Phase Sensitivity Function in Oscillator Dynamics

• Phase Sensitivity Function (PSF) is a measure that quantifies the response of a limit-cycle oscillator’s phase to small perturbations, offering insights into synchronization and bifurcation behaviors.
• It distinguishes between time-parameterized (tPSF) and spatial (sPSF) approaches, with sPSF capturing the geometric localization of phase sensitivity on the oscillator’s cycle.
• PSF analysis under stochastic forcing reveals how spatially targeted perturbations and noise levels shape phase locking, which is critical in fields like neuroscience and physics.

A phase sensitivity function (PSF) quantifies the linear response of the asymptotic phase of a stable limit-cycle oscillator to infinitesimal perturbations. The PSF is a central tool in the analysis of oscillator synchronization and phase reductions, with prominent roles in neuroscience, physics, and applied mathematics. Traditional PSF definitions are most effective when perturbations are globally homogeneous or time-dependent, but spatially inhomogeneous or geometrically localized perturbations necessitate extensions, leading to the formulation of the spatial phase sensitivity function (sPSF). Both tPSF and sPSF elucidate different structural features of phase sensitivity, with recent developments highlighting subtle distinctions near critical bifurcations such as the saddle-node homoclinic (SNH, or saddle-node on invariant circle, SNIC) regime (Zhu, 2021).

1. Traditional Phase Sensitivity Function: Definition and Computation

For a smooth dynamical system $\dot{X} = F(X)$, $X \in \mathbb{R}^n$, admitting a stable $T$-periodic limit cycle $X_0(t) = \varphi(\omega t)$, $\omega=2\pi/T$, a phase map $\theta(X) \in S^1$ is constructed so that $\dot{\theta} = \omega$ on the cycle. The phase sensitivity function, or infinitesimal phase response curve (iPRC), is defined as

$$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$

Under weak perturbations,

$\dot{X} = F(X) + \epsilon P(X, t),$

the evolution of phase to first order in $\epsilon$ follows (see Eq. (6) of (Zhu, 2021)):

$X \in \mathbb{R}^n$0

The adjoint equation required to compute $X \in \mathbb{R}^n$1 is (Eq. (2.1)):

$X \in \mathbb{R}^n$2

subject to normalization (Eq. (2.2)):

$X \in \mathbb{R}^n$3

and periodicity $X \in \mathbb{R}^n$4. Numerical approaches typically integrate this adjoint equation backward in $X \in \mathbb{R}^n$5 and normalize at each step to ensure the condition is met.

Example: SNH Oscillator

The SNH oscillator in Cartesian coordinates (Eq. (1)): $X \in \mathbb{R}^n$6 On its limit cycle ($X \in \mathbb{R}^n$7 in polar coordinates), $X \in \mathbb{R}^n$8 is evaluated by integrating the adjoint equation and normalizing. Both direct perturbative and adjoint approaches for $X \in \mathbb{R}^n$9, $T$0 for a representative parameter set (e.g., $T$1) yield consistent results (see Fig. 3 in (Zhu, 2021)).

2. Spatial Phase Sensitivity Function: Geometric Perspective

When perturbations act spatially along the limit cycle rather than purely as functions of the phase variable $T$2, it is beneficial to reparameterize the cycle by arc length or a geometric angle $T$3, particularly for cycles such as the unit circle. The spatial phase sensitivity function (sPSF) is then

$T$4

coinciding with the original PSF but parameterized by geometric location.

For the SNH oscillator expressed in polar form (Eq. (2)): $T$5 with $T$6 on the cycle, so $T$7 and phase speed fixed as $T$8. The chain rule yields the transformation:

$T$9

and thus

$X_0(t) = \varphi(\omega t)$0

Due to the nonlinear relationship between $X_0(t) = \varphi(\omega t)$1 and $X_0(t) = \varphi(\omega t)$2, $X_0(t) = \varphi(\omega t)$3 is generally plotted as $X_0(t) = \varphi(\omega t)$4 against the geometric angle.

3. Distinctions Between tPSF and sPSF Near the SNIC Bifurcation

In proximity to the SNIC bifurcation ($X_0(t) = \varphi(\omega t)$5), the time-parameterized PSF $X_0(t) = \varphi(\omega t)$6 attains the canonical Type I phase response curve (PRC) forms:

$X_0(t) = \varphi(\omega t)$7

with $X_0(t) = \varphi(\omega t)$8 for all $X_0(t) = \varphi(\omega t)$9 (Fig. 4). However, the geometric accumulation of phase is concentrated near $\omega=2\pi/T$0, i.e., close to the saddle-node, so

$\omega=2\pi/T$1

becomes sharply peaked near $\omega=2\pi/T$2, while $\omega=2\pi/T$3 remains broad. Thus, tPSF expresses temporal sensitivity, whereas sPSF demarcates localizations of phase sensitivity where the geometric speed $\omega=2\pi/T$4 is smallest, highlighting spatial features invisible to tPSF alone.

