tACS Entrainment of Cortical Alpha Oscillations

Transcranial Stimulation

Stimulation frequency f_s 10.0 Hz

Stimulation amplitude K_s 0.60

Cortical coupling K_c 0.40

Frequency dispersion σ 0.8 Hz

Population State

Phase-Locking Value R0.00

Mean phase ψ

0°

Detuning Δf

0.0 Hz

Entrainment

—

Mean intrinsic f

10.0 Hz

How it works

Each of N cortical neural-mass oscillators has its own natural alpha-band frequency ω_i, drawn from a Lorentzian spread of width σ around a mean f₀ = 10 Hz. They interact through local (Kuramoto) coupling K_c and are all driven by the same external tACS electric field oscillating at f_s with strength K_s:

dθᵢ/dt = ωᵢ + (Kc/N) Σⱼ sin(θⱼ − θᵢ)  +  Ks · sin(θs(t) − θᵢ)
θs(t) = 2π fs t          (external stimulation phase)
R e^(iψ) = (1/N) Σᵢ e^(iθᵢ)      (Kuramoto order parameter)

R (0–1) is the phase-locking value: R ≈ 0 means the population fires with random relative phase, R → 1 means the whole cortical patch has synchronized. Two distinct mechanisms raise R here:

Endogenous synchrony — cortical coupling K_c alone can synchronize the population once it exceeds a critical value (the classic Kuramoto transition), independent of any stimulation.
Entrainment — the external field pulls the whole ensemble to lock onto f_s itself (θᵢ tracks θs). This only succeeds within an "Arnold tongue": entrainment requires K_s large enough relative to |f_s − f₀| and to the dispersion σ — a stronger, more detuned stimulus needs more current to entrain the same tissue.

This is the accepted mechanistic model for non-invasive tACS neuromodulation used in sleep, memory and depression research: a weak (1–2 mA) sinusoidal scalp current cannot directly drive neurons to fire, but it can nudge the phase of an already-oscillating population, and if the drive is close enough to the tissue's natural rhythm and coupling is high enough, the whole patch locks to the stimulation frequency.

Cortical oscillator External tACS field Population mean phase

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