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Critical Dynamics and Cyclic Memory Retrieval in Non-reciprocal Hopfield Networks.

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This study explores Hopfield networks with non-reciprocal coupling, revealing dynamical phase transitions between memory retrieval, point-attractors, and limit-cycle attractors near critical thresholds.

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Area of Science:

  • Computational neuroscience
  • Complex systems dynamics
  • Statistical physics

Background:

  • Hopfield networks are foundational models for associative memory.
  • Non-reciprocal coupling introduces novel dynamics not present in traditional models.
  • Understanding phase transitions is crucial for complex system behavior.

Purpose of the Study:

  • To investigate the impact of non-reciprocal coupling on Hopfield network dynamics.
  • To identify and characterize different phases of memory retrieval.
  • To analyze the critical behavior at the boundaries of the limit-cycle phase.

Main Methods:

  • Analytical derivation of network dynamics.
  • Numerical verification using a Master Equation approach.
  • Analysis of critical exponents and sensitivity to perturbations.

Main Results:

  • Identified three distinct dynamical phases: no memory, point-attractor memory, and limit-cycle memory.
  • Characterized the limit-cycle phase boundaries by Hopf and fold bifurcation lines.
  • Confirmed analytical predictions of critical behavior through numerical simulations.

Conclusions:

  • Non-reciprocal coupling in Hopfield networks leads to rich dynamical transitions.
  • The identified critical regions offer insights into systems near cyclic instability.
  • These findings have potential applications in modeling biological processes exhibiting multi-step transitions.