On a Neural Phonon Model of EEG Brain Dynamics

Abstract

Neuronal oscillations are a ubiquitous feature of brain activity, indexing functions from sensory selection to memory formation. Yet a unified framework that (i) accommodates the nonlinear, noise-driven nature of cortical dynamics and (ii) explains standard empirical measures—power, spectral entropy, coherence, Phase-Locking Value (PLV), Phase-Amplitude Coupling (PAC), and envelope correlations—remains elusive. A natural candidate is the noisy Stuart–Landau (SL) oscillator, whose deterministic form models cortical rhythms as limit cycles, while additive noise induces stochastic phase and amplitude fluctuations. Prior work has shown that networks of SL oscillators can replicate burst statistics, multistability, and cross-frequency modulation in electroencephalography/magnetoencephalography (EEG/MEG). However, an analytical framework linking these models directly to observed connectivity metrics has been lacking. Here we derive such a framework by mapping the Fokker–Planck equation (FPE) of each SL oscillator to an imaginary-time Schrödinger operator via a classical similarity transform. A second-order expansion around the limit-cycle amplitude yields a quadratic Hamiltonian whose ladder operators describe quantised fluctuations—neural phonons—in oscillatory power. Bilinear coupling terms inherited from diffusion give rise to analytically diagonalisable bosonic interactions. This construction yields closed-form expressions for spectral observables and their dynamics, including Green-function-derived coherence and PLV, perturbative PAC, and a five-parameter “personality map” linking microscopic physics to macroscopic brain states. By unifying noisy limit-cycle theory with operator methods from statistical physics, we introduce a tractable, interpretable formalism for understanding neural coherence as the dynamics of quantised phonons.