Coherence resonance and energy dynamics in a memristive map neuron

Nonlinear circuits can be tamed to produce similar firing patterns as those detected from biological neurons, and some suitable neural circuits can be obtained to propose reliable neuron models. Capacitor C and inductor L contribute to energy storage while resistors consume energy, and the time constant RC or L/R provides a reference scale for neural responses. The inclusion of memristors introduces memory effects by coupling energy flow with the historical states of the circuit. A nonlinear resistor introduces nonlinearity, enriching the circuit's dynamic characteristics. In this work, a neural circuit is constructed and one branch circuit contains a constant voltage source E. The relation between physical variables is confirmed and a memristive oscillator with an exact energy function is proposed. Furthermore, an equivalent map neuron is derived when a linear transformation is applied to the sampled variables of the oscillator-like neuron. The energy function for the memristive oscillator is calculated following Helmholtz's theorem, and the memristive map is expressed with an energy description. It is found that the energy of the periodic state is higher than that of the chaotic state, which highlights the key role of energy in mode conversion. Noise-induced coherence resonance or stochastic resonance is induced under an external field. The adaptive control mechanism influenced by Hamilton energy is investigated, revealing its impact on neural mode transitions. These findings bridge the gap between physical circuit design and neural modeling, providing theoretical insights into applications in neuromorphic computing, signal processing, and energy-efficient control systems.