Effective neural response function for collective population states

Collective behaviour of neural networks often divides the ensemble of neurons into sub-classes by neuron type; by selective synaptic potentiation; or by mode of stimulation. When the number of classes becomes larger than two, the analysis, even in a mean-field theory, loses its intuitive aspect because of the number of dimensions of the space of dynamical variables. Often one is interested in the behaviour of a reduced set of sub-populations (in focus) and in their dependence on the system's parameters, as in searching for coexistence of spontaneous activity and working memory; in the competition between different working memories; in the competition between working memory and a new stimulus; or in the interaction between selective activity in two different neural modules. For such cases we present a method for reducing the dimensionality of the system to one or two dimensions, even when the total number of populations involved is higher. In the reduced system the familiar intuitive tools apply and the analysis of the dependence of different network states on ambient parameters becomes transparent. Moreover, when the coding of states in focus is sparse, the computational complexity is much reduced. Beyond the analysis, we present a set of detailed examples. We conclude with a discussion of questions of stability in the reduced system.