Identifying delayed directional couplings with symbolic transfer entropy

We propose a straightforward extension of symbolic transfer entropy to enable the investigation of delayed directional relationships between coupled dynamical systems from time series. Analyzing time series from chaotic model systems, we demonstrate the applicability and limitations of our approach. Our findings obtained from applying our method to infer delayed directed interactions in the human epileptic brain underline the importance of our approach for improving the construction of functional network structures from data.

Figure 1

Schematics of flow of information between two unidirectionally delay-coupled systems $(X$ drives $Y)$ together with the procedure of symbolizing the time series and of estimating the delayed flow of information. The value of each element $x_i \left(y_i\right)$ in the time series is encoded in the vertical position of the respective box. Flow of information from the system's own past to the current state $i$ is indicated by gray arrows (the transition from dark to light coloring indicates the loss of previous information, as the system evolves). The delayed flow of information (here ${\Delta }_X=10$) from states of system $X$ to the current state of system $Y$ is indicated by red arrows (transition from dark to light coloring as before). Blue arrows exemplify the procedure to estimate the flow of information: Arrows point from symbols, composed of exemplary previous states of either system $X$ or $Y$, to the actual symbol ${\widehat{y}}_i$, which is marked in blue. Here, the previous states ${\widehat{x}}_{i-1}$ or ${\widehat{x}}_{i-11}$ are ${\tau }_2$ time steps and ${\widehat{y}}_{i-1}$ is ${\tau }_1$ time steps past the actual symbol ${\widehat{y}}_i$. The derivation of these permutation symbols is exemplarily shown for an embedding dimension $m=3$ and lag $l=2$, e.g., ${\widehat{x}}_{i-11}=(2,1,3)$.

Figure 2

Color-coded estimates of delayed symbolic transfer entropies ${\mathcal{T}}_{X\to Y}({\tau }_1,{\tau }_2)$ (top) and ${\mathcal{T}}_{Y\to X}({\tau }_1,{\tau }_2)$ (middle) and of the directionality index $\mathcal{T}({\tau }_1,{\tau }_2)$ (bottom) for unidirectionally delay-coupled logistic maps ($c_{Y\to X}=0,c_{X\to Y}=0.45,{\Delta }_X=10$; embedding parameters $m=3,l=1$; $N=100$ data points). Positive values of $\mathcal{T}({\tau }_1,{\tau }_2)$ (particularly the resonance-like pattern and the lower secondary diagonal) indicate the driving behavior of system $X$. Negative values of $\mathcal{T}({\tau }_1,{\tau }_2)$ (particularly the upper secondary diagonal) indicate the responding behavior of system $X$. Amplitude values are scaled linearly in $[-0.25,0.25]$ and logarithmically otherwise.

Figure 3

Color-coded estimates of the directionality index $\mathcal{T}({\tau }_1,{\tau }_2)$ for unidirectionally delay-coupled logistic maps with delay ${\Delta }_X=10$ and coupling strength $c_{X\to Y}=0.45$. Embedding parameters: (top) $m=3,l=1$, (bottom) $m=4,l=1$. Left to right: Increasing number of data points $N$. Positive values of $\mathcal{T}({\tau }_1,\tau 2)$ (particularly the resonance-like pattern and the lower secondary diagonal) indicate the driving behavior of system $X$. Negative values of $\mathcal{T}({\tau }_1,\tau 2)$ (particularly the upper secondary diagonal) indicate the responding behavior of system $X$.

Figure 4

Color-coded estimates of the directionality index $\mathcal{T}({\tau }_1,{\tau }_2)$ for unidirectionally delay-coupled logistic maps with delay ${\Delta }_X=10$ and coupling strength $c_{X\to Y}=0.45$ estimated with $N={10}^{m+1}$ data points. Embedding dimension: (top) $m=3$ (bottom) $m=4$. Left to right: Increasing embedding lag $l$. Positive values of $\mathcal{T}({\tau }_1,{\tau }_2)$ (particularly the resonance-like pattern and the lower secondary diagonal) indicate the driving behavior of system $X$. Negative values of $\mathcal{T}({\tau }_1,{\tau }_2)$ (particularly the upper secondary diagonal) indicate the responding behavior of system $X$.

Figure 5

Means and standard deviations of delayed symbolic transfer entropies for the directions $X\to Y$ (purple (dark gray) solid line and shaded area) and $Y\to X$ (orange (light gray) solid line and shaded area) depending on the coupling strength $c_{X\to Y}$ for 20 realizations of delay-coupled logistic maps with delay ${\Delta }_X=10$ and $N=100$ (left) and $N=1000$ data points (right). Embedding parameters: $m=3$ and $l=1$. Upper row: Estimates ${\mathfrak{T}}^R$ from the resonance-like pattern with ${\tau }_1=11$ and ${\tau }_2={\Delta }_X=10$; middle row: Estimates ${\mathfrak{T}}^U$ from the upper secondary diagonal (cf. Fig. 2) with ${\tau }_1=11$ and ${\tau }_2=21$; lower row: Estimates ${\mathfrak{T}}^L$ from the lower secondary diagonal with ${\tau }_1=11$ and ${\tau }_2=1$. In all plots, averaged estimates from the background ${\overline{\mathfrak{T}}}^B$ (cf. Fig. 2) with ${\tau }_1={\tau }_2=25$ are shown as gray dotted lines and gray-shaded areas.

Figure 6

Means of delayed symbolic transfer entropies for the directions $X\to Y$ (purple solid lines, upper and middle row) and $Y\to X$ (orange solid lines, lower part) depending on the coupling strength $c_{X\to Y}$. Twenty realizations of noisy delay-coupled logistic maps with delay ${\Delta }_X=10$ and $N=100$ (left) and $N=1000$ data points (right). Averaged estimates from the background ${\overline{\mathfrak{T}}}^B$ are shown in gray dotted lines. The transitions from dark to light coloring encodes a decreasing signal-to-noise ratio $(128,32,8,2)$. Embedding parameters and choice of pairs $\left({\tau }_1,{\tau }_2\right)$ for delayed symbolic transfer entropies $\mathfrak{T}$ as in Fig. 5.

Figure 7

Upper left: Schematics of electrode strips placed over the left and right temporal lateral neocortex and of bilateral intrahippocampal depth electrodes. Recording sites that were used for analyses are marked red. Upper right to lower right: Color-coded estimates of the mean directionality index $\mathcal{T}({\tau }_1,{\tau }_2)$ for interactions between various brain regions estimated from intracranial EEG recorded during day times and night times. Interactions between anterior and posterior sites within the nonepileptic mesial temporal brain structures (TL01-TL05, upper right), between homologous sites within the epileptic brain hemisphere (TR01-TR05, lower left), and between regions in the left and right temporal lateral neocortex (TLL04-TLR04, lower right). The horizontal stripes that can be observed for ${\tau }_1$ taking on integer multiples of the embedding lag $l$ can be related to the applied symbolization (cf. Fig. 4).