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Complex dynamics in adaptive phase oscillator networks.

Benjamin Jüttner1, Erik A Martens2,3

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Adaptive coupling in oscillator networks enhances synchronization. Even minimal models with adaptive learning rules exhibit complex dynamics, including multi-stability and chaos, improving collective behavior.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Computational Neuroscience

Background:

  • Coupled dynamical units, like neurons, exhibit collective behaviors such as synchronization.
  • Adaptive coupling, where connection strengths change with activity, is crucial in systems like neural plasticity.
  • The classical Kuramoto model assumes fixed coupling, limiting the study of adaptive network dynamics.

Purpose of the Study:

  • To investigate the impact of adaptive learning rules on the collective dynamics of coupled oscillators.
  • To analyze a minimal Kuramoto model with tunable adaptivity parameters, mimicking spike-time-dependent plasticity.
  • To systematically explore how varying the strength of adaptivity influences network behavior.

Main Methods:

  • Bifurcation analysis of a minimal Kuramoto model with N=2 phase oscillators.
  • Introduction of a general adaptive learning rule with three parameters: strength, offset, and shift.
  • Numerical simulations for a larger system of N=50 oscillators to compare dynamics.

Main Results:

  • Non-adaptive Kuramoto models show simple dynamics (drift, frequency-locking).
  • Exceeding a critical adaptivity threshold reveals complex bifurcation structures.
  • Symmetric adaptation leads to multi-stability; asymmetric adaptation results in period-doubling cascades to chaos and mixed oscillations (librations/rotations).
  • Adaptation generally improves oscillator synchronizability.

Conclusions:

  • Adaptive learning rules introduce rich, complex dynamics not present in classical models.
  • The strength of adaptivity is a critical parameter controlling system behavior.
  • Adaptation enhances the synchronizability of coupled oscillator networks, with implications for neural systems.