Recent work in dynamical systems theory has shown how a behavior, called mutual stabilization, can arise. Mutual stabilization occurs when two interacting chaotic systems, communicating bidirectionally, can transition from chaotic to periodic dynamics. A version of mutual stabilization was demonstrated in a two-cell bidirectional neural network with each neural oscillator modeled by Fitzhugh-Nagumo dynamics. This mutual stabilization is simpler than the original version that was first demonstrated that used two communicating double scroll systems. However, the existence of mutual stabilization in coupled Fitzhugh-Nagumo neurons gives credence to the idea that it might be a property of neural systems. Therefore, the work presented here seeks to establish how mutual stabilization might occur in more complex neural models, specifically the Hindmarsh-Rose neural model. In order to establish this behavior more robustly in a coupled Hindmarsh-Rose neuron it first needs to be demonstrated how a chaotic Hindmarsh-Rose system can stabilize into periodic behavior. This work primarily discusses how a control scheme, the same scheme that was used for mutual stabilization in the double scroll oscillator, can stabilize the chaotic Hindmarsh-Rose system into cupolets (Chaotic Unstable Periodic Orbit-LETS). Then, these cupolets can be used to interact in a way such that mutual stabilization may occur between two coupled Hindmarsh-Rose neurons.
