i L]R The model
ity, the exponential decay of the post-synaptic potentials, and the exponential decay of the devia-
tion of the membrane potential from the rest potential. As an example, I take a model that has
been introduced by Kryukov 1978 under the assumption that the electrical activity the mem-
brane potential of a single independent neuron can be described by some Markov process. In
fact, a model for a single neuron similar to that described in Section 2.1 appears already as a
‘realistic neuronal model’ in the monograph by Griffith 1971. In the form presented here, the
model was formulated and studied by Turova 1996, 1997.
2
.
1
. A model neuron The neuron is modelled as one point neglecting
the propagation of membrane current along the axons and the dendrites. The activity of each
neuron is described by the stochastic point pro- cess of the consecutive firing moments spikes.
Assume that any individual neuron is ‘active’, i.e. the sequence of its spikes forms a renewal process.
More exactly, in the absence of interactions, let the inter-spike intervals be independent random
variables with a generic variable
Y =
d
inf{r \ 0: hr = ye
− a r
}, 1
where y \ 0, a \ 0, and ht is Ornstein – Uhlen- beck process defined by
dht = − aht dt + dWt, with h0 = 0. In this case, the density of Y is
p
Y
6 = 2a
32
e
2a6
y pe
2a6
− 1
3
exp −
a y
2
e
2a6
− 1
, 6
\ 0.
2 Varying parameter y, one can adjust the mean of
the inter-spike interval to the data. Roughly speaking, the mean inter-spike interval is an in-
creasing function of y.
2
.
2
. A network The neurons of the network are numerated by
the sites i L¤Z
2
. The evolution of the ith neu- ron is described by a membrane potential x
i
t R
and a threshold function y
i
t R
+
, t ] 0, that are continuous stochastic processes except for, at
most, a countable number of points of discontinu- ity. It is assumed that x
i
t B y
i
t for all t ] 0, except for the random moments 0 B t
i 1
B t
i 2
, …, when x
i
t
i n
] y
i
t
i n
, n ] 1. We call these moments t
i n
, n ] 1, the moments of ‘firing’ of the ith neu- ron. At any moment of firing of the ith neuron,
the membrane potential x
i
t and the threshold y
i
t are reset jumpwise to the values 0 and y, respectively, i.e.
lim
o ¡0
x
i
t
i n
+ o = 0,
lim
o ¡0
y
i
t
i n
+ o = y,
and the process of the membrane potential accu- mulation is repeated until the next firing. Set
t
i
0 for all i L, and define for t]0 S
i
t: = t − t
i n
, if t t
i n
, t
i n + 1
]. Thus S
i
t denotes the time elapsed since the last firing of the ith neuron until the moment t. The
threshold function y
i
t is defined by y
i
t: = ye
− a S
i
t
, t ] 0.
The evolution of the membrane potentials of the interacting neurons is given by the following
system x
i
t: = x
i
0 + h
i n
[S
i
t] + It, i
L, if t t
i n
, t
i n + 1
], 3
where x0 − ;y
L
is the initial state, h
i n
t, i
L, n]1 are independent copies of ht, and the interaction term is
It =
k ] 1, i j L:t
i n
B t
j k
5 t
a
ij
e
− a t − t
j k
, if t t
i n
, t
i n + 1
]. 4
This form of interaction suggested by Kryukov et al. 1990 has the advantage that being mathemat-
ically tractable, as indeed it still resembles physio- logical data: it takes into account the exponential
decay of the post-synaptic potentials, and the connection constants a
ij
can be chosen as positive as well as negative to model excitatory and in-
hibitory connections, correspondingly. In order to study the dynamics of the spike
trains generated by the model of Eq. 3, I
introduce an
embedded process
Rt = [R
i