| Abstract: |
| We investigate a class of two-dimensional hybrid non-autonomous neuron models with a dynamical threshold, focusing on the interplay between continuous subthreshold dynamics and discrete reset mechanisms. The model consists of a linear voltage equation coupled with a threshold variable evolving according to a nonlinear function of the voltage, and incorporates periodic external input in the form of either pulse or rectified sinusoidal currents.
In particular, we study periodic solutions of the system and prove the existence and uniqueness of a globally attracting periodic orbit in the non-hybrid case and show that the hybrid system admits at most one reset-free periodic orbit, while numerical evidence suggests the possible coexistence of reset-free and resetting periodic attractors.
These results provide a rigorous mathematical framework for understanding dynamical threshold mechanisms in neuron models and contribute to the theoretical characterization of spiking versus non-spiking behavior under time-dependent forcing. |
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