The Slowest Point

Pathological neural synchronization drives the symptoms of Parkinson’s disease and epilepsy. Populations of neurons that should fire independently instead lock into collective oscillation. The synchronization is robust — it resists perturbation and rebuilds after disruption. Breaking it requires not just a push but the right push in the right direction.

The authors identify where to push. The target is the centroid of the limit cycle — the geometric center of the closed orbit that each neural oscillator traces in phase space. A brief electrical pulse aimed at the centroid does not stop the oscillation. Each neuron continues to oscillate. But the collective synchronization is destroyed because the pulse scatters the neurons’ phases across the cycle.

The centroid works because it lies in the region of minimal return times. Trajectories that pass near the centroid take the longest to return to the limit cycle. A pulse that sends the system toward the centroid maximizes the time the neuron spends off its usual rhythm, which maximizes the phase disruption across a population of coupled oscillators.

The method does not require knowledge of the coupling between neurons. It does not require a detailed model of the neural dynamics. It requires only the limit cycle itself — the shape of the oscillation in phase space — from which the centroid can be calculated directly. The geometric property is robust under changes in coupling strength.

The inverse problem — determining exactly when and how hard to pulse — is converted into a geometric problem: aim for the slowest point. The control is in the geometry, not in the timing.

To desynchronize a population of oscillators, push each one toward the place where it moves most slowly. The slowest point is the most disruptive destination.