Directionality indices: Testing information transfer with surrogate correction

Directionality indices can be used as an indicator of the asymmetry in coupling between systems and have found particular application in relation to neurological systems. The directionality index between two systems is a function of measures of information transfer in both directions. Here we illustrate that before inferring the directionality of coupling it is first necessary to consider the use of appropriate tests of significance. We propose a surrogate corrected directionality index which incorporates such testing. We also highlight the differences between testing the significance of the directionality index itself versus testing the individual measures of information transfer in each direction. To validate the approach we compared two different methods of estimating coupling, both of which have previously been used to estimate directionality indices. These were the modeling-based evolution map approach and a conditional mutual information (CMI) method for calculating dynamic information rates. For the CMI-based approach we also compared two different methods for estimating the CMI, an equiquantization-based estimator and a $k$-nearest neighbors estimator.

Figure 1

Comparison of surrogate data regions for $D$ and $I_{1\to 2}$ and $I_{2\to 1}$.

Figure 2

Directionality indices averaged over 100 initializations for system 1: Rössler oscillators with unidirectional coupling from oscillator 1 to oscillator 2 (positive directionality indices) using a range of parameters of the different estimation methods. The left-hand plots display the original directionality indices, while the right-hand plots are the recalibrated directionality indices. Top row: EMA; middle row: CMI; bottom row: CMI-$k\mathrm{NN}$.

Figure 3

Information transfer using EMA, $\text{order}=3$ for coupled Rössler systems with a frequency ratio of 1:5. Plots on the left are coupling from fast to slow (positive directionality index) and on the right from slow to fast (negative directionality index). Top row shows the average recalibrated coupling in each direction. Blue (crosses) solid line: average recalibrated coupling in the direction $1\to 2$; dashed line: average coupling from surrogates in the direction from $1\to 2$; shaded area: 95% range of the surrogate data. Red (squares) lines in the direction $2\to 1$. Bottom row shows the average original directionality indices. Solid line: average original directionality index; dashed line: average directionality index from the surrogates; shaded area: 95% range of the surrogate data.

Figure 4

Information transfer using CMI, $q=4$ for coupled Rössler systems with a frequency ratio of 1:5. Plots on the left are coupling from fast to slow (positive directionality index) and on the right from slow to fast (negative directionality index). Top row shows the average recalibrated coupling in each direction. Blue (crosses) solid line: average recalibrated information in the direction $1\to 2$; dashed line: average information from surrogates in the direction from $1\to 2$; shaded area: 95% range of the surrogate data. Red (squares) lines in the direction $2\to 1$. Bottom row shows the average original directionality indices. Solid line: average original directionality index; dashed line: average directionality index from the surrogates; shaded area: 95% range of the surrogate data.

Figure 5

Information transfer using CMI-$k\mathrm{NN}$, $k=16$ for coupled Rössler systems with a frequency ratio of 1:5. Plots on the left are coupling from fast to slow (positive directionality index) and on the right from slow to fast (negative directionality index). Top row shows the average recalibrated coupling in each direction. Blue (crosses) solid line: average recalibrated information in the direction $1\to 2$; dashed line: average information from surrogates in the direction from $1\to 2$; shaded area: 95% range of the surrogate data. Red (squares) lines in the direction $2\to 1$. Bottom row shows the average original directionality indices. Solid line: average original directionality index; dashed line: average directionality index from the surrogates; shaded area: 95% range of the surrogate data.