Periodic and chaotic spiking behaviors in a simplified memristive Hodgkin-Huxley circuit

The famous Hodgkin-Huxley circuit contains two time-varying resistors to describe the electrophysiological characteristics of sodium and potassium ion channels. But the time-varying resistors are expressed by mixed exponential equations, which are unavailable to directly implement in analog circuit and hinders hardware application of the Hodgkin-Huxley circuit. To hit this issue, a simplified memristive Hodgkin-Huxley (m-HH) circuit is proposed, which only involves two locally active memristors (LAMs) to depict the sodium and potassium ion channels, a capacitor to describe the neuron membrane, a DC current to represent external stimulus, and two DC voltages to express the reversal potentials. MATLAB-based numerical simulations and analog circuit-based hardware measurements display that the simplified m-HH circuit can generate memristor parameter- and DC current-related periodic spiking behaviors with different periodicities and chaotic spiking behavior. This delights that electrophysiological characteristics of ion channels and external stimulus can be employed for regulating these periodic and chaotic spiking behaviors. It is interesting that the simplified m-HH circuit can generate frequency self-adaptation and coexisting firing patterns. The inter-spike interval bifurcation diagram shows that the frequency of the periodic spiking behavior increases with the increase of externally applied DC current. Besides, the hybrid parameter bifurcation diagram displays that the simplified m-HH circuit can generate memristor initial state-related coexisting firing patterns of period-6 with chaos, period-3 with chaos, and period-3 with period-8. These offer us the unique insight into explaining some biological firing patterns. The most significance is that the analog hardware circuit is feasible in reproducing these periodic and chaotic spiking behaviors and benefits for developing spiking-based neuromorphic hardware.