a computer model of cerebellar purkinje cells. pellionisz a ...38 a. pellionisz and r. llinas active...

A computer model of cerebellar Purkinje cells. Pellionisz A, Llinás R. Neuroscience. 1977;2(1):37-48. PMID: 199854 See Pubmed Reference at the link http://www.ncbi.nlm.nih.gov/pubmed/199854 and searchable full .pdf file below

Nuuro.~ienw. 1977 Vol 2. pp. 37 48. Pergamon Press. Prmted in Great Britam.

A COMPUTER MODEL OF CEREBELLAR

PURKINJE CELLS

A. PELLIONISZ and R. LLINP;S

Department of Physiology and Biophysics, New York University School of Medicine,

New York, NY 10016

Abstract-A mathematical computer model of frog Purkinje cells is generated on the basis of present-

day morphological and physiological data. The mode1 utilizes passive cable equations and the Hodgkin

& Huxley equations describing ionic conductances in excitable membranes. It comprises 62 compart-

ments, the active and passive properties being specified independently for each compartment. The

morphological properties of the mode1 were obtained from computer reconstruction and direct observa-

tion of Golgi-stained Purkinje cells. Three forms of activation were utilized to test the adequacy of

the model: (1) antidromic invasion, (2) orthodromic invasion via the parallel fiber-Purkinje cell synapse,

and (3) climbing fiber activation. It was shown that the electrophysiological parameters available in

the literature make it possible to construct a model capable of demonstrating most of the electrical

properties of Purkinje cells, such as antidromic invasion and the ability to generate simple spikes

and spike bursts. Questions such as the mechanism of generation of climbing fiber bursts were analyzed. The model may be used as a heuristic tool to help in the analysis and interpretation of electrophysiology

and as a prototype element in the construction of more complex computer simulations of neuronal

circuits.

RIGOROUS mathematical models of the electrical activity of central neurons, besides being a powerful tool to test and interpret experimental data, can also be an essential component in the development of computer simulation of neuronal networks. Our interest in the Purkinje cell stems from its well understood morphology and function. In fact, this cell is unique in that its input, the climbing and the mossy-parallel fiber systems, have been totally described anatomi- cally and functionally. Further, these inputs are ex- tremely different from each other and represent the limits of afferent innervation. On the one hand, the climbing fiber input establishes a one-to-one pattern of innervation with the Purkinje cell and is probably the most specific afferent in the central nervous system. In contrast, at the other extreme of the con- tinuum, the mossy fiber-parallel fiber complex represents the largest and most dispersed input to any

central cell (RAY&N Y CAJAL, 1911; cf. PALAY & CHAN-PALAY, 1974). From an electrophysiological point of view, the Purkinje cell is probably one of the most exhaustively studied neurons (cf. ECCLES, ITO & SZENTAGOTHAI, 1967; LLINAS, 1969).

On a more general level the integrative properties of the Purkinje cell have gained further interest since the demonstration of dendritic electroresponsiveness (cf. LLINP;S & NICHOLSON, 1969, 1971; LLINAS & HESS,

1976). This latter variable emphasizes the need for a modeling technique capable of handling a partially or totally active dendritic tree. The developing of such a mode1 for the Purkinje cell offers the particular advantage that it must reproduce the responses to three distinctly different forms of activation, (1) the

Abbreviation: EPSP, excitatory postysnaptic potential.

antidromic invasion, (2) orthodromic parallel fiber

mediated single spikes, and (3) the climbing fiber-activated spike burst. Meeting these stringent require- ments provides an assurance of adequacy and a clari-

fication of the mechanisms generating the complex bursting discharges which thus far have been purely

speculative (ECCLES, LLINAS & SASAKI, 1966; MAR- TINEZ, CRILL & KENNEDY, 1971).

Given the morphological richness of the Purkinje

cell dendritic arborization, the very useful equivalent cylinder assumption developed for the spinal

motoneuron (Rall, 1962a,b) will not be used here. While this decision introduces significant computa-

tional difficulties, it does, however, extend the mode1 to trees not following the 3/2 constraint implicit in this concept. In fact, a preliminary quantitative analysis on frog Purkinje cells shows that the actual branching power averages 1.75 (J. MCLAREN, personal communication).

In an important new development in neuronal modeling for the spinal motoneuron, DODGE & Coo-

LEY (1973) combined the idealized equivalent cylinder cable model of Rall with the mathematical formula- tion of HODGKIN & HUXLEY (1952) for membrane ionic permeabilities. The model is composed of non-

uniform cable segments representing the cellular morphology. In the present paper Dodge & Cooley’s mode1 is further developed and applied to Purkinje cells. The neuron is divided into a multitude of spatial components lumped together by the cable equations; the electrical phenonena in each compartment are governed by either the Hodgkin-Huxley equations (in the compartments representing excitable sections of the neuron membrane) or by passive R.C. properties. In this paper only passive dendritic arborizations will be considered; a future publication will consider

_~ .31

38 A. PELLIONISZ and R. LLINAS

active dendritic properties. A preliminary note (PEL-

LIONISZ & LLIN& 1975) has been published on the

present study.

