In this chapter we have studied an integral equation for the population dynamics. and discussed its relation to density methods. The validity of the population equations relies on three assumptions: (i) a homogeneous population of (ii) an infinite number of neurons which show (iii) no adaptation.
It is clear that there are no large and completely homogeneous populations in biology. The population equations may nevertheless be a useful starting point for a theory of heterogeneous populations (Brunel and Hakim, 1999; Chow, 1998; Senn et al., 1996; Tsodyks et al., 1993; Pham et al., 1998). We may distinguish between heterogeneity in the coupling weights wij and heterogeneity in the local parameters of the neurons, e.g., the threshold or reset value. The case of randomly chosen weights has been discussed in Section 6.4.3. In the stationary case, the population activity equations can be discussed by solving simultaneously for the mean activity A0 and the noise amplitude . The form of the population activity is similar to that of a homogeneous network. In order to treat heterogeneity in local neuronal parameters, the variability of a parameter between one neuron and the next is often replaced by slow noise in the parameters. For example, a population of integrate-and-fire neurons where the reset value ur is different for each neuron is replaced by a population where the reset values are randomly chosen after each firing (and not only once at the beginning). Such a noise model has been termed `noisy reset' in Chapter 5.4 and discussed as an example of slow noise in parameters. The replacement of heterogeneity by slow noise neglects, however, correlations that would be present in a truly heterogeneous model. To replace a heterogeneous model by a noisy version of a homogeneous model is somewhat ad hoc, but common practice in the literature.
The second condition is the limit of a large network. For N the population activity shows no fluctuations and this fact has been used for the derivation of the population equation. For systems of finite size fluctuations are important since they limit the amount of information that can be transmitted by the population activity. For a population without internal coupling (J0 = 0), fluctuations can be calculated directly from the interval distribution PI(t |); cf. Chapter 5. For networks with internal coupling, an exact treatment of finite size effects is difficult. For escape noise first attempts towards a description of the fluctuations have been made (Spiridon et al., 1998; Meyer and van Vreeswijk, 2001). For diffusive noise, finite size effects in the low-connectivity limit have been treated by Brunel and Hakim (1999).
The limit of no adaptation seems to be valid for fast-spiking neurons (Connors and Gutnick, 1990). Most cortical neurons, however, show adaptation. From the modeling point of view, all integrate-and-fire neurons that have been discussed in Chapter 4 are in the class of non-adaptive neurons, since the membrane potential is reset (and the past forgotten) after each output spike. The condition of short memory (= no adaptation) leads to the class of renewal models (Perkel et al., 1967a; Cox, 1962; Stein, 1967b) and this is where the population equation applies; cf. (Gerstner, 1995,2000b). A generalization of the population equation to neuron models with adaptation is not straightforward since the formalism assumes that only the last spike suffices. On the other hand, adaptation could be included phenomenologically by introducing a slow variable that integrates over the population activity in the past. A full treatment of adaptation would involve a density description in the high-dimensional space of the microscopic neuronal variables (Knight, 2000).
© Cambridge University Press
This book is in copyright. No reproduction of any part of it may take place without the written permission of Cambridge University Press.