Quantitative analysis of a neuron model and its application to directional selectivity in the retina

AND ITS APPLI CATI ON TO

DI RECTI ONAL SELECTI VI TY IN THE RETINA

BY

ROMAN RICHARD POZNANSKI, M. S c . (

M

onash )

(

jBfvtran ‘vj

A THESIS SUBMITTED FOR THE DEGREE OF

D

octor

of Ph i l os o p h y

of

The Au s t r a l i a n

Na t i o n a l

U

n i v e r s i t y

STATEMENT

i

ACKNOWLEDGEMENTS

ii

PREFACE

iii

ABSTRACT

v

NOMENCLATURE

viii

LIST OF FIGURE LEGENDS

xiv

CHAPTER 1

MEMBRANE VOLTAGE

CHANGES

IN PASSIVE

DENDRITIC

TREES: A GENERAL

TAPERING EQUIVALENT

CABLE

MODEL

1.1

Inroduction

1

1.2

Derivation of the governing equation

6

1.3

Condition for reduction to a tapering

9

equivalent cable

1.4

Condition for reduction to a tapering

16

equivalent cable with electrical and

geometrical nonuniformities

TAPERING EQUIVALENT CABLE MODEL

2 . 1 G r e e n ’s function for a tapering equivalent 28

cable with sealed ends

2 . 2 Time course to a 6-current pulse 34

2.3 Asymptotic solutions at relatively short 41 and long time periods

2.4 Time course to a long-lasting current 47 pulse

2.5 Transient voltage decrement from a clamped 56 soma and steady-state somatopetal voltage

decay

2 . 6 Estimation of the membrane time constant 61

(t ) and electrotonic length parameter (L) m

CHAPTER 3 ANALYSIS OF A POSTSYNAPTIC SCHEME BASED ON A SINUSOIDALLY TAPERING EQUIVALENT CABLE MODEL

3.1 Introduction 70

3.2 Synaptic conductance change and 71

Volterra expansion

3.4

Interaction between excitation and

84

shunting inhibition

3.5

Application of the postsynaptic scheme to

89

directional selectivity in the retina

CHAPTER 4

MODELLING THE ELECTROTONIC STRUCTURE OF A

STARBURST AMACRINE CELL IN THE RABBIT

RETINA

4.1

Introduction

95

4.2

Experimental methods

99

4.3

Results and discussion

106

4.4

Directional selectivity in the retina:

111

a presynaptic cotransmission model

BIBLIOGRAPHY

119

APPENDIX I

129

This

thesis

contains

no

material

which

has

been

accepted for the award of any other degree or diploma

in any university, and to the best of my knowledge and

belief,

contains

no

material

previously

published

or

written by another person, except when due reference is

made in the t ext.

Roman R Poznanski

Canberra, Australia

A C K N O W L E D G E M E N T S

I am extremely indebted to my supervisor Professor

W.R. Levick,FRS, for his guidance and assistance throughout

the duration of this work.

I am additionally indebted to Dr I.G. Morgan for

his role as Chair of the Supervisory Committee and his

constant help and guidance especially in the final stages

of my research

I am also very grateful to Professor G.A. Horridge,

FRS,

and Professor A.W. Snyder,FRS, for their encouragement

which they have given me over the

last three years.

I would

like to thank Professor S.J.

Redman who

provided the initial idea behind the tapering equivalent

cable model developed in this thesis, and who acted as

an advisor during the early stages of my research.

I would also like to thank Dr D. Osorio for his advise

on the photographic techniques used in this thesis and

to Dr D.I. Vaney (University of Queensland) for kindly

providing the histological material used in this thesis.

Financial support has been provided by an Australian

P R E F A C E

My candidature as a Ph.D. student formally commenced

in February 1988. Since this time presentations have been

made at the Australian Neuroscience Society meetings in

Brisbane

(Australia)

and

Dunedin

(New

Zealand).

In

conjunction with these presentations the following papers

have also been published:

1. Poznanski, R.R. (1988) Membrane voltage changes

in

passive

dendritic

trees:

a

tapering

equivalent

cylinder

model.

IMA

Journal

of

Mathematics Applied in Medicine and Biology.

5

,

113-145.

2. Poznanski, R.R. (1990) Analysis of a postsynaptic

scheme based on a tapering equivalent cable

model.

IMA

Journal

of

Mathematics

Applied

in Medicine and Biology.

7

,

175-197.

3.

Poznanski,

R.R.

(1991)

A generalized

tapering

equivalent cable model for dendritic neurons.

accepted for publication:

4. Poznanski, R.R. (1992) Modelling the electrotonic

structure of starburst amacrine cells in the

rabbit

retina:

a

functional

interpretation

of

dendritic

morphology.

Bulletin

of

Mathematical Biology.

The content of this thesis has been contructed from the

above mentioned material, with the policy

that the thesis

was

written

first,

and

thereby

avoiding

reference

to

the above papers in the text of this thesis.

Roman R. Poznanski

Canberra (Australia)

A B S T R A C T

A new mathematical model has been developed which

collapses a dendritic neuron of complex geometry into

a single electrotonically tapering equivalent cable. The

modified

cable

equation

governing

the

transient

distribution

of

subthreshold

membrane

potential

in

a

branching

tree

was

transformed,

becoming

amenable

to

analytic

solution.

This

transformation

resulted

in

a

Riccati

differential

equation

whose

six

solutions

(expressed

in

terms of elementary

functions)

express

the amount and degree of taper found in the equivalent

cable model.

The Laplace

transform method

was used

to obtain

analytic expressions for the Green’s function corresponding

to an

instantaneous pulse

of current

injected

at a single

point

along a tapering

equivalent

cable with

sealed

ends .

The time course

of

the voltage in

response to

an arbitrary input was computed using the Green’s function

in a convolution integral. Infinitesimally brief (Dirac

6-function) pulses and

step pulses were considered. It

has

been demonstrated that

inputs located on a

tapering

equivalent

cable

are

more

effective

at

the

cylinder. Asymptotic solutions were also derived to enable

the voltage response behaviour over both relatively short

and

long time periods to be analysed.

Semilogarithmic

plots of these solutions provided a basis for estimating

the membrane time constant x from experimental transients.

m

A formula was derived which showed that tapering tends

to

increase

the

estimate

of

the

electrotonic

length

parameter L. Transient voltage decrement from a clamped

soma revealed that tapering tends to reduce the error

associated

with

inadequate

clamping

of

the

dendritic

membrane.

An analytic solution of the modified cable equation

with reversal potentials was used to explore nonlinear

synaptic effects in passive dendritic trees of arbitrary

geometry. To illustrate the theory, a sinusoidal equivalent

cable representing the dendritic arbor of

a cat retinal

6-ganglion cell was used to show that shunting inhibition

can be effective when located off the direct-path between

the excitation and the soma. In particular it was shown

that a peripherally placed excitatory input juxtaposed

with

a

shunting

inhibitory

input

could

produce

a

voltage-peak minimum at the soma in order to suppress

the initiation of an action potential at the axon hillock.