METHODS

The Purkinje cells of the model are represented by differ-

ent types of dichotomous tree dendritic structures. The

morphogenesis of these trees provides a variant of the

‘skeleton’ of branch-arborization in respect of lengths and

angular deviations of branches on the 29 x 29 two-dimen-

sional lattice (of 10pm spacing). As is shown in Fig. 1 A,

which demonstrates the basic neuronal elements of the cir-

cuitry model (cf. PELLIONISZ, LLINAS & PERKEL. 1977). the

Purkinje cell bodies are distributed on a staggered arrange-

ment, The Purkinje cell dendritic trees are gradually

thinned towards the distal end to provide a clarity of the

positioning of trees and an impression about the density

of the dendritic structure (Fig. 1 B). For illustrative reasons,

three representative dendritic arbors are depicted in Fig.

1 C-E. Close to one-half of the 29 x 29 matrix points are

marked by dots. Each dot represents one complete spiny

branchlet on the tree. Variation of the branch distribution

in these dendritic trees is indicated in Fig. 1. In C mini-

mum dendritic overlapping is observed, and in D a large

degree of overlapping is shown. In E dendritic trees with

clear dendritic asymmetry is illustrated. These skeleton

trees represent the distribution of the Purkinje cell’s

smooth dendrites and are considered an adequate model

for the development of a connectivity circuit in which

a rigid characterization of the neuronal connectivity is

largely avoided.

Morphological structure of’ the Purkinje cells

All Purkinje cell dendritic arborizations are represented

as a set of cable segments. Each segment has a uniform

diameter but the different segments have different dia-

meters (Fig. 1 F). The basic dimensions were obtained from

Golgi-stained preparations (HILLMAN, 19696) and electron

micrographs (SOTELO. 1969). In establishing the diameters

of the dendritic segments a crucial parameter is the

exponent p in the equation for parent and daughter

branches (RALL, 1962a).

af + a4 = a{

which for the equivalent cylinder model is 3/2. However,

since preliminary measurements (J. MCLAREN, personal

communication) indicate that this figure for first bifurca-

tion of the Purkinje cell dendritic tree averages 1.75, this

latter figure is used in the models presented in this paper

unless otherwise noted. The total morphological structure

of the cell is constructed in the following way. Drndritic tree. (i) The length of the ‘trunk’ is separately

set [normally 70pm (HILLMAN, 1969a)]. (ii) The lengths

of dendritic branches are determined by the morphogenesis of skeleton trees (PELLIONISZ et al., 1977) (iii) The branch

point diameter (initially 6.0 pm) is established for the trunk.

The diameters for each successive bifurcation, the ai and

a2 daughter branch and a0 parent diameter, follow the

equations a? + a{ = a6 and al/a2 = L,/Lz, where L, is

the combined length of the remainder dendritic tree sup-

ported by the daughter branch with diameter a,, while L, is that for the other daughter branch.

Sonla. The soma of the cell is assumed to be a sphere

(with diameter normally set at 15 pm).

Alon. The axon is constructed as consisting of three

different structural parts. each being a uniform cylinder:

(i) Initial segment of axon (normally with length of 25 and

diameter of 2 pm). (ii) Myelinated segments (of 200ltm

length and 3 pm diameter). (iii) Axonal node of Ranvier

(cylinder of 3 grn length and 2 ILrn diameter).

The morphological structure of such a Purkinje cell is

shown in Fig. 1 F. Note that different scales apply for den-

dritic diameter and branch length and for the length and

diameter of axonal segments.

Compartmentalizution of’ the Purkinje cell model

Compartmentalization of the Purkinje cell model is

shown in Fig. 2. As seen in A. each of the seven myelinated

segments, the seven Ranvier nodes, and the axonal initial

segments is considered as a separate spatial compartment.

The electrical parameters utilized were lOOR/cm. for the

intracellular medium, 6000 R/cm* and I pF/cm’ (DODGE

& COOLEY. 1973) for the specific resistivity and capacitance

of the soma, dendrites and nodes and infinite resistance,

and 0.05 PFicm’ for resistance and capacitance of the in-

ternode respectively. The spherical soma is equated to a

cylinder of identical volume and is represented as a separ-

ate compartment. The dendritic branches are broken into

thirds where the midsection represents a simple cylindrical

compartment. while the adjoining three sections. at each

branch, represent a combined spatial compartment. Thus.

while the Purkinje cell model had a variety of dendritic

branching patterns, it consisted. in every case. of 62 spatial

compartments.

Each cylindrical compartment (e.g. axona! compart-

ments, dendritic midsections. or the two adjoining thirds

at the upper end of the top branches) is equated to an

equivalent cylinder of two halves while the branching com-

partments have a three cylindrical equivalent (Fig. 2 B).

If in the i neighbor compartments the diameters of cylin-

drical segments are di, the lengths are Li, and the compart-

ment receives I, activated spines, then the V membrane

potential in the equivalent circuit (C in Fig. 2) is deter-

mined by the following equations:

(where ri is the specihc resistance of the internal medium

c is the specific cross-membrane capacitance of the mem-

brane and R,,,,,, is the resistance represented by one spine.

