Acta Biochim. et Biophys. Acad. Sci. Hung. Vol. 5 (1), pp. 71-79
(1970)
Computer
Simulation of the Pattern Transfer of Large Cerebellar Neuronal Fields
A.
PELLIONISZ
Institute
of Anatomy, Medical University, Budapest
(Received
August 11, 1969)
A
computer simulation method is applied to the cerebellar neuron circuit, giving
an opportunity to study the activity of many neurons simultaneously. As a
morphological basis a schematic connection chart is deduced from the original
neural net. Each neuron is supposed to fire if the number of their excited
input channels reaches a threshold value. The patterns of excited neurons at a
particular instant of time are computed and displayed. Four types of cerebellar
neurons are taken into consideration in the model. The simulated structure
contains altogether 64,260 neurons. Through the patterns displayed one can get
an insight into the possible activities in neuronal fields composed of 31,510
neurons, where the complete set of connections of each element has been
considered.
I. Introduction
This
work aims at simulating the holistic behaviour of neural fields of realistic
structure. Earlier attempts at modelling hypothetical networks built up of
"formal neurons" show that their study requires simulation methods
that permit to consider the networks as a whole.
The
earliest discrete representation of a network was made by Rochester et al.
(1956) to test the postulates of Hebb and Milner on a quasi-random connected
net of 512 formal neurons. Farley and Clark (1961) simulated the propagation of
activity-spots in a planar net composed of 1296 elements with interconnections
specified by two dimensional probability distributions.
The
attempt, to be reported in the present paper, at simulating the neuron network
of the cerebellar cortex difl'ers from preceding studies in two important
aspects:
(1) The
structural characteristics of the network are deduced from the known
histological structure of the cerebellar cortex.
(2) The
smallest part of the cerebellar cortex that might be considered as a functional
unit of higher order and hence worth while to be simulated is composed of far
more (in the order of 10000) neurons even if the model were drastically
simplified.
Owing
to limitations of the computer available the network, although conforming in
principle to the real neuron network of the cerebellum, has been reduced in
neuron numbers. It has additionally been simplified by placing all kinds of the
neurons considered into two-dimensional fields and by considering their state
of
activity
for a particular moment only. In this respect the elements of the model are
typical McCulloch-Pitts neurons. This implies that no other functional
parameter of the neurons can be taken into consideration in this model, than a
threshold value for each one: i.e. the units are necessarily considered
threshold elements for the time being.
II.
Simulation of the Cerebellar Neuronal Network
The
neuronal network of the cerebellar cortex offers considerable advantages for
the construction of simplified wireing models:
(1) It
consists of 5 types of neurons only.
(2)
Their connections form a three-dimensional rectangular lattice (if the
curvature of the surface is neglected).
(3) The
distances bridged by various types of connections, and
(4) the
numbers of other elements with which any given type of connection is
established, and finally
(5) the
physiological properties of all known elements are fairly well understood
(Eccles et al., 1967).
(6) It
has two input channels.
The
connectivity model and its functional interpretation for one of the two input
channels - the mossy afferents - has been proposed by Szentágothai (1963). The
present attempt of computer simulation is based on his model, shown in its
essential features in Fig. 1. In this model the second input channel, the
climbing fibres, are neglected for the time being. This is thought permissible
as the climbing fibres contact the Purkinje cells directly, so that whatever
occurs between the mossy afferent input and the Purkinje cell is not influenced
by the climbing fibres.
(Footnote:
This is not quite true, but for this first approach this simplification may be
justified).
The
present simulation considers only (part of) what may happen if an input reaches
the cerebellar cortex through the mossy fibres: the pattern of excitation and
inhibition that is set up by this input in the matrix of Purkinje cells.
________
Fig. 1.
