Frustration-Induced Collective Dynamical States in Pulse-Coupled Adaptive Winfree Networks

TL;DR

This study explores how frustration and adaptive coupling in pulse-coupled Winfree networks lead to diverse collective states, including novel spontaneous entrainment and bump states, characterized through new measures and analytical stability analysis.

Contribution

It introduces the first observation of spontaneous entrainment and bump states in adaptive networks without external forcing, and develops measures and analytical tools for characterizing these dynamics.

Findings

Discovery of spontaneous entrainment, bump, and bump--frequency cluster states without external forcing.

Development of three measures for characterizing incoherence in network dynamics.

Analytical derivation of stability conditions matching numerical simulations.

Abstract

We investigate collective dynamics in a pulse-coupled adaptive Winfree network under the influence of a frustration (phase-lag) parameter. The coupling strengths coevolve according to a Hebbian adaptation rule and self-organize to support a wide variety of collective states. We observe frequency-clustered states, entrainment, bump states, bump--frequency cluster states, antipodal and multi-antipodal cluster states, chimera states, and incoherent dynamics. Notably, we report for the first time the spontaneous emergence of entrainment, bump, and bump--frequency cluster states in an adaptive network {\it without} any external forcing. To systematically characterize these regimes, we introduce three complementary measures of incoherence based on (i) time-averaged frequencies, (ii) instantaneous phases, and (iii) mean frequencies per bin. These measures enable the construction of one- and two-parameter phase diagrams that clearly delineate transitions between distinct dynamical states. Furthermore, we analytically derive the stability condition for the frequency-entrained state, which shows excellent agreement with numerical simulations. Our results highlight the crucial role of frustration-mediated plasticity in shaping rich self-organized dynamics in pulse-coupled adaptive networks.