🧠 Neural Oscillators — Gamma, Theta & Brain Rhythms
Kuramoto oscillators synchronise into brain-like rhythms — watch order parameter and phase transitions
Presets
Kuramoto Model
Live Stats
Physics
r = |1/N·Σe^(iθⱼ)|
K_c = 2/(π·g(0))
Sync when K > K_c
γ: 30-80Hz E-I balance
About Neural Oscillators
The Kuramoto Model
The Kuramoto model describes N coupled oscillators each with a natural frequency ωᵢ drawn from a distribution g(ω). Coupling K drives phases together: dθᵢ/dt = ωᵢ + K/N·Σsin(θⱼ−θᵢ). The order parameter r = |1/N·Σe^(iθⱼ)| measures synchrony: r≈0 is incoherent, r≈1 is fully synchronised. A phase transition occurs at the critical coupling K_c = 2/(πg(0)) — analogous to a magnetic phase transition.
Brain Rhythms
The brain generates characteristic oscillatory patterns measurable by EEG. Delta (1-4Hz) dominates deep sleep; theta (4-8Hz) appears during memory encoding and REM sleep; alpha (8-13Hz) reflects relaxed wakefulness; beta (13-30Hz) accompanies active thinking; gamma (30-80Hz) correlates with perception and consciousness. These rhythms emerge from synchronised neuronal populations — much like the Kuramoto model.
Gamma and E-I Balance
Gamma oscillations (30-80Hz) arise from a balance between excitatory (AMPA, NMDA) and inhibitory (GABA) synaptic currents. Fast-spiking interneurons provide precisely timed inhibition that paces excitatory cells into synchrony. This Excitatory-Inhibitory (E-I) balance is critical: too much excitation causes epileptic seizures; too much inhibition suppresses all activity. Disruption of gamma is implicated in schizophrenia and Alzheimer's disease.