Attractors in Networks of Bistable Neuronal Units with Depressing Synapses

Populations of neurons with strong excitatory recurrent connections can exhibit bistability in their mean firing rate. This leads to multiple fixed points in a network of weakly-coupled bistable units. Short-term synaptic depression induced by neuronal activity may change the stability of fixed points, enabling transitions between different states in both a stimulus-dependent and history-dependent manner. A sequence of such state-transitions, similar to those seen in neural data, allows the network to encode the history of time-varying information. Therefore, it is of great interest to characterize the fixed points (activity states) and the transitions between them in large networks in response to diverse stimuli. To this end, here we apply the Wilson-Cowan equation to model bistable units with depressing synapses under time-dependent input. For small networks, we analyze the attractors and bifurcations. With biologically relevant parameters, we uncover an invariant subspace where bistable fixed points are bounded by a limit cycle. This subspace provides us with a basis for developing a dimensionality-reduction formalism, which will allow us to compare model results with recorded neural data.