Such a postsynaptic scheme was shown to be applicable to

F i n a l l y , a d e t a i l e d m o r p h o m e t r i c a n a l y s i s of a

L u c i f e r y e l l o w - f i l l e d Cb a m a c r i n e c e l l u n d e r t a k e n to

p r o v i d e r a w d a t a fo r the c o n s t r u c t i o n of a t a p e r i n g c a b l e

m o d e l w i t h b o t h e l e c t r i c a l a n d g e o m e t r i c a l n o n u n i f o r m i t i e s

w a s e m p l o y e d to d e t e r m i n e w h e t h e r d i s t a l i n p u t - o u t p u t

r e g i o n s of d e n d r i t e s w e r e e l e c t r i c a l l y i s o l a t e d f r o m t he

s o m a a n d e a c h o t h e r . C a l c u l a t i o n s of s t e a d y - s t a t e

e l e c t r o t o n i c c u r r e n t s p r e a d s h o w e d t h a t d e n d r i t e s of

s t a r b u r s t a m a c r i n e c e l l s p r o c e s s i n f o r m a t i o n

e l e c t r o t o n i c a l l y w i t h a b i a s t o w a r d s the s o m a t o f u g a l

d i r e c t i o n a n d f or a p a r t i c u l a r c h o i c e of m e m b r a n e

r e s i s t a n c e R v a l u e s t he v o l t a g e a t t e n u a t i o n in the m

s o m a t o p e t a l d i r e c t i o n r e v e a l e d t h a t the a c t i o n of t h e s e

d e n d r i t e s c o u l d be c o n f i n e d l o c a l l y . A f u n c t i o n a l

i n t e r p r e t a t i o n of t h e s e r e s u l t s f a v o u r s a p r e s y n a p t i c

v e r s i o n of t h e c o t r a n s m i s s i o n m o d e l w h i c h a t t e m p t s to

e x p l a i n h o w t he n e u r a l n e t w o r k of s t a r b u r s t a m a c r i n e c e l l s

c o u l d a c c o u n t f or d i r e c t i o n a l l y s e l e c t i v e r e s p o n s e s

N O M E N C L A T U R E

The following symbols are used throughout the whole

thesis, unless stated otherwise.

V

=

V -E

m

r

Electrotonic

potential,

as

deviation

of

membrane potential from resting value (mV).

V .

reV =

E .

-J

J

-E

Synaptic reversal potential, as deviation

of the synaptic equilibrium potential from

resting value (mV).

G

r

Resting membrane conductance per unit area

(fi_ 1cm"2 ) .

R.

l

Resistivity of the intracellular material

( ftcm ).

R

=G

~ 1

m

r

Resistance across a unit area of membrane

(ft cm2 ) .

C

m

Capacitance per unit area of membrane

( F cm 22 ) .

I

I

m Total membrane current per unit area

(A cm ^ ) .

t Time (m s ).

T =C /G Resting membrane time constant (ms),

m m r v '

T=t/x Dimensionless time variable,

m

G. Input conductance (fi ).

d .

J Diameter of the j ^ branch element, assumed

to be constant (cm).

G .

J Synaptic conductance per unit area

,0 -l

( u cm ) .

Ax

j Distance increment for each different branch

element encountered along the path to the soma (c m ) .

= 1 Ax . j

R

.

mj R e s i s t a n c e a c r o s s a u n i t a r e a o f m e m b r a n e o f t h e j ^ b r a n c h e l e m e n t

(

ß

c m ^ )

.

V {<Rmj V /

4

Ri}i

C h a r a c t e r i s t i c l e n g t h p a r a m e t e r f o r e a c h d i f f e r e n t b r a n c h e l e m e n t

e n c o u n t e r e d a l o n g t h e p a t h t o t h e

s o m a , a s s u m e d t o c h a n g e a t e a c h b r a n c h p o i n t ( c m ) .

Z = £ ( A x

j

/ Aj ) - / ( d c / X J.) j

E l e c t r o t o n i c d i s t a n c e f r o m t h e s o m a ( Z = 0 ) i s t h e s u m o f t h e A x . ,

e a c h d i v i d e d b y t h e A. f o r t h a t b r a n c h J

e l e m e n t ( d i m e n s i o n l e s s ) .

n N u m b e r o f b r a n c h e l e m e n t s a t a n e l e c t r o t o n i c d i s t a n c e Z f r o m t h e s o m a ( d i m e n s i o n l e s s ) .

n

0 N u m b e r o f p r i m a r y t r u n k s ( d i m e n s i o n l e s s ) .

D = ( I d . 3 / 2 ) 2 / 3 t a p e r ^ j

j = l

n o

D = ( [d.

3

/

2^273

j=l

D i a m e t e r of a u n i f o r m e q u i v a l e n t cable, with D=d^ if there is

only a s i n g l e p rimary trunk (cm).

F = D

3/2

-

3/2

taper G e o m e t r i c ratio factor ( d i m e n s i o n l e s s ) .

F = D 3 ^ 2 D 3//2 taper

x {

I

R . 1/2 /

I

R . 1/2

L

mj

L

mj

j=l j = l

E l e c t r o g e o m e t r i c ratio factor ( d i m e n s i o n l e s s ) .

\ a p e r = ^ ^ m ^ t a p e r )/^ R i } 2 C h a r a c t e r i s t i c length p a r a m e t e r of a t a p e r i n g e q u i v a l e n t cable

( cm ) .

A = tt D A.

taper taper Unit area of m e m b r a n e

(

2

.

(cm ) .

A= [ ( Rm D )/4R_^ ]

*

C h a r a c t e r i s t i c l e n g t h p a r a m e t e r of a u n i f o r m e q u i v a l e n t cable ( cm ) .

r = 4 R . / ( 7T a , t a p e r 1

2

D ^ ) I n t r a c e l l u l a r r e s i s t a n c e p e r t a p e r

u n i t l e n g t h o f a t a p e r i n g e q u i v a l e n t c a b l e ( f i c m ^ ).

r = 4 R . / (tt D 2 )

a l I n t r a c e l l u l a r r e s i s t a n c e p e r u n i t l e n g t h o f a u n i f o r m e q u i v a l e n t c a b l e ( f i c m ^ ).

Q o C h a r g e ( C ) .

I A p p l i e d C u r r e n t ( A ) .

I

o M a g n i t u d e o f a p p l i e d c u r r e n t a t t h e s o m a ( A ).

L E l e c t r o t o n i c l e n g t h o f a t a p e r i n g e q u i v a l e n t c a b l e ( d i m e n s i o n l e s s ) .