Diagram illustrating the main neuronal circuit of the cerebellar cortex,
detailed explanation in the text. After Szentágothai (1963, 1965)
Fig. 1
shows a composite diagram of a small piece (top of a so-called folium) of the
cerebellar cortex. Part A of the diagram shows a transverse section of the
folium with mossy afferents (M) entering from below. They establish synaptic
contacts with the claw-shape dendrites of the small granule neurons (G). Axons
of these neurons ascend into the superficial layer of the cortex where they divide
in T-shape fashion to form the parallel fibres. This is seen in part D of the
diagram representing a longitudinal section of the cortical folium. A
comparison between the shapes of the Purkinje cells (P; drawn in outlines) as
they present themselves in the transverse (A) and the longitudinal (D)
sections, explains that the dendritic trees of these cells are forming flat
sheets that are pierced at right angles by the parallel fibers. Part C of the
diagram shows the cerebellar cortex as it would appear if looked at from above.
The Purkinje cells are shown here in highly schematic way only as a circle (for
the cell body) and a transverse bar (for the dendritic sheet). Only some
representative cells are indicated in the diagram, in reality the space available
is fully filled by similar cells in the density and arrangement as can be
deduced from the drawing. Inhibitory basket cells are shown in full black in
parts A and C in the Figure. Their axons extend in transversal direction and
establish inhibitory synapses with the Purkinje cell bodies. The functional
operation of the model is explained as follows: let us assume that a number of
mossy afferents terminating in the regions of the dashed circles become excited
simultaneously and set up an excitatory volley in a beam of parallel fibres
that arise from this excited focus. Consequently all Purkinje cells the
dendritic trees of which are crossed by this excited beam of parallel fibres
(white in part C) would also be excited. This is indicated by a curved arbitrary
scale drawn above part A, in which the thick line represents the state of
excitation (or inhibition) of the Purkinje cells situated below. A similar
arbitrary scale and curve (at the extreme right) indicates the state of
excitation of the Purkinje cells along the axis of the excited parallel fibre
beam. As some of the parallel fibres terminate towards both ends of the beam
the state of excitation of the Purkinje cells falls off at both ends. As can be
deduced from parts A and C also the basket cells situated in the beam of
excited parallel fibres would become excited. As their axons are directed
laterally in either direction, and as their function is known to be an
inhibitory one, they can be assumed to set up inhibition of Purkinje cells on
both sides of the excited parallel fibre beam. This is indicated by gray
shadowing in part C and by the thick line in part A. Part B shows the supposed
state of inhibition of Purkinje cells in a longitudinal section along the
dotted line in the left part of C. This is logically a mirror image of the
curve of excitation as assumed to prevail along the axis of the excited beam of
parallel fibres as indicated at extreme right of part D. Inhibition could be
supposed to fall off in correspondence to both ends of the beam.
_________
As seen
in Fig. 1 (A and D) mossy afferents establish contacts with the clawshape
dendrites of granule cells. Each granule cell has four dendrites, and each of
them becomes involved with a terminal thickening of a mossy afferent. The
granule cell gives origin to an ascending axon that upon reaching the
superficial (molecular) layer of the cortex divides in T-shape fashion into
parallel fibres (Fig. 1. D). The parallel fibres run for about 2-3 mm in the
longitudinal direction of the folia and make excitatory contacts with the flat
dendritic trees of all Purkinje cells, which they cross. As between two
neighbouring Purkinje cells inhibitory interneurons (so-called basket cells)
are positioned, the parallel fibres excite also the basket cells. The basket
cells, in turn, establish inhibitory connections with Purkinje cells, which are
situated in lateral position from the group of parallel fibres that have been
excited simultaneously through an incoming mossy fibre volley. The concept of
Szentágothai (1963), which has been substantiated by the studies of Eccles and
co-workers (summarized by Eccles et al., 1967) is based on the assumption that
if a beam of parallel fibres would become excited by a near simultaneous volley
of mossy fibres, the Purkinje cells crossed by this beam would be excited (Fig.
1 C) and the longitudinal row of excited Purkinje cells would be flanked from
both sides by a fringe of inhibited rows of Purkinje cells.