Time functio,l ofmrmhrunc potential and Hodgkin & Huzlr) cquatiorls

The following equations apply for the membrane poten-

tial c’ as determined by the I,, cross-membrane current,

I, longitudinal current (inside the cylindrical segment) and

the I, synaptic current.

o=c;;. + I,, + I, + I,s where

’ I I,= 1 R (b’ - b;, 1 and I, = i’~m (V - Vypsp)

*=I n \P

Where V,,. k& and V,,,,, arc the equilibrium potential values for the Na+ and K+ ions and for the synaptic depo-

larization (with the assumed values of 115 mV, - 5 mV and

i i i

0 0

0 0

- 0

o o o

U ~ w

~ o

. ~ o

~ . ~

~ °.~.~

. ~ ~-~.~ o ~ ~ - ~

~ ~ ~ o

~-~

"-~ ~ ~ . ~ o ~ o n

o . - - ~

° "0 . ° "~ ~

n ~

o . 0 ~ ' ~

Purkinje cell model 41

C Ri -_i

1 mrec SOOpA/cm=

FIG. 2. Multicompartmental model of the Purkinje cell. The model consists of 31 dendritic branches, 15 bifurcation points, a soma, an initial segment and 7 nodes and myelinated segments. Each of these 62 compartments are represented by a cylindrical spatial segment of uniform diameter; or, for the branching compartments, a combination of three cylinders as shown in B. The equivalent electrical circuit is shown in C, where Ri represents the longitudinal resistance of the cylindrical compartment and the rest represents the electrical equivalent of the membrane as described by HODGKIN & HUXLEY (1952) all of the parameters being set individually for each compartment. D-E-F show the assumed electrical properties of three representative compartments: a low excitability branching point (upper row in DE-F), the soma (second row) and nodes of Ranvier (lowest row) being of increasing excitability. Numerical solutions expressing the waveform of membrane potential change (D), the membrane currents (E), and the m, n, h variables in Hodgkin-Huxley equations (F) are shown as a function

of time following the brief current pulse injection illustrated in F.

90 mV respectively), VL is the e.m.f. (0 mV) for ‘leakage’ current. The neighboring compartmental potential values are denoted by I$

The RNa, RK and R, cross-membrane resistance values for Na+, K+ and leakage currents are determined as fol- lows:

t/R,, = gNa.s

l/R, = gK.s

R, = RJs

where S is the combined surface area of the i cylinders involved in one compartment:

s = 1 nd,L,. II=,

While the specific cross-membrane resistance (R,) is assumed to be constant (normally taken as 6C00Q/cm2), the Na and K conductances follow those of HODGKIN & HUXLEY (1952), in which the actual gNa and gK values are at all times determined by the average gNa and gK but are changing with the Vmembrane potential and time as determined by the equations:

gNa = msh &

gk = n4&.

Here the dimensionless variables of k = m, n and h follow the equation

dk/dt = ~(1 - k) - &. k.

The c( and B constants are deduced from those in FRANKENHAEUSER & HUXLEY (1964) with only minor modifications:

a,,, = 0.36 v - 22.0

1 - exp(22.0 - V)/3

/I, = 0.40 13.0 - v

1 - exp(V - 13)/30

a, = 0.02 v - 35.0

1 - exp(45.0 - V)/lO

p, = 0.25 10.0 - v

1 - exp(V - lO.O)/lO

ah = 0.35 -10.0 - v

1 - exp(V + 10.0)/7

/?* = 0.05 v - 45.0

I - exp(45.0 - V)/lO

These sets of parameters were established through the pro- cedure of requiring an adequate rising time and propagation of the action potential (see HARDY, 1973) and also a capability of firing with a frequency exceeding 5OOHz, as it can be recorded experimentally (LLIN& BLOEDEL & HILLMAN, 1969). Since the morphological and electrical properties of any of the 62 compartments could be adjusted independently, this provided the possibility of studying the significance of the different parameters in overall integrative properties. Especially important were


(hose relating to & and gK. An example of this versatility

is illustrated in Fig. 2 D-F where the excitability is in-

creased from close to zero at the upper portion of the

dendritic tree (top row) to a moderate level at the somatic

compartment (middle row). The axonal nodes exhibit their

usual high excitability (lower row). This individual treat-

ment of spatial compartments was considered essential in

order to study potential inhomogeneities of the excitability

on the dendritic tree (so-called ‘hot spots’) associated with

dendritic spikes (LLINAS & NICHOLSON, 1971). The present

paper aims at modeling the frog Purkinje cell in which

dendritic spikes are not prominent, the dendritic compart-

ments (unless otherwise specified) being considered non-

excitable. passive cables.

axon and then the soma itself. The depolarization. however. should rapidly decrease upward on the dendritic structure (cf. LLINAS et ul., 1969; FREEMAN &

NICHOLSON, 1975). The orthodromic activation via parallel fiber synaptic input must be graded and must generate unitary simple action potentials. The model should also be able to generate the complex spike bursts which follow the climbing fiber activation

(ECCLES et al., 1966; LLINAS et a/.. 1969) without any parameter modification.

Antidromic actiaution

Cornputu tionul methods

The set of equations was solved numerically using a

PDP- I5 computer (Digital Equipment Corp.) with floating

point processor and 32K lb-bit core memory, and the data

monitored on a Tektronix 611 storage oscilloscope and

stored via DEC-pack cartridge disc. The data later could

be displayed using either the passive storage oscilloscope

or the dynamic graphic-15 display, therefore enabling both

Ihe selecting and composing of static pictures or produc-

lion of spatio-temporal display by computerized cinemato-

graphic methods.

The numerical integration was chosen as the classical

Euler (explicit) integration form which requires suitably

limited du and dt to avoid instabilities of the computation.

