Jeff Moehlis

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  Controlling Biological Oscillators
  Jeff Moehlis

  Nonlinear oscillators - dynamical systems with stable periodic solutions - arise in many systems of physical, technological, and biological interest. Examples from biology include pacemaker cells in the heart, the firing of action potentials in neurons, and circadian rhythms. There are situations in which it is desirable to control biological oscillators, for example changing the phase of the circadian rhythm in order to adjust to a new time zone. With this in mind, we have developed an optimal control algorithm to change the phase of a periodic orbit using a minimum energy input, which also minimizes the transversal distance to the uncontrolled periodic trajectory. Our algorithm uses a two-dimensional augmented phase reduction technique based on both isochrons and isostables. This control algorithm is effective even when a large change in time period is required or when the nontrivial Floquet multiplier of the periodic orbit is close to one; in such cases, an analogous control algorithm based on standard phase reduction fails. Inspired by deep brain stimulation treatment of Parkinson's disease, we have also developed control algorithms for desynchronizing populations of oscillators, for example by maximizing the Lyapunov exponent associated with their phase dynamics, and through optimal phase resetting.