Synchrony Breaking Bifurcations in Small Neuronal Networks

We investigate synchrony breaking bifurcations in neuronal networks. These bifurcations occur from synchronous steady-states. In the mutual dyad and a three-neuron feed-forward chain we show that the generic bifurcation behaviour can be derived from the physical modelling parameters, in particular from the sign of the interaction between neurons. Each neuron is equipped with a simplified FitzHugh-Nagumo model and the coupling is based on synaptic coupling. An inhibitory or excitatory coupling can determine if the bifurcation is ‘soft’ (supercritical) or ‘hard’ (subcritical). For the analysis of the three-neuron feed-forward chain we follow the work of Rink and Sanders (2013): we can relate excitatory and inhibitory coupling to a ‘soft’ and a ‘hard’ transition, respectively. For the mutual dyad system we make use of a centre manifold reduction to find the type of pitchfork bifurcation. As we find an expression in terms of physical parameters, we can state whether the bifurcation is subcritical or supercritical in the weak coupling limit, and for slow and fast input.