While the program automatically monitored the dVincre-

ment in every compartment and accordingly relaxed or

restricted the time increment of the integration (within the

extremes of 20011s and 25,~s) separately, in an asyn-

chronous manner for every compartment, still the compu-

tation involved in the solution of Hodgkin-Huxley and

cable equations makes this 62 compartment model work

about 5 x IO” times slower than the real time activity of

Purkinje cells.

The spatio-temporal display of the membrane potential waveforms throughout the 62 compartment

model in the case of antidromic activation is shown in Fig. 3. In this particular example the main trunk and all dendritic branches are passive RC cables. The

specific Na and K conductance figures were chosen in order to arrive at an appropriate spike propagation speed through the axon and a proper waveform for the action potential. Thus &, = 600R~‘,Jcm2 and

i& = IO0 rnQ_ ‘/cm2 were assumed for the excitable Ranvier nodes and soma. A SOS:; increase of these values was assumed for the initial segment. With these values a 10 nA stimulus applied to the sixth Ranvier node generated the antidromic invasion seen in Fig. 3. Display of the intracellular voltage across the different segments indicates a rapid reduction of r/; with distance from the soma, as expected from the analysis of field potentials (LLINAS et al.. 1969; FREEMAN &

NICHOLSON. 1975).

Orthodromic uctication hi, purullr/ fibers

The traditional questions of boundary problems in a

spatially limited model were handled in two different ways:

the axonal end of the distal compartment could either have

the option of ‘open end’ (connected to the isopotential

extracellular fluid by short circuit) or ‘sealed end’, in which

the next to last compartment echoed the potential of the

last compartment with an identical value. While the upper-

most dendritic branches were always considered sealed, the

axonal ending could be set either way. The short circuited

‘cur’ ending of axon introduces a clearly visible distortion

of action potential on the last nodes which is the reason for using a relatively long axon consisting of seven myelinated segments.

RESULTS

The orthodromic activation of Purkinje cells following parallel fiber stimulation via surface electrodes has been carefully investigated in many different preparations (ECCLES et ul., 1966; LLINAS et al., 1969) and some theoretical consideration has been given to the possible impact of differences in integration of input

carried by different spatial patterns of parallel fiber input (LLINAS, 1971). Simulation of this type of Pur- kinje cell activity was obtained by current injections distributed over the dendritic tree via the dendritic spines. Thus, as seen in Fig. 2 B, the cylindrical spatial compartments may have different numbers of activated spines. The activated spine is represented by

a lOOR_’ conductance increase allowing a synaptic current to be injected as shown in the equivalent circuit of Fig. 2 C.

The simulation of electrical activity of Purkinje Thus for orthodromic activity evoked by, let us say, cells is particularly advantageous since this cell pro- a horizontal strip of activated parallel fibers, we duces three markedly different types of activation assumed that each of the I6 uppermost branches all of which have been studied in great detail. Thus, received five continuously ‘open’ synapses during a any Purkinje cell model which claims to be adequate, 0.5 ms time interval. This case is shown in Figs. 3 B at least at first approximation, must satisfy the rather and 4. The off-line display of data provided a good stringent requirement of producing appropriate opportunity to compare the results in the conven- potential waveforms for all of these three forms of tional electrophysiological manner (as two-dimen- activation. Namely, the propagation of the action sional graphs of time-function of membrane potential potential evoked by antidromic electrical stimulation at one spatial location) with the simultaneous visuali- must result in a firing of the initial segment of the zation of the depolarization wave throughout the den-


A B

lOOp

~u

10

lOO~

71

l 0(,~,

P

10

lOOp

I'msec 50mV

1 msec 50 mV

Fxo. 3. A: Numerical solution for membrane potential against time throughout the 62 compartmental model in the case of simulated antidromic invasion of the Purkinje cell. All compartments of the dendritic tree are passive. Since the lowermost node in the axon has a boundary condition of an 'open end' (short circuited to the equipotential extracellular field), the action potential of the distal three nodes is distorted. The invasion of the soma and lower dendritic branches is clearly illustrated. As expected, the invasion is blocked at the base of the dendritic arbor and produces a slight depolarization on the most peripheral branches. This remote electrotonic invasion produces an after-depolarization at somatic level. B: Orthodromic activation of the model by parallel fiber synaptic input on the uppermost dendritic branches of the tree (horizontal distribution). At the right, numerical solution for membrane potential against time indicates a large depolarization at the peripheral dendritic branches which produces a prolonged EPSP at somatic level. A second slightly longer input (0.05 ms) produced

an orthodromic action potential with 2 ms latency which propagated along the axon.

dritic tree. This possibility allowed us to obtain cine- matographic display for visual analysis of spatio-temporal events in different situations. Figure 4 shows the pictorial display of membrane poterxtial at 1.0 ms point of time, and on the right side the conventional electrophysiological displays of membrane potential time courses at each neuronal compartment.

The large depolarization of the uppermost branches is evident in Figs. 3 B and 4. As shown on the right, the EPSP generated at lower spatial compartments shows the expected changes in amFlitude and time course. In this case the input produced a just threshold EPSP at the soma which fired after a delay of about 2 ms from the onset of parallel fiber input.

