made because the Q^0  values were shown to be constant  over a range of two decades i n Frankenhaeuser' s data.  The simulation of two  adjacent passive nodes resulted i n a s i g n i f i c a n t increase i n model-relative threshold at low stimulus durations, y i e l d i n g threshold values much closer to those of the transcutaneous scenario.  106  F i g u r e 4.9 C l i n i c a l Thresholds for Percutaneously and T r a n s c u t a n e o u s l y A p p l i e d S t i m u l i Comparison of r e l a t i v e threshold amplitudes for c l i n i c a l l y applied transcutaneous and percutaneous ramped c o n s t a n t current stimuli. The t h r e s h o l d l o c i a r e phase s e p a r a t i o n s of 20 ps and 80 u s .  107  4.6  CONCLUSIONS  In order to use the Frankenhaeuser-Huxley model of myelinated nerve to simulate and  in-vivo human median nerve thresholds, discrepancies between model  experiment r e l a t i n g  must be addressed.  to four groupings of threshold influencing factors  The differences may be accounted f o r by extending the  temperature range of the FH model to encompass t y p i c a l temperatures of human nerve.  The temperature  extension  i s achieved  through  introduction of a  constant chloride system which allows the resting potential to be expressed as a function of temperature.  This resulted i n a value of Vr = -75.14 at the  c l i n i c a l temperature of 33°C.  Such a p o t e n t i a l change, although not large i n  magnitude, w i l l strongly a l t e r the threshold c h a r a c t e r i s t i c s of the FH system.  The mechanism of these threshold changes can be understood by examining  the e f f e c t s of a changed V"r on the FH phase plane [15]. For transcutaneous stimulation scenarios, where direct current i n j e c t i o n at a single active node i s not a good model approximation, the model must be extended to include the e f f e c t s of adjacent nodes and passive t i s s u e s .  In  a d d i t i o n , d e f i n i t i o n of an experimental protocol which yields threshold data relative  to an a r b i t r a r i l y  chosen reference  stimulus, eliminates threshold  influencing factors such as stimulus and recording electrode placement, and the uniform saltatory conduction  scale f a c t o r .  With the above model enhancements and extensions, percutaneous  threshold  data  collected  transcutaneous and  from human subjects  under the 'set  point / reference' protocol, c l o s e l y matches the thresholds predicted by the FH model for biphasic ramped current pulses.  The discrepancies between model  108  and  experimental results may  influence  be  due  to stimulus current shunting and  of nearby nodal membranes f o r the percutaneous  and  the  transcutaneous  cases, r e s p e c t i v e l y . Given the nature of the stimulus waveform d i s t o r t i o n s caused by adjacent nodes and  passive surrounding  possible are preferred.  t i s s u e s , s t i m u l i with as  little  as  Square s t i m u l i undergo l i t t l e d i s t o r t i o n compared to  ramped s t i m u l i , and hence the square pulses are more e f f i c i e n t . a factor of significance i n stimulator  Consequently  design i s the r i s e time parameter of  the power transistors i n the output stage of the stimulator should be minimized.  ramping  - t h i s r i s e time  109  5.  SIMULATION OF NODAL POTASSIUM ACCUMULATION  5.1  INTRODUCTION  It i s well known that the outward potassium current which occurs during the r e p o l a r i z a t i o n phase of an action potential can give r i s e to an accumulat i o n of potassium ions i n the immediate v i c i n i t y of the active membrane [66], [67].  Although the phenomenon was i n i t i a l l y noticed i n the squid giant axon,  the same effect has also been observed i n myelinated nerve from frog (Rana esculenta) [68], [69], [70], and i n the s n a i l (Helix pomatia) [71]. ences  i n the mechanism and parameters  of K+  ion accumulation and  Differ-  diffusion  have been proposed i n [69] as one of the reasons for the observable d i f f e r ences between motor and sensory nerve action p o t e n t i a l s . Two [66],  accumulation  i n the  models were proposed  original  work  on  1  K"  by  Frankenhaeuser  accumulation.  Both  and  Hodgkin  models are  accepted, with minor a l t e r a t i o n s , i n more recent l i t e r a t u r e [68].  