Summary Table: Distinction Between tPSF and sPSF

Function	Parameterization	Key Sensitivity
tPSF	$\omega=2\pi/T$5 (time)	Temporal sensitivity
sPSF	$\omega=2\pi/T$6 (space)	Spatial localization

The spatial localization accentuated by sPSF is critical for characterizing the response to spatially patterned stimuli, especially near bifurcation points.

4. Universality of Type II PRC in Smooth 2D Limit Cycles

A Type I PRC is defined by $\omega=2\pi/T$7 for all $\omega=2\pi/T$8. Section 3.2 of (Zhu, 2021) rigorously proves that for any smooth two-dimensional dynamical system with a limit cycle, it is impossible to have a strictly nonnegative PSF. The argument proceeds from the normalization condition (Eq. (4.1)):

$\omega=2\pi/T$9

For any unit vector $\theta(X) \in S^1$0 in $\theta(X) \in S^1$1, as the cycle traces the plane, it must cross the direction $\theta(X) \in S^1$2 at least twice, requiring $\theta(X) \in S^1$3 to change sign. By (4.1), $\theta(X) \in S^1$4 must likewise change sign over the cycle. Continuity then ensures that $\theta(X) \in S^1$5 attains both positive and negative values. Hence, no smooth 2D oscillator can exhibit a Type I PRC—all such systems have Type II PRCs.

A plausible implication is that classical classifications of oscillator response curves based on the sign of the PRC must be revisited for higher dimensions and smoothness constraints.

5. Phase Sensitivity in Synchronization Under Stochastic Forcing

Consider an ensemble of $\theta(X) \in S^1$6 uncoupled SNH oscillators subjected to both common and independent white noise:

$\theta(X) \in S^1$7

with $\theta(X) \in S^1$8, $\theta(X) \in S^1$9 controlling noise strengths, $\dot{\theta} = \omega$0 the common and $\dot{\theta} = \omega$1 the independent zero-mean Gaussian white noises. Applying phase reduction, the stationary distribution $\dot{\theta} = \omega$2 of phase difference $\dot{\theta} = \omega$3 satisfies (Eq. (5.1)):

$\dot{\theta} = \omega$4

normalization $\dot{\theta} = \omega$5.

Here,

$\dot{\theta} = \omega$6

$\dot{\theta} = \omega$7, $\dot{\theta} = \omega$8 yields $\dot{\theta} = \omega$9 (complete synchronization); $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$0, $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$1 makes $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$2 uniform. The interplay of $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$3, $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$4, and the spatial structure of $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$5, $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$6 shapes the observed clustering.

Influence of sPSF on Synchronization

For the SNH example, spatial kernels $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$7 can be selected to target geometric points (e.g., $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$8 localizes coupling), revealing that synchronization strength peaks when stimulation aligns with the geometric position where sPSF is maximal (near $$Z(\theta) = \nabla_X \theta \,\Big|_{X=X_0(\theta/\omega)}.$$9), as empirically validated by agreement of analytical $\dot{X} = F(X) + \epsilon P(X, t),$0 with Monte Carlo simulations (Figs. 8–10 in (Zhu, 2021)). The geometry of the cycle and spatial phase sensitivity, rather than merely temporal snapshots, control noise-induced phase locking.

6. Summary and Implications for Oscillator Theory

The phase sensitivity function, both in time and space, is fundamental to the quantitative understanding of oscillator response and synchronization under weak perturbations. The spatial PSF extends the standard framework by capturing the spatial localization of phase sensitivity, revealing critical features near bifurcations (notably SNIC) that tPSF overlooks. No smooth two-dimensional oscillator possesses a strictly nonnegative PSF—Type II behavior is universal in this context. The geometric structure of the cycle, as encoded by sPSF, directly influences synchronization properties under stochastic forcing, especially when noise or coupling is spatially inhomogeneous. These results underscore the necessity of considering both temporal and spatial parametrizations for a comprehensive theory of phase response in nonlinear oscillators (Zhu, 2021).