6. Population Equations

In many areas of the brain neurons are organized in populations of units with
similar properties. Prominent examples are columns in the somatosensory and
visual cortex (Hubel and Wiesel, 1962; Mountcastle, 1957) and pools of motor neurons
(Kandel and Schwartz, 1991). Given the large number of neurons within such a column or
pool it is sensible to describe the mean activity of the neuronal population
rather than the spiking of individual neurons. The idea of a population
activity has already been introduced in Chapter 1.4. In a
population of *N* neurons, we calculate the proportion of active neurons by
counting the number of spikes
*n*_{act}(*t*;*t* + *t*) in a small time
interval *t* and dividing by *N*. Further division by *t*
yields the *population activity*

where denotes the Dirac function. The double sum runs over all firing times

Theories of population activity have a long tradition
(Nykamp and Tranchina, 2000; Treves, 1993; Wilson and Cowan, 1972; Amari, 1974; Wilson and Cowan, 1973; Gerstner and van Hemmen, 1992; Brunel, 2000; Abbott and van Vreeswijk, 1993; Gerstner, 1995; Fusi and Mattia, 1999; Brunel and Hakim, 1999; Gerstner, 2000b; Amit and Brunel, 1997a,b; Eggert and van Hemmen, 2001; Knight, 1972a; Omurtag et al., 2000).
In this chapter we study the properties of a large and homogeneous population
of spiking neurons. Why do we restrict ourselves to large populations? If we
repeatedly conduct the same experiment on a population of, say, one hundred
potentially noisy neurons, the observed activity *A*(*t*) defined in
Eq. (6.1) will vary from one trial to the next. Therefore we cannot
expect a population theory to predict the activity measurements in a single
trial. Rather all population activity equations that we discuss in this
chapter predict the *expected* activity. For a large and homogeneous
network, the observable activity is very close to the expected activity. For
the sake of notational simplicity, we do not distinguish the observed activity
from its expectation value and denote in the following the expected activity
by *A*(*t*).

After clarifying the notion of a homogeneous network in
Section 6.1, we derive in Section 6.2 population
density equations, i.e., partial differential equations that describe the
probability that an arbitrary neuron in the population has a specific internal
state. In some special cases, these density equations can be integrated and
presented in the form of an integral equation. In Section 6.3
a general integral equation for the temporal evolution of the activity *A*(*t*)
that is exact in the limit of a large number of neurons is derived. In
particular, we discuss its relation to the Wilson-Cowan equation, one of the
standard models of population activity. In Section 6.4 we
solve the population equation for the fixed points of the population activity
and show that the neuronal gain function plays an important role. Finally, in
Section 6.5 the approach is extended to multiple
populations and its relation to neuronal field equations is discussed.

Most of the discussion in part II of the present book will be based upon the population equations introduced in this chapter. The population activity equations will allow us to study signal transmission and coding (cf. Chapter 7), oscillations and synchronization (cf. Chapter 8), and the formation of activity patterns in populations with a spatial structure (cf. Chapter 9). The aim of the present chapter is two-fold. Firstly, we want to provide the reader with the mathematical formalism necessary for a systematic study of spatial and temporal phenomena in large populations of neurons. Secondly, we want to show that various formulations of population dynamics that may appear quite different at a first glance, are in fact closely related. Paragraphs that are more mathematically oriented are marked by an asterix and can be omitted at a first reading.

- 6.1 Fully Connected Homogeneous Network
- 6.2 Density Equations
- 6.2.1 Integrate-and-Fire Neurons with Stochastic Spike Arrival
- 6.2.2 Spike Response Model Neurons with Escape Noise
- 6.2.3 Relation between the Approaches

- 6.3 Integral Equations for the Population Activity

- 6.4 Asynchronous firing
- 6.4.1 Stationary Activity and Mean Firing Rate
- 6.4.2 Gain Function and Fixed Points of the Activity
- 6.4.3 Low-Connectivity Networks

- 6.5 Interacting Populations and Continuum Models

- 6.6 Limitations
- 6.7 Summary

Cambridge University Press, 2002

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