Existence of periodic solutions for the Voltage–Conductance kinetic equation

Prove the existence of time-periodic solutions for the Voltage–Conductance kinetic Fokker–Planck system (equation (VC) with its boundary conditions) that models networks of integrate-and-fire neurons with voltage–conductance dynamics. Numerical simulations indicate the presence of periodic oscillations for certain parameter regimes, but a rigorous analytic proof of periodic solutions for this PDE system is currently lacking.

Background

The Voltage–Conductance kinetic model (equation (VC)) describes the joint evolution of membrane voltage and synaptic conductance in large neuronal networks via a degenerate parabolic Fokker–Planck PDE with nontrivial boundary conditions. It was derived from conductance-based integrate-and-fire models and studied both analytically and numerically.

While linear well-posedness, stationary states, and various qualitative properties are known in certain regimes, numerical experiments show stable time-periodic behavior (self-sustained oscillations) for specific parameters. However, a rigorous mathematical existence theory for periodic solutions of the full Voltage–Conductance kinetic PDE remains unestablished.