Phase-amplitude coupling in neuronal oscillator networks

Cross-frequency phase-amplitude coupling (PAC) describes the phenomenon where the power of a high-frequency oscillation evolves with the phase of a low-frequency one. It has been widely observed in the brain and linked to various brain functions. In this paper, we show that Stuart-Landau oscillators coupled in a nonlinear fashion can give rise to PAC in two commonly accepted architectures, namely, (1) a high-frequency neural oscillation driven by an external low-frequency input and (2) two interacting local oscillations with distinct, locally generated frequencies. We characterize the parameters that affect PAC behavior, thus providing insight on this phenomenon observed in neuronal networks. Inspired by some empirical studies, we further present an interconnection structure for brain regions wherein cross-region interactions are established only by low-frequency oscillations. We then demonstrate that low-frequency phase synchrony can integrate high-frequency activities regulated by local PAC and control the direction of information flow between distant regions.

Figure 1

An overview of PAC. (a) Illustration of PAC. (b), (c) Different architectures for the emergence of PAC: (b) A fast population with an external slow input (e.g., sensori stimulus); (c) Two interacting populations with distinct local oscillations (which also constitute a simplified brain region). (d) Cross-region interconnection structure.

Figure 2

PAC by exogenous inputs. (a), (b) A weaker input ($k_{\mathrm{I}}=3$) yields continuous PAC, while a strong one ($k_{\mathrm{I}}=18$) yields intermittent PAC, where ${\delta }_{\mathrm{f}}=6$ for both. (c) The ${\delta }_{\mathrm{f}}$ in (a) is reduced from 6 to 0.3, and a noticeable phase lag appears between peaks (or valleys) of the fast amplitude oscillation and the slow phase oscillation. (d)–(f) The dependence of PAC intensity (quantified by the MI) on the input strength (d), the input frequency (e), and the fast frequency (f). (Parameters adopted in this figure: If not specified otherwise, ${\delta }_{\mathrm{f}}=0.3$, $k_{\mathrm{I}}=3{\omega }_f=3$ Hz, and ${\omega }_{\mathrm{I}}=0.5$ Hz.)

Figure 3

PAC by local interactions. (a) $D_s/{\delta }_{\mathrm{s}}$ with $D_s={sup}_t\left|\right|z_{\mathrm{s}}\left(t\right)-{\widehat{z}}_{\mathrm{s}}\left(t\right)\left|\right|$ measures the derivation of $z_{\mathrm{s}}\left(t\right)$ from the solution of its decoupled counterpart. As $v_{\mathrm{f}}/v_{\mathrm{s}}$ increases, the derivation shrinks dramatically ($k=5,{\delta }_{\mathrm{f}}=0.3$). (b) PAC emerges by local interaction between two populations as in Fig. 1 (c), where the slow frequency is 6.5 Hz ($\theta$ band), and the fast one is 30Hz ($\gamma$ band). (c) PAC intensity (measured by the MI) increases with $k{\delta }_{\mathrm{s}}/{\delta }_{\mathrm{f}}$. Given a fixed ratio $k{\delta }_{\mathrm{s}}/{\delta }_{\mathrm{f}}$, PAC becomes more intense for larger ${\delta }_{\mathrm{f}}$.

Figure 4

General structure for a brain region. (a) Heterogeneous oscillators constitute each frequency band. (b) The dependence of the PAC intensity (quantified by the MI) on the intracluster coupling strengths $k_{\mathrm{s}}$ and $k_{\mathrm{f}}$. For simplicity, we consider $n=m=20$, and the within-cluster connections and the cross-cluster interconnections are both assumed to be all-to-all. Parameters ${\delta }_{\mathrm{s}}^i,{\delta }_{\mathrm{f}}^p,v_{\mathrm{s}}^i$, and $v_{\mathrm{f}}^p$ are randomly drawn from normal distributions $\mathcal{N}({\delta }_{\mathrm{s}},{\mathrm{\Delta }}_{\mathrm{s}})$, $\mathcal{N}({\delta }_{\mathrm{f}},{\mathrm{\Delta }}_f)$, $\mathcal{N}(v_{\mathrm{s}},V_{\mathrm{s}})$, and $\mathcal{N}(v_{\mathrm{f}},V_{\mathrm{f}})$, respectively. Each cluster is considered to consist of 20 oscillators, and $K=1,({\delta }_{\mathrm{s}},{\mathrm{\Delta }}_{\mathrm{s}})=(8,0.8),({\delta }_{\mathrm{f}},{\mathrm{\Delta }}_{\mathrm{f}})=(1,0.1),(v_{\mathrm{s}},V_{\mathrm{s}})=(4,0.4)$, and $(v_{\mathrm{f}},V_{\mathrm{f}})=(40,2)$. For each pair of $k_{\mathrm{s}}$ and $k_{\mathrm{f}}$, the MI value in the figure is the mean for 1000 random trials.

Figure 5

Roles of low-frequency phase synchrony. (a) PLV of the fast amplitudes in the two regions against the ratio $K_c/\mathrm{\Delta }{\omega }_{\mathrm{s}}$. The line indicated the mean of 1000 random realizations contained in the shaded area. It can be inferred that the fast populations becomes more correlated as the synchrony of the slow oscillations increases. (b), (c) Lead-lag relationship between the two signals: $z_{\mathrm{s}}^1$ precedes $z_{\mathrm{s}}^2$ if $v_{\mathrm{s}}^1>v_{\mathrm{s}}^2$ and vice versa.