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| 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. |
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