Synchrony Breaking Bifurcations in Small Neuronal Networks

Abstract

Abstract. 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.1. Introduction. We will consider neuronal networks as dynamical systems with a network structure. The nodes of the network model the neurons. Real life neuronal networks are immensely complex but in this paper we will consider small networks. Such small networks are called 'motifs' and are often present as sub-networks of large neuronal networks. In this paper we consider the mutual dyad and a feedforward chain. The mutual dyad contains two identical neurons, mutually coupled by identical coupling. The feed-forward chain contains three identical neurons with a specific feed-forward coupling. These two motifs are shown in figures 1.1 and 1.2.

The lifting bifurcation problem on feed-forward networks

“…Examples of modeling through networks include biological, computational and physical real-world applications, see e.g. [4,12,16,25]. In applications, networks are commonly used to describe properties of dynamical systems formed by interacting individual dynamical systems.…”

“…We consider feed-forward networks where each cell in the first layer only receives inputs from itself and cells in the other layers only receive inputs from cells in the previous layer. Feed-forward networks have been used, for example, to design machine learning networks and neuronal networks, [12,16]. In those applications, the cells emulate neurons and each edge corresponds to a unidirectional connection between two neurons.…”