Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models with Periodic Drive

Abstract

We study one-dimensional integrate-and-fire models of the general type x˙=F (t, x) and analyze properties of the firing map which iterations recover consecutive spike timings. We impose very week constraints for the regularity of the function F (t, x), e.g. often it suffices to assume that F is continuous. If additionally F is periodic in t, using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide a detailed description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution.

Keywords:

• integrate-and-fire neuron
• firing map
• interspike intervals
• ordinary differential equations
• circle maps

AMS Subject Classification:

• 37E10
• 37E45
• 37M25
• 37A05
• 92B20