On the interspike-intervals of periodically-driven integrate-and-fire models

Abstract

We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input functions, we generalize some results of other authors. In particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces. Using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide also a complete description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution. Our results allow to explain some facts concerning this distribution observed numerically by other authors. These theoretical findings are illustrated by carefully chosen computational examples.

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

“…Since the equation (2.1) is a Riccati type equation, sometimes we can solve it, get the implicit formula for the firing map and use similar technics as in case of the LIF-model in [20] to prove desired properties of the firing map and the sequence of interspikeintervals. However, we will not discuss QIF in details.…”

“…In this work we extent results for the LIF and PI models obtained for locally integrable input functions f (t) in [20] to the general modelẋ = F (t, x) at the cost of more restrictive constraints on F . Here the technical problem remains often to assure that the firing map Φ(t) is defined for all t ∈ R.…”

“…[4,6,10,20]. Combination of analytical and numerical approach to the firing map was taken, for example, in [7] (phase-locking and Arnold tongues), [16] (LIF model with sinusoidal input), [21] (LIF with periodic input) and [25] (LIF with periodic input and noise).…”

“…Analytical results concerning the firing map Φ were obtained assuming that F (t, x) is regular enough (always at least continuous) and often periodic in t. However, in [20] (and in the thesis [24]) it was proved that for the linear models, LIF and PI, many required properties (e.g. continuous dependence of the firing map on the input function f in appropriate topology), hold even if f is just locally-integrable (and thus not-necessary continuous).…”

“…This work together with our previous paper [20] provides detailed and complete description of interspike-intevals for periodically driven one-dimensional integrate-and-fire models. This is the natural extension of [20] where we studied only linear models.…”