Biomedical Engineering Reference
In-Depth Information
However, in order for us to be able use the input-output relationship found in
Section 15.2, the dynamics of the network should be such that the network properties
in this stationary states are consistent with the assumptions we made in Section 15.2.
Thus, these assumptions impose several additional conditions that the steady-states
should obey to be truly self-consistent:
•
The fluctuations in the inputs must be approximately independent from neuron
to neuron. This condition will be trivially satisfied when the major part of
the noise comes from external independent sources. It will also be satisfied
when the network is sparsely connected, i.e., when the connection probability
between any pair of neurons is weak. In this case, the 'noise' term coming
from the recurrent network itself becomes uncorrelated from neuron to neuron
[19, 24, 118, 119].
•
The probability of a spike being emitted in the network at any moment must
be constant in time. Thus, the steady state must be stable with respect to
any instability that leads to non-stationary global network activity, such as
synchronized oscillations.
•
The neurons must emit approximately as Poisson processes for the input-
output relationship to be valid. This is in general expected to be true when
the average total input to the neurons is sub-threshold, which will be the situ-
ation of interest in our discussion.
Several types of local network connectivity and synaptic structure are conceivable.
They differ mainly in the source of the fluctuations in the synaptic current to the
neurons. One approach is to investigate the behavior of the network as a function of
its size
N
and of the number of connections per cell
C
. The strategy is to scale the PSP
size
J
C
m
(a measure of the synaptic strength) with
C
, and study the behavior of
the network as
C
≡
J
/
•. An advantage of this procedure is that
a)
the behavior of the
network is much simpler and easier to analyze in the
C
→
• limit, and
b)
network
behaviors which are only quantitatively different for finite
C
, become qualitatively
different as
C
becomes infinite. Additionally several of the technical assumptions we
had to make in Section 15.2 become exact in this limit.
Alternatively, one can assume that
N
and
C
are large but finite. In this case one
does not assume any specific scaling of the PSP size with
C
, but rather uses the for-
mulas for arbitrary values of these parameters (as in the previous sections) and stud-
ies the behavior of the resulting equations when they take realistic values informed
by the available experimental data. Some of the hypothesis made in the calculations
will only be verified approximately, but the theory will be more directly comparable
to experiments, where
C
and
J
=
C
m
are finite.
In the following subsections we describe the self-consistency equations obtained
in each of these scenarios. For ease of exposition, our discussion will be carried out
in the simplest case of neurons connected through instantaneous synapses, unless
specified otherwise.
/
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