For comparative purposes a second parallel fiber input of identical strength and duration, but with markedly different spatial pattern, was applied as shown in Fig. 5. In this case, integration of an asym- metrical vertical input (as an idealized case of an 'off beam' Purkinje cell activation from a parallel fiber strip; LLIN~,S, 1971) was produced by the activation of five spines, for 0.5 ms, on the main trunk and each right side branch. The overall spatial pattern of depolarization displayed on the left in Fig. 5 is self- explanatory and was taken at 2.5 ms after parallel fiber activation: The conventional electrophysiological display of the membrane potential demonstrates the functional properties of such input distribution. First of all, as analyzed in detail by PALL (1970), the

] m s e c 5 0 m V

FIG. 4. Numerical solution for a just threshold orthodromic activation of Purkinje cell model. At the left schematic representation of the Purkinje cell, the density of 'hatching' is a function of the spatial distribution of potential at 1 ms after the onset of computation. On the right is continuous display of computed potential against time at the different neuronal compartments. Time and voltage calibration as

indicated.

Nsc 2/I---c

44 A. PELLIONISZ and R. LLIN.~S

! .... i

~i~:::ii~iii!ii!i~ t

FIG. 5. Numerical solution illustrating vertical integration of an orthodromic input applied to one side of the dendritic tree (vertical distribution). The amplitude and duration of the input are identical to that in Fig. 4, the only difference being the spatial distribution of the input. The schematic diagram to the left was taken at 2.5 ms from the

onset of computation (arrow).

EPSP evoked at somatic level differed markedly both in amplitude and rising phase with different localiza- tion of parallel fiber input in varying distance from the soma. Accordingly the EPSP evoked at somatic

A

lOOp

level produced a practically immediate action potential of soma and initial segment, and the spike propagated along the axon. Moreover, the initial spike of short latency was followed, slightly more than 2 ms later, by a second propagating spike. Only after this short 'burst' was a general decay of the depolarization visible throughout the model. Interestingly, the second spike was initiated almost simultaneously at the initial segment and soma, followed by outward propagation towards the dendrites and axon.

Activation of Purkinje cell by climbing fiber

The distribution of membrane potential in the case of a climbing fiber activation of the same model Pur- kinje cell is shown in Fig. 6 B and compared with the actual record from a frog Purkinje cell (Fig. 6 A). This highly stereotyped all-or-none spike, which is followed by several (normally 3-5 but in some cases up to 12) oscillatory wavelets, has been observed fre- quently in the cerebellum of the frog (LLINLS et al., 1969; FREEMAN, 1969; FABER & KORN, 1970; RUSHMER & WOODWARD~ 1971; LLrNLs, PRECX-rr & CLARKE, 1971; HACKETT, 1972, 1976). While the climbing fiber-Purkinje cell synapse is well understood morphologically (LARRAVmNDI & VICTOR, 1967; HmLMAN, 1969b) and electrophysiologically (EcCLES et al., 1966; LLINAS & NICHOLSON, 1976), the exact mechanism which generates this characteristic spike burst has remained unclear hitherto and only speculative statements are available (EccLES et al., 1966; MARTINEZ et al., 1971).

Climbing fiber activation of the Purkinje cell model was simulated by activating five synapses on each

B i i i i i i i i i i

i ii iiiiiiiii

raV

i i i i i i ! i ! " ~ 1 . . . .

J l m s e ¢ 50mV

FIG. 6. Climbing-fiber activation of Purkinje cell. In A, sample of actual ifitracellular recording of the climbing fiber evoked spike burst in a frog Purkinje cell. The first spike is followed by a highly stereotyped pattern of wavelets. B: Numerical solution illustrating action potential waveform at Pur- kinje cell somatic level by a synaptic input equivalent to the distribution of climbing fiber synapses. The similarities with the experimental results are evident. In C similar activation as in B illustrating

membrane potential throughout the 62 compartments. For further explanation, see text.


dendritic branch for a duration of 0.5 ms with a linear shift in the onset with respect to distance. The somatic onset of depolarization was defined as zero time and 0.5 m s a t the uppermost branches. The overall membrane potential waveform at somatic level is shown in Fig. 6 B and was considered an adequate analog of the experimental data. Thus, the large initial spike is followed by a second depolarization of about 50-60~ amplitude and a set of such wavelets which wax and wane.

Display of membrane potential change along the cell puts in evidence a possible mechanism for the generation of t he waveform. Apparently the peaks of the burst are generated by the repetitive firing of the initial segment (followed shortly by the somatic firing) rather than in the dendrites as originally proposed (EccLES et al., 1966; MARTINEZ et al, 1971). It is clear that in this model the firing pattern is evoked by the delayed repolarization of the dendritic tree which drives the somatic and initial segment compartments into a short burst of repetitive firing. The model also suggestS a possible explanation for the fragile nature of some of the spike peaks, as opposed to the stable components of this burst. Furthermore, the results are consistent with similar findings in the cat Purkinje cells. For instance, the last of the three spike wavelets in the climbing fiber bursts shows the most variability (also in extracellular records) (cf. LLINAS & VOLKIND, 1973). As indicated by the model, this phenomenon may be produced by the antidromic invasion of the propagating axonal action potential into the soma. This feature appears to result from the impedance mismatch and loose coupling of the somatic and the initial segment and axon regions, due to morphological and excitability differences.

Similar phenomenology may also be found using a different dendritic branch configuration with parameters otherwise identical to those of the model used

in Fig. 6. One such example is illustrated in Fig. 7 A and B, where the basic pattern of complex spike is similar to the previous case. This is in accordance with the fact that the complex spike waveform is a highly stereotyped response, quite similar from cell to cell, although these cells exhibit more than trivial morphological differences.