still  Using the  nomenclature of Moran e t . a l . [68], the two mechanisms are i d e n t i f i e d as the 'three compartment model' (TCM)  (or multi compartment model), and  'diffusion i n an unstirred layer' (DUSL) model.  these accumulation models simulate K  in  to  a  way  as  Frankenhaeuser-Huxley  effect  axon.  the  the  The work presented here w i l l  be concerned with how such  as  threshold  +  ion accumulation  characteristics  of  the  110  5.2 5.2.1  The Three Compartment Model Originally  axon  MODELS OF K+ ACCUMULATION  in  [66]  proposed by as  Frankenhaeuser and  'Hypothesis  d i f f u s i o n ' , i t has  1:  finite  Hodgkin for the  space  and  very  squid  thin  barrier  become known i n the most recent l i t e r a t u r e as the  compartment model of d i f f u s i o n (TCM). the external space and  The TCM,  giant to  three  which consists of the f i b r e , 1  the bulk s o l u t i o n , explains K"" accumulation i n terms  of d i f f u s i o n obstruction caused by other anatomical structures (membranes). In  this  model, the  outward flowing  K  +  ions responsible for the  potassium  current are blocked by a membrane, some distance 8 from the axon, which has a permeability, PK,  to potassium such that ions escape from the  external  compartment at a rate slower than their rate of entry into the compartment. The result i s that the active membrane 'sees' an i n f l a t e d external potassium ion concentration, with a subsequently increased K  +  Nernst p o t e n t i a l .  Analysis of their squid giant axon data led Frankenhaeuser and Hodgkin to the conclusion that the TCM was situation.  They  calculated  the best description of t h e i r experimental  the  space  thickness,  9,  of  the  external 5  compartment to be 300 Angstroms, with an apparent K+ permeability of 6*10" cm/s.  In  [72], Adelman and  P a l t i reported  a space thickness of about  1  Angstroms with a permeability of 3.2*!©" * cm/s. to  support  the  membranes as  assumption that  the  9 values  dimensions i n the squid.  the  K+  correspond  There i s anatomical evidence  ion accumulation i s due fairly  360  well  to nearby  to known Schwann space  Ill  5.2.2  Diffusion i n an Unstirred Layer Model This  model  'Hypothesis 2: hypothesis. centration  was  put  forth  by  Frankenhaeuser  and  Hodgkin  [66]  as  f i n i t e d i f f u s i o n barrier and no space', along with their  TCM  In the DUSL model, an elevation i n external potassium ion coni s observed  because of the  relatively  slow mixing  of the bulk  solution with the contents of an aqueous layer adjacent to the active membrane patch from which the K+ ions emerge.  This model, thermodynamic rather  than anatomical i n o r i g i n , requires two parameters tion  phenomenon -  the unstirred  layer  to describe the accumula-  depth, and  the e f f e c t i v e  unstirred  1  layer K"" d i f f u s i o n constant.  5.3  CHOICE OF A K+ ACCUMULATION MODEL FOR THE FH SYSTEM  Based upon the recent findings of Moran e t . a l . [68], the DUSL model of accumulation was  chosen to simulate JC*" accumulation i n the FH model.  In  their experiments, Moran e t . a l . found that data for myelinated frog nerve (Rana esculenta) was well f i t t e d by a TCM - this was determined by p l o t t i n g ' TCM  theoretical  potassium  Nernst  potentials,  for  various voltage  clamp  depolarizations, against the duration of depolarization, and comparing these values to the experimentally measured p o t e n t i a l s .  However, the anatomical  basis for this model i s not obvious  s i n c e , for a frog myelinated axon, a  continuous  the axon has not been described - as  structure which surrounds  reported by Moran e t . a l . Since the TCM had no apparent s t r u c t u r a l b a s i s , Moran e t . a l . postulated that  the IT*" accumulation was  due  '...solely  to the discontinuity of ion-  transport number i n the path of e l e c t r i c a l current flow, accompanied by slow  112  mixing  at the (nodal) surface' .  They tested this hypothesis by comparing  potassium Nernst potential s h i f t s as functions of depolarization duration to DUSL c a l c u l a t i o n s , parameters  and concluded  which best  that  corresponded  f o r 17 f i b r e s  to the experimental  the averaged data 6  DUSL  were a DUSL  2  thickness of 1.4 um, and a d i f f u s i o n constant of 1.8*10~ cm /s.  