Sinusoidal Winfree Model Dynamics

• The sinusoidal Winfree model is a mathematical framework for coupled oscillators, featuring mean-field dynamics that capture synchronization and oscillator death.
• It employs order-parameter bootstrapping, volumetric analysis, and the Ott–Antonsen reduction to determine critical coupling thresholds and analyze bifurcation structures.
• The model also clarifies its connection to the Kuramoto–Sakaguchi framework, demonstrating analytical robustness and practical insights into phase dynamics in complex systems.

The sinusoidal Winfree model is a paradigmatic mean-field system of coupled phase oscillators, central to the mathematical theory of collective synchronization and oscillator death. It combines explicit biologically inspired pulse-coupled dynamics with tractable, harmonic coupling—serving as a rigorous bridge between the original Winfree model and the archetypal Kuramoto model. The model admits both an exactly solvable mean-field limit and a set of robust results concerning global convergence, bistability, and sharp thresholds for the loss of oscillatory dynamics.

1. Mathematical Formulation

The standard sinusoidal Winfree model for $N$ oscillators with phase variables $\theta_i\in[0,2\pi)$, intrinsic frequencies $\omega_i\in\mathbb{R}$, and coupling strength $K\geq 0$ is defined by

$$\dot{\theta}_i = \omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j-\theta_i), \qquad i=1,\ldots,N.$$

This is mathematically equivalent to the Kuramoto model with pure sine coupling but arises from Winfree’s framework by selecting a sinusoidal phase response curve and pulse waveform. More generally, the Winfree model reads

$$\dot{\theta}_i = \omega_i + \frac{\kappa}{N} \sum_{j=1}^N I(\theta_j) S(\theta_i),$$

where $I(\cdot)$ is the influence function (pulse), and $S(\cdot)$ encodes oscillator sensitivity (PRC).

For the “standard” sinusoidal Winfree case, $I(\theta)=1+\cos\theta$, $S(\theta)=-\sin\theta$. The system admits a decomposition via the complex order parameter

$Re^{i\Psi} = \frac{1}{N}\sum_{j=1}^N e^{i\theta_j}$

yielding the compact mean-field form

$$\dot{\theta}_i = \omega_i + K R \sin(\Psi-\theta_i)$$

with $R$ quantifying the degree of synchrony.

2. Critical Coupling, Oscillator Death, and Global Convergence

The main result for the standard sinusoidal Winfree model states that for fixed $(\omega_1,\ldots,\omega_N)$,

$K > 2\max_i |\omega_i|$

guarantees, for Lebesgue-almost-all initial phases, that every oscillator’s phase $\theta_i(t)$ converges to a stationary value—demonstrating oscillator death—and $\dot{\theta}_i(t)\to 0$ as $t\to\infty$. Furthermore, the limiting order parameter $R_\infty$ is strictly bounded away from zero: $$R_\infty \geq \sqrt{\frac{1}{2} + \sqrt{\frac{1}{4} - (\| \Omega \|_\infty/K)^2}} > 0$$ where $\| \Omega \|_\infty = \max_i |\omega_i|$.

The proof is twofold:

• Order-parameter bootstrapping: If $R(t_0)\geq R_0>0$ at some time $t_0$, then $R(t)\geq R_1>0$ for all later times, and the phase differences remain bounded; this leads to convergence.
• Volumetric argument: The flow divergence is positive when $R<1$, implying phase-space regions with persistently low $R$ contract exponentially in volume. Consequently, almost no initial condition remains trapped with low synchrony indefinitely.

This analysis quantifies and rigorously establishes the “oscillator death” regime previously observed numerically (Ryoo, 3 Jan 2026).

3. Ott–Antonsen Reduction and Low-Dimensional Dynamics

For the $N\to\infty$ limit, the Ott–Antonsen (OA) ansatz enables an exact low-dimensional reduction of the continuum Winfree model. With Lorentzian-distributed frequencies, the OA ansatz leads to a closed system for the order parameter $Z=R e^{i\Psi}$: $$\dot{Z} = (i\omega_0 - \Delta) Z + \frac{K}{2}(1-|Z|^2)Z,$$ for the standard sinusoidal case, with possible extensions to incorporate phase-lag, higher harmonics (general pulse/PRC shape), or symmetry-breaking terms. For sinusoidal coupling with explicit symmetry breaking (as in generalized Winfree models),

$$\dot\theta_i = \omega_i + \varepsilon\frac{1}{N}\bigg[ \sum_j \sin(\theta_j - \theta_i) + q\sum_j \sin(\theta_j + \theta_i) \bigg ]$$

with $q$ modulating from pure Kuramoto ($q=0$) to Winfree ($q=1$). The OA-reduced mean-field dynamics admit analytic bifurcation and stability analysis for both unimodal and bimodal frequency distributions (Manoranjani et al., 2023, Gallego et al., 2017).