The
model simplifies this structure by placing each kind of elements into a
two-dimensional field. The interconnections of the fields are determined in
accordance with the actual relative distances, directions, and numbers of
connections as they are found in the real cerebellar neuronal net.
As an
input to the system a random binary pattern is assumed to enter through the
mossy afferents and to create at each mossy terminal either 1 for being in
excited state or 0 for not being excited. Arbitrary assumptions are made for
the conditions under which any further element is excited above threshold.
These assumptions concern the fraction of synaptic contacts that have to be
excited simultaneously from the total number of the contacts of each respective
neuron type.
III.
The Geometrical Structure of the Model
Consider
a set of matrices having r rows and c columns each. Every matrix represents a
different neuronal field consisting of one kind of neuron only. The neurons are
placed at particular matrix points, so that the arrangement should be similar
to the original neuronal network as much as possible. The value of the matrix
elements is 1 if the neuron at that place is in an excited state, else 0. Every
matrix is denoted by a capital referring to the name of the neuron type
concerned. The neurons are denoted by the same capital having an r, c pair of
subscripts which indicates the row and column where the neuron is located in
the matrix.
The
four types of neurons considered in the model are the following:
M:
Mossy fibre terminal
G:
Granule cell
P:
Purkinje cell
B:
Basket cell
The
structure of the model is characterized by the following list, which indicates
the four sets of matrix points containing neurons:
The
above formula - also used in the following - means that the i, j subscripts
take all the values between the lower a and upper b limits (for i) or between c
and d limits (for j) respectively.)
This
means that mossy fibre terminals can be found at every matrix point and so can
the granule cells. The number of Purkinje cells and that of basket cells is
supposed to be equal, but it is five times less than the number of granule
cells. Every fifth column only contains Purkinje (and basket) cells. (The first
column which contains Purkinje and basket cells has the column subscript 3.)
IV.
Interconnections among the Neurons in the Model
A
granule cell is connected with 4 mossy fibre terminals and every mossy fibre
terminal receives also 4 granule cell dendrites, so both the divergence from
the mossy fibre terminals and convergence to the granule cells are 4. The
system of connections is:
(The
directions of the interconnections are indicated by arrows.) Every granule cell
is connected by its longitudinal (columnal), bifurcated parallel fibre with 51
Purkinje cells situated in a column. The Purkinje cell dendritic trees are
supposed to span 5 matrix columns in width and not to overlap one another. In
such a way 51 x 5 = 255 granule cell parallel fibres get through each Purkinje
dendritic tree. An important assumption is that all the parallel fibres getting
through a given Purkinje cell have functional connection with that cell. The
system of the connections among granule and Purkinje cells is the following:
(Where
k is the subscript of the nearest column containing Purkinje and basket cells
to the column j.)
Basket
cells are connected with 9-9 Purkinje cells placed symmetrically in both
lateral sides:
V. The
Logic of the Model
The conditions
(thresholds) which determine whether a granule-, Purkinje-, or basket cell in
the model is excited or not, are summarized in the following equations :
The
infiuence of the basket cell's inhibition upon a Purkinje cell is expressed by
an l;,j factor, namely:
Therefore
an element of the P matrix after being modified by the basket inhibition system
(denoted by PB;) can be calculated by
VI.
Results
The
patterns of the excited mossy terminals, granule cells, Purkinje cells having
been computed (without and with considering the basket cell inhibitory system)
the G, P, and PB matrices can be displayed as follows:
* The
nearest majority of the 4 granule cell dendrites.
** This
threshold is chosen in order to have an output Purkinje pattern with 50 per
cent rate of activity.
*** A
simple majority of the 18 basket cell axons arriving at a Purkinje axon
hillock.
In the
matrices having r rows and c columns an x represents the elements of the matrix
with the value I, and a space the element of the matrix with the value 0.
(According to the excited and not excited points respectively.)