The DUSL  data so calculated was observed to f i t the transient portion of the K+ Nernst potential s h i f t (for up to 10 ms at 15°C), but to exhibit less agreement with the steady state experimental value. of Frankenhaeuser  This i s i n contrast to the observation  and Hodgkin that the DUSL model did not accurately describe  K+ accumulation around the squid giant axon f o r short time periods, but was more accurate i n the l i m i t of long depolarizations. The because  DUSL model was selected of the observed  f o r this  agreement  with  simulation  experiment  of  at short  accumulation (<  10 ms)  depolarization times, and because of lack of anatomical evidence for a TCM.  5.4 The  THE AFFECTS OF K  extent  of potassium  +  DUSL ACCUMULATION ON THE FH SYSTEM i o n accumulation  during  the 'standard' FH 2  response to a 120 us monophasic current stimulus of 1 mA/cm i s summarized i n F i g s . 5.1 and 5.2.  In F i g . 5.1, the time course of the outward potassium  current i s shown, while F i g . 5.2a d e t a i l s the resulting increase i n external potassium ion concentration.  The corresponding changes i n resting potential  are i l l u s t r a t e d i n F i g . 5.2b. For the c a l c u l a t i o n of F i g . 5.2, the parameter values of Moran e t . a l . are used, as well as parameter values f o r the DUSL system  i n which no IC*" ions are allowed to escape from the unstirred  Assuming the standard response  layer.  to be quite t y p i c a l , the l a t t e r curves show  113  the  theoretical  maximum  potassium  accumulation, and  corresponding  maximum  e f f e c t , for a single action p o t e n t i a l . The external potassium ion concentration appears d i r e c t l y i n only one of the standard FH equations - that describing potassium current:  (V+Vr)  h  =  P  K  (V)F  ~RT~  +  - [K ] o *e(V+V r )F/RT 1 - e(V-W  JF/RT  However, as previously mentioned, the constant -70  mV  resting potential may  be r e p l a c e d by an expression which allows c a l c u l a t i o n of ture T based upon permeabilities and  at some tempera-  ionic concentrations of sodium, potas-  sium and chloride ions:  V r  (T  ,[K+] ) '  °  =  —* F  l n  ^ P ^ ^ N a ^ ^ t ^ l o ^ ^ ^ o ^ q C ^ t  0 1  ^ !  ,  (P p (T)+P N a (T))[N a +] 1 +P K (T)[K+] 1 +P c l (T)[Cl-] o  1  This expression of Vr as a function of external K"" concentration i s e s s e n t i a l if The  the f u l l  1  influence of K"" DUSL accumulation i s to be taken into account.  global effects of increased external K+  concentration on the FH system  are best observed i n the reduced FH phase plane;  t h i s i s shown i n F i g . 5.3,  where the effect of a 20% increase i n external K**" concentration upon a membrane i n i t s resting  state i s demonstrated.  As far as neural threshold i s  concerned, F i g . 5.3  shows that an increase i n  w i l l have the effect of  r a i s i n g threshold by v i r t u e of the translocation of the unstable saddle point  114  8  Figure 5.1 Potassium Current Flow i n Response to Standard Stimulus Membrane2 potassium current flow i n response to the 'standard* 1 mA/cm 120 us duration monophasic constant current sitmulus.  FH  stimulus,  115  2.40  .00  oT*©Teoi;»  Tiie (aiHi-seconds) Figure 5.2  FH Variables Affected by Potassium Ion Accumulation a)  b)  External potassium i o n concentration during FH standard response using DUSL data of Moran e t . a l . , and assuming no d i f f u s i o n from the unstirred layer. Resting potential changes during FH standard response using DUSL data of Moran e t . a l . , and assuming no d i f f u s i o n from the unstirred l a y e r .  116  0.08 (volts)  Figure 5.3 Phase Plane Changes Due to Potassium Ion Accumulation The quiescent FH Vm phase plane (thick l i n e s ) showing phase plane changes brought about by a 20% Increase i n external potassium i o n accumulation (thin lines).  117  (and  i t s associated separatrix) towards larger values  translocation i s shown i n greater d e t a i l i n F i g . 5.4. excited  node of  the  system  i s shifted  to a  m.  This  Further, the unstable  smaller  reduced maximum amplitude of the action p o t e n t i a l .  