Biological Cybernetics

, Volume 113, Issue 1–2, pp 161–178 | Cite as

Optimal phase control of biological oscillators using augmented phase reduction

  • Bharat MongaEmail author
  • Jeff Moehlis
Original Article


We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator’s transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails.


Optimal control Phase reduction Alternans Circadian rhythms Spike timing control 



This work was supported by National Science Foundation Grants Nos. NSF-1363243 and NSF-1635542. We thank Dan Wilson for helpful discussions on numerical computation of the augmented phase reduction.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA

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