Chapter 2
Phase and Amplitude Description
of Complex Oscillatory Patterns
in Reaction-Diffusion Systems
Hiroya Nakao
Abstract Spontaneous rhythmic oscillations are widely observed in various realworld systems. In particular, biological rhythms, which typically arise via synchronization of many self-oscillatory cells, often play important functional roles in living systems. One of the standard theoretical methods for analyzing synchronization
dynamics of oscillatory systems is the phase reduction for weakly perturbed limitcycle oscillators, which allows us to simplify nonlinear dynamical models exhibiting
stable limit-cycle oscillations to a simple one-dimensional phase equation. Recently,
the classical phase reduction method has been generalized to infinite-dimensional
oscillatory systems such as spatially extended systems and time-delayed systems, and
also to include amplitude degrees of freedom representing deviations of the system
state from the unperturbed limit cycle. In this chapter, we discuss the method of phaseamplitude reduction for spatially extended reaction-diffusion systems exhibiting stable oscillatory patterns. As an application, we analyze entrainment of a reactiondiffusion system exhibiting limit-cycle oscillations by an optimized periodic forcing
and additional feedback stabilization.
2.1 Introduction
There are abundant examples of spontaneous rhythmic oscillations in living systems, ranging from microscopic oscillations of cardiac cells and spiking neurons
to macroscopic oscillations of heartbeats and brainwaves [5, 6, 28, 32, 34, 39]. In
many cases, macroscopic oscillations result from synchronized collective dynamics
of many microscopic cells and play essentially important functional roles in the survival of living systems. Regular rhythmic dynamics of the cells are typically modeled
as limit-cycle oscillations in nonlinear dynamical systems. A representative example
of such dynamical systems is the Hodgkin-Huxley model of spiking neurons, which
H. Nakao (B)
Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo
152-8552, Japan
e-mail: nakao@sc.e.titech.ac.jp
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_2
11
Phase and Amplitude Description
of Complex Oscillatory Patterns
in Reaction-Diffusion Systems
Hiroya Nakao
Abstract Spontaneous rhythmic oscillations are widely observed in various realworld systems. In particular, biological rhythms, which typically arise via synchronization of many self-oscillatory cells, often play important functional roles in living systems. One of the standard theoretical methods for analyzing synchronization
dynamics of oscillatory systems is the phase reduction for weakly perturbed limitcycle oscillators, which allows us to simplify nonlinear dynamical models exhibiting
stable limit-cycle oscillations to a simple one-dimensional phase equation. Recently,
the classical phase reduction method has been generalized to infinite-dimensional
oscillatory systems such as spatially extended systems and time-delayed systems, and
also to include amplitude degrees of freedom representing deviations of the system
state from the unperturbed limit cycle. In this chapter, we discuss the method of phaseamplitude reduction for spatially extended reaction-diffusion systems exhibiting stable oscillatory patterns. As an application, we analyze entrainment of a reactiondiffusion system exhibiting limit-cycle oscillations by an optimized periodic forcing
and additional feedback stabilization.
2.1 Introduction
There are abundant examples of spontaneous rhythmic oscillations in living systems, ranging from microscopic oscillations of cardiac cells and spiking neurons
to macroscopic oscillations of heartbeats and brainwaves [5, 6, 28, 32, 34, 39]. In
many cases, macroscopic oscillations result from synchronized collective dynamics
of many microscopic cells and play essentially important functional roles in the survival of living systems. Regular rhythmic dynamics of the cells are typically modeled
as limit-cycle oscillations in nonlinear dynamical systems. A representative example
of such dynamical systems is the Hodgkin-Huxley model of spiking neurons, which
H. Nakao (B)
Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo
152-8552, Japan
e-mail: nakao@sc.e.titech.ac.jp
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_2
11