Brain Research, 49 (1973) 83-99 page 83 © Elsevier Scientific Publishing Company, Amsterdam - Printed in the Netherlands

© 1952 IBR0

DYNAMIC SINGLE UNIT SIMULATION OF A REALISTIC CEREBELLAR NETWORK MODEL

A. PELLIONISZ AND J. SZENTÁGOTHAI

Department of Anatomy, Semmelweis University Medical School, Budapest (Hungary)
(Accepted September l2th, 1972).


A SUMMARY

On the basis of recent quantitative histological analysis of the cerebellum a single unit simulation model of the neuronal network was developed to study some spatio-temporal aspects of its activity. The hypothesis that Golgi inhibition serves to keep the density of excited parallel fibers in a narrow range both in time and in space has been tested, with special attention to`the possible role of the 2-fold input to the Golgi cells. The result of the simulation supports the hypothesis in showing that: (i) marked `leveling' of activity occurs, basically as a result of mossy fiber-driven Golgi cell action, (ii) the indirect (parallel fiber) input appears to improve the efficiency of the mechanism by increasing the speed of development and stabilizing the long-term accuracy of the leveling, and (iii) a precondition of such an effect is, however, that Golgi cells do not act as coincidence detectors, but can be fired independently either by direct or indirect inputs.

INTRODUCTION

The neuronal network of the cerebellar cortex offers challenging theoretical problems regarding structuro-functional relationships, which are too complex to be resolved without the help of computer simulation. Fortunately, the network has unique structural properties that favor such modeling. It (1) is built up of only 5 different types of neuron; (2) has two characteristically different input channels; and (3) is connected into a rectangular spatial lattice of very high regularity. Speculations on structuro-functional relationships of the network began with the assumption of the specific inhibitory nature of interneuronsl9, and led to concepts (l5,l6) of the lateral inhibition effected by basket cells, which were borne out well by electrophysiological analysis (l,2).

Earlier attempts at simulating the cerebellar cortex neuronal network (l2,l3) were seriously limited on the one hand by being entirely static, and on the other by being based on rather unrealistic earlier concepts of the numerical parameters of the network. Mortimer (7) simulated the dynamic interactions of assemblies of several hundred neurons each, interconnected in quasi-realistic fashion. This opened. new possibilities for modeling in the time domain, although the assembly-level approach made it difficult to make safe inferences about the behavior of single cells. A spatio-temporally continuous analysis was attempted by Meno (6), although with the serious limitation that the behavior of the network had to be assumed as linear.

The present paper attempts to answer certain specific theoretical questions by means of a model consisting of discrete single units, assembled according to realistic numerical, geometrical and connectivity relations. This has been rendered possible by a recent systematic analysis of the entire neuronal network of the cerebellar cortex (8-ll). With this d.egree of realism, it is possible to investigate in some detail the consequences of 4 alternative hypotheses as to the mode of interaction betwecn mossy fiber and parallel fiber inputs to Golgi cells.

THE MODEL

The model simulates the inhibitory action of a large array of Golgi cells upon the mossy fiber input, by showing how an arbitrary spatial. input pattern of activation is transformed at successive points in time, while traversing the direct route from mossy fibers to Golgi cells, and/or the indirect path over granule cells and parallel fibers. A cinefilm of these transformations has been made to enable the dynamic process to be better visualized.

Structural features of the network model

The simulated neuronal network is based essentially on the idealized Purkinje cell arrangement of Palkovits et al. (8). As the function of the Purkinje cells is not simulated at this stage of the studies, it would not have been necessary to include them in Fig. 1. However, since the arrangement of the Golgi cells has been given by Palkovits et al. () only with reference to the density and arrangement of the Purkinje cells, it was necessary to have the idealized Purkinje cell arrangement as a background upon which the Golgi cell arrangement pattern could be superimposed in appropriate density. The model considers the several neuronal types of the cerebellar cortex as if they were lying in successive separate planes (Fig. 2). These neuronal fields (one for each type of neuron) can thus be considered as matrices, in which the neurons represent the matrix elements. The stimulus patterns traversing the several fields can thus be described as simple binary matrices.

The structural and numerical relations of the model network are matched with the real data from the adult cat cerebellar cortex (8-11) as closely as possible, allowing for less than 10 % deviation in general. There is, however, one major exception, in that the number of granule neurons has been made equal to that of the mossy fiber terminals, whereas in reality the ratio is 28 granule cells per mossy terminal. This, which had to be done to keep the number of elements within reasonable limits, was deemed permissible, since a previous studyl3 showed that the excess number of granule cells per mossy terminal could be taken into account as structural redundancy.

