6.3 Integral Equations for the Population Activity

In this section, we derive, starting from a small set of assumptions, an
integral equation for the population activity. The essential idea of the
mathematical formulation is that we work as much as possible on the
macroscopic level without reference to a specific model of neuronal dynamics.
We will see that the interval distribution
*P*_{I}(*t*|) that has already
been introduced in Chapter 5 plays a central role in the
formulation of the population equation. Both the activity variable *A* and
the interval distribution
*P*_{I}(*t*|) are `macroscopic' spike-based
quantities that could, in principle, be measured by extracellular electrodes.
If we have access to the interval distribution
*P*_{I}(*t*|) for arbitrary
input *I*(*t*), then this knowledge is enough to formulate the population
equations. In particular, there is no need to know anything about the
internal state of the neuron, e.g., the current values of the membrane
potential of the neurons.

Integral formulations of the population dynamics have been developed by (Gerstner, 1995,2000b; Wilson and Cowan, 1972; Gerstner and van Hemmen, 1992; Knight, 1972a). The integral equation (6.44) that we have derived in Section 6.2 via integration of the density equations turns out to be a specific instance of the general framework presented in this section.

We consider a homogeneous and fully connected
network of spiking neurons in the limit
of
*N*.
We aim for a dynamic equation that
describes the evolution of the population activity *A*(*t*)
over time.

We have seen in Eq. (6.8) that,
given the population activity *A*(*t'*)
and the external input
*I*^{ext}(*t'*) in the past,
we can calculate the current input potential
*h*_{PSP}(*t*|)
of a neuron that has fired its last spike at ,
but we have no means yet to transform the
potential
*h*_{PSP} back into a population activity.
What we need is therefore another equation which allows us to
determine the present activity *A*(*t*) given
the past.
The equation for the activity dynamics
will be derived from three observations:

- (i)
- The total number of neurons in the population remains constant.
We exploit this fact to derive a conservation law.
- (ii)
- The model neurons are supposed to show no adaptation.
According to Eq. (6.6), the state of neuron
*i*depends explicitly only on the most recent firing time (and of course on the input*h*_{PSP}), but not on the firing times of earlier spikes of neuron*i*. This allows us to work in the framework of an input-dependent renewal theory; cf. Chapter 5. In particular, the probability density that neuron*i*fires again at time*t*given that its last spike was at time and its input for*t'**t*was*I*(*t'*) is given by the input-dependent interval distribution*P*_{I}(*t*|). - (iii)
- On a time scale
*t*that is shorter than the axonal transmission delay, all*N*neurons in the population can be considered as independent. The number of spikes*n*_{act}(*t*;*t*+*t*) that the network emits in a short time window*t*is therefore the sum of independent random variables that, according to the law of large numbers, converges (in probability) to its expectation value. For a large network it is thus sufficient to calculate expectation values.

6.3.2 Integral equation for the dynamics

Because of observation (ii) we know that the input-dependent interval
distribution
*P*_{I}(*t* | ) contains all relevant information.
We recall that
*P*_{I}(*t* | ) gives
the probability density that a neuron fires at time *t*
given its last spike at and an input *I*(*t'*) for *t'* < *t*.
Integration of the probability density
over time
*P*_{I}(*s* | ) d*s*
gives the probability
that a neuron which has fired at
fires its next spike at some arbitrary time between
and *t*. Just as in Chapter 5,
we can define
a survival probability,

i.e., the probability that a neuron which has fired its last spike at `survives' without firing up to time

We now return to the homogeneous population of neurons
in the limit of
*N* and use
observation (iii).
We consider the network state at time *t*
and label all neurons by their last firing time .
The proportion of neurons at time *t* which have fired
their *last* spike between
*t*_{0} and *t*_{1} < *t* (and have not fired since) is expected to be

For an interpretation of the integral
on the right-hand side of Eq. (6.72), we recall
that
*A*() is the fraction
of neurons that have fired in the interval
[, + ].
Of these a fraction
*S*_{I}(*t*|) are expected
to survive from to *t* without firing.
Thus among the neurons that we observe at time *t*
the proportion of neurons that have fired their
last spike between *t*_{0} and *t*_{1} is
expected to be
*S*_{I}(*t* | ) *A*() d ;
cf. Fig. 6.7.

