Display Abstract

Title Piecewise-smooth stroboscopic maps in periodically driven spiking models

Name Albert Granados
Country France
Email albert.granados@inria.fr
Co-Author(s) Martin Krupa, Fr\'ed\'erique Cl\'ement
Submit Time 2014-01-13 05:28:51
Special Session 112: Nonlinear dynamics in neuroscience
In this work we consider a general non-autonomous spiking model based on the integrate-and-fire model, widely used in neuronal modeling. Our unique assumption is that the system is monotonic, possesses an attracting subthreshold equilibrium point and is forced by means of periodic pulsatile (square wave) function.\\ In contrast to classical methods, in our approach we use the stroboscopic map instead of the so-called firing-map, and becomes a discontinuous map. By applying theory for piecewise-smooth systems, we avoid relying on particular computations and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions.\\ We rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude and the duty cycle of the pulse. We show that it is covered by regions of existence of periodic orbits given by period adding structures. They completely describe all the possible spiking asymptotic dynamics and the behavior of the firing rate, which is a devil's staircase. Our results allow us to show that the firing-rate also follows a devil's staircase with non-monotonic steps when the frequency of the input is varied, and that there is an optimal response in the whole frequency domain.