Heterogeneous noise-induced extreme events and synchronization in a globally coupled network of FitzHugh-Nagumo oscillators

TL;DR

This work demonstrates that heterogeneity in stochastic inputs can induce and synchronize extreme events in a globally coupled network of FitzHugh-Nagumo oscillators. By defining a threshold-based EE metric $H_T$ and a spatial-extents measure $Z$, and by analyzing phase coherence with a Hilbert-transform-derived phase and the Kuramoto order parameter $r$, the authors classify five dynamical regimes (QS, LEE, PCEE, GCEE, NES) and reveal noise- and coupling-dependent transitions between them. A minimal two-oscillator reduction explains how local noise variance drives EE onset and global synchronization, showing that heterogeneous noise alone can organize EE across the network. These findings advance understanding of noise-driven collective dynamics in excitable systems and have potential implications for neuronal dynamics and other complex networks where stochastic forcing and heterogeneity interact. The study also points to future directions in multilayer and time-delayed networks to deepen insights into sporadic, noise-driven extreme phenomena.

Abstract

This study investigates the dynamics of a globally coupled network of heterogeneous FitzHugh Nagumo (FHN) oscillators under stochastic influences, with particular emphasis on the emergence of extreme events (EE). While previous studies explored FHN networks subjected to homogeneous noise, revealing behaviors such as noise-induced synchronization, stochastic resonance, and coherence resonance, the impact of noise heterogeneity remains poorly understood. Moreover, the emergence of EE under heterogeneous stochastic excitation has largely been overlooked. To address these gaps, we capture the natural variability in neuronal responses to external stimuli by introducing nonidentical noise sources, thereby reflecting diversity across the network. Our results reveal that EE can arise both globally, where large excursions occur collectively across the entire network, and partially, where only a subset of oscillators exhibits extreme activity depending on the interplay between noise intensity and coupling strength. We further identify three distinct classes of EE that enrich the system's dynamical repertoire and propose a quantitative metric capable of distinguishing between global and local occurrences. Remarkably, we demonstrate that even under heterogeneous noise inputs, noise can synchronize EE across the network, underscoring the robustness of collective dynamics in stochastic regimes. Furthermore, causal interaction analysis between oscillator pairs provides mechanistic insights into the initiation and propagation of EE. To the best of our knowledge, this constitutes the first demonstration of both partially and globally synchronized EE triggered solely by noise in a network of coupled oscillators. These findings enhance our understanding of noise-driven collective behavior in complex systems and provide new insights into neuronal dynamics under random influences.

Figures (9)

• Figure 1: Sensitivity analysis of EE with respect to noise intensity $D$. For each value of $D$, the coupling strength $\kappa$ is varied within the range $(0, 0.1)$ using 50 uniform divisions, and the corresponding averaged value is used as the sensitivity measure. Each data point represents the mean of 50 independent realizations, with error bars indicating the standard deviation across simulations.
• Figure 2: Time series and probability density function (PDF) profiles of the global average ($\bar{x}$) for a globally coupled network of FHN oscillators subjected to non-identical noise sources, shown for various noise intensities: (a1–a2) $D$ = 0.02, (b1–b2) $D$ = 0.03, and (c1–c2) $D$ = 0.1. The coupling strength is fixed at $\kappa$ = 0.1. In each subplot, the red dashed horizontal and vertical lines denote the threshold level $H_T$ used to identify EE, as determined from the corresponding time series data.
• Figure 3: Extreme event measure $Z$ for (a) varying coupling strengths $\kappa$ as a function of noise intensity $D$, and (b) varying $\kappa$ for different fixed values of $D$. Each data point represents the mean of 50 independent realizations, with error bars showing the standard deviation across trials. The plots illustrate how the interplay between noise intensity and coupling strength influences the emergence of EE in the network (\ref{['eq:e2']}).
• Figure 4: Two-parameter phase diagram illustrating the emergence of five distinct dynamical states in the $(\kappa, D)$ parameter space, where $\kappa \in (0, 0.1)$ is the coupling strength and $D \in (0, 0.1)$ is the noise intensity. The diagram delineates regions corresponding to: quiescent/steady state (QS, black), partially coherent extreme events (PCEE, red), globally coherent extreme events (GCEE, green), localized extreme events (LEE, white), and non-extreme spiking (NES, grey). The boundaries highlight how the interplay between noise and coupling determines whether EE occurs locally, globally, or is absent altogether.
• Figure 5: Distinct dynamical states of the oscillator network for different combinations of coupling strength ($\kappa$) and noise intensity ($D$). Each row shows (left) the time series of the global average variable $\bar{x}$, where the dashed red line denotes the EE threshold $H_T$, and (right) the corresponding spatiotemporal plot indicating oscillators whose amplitudes exceed $H_T$. The panels illustrate: (a) Quiescent state (QS): subthreshold oscillations or near-steady behavior ($\kappa = 0.05$, $D = 0.008$); (b) Partially coherent EE (PCEE): intermittent EE visible both at the individual and global levels ($\kappa = 0.02$, $D = 0.03$); (c) Globally coherent EE (GCEE): large, synchronized EE across all oscillators ($\kappa = 0.1$, $D = 0.03$); (d) Localized extreme events (LEE): EE confined to subsets of oscillators without global excursions ($\kappa = 0.02$, $D = 0.08$); and (e) Non-extreme spiking (NES): frequent large-amplitude oscillations that fail to exceed $H_T$ ($\kappa = 0.05$, $D = 0.08$). Panels (a2) and (e2) appear blank because they correspond to the QS and NES regimes, respectively, where no oscillator surpasses the predefined extreme-event threshold $H_T$; hence, no activity is displayed in the spatiotemporal plots.