Z . J

L o c a t i o n o f s y n a p t i c i n p u t

( d i m e n s i o n l e s s ) .

^(Zp’Zj;T)=Kpj(T)

G r e e n ' s f u n c t i o n d e f i n e d a s t h e v o l t a g e t r a n s i e n t a t Z = Z a n d

P

t i m e T i n a t a p e r i n g e q u i v a l e n t c a b l e a s a r e s u l t o f a v e r y b r i e f

( D i r a c d e l t a - f u n c t i o n ) c u r r e n t p u l s e d e l i v e r e d a t Z = Z^. ($7).

U n E i g e n v a l u e s o f t h e G r e e n ' s f u n c t i o n .

E i g e n f u n c t i o n s c o r r e s p o n d i n g t o t h e e i g e n v a l u e s .

g •

J »m a x M a x i m u m a m p l i t u d e ( p e a k ) o f s y n a p t i c c o n d u c t a n c e (fi ^ ) .

T 1

T . = a .

J ,m a x j T i m e t o p e a k o f t h e s y n a p t i c c o n d u c t a n c e c h a n g e

( d i m e n s i o n l e s s ) .

8 j ( T ) = g j , m a x a j T e x p ( 1 '-otjT) T i m e c o u r s e o f t h e s y n a p t i c c o n d u c t a n c e c h a n g e ( f 2 ^ ) .

L I S T

OF

F I G U R E

L E G E N D S

1.1 A schematic illustration of two single dendrites showing a wide range of profuseness and paucity in their branching pattern. The dashed line divides each dendrite at points of equal electrotonic distance mapping onto a sinusoidal equivalent "dendrite" cable.

1.2 Longitudinal sections of equivalent cables derived from Table 1.1 whose inhomogeniety is governed by the decline of the factor F(Z) with electrotonic distance Z. The diameters of the equivalent cables obey equation (1.20) and are simply two-thirds root of the combined DTP. Parameter v a l u e s : ot=l . 5 and u)=-1.047, except for the sinusoidal cable where the chosen values were ot = 0.15 and

I co I =1 . 1 63 .

2.1 Time course of voltage in response to a very brief (Dirac delta function) current pulse delivered at various distances along an open-circuit finite equivalent cable, whose diameter decreases exponentially at a rate determined by the constant k:(a) k = 0.0; (b) k = — 1.1; (c) k = — 1.8 ; (d) k = -4.0. The peak response for input injected at the soma

to a very brief (Dirac delta-function) current pulse, delivered at various distances from the soma, of an open-circuited finite equivalent cable, whose diameter decreases exponentially at a rate determined by the constant k:(a) k=0.0; (b) k=-l.l;(c) k=-1.8;(d) k = - 4 .0. The responses have been normalized in terms of their peaks and the abscissa begins at the peak time in each case. The continuous lines correspond to an electrotonic length L of the equivalent cable equal to 1.0, while the curves shown by the interrupted lines correspond to L=°°.

2. 4 T r a n s i e n t v o l t a g e d e c r e m e n t in a n o n t a p e r i n g e q u i v a l e n t

c a b l e ( --- ) a n d a t a p e r i n g e q u i v a l e n t c a b l e ( --- ). The

t a p e r i n g is g o v e r n e d by (a) e x p ( - l . l Z ) , (b) e x p ( - 1 . 8 Z ) ,

a n d (c) e x p ( - 4 . 0 Z ) . V a l u e s w e r e c o m p u t e d f r o m e q u a t i o n

( 2 . 2 7 ) w i t h L = 1.0; T = 0.1, 0.2, 0.5, a n d a n d n v a l u e s

f r o m 1 to 50. R o o t s of t he t r a n s c e n d e n t a l e q u a t i o n ( 2 . 2 9 )

w e r e e v a l u a t e d n u m e r i c a l l y u s i n g a N e w t o n - R a p h s o n i t e r a t i o n

s c h e m e .

2 .5 A p l o t of ß v e r s u s L for a c u r r e n t s t e p r e c o r d e d and

a p p l i e d at the s o m a of an e x p o n e n t i a l l y t a p e r i n g e q u i v a l e n t

c a b l e .

3.1 T h e F f a c t o r b a s e d on the b r a n c h i n g p a t t e r n of a cat

r e t i n a l d e l t a - g a n g l i o n c e l l v e r s u s the e l e c t r o t o n i c

d i s t a n c e . T h e s i n u s o i d a l e q u i v a l e n t c a b l e r e f e r s to the

p r o f i l e of t h i s c u r v e . A d e l t a - c e l l , t a k e n f r o m B o y c o t t

a n d W a s s l e ( 1 9 7 4 ) , is s h o w n in t he t o p c o r n e r .

3 .2 T i m e c o u r s e of t he e x c i t a t o r y an d i n h i b i t o r y

c o n d u c t a n c e c h a n g e r e p r e s e n t e d by e q u a t i o n g. oi.T e x p ( l

-1 , IT-13 X 1

a . T ) a n d g 0 a 0 T e x p ( l - a 0T), r e s p e c t i v e l y , w i t h a = 1 . 5

1 Z ,max Z Z i

a n d a 0 = 1 . 2 5 . T h e o r d i n a t e is e x p r e s s e d in t e r m s of g,

Z 1 , ma x

w h i c h is a s s u m e d to v a r y in t h e a n a l y s i s , w h i l e the

a b s c i s s a r e p r e s e n t s the t i m e s c a l e in u n i t s of the

3.3 T h e i n p u t c o n d u c t a n c e G ^ n a l o n g the s i n u s o i d a l

e q u i v a l e n t c a b l e is o b t a i n e d f r o m e q u a t i o n ( 3 . 1 0 ) and

e x p r e s s e d as the d i m e n s i o n l e s s v a r i a b l e r AG. a m

3 . 4 E l e c t r i c a l e q u i v a l e n t c i r c u i t f or a s m a l l p a t c h of

s y n a p t i c m e m b r a n e of a d i r e c t i o n a l l y s e l e c t i v e g a n g l i o n

c e l l r e c e i v i n g t wo d i s t i n c t , c l o s e l y a d j a c e n t s y n a p s e s

(a d y a d ) . T h e e x c i t a t o r y b a t t e r y h a s i t s p o s i t i v e p o l e

f a c i n g the i n t e r i o r of the p o s t s y n a p t i c m e m b r a n e b e c a u s e

the e q u i l i b r i u m p o t e n t i a l for e x c i t a t i o n is p o s i t i v e i n s i d e

(i.e. E e = 1 0 mV); t he i n h i b i t o r y a n d r e s t i n g m e m b r a n e

b a t t e r i e s h a v e t h e i r n e g a t i v e p o l e f a c i n g the i n t e r i o r

of t he p o s t s y n a p t i c m e m b r a n e b e c a u s e t h e i r e q u i l i b r i u m

p o t e n t i a l s a r e n e g a t i v e i n s i d e (i.e. E ^ - E - - 7 0 m V ) . Th e

c o n d u c t a n c e f o r e x c i t a t i o n a n d i n h i b i t i o n h a v e a r r o w s

i n d i c a t i n g t h a t t h e y c a n be c h a n g e d , but the r e s t i n g

m e m b r a n e c o n d u c t a n c e r e m a i n s f i x e d .

