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Dynamics of one-dimensional spiking neuron models. (English) Zbl 1050.92007
Summary: We make a rigorous mathematical analysis of one-dimensional spiking neuron models in a unified framework. We find that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior. A rigorous expression of the Lyaponov exponent is derived. Finally, we analyse the periodically driven perfect integrator and show that the restriction of the phase map to its range is always conjugated to a rotation, and we provide an explicit expression of the invariant measure.

92C20 Neural biology
34D20 Stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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