Phase reduction

Phase reduction is a method used to reduce a multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator into a one-dimensional phase equation. Many phenomenon in our world such as chemical reactions, electric circuits, mechanical vibrations, cardiac cells, and spiking neorons are examples of rhythmic phenomena, and can be considered as nonlinear limit cycle oscillators.

History

The theory of phase reduction method was first introduced in the 1950s, the existence of periodic solutions to nonlinear oscillators under perturbation, has been discussed by Malkin in , in the 1960s, Winfree illustrated the importance of the notion of phase and formulated the phase model for a population of nonlinear oscillators in his studies on biological synchronization. Since then, many researchers have discovered different rhythmic phenomena related to phase reduction theory.

Phase dynamics

Consider the dynamical system of the form

${\frac {dx}{dt}}=f(x),$

where $x\in \mathbb {R} ^{n}$ is the oscillator state variable, $f(x)$ is the baseline vector field. Let $\varphi :\mathbb {R} ^{n}\times \mathbb {R} \rightarrow \mathbb {R} ^{n}$ be the flow induced by the system, that is, $\varphi (x_{0},t)$ is the solution of the system for the initial condition $x(0)=x_{0}$. This system of differential equations can describe for a neuron model for conductance with $x=(V,n)\in \mathbb {R} ^{N}$, where $V$ represents the voltage difference across the membrane and $n$ represents the $(N-1)$-dimensional vector that defines gating variables. When a neuron is perturbed by a stimulus current, the dynamics of the perturbed system will no longer be the same with the dynamics of the baseline neural oscillator. Assuming that the baseline (unperturbed) neural oscillator has an attracting limit cycle $\gamma$ with period $T_{0}$ (example, see Figure xx) $\gamma$ that is normally hyperbolic, one can show that $\gamma$ persists under small perturbations. This implies that for a small perturbation, the perturbed system will remain close to the limit cycle. Hence we assume that such a limit cycle always exists for each neuron.

The target here is to reduce the system by defining a phase for each point in some neighborhood of the limit cycle. The allowance of sufficiently small perturbations (e.g. external forcing or stimulus effect to the system) might cause a large deviation of the phase, but the amplitude is perturbed slightly because of the attracting of the limit cycle. Hence we need to extend the definition of the phase to points in the neighborhood of the cycle by introducing the definition of asymptotic phase. So for all point $x$ in some neighbourhood of the cycle, the evolution of the phase $\phi =\Phi (x)$ can be given by the relation ${\frac {d\phi }{dt}}=\omega$, where $\omega ={\frac {2\pi }{T_{0}}}$ is the natural frequency of the oscillation. By the chain rule we then obtain an equation that govern the evolution of the phase of the neuron model is

${\frac {d\phi (\gamma }{dt}}=\nabla _{x}\Phi (x)\cdot f(x)=\omega ,$

where $\nabla _{x}\Phi (x)$ is the gradient of the phase with respect to the vector of the neuron's state vector $x$, for the derivation of this result, see