3 .5 T h e i m p u l s e r e s p o n s e f u n c t i o n K ( Z , Y ; T ) / r A as a r e s u l t a

of a v e r y b r i e f ( D i r a c d e l t a - f u n c t i o n ) c u r r e n t p u l s e

d e l i v e r e d at v a r i o u s l o c a t i o n s a l o n g the s i n u s o i d a l

e q u i v a l e n t c a b l e w a s c o m p u t e d f r o m e q u a t i o n ( 3 . 1 2 ) w i t h

n v a l u e s f r o m 1 to 50. T h e e i g e n v a l u e s w e r e n u m e r i c a l l y

scheme. The value of Y associated with each curve is shown.

In (a) the impulse response function is measured at the

same position as the input. The ordinate in (b) is 1/200

of the value shown in (a). The peak response at Z

=

Y

=

L

is approximately 5335. The ordinate is in dimensionless

units, while the abscissa is the time in units of the

dimensionless time variable T.

3.6 Effect of the peak synaptic conductance strength on

the peak somatic depolarization caused by the excitatory

input being activated at (a)

the soma, (b) the centre,

and (c) the distal tip of the sinusoidal equivalent cable.

Parameter values: r A=50 Mf2; V, rev =

90 mV.

a

1

3.7

Deviation of

the somatic

potential from an initial

voltage of 17 mV caused by the activation of a shunting

inhibitory conductance change at three distinct locations

along

the sinusoidal

equivalent

cable.

The excitatory

and

inhibitory

conductance

changes

are

activated

simultaneously

(as expected for stimuli moving in the

null

direction) at (a) the soma, (b) the centre, and (c)

the

distal

tip. The dashed

line

corresponds to

the

threshold level for action potential (spike) initiation.The

abscissa represents the time in units of the dimensionless

time variable

T.

Parameter

values:

(a)

g.

=10nS,

(b)

8 2 , 8

m a x

1 , m a x

= 2 0 n S .

1 5 nS , ( c ) g 1 = 1 8n S ; r A=50Mft; V , r e v = 9 0 mV;

1 , m a x a 1

4.1 D e n d r o g r a m s of f i v e d e n d r i t e s f r o m the Cb s t a r b u r s t

a m a c r i n e c e l l p l o t t e d in t e r m s of a n a t o m i c a l p a t h d i s t a n c e

f r o m the c e l l b o d y ( a b s c i s s a ) . E a c h h o r i z o n t a l l i n e

r e p r e s e n t s a d e n d r i t i c b r a n c h . T h e t h i c k n e s s of e a c h

h o r i z o n t a l l i n e r e p r e s e n t s the a v e r a g e d i a m e t e r of the

b r a n c h ( c a l i b r a t i o n ba r on l o w e r l e f t r e p r e s e n t s 1.2 ym).

4 . 2 A c u r v e s h o w i n g t he a s s u m e d s i g m o i d a l R i n c r e a s e m

( l e f t o r d i n a t e ) a n d the c u m u l a t i v e m e m b r a n e a r e a ( r i g h t

o r d i n a t e , s c a l e d as % of t o t a l a r e a ) b o t h p l o t t e d as

f u n c t i o n s of d e n d r i t i c p a t h d i s t a n c e ( a b s c i s s a ) . H i s t o g r a m

of m e m b r a n e a r e a s in s u c c e s s i v e 10 ym b i n s is s h o w n at

b o t t o m ( C a l i b r a t i o n b ar on l o w e r r i g h t r e p r e s e n t s 90 y m 2 ).

4 . 3 T h e p a r a m e t e r (F) of the s t a r b u r s t a m a c r i n e c e l l

p l o t t e d as a f u n c t i o n of the e l e c t r o t o n i c d i s t a n c e (Z)

as c a l c u l a t e d f r o m th e d e t a i l e d g e o m e t r y of t he d e n d r i t i c

t r e e a n d a s s u m i n g a n o n u n i f o r m ' s i g m o i d a l ' R m d i s t r i b u t i o n

w i t h a s p e c i f i c a x i a l r e s i s t i v i t y R_^ of 300 f2cm. T h e d a s h e d

c u r v e r e p r e s e n t s an a p p r o x i m a t i o n of the p a r a m e t e r by

state voltage along a tapering (exponentially) equivalent

cable model plotted as a function of electrotonic distance

(Z) with Z=0 and Z=0.8 representing the soma and dendritic

terminal,

respectively.

Arrowheads denote direction of

current flow.

4.5

Tip-to-tip

signal

decrement

along

the

starburst

amacrines dendritic arbor shown as a percentage of voltage

plotted

against

the

maximum

value

of

the

spatially

nonuniform R

distribution for two different values of

m

R. shown labelled.

l

4.6 Somatofugal(--- )

and somatopetal(----)

attenuation

of

steady-state voltage along

the starburst amacrines

dendritic arbor represented as an exponential equivalent

dendrite having a specific axial resistivity (R^) of 300

ficm plotted against the maximum value of the spatially

nonuniform R distribution,

m

4.7(a)

The model predicts that a single amacrine cell

provides

excitatory

input

to

directionally

selective

subunits from 4 different groups of ganglion cells whose

preferred directions are anisotropic with respect to each

w i t h i n a s i n g l e s u b u n i t t h a t is r e d u p l i c a t e d 9 or m o r e

t i m e s t 0 c o v e r the t o t a l a r e a of a p a r t i c u l a r r e c e p t i v e

f i e l d in a g r e e m e n t w i t h the p h y s i o l o g i c a l o b s e r v a t i o n s

of B a r l o w a n d L e v i c k ( 1 9 6 5 ) . T h e r e c e p t i v e f i e l d of e a c h

d i r e c t i o n a l l y s e l e c t i v e u n i t is s h o w n to p o s s e s s

p r e f e r r e d d i r e c t i o n t h a t i s r e s t r i c t e d to on e of

n o n o v e r l a p p i n g d i r e c t i o n s in a c c o r d a n c e w i t h O y s t e r

o b s e r v a t i o n s ( O y s t e r a n d B a r l o w , 1 967; O y s t e r , 1 9 6 8 ) . E a c h

i n d i v i d u a l d i r e c t i o n - s e l e c t i v e g a n g l i o n c e l l is s h o w n

to h a v e th e s a m e a x i s of p r e f e r e n t i a l r e s p o n s e ( i n d i c a t e d

by a r r o w ) t h r o u g h o u t its r e c e p t i v e f i e l d in a c c o r d a n c e

w i t h th e e x p e r i m e n t a l r e s u l t s ( B a r l o w a nd H i l l , 1963; B a r l o w

et al . , 1 9 6 4 ) .

