Theory Note: Emergence of a Proportional-Derivative Control Law from Two Coupled Oscillating Brain Circuits Near Synchrony

Oscillations and oscillatory synchronization are pervasive in motor circuits, where their role in rhythm generation and entrainment is well established but their role in feedback control of movement remains unclear. Here I show analytically that two oscillators of any type, coupled through a delayed interaction that is an odd function of their phase difference, necessarily implement a proportional-derivative (PD) control law in the near-synchrony limit. The proportional gain follows from the slope of the coupling function and the derivative gain is set by the coupling delay, so that PD control emerges with no additional machinery. Simulations confirm that such oscillators reproduce ideal PD step responses near synchrony and that control quality degrades systematically away from it. This establishes a direct, model-independent bridge between oscillatory synchronization and feedback control, and suggests concrete experimental signatures for candidate systems.