Reliability and energy function of an oscillator and map neuron

External physical and chemical stimuli can be perceived and encoded in biological neurons, and then synaptic couplings guide neurons to present appropriate firing modes in electrical activities. Oscillator-like and map neurons can produce similar deriving-responses while the working mechanism is open before considering the effect of membrane properties and channels function. In this study, a theoretical neuron model is proposed by involving two capacitive variables and a memristive channel sensitive to external electric field, and the double-layer membrane property is relative to temperature. During circuit approach, two capacitors are connected via a thermistor, and a charge-dependent memristor (CDM) is connected into one branch circuit of the neural circuit. The temperature-dependent and memristive neuron model is described by a nonlinear oscillator containing four variables and energy function is defined from physical aspect. Furthermore, linear transformation is applied to the sampled time variables from the oscillator neuron, and an equivalent map neuron following covariance is obtained for dynamical analysis, energy definition and adaptive control, and similar coherence resonance is detected under noisy excitation. The results show how to obtain reliable map neurons with exact energy function, and adaptive control law under energy flow becomes reasonable.