4. Bifurcation Structure and Macroscopic Phase Diagram

Analytical and numerical investigation identifies several qualitatively distinct macroscopic regimes:

• Incoherence (IC): $R=0$
• Synchronized stationary state (SS): $R>0$, stationary order parameter
• Standing wave (SW): $R>0$, order parameter rotates with $\xi>0$
• Bistability: coexistence of pairs drawn from IC, SS, SW

The boundaries between these regimes are determined by Hopf, pitchfork, and saddle-node bifurcations of the mean-field-reduced ODEs. For sinusoidal Winfree models, the critical coupling for the loss of incoherence and emergence of SS or SW can be given explicitly. For example, in the unimodal frequency case,

$$\varepsilon_{HB} = 2\gamma \quad (\text{Hopf}), \qquad \varepsilon_{PF} = 2\sqrt{\frac{\gamma^2 + \omega_0^2}{1+q^2}}$$

The introduction of symmetry-breaking harmonics ($q>0$) opens up the stationary synchronized regime and enhances bistability (Manoranjani et al., 2023).

5. Robustness to General Coupling and Exact Equilibria Counting

The order-parameter bootstrapping and volumetric instability arguments generalize to all Winfree-type models of the form

$$\dot{\theta}_i = \omega_i + (\kappa/N) \sum_j I(\theta_j) S(\theta_i)$$

provided $I\geq 0$, $S$ changes sign once, and both admit well-controlled (power-law) scaling near their zeros. Under these conditions, oscillator death and convergence persist for $\kappa = O(\|\Omega\|_\infty)$. The proof further extends to derive a finite upper bound on the total number of equilibria of the system: for the standard sinusoidal Winfree model, the number of equilibria (modulo $2\pi$) does not exceed $2^{N+1}$. This result is established via a constructive polynomial equation for the order parameter, indexed by sign patterns, demonstrating the sharpness of the oscillator death threshold (Ryoo, 3 Jan 2026).

6. Comparison to Kuramoto Model and Averaging Regimes

In the limit of weak coupling and sharp frequency distributions, the sinusoidal Winfree model reduces to the Kuramoto–Sakaguchi model with a phase lag determined by the pulse and PRC properties: $$\dot{\theta}_i = \omega_i + K R \sin(\Psi - \theta_i + \alpha)$$ with $K$ and $\alpha$ determined by the first Fourier coefficient of the pulse and the PRC offset. The onset of synchrony is then given by

$$K_c = \frac{2\Delta}{\cos\alpha}, \qquad |\alpha| < \frac{\pi}{2}$$

Discrete discrepancies of $O(\Delta)\,$odd-in-PRC-offset arise beyond leading order, revealing how the Winfree model interpolates and extends classical Kuramoto dynamics, especially when higher harmonics or PRC heterogeneity are present (Pazó et al., 2018, Gallego et al., 2017).

7. Phase-Response Curve Heterogeneity and Synchronization Thresholds

Extending the sinusoidal Winfree model to heterogeneous phase-response curves and frequency distributions, the onset and stability of collective states depend sensitively on the degree and structure of heterogeneity. Analytical results show that above a critical PRC heterogeneity, the incoherent state is globally stable. In the case of Lorentzian distributions (for both frequencies and PRC parameters), the OA reduction applies and critical thresholds for synchrony can be computed exactly. The effect of PRC phase shift is manifested mainly as an effective phase lag in the Kuramoto mapping; higher-order corrections (and thus discrepancies between Winfree and Kuramoto thresholds) depend on the exact distribution of phase-response properties (Pazó et al., 2018).

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References:

(Ryoo, 3 Jan 2026, Manoranjani et al., 2023, Pazó et al., 2018, Gallego et al., 2017)