Dynamical analysis and hardware implementation of Hopfield Neural Networks based on fractional calculus and memristor coupling

In view of the limitations of traditional Hopfield Neural Networks in capturing the complex interactions among neurons, this paper proposes a novel Hopfield Neural Network model based on fractional calculus and memristor coupling. The model employs the Caputo fractional derivatives to characterize the inherent memory effects in the temporal evolution of neurons and introduces hyperbolic tangent memristor to model synaptic crosstalk, thereby enabling bidirectional coupling between neurons. Theoretical analysis involves determining equilibrium points, constructing Jacobian matrix, and applying Lyapunov methods to examine the stability and bifurcation behavior of the system, revealing rich dynamic features such as periodicity, multistability, and chaos. The theoretical findings were validated through numerical simulations, using bifurcation diagrams, phase portraits, Poincaré maps, and 0–1 tests, validate the theoretical findings. Finally, a hardware circuit based on the NI PXIe platform was developed, and the experimental results were highly consistent showing strong agreement with the simulation outcomes.