Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models

Arnaud Tonnelier (1) , Gerstner Wulfram (1)

1 LCN - EPFL - Laboratory of Computational Neuroscience

We derive and study two-dimensional generalizations of integrate-and-fire models which can be found from a piecewise linear idealization of the FitzHugh-Nagumo or Morris-Lecar model. These models give rise to new properties not present in one-dimensional integrate-and-fire models. A detailed analytical study of the models is presented. In particular, (i) for the piecewise linear FitzHugh-Nagumo model, we determine analytically the bistability regime between stationary solutions and oscillations, that is, typical for class-II models. (ii) In the piecewise Morris-Lecar model, we find a noncanonical class-I transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate-and-fire models. (iii) Furthermore, we establish a relation to the recently proposed resonate-and-fire model and show that a short input current pulse can trigger several spikes.

Mots clés: Neuronal models; piecewise linear models; phase plane; integrate-and-fire neurons

Domaines: Systèmes dynamiques [math.DS]; Neurosciences [q-bio.NC]

Arnaud Tonnelier, Gerstner Wulfram. Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2003, pp.021908. ⟨hal-00393720⟩