Subthreshold oscillations and spiking in hybrid neuron models with a dynamic threshold

Abstract

We investigate a class of two-dimensional hybrid non-autonomous periodically forced neuron models known as the Meng-Huguet-Rinzel neuron model, with a dynamic threshold and reset mechanism. We focus on the interplay between the continuous subthreshold dynamics and discrete spike-induced resets. The model consists of a linear voltage equation coupled with a threshold variable governed by a nonlinear function of the membrane potential (voltage), and incorporates periodic external forcing in the form of either pulse or rectified sinusoidal currents. We analyze the existence and uniqueness of periodic solutions and prove that the non-hybrid version of the system possesses a unique globally attracting periodic orbit. For the hybrid system, we show that there exists at most one periodic orbit without resets, while numerical simulations indicate the possibility of coexistence of a reset-free periodic orbit and periodic spiking attractors. We also prove that some natural polygons contained in the phase space of the hybrid system form compact positively invariant globally attracting sets of this system. These results provide a rigorous mathematical framework for the analysis of dynamic threshold mechanisms in neuron models and contribute to the theoretical understanding of transitions between subthreshold oscillations and spiking behavior under a time-periodic forcing.