Reversal of the climbing fiber evoked EPSP

The reversal properties of EPSP evoked by a climbing fiber-like activation is illustrated in Fig. 8 B. Figure 8 A shows actual intracellular recordings from a cat Purkinje cell, the EPSP showing a characteristic biphasic reversal as a depolarizing current of increasing amplitude is applied across the somatic membrane (LLIN~.S & NICHOLSON, 1976). As seen in Fig. 8 B in a purely passive model with an gEPSp of --10mV, the results from the model show striking similarities with the experimental results, even in quantitative terms. The biphasic reversal of climbing fiber-evoked EPSP at 11.7 nA outward current injection is shown in detail in Fig. 8 C. As expected from theoretical treatment of distributed synaptic inputs (RALL, 1967; CALVIN, 1969), the EPSP in the upper branches is not reversed, producing the late depolarizing portion of the biphasic EPSP reversal.

Modification of branching and internal resistance parameters

The functional importance of different parameters in this model can also be studied with great ease. Given the large number of possible permutations of these parameters, only one representative example will be illustrated. The antidromic invasion of a Pur- ' kinje cell is shown after the introduction of two small changes in the model (Fig. 9 A), and the results may be compared to those shown in Fig. 3. In the present case the branching power is set to 1.25 and the inter-

l I

I"

"1 mse¢ 50 mV

Fxr. 7. Modification of climbing fiber response of Purkinje cell with slightly different excitability of soma (A) or slightly different arborization of dendritic tree (B). As shown by both numerical solutions, the general pattern of complex spike is remarkably insensitive to these changes. The only difference

(as compared to Fig. 6 B) is the lack of the last wavelet.

46

16.0 - - ~

A. PELLIONISZ and R. LLIN.~S

B C

FIG. 8. Reversal of climbing fiber evoked EPSP. A: Experimental results obtained in a mammalian Purkinje cell. The climbing fiber EPSP is reversed by membrane depolarization (from LLINLS & NICHOLSON, 1976). In B, calculated climbing fiber EPSP at the Purkinje cell soma during trans-somatic current'injections of increasing amplitudes. These results may be compared with those in A. In C, calculated distribution of potential at different portions of the Purkinje cell model for an intrasomatic current injection of 11.7 nA (time and voltage calibration as indicated). The parallel recording demonstrates that while the EPSP would be totally reversed at somatic level (due to depolarization surpassing the EPSP equilibrium potential) the upper branches are less depolarized by the applied current and therefore the CF-evoked EPSP does not reverse there. This produces the biphasic character of EPSP

at somatic level.

nal resistance changed to 250 t~/cm. Note that even this small set of parameter changes produces a marked difference in the degree of passive antidromic invasion into the dendritic tree. Similarly a reduction of initial segment excitability of 30~o (Fig. 9 B) gener- • ates a functional state such that, while the soma produces repetitive firing, the action potentials do not propagate into the axon following the initial spike. Thus, the relative importance of the excitability o f the initial segment becomes evident.

DISCUSSION

The Purkinje cell model described aaere exemplifies the increasing interest toward using computer models as heuristic tools to clarify special problem s regarding the electrophysiology of single cells. Given the immediate possibility of three-dimensional quantitative

l 0

_ J?__~ ..... r ~

1 0 0 ~

reconstruction of nerve elements (GLASER & VAN DER LOOS, 1965; LEVIrCrrIAL & WARE, 1972; WANN, WOOL- SEY, DIERKER & COWAN, 1973; LEVINTHAL, MACAGNO & TOUNTAS, 1974; LLINAS & HILLMAN, 1975; HILL- MAN, LLINLS & CHUJO, 1976), analysis of the biological significance of this morphological information demands the development of adequate techniques to allow the functional correlation of morphology and function at a detailed level. In the past, modeling techniques tended to reduce the morphological com- plexity of the neurons by introducing important sim- plifications in order to be able to handle the problem in analytical form.

In principle the relative importance of a particular morphological or physiological set of parameters is often difficult to appreciate intuitively. Computer models, on the other hand, not only allow a wide

r I I l l

! ! : ~ r l i

1 0 0 m V

m s e c 1 m s e c

FIG. 9. Functional implications of varying morphological and physiological parameters: Numerical solutions for antidromic (A) or climbing fiber (B) activations. In A the internal resistance of the branches is increased by assuming a faster taper of dendritic diameter with distance. As a result, the antidromic invasion is delayed and the dendritic invasion minimal. In B the excitability of initial segment is reduced by 30~ and a climbing fiber activation is simulated, as in Fig. 3. Note that while the general pattern of complex spike at somatic level is retained, these spikes do not propagate along the axon.


analysis of morphofunctional states but also the possible implication of certain pathological conditions.

This means that rather than idealizing the rich variety of neurons into a single model, the functional features of neurons with special geometry of dendritic trees

may be approached. Obviously some of the most important problems

to be tackled relate to the significance of possible spatial inhomogeneities in excitability. The number of instances where more than one site of spike initiation

appears to be present in central neurons is quite

appreciable (ECCLES, LIBET & YOUNG, 1958; LORENTE DE N6 & CONDOURIS, 1959; SPENCER & KANDEL, 1961; PURPURA, 1967; LLIN~~S & NICHOLSON, 1969,

1971; KIDOKORO, 1969; KUNO & LLIN~, 1970; KORN & BENNETT, 1971; BAKER & PRECHT, 1972; CZ~H, 1972; PRECHT, 1975; LLINA~ & HESS, 1976; ZIPSER

& BENNE~, 1976). The problems relating to such parameters and their probable functional nuances as circuit modifiers may become of significance in the study of such important functions as learning, which may in fact be subserved by subtle modifications of neur-

onal excitability or geometry.