for V and  V value, implying  a  Such amplitude reductions  have been observed i n action potentials of the squid giant axon [66], [67]. It  is likely  inactivation  that  other  FH  variables,  such  as  some of  r a t e c o n s t a n t s , are i n f l u e n c e d by changing  the +  [K ]Q  sodium values.  However, these dependencies must be illuminated through further voltage clamp studies as voltage  they cannot be deduced from existing FH data.  clamp  studies have been done  for the  Such additional  Hodgkin-Huxley  system,  and  dependencies of the  sodium i n a c t i v a t i o n variable (h) rate parameters, time  constant  state values  and  steady  have been revealed  [68], [73].  Similar  a l t e r a t i o n s to the FH system without benefit of further voltage clamp data could only be made i n a speculative manner, and hence the present modifications of the FH system to account for K+ accumulation should be considered as only f i r s t order i n extent.  5.5 The  THE INFLUENCE OF K+ ACCUMULATION ON MULTIPLE PULSE THRESHOLDS -  phenomenon of K*  brane potassium never  accumulation  can only occur when the outward mem-  current becomes s i g n i f i c a n t .  influence the  threshold  characteristics  stimulated from i t s quiescent state since  Hence, K+ of an  accumulation  will  excitable node being  current becomes s i g n i f i c a n t only  during the l a t t e r (repolarization) phase of the action p o t e n t i a l .  Potassium  accumulation w i l l thus only e f f e c t neural threshold i n a multiple stimulus multiple  response  s i t u a t i o n , where the  accumulted outside a node as a  118  Figure 5.4 Phase Plane Detail Detail of phase plane of Fig. 5.3 showing unstable saddle point  119  result of a primary response may node with respect to t r a i l i n g  a l t e r the threshold c h a r a c t e r i s t i c s of the  stimuli.  The simplest means by which to test the degree of potassium accumulation influence i s to f i r s t  condition the FH system by stimulation with a supra-  threshold pulse, then to c a l c u l t e the minimum (threshold) delay time required in  order  that  a  trailing  successfully activate  stimulus, i d e n t i c a l  the model node a second  to  the  time.  conditioning  pulse,  Since accumultion of  extra-nodal potassium has the e f f e c t of r a i s i n g the required threshold amplitude  at  any  increasing approached  given  threshold  stimulus parameter delay times  as  s e t , one  the  would  amplitude  of  expect the  to  trailing  observe pulse  the absolute threshold amplitude f o r the pulse parameter set under  consideration.  The  threshold  delay  times  calculated  with  simulation  of  external potassium accumulation would be expected to be larger than the delay times determined  during the r e l a t i v e refractory period of a nodal membrane 1  without simulation of K"" DUSL accumulation.  This i s shown i n F i g . 5.5, where  the threshold delay time difference between FH systems with and without DUSL simulation i s plotted as a function of the percentage threshold amplitude of the conditioning current s t i m u l i . as  stimulus  and  trailing  pulse f o r 100  us monophasic square constant  As a n t i c i p a t e d , the required minimum delay times decrease  amplitude  increases  above  threshold.  A  vertical  asymptote  appears at a threshold amplitude f r a c t i o n of 1.0 since any amplitude f r a c t i o n less than this value i s subthreshold and hence can never activate the e x c i t able node.  For pulse amplitudes closer to threshold, the delay difference i s  quite pronounced - on the order of 1 ms.  As pulse amplitudes become l a r g e r ,  the small threshold Increase due to IC*" accumulation becomes less s i g n i f i c a n t ,  120  Figure 5.5 Threshold Delay Times Threshold delay times of t r a i l i n g 100 us duration monophasic constant current 2 stimulus (threshold amplitude 0.8672 mA/cm ) after an i d e n t i c a l conditioning stimulus; as a function of threshold amplitude f r a c t i o n .  121  and  the delay difference decreases.  For an FH system with no DUSL simula-  t i o n , a t r a i l i n g pulse with an amplitude within 1% of absolute threshold has a threshold delay of 15.51 ms. 16.406 ms.  The  With DUSL simulation, the delay threshold was  largest amplitude  pulse tested, about 1.7  times threshold,  had a threshold delay time of only 1. 798 ms, with DUSL simulation yielding a 1.803  ms delay.  