4 . 7 ( b ) T h e m o d e l p r o p o s e s t h a t a n y c h o s e n l o c a l r e g i o n

( s u b u n i t ) in t he r e c e p t i v e f i e l d of a s i n g l e d i r e c t i o n a l l y

s e l e c t i v e u n i t r e c e i v e s i n p u t f r o m the d i s t a l d e n d r i t e s

of s t a r b u r s t a m a c r i n e s . T h e d e n d r i t e s of s t a r b u r s t

a m a c r i n e s w i t h s o m a t a s h o w n as u n f i l l e d c i r c l e s p r o v i d e

c h o l i n e r g i c i n p u t to t he c h o s e n s u b u n i t , w h i l e the

d e n d r i t e s w h i c h m e d i a t e th e r e l e a s e of G A B A a r e s p a t i a l l y

a r r a n g e d so as to p r o d u c e a " s i l e n t " i n h i b i t o r y s u r r o u n d

( s h o w n by the d a s h e d l i n e ) t h a t is s i m i l a r in a p p e a r a n c e

to the c a r d i o i d s h a p e o b s e r v e d e x p e r i m e n t a l l y by W y a t t

t he l a t e r a l e x t e n t of t h i s i n h i b i t o r y a r e a is j u s t u n d e r

h a l f the s i z e of t he r e c e p t i v e f i e l d d i a m e t e r of the

d i r e c t i o n a l l y s e l e c t i v e g a n g l i o n c e l l a n d i t can a l s o

p r o j e c t s o m e d i s t a n c e i n t o t h e s u r r o u n d of the r e c e p t i v e

f i e l d , d e p e n d i n g on the p o s i t i o n of t h e l o c a l r e g i o n

( s u b u n i t ) in the r e c e p t i v e f i e l d ( W y a t t a n d D a w , 1975).

F u r t h e r m o r e , g i v e n t h a t the d i r e c t i o n a l l y s e l e c t i v e u n i t

h a s a r e c e p t i v e f i e l d s l i g h t l y g r e a t e r t h a n the d e n d r i t i c

f i e l d of a s t a r b u r s t a m a c r i n e c e l l at e a c h e c c e n t r i c i t y ,

t he s i l e n t - i n h i b i t o r y s u r r o u n d s h o u l d e x t e n d for o n l y

h a l f the w i d t h of a s t a r b u r s t a m a c r i n e c e l l ( V a n e y et

a l . , 1 9 8 9 ; V a n e y , 1 9 9 0 ) . T h e s i z e of t h e s i l e n t i n h i b i t o r y

s u r r o u n d w o u l d be d e p e n d e n t on t h e e c c e n t r i c p l a c e m e n t

of the s t a r b u r s t a m a c r i n e c e l l w i t h r e s p e c t to the v i s u a l

s t r e a k in r a b b i t or a r e a c e n t r a l i s in c a t . For e x a m p l e ,

in the v i s u a l s t r e a k t he d e n d r i t i c f i e l d d i a m e t e r s are

in the r a n g e of 2 5 0 jjm > w h i l e in t he p e r i p h e r a l r e t i n a

t h e y r a n g e up to 8 00 ym, d e m o n s t r a t i n g a l a r g e v a r i a t i o n

w i t h r e t i n a l e c c e n t r i c i t y ( T a u c h i a n d M a s l a n d ,1984) . T he

p r e f e r r e d d i r e c t i o n of the d i r e c t i o n a l l y s e l e c t i v e u n i t

e n c o m p a s s e d by t he n e t w o r k of s t a r b u r s t a m a c r i n e c e l l s

4 . 8 T h e a n a t o m i c a l s t r u c t u r e of the m o d e l c o n s i s t s of

a d i r e c t i o n a l l y s e l e c t i v e g a n g l i o n c e l l w i t h a r e c e p t i v e

f i e l d s u b t e n d i n g 4^° a n d s e q u e n c e - d i s c r i m i n a t i n g s u b u n i t s

m a d e up f r o m the d i s t a l d e n d r i t e s of s t a r b u r s t a m a c r i n e

c e l l s a c t i v a t e d by c o n e b i p o l a r c e l l s . T h e i n h i b i t o r y

m e c h a n i s m for i m p l e m e n t i n g the s e q u e n c e - d i s c r i m i n a t i o n

c o r r e s p o n d i n g to a s i n g l e s u b u n i t is s h o w n to be e x c i t a t e d

by b i p o l a r i n p u t s 1 7 ’ a p a r t , w h i l e s u c c e s s i v e s u b u n i t s

a r e s e p a r a t e d by b i p o l a r i n p u t s 12' a p a r t . T h i s is in

a g r e e m e n t w i t h th e s i n g l e - s l i t e x p e r i m e n t , w h e r e b y the

c o m p l e t e m e c h a n i s m of d i r e c t i o n a l s e l e c t i v i t y w a s f o u n d

to be c o n t a i n e d w i t h i n a s m a l l s u b u n i t 1/9 to 1 / 1 8 of

th e e n t i r e r e c e p t i v e f i e l d ( B a r l o w a n d L e v i c k , 1 9 6 5 ) . T h e

e x i s t e n c e of an " i n h i b i t i o n - f r e e " z o n e a d j a c e n t to the

e d g e of t h e r e c e p t i v e f i e l d t h a t is f i r s t c r o s s e d w h e n

m o t i o n is in t he p r e f e r r e d d i r e c t i o n e x t e n d s a p p r o x i m a t e l y

i° f r o m t h e e d g e . A s e r i a l s y n a p s e is s h o w n e n l a r g e d w i t h

(+) d e n o t i n g an e x c i t a t o r y s y n a p s e a n d (-) an i n h i b i t o r y

s y n a p s e . T h e s e r i a l s y n a p s e s a r e a r r a n g e d s u c h t h a t an

a m a c r i n e - a m a c r i n e s y n a p s e u s e s G A B A to i n h i b i t the

a m a c r i n e - g a n g l i o n c e l l s y n a p s e . W i t h the s e r i a l s y n a p s e s

o r g a n i z e d as in the d i a g r a m , a s p o t m o v i n g f r o m l e f t to

r i g h t ( n u l l d i r e c t i o n ) w i l l r e s u l t in no f i r i n g in t he

g a n g l i o n c e l l (cf. D o w l i n g , 1 9 7 0 ) . (A) d e n o t e s s t a r b u r s t

a m a c r i n e c e l l s ; (B) d e n o t e s c o n e b i p o l a r c e l l s ; (G) d e n o t e s