Acknowledgement-Research was supported by United States Public Health Service grant NS-09916 from the National Institute of Neurological and Communicative Disorders and Stroke.

REFERENCES

BAKER R. & PRECHT W. (1971) Electrophysiological properties of trochlear motoneurons as revealed by IVth nerve stimulation. Expl Bruin Res. 14, 127-157.

CALVIN W. H. (1969) Dendritic synapses and reversal potentials: Theoretical implications of the view from the soma. Expl Neural. 24, 248-264.

CZ~H G. (1972) The role of dendritic events in the initiation of monosynaptic spikes in frog motoneurons. Brain Res. 39, 505-509

DODGE F. A. JR & C~~LEY J. W. (1973) Action potential of the motorneuron. IBM 1. Res. Develop. May 219-229. ECCLES J. C., ITO M. & SZENT~GOTHAI, J. (1967) The Cerebellum as a Neuronal Machine. Springer, Berlin. ECCLES J. C., L~ET B. & YOUNG R. R. (1958) The behaviour of chromatolysed motoneurones studied by intracellular

recording. J. Physiol., Lond. 143, 1140. ECCLES J. C., LLINAS R. & SASAKI K. (1966) The excitatory synaptic actions of climbing fibres on the Purkinje cells

of the cerebellum. J. Physiol., Lond. 182, 268-296. FABER D. S. & KORN H. (1970) Inhibition in the frog cerebellar cortex following parallel fiber activation. Brain Res.

17, 506510. FRANKENHAEUSER B. & HUXLEY A. F. (1964) The action potential in the myelinated nerve fibre of Xenopus Laevis

as computed on the basis of voltage clamp data. J. Physiol., Lond. 171, 302-315. FREEMAN J. A. (1969) The cerebellum as a timing device: An experimental study in the frog. In Neurobiology of Cerebellar

Evolution and Development (ed. LLINAS R.), pp. 397420. Am. Med. Assn., Chicago. FREEMAN J. A. & NICHOLSON C. (1975) Experimental optimization of current source-density technique for anuran cerebel-

lum. .I. Neurophysiol. 38, 369-382. GLASER E. M. & VAN DER Loos H. (1965) A semi-automatic computer-microscope for the analysis of neuronal morpho-

logy. IEEE Trans. Biomed. Eng. 12, 22-31. HACKETT J. T. (1972) Electrophysiological properties of neuronal circuits in the frog cerebellum in oitro. Brain Res.

48, 385-389. HACKETT J. T. (1976) Calcium dependency of excitatory chemical synaptic transmission in the frog cerebellum in vitro.

Brain Res. 114, 35-16. HARDY W. L. (1973) Propagation speed in myelinated nerve-II. Theoretical dependence on external Naf and on

temperature. Biophys. J. 13, 1071-1089. HILLMAN D. E. (1969a) Neuronal organization of the cerebellar cortex in amphibia and reptilia. In Neurobiology of

Cerrbellar Euolution and Development (ed. LLINAS R.), pp. 2799325. Am. Med. Assn., Chicago. HILLMAN D. E. (19696) Morphological organization of the frog cerebellar cortex: A light and electron microscopic

study. J. Neurophysiol. 32, 818-846.

HILLMAN D. I?., LLIN~S R. & CHUJO M. (1976) Automatic and semi-automatic analysis of nervous system structure. In Computer Analysis of Neuronal Structures (ed. LINDSAY R. D.), Plenum, New York.

HODGKIN A. L. & HUXLEY A. F. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., Lond. 117, 50&544.

KI~KORO Y. (1969) Cerebellar and vestibular control of fish oculomotor neurones. In Neurobiology qf Cerrhellar

Evolution and Deoelopment (ed. LLINAS R.), pp. 257-276. Am. Med. Assn., Chicago. KORN H. & BENNETT M. V. L. (1971) Dendritic and somatic impulse initiation in fish oculomotor neurons during

vestibular nystagmus. Brain Res. 27, 169-175. KUNO M. & LLIN~S R. (1970) Enhancement of synaptic transmission by dendritic potentials in chromatolysed motoneur-

ones of the cat. J. Physiol., Lond. 210, 807-821. LARRAMENDI L. M. H. & VICTOR T. (1967) Synapses on spines of the Purkinje cell of the mouse. An electron microscopic

study.Brain Res. 5, 15-30. LEVINTHAL C., MACAGNO E. & TOUNTAS C. (1974) Computer-aided reconstruction from serial sections. Fedn Proc.

F&I Am. Sots exp. Biol. 33, 2336-2340.


LEVINTHAL C. & WARE R. (1972) Three-dimensional reconstruction from serial sections. Nature. Lend. 236, 207-210.

LLINAS R. (ed.) (1969) Neurobiology of Cerebellur Erolutiorl ur~d De~~~lop~net~t. Am. Med. Assn.. Chicago.

