Frequency decoding subject to dose conservation: firing rate response of periodically driven spiking models

In this work we consider a periodically forced generic integrate-and-fire model with a unique attracting equilibrium in the subthreshold dynamics and study the dependence of the firing rate on the frequency of the drive. The biological motivation underlying this work is the differential control exerted by the hypothalamic gonadotropin releasing hormone (GnRH), which is secreted in a pulsatile manner, on the secretion of the pituitary hormones FSH (follicle stimulating hormone) and LH (lutenizing hormone) according to GnRH pulse frequency. Our forcing satisfies the constraint of dose conservation, which can be interpreted as fixing the average GnRH concentration, while the frequency, amplitude and duty cycle are allowed to vary. The bifurcation structure in the parameter space formed by these last ones has been rigorously studied in \cite{GraKruCle13} where the firing rate was related with the rotation number of the existing periodic orbits. In this work we rigorously study how these bifurcation structures behave upon frequency variation. This allows us to show that the dependence of the firing rate on frequency of the drive follows a devil's staircase with non-monotonic steps and that there is an optimal response in the whole frequency domain. We also characterize certain bounded frequency windows in which the firing rate exhibits a bell-shaped envelope with a global maximum.