Phase Response Curves: Overview

Abstract

Phase response curves (PRCs, alternatively phase-resetting curves) are a powerful way of characterizing and explaining the behavior of nonlinear oscillators without knowing anything about their specific internal dynamics. The phase response curve represents the shortening or lengthening of the cycle period caused by an input depending upon at what point (phase) within the cycle an input is received. In contrast, for a linear time invariant system, the effect of an input is independent of when it is applied. No matter whether the oscillations represent the flashing of fireflies, a pendulum-based clock, a cardiac cell, or a neural oscillator, the phase response curve predicts the phasic relationship of the oscillator to periodic forcing or to coupling within a network of other oscillators. Each oscillator can be reduced to a phase oscillator whose angular velocity on a circle is constant, except when it receives an input that resets (advances or delays) its phase on the circle. In a neural context, there is usually a threshold event, often the action potential or spike, which is used to demarcate the boundaries of a cycle. If an input shortens the cycle, the next spike occurs sooner than it otherwise would have, and so the next spike is advanced. Conversely, if the next spike occurs later than it otherwise would have, the spike is delayed. The phase-resetting curve is sometimes presented as the response to a specific input. We have called this the general PRC. One example might be to perturb the oscillator underlying the circadian rhythm by exposing an animal to a period of light during its usual period of darkness; another example might be to stimulate a particular synaptic input or set of synaptic inputs to a neural oscillator. The “▶ Spike-Time Response Curve” is a way to plot the phase response to a specific input, that is, a spike in the presynaptic neuron, in a way that preserves information about time intervals. Phase-resetting theory is generally applied to neural circuits under a simplifying assumption; two common assumptions are that of pulsatile (brief) coupling or weak coupling. The entry on “▶ Pulse-Coupled Oscillators” uses the information in the spike-time response curve (or the information in the general phase-resetting curve combined with the intrinsic period information) to predict the response of an oscillator to periodic forcing or mutual coupling, under the pulse-coupled assumption that the effect of each input dissipates quickly compared to the cycle period. A second use of the term phase-resetting curve is to represent the response of the oscillator to an infinitesimal input, in other words as a change in the period per unit of the input, called the infinitesimal PRC or iPRC. The entry on “▶ Phase Response, Measurement of the Infinitesimal” explains how to measure the infinitesimal phase response curve (iPRC). The iPRC is useful mostly in the context of the weak-coupling assumption, which assumes that the phase resetting due to two brief, sequential pulses summates linearly, and the phase resetting due to a single brief pulse rescales linearly with the height of the pulse. Thus, a complicated input like a postsynaptic current waveform can be conceptually broken up into a series of shifted and scaled pulses (Dirac delta functions). The entry on “▶Weak Coupling Theory” explains the connection between the iPRC and the general PRC. That is, if the coupling is sufficiently weak, the general phase resetting due to any arbitrary