C H A P T E R

1

M E M B R A N E

V O L T A G E

C H A N G E S

IN

P A S S I V E

D E N D R I T I C

T R E E S :

A

T A P E R I N G E Q U I V A L E N T C A B L E M O D E L

§ 1 . 1 I n t r o d u c t i o n

A particular class of dendritic tree can be represented by a single equivalent cylinder, if several symmetry requirements hold [for a comprehensive summary see Rail ( 1977, 1989) and the lucid monograph by Jack

et al. (1975)]. The equivalent cylinder concept was formulated by Rail ( 1962a,b) over two decades ago and it provided neurophysiologists with tremendous insight into the role dendrites play in neuronal functioning (see e.g. Redman, 1976). It is based on four symmetry requirements, which restrict its application in neuronal modelling. In particular, the assumption that synaptic input must be equal at all points that are at the same electrotonic distance from the soma is a significant restriction, since local interactions between inputs on different branches cannot be investigated. In view of

the above limitation more recent models of the equivalent cylinder type have been constructed by Redman (1973) and

The 3/2 power law is based on the following two conditions, which must hold if a dendritic tree is to be transformed into an electrotonically equivalent cylinder :

(1) At every branch point, the parent branch diameter (assumed constant) raised to the 3/2 power

must equal the sum of the daughter branch

diameters (not necessarily equal, but assumed to be constant) each raised to the 3/2 power.

(2) The sum of the 3/2 power of all branch diameters at any given electrotonic distance from the soma must remain constant to the point of termination.

Implicit in these two conditions is a further symmetry requirement that all terminal branches must end at the same electrotonic distance from the soma.

As most neurons are not of the equivalent cylinder class, but instead show a decline in the dendritic trunk parameter, caused by a deviation from the 3/2 power law at branch points, completely new models not dependent on any of the symmetry requirements have been developed (see e.g. T u r n e r , 1984; Koch and Poggio, 1985; Segev et a l ., 1985; Holmes, 1986; Carnevale and L e b e d a ,1987). However, all these models have the disadvantage that analytic solutions are not directly available.

assumption can be modified by introducing the notion of taper into the model. This modification to the 3/2 power law enables a variety of different dendritic trees with a relative paucity of branching (caused by the early termination of individual branch segments) to be treated, and the symmetry requirement that terminal branches must end at the same electrotonic distance from the soma is therefore no longer required.

A dendritic tree can have many forms and still be equivalent to the same equivalent cable. An example of a dendritic tree that maps onto a sinusoidal equivalent cable is shown in Fig. 1.1. However, there is a large

difference between dendritic trees with dendrites

SOMA

Electrotonic

Distance

Sinusoidal Equivalent Dendrite

Dendrite

(X F[Z]

SOMA

Electrotonic

Distance

Sinusoidal Equivalent Dendrite

T h e a d v a n t a g e i n u s i n g t h i s m o d e l is t h a t o n l y a

s i n g l e c a b l e e q u a t i o n n e e d s to be s o l v e d an d i t is

t h e r e f o r e e a s y to i m p l e m e n t . F or e x a m p l e , the G r e e n Ts

f u n c t i o n c a n be o b t a i n e d by a n a l y t i c a l r a t h e r t h a n

c o m p u t a t i o n a l m e t h o d s , a n d as a r e s u l t t h e r e is no r i s k

of e r r o r c a u s e d by n u m e r i c a l i n v e r s i o n of the t r a n s f e r

f u n c t i o n (s e e e.g. K o c h a n d P o g g i o , 1 9 8 5 ) . A n o t h e r

i m p o r t a n t a d v a n t a g e is its a b i l i t y to a n a l y t i c a l l y

i n v e s t i g a t e t he e f f e c t s of p a r a m e t e r c h a n g e s i n c o r p o r a t e d

in the m o d e l , a n d a l s o t he l u x u r y of h a v i n g c l o s e d f o r m

s o l u t i o n s a v a i l a b l e . H e n c e , the a n a l y t i c a l t r a c t a b i l i t y

of the m o d e l p r o v i d e s t he i m p e t u s for f u r t h e r w o r k in

u n d e r s t a n d i n g the f u n c t i o n a l r o l e d e n d r i t e s p l a y in v a r i o u s

i n f o r m a t i o n p r o c e s s i n g t a s k s , s u c h as d i r e c t i o n a l

s e l e c t i v i t y in t he m a m m a l i a n r e t i n a . T h i s is t h e m a j o r

a i m of t h i s t h e s i s .

A l t h o u g h , R a i l r e c o g n i s e d t h a t h i s t h e o r y c o u l d

be e x t e n d e d to e x p o n e n t i a l l y t a p e r e d e q u i v a l e n t c a b l e s ,

he did n o t d e v e l o p t he t h e o r y to c o v e r s u c h c a s e s w i t h

e x p l i c i t s o l u t i o n s , a n d f u r t h e r m o r e the t h e o r y f or the

m o r e g e n e r a l c l a s s of t a p e r i n g e q u i v a l e n t c a b l e s r e m a i n s

u n p u b l i s h e d (but s e e S c h i e r w a g e n , 1 9 8 9 ) .

In o r d e r to u n d e r s t a n d th e r o l e c h a n g i n g g e o m e t r y

p l a y s in c o n t r o l l i n g n e u r o n a l a c t i v i t y t h e o r e t i c a l w o r k s

d e a l i n g w i t h t a p e r s h a v e b e e n p u b l i s h e d by s e v e r a l a u t h o r s

d e n d r i t i c b r a n c h e s p a r a m e t e r is e q u a l to u n i t y (s e e

G o l d s t e i n a n d R a i l , 1974; S t r a i n a n d B r o c k m a n , 1975; K e l l e r

a n d Lai, 1976; B r o c k m a n , 1981; E l l i a s a n d S t e v e n s , 1983;

R o s e a n d D a g u m , 1 9 8 8 ) .

T h e c l a s s i c a l w o r k of G o l d s t e i n a n d R a i l ( 1 9 7 4 ) d e a l t

w i t h the e f f e c t s of c h a n g e s in the s h a p e a n d v e l o c i t y

of i m p u l s e s as t h e y p r o p a g a t e d a l o n g n o n c y l i n d r i c a l a x o n s

w h i c h t a p e r e d e x p o n e n t i a l l y w i t h d i s t a n c e . H o w e v e r , the

f i r s t m a j o r w o r k to a p p e a r in t he l i t e r a t u r e w h i c h i n v o l v e d

n o n a x o n i c n e r v e p r o c e s s e s w i t h g e o m e t r i c a l i n h o m o g e n e i t i e s

in t he f o r m of t a p e r i n g w a s by S t r a i n a n d B r o c k m a n ( 1 9 7 5 ) .

