Dynamics of one-dimensional spiking neuron models
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Updated: Aug 29, 2026

Real-time Electrophysiology: Using Closed-loop Protocols to Probe Neuronal Dynamics and Beyond
Published on: June 24, 2015
1Centre de Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan, 61, avenue du Président Wilson, 94230 Cachan, France. brette@ccr.jussieu.fr
Abstract:
In this paper 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 Lyapunov 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.
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