Switching dynamics analysis and synchronous control of a non-smooth memristive Hindmarsh-Rose neuron model

A reliable neuron model can accurately estimate and predict complex bioelectrical activities, which provides potential guidance and assistance for designing neurocomputing science and intelligent sensors. To disclose the regulation mechanism of heterogeneous electromagnetic fields on action potential, a non-smooth memristive Hindmarsh-Rose (HR) neuron model is established by introducing a three-stage flux-controlled memristor. Based on the theory of flow switchability, the sufficient and necessary conditions for crossing, grazing and sliding motions of the system are presented and verified. It is particularly interesting that the existence and stability of the equilibrium point of the system by exciting bipolar pulse current have time-varying behaviour. Further, extensive coexisting attractors are discovered by multiple numerical tools. Importantly, the switching dynamics of the coexisting firing modes involving period and chaos are qualitatively discussed by calculating the normal vector field G-functions. Moreover, a Hamilton energy adaptive controller is designed to efficiently achieve the complete synchronisation of the coupling neurons consisting of the non-smooth system via unidirectional synaptic connections under mismatched parameters. These obtained results provide potential theoretical guidance for the lesion, control and treatment of neuron-related diseases.