Biological Cybernetics

, Volume 113, Issue 1–2, pp 11–46 | Cite as

Phase reduction and phase-based optimal control for biological systems: a tutorial

  • Bharat Monga
  • Dan Wilson
  • Tim Matchen
  • Jeff MoehlisEmail author


A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.


Phase reduction Optimal control Nonlinear oscillators Control of biological systems 



Support for this work by National Science Foundation Grants NSF-1635542 and NSF-1602841 is gratefully acknowledged.


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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of Electrical Engineering and Computer ScienceUniversity of TennesseeKnoxvilleUSA

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