Transcript-based estimators for characterizing interactions

TL;DR

The concept of transcripts is revisited and the usage of transcript-based estimators for a time-series-based investigation of interactions between coupled paradigmatic dynamical systems of varying complexity is showcased.

Abstract

The concept of transcripts was introduced in 2009 as a means to characterize various aspects of the functional relationship between time series of interacting systems. Based on this concept that utilizes algebraic relations between ordinal patterns derived from time series, estimators for the strength, direction, and complexity of interactions have been introduced. These estimators, however, have not yet found widespread application in studies of interactions between real-world systems. Here, we revisit the concept of transcripts and showcase the usage of transcript-based estimators for a time-series-based investigation of interactions between coupled paradigmatic dynamical systems of varying complexity. At the example of a time-resolved analysis of multichannel and multiday recordings of ongoing human brain dynamics, we demonstrate the potential of the methods to provide novel insights into the intricate spatial-temporal interactions in the human brain underlying different vigilance states.

Figures (8)

• Figure 1: Illustrative example for calculating the transcript $\tau$ between two time-series-derived ordinal patterns $\mu$ and $\nu$ as well as the order class $C_n$ to which $\tau$ belongs. a) Schematic of time-resolved derivation of ordinal-pattern (symbol) sequences ($d=4$) from two time series. b) Exemplary calculation of transcript $\tau$ from the composition of ordinal pattern $\nu$ with the inverse of ordinal patterns $\mu$ denoted as $\mu^{-1}$ (cf. Eq. \ref{['eq:transcript']}). Colors highlight the entries of $\nu$ that are composed elementwise with $\mu^{-1}$ to obtain $\tau$. For example, $\textit{2}_\textrm{\bf {\color{red} 3}}$ means: exchange entry $\textit{2}$ (at position 0 in $\mu^{-1}$) with the entry at position 3, i.e., $\textit{0}$. Note that the composition $\mu$ with $\nu^{-1}$ can yield a different transcript $\tau$. c) Exemplary derivation of an order class. The transcript $\tau = \left[\textit{0,2,3,1}\right]$ needs to be composed with itself three times to achieve the unit symbol $\pazocal{I}$. This transcript thus belongs to order class $C_3$.
• Figure 2: Dependence of estimators for the strength (top left), directionality (top right) and complexity (bottom left) of interaction on coupling strength $k$ for unidirectionally coupled Hénon maps. The dependence of the probability densities of order classes on coupling strength is shown in bottom right part of the figure. We generated time series of length $N=2^{17}$ of the $x$-components and derived ordinal patterns using an embedding dimension $d=4$ and an embedding delay $m=1$ (first zero-crossing of the autocorrelation function).
• Figure 3: Same as Fig. \ref{['fig:measureshenon']} but for unidirectionally coupled Rössler oscillators. We generated time series of length $N=2^{18}$ of the $x$-components and derived ordinal patterns using an embedding dimension $d=6$ and an embedding delay $m=144$.
• Figure 4: (From top to bottom) Time-evolution of estimated interaction strength, direction and complexity for short-range (left) and long-range (right) interactions in a human brain. Dark lines and shaded areas indicate mean and standard deviation over successive $30$ min time windows. We derived ordinal patterns using an embedding dimension $d=6$ and an embedding delay $m=1$.
• Figure 5: Spatial distributions of transcript-based estimators for strength (top), preferred direction of information flow (middle) and complexity $\pazocal{C}$ (bottom) of brain-wide interactions during day- and nighttime projected onto the surface of the head (cubic polynomial interpolation). Grand averages over all subjects of the respective temporal means of mean estimators from each recording site to the remaining sites (cf. Eq. \ref{['eq:sumrec']}). Middle left plot highlights electrode positions and labels.