Burst-firing and extreme multistability in a dual-neuron fractional-order memristive HNN with hardware implementation

In neuromorphic circuits, memristors are ideal devices for mimicking biological synapses, while fractional-order systems, described by fractional-order integrals and derivatives, can capture past, present, and future states, making them an ideal choice for modeling complex biological systems. This work proposes a dual-neuron fractional-order memristive Hopfield neural network (FOMHNN-DN), incorporating a single memristive synapse, to investigate the dynamics of fractional-order memristive neural networks. Research results show that the dynamics of FOMHNN-DN is profoundly influenced by the memristor's internal parameters and the system's fractional order. Furthermore, the initial conditions v1(0), v2(0), and ϕ(0) of FOMHNN-DN modulate attractor positions, demonstrating a special extreme multistability. Additionally, FOMHNN exhibits rare chaotic scroll-growth attractors and burst-firing modes, both of which are modulated by initial offset-boosted effects, potentially yielding infinite dynamic variations. Finally, based on the scroll-growth situation, FOMHNN-DN is discretized using the Adomian decomposition method and implemented on a field- programmable gate array for hardware validation. By exploring FOMHNN-DN, this work lays a foundation for further research into biomimetic networks with complex nonlinear dynamic behaviors and their applications.