Collective behaviour of neural networks often divides the ensemble of neuro
ns 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 beca
use 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 competit
ion 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 s
ystem to one or two dimensions, even when the total number of populations i
nvolved 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 focu
s is sparse, the computational complexity is much reduced. Beyond the analy
sis, we present a set of detailed examples. We conclude with a discussion o
f questions of stability in the reduced system.