Finally, we make use of observation (i).
All neurons have fired at *some* point in
the past^{6.2}. Thus, if we extend the lower bound
*t*_{0} of the integral on the right-hand side
of Eq. (6.72) to -
and the upper bound to *t*,
the left-hand side becomes equal to one,

because all

Since Eq. (6.73) is rather abstract,
we try to put it into a form that is easier
to grasp intuitively. To do so, we take
the derivative of Eq. (6.73)
with respect to *t*.
We find

0 = S_{I}(t| t) A(t) + A() d . |
(6.74) |

We now use

A different derivation of Eq. (6.75) has been given in Section 6.2; cf. Eq. (6.44).

Eq. (6.75) is easy to understand.
The kernel
*P*_{I}(*t* | ) is the probability density
that the next spike of a neuron which is
under the influence of an input *I*
occurs at time *t* given that its last
spike was at .
The number of neurons which have fired at is proportional
to
*A*() and the integral runs over all times in the past.
The interval distribution
*P*_{I}(*t*|) depends upon the total
input (both external input and synaptic input
from other neurons in the population)
and hence upon the postsynaptic potential (6.8).
Eqs. (6.8) and (6.75) together
with an appropriate noise model yield
a closed system of equations for the population dynamics.
Eq. (6.75) is exact in the limit of
*N*.
Corrections for finite *N*,
have been discussed by Meyer and van Vreeswijk (2001) and
Spiridon and Gerstner (1999).

An important remark concerns the proper normalization
of the activity.
Since Eq. (6.75) is defined as the *derivative*
of Eq. (6.73), the integration constant on the left-hand side
of (6.73) is lost.
This is most easily seen for constant activity
*A*(*t*) *A*_{0}.
In this case the variable *A*_{0}
can be eliminated on both sides
of Eq. (6.75)
so that Eq. (6.75) yields the trivial
statement that the interval distribution is normalized to unity.
Equation
(6.75)
is therefore invariant under a rescaling
of the activity
*A*_{0} *c* *A*_{0}.
with any constant *c*.
To get the correct normalization
of the activity *A*_{0}
we have to go back to
Eq. (6.73).

We conclude this section
with a final remark on the form of Eq. (6.75).
Even though
(6.75) looks linear, it is in fact
a highly non-linear equation
because the kernel
*P*_{I}(*t* | )
depends non-linearly on
*h*_{PSP}, and
*h*_{PSP}
in turn depends on
the activity via Eq. (6.8).

6.3.2.1 Absolute Refractoriness and the Wilson-Cowan integral equation

Let us consider a population of Poisson neurons with
an absolute refractory period
.
A neuron that is not refractory,
fires stochastically with a rate *f*[*h*(*t*)]
where *h*(*t*) is the total input potential,
viz., the sum of the postsynaptic potentials
caused by presynaptic spike arrival.
After firing, a neuron is inactive
during the time
.
The population activity of
a homogeneous group of Poisson neurons
with absolute refractoriness is
(Wilson and Cowan, 1972)

We will show below that Eq. (6.76) is a special case of the population activity equation (6.75).

The Wilson-Cowan integral equation
(6.76) has a simple interpretation.
Neurons stimulated by a total postsynaptic potential
*h*(*t*) fire with an instantaneous rate *f*[*h*(*t*)].
If there were no refractoriness, we would
expect a population activity
*A*(*t*) = *f*[*h*(*t*)].
Not all neurons may, however, fire since some of the neurons are
in the absolute refractory period.
The fraction of neurons that participate in firing
is
1 - *A*(*t'*) d*t'*
which explains the factor in curly brackets.

For constant input potential,
*h*(*t*) = *h*_{0},
the population activity has a stationary solution,

For the last equality sign we have used the definition of the gain function of Poisson neurons with absolute refractoriness in Eq. (5.40). Equation (6.77) tells us that in a homogeneous population of neurons the population activity in a stationary state is equal to the firing rate of individual neurons. This is a rather important result since it allows us to calculate the stationary population activity from single-neuron properties. A generalization of Eq. (6.77) to neurons with relative refractoriness will be derived in Section 6.4.

The function *f* in Eq. (6.76)
was motivated by an instantaneous `escape rate'
due to a noisy threshold
in a *homogeneous* population.
In this interpretation,
Eq. (6.76) is the exact equation for
the population activity of
neurons with absolute refractoriness.
In their original paper,
Wilson and Cowan motivated the function *f*
by a distribution of threshold values
in an *inhomogeneous* population.
In this case, the population equation
(6.76) is an approximation,
since correlations are neglected
(Wilson and Cowan, 1972).

