The brain’s rhythms

Electrodes on the scalp pick up rhythmic fluctuations produced by large numbers of neurons acting together. Since Hans Berger recorded the alpha rhythm in the 1920s, these rhythms have been sorted into bands, roughly:

The boundaries differ between laboratories, and the “associations” are tendencies, not one-to-one mappings. A useful way to think about these rhythms is that oscillations let groups of neurons coordinate their timing: neurons that fire in phase can influence each other more effectively than neurons that fire at random (Buzsáki and Draguhn 2004).

Coupled oscillators

How do many neurons, each with its own natural rhythm, come to fire together? The same question arises for fireflies flashing in unison, pacemaker cells in the heart and the footfall of people crossing a bridge. In 1975 the physicist Yoshiki Kuramoto proposed a model simple enough to solve and rich enough to show the essential phenomenon (Kuramoto 1984). Each oscillator $i$ has a phase $\theta_i$ and a natural frequency $\omega_i$, and is pulled towards the others:

$$\dot{\theta}_i = \omega_i + \frac{K}{N}\sum_{j=1}^{N}\sin(\theta_j – \theta_i)$$

where $K$ is the coupling strength. The degree of synchrony is measured by the order parameter $r$, the length of the average of the unit vectors $e^{i\theta_j}$: $r = 0$ when the phases are scattered, $r = 1$ when all are locked together. With the definitions $r e^{i\psi} = \frac{1}{N}\sum_j e^{i\theta_j}$, the equation can be rewritten as $\dot{\theta}_i = \omega_i + K r \sin(\psi – \theta_i)$: each oscillator feels only the mean field, whose strength itself depends on how synchronised the population is. That feedback is what produces a sudden transition.

For natural frequencies spread according to a Lorentzian distribution of half-width $\gamma$, Kuramoto showed that synchrony is impossible below a critical coupling $K_c = 2\gamma$ and grows as $r = \sqrt{1 – K_c/K}$ above it (Strogatz 2000).

import math, random

def kuramoto_r(K, N=400, gamma=0.5, dt=0.05, steps=1600, seed=1):
    """Order parameter r after the transient, for N oscillators whose natural
    frequencies follow a Lorentzian of half-width gamma (rad/s), measured in
    the rotating frame. Mean-field form of the Kuramoto equation:
    dθi/dt = ωi + K·r·sin(ψ − θi)."""
    omega = [gamma * math.tan(math.pi * ((i + 0.5) / N - 0.5)) for i in range(N)]
    rnd = random.Random(seed)
    th = [rnd.uniform(0, 2 * math.pi) for _ in range(N)]
    rs = []
    for step in range(steps):
        c = sum(math.cos(t) for t in th) / N
        s = sum(math.sin(t) for t in th) / N
        r, psi = math.hypot(c, s), math.atan2(s, c)
        th = [t + dt * (w + K * r * math.sin(psi - t)) for t, w in zip(th, omega)]
        if step >= steps // 2:
            rs.append(r)
    return sum(rs) / len(rs)

gamma = 0.5
print(f"critical coupling Kc = 2γ = {2 * gamma:.1f}")
print("  K    r (simulated)   r (theory, √(1 − Kc/K))")
for K in (0.5, 0.9, 1.1, 1.5, 2.0, 3.0):
    theory = math.sqrt(1 - 2 * gamma / K) if K > 2 * gamma else 0.0
    print(f"{K:4.1f}   {kuramoto_r(K):13.2f}   {theory:10.2f}")
# critical coupling Kc = 2γ = 1.0
#   K    r (simulated)   r (theory, √(1 − Kc/K))
#  0.5            0.08         0.00
#  0.9            0.09         0.00
#  1.1            0.29         0.30
#  1.5            0.57         0.58
#  2.0            0.70         0.71
#  3.0            0.82         0.82

Below the threshold the simulated $r$ is not exactly zero but about 0.08, roughly what one expects from the random scatter of 400 phases ($1/\sqrt{N} = 0.05$). Above it the simulation follows the theory closely. The transition is a phase transition in the physicist’s sense: a small change in a control parameter produces a qualitative change in the collective state.

From the model to the brain

The Kuramoto model has been used as a starting point for models of cortical rhythms and of large-scale brain dynamics, with oscillators standing for brain regions and coupling given by anatomical connections and delays (Breakspear et al. 2010). Such models have also been used to explore an idea that is still debated: that the healthy brain operates near a transition between order and disorder, where it can switch flexibly between more and less synchronised states.