Statistical modeling approach for detecting generalized synchronization

Detecting nonlinear correlations between time series presents a hard problem for data analysis. We present a generative statistical modeling method for detecting nonlinear generalized synchronization. Truncated Volterra series are used to approximate functional interactions. The Volterra kernels are modeled as linear combinations of basis splines, whose coefficients are estimated via $l_1$ and $l_2$ regularized maximum likelihood regression. The regularization manages the high number of kernel coefficients and allows feature selection strategies yielding sparse models. The method's performance is evaluated on different coupled chaotic systems in various synchronization regimes and analytical results for detecting $m:n$ phase synchrony are presented. Experimental applicability is demonstrated by detecting nonlinear interactions between neuronal local field potentials recorded in different parts of macaque visual cortex.

Figure 1

Identification of nonlinear interaction in a coupled Rössler-Lorenz system. (a1) Nonlinear synchronization manifold between original sampled data $x$ and $y$ (the systems' third coordinates) in generalized synchronization with correlation $r(x,y)=-0.168$. (a2) Linearized manifold between $y_E$ and $y$, where $y_E\left(t\right)=F\left[x\right]\left(t\right)$ is the output of a second-order Volterra model, yielding $r(y_E,y)=0.98$. (b1) Delay-shifted (by $\tau$) correlation coefficients. (b2) Second-order kernel corresponding to (a2). (c1) Set of cubic $b$ splines corresponding to (b2), used in Eq. (3). (c2) Performance of the method $($FSM, $r(y_E,y)\in [-1,1])$ for Rössler-Lorenz system with additive white noise over increasing variance ${\sigma }^2$, compared against correlation $r(x,y)$, as well as the $\text{JPR}\in [0,1]$.

Figure 2

Performance on generalized synchronized Mackey-Glass delay rings. (a1) Nonlinear time-embedded GS manifold of ring with two nodes $x$ and $y$. (a2) Linearized synchronization manifold between $y_E$ (second-order model) and $y$. (b) Results for Mackey-Glass rings of varying size. Shown are prior correlation in data [$r(x,y)$, dashed], JPR (light dashed) and our method (FSM, solid) using Volterra series models up to order 3.

Figure 3

Identification of nonlinear interaction between coupled Rössler systems. (a1) Nonlinear synchronization manifold between original sampled data $x$ and $y$ (the systems' first coordinates) at onset of phase synchronization ($\mu =0.034$) with correlation $r(x,y)\approx 0$. (a2) Linearized manifold between $y_E$ and $y$, where $y_E\left(t\right)=F\left[x\right]\left(t\right)$ is the output of a second-order Volterra model, yielding $r(y_E,y)=0.97$. (b1) First-order kernel corresponding to (a2). (b2) Second-order kernel corresponding to (a2). (c) As $\mu$ increases, the system transitions from unsynchronized, via phase ($\mu >0.04$) to generalized chaotic synchronization ($\mu >0.08$). Performance of the method $($FSM, $r(y_E,y)\in [-1,1])$ for various coupling strengths $\mu$ is compared to correlation of the raw data $r(x,y)$, as well as the $\text{JPR}\in [0,1]$.

Figure 4

Identification of interaction between unidirectionally coupled Rössler systems in $4:1$ phase synchronization [Eq. (15)]. (a1) Nonlinear synchronization manifold projected onto first coordinates $x_1,x_2$ of the two systems (brown). (a2) Linearized synchronization manifold after application of a fourth-order Volterra series operator (blue). (b1) Time domain plots of original signals $x\left(t\right)$ [green (light gray)] and $y\left(t\right)$ [orange (gray)] compared to the fourth-order model prediction $y_E\left(t\right)$ [red (dark gray)]. (b2) Delay-shifted (by $\tau$) correlation plots of original signals $r(x,y)$ (brown, thick) and model performance $r(y_E,y)$ (blue, thin). Bootstrapped confidence intervals are shown as dashed lines in light blue.

Figure 5

Two macaque V1 LFP recordings $x$ and $y$ recorded from electrodes with different retinotopy. (a) Interaction manifolds of the EPs. (a1) Nonlinear manifold between $x$ and $y$. (a2) Linearized manifold corresponding to $r_{\mathit{EP}2}(y_E,y)$ in (b1). (b1) Lagged correlation coefficient between EPs of $x$ and $y$ [$r_{\mathit{EP}}(x,y)$], and between a first-order [$r_{\mathit{EP}1}(y_E,y)$] and second-order [$r_{\mathit{EP}2}(y_E,y)$] model $y_E=F\left[x\right]$ and predicted tetrode $y$. For the IPs correlations are shown between a second-order model $y_E$ and $y$ [$r_{\mathit{IP}2}(y_E,y)$]. Lighter colored areas show the bootstrapped confidence intervals of the respective models. (b2) Shows the second-order kernel.