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

TL;DR

A novel method for PAC detection and characterisation based on nonlinear system identification that accounts for harmonic-induced spurious couplings through empirically derived criteria and remains robust to high noise levels and variations in slow-frequency power is proposed, offering an accurate and interpretable framework for PAC analysis.

Abstract

Phase-amplitude coupling (PAC), a form of cross-frequency interaction, has been implicated in various cognitive functions and, by extension, in neural communication and information integration. Accurately detecting and characterising PAC is essential for understanding its role in processes such as memory and attention. However, this remains a significant challenge. Most existing methods rely on variations in the temporal profile to detect PAC, but they often suffer from key limitations, most notably, their sensitivity to filter bandwidth selection and their susceptibility to detecting spurious couplings. Previous studies have suggested that approaches grounded in the actual generative dynamics of PAC may offer improved accuracy. In this study, we adopt a dynamical systems perspective and propose a novel method for PAC detection and characterisation based on nonlinear system identification. This approach involves identifying a nonlinear dynamical model that captures the temporal dynamics underlying PAC. The resulting generative model enables noise-free simulation of estimated PAC signals, facilitating detailed analysis of modulation strength and the low-frequency phase at which the high-frequency bursts occur. The proposed method accounts for harmonic-induced spurious couplings through empirically derived criteria and remains robust to high noise levels and variations in slow-frequency power, offering an accurate and interpretable framework for PAC analysis. The performance of the proposed approach is illustrated using several simulated examples and a real case using local field potentials (LFP) data. The results are compared with several popular methods.

Figures (20)

• Figure 1: Decomposition of a general PAC signal. A PAC signal, $z(t)$, can be decomposed into two components: a low-frequency oscillation, $x(t)$, and a high-frequency oscillation, $y(t)$, whose amplitude is modulated. The amplitude envelope of $y(t)$ ($A_{y}(t)$) is coupled to the phase of $x(t)$ ($\varphi_{x}(t)$).
• Figure 2: Mechanisms and models for generating PAC.A and B illustrate two model classes capable of generating PAC dynamics. A shows a simple analytical model of a PAC signal. In these models, a slow oscillation, $x(t)$, and a fast oscillation, $h(t)$, are combined through a nonlinear function in model $\mathcal{M}$ to produce the PAC, $z(t)$. B illustrates coupled differential equations that generate PAC signals resembling those observed in neural activity. Here, the mean activity of two interacting neural populations, $E(t)$ and $I(t)$, is modelled using differential equations, $\mathcal{E}\left( \ \right)$ and $\mathcal{I}\left( \ \right)$, respectively. $x_E(t)$ and $x_I(t)$ denote the slow oscillation inputs to the neural populations $E$ and $I$. The fast oscillation dynamics are generated internally due to the activity between $E(t)$ and $I(t)$. An explicit link can be established between these two types of models as described in Section \ref{['sec:NMM']}.
• Figure 3: Effects of varying modulation parameters in illustrative PAC models. The generated PAC signals, $z(t)$, are shown in green. The $10$Hz slow oscillation, $x(t)$, is shown in red, while the corresponding amplitude-modulated $50$Hz fast oscillations, $y(t)$, are shown in blue, with their amplitude envelope indicated by dotted red lines. The magnitude spectrum of $z(t)$, $Z(\omega)$ is shown in orange. A and B illustrate the effect of varying the parameter $m$ (lower vs. higher) in the basic PAC model, equations \ref{['eq:PAC_math']} and \ref{['eq:simple_PAC']}. Intermodulation components can be seen at $40$Hz and $60$Hz, with $m$ directly controlling the modulation depth (amplitude of the red dotted envelope). Comparing A and B shows that the depth of the modulation is proportional to the magnitude of the intermodulation frequencies. C and D show the effect of varying $\alpha$ and $c$ in the more complex PAC model, as given in equations \ref{['eq:PAC_math']} and \ref{['eq:nonsine_PAC']}. Due to the more complex modulation, the amplitude modulation is non-sinusoidal in D compared to A -- C, resulting in additional intermodulation components. Parameters $\alpha$ and $c$ can influence both the modulation depth (envelope amplitude in the red dotted line) and its complexity (generation of extra intermodulation components).
• Figure 4: Effects of non-sinusoidal input and modulation parameters in simple PAC models. The generated PAC signals, $z(t)$, are depicted in green, while the $10$Hz non-sinusoidal slow oscillation, $x(t)$, is shown in red. The corresponding amplitude-modulated $50$ Hz fast oscillations, $y(t)$, with the amplitude envelope shown as dotted red lines, are depicted in blue. The magnitude spectrum of $z(t)$, $Z(\omega)$ is shown in orange. This figure is generated with the same corresponding modulation parameters as in Figure \ref{['fig:simple_gen_model']}, but with a non-sinusoidal slow oscillation. A and B show the variation of $m$ in the basic PAC model, equations \ref{['eq:PAC_math']} and \ref{['eq:simple_PAC']}. In comparison to Figure \ref{['fig:simple_gen_model']}A and Figure \ref{['fig:simple_gen_model']}B, the additional frequencies are because of the harmonics in the non-sinusoidal slow oscillation and because of this, extra intermodulations are produced by the resulting non-sinusoidal amplitude modulation (see envelope of $y(t)$ in A and B). The contribution of the extra intermodulations is more apparent in B as the modulation depth (envelope of $y(t)$) is directly proportional to the magnitudes of all the intermodulations. C and D show the variations of $\alpha$ and $c$ in the more complex PAC model shown in equations \ref{['eq:PAC_math']} and \ref{['eq:nonsine_PAC']}.
• Figure 5: Monophasic and biphasic PAC. This figure illustrates the transition from monophasic (A) to biphasic (B and C) PAC using the basic PAC model, equations \ref{['eq:PAC_math']} and \ref{['eq:simple_PAC']}. The generated PAC signals, $z(t)$, are depicted in green, while the $10$Hz sinusoidal slow oscillation, $x(t)$, is shown in red. The corresponding amplitude-modulated $75$Hz fast oscillations, $y(t)$, with the amplitude envelope shown in dotted red lines, are depicted in blue. The magnitude spectrum of $z(t)$, $Z(\omega)$ is shown in orange. The modulation parameter, $m$, in equation \ref{['eq:simple_PAC']} is increased from its level in A to transition from monophasic to biphasic PAC in B. After this point, a further increase in $m$ increases the depth of modulation at the second phase. This is shown in C.