LLIN~S R. (1971) Frog cerebellum: Biological basis for a computer model. Math. Biosci. 11, 137-151.

LLIN.&S R., BLOEDEL J. R. & HILLMAN D. E. (1969) Functional characterization of the neuronal circuitry of the frog

cerebellar cortex. J. Neurophysiol. 32, 847-870.

LLIN~S R. & HESS R. (1976) Tetrodotoxin-resistant dendritic spikes in avian Purkinje cells. Proc. mtn. Acud. Sci. U.S.A.

73, 252@2523.

LLIN~S R. & HILLMAN D. E. (1975) A multipurpose tridimensional reconstruction computer system for neuroanatomy.

In Golgi Crrltemial Symposium (ed. SANTTNI M.), pp. 519-528. Raven Press, New York. LLIN,&S R. & NICHOLSON C. (1969) Electrophysiological analysis of alligator cerebellum: A study on dendritic

spikes, In Neurobiology qf‘ Cerebellur Eoolutior~ am/ De~rloprwr~t (ed. LLINAS R.). pp. 431 465. Am. Med. Assn..

Chicago. LLINAS R. & NICHOLSON C. (1971) Electrophysiological properties of dendrites and somata in alligator Purkinje cells.

J. Neurophysiol. 34, 532-551.

LLIN~S R. & NICHOLSON C. (1976) Reversal properties of climbing fiber potential in cat Purkinje cells: An example

of a distributed synapse. J. Neurophysiol. 39, 311-323.

LLINAS R.. PRECHT W. & CLARKE M. (1971) Cerebellar Purkinje cell responses to physiological stimulation of the

vestibular system in the frog. Expl Bruin Rex. 13, 408-431.

LLINAS R. & VOLKIND R. A. (1973) The olive-cerebellar system: Functional properties as revealed by harmaline-induced

tremor. Expl Brain Res. 18, 69-87. LORENTE DE N6 R. & CONDOURIS G. A. (1959) Decremental conduction in peripheral nerve. Integration of stimuli

in the neuron. Proc. natn. Acud. Sci.. U.S.A. 45, 592 617.

MARTINEZ F. E., CRILL W. E. & KENNEDY T. T. (1971) Electrogenesis of the cerebellar Purkinje cell response in

cats. J. Neurophysiol. 34, 348-356.

PALAY S. L. & CHAN-PALAY V. (1974) Cerebrkr Corrr.x. Cytology, tmd Orgurlixtion. Springer, Berlin.

PELLIONISI. A. & LLIN& R. (1975) Simple and complex spike generation in a computer model of cerebellar Purkinje

cells. Neurosci. Ahst. 1. 203.

P~LLIONISZ A.. LLINAS R. & PERKEL D. H. (I 977) A computer model of the ccrebellar tortes of the frog. Newoscirrw

2. 19-32.

PRECHT W. (1975) Some dendritic properties of amphibian neurons. In Adt>clilcr,s iti ,L wrology (ed. KRI:UT~~I-.~c; G.

W.). Vol. 12, pp. 15-27. Raven Press, New York.

PURPURA D. P. (1967) Comparative physiology of dendrites. In The Neurosciences: ,4 S&y Progrum (eds QUART~N

G. C.. MELNECHUK T. & SCHMITT F. 0.) pp. 372-393. Rockefeller Univ. Press, New York.

RALL W. (1962~) Theory of physiological properties of dendrites. 4nn. N.Y. A&. Sc,i. 96. 1071~1092.

RALL W. (19626) Electrophysiology of a dendritic neuron model. Bioph~s. J. 2 (2. 2), 145-167.

RALL W. (1967) Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of

synaptic input. J. Neurophysiol. 30, 1138-l 168.

RALI. W. (1970) Cable properties of dendrites and effects of synaptic location. In E.witoror) Synuptic Meckunisms

(eds ANDERSEN P. & JANSEN J. K. S.) pp. 175.-187. Universitetsforlaget, Oslo.

RAM& y. CAJAL S. (191 1) Histologic> du Syst&w Nerreu\- de I’Homw ct dcs I ‘ktihrcJ.s. Maloine. Paris.

RUSHMER D. S. & WCIODWARD D. J. (1971) Responses of Purkinje cells in the frog cerebellum to electrical and natural

stimulation. Bruin Rex 33, 315.-335.

SIXELO C. (1969) Ultrastructural aspects of the cerebellar cortex of the frog. In Nrurohiolog~~ of Cwehrllur Eaolufion

cold Derrloprmwt (ed. LLINAS R.), pp. 327-371. Am. Med. Assn.. Chicago.

SPENCER W. A. & KANDEL E. R. (1961) Electrophysiology of hippocampal neurons~~~~I\i. Fast prepotentials. J. Neurophy-

siol. 24, 272-285.

WANN D. F.. WOOLSEY T. A., DIERKER M. L. & COWAN W. M. (1973) An on-line digital computer system for the

semi-automatic analysis of Golgi-impregnated neurons. IEEE Truns. BME 20, 233- 248.

ZIPSER B. & BENNETT M. V. L. (1976) Responses of cells of posterior lateral line lobe to activation of electroreceptors

in a mormyrid fish. J. Neurophysiol. 39, 693-721.

(Accepted 25 October 1976)

a computer model of cerebellar purkinje cells. pellionisz a ...38 a. pellionisz and r. llinas active...

Documents