Multiscale dynamics of special memristive ion channels in a neural circuit

The multiscale dynamics of the nervous system are crucial for functions such as computation and memory. This paper investigates the multiscale dynamics of special memristive ion channels containing T-type calcium and afterhyperpolarization (AHP) in a neural circuit. The T-type calcium channel, AHP channel, and potassium channel are represented by a first-order locally active memristor (LAM), a second-order LAM, and a first-order passive memristor, respectively. The sodium channel is represented by a voltage-dependent nonlinear resistor. The results show that two resistance parameters regulating the timescales of T-type calcium and AHP ion channel memristors modulate rich dynamical behaviors including fast spiking, chaotic spiking (CS), continuous bursting (CB), intrinsic bursting (IB), and chaotic bursting (CB). Additionally, the dynamics of two types of coexisting behaviors are obtained: the coexistence of two periodic bursting patterns, and the coexistence of periodic and chaotic bursting. Subsequently, using fast-slow variable dissection with two slow variables, the fast-slow dynamics and bifurcation mechanisms of CB and IB are identified in the codimension-2 plane. Specifically, within one period, CB initiates at a saddle-node bifurcation on an invariant circle and terminates at a Hopf bifurcation, while IB both initiates and terminates at Hopf bifurcations. Finally, the analog circuit of the neural circuit is designed and implemented on the LTspice simulation platform and digital circuits based on CH32 microcontrollers, validating the simulation and theoretical results of multiscale dynamics. These findings elucidate the multiscale dynamics of memristive ion channels influencing neuronal firing patterns, providing a theoretical and hardware foundation for neuromorphic computing.