It can
be easily understood that the neurons situated near to the margins of the
fields cannot have all their inputs because some of their input elements lie
outside of the matrix. That is why in the following all the patterns presented
display
Fig. 2.
An input pattern of the system, supposed to be existing in the field of the
mossy fibre terminals at a particular moment. Every point of the 101 row-, 130
column matrix represents a mossy fibre terminal marked with x when excited. The
states of the mossy fibre endings in this random binary pattern are considered
independent of each other. The pattern of granule cells is transformed from
this field: the granule cell, for example, connected with 4 glomeruli lying in
the frame will be excited if 3 of its inputs are excited
Fig. 3.
The pattern of the excited granule cells, transferred from the previous M
matrix. The emergence of concentrated excitator spots as a result of the local
averaging effected by the granule cells is worthy of mention. The granule cells
being connected with particular Purkinje cell can be seen in the frame. The
Purkinje cell itself is in the middle of the frame (in black) and its dendritic
tree is also indicated, spanning five granule cells in width
the
central part of the matrices only, where the neuronal interconnections are
complete from this point of view. This means that the patterns (which have 101
rows and 130 columns) display the array of the matrices from the row subscript
27 till 127 and from column subscript 2l till 150. The general formula for the
matrices is the following:
N[27:127,
21 :150]
The
vertical directions in the pictures correspond to the longitudinal axis of the
cerebellar folium.
The
input pattern of the system is shown in Fig. 2 and the G, P, PB matrices are
presented in Figs 3 - 5.
Fig. 4.
The pattern of the excited Purkinje cells, transformed from the G granule cell
matrix. The Purkinje cells to be inhibited by a single basket cell (marked with
a circle) are framed by two rectangles.
Fig. 5.
The pattern of the excited Purkinje cells after being modified by the basket
cell inhibitory system. Besides the remarkable columnal activity of the
Purkinje cells, the single or very short columns of excited Purkinje cells are
also
worth while noting
Although
it is not proposed to discuss here the special functional meaning of these
patterns, some characteristic features of the transfers are remarkable at the
first sight. For example, I wish to draw attention to the emergence of
concentrated excitatory spots as a result of the local averaging effected by
the granule cells (Fig. 2). The arrangement of the active Purkinje columns
separated by inactive or inhibited surrounding is also worthhile studying (Fig.
4). It is of particular interest that no such pattern could be observed if the
histologically established fact were not taken into consideration that the
basket cell has little if any inhibitory effect on Purkinje cells situated in
immediately neighbouring rows, whereas the inhibitory effect becomes pronounced
in the second neighbouring row. It is also interesting to note that in contrast
to the speculations of Szentágothai (1963) single Purkinje cells or very short
rows of Purkinje cells could be set into action by such a mechanism. This
result could not have been expected by conventional reasoning.
***
The
simulation has been carried out with a computer of ODRA 1013 type. The time
required has been granted by the Chair of Numerical and Computer Mathematics of
the Roland Eotvos University. It is gratefully acknowledged to the Head of the
Chair: Dr. J. Mogyoródi. The author is indebted to Dr. A. Meskó as well, for
his valuable assistance in the programming.
References
Eccles,
J. C., Ito, M., Szentágothai, J. (1967) The Cerebellum as a Neuronal Machine.
Springer-Verlag, New York, Inc.
Farley,
B. G., Clark, W. A. (1961) In: Information Theory, ed. by C. Cherry London,
Butterworths, p. 242
McCulloch,
W. S., Pitts, W. (1943) Bull. Math. Biophys. 5 115
Rochester,
N., et al. (1956) IRE Trans. Inform. Theory IT-2 p. 80
Szentágothai,
J. (1963) Magy. Tud. Akad. Biol. Oszt. Közl. 6 217
Szentágothai,
J. (1965) In: Progress in Brain Research, Vol. 14. Eds. M. Siger and J. P.
Schadé, Elsevier, Amsterdam-London-N. Y. p. 1
