Analysis and Control of the Dopamine Circadian Rhythms Model

Abstract Background: The high nonlinearity of the dopamine circadian rhythms model is seen in the presence of limited cycles that disrupt the circadian rhythms. Limit Cycles originate from Hopf bifurcation points. Bifurcation analysis and Multiobjective nonlinear model predictive control are performed on the dopamine circadian rhythms model. Methods: The MATLAB software MATCONT was used to perform the bifurcation analysis. The Multi-objective Nonlinear Model Predictive Control was performed using the optimization language PYOMO. Results: The Bifurcation analysis reveals Hopf Bifurcation points that produce limit cycles. To eliminate the rhythm disturbing limit cycles the bifurcation parameter is multiplied by an activation factor involving the tanh function. The nonlinearity of the dopamine circadian rhythms model also causes spikes in the control profiles when multiobjective nonlinear model predictive control calculations are performed. The spikes are also eliminated when the control variable is multiplied by the same activation factor. Conclusion: The dopamine circadian rhythms model is shown to have two Hopf bifurcations, which cause limit cycles that can disrupt the circadian rhythms. An activation factor involving the tanh function eliminates the limit cycle causing Hopf bifurcations. This activation factor also removes the spikes that occur in the control profile.