Reducing neuronal networks to discrete dynamics

We consider a general class of purely inhibitory and excitatory–inhibitory neuronal networks, with a general class of network architectures, and characterize the complex firing patterns that emerge. Our strategy for studying these networks is to first reduce them to a discrete model. In the discrete model, each neuron is represented as a finite number of states and there are rules for how a neuron transitions from one state to another. In this paper, we rigorously demonstrate that the continuous neuronal model can be reduced to the discrete model if the intrinsic and synaptic properties of the cells are chosen appropriately. In a companion paper [W. Just, S. Ahn, D. Terman. Minimal attractors in digraph system models of neuronal networks (preprint)], we analyse the discrete model.