Bifurcation Analysis Of A Sparse Neural Network With Cubic Topology

We study analytically the changes of dynamics of a firing-rate network model with cubic topology. The present study is performed by extending to this sparse network a formalism we previously developed for the bifurcation analysis of fully-connected circuits. In particular we prove that, unlike the fully-connected model, in the cubic network the neural activity may undergo spontaneous symmetry-breaking even if the network is composed exclusively of excitatory neurons. Moreover, while in the fully-connected topology the symmetry-breaking occurs through pitchfork bifurcations, in the excitatory cubic network it occurs through complex branching-point bifurcations with five branches. These results lead to the conclusion that the sparseness of the synaptic connections may increase the complexity of dynamics compared to dense networks.