The essentials of neuronal behavior are the absolute and relative
refractory period, the response at the soma resulting from synaptic input
(usually described by an alpha function), the omnipresent delays, and
noise. All these ingredients have been incorporated in the spike response
model (Gerstner and van Hemmen 1992, 1993; Gerstner et al. 1993)
[5,7,6]
.
It presents a faithful but simplified description of the neurons themselves
without taking recourse to differential equations. This is
essential since we have to study the spatial activity of a large system of
neurons (say 
) over a long period of time.
We discretize time by units 
ms, the width of a spike, and
label the neurons on a 2-dimensional square lattice by the index i.
The state of a neuron is described by 
. If the
potential 
at the hillock of neuron i reaches the threshold
, then the neuron is expected to fire. We describe this
stochastic behavior through a noise parameter 
in the transition
probability
This is the conditional probability that neuron i fires at time
given 
. In the noise-free limit 
we get 
where 
is
the Heaviside step function: 
for 
and 
for x < 0. In the numerics to be described below we have taken
and 
.
The spike response model describes the response of a neuron -- both the
sender and the receiver -- to a spike. If a neuron has fired a spike, it
exhibits refractory behavior for a while, i.e., it cannot or hardly
spike. This is taken care of by the refractory function 
, which is 
during the absolute refractory period and negative but increasing
to zero thereafter,
Here we take 
for 
and zero elsewhere.
The spike travels along an axon and reaches a synapse on the dendritic
tree of neuron i after 
ms. Let the synaptic strength be
and denote the alpha function by 
. Then we obtain
for the total input at the hillock of neuron i
where 
so that 
; here
ms.
For the sake of computational simplicity we have assumed that the
delays 
depend on i (instead of, say, j). In this work
the 
are taken from 
with equal probability.
Furthermore, 
always vanishes.
The neurons considered so far are pyramidal cells. The stellate cells are modeled by an inhibitory loop which is assigned to each neuron,
where 
first assumes a strongly negative value
during 5ms ( shunting inhibition) and then decays exponentially with
a time constant 
ms. Moreover,
is a uniformly distributed
random variable. It is known that stellate cells operate locally. This we
have simplified to a strictly local interaction; for details, see
Gerstner et al. (1993)
[6]
. Putting things together we find
which is to be substituted into (1). What is left is specifying the
in (3).
Since we are concerned with visual percepts such as hallucinations
it seems natural, even imperative (Zeki 1993)
[16]
, to model the primary visual
cortex. We will work with a simplified model of cortical connectivity.
Inside a column the pyramidal cells experience an excitatory interaction.
Different columns with strongly different direction preferences are
expected to inhibit each other. The upshot is a ``mexican
hat''
,
with 
and 
. Here 
is the
Euclidean distance between i and j. A second possibility, which has
also been studied, is
with 
and, again, 
. We use
free boundary conditions. In our
numerical simulations we have seen no difference between (6)
and (7). Alternatively and giving rise to the very same scenarios,
one can replace 
in (3) by 
where 
with probability 
or 
; otherwise
vanishes. Typical values for the 
s are in the range
between 2 and 5. The probabilities have been chosen in such a way that
that nearest neighbors (
) are always connected. D is a drug
parameter.
Summarizing, we have explicitly modeled the various interactions including the stellate cells, the delays which are abundantly present in the cortex, and the noise. We now turn to the network behavior itself.
