Abstract
Mathematics has changed over time to comprise interdisciplinary fields of research, and considering this, biomathematics has arisen as an interface study. In this work, we analyze the dynamical behavior of the Hindmarsh-Rose (HR) neuron model, which describes the neuronal bursting in a single neuron. A stability study through the Lyapynov exponents method is proposed and evidence of a chaotic dynamics is presented. Therefore, a control design based on the State-Dependent Ricatti Equation (SDRE) is proposed aiming to reduce the oscillation of the system to a desired orbit. The results show that the controller is efficient and robust as a method for preventing epileptic seizures.
Highlights
In mechanical engineering, there is a great interest in controlling chaotic behaviors
Based on the results presented, the presence of chaos for all the variation of the membrane potential was verified, since the depolarization process until repolarization, representing an individual undergoing an epileptic seizure; this fact highlights the need of applying a control law to stabilize the whole process
The dynamics of a model called the Hindmarsh-Rose System with chaotic behavior is analyzed through the Lyapunov exponents
Summary
The nerve impulse is characterized by a propagation of such depolarization through the neuron and after the impulse passes, the membrane undergoes a repolarization, recovering its normal quiescent state and ending impulse transmission [2,19]. Bursting generation has been extensively studied in the context of the Hindmarsh-Rose model, which establishes a dimensionless state variable for the membrane potential x(t), and other two ( dimensionless) variables, namely y(t), associated with the fast ion flows generated by the transfer of Na+ e K+ and z(t), which describes the dynamics of other low channels. Where a, b, c, d, s, r, xr and Iare parameters of the system, which, depending on their values, can simulate a wide range of dynamical behaviors that are topologically equivalent to the ones observed experimentally This makes the Hindmarsh-Rose model one of the most emblematic models concerning the qualitative study of neuronal bursting [1]. The system presents many behaviors, and one of them is a typical chaotic behavior of the membrane potential, that is, a behavior that is aperiodic and sensitive to the initial conditions adopting the external stimulation (variable I) as a control parameter
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