Von Mises Phase Densities in Coupled Oscillator–Neuron Systems: A Zero-Mode Theorem, Respiratory Modulation, and Theta-Driven Amplification of Phase Locking

Neural oscillators embedded in rhythmic environments exhibit structured phase distributions that govern information encoding. Here we derive a closed-form expression for the phase density function σ(ϕ) of periodically driven spiking neurons, showing that σ converges to a von Mises distribution whose concentration parameter κ is determined by the ratio of oscillatory drive strength to noise level. We prove a zero-mode theorem establishing that phase density perturbations decouple from oscillation amplitude dynamics to leading order, enabling independent analysis of phase and amplitude degrees of freedom. A quartic approximation relating κ to the phase–amplitude coupling (PAC) coefficient is derived with <2.5% error for |PAC| < 0.6. Fokker–Planck analysis of the spike-generation process yields the von Mises form as the unique steady-state solution, with a relaxation timescale of ∼ 50–65 ms for cortical parameters—much faster than the respiratory cycle (∼ 3–5 s), validating the adiabatic approximation. A theta-driven amplification law is derived showing that hippocampal theta-band synaptic drive amplifies the single-neuron κ by a factor of ∼ 5, quantitatively closing the gap between simulated single-neuron κ (∼ 0.18) and empirically observed cortical κ (∼ 0.9). Fisher information analysis of the von Mises family yields a natural distance metric on phase density space, enabling classification of neural states (wakefulness, sleep stages, anesthetic states) with resolution exceeding conventional scalar indices. Cross-species analysis predicts a monotonic relationship between respiratory coupling strength and κ across Human, Drosophila, C. elegans, and Octopus.