The basis of the structure is a matrix of 114 X 142 mossy terminals symbolized by the background pattern of small asterisks in Fig. 1. This matrix would correspond to an area of 444 microns X 1255 microns of the real cerebellar cortex. The granule cells are represented by the same 114 X 142 asterisks, since their number is arbitrarily set to be equal to that of the mossy terminals. Each granule cell is assumed to be connected by 4 dendritesll with 4 of its neighboring mossy terminals. The parallel fibers (granule cell axons, see Fig. 2) run longitudinally (i.e. vertically in Fig. 1) for 1.1 mm in both directions. (To avoid `edge effects' in this restricted model, the two longitudinally split halves of Fig. 1 are functionally connected to form a closed ring, in which the parallel fibers present in the 6 continuous stripes established by the Golgi cells (squares in Fig. 1) have to be imagined as running continuously through stripes 1,4, 2,5,3,6,1. . .). The territories occupied by the Purkinje cells in the plane view of the cerebellar cortex are symbolized in the upper and lower parts of the figure by large circles for the cell bodies and by long transverse bars for the dendritic trees, arranged in the staggered manner assumed by Palkovits et al. (8). At lower center a single Purkinje cell is shown separately. The territories available to each Golgi cell (their bodies indicated by smaller black circles) have been considered, for the sake of simplicity, as squares instead of hexagons9. Each Golgi cell is assumed to receive inputs over its lower dendrites (Fig. 2) from all mossy terminals within its territory, and additional indirect inputs from all parallel fibers traversing the territory in the molecular layer above the cell (Fig. 2), i.e. from 21 X 228 parallel fibers. Each cell, in turn, controls the transmission from all mossy terminals to granule cells within its territory. (To avoid too many complications in the diagram, this latter connection is not indicated; for details consult Eccles et al.l).

Functional assumptions

Input pattern. Unfortunately no natural input patterns are available that would lend themselves for modeling. The analysis has therefore been carried out for a simple step function input through the mossy fibers, spatially characterized by a centrally symmetrical Gaussian distribution. Thus, the probability that a mossy terminal at a distance from the center of Q ,um will be excited at time t is

(see equations in facsimile)

This input pattern is shown in Fig. 4A.

Threshold variation. At the first step of transformation - during transmission from mossy terminals to the granule cells - the threshold of the granule cells becomes crucial. While in an earlier investigationl2 the threshold was considered as fixed, it was pointed out laterl3 that the Golgi cells, which are known to inhibit the transmission from mossy afferents to granule cellsl, might control the activity of their fields of granule cells by resetting their thresholds. Advantage can be taken of the fact mentioned earlier that the granule cell has only 4 dendrites (4.17 in realityli) ~ a structural feature almost unique in the nervous system. Each dendrite is engaged in synaptic contact with one mossy fiber terminal of different originl~. It is thus useful to distinguish between 4 conditions of the granule cell, in which 1, 2, 3 or 4 simultaneous mossy inputs, respectively, would be sufficient to fire it. Let M be the average local density of active mossy terminals, G that of the granule cells and T the threshold value of the granule cells. Then G can be expressed as follows:

4-T G(T'M) ~ rnl M4 n (1 ~ M)n n=0

As shown in Fig. 3, under the 4 conditions mentioned the same random mossy input of 50 % density (inset below) can be transformed into a granule output density varying from 94 to 6 % (insets at left).

Since each mossy fiber-granule cell synapse has an inbuilt inhibitory mechanism furnished by the terminals of the Golgi cell axonsl, it seems reasonable to assume that in the absence of Golgi inhibition the granule cell threshold will be minimal (i.e. level l.). Depending on the effectiveness of the Golgi inhibition the granule cell might then need simultaneous excitation through 2, 3 or 4 of its contacts with mossy terminals. As there is no information available on the spatial density of the mossy fiber input at which the Golgi cells start becoming active, their threshold is set arbitrarily in Fig. 3 (and in the model) to a value at which the output is in the medium range. In this case the output characteristic would correspond to the heavy parts of the constituent curves. Suppose, further, that the threshold in a population of granule cells controlled by a Golgi cell were inhomogenous, i.e. some of lower

[excerpts from the Article is in Facscimile]