T h e y c a l c u l a t e d t he s t e a d y - s t a t e v o l t a g e d e c a y in a p a s s i v e

n e r v e c y l i n d e r for t h r e e d i f f e r e n t g e o m e t r i e s by

n u m e r i c a l l y i n t e g r a t i n g the d i f f e r e n t i a l e q u a t i o n . F or

a l i n e a r t a p e r , t h e d e c a y of t he v o l t a g e w a s f o u n d be

s l o w e r t h a n t h a t of a u n i f o r m c y l i n d e r . U n f o r t u n a t e l y ,

a n a l y t i c a l s o l u t i o n s f or t he t i m e - v a r i a n t c a s e w e r e not

o b t a i n e d , b u t n u m e r i c a l m e t h o d s h a v e b e e n u t i l i z e d t h a t

e n a b l e t h e t r a n s i e n t r e s p o n s e in g e o m e t r i c a l l y

i n h o m o g e n e o u s n e r v e c y l i n d e r s to be d e t e r m i n e d (s e e e.g.

K e l l e r a n d Lai, 1976; B r o c k m a n , 1981; E l l i a s a n d S t e v e n s ,

§1.2 Derivation of the governing equation

The formulation given here of the equation associated with a tapering equivalent cable in the presence of

synaptic input proceeds along different lines from that previously presented by Jack et al.(1975). All the quantities are expressed in terms of the generalized electrotonic distance Z, under the assumption that all branch elements have constant but not identical diameters. In addition, the factor F which imposes a nonuniformity on the equivalent cable is incorporated into the equation, together with effects of synaptic action modelled by a change in postsynaptic membrane conductance.

In a nonuniform structure, the membrane current per unit area is given by

-1 3 la

Im = [ttADF(Z) ] { } 3Z

(

1

.

1

)

where the total axial current is expressed as

la = [ra,taper \ a p e r ] { }

(

1

.

2

)

3Z

y i e l d s t he f o l l o w i n g r e s u l t

Im =

-1

F dF av 3 V dZ 3Z 3 Z 2

(1.3)

T h e t o t a l m e m b r a n e c u r r e n t per u n i t a r e a of the

s y n a p t i c m e m b r a n e c a n be w r i t t e n as the s u m of the i o n i c

a n d c a p a c i t i v e c u r r e n t s , t h a t is,

av

rev

rev

Im = C m — + G r V + G e ( V - V e ) + G i ( V - V i ) (1.4) 31

w h e r e it is a s s u m e d t h a t s y n a p t i c a c t i v i t y a s s o c i a t e d

w i t h e x c i t a t i o n or i n h i b i t i o n on a d e n d r i t i c t r e e o c c u r s

on a l l of t h e b r a n c h e l e m e n t s t h a t a r e at the s a m e

e l e c t r o t o n i c d i s t a n c e f r o m t he s o m a . It is i m p o r t a n t to

s t r e s s t h a t t he m e m b r a n e r e s i s t a n c e a c r o s s a u n i t a r e a

of s y n a p t i c m e m b r a n e is no l o n g e r p a s s i v e b e c a u s e it is

a l s o d e p e n d e n t on t he s y n a p t i c c o n d u c t a n c e c h a n g e s , t h a t

is

R m

*

+ G + G. )

e l ( 1.5)

but, if e a c h s y n a p t i c c o n d u c t a n c e c h a n g e is a s s u m e d to

o c c u p y an i n f i n i t e s i m a l r e g i o n r e p r e s e n t e d s y m b o l i c a l l y

p r o p e r t i e s will not c h a n g e (Poggio and Torre, 1978). By p a s s i v e it is a s s u m e d that for a unit area of m e m b r a n e the c a p a c i t a n c e and the r e s i s t a n c e in p a r a l l e l are i n d e p e n d e n t of both time and voltage.

E q u a t i n g e q u a t i o n s (1.3) and (1.4), and m u l t i p l y i n g both s i d e s by , a m o d i f i e d cable e q u a t i o n d e s c r i b i n g

m e m b r a n e v o lt a g e c h a n g e s in the p r e s e n c e of s y n a p t i c input is o b t a i n e d :

av = a_^_v +

xdF av

3T

az2

dz az

2

v ,

l

r „ A 6 ( Z - Z .)g.(T) V +

L

a , t ap e r taper j j

j = l

x [V . rev - V] (1.6)

w h e r e the s u m m a t i o n s u b s c r i p t j = l , 2 d e n o t e s e x c i t a t i o n and i n h i b it i o n , r e s p e c t i v e l y , and time has been scaled, or cast in d i m e n s i o n l e s s form, in terms of the m e m b r a n e time c o n s t a n t . Note that the s y n a p t i c c o n d u c t a n c e c h a n g e s per unit area G and G.

e l have been c o n v e r t e d into g^ = AGe and 8 2 = A G . , r e s p e c t i v e l y , w h ere A is the unit areal o f

m e m b r a n e . In the a b s e n c e of s y n a p t i c input the last term on the right hand side of e q u a t i o n (1.6) v a n i s h e s and

the e q u a t i o n becomes:

a2v

az2

- l

av

3T

V + F dF

av

dz az

It is emphasized that equation (1.7) also arises in circuit theory as an equation representing an inhomogeneous (nonuniform) RC network or transmission line. As a result, similar solutions have previously been derived by Kelly and Ghausi (1965).

§1.3 Condition for reduction to a tapering equivalent

cable

The reduction of a dendritic tree to a single tapering equivalent cable is analogous to the approach used by Rail (1962a) in constructing an equivalent cylinder representation of the dendritic tree, with the exception that the so-called 3/2 power law at branch points need not hold.

The following parabolic partial differential equation describes passive membrane potential distribution in a dendritic tree with noncylindrical branches (Rail, 1962a, equations 20 and 23; Jack et al., 1975, equation 7.42):

9 2 V 9Z 2

3V

3Z

[(—

)

-1

d x

d_ d x

3/2

ln{ r n

[i+ (—

>2 ]4} ]

d x

(

1

.

8

)

3V_ 9T

where the radius (r) of all branch segments and the number of dendritic branches (n) are both functions of actual distance (x) from the soma, and

Z

X

/ d?/A

0

taper (1.9)

defines the electrotonic distance for situations where there is a continuously changing characteristic length parameter (A ) :

13 p g r

taper (r/ro ) [1 + ( — ) 2

d x

- i

(

1

.

