Neural Oscillations: The Kuramoto Model of Brain Rhythms

Brain rhythms are populations, not single neurons

An EEG trace doesn't record one neuron — it records the synchronised summed activity of millions of them. This simulation models that synchronisation with the Kuramoto model, the standard mathematical tool for coupled oscillators: each unit has its own natural frequency, and units nudge each other's phase through a coupling term. The headline result, first shown by Yoshiki Kuramoto in 1975, is that above a critical coupling strength a population of independent oscillators with a spread of natural frequencies spontaneously locks into a shared rhythm — exactly the kind of large-scale synchrony an EEG electrode picks up as a clean brain wave.

dθ_i/dt = ω_i + (K/N) · Σ_j sin(θ_j - θ_i)
// θ_i: phase of oscillator i    ω_i: its natural frequency
// K:   coupling strength        N:  number of oscillators
// order parameter r·e^{iψ} = (1/N)Σ e^{iθ_j} measures synchrony (0..1)

The classic frequency bands

By tuning each population's mean natural frequency and coupling strength you reproduce the five canonical EEG bands neuroscientists name by frequency: delta (0.5–4 Hz, deep sleep and slow-wave states), theta (4–8 Hz, drowsiness and memory encoding), alpha (8–13 Hz, relaxed wakefulness with eyes closed, strongest over occipital cortex), beta (13–30 Hz, active concentration and motor planning) and gamma (30–100+ Hz, associated with feature binding and focused attention). None of these bands is a single pure sine wave in a real brain — they are the dominant peaks in a noisy power spectrum, which is why this simulation's output looks more like an EEG when you sum several coupled populations rather than plotting one clean oscillator.

Synchrony as an order parameter

Kuramoto's trick for measuring "how synchronised" a population is comes from treating each oscillator's phase as a point on the unit circle and averaging the resulting vectors. The magnitude of that average, the order parameter r, runs from 0 (phases scattered randomly, no coherent signal) to 1 (every oscillator in lockstep, a strong coherent wave). Below the critical coupling K_c the population sits near r ≈ 0 — incoherent, individually noisy firing that doesn't add up to anything an electrode could detect. Cross K_c and a growing fraction of oscillators phase-lock together, r rises, and a macroscopic rhythm emerges from microscopic disorder with no central conductor — synchrony is an emergent, self-organised property of the coupling, not something imposed from outside.

Why this matters clinically

Real brain rhythms are not just curiosities — their balance is diagnostic. Excess low-frequency synchrony (delta/theta dominating where alpha or beta should) shows up in encephalopathy and some dementias; pathologically strong beta-band synchrony in the basal ganglia is a hallmark of Parkinson's disease and the target of deep-brain-stimulation therapy, which works in part by disrupting that synchrony; and hypersynchronous firing across a large cortical population is more or less the definition of an epileptic seizure. The Kuramoto framework, despite being a drastic simplification of real neural biophysics, captures the essential mechanism common to all of these: synchrony is a knob, coupling strength moves it, and both too little and too much synchrony are pathological.

Frequently asked questions

Try it live

Everything above runs in your browser — open Neural Oscillations and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.