A Dynamical Systems and System Identification Framework for Phase Amplitude Coupling Analysis

Abstract

Phase-amplitude coupling (PAC), a form of cross-frequency coupling, is involved in diverse cognitive functions and neural communication, making its accurate detection and characterisation essential yet challenging. Most existing methods infer PAC from variations in instantaneous phase and amplitude profiles, but are limited by sensitivity to filter bandwidths, inconsistent performance across noise levels and data lengths, and vulnerability to spurious couplings. Here, we formulate PAC as a nonlinear dynamical-systems phenomenon characterised by quadratic phase coupling (QPC), and propose a nonlinear system identification framework that directly models the dynamics generating PAC. Rather than relying on filtered phase and amplitude fluctuations, the proposed method identifies a generative nonlinear model, enabling noise-free simulation of the estimated dynamics and model-based characterisation of coupling strength and preferred phase. It also provides dynamically grounded criteria for identifying harmonic- and intermodulation-related false detections, remains robust under high noise, and yields consistent characterisation despite changes in slow-frequency power. In simulations and rat hippocampal local field potentials (LFP) recordings, the proposed method produced more sharply localised and frequency-specific PAC estimates than benchmark filtering-based methods, while preserving detailed preferred-phase information. In simulations, it rejected harmonic-related spurious coupling, remained robust at a signal-to-noise ratio (SNR) of 2 and reasonably robust at an SNR of 1, and reliably characterised PAC using 5-second analysis windows. These results establish nonlinear system identification as a complementary dynamical framework for detecting and characterising PAC, with particular advantages for noisy and short-duration neural recordings susceptible to spurious coupling.

39 pages, 26 figures