6.3.2.2 Derivation of the Wilson-Cowan integral equation (*)

We apply the population equation
(6.75) to SRM_{0} neurons with
escape noise; cf. Chapter 5.
The escape rate
*f* (*u* - ) is
a function of the distance between the membrane
potential and the threshold. For the sake
of notational convenience, we set
= 0.
The neuron model is specified by a refractory function
as follows.
During an absolute refractory period
0 < *s*,
we set (*s*) to - .
For
*s*, we set
(*s*) = 0.
Thus the neuron exhibits absolute refractoriness only;
cf. Eq. (5.57).
The total membrane potential is
*u*(*t*) = (*t* - ) + *h*(*t*)
with

Given it seems natural to split the integral in the activity equation (6.75) into two parts

The interval distribution

P_{I}(t | ) = f[h(t) + (t - )] exp - f[h(t') + (t' - )]dt' . |
(6.80) |

Let us evaluate the two terms on the right-hand side of
Eq. (6.79).
Since
spiking is impossible during the absolute refractory time,
i.e.,
*f*[- ] = 0, the second term in Eq. (6.79)
vanishes.
In the first term we can move a factor
*f*[*h*(*t*) + (*t* - )] = *f*[*h*(*t*)]
in front of
the integral since vanishes for
*t* - > .
The exponential factor is
the survivor function
of neurons with escape noise as
defined in Eq. (5.7); cf. Chapter 5.
Therefore Eq. (6.79) reduces to

Let us now recall the normalization of the population activity defined in Eq. (6.73), i.e.,

The integral in (6.81) can therefore be rewritten as

During the absolute refractory period we have a survival probability

which is the Wilson-Cowan integral equation (6.76) for neurons with absolute refractoriness (Wilson and Cowan, 1972).

6.3.2.3 Quasi-stationary dynamics (*)

Integral equations are often difficult to handle. Wilson and Cowan aimed therefore at a transformation of their equation into a differential equation (Wilson and Cowan, 1972). To do so, they had to assume that the population activity changes slowly during the time and adopted a procedure of time coarse-graining. Here we present a slightly modified version of their argument (Pinto et al., 1996; Gerstner, 1995).

We start with the observation that the total postsynaptic potential,

contains the `low-pass filters' and . If the postsynaptic potential is broad,

In other words, time coarse-graining implies that we replace the instantaneous activity

As a second step, we transform Eq. (6.85) into
a differential equation. If the response kernels are exponentials,
i.e.,
(*s*) = (*s*) = exp(- *s*/),
the derivative of Eq. (6.85) is

The evolution of the activity

Equation (6.87) is a differential
equation for the membrane potential.
Alternatively, the integral equation
(6.76)
can also be approximated by a differential
equation for the activity *A*.
We
start from the observation
that in a stationary state
the activity *A*
can be written as a function of the input *current*, i.e,
*A*_{0} = *g*(*I*_{0}) where
*I*_{0} = *I*^{ext} + *J*_{0} *A*_{0}
is the sum of the external driving current
and the postsynaptic current caused by lateral
connections within the population.
What happens if the input current changes?
The population activity of neurons with a large amount
of escape noise will not
react to rapid changes in the input instantaneously,
but follow slowly with a certain delay similar to
the characteristics of a low-pass filter.
An equation that qualitatively reproduces
the low-pass behavior is

This is the Wilson-Cowan differential equation. Note that, in contrast to Eq. (6.87) the sum

The Wilson-Cowan integral equation that has been discussed above is valid for
neurons with absolute refractoriness only. We now generalize some of the
arguments to a Spike Response Model with relative refractoriness. We suppose
that refractoriness is over after a time
so that
(*s*) = 0 for
*s*. For
0 < *s* < , the
refractory kernel may have any arbitrary shape. Furthermore we assume that
for
*t* > the kernels
(*t* - , *s*) and
(*t* - , *s*) do not depend on *t* - . For
0 < *t* - < we allow for an arbitrary time-dependence. Thus, the
postsynaptic potential is

and similarly for . As a consequence, the input potential

We start from the population equation (6.75) and split the integral into two parts

Since (

cf. Wilson and Cowan (1972), Gerstner (2000b). Here we have used

The integrals on the right-hand side of (6.91) have finite support which makes Eq. (6.91) more convenient for numerical implementation than the standard formulation (6.75). For a rapid implementation scheme, it is convenient to introduce discretized refractory densities as discussed in Section 6.2; cf. Eq. (6.57).

Cambridge University Press, 2002

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