5.6 The  DISCUSSION  simulation chosen to demonstrate the effect of potassium accumula-  t i o n has no experimental twin for which data has been published.  The usual  method of experimental evaluation of potassium  accumulation i s to stimulate  over  observe  extremely  long  brought about by experiments  time  potassium  intervals,  and  accumulation.  to The  the  gradual  stimulation  times  changes for such  are on the order of hundreds of milli-seconds, and hence model  emulation of these r e s u l t s i s not p r a c t i c a l as c a l c u l a t i o n of a single 200 ms model response would consume approximately 10 hours of computer time on a PDP 11/23. In are  [66], Frankenhaeuser and Hodgkin performed a set of experiments which  quite  amenable  to  simulation.  They measured  the change i n  trailing  action potential positive phase amplitude at varying time i n t e r v a l s following a conditioning  stimulus.  Unfortunately, similar experiments  have not been  reported using myelinated f i b r e s . The modelling of the potassium  phenomenon reported here should be con-  sidered only as a preliminary attempt to account f o r the influence of i o n i c accumultion upon the FH system.  In as much as this phenomenon produces a  122  long  term multiple-pulse/multiple-response influence  on  the  FH  system, i t  represents a departure from the shorter term phenomena with which this work i s p r i n c i p a l l y concerned. further  experimentation  Much more model modification i n conjunction with is  necessary  +  parameters have [ K ] Q dependencies. these  r e s u l t s , while  potassium  serving  to  to  discover  what  additional  model  U n t i l such work has been accomplished,  indicate  the  trends of  influence  of  the  assumulation extension to the FH model, do not represent absolute  data intended for direct comparison with p a r a l l e l experimental work.  123  6.  SUMMARY  This thesis has studied the threshold behaviour of myelinated nerve by using a computational model, the Frankenhaeuser-Huxley  (FH) model.  The ques-  t i o n of a p p l i c a b i l i t y of the model results to threshold properties of human peripheral nerve was also addressed. Chapter  2 began by examining  the mathematical  properties  of the FH  system of d i f f e r e n t i a l equations i n terms of s t a b i l i t y and e x c i t a b i l i t y , with p a r t i c u l a r reference to the nature of threshold i n the system.  This examina-  t i o n of the FH system used techniques of phase-space a n a l y s i s , and suggested the form of a computational algorithm for use i n threshold calculations on a d i g i t a l computer.  With knowledge of the FH system's threshold properties, i t  was possible to suggest methods to improve the modelling of nerve excitation i n a way which accounted  f o r i n t r i n s i c membrane dynamics, but yet which was  not excessively complicated.  The performance of the model was then evaluated  against c l i n i c a l l y derived thresholds from human median nerve.  To effect the  comparison with c l i n i c a l data, i t was necessary to enhance the computational c a p a b i l i t i e s of the FH system, and to introduce a protocol for c o l l e c t i n g i n vivo human threshold data.  The comparison  of model to c l i n i c a l  data was  found to be favourable, with the model predictions l y i n g within experimental error  of  the c l i n i c a l l y  measured  thresholds.  Finally,  a  preliminary  mechanism was provided for enhancing the computational c a p a b i l i t i e s of the FH system to include the a f f e c t s of extra-nodal potassium ion accumulation. Phase-space analysis of the FH system revealed i t to be a member of the quasi-threshold class of excitable systems f o r physically meaningful tempera-  124  tures.  Using the sharpness d e f i n i t i o n of Fitzhugh to quantify the degree of  excitability orders  of  sharpness,  magnitude  i t was  possible to show that the FH system i s two  'sharper'  than  another  familiar  Hodgkin-Huxley model of unmyelinated nerve. was  shown  to  be  such  that  no  gradation  observed up to temperatures of 40°C  nerve model -  the  The sharpness of the FH system of  stimulus-response  curves  was  Analysis of a reduced FH phase space,  the plane of the variables V (transmembrane potential) and m (sodium activation v a r i a b l e ) , revealed that the e x c i t a b i l i t y of the FH system was dependent upon the location of a saddle point and i t s associated separatrix i n the  Vm  phase plane.  