Energy computation, infinitely coexisting patterns and their control from a Hindmarsh-Rose neuron with memristive autapse: Circuit implementation

An autapse is a particular synapse with its axon and dendrites interconnected. In this contribution, the effect of a memristive autapse is investigated on the dynamics of a 2D Hindmarsh-Rose neuron. The Hamilton energy of the proposed model is established using the Helmholtz theorem. It is found after monitoring the parameters that the frequency of the external current has no effect on the energy while only the memristive autapse strength and the amplitude of the stimuli are able to affect the energy released by the considered neuron. The study of the dynamics of the proposed model revealed exotic neuronal behaviors such as reverse period doubling route to chaos, and the phenomenon of interior and exterior crises. In addition, the striking phenomenon of homogeneous extreme multistability is captured. It is characterized by the coexistence of firing patterns with identical shapes for the same set of parameters. Of particular interest, a non-invasive control method exploiting a temporal feedback term is applied to enable the basin of attraction of all the undesired dynamics to merge with the desired one for a suitable value of the controller coupling strength. The electronic circuit of the proposed model is constructed and simulated in the PSpice environment. The obtained results match well with those from the direct investigations of the mathematical model of the introduced neuron.