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

R. Anand, V. K. Chandrasekar, R. Suresh

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.

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

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.
Paper Structure (14 sections, 30 equations, 18 figures, 1 table)

This paper contains 14 sections, 30 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Pulse shape
  4. Phase response curve (PRC)
  5. Methods and measures
  6. Frequency-based strength of incoherence
  7. Phase-based strength of incoherence
  8. Mean-frequency-based strength of incoherence
  9. Results and Discussion
  10. Negative PRC offset $(q<0)$
  11. Balanced PRC $(q=0)$
  12. Positive PRC offset $(q>0)$
  13. Analytical Approach
  14. Conclusion

Figures (18)

  • Figure 1: (a) The influence function $P(\theta)$ for $n=1$. (b) Phase response curves $Q(\theta)$ for different values of $q = 1$, $0$, and $-1$.
  • Figure 2: Frequency cluster (FC) state for $\alpha = 0$ and $q = -1$: (a) snapshot of instantaneous phases, (b) space--time evolution of the oscillators, (c) time-averaged frequency distribution, and (d) coupling matrix $k_{ij}$. Other parameters are fixed at $\omega = 1$, $\sigma = 1$, and $\epsilon = 0.01$..
  • Figure 3: Entrainment (ENT) state for $\alpha = 0.15\pi$ and $q = -1$: (a) snapshot of instantaneous phases, (b) space--time evolution of the oscillators, (c) time-averaged frequency distribution, and (d) coupling matrix $k_{ij}$. All other parameters are the same as in Fig. \ref{['FC1']}.
  • Figure 4: Bump–frequency cluster (BFC) state for $\alpha = 0.3\pi$ and $q = -1$: (a) snapshot of instantaneous phases, (b) space--time evolution of the oscillators, (c) time-averaged frequency distribution, and (d) coupling matrix $k_{ij}$. All other parameters are the same as in Fig. \ref{['FC1']}.
  • Figure 5: Bump state (BS) for $\alpha = 0.5\pi$ and $q = -1$: (a) snapshot of instantaneous phases, (b) space--time evolution of the oscillators, (c) time-averaged frequency distribution, and (d) coupling matrix $k_{ij}$. All other parameters are the same as in Fig. \ref{['FC1']}.
  • ...and 13 more figures