A least-squares surface was f i t t e d to the separatrix as a func-  tion  initially  of  the  inactivation  most active  variable).  This  surface  threshold d e f i n i t i o n , which was ous  definitions,  the  level  FH  variables V, was  used  to  m  and  h  automate  (the  sodium  a separatrix  found to be unambiguous i n cases where previ-  and  inflection  definitions, failed.  Computer  implementation of the separatrix d e f i n i t i o n was found to s i g n i f i c a n t l y reduce computation time when i t e r a t i n g to a value of threshold stimulus  amplitude.  An improvement on the modelling of nerve excitation was suggested by the nature  of  the  excitability  of  the  FH  system.  While previous  excitation  models assumed a constant potential threshold, the FH system indicated that potential thresholds were not constant, but rather were strong functions of stimulus parameter s e t s .  It was  shown that the FH variable thresholds could  be represented by exponential curves, with three parameters, for biphasic and monophasic square constant current s t i m u l i .  The general expression for the  variable  conjunction  passive  potential RC  model of  threshold nerve  to  was  used  in  produce analytic  with  expressions  a  simplified  for threshold  125  current amplitudes of square stimuli as functions of stimulus inter-phase  delay.  minimize various  It was  then shown how  such expressions  duration  and  could be used to  'cost' or 'damage' functions, which yielded expressions for  minimum charge and  amplitude-squared (proportional to ohmic heating) damage  components of varying proportion. analytic expression  The  optimizing procedure resulted i n an  for an exponentially r i s i n g  stimulus wave shape with  a  pulse duration setting parameterized by phase separation for various 'weightings'  of  the  charge  and  heating  damage components.  solved to produce a chart of optimal pulse widths. ing  How  This  expression  was  the derived optimiz-  procedure could be applied to optimize any cost function was  then shown,  and as an example, the optimization of a cost function containing components for  charge and heating damage as well as a stimulator compliance components  was  demonstrated.  indicated  by  The c l i n i c a l relevance of this mathematical procedure  illustrating  how  to c l i n i c a l l y determine the  parameters used i n the optimizing process. to be  related to  the  familiar  'theta current' which was current was  The  and  threshold  three parameters were shown  rheobase current, the  introduced  three  was  chronaxie,  defined i n Chapter 5.  and  to  a  The  theta  shown to bear a r e l a t i o n to the charge injected through the nerve  membrane, as capacitive displacement current, i n an element of time which was short i n comparison with the membrane time constant. Clinically,  i t was  demonstrated  that  the  FH  model could  predict  the  threshold functions of in-vivo human median nerve, within experimental e r r o r , in The  response to ramped current  s t i m u l i generated by  s t i m u l i were applied both transcutaneously  and  a DISA Neuro-Myograph. percutaneously,  and  the  r e s u l t i n g compound action potentials were measured using a near-nerve record-  126  ing  technique.  In order to compare the model to c l i n i c a l threshold data, i t  was  shown that threshold-influencing factors from four groups  (functional,  p h y s i c a l , geometrical and environmental) had to be taken into account. was  This  done by enhancing and extending the computational c a p a b i l i t i e s of the FH  model, and clinical  by defining a set-point/reference (SPR)  data  was  collected.  The  SPR  protocol under which the  protocol assumed that active axons  generate potential f i e l d s which are l i n e a r l y superimposed to form the c l i n i cally  measured  recruitment  compound action  order  are  potential.  