1 0

)

where A =[R r / 2 R . 1 2 is the characteristic length parameter o m o l

for a cylinder with radius rQ (taken as the initial radius at x=0). The generalized electrotonic distance and

characteristic length Z and A respectively, are

L cL p 0 IT

concepts first introduced by Rail ( 1962a,b) and illustrated in some detail for an exponentially tapering core conductor by Goldstein and Rall(1974).

single one-dimensional cylinder. The following condition between r and n must hold for such a branching pattern

(see also Jack et al., 1975, equation 7.43):

3/2

2

i

nr [1 + ( — — ) ] = constant (1.11) d x

However, since our aim is to determine combinations of tapering and branching which will allow the dendritic

tree to be electrotonically equivalent to a tapering cable [represented by equation (1.7)], the following condition between r and n must hold (cf. Rail, 1962a, equation 21; and also Jack et al., 1975, equation 7.59 for the special case of exponential taper):

3/2 ckr 2 i 3/2

n r [ l + ( ) ] = n r F(Z;k) (1.12)

a „ o o

where n is the number of branches at x=0 and F is a factor o

which imposes taper on the equivalent cable. As a result, the coefficient of 3V/3Z in equation (1.8) simplifies to F 1(dF/dZ) and the equation describing passive membrane

electrotonus in a branching dendritic tree (with

It has been shown that the class of dendritic trees which are represented as equivalent cylinders , satisfy the property that the rate of change of dendritic surface ar ea , with respect to x, remains proportional to the rate of change of electrotonic d i s t a n c e , with respect to x

( R a i l , 1962a, equation 25 ; Jack et al. ,1975 , equation 7.45):

d A dZ

oc _(1.13)

d x d x

However, for the larger class of dendritic trees represented by tapering equivalent cables, the following property must hold:

dA dZ

- F (Z ;k )[ ] (1.14)

d x d x

or alternatively

d A

« F (Z ;k ) (1.15)

dZ

If for convenience, the analysis can be restricted to the case where each dendritic branch is represented by a cylinder of constant diameter [dr/dx=0] and each individual branch at any given value of x or Z is characterized by a different diameter, then after replacing radius with diameter, together with the above assumptions, the condition on r and n governed by equation (1.12)

becomes (cf. Rail, 1962b. p.149 for the special case of exponential taper):

[ X d j 3/2 ] F(Z;k) j = l

n ( x )

: I d

j=l

3/2

(1.16)

which defines the combined DTP, where d^. represents the diameter of the branch at distance x from the soma, with Z=0 when x=0. Alternatively, if x represents the actual distance measured along successive branch points and branching occurs at distances 0 =x o < #,,<Xp with n^ branches between x. and x . (1, where x.^x^x..,, then an

l l+l l l+l

equivalent representation of equation (1.16) in more standard notation is (Goldstein and Rail, 1974, equation 32) :

F (Z ;k ) =

I d± .3/2

[

I

do j 3/2 ] 1 i=0,l,...,p (1.17)

Equation (1.17) simply states that the combined DTP at any given electrotonic distance from the soma is proportional to the factor F. It should be emphasized that the assumption [dr/dx=0] imposes a physical limitation in interpreting equation (1.17), because F is assumed to decrease continuously with Z rather than in discrete steps at each branch point (cf. Jack et al., 1975, Fig. 7.11). However, the theory can be extended to branches which taper at a steady rate and satisfy the condition [d r/d x ]2« i , but only if all the branches are assumed to be equal in diameter (see Jack et al., 1975, p.156).

If F is unity then the combined DTP is constant, permitting a dendritic tree to be transformed into an equivalent cylinder that has a diameter equal to

n

D = [

l

o J 3/2 ]2/3 d (1.18)

j=l

or D=dQ^ if there is only a single dendrite emanating from the soma. Also, the characteristic length parameter

is given by

A [R D/4R. ]

m l

(1.19)

of the equivalent cable D changes continuously with

l0. p 0 r electrotonic distance from the soma

Dta per=D[F(Z;k)l2/3=[^d ij3/2]2/3

i = 0 ’1 .... P

(1-20)

j-1

and the characteristic length parameter, reduces from equation (1.10) to become

X

taper

[ R D

m taper

/4iq J

*

(

1

.

2 1

)

If F is greater than unity the branching pattern will exhibit a wide range of profuseness, and if F is less than unity there will be a relative paucity of branching in comparison with the branching pattern of a dendritic tree which can be reduced to an equivalent cylinder. Note that at Z=0, equation (1.20) reduces to D =D, and

18 p g r

§1.4 Condition for reduction to a tapering equivalent cable with electrical and geometrical nonuniformities

The aim of this section will be to generalize the condition for reduction to a geometrically tapering equivalent cable, to include electric nonuniformities such as a spatially nonuniform R distribution, which

m

has recently attracted a lot of attention from several workers (see Rail, 1982; Redman et al . , 1987; Fleshman et al.,1988). It should be mentioned that numerical methods employed in compartmental models of neurons with different Rm values in each branch segment is an alternative approach that will not be discussed here [see Rail (1990) for a critical analysis of this technique with regard to the problem of nonuniqueness].

The derivation will be carried out in the steady- state domain, but extension to the time-dependent domain can be carried out along the same lines as sketched by Leibovic (1972) for a symmetrical tree.

[r ( x )R m (x ) / 2Ri ]

ll +

II (II) :I_

dx d Z 2 dZ dx dx

In [ r 2 ( x )n( x )— ]} = d x

V

(

1

.

2 2

)

w h e r e t he r a d i u s (r) of a ll b r a n c h s e g m e n t s , the n u m b e r

of d e n d r i t i c b r a n c h e s (n) a n d R a r e a ll f u n c t i o n s of m

a c t u a l d i s t a n c e (x) f r o m the s o m a ( a s s u m e d to be the p o i n t

x = 0 ) . T h e v a r i a b l e Z is d e f i n e d by e q u a t i o n ( 1 . 9 ) , e x c e p t

t h a t t he c h a r a c t e r i s t i c l e n g t h p a r a m e t e r ^ t a p e r ^ as the

f o l l o w i n g f o r m :

t a p e r [ R ( x ) r ( x ) / 2R m l ] ( 1 . 2 3 )

w h e r e R^ is the a x i a l r e s i s t i v i t y a s s u m e d to be c o n s t a n t .

L i k e w i s e , t he g e n e r a l e q u a t i o n for s t e a d y - s t a t e

v o l t a g e d i s t r i b u t i o n in a s i n g l e o n e - d i m e n s i o n a l u n i f o r m

c a b l e h a s the f o l l o w i n g f o r m ( J a c k et al., 1975, p . 1 4 8 ) :

( 1 . 2 4 )

T h e a i m w i l l n o w be to r e l a t e e q u a t i o n ( 1 . 2 4 ) w i t h

e q u a t i o n ( 1 . 2 2 ) by s o m e k i n d of p o w e r law. I n t e g r a t i n g

e q u a t i o n (1 . 9 ) w i t h r e s p e c t to x a n d s u b s t i t u t i n g t he

r e s u l t i n t o e q u a t i o n ( 1 . 2 2 ) y i e l d s th e f o l l o w i n g