Statistics

White noise

Physics

Hamiltonian system

Statistical physics

Mathematical optimization

Hamiltonian (control theory)

Memristor

Artificial intelligence

Nonlinear system

Logistic function

Biological neuron model

Electrical network

Helmholtz free energy

Control (management)

Bifurcation

Electromagnetic field

Mathematics

Computer science

Control theory (sociology)

Artificial neural network

Quantum mechanics

Classical mechanics

Telecommunications

Applied Mathematical Modelling

It has been extensively studied to employ memristors to model the relationship between the electromagnetic field and the membrane potential, especially for the research of modeling and dynamical analyses of electrical activity using HR neurons with memristors. This paper proposes a novel 4D HR model with a threshold flux-controlled memristor (MHR), which describes the electromagnetic induction effect. The proposed 4D HR model retains the original HR properties and can describe the complex dynamics of neurons' electrical activities with fewer parameters than the existing models. Due to the particularity of the no equilibrium point of the MHR model, the hidden dynamics are found in the proposed MHR model. The generalized Hamiltonian function is fully derived for the MHR neuron model using Helmholtz's theorem. The simplest form of the Hamiltonian form is given by assigning special values. The average Hamiltonian energy and its bifurcation are employed to find the connection between energy and firing patterns. The band-limited white noise is also studied, and it is found that it positively influences the electrical activities in the proposed MHR system.

Modeling of memristor-based Hindmarsh-Rose neuron and its dynamical analyses using energy method