The Spike-and-reset dynamics for non-linear integrate-and-fire neuron models

Jonathan Touboul (1) , Romain Brette (1)

1 ODYSSEE - Computer and biological vision

The class of non-linear integrate and fire neuron models introduced in (Touboul, 2008), containing such models as the Izhikevich and the Brette-Gerstner ones, are hybrid dynamical systems, defined both by a continuous dynamics, the subthreshold behavior, and a discrete dynamics, the spike and reset process. Interestingly enough, the reset induces in bidimensional models behaviors only observed in higher dimensional continuous systems (bursting, chaos, ...). The subthreshold behavior (continuous system) has been studied in previous papers. Here we study the discrete dynamics of spikes. To this purpose, we introduce and study a Poincaré map which characterizes the dynamics of the model. We find that the behavior of the model (regular spiking, bursting, spike frequency adaptation, bistability, ...) can be explained by the dynamical properties of that map (fixed point, cycles...). In particular, the system can exhibit a transition to chaos via period doubling, which was previously observed in Hodgkin-Huxley models and in Purkinje cells.

Domaines: Dynamique Chaotique [nlin.CD]; Systèmes dynamiques [math.DS]; Neurosciences [q-bio.NC]

Jonathan Touboul, Romain Brette. The Spike-and-reset dynamics for non-linear integrate-and-fire neuron models. [Research Report] RR-6531, 2008. ⟨inria-00276924v2⟩