Splay States in Finite Pulse-Coupled Networks of Excitable Neurons

The emergence and stability of splay states is studied in fully coupled finite networks of $N$ excitable quadratic integrate-and-fire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (pulse emission and termination), we were able to write down an exact event-driven map for the system and to evaluate the splay state solutions. For $M$ overlapping postsynaptic potentials, the linear stability analysis of the splay state should also take in account, besides the actual values of the membrane potentials, the firing times associated with the $M$ previous pulse emissions. As a matter of fact, it was possible, by introducing $M$ complementary variables, to rephrase the evolution of the network as an event-driven map and to derive an analytic expression for the Floquet spectrum. We find that, independently of $M$, the splay state is marginally stable with $N-2$ neutral directions. Furthermore, we have identified a family of periodic solutions surrounding the splay state and sharing the same neutral stability directions. In the limit of $\delta$-pulses, it is still possible to derive an event-driven formulation for the dynamics; however, the number of neutrally stable directions associated with the splay state becomes $N$. Finally, we prove a link between the results for our system and a previous theory [S. Watanabe and S. H. Strogatz, Phys. D, 74 (1994), pp. 197--253] developed for networks of phase oscillators with sinusoidal coupling. [PUBLICATION ABSTRACT]