Each neuron is a phase oscillator. The Kuramoto coupling term synchronises their rhythms β the order parameter r measures global coherence.
Delta (0.5β4 Hz) dominates deep sleep; theta (4β8 Hz) links to memory; alpha (8β12 Hz) to relaxed states; beta (12β30 Hz) to active thought.
r = 0 means random phases (noise-like EEG); r = 1 means all oscillators locked in perfect sync β as seen during epileptic seizures.
This simulation models neural rhythms using the Kuramoto coupled-oscillator framework. Twelve phase oscillators represent neuronal populations; each oscillator has a natural angular frequency drawn from a Gaussian distribution centred on the selected brain-wave band. The coupling term pulls oscillators toward a common phase, while a tunable noise term mimics the stochastic nature of synaptic input. The result is an emergent collective rhythm that matches real EEG-like waveforms: when coupling is weak, phases wander and the mean field is noisy; when coupling exceeds a critical threshold, the population locks into a coherent oscillation visible as a clean wave in the mean-field trace.
Brain waves are not simply clock-like signals; they arise from millions of neurons competing and cooperating. The Kuramoto order parameter r (shown in the right panel) quantifies this competition: r close to zero indicates a desynchronised state as in normal waking alpha, while r near one indicates the extreme synchrony associated with epileptic seizures. By adjusting frequency, coupling, and noise you can explore how each brain-wave band transitions from irregular to synchronised activity, and observe the polar phase-space plot to see individual oscillators cluster together as coupling strengthens.
What are neural oscillations?
Neural oscillations are rhythmic fluctuations in electrical activity generated by large groups of neurons firing together. They are measured on the scalp as EEG signals and are classified into frequency bands β delta, theta, alpha, beta, and gamma β each associated with distinct cognitive states and brain functions.
What is the Kuramoto model?
The Kuramoto model is a mathematical framework describing a population of coupled phase oscillators. Each unit has its own natural frequency; a coupling term proportional to the sine of the phase difference pulls oscillators together. Above a critical coupling strength the population spontaneously synchronises, producing a sharp collective rhythm from disordered initial conditions.
What does the order parameter r represent?
The order parameter r is a number from 0 to 1 measuring global synchrony. r = 0 means all oscillators have random phases with no net coherence; r = 1 means every oscillator is perfectly phase-locked. Real EEG rhythms typically have r in the range 0.3β0.7, reflecting partial synchrony across cortical regions.
Delta waves (0.5β4 Hz) dominate during slow-wave sleep (stages N2 and N3). High-amplitude delta activity reflects widespread synchrony in thalamocortical circuits and is essential for memory consolidation and tissue repair. Theta waves (4β8 Hz) appear during light sleep and drowsiness and are strongly associated with hippocampal memory processing.
When visual input ceases, the primary visual cortex shifts from processing incoming signals to an idling or inhibitory state. This allows large thalamocortical loops to resonate at roughly 8β12 Hz, producing the characteristic posterior alpha rhythm. Opening the eyes suppresses it almost instantly β a phenomenon called alpha blocking.
Excessive coupling drives all oscillators into a single phase-locked state, mimicking pathological hypersynchrony. In the brain this corresponds to epileptic seizure activity, where normal desynchronised processing breaks down and large neuronal populations fire simultaneously, overriding normal computation.
Noise represents random synaptic fluctuations and background input variability. Moderate noise is beneficial β it prevents pathological locking and keeps the system at the edge of synchrony, where information processing is most efficient. Too much noise destroys coherence; too little noise combined with strong coupling drives the system into seizure-like hypersynchrony.
Beta oscillations (12β30 Hz) are linked to active thinking, motor preparation, and maintained cognitive states. They are prominent in sensorimotor cortex during sustained muscle contractions and are often associated with the maintenance of the current mental state. Reduction of beta (event-related desynchronisation) signals the start of movement or cognitive engagement.
The Kuramoto model captures the essential phase-coupling dynamics of neural populations and reproduces qualitative EEG features such as synchrony transitions and frequency-band structure. However, it simplifies away detailed ionic mechanisms, spatial organisation, and multi-scale coupling. Real neural circuits involve excitatory and inhibitory populations, axonal delays, and non-linear membrane dynamics not modelled here.
Start with the coupling slider at zero β you will see noisy, incoherent waveforms and r near 0. Slowly increase coupling while watching the order-parameter gauge and the mean-field trace. Around K = 0.3β0.5 (depending on noise) the system undergoes a phase transition: the mean field sharpens dramatically and r jumps upward. This is the Kuramoto synchronisation transition.