required  to  yield  Further  the  final  assumptions  regarding  protocol which enables  c l i n i c a l thresholds to be compared to model thresholds r e l a t i v e to an a r b i trarily  chosen  reference  resting  potential to be  stimulus. expressed  Extensions as  to  the  model allowed  a function of temperature and  the  ionic  concentrations, and enabled the e f f e c t s of adjacent passive nodes to be taken into account. current  The c l i n i c a l and model threshold data were compared for ramped  stimuli with durations  from 20 us to 80  us.  from 20 us to 100  us and  inter-phase  For percutaneous s t i m u l i , the greatest  delays  discrepancies  between c l i n i c a l data and model predictions occurred at low pulse widths with low  phase  assumption  separation. of  complete  These  discrepancies  stimulus  injection  are into  thought the  to a r i s e  a c t i v e node.  from  an  In  the  transcutaneous case, s t i m u l i with large durations and large phase separations were found to be  the only settings where model and  compatible within experimental e r r o r .  clinical  data were not  It Is possible that neglecting  the  presence of a l l but two adjacent excitable nodes r e s u l t s i n the observed long duration discrepancies.  127  Chapter  5 showed how  to modify the FH equations to account  accumulation of potassium ions in the extra-nodal space.  for the  The modification  involved expression of the transmembrane potential as a function of the 1  +  internal and external concentrations of three ionic species, K"", Na  and C l ~ .  A thermodynamic model for IC*" accumulation, the 'diffusion in an unstirred layer' (DUSL) model, was model the affects of  incorporated into the FH system and was accumulation.  used to  The affects of K+ accumulation for a  single response were shown to be in terms of both increased external IC*" concentration  and  increased resting  potential.  The  affects with regard to model excitability and phase-space analysis.  Increasing threshold was  implications of these  stability were shown with shown to result from  K  +  accumulation, and was demonstrated by measuring the threshold of a monophasic square  stimulus which was  applied  after  a  supra-threshold conditioning  stimulus• In summary, the main contributions of this thesis are: 1.  Derivation of the non-constant threshold potential equation, and use of this equation to optimize neural stimuli with respect to a 'cost' function (Chapter 3).  2.  Definition of the set-point/reference experimental protocol (Chapter 4).  3.  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P r o c , Vol. 34, pp. 1322-1329, 1975.  134  APPENDIX I  THE FRANKENHAEUSER-HUXLEY EQUATIONS The model equations as reported by Frankenhaeuser and Huxley [1], and a l l model constants are repeated for convenience i n this Appendix.  The equation for membrane voltage:  51. dt  - A Lf i C  ^  Na  IR  Ip  =  =  K  P  %)  (mA/cm )  . P  1  i - i - i - i Na 2  Ionic currents:  =  -  stim  • m  h  ^  r  2  PK n  ^F  2  V + V  2  f  [Na+]Q - [Na*],  r ) Rf <  ,F  2  (V+V r ) ^  _ ^p 2 2 P p p (V+V r ) ^  , (  (V-V JF/RT r 1 - e  -  V\  1 - e  r  .  )F/RT  (V-Vr JF/RT  [Na+]o e ( ™ r ^ ( (v-f-V JF/RT r 1 - e  T  135  The equations f o r m, n, p and h  dT - .V -*) - h  1  1  -  The rate constants (ms *)  v+10  -  -0.1 CV+10) / ( i - e (  Ph  -  4.5 / ( l + e <  affl  -  0.36 (V-22) / ( l - e  -  0.4 (13-V) / ( l - e < - > ° )  an  =  0.02 (V-35) / ( l - e  8n  =  0.05 (10-V) / (1_e(V-10)/10)  ap  -  0.06 (V-40) / ( i - e < ° -  Bp  -  -0.09 (V+25) / ( l - e  P m  45  V  - >  /10  V  )  )  ( 2 2  ~  13  ( 3 5  4  >  /6  «h  V ) / 3  )  /2  "  V ) / 1 0  V)/l  )  °)  (V+25)/2  °)  136  Constants: 3  sodium permeability  P„ Na  =  8*10~  F  =  1.2*10  =  0.54*10"  =  0.0303 mho/cm  leakage conductance  =  0.026 mV  leakage current equilibrium potential  [Na+]Q  =  114.5 mM  external sodium concentration  [Na^  =  13.7 mM  internal sodium concentration  [K+]  =  2.5 mM  external potassium concentration  =  120.0 mM  external potassium concentration  F  =  96514.0 C/g/M  Faraday's constant  R  =  8.3144 J/K°/M  gas constant  T  =  295.18 K  P  K  v  P  g^  cm/s -3  potassium permeability  cm/s 3  cm/s  2  e  unspecific permeability  absolute temperature  I n i t i a l conditions:  m(0)  =  0.0005  h(0)  =  0.8249  n(0)  -  0.0268  p(0)  =  0.0049  V(0)  =  0  dV dt  t=0, I « , =0 ' stim  

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