6.5 Interacting Populations and Continuum Models

In this section we extend the population equations from a single homogeneous population to several populations. We start in Section 6.5.1 with interacting groups of neurons and turn then in Section 6.5.2 to a continuum description.

6.5.1 Several Populations

Let us consider a network consisting of several populations; cf. Fig. 6.14. It is convenient to visualize the neurons as being arranged in spatially separate pools, but this is not necessary. All neurons could, for example, be physically localized in the same column of the visual cortex. Within the column we could define two pools, one for excitatory and one for inhibitory neurons, for example.

We assume that neurons are homogeneous within each pool.
The activity of neurons in pool *n* is

where

We use Eq. (6.113) to replace the sum on the right-hand side of Eq. (6.114) and obtain

We have dropped the index

In case of several populations, the dynamic equation (6.75) for the
population activity is to be applied to each pool activity separately, e.g.,
for pool *n*

Equation (6.116) looks simple and we may wonder where the interactions between different pools come into play. In fact, pool

with

The fixed points of the activity in a network consisting of several
populations can be found as in Section 6.4. First we
determine for each pool the activity as a function of the total input *I*_{m}

where

Inserting Eq. (6.119) in (6.118) yields the standard formula of artificial neural networks,

derived here for interacting

6.5.2 Spatial Continuum Limit

The physical location of a neuron in a population often reflects the task of a neuron. In the auditory system, for example, neurons are organized along an axis that reflects the neurons' preferred frequency. A neuron at one end of the axis will respond maximally to low-frequency tones; a neuron at the other end to high frequencies. As we move along the axis the preferred frequency changes gradually. For neurons organized along a one-dimensional axis or, more generally in a spatially extended multidimensional network, a description by discrete pools does not seem appropriate. We will indicate in this section that a transition from discrete pools to a continuous population is possible. Here we give a short heuristic motivation of the equations. A thorough derivation along a slightly different line of arguments will be performed in Chapter 9.

To keep the notation simple, we consider a
population of neurons that extends
along a one-dimensional axis; cf. Fig. 6.16.
We assume that the interaction
between a pair of neurons *i*, *j* depends only
on their location *x* or *y* on the line.
If the location of the presynaptic neuron is
*y* and that of the postsynaptic neuron is *x*,
then
*w*_{ij} = *w*(*x*, *y*).
In order to use Eq. (6.115),
we discretize space in segments of size *d*.
The number of neurons in the interval
[*n* *d*,(*n* + 1) *d*] is
*N*_{n} = *d*
where is the spatial density.
Neurons in that interval form the group .

h_{n}(t|) |
h(n d, t|) = h(x, t|) |
(6.121) | |

A_{n}(t) |
A(n d, t) = A(x, t) |
(6.122) |

Since the efficacy of a pair of neurons with

h(n d, t|) = d w(n d, m d )(t - , s) A(m d, t - s) ds . |
(6.123) |

For

which is the final result. The population activity has the dynamics

where

If we are interested in stationary states of asynchronous firing,
the activity
*A*(*y*, *t*) *A*_{0}(*y*) can be calculated as before
with the help of the neuronal gain function *g*. The result is
in analogy to Eqs. (6.118) and (6.120)

In the case of SRM_{0} neurons, the input potential
*h* does not depend on the last firing time
so that Eq. (6.124) reduces to

We assume that the postsynaptic potential can be approximated by an exponential function with time constant , i.e., (

If we make the additional assumption that the activity

If we insert
*A*(*y*, *t*) = *g*[*h*(*y*, *t*)] in Eq. (6.128),
we arrive at an integro-differential equations for
the `field' *h*(*x*, *t*)

We refer to Eq. (6.129) as the neuronal field equation (Amari, 1977a; Feldman and Cowan, 1975; Wilson and Cowan, 1973; Ellias and Grossberg, 1975). It will be studied in detail in Chapter 9.

Cambridge University Press, 2002

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