Firing behaviors of memristor-based Rulkov neuron map using energy method

Abstract

Neural firing behavior is essentially the energy transmission between neurons, which can represent the information process of nervous system. Consequently, it is crucial to delve deeply into the intricate firing activity and energy characteristics of neurons. The Helmholtz theorem offers an accurate energy description for continuous neurons. However, the energy of discrete neurons remains under-explored due to lacking an effective theoretical framework. In this paper, we introduce a novel Rulkov neuron map that incorporates a charge-controlled memristor for imitating the effect of electromagnetic radiation. The Hamilton energy method is utilized to investigate the firing behaviors of this neuron map. A pivotal step in our analysis is meticulously scaling the parameters and variables to transform the neuron map into a continuous oscillator, which is essential for obtaining its energy function. In the exploration of the Rulkov neuron map, a range of intricate firing behaviors, including mixed-mode firing, bursting firing and coexistence firing, are observed. Notably, the occurrence of coherence resonance is detected under noise excitation. These behaviors are characterized by the time evolution of membrane potential and Hamilton energy spectrum. Furthermore, FPGA-based experiments are conducted to confirm the numerical results.