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Governing EquationsđŸĒļ

🍨 Kuramoto ModelđŸĒļ

The time-evolution of the phase \(\theta_i\) of the \(i^\text{th}\) oscillator in the ensemble is governed by the Kuramoto model (vanilla version with noise term):

\[ \dot{\theta}_i = \omega_i + \frac{K}{N}\sum_{j=1}^{N}{A_{i,j} \, \sin(\theta_j - \theta_i)} + \xi_i(t) \, .\]
Variable ✨ Interpretation 📝 Notes
\(\omega_i\) intrinsic frequency of the \(i^\text{th}\) oscillator in the network Typically drawn from a Lorentzian or Gaussian distribution \(g(\omega)\)
\(K\) global coupling strength
\(A_{i,j}\) adjacency matrix topology
\(\xi_i(t)\) stochastic (noise) term Often modeled as additive Gaussian white noise (AGWN): $$ \big\langle \xi_i(t) \, \xi_j(t')\big\rangle = 2D\delta_{ij}\delta(t-t') \, .$$

The Order ParameterđŸĒļ

Concept OverviewđŸĒļ

The order parameter \(r(t)\) is the primary metric for quantifying synchronization in the Kuramoto model. It acts as a macroscopic observable that emerges from the collective microscopic interactions among the oscillators.

In the context of our benchmark suite, \(r(t)\) quantifies the degree of synchronization among the oscillator population at any given moment.

Mathematical DefinitionđŸĒļ

The order parameter is defined via the complex mean phasor:

\[ z(t) = r(t) e^{i\psi(t)} = \frac{1}{N} \sum_{j=1}^N e^{i \theta_j(t)} \]

Where:

  • \(N\) is the total number of oscillators in the system.
  • \(\theta_j(t)\) is the phase of the \(j^\text{th}\) oscillator.
  • \(r(t) = |z(t)|\) is the magnitude of the complex mean phasor and represents the degree of synchronization at time \(t\).
  • \(\psi(t) = \arg{z(t)}\) is the average phase (center-of-mass of the phases).

Need to double-check with sources.

Python ImplementationđŸĒļ

In the source code for the Kuramoto benchmark, \(|r(t)|\) is computed as:

\[ r(t) = \left| \frac{1}{N} \sum_{j=1}^N e^{i \theta_j(t)} \right| \]

Need to double-check with sources.

✅ To Do

  • Include code snippet of implementation in src/kuramoto/order_parameter.py.

InterpretationđŸĒļ

Limits of the order parameterđŸĒļ

Value Interpretation
\(r \approx 1\) Complete synchronization
\(r \approx 0\) Incoherence
\(0 < r < 1\) Partial synchronization

Connection to NeuroscienceđŸĒļ

💡 Big Idea

The emergence of \(r(t)\) from \(N\) independent units corresponds with the binding problem in consciousness studies:

  • How do discrete neural oscillators (micro) give rise to a unified conscious experience (macro)?
  • The transition from \(r\approx 0\) to \(r\approx 1\) represents a phase transition, analogous to symmetry breaking in physics.
  • In the conventional Kuramoto model, this transition occurs at a critical coupling \(K_c\) where \(r(t) < K_c\) corresponds with a disordered system and \(r(t) > K_c\) with spontaneously emerging synchronization (ordered system).

Thus, the order parameter is not only a statistical tool, but a potential bridge between statistical mechanics and theories of integrated information.

Personal Reflection

This is a personal reflection. The connection to neuroscience and consciousness is speculative and ill-defined. It is an area of ongoing research and debate.

ReferencesđŸĒļ

  1. Strogatz, S. H. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143(1–4), 1–20. https://doi.org/10.1016/s0167-2789(00)00094-4
  2. Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: A universal concept in nonlinear sciences. Cambridge University Press.