🧠 Neural Oscillators

It is a mathematical model of N coupled phase oscillators, each with its own natural frequency ωᵢ drawn from a distribution. Every oscillator's phase θᵢ evolves according to dθᵢ/dt = ωᵢ + K/N·Σⱼsin(θⱼ−θᵢ), so each oscillator is pulled toward the average phase of all the others with strength proportional to the coupling constant K. It is one of the simplest solvable models of spontaneous synchronisation and is used across physics, biology and neuroscience.

r = |1/N·Σe^(iθⱼ)| treats each oscillator's phase as a unit vector on the complex plane and averages them. If phases are scattered randomly the vectors cancel out and r stays near 0; if oscillators lock into the same phase the vectors add up and r approaches 1. The simulation draws this resultant vector directly on the phase-diagram ring, so you can watch synchrony build in real time as K increases.

The Kuramoto model has an analytically derived critical coupling K_c = 2/(π·g(0)), where g(0) is the value of the natural-frequency distribution at its centre. Below K_c the spread in natural frequencies σ dominates and oscillators drift apart, keeping r near 0. Above K_c a growing fraction of oscillators lock to a common frequency, and r rises sharply — a phase transition mathematically similar to magnetisation appearing in a ferromagnet as temperature drops.

Each preset sets a mean frequency f0 and a coupling/spread combination tuned to sit in a classic EEG band: Theta targets 6Hz (memory encoding, REM sleep), Alpha targets 10Hz (relaxed wakefulness), Gamma targets 40Hz (perception, high coupling K=8 for tight synchrony), and Deep Sleep targets 2Hz with strong coupling and low spread, mimicking the highly synchronised slow-wave activity of deep non-REM sleep. The Desynchronised preset uses low K and high σ so the population never locks, illustrating an incoherent baseline state.

The simulation keeps a rolling history of the population's mean field (the real part of the order-parameter vector) and runs a discrete Fourier transform over the most recent 512 samples. The frequency bin with the largest magnitude becomes the "dominant frequency" readout, which is then classified into a band — δ below 4Hz, θ below 8Hz, α below 13Hz, β below 30Hz, or γ above that — mirroring how neuroscientists classify EEG power spectra into canonical rhythm bands.