Inferring direct directed-information flow from multivariate nonlinear time series

Estimating the functional topology of a network from multivariate observations is an important task in nonlinear dynamics. We introduce the nonparametric partial directed coherence that allows disentanglement of direct and indirect connections and their directions. We illustrate the performance of the nonparametric partial directed coherence by means of a simulation with data from synchronized nonlinear oscillators and apply it to real-world data from a patient suffering from essential tremor.

Figure 1

(Color online) Percentage of detections for (a) van-der-Pol oscillators, and (b) Rössler oscillators. Bold line: $x_1\left(t\right)$ to $x_2\left(t\right)$ (true positives), thin line: $x_2\left(t\right)$ to $x_1\left(t\right)$ (false positives). With increasing coupling strength the connection $x_1\left(t\right)$ to $x_2\left(t\right)$ is detected more often but the absent connection $x_2\left(t\right)$ to $x_1\left(t\right)$ is detected with an error level of 5%, which corresponds to the significance level (indicated by a gray dashed line).

Figure 2

(Color online) Nonparametric PDC of the Rössler system. Gray lines indicate bootstrapped significance levels. The connections $1\leftarrow 2$, $2\leftarrow 1$, $4\leftarrow 2$, $3\leftarrow 4$, and $2\leftarrow 3$ are significant. The spectra $\widehat{S}$ are displayed in arbitrary units.

Figure 3

Tremor data, zoomed segment of 5 s. The first and second traces are the EEG signals while the third trace is the rectified and filtered EMG. The traces are rescaled to equalize their variances and are depicted in arbitrary units.

Figure 4

(Color online) Nonparametric PDC of the tremor data set and the block-averaged periodograms in logarithmic scale. It shows a connection from the left extensor to the right EEG, especially at the tremor frequency of 5 Hz. At 10 Hz, there is a first harmonic of the tremor EMG indicating a nonlinear signal generation mechanism.