Fig 1.
Spectral TE algorithm pipeline.
(A) The neural signal (blue) is converted to a time-frequency representation (grey) using the invertible maximum overlap discrete wavelet transform (MODWT). (B) At a frequency (wavelet scale) of interest in the source (or the target) the wavelet coefficients are shuffled in time, destroying its connection to the target (or source). (C) The signal is recreated by the inverse MODWT. (D) The transfer entropy for the original and many shuffled signals is computed. (E) A statistical test determines whether the shuffling reduced the information transfer, indicating that the transferred information was indeed encoded at the specific frequency. Each panel here shows the distribution of mTE′ values (vertical bars) obtained from surrogate data where the wavelet coefficients of the scale of interest were shuffled, the median of this distribution (red line), and the original transfer entropy (black line). The analysis and the testing is repeated for all scales of interest (here 4, 5, 6).
Fig 2.
Three systems with the same identified sending and receiving frequencies (indicated by the darker blue and red colors), but a different structure of information transfer.
In system A one source and one target frequency take part in a direct transfer of information between them. In system B one source frequency sends information to all target frequencies except the identified target frequency. This one target frequency, in turn, receives other information from all source frequencies except the identified source frequency. In system C the same source frequency sends information redundantly into all target frequencies, while one target frequency receives (partially different) information redundantly from all source frequencies.
Fig 3.
Algorithm to determine whether information transfer exists from an identified information source scale to an identified target scale. (A) Results from the initial analysis using Algorithm I indicating significant information transfer emanating from one scale (source scale j) and significant information reception at a target scale (target scale r). (B) To test if the information send from the source scale is indeed the information that is received at the target scale do the following: scramble the target at the relevant scale N times and note the values. For each such scrambled target then apply algorithm I for the source, i.e. scramble the relevant source scale K times and note the distribution of the
values. Compute the drop in mTE obtained for the n − th target shuffling with respect to the median
of the distribution of source-and-target shuffled
values,
. (C) Statistically test the original target drop
against the distribution of the
. A significantly larger value of
indicates that information send by the source scale is indeed received by the target scale.
Fig 4.
Spectrally resolved transfer entropy for the null-case (example 1).
(A) Top, a ‘source’ S0 and a ‘target’ T1 of an uncoupled system. Bottom, power spectra of S0 and T1. (B) Spectrally resolved Transfer Entropy. Each panel, except those at the bottom, shows the mTE′ distribution obtained from the surrogate datasets with shuffled coefficients at the scale indicated to the left, or, equivalently, the frequency band indicated at the top of each panel. White bars represent histograms of surrogate data, i.e. relative frequencies in (a.u.), the red dashed line is the median of the surrogate mTE′ distribution, the black dashed line is the original mTE value. The horizontal black line indicates the distance δTE between the original mTE and the median of the surrogate distribution (**, p < 0.005; *, p < 0.05). These display conventions will be kept for figures displaying spectrally resolved TE analyses. The temporal surrogate analysis using surrogates constructed by permuting blocks of samples in the time-domain is shown in the bottom row. No significant drop of the shuffled wavelet coefficients could be found, since no information transfer occurred between a putative source and the target site. (Note that the choice of source or target here is arbitrary since no coupling was simulated).
Fig 5.
Spectrally resolved transfer entropy for example 2.
(A) Left, a source S0 is unidirectionally coupled, at scale j = 4 (frequency band 4 − 7 Hz), with a target T1 at scale j = 1 (frequency band 31 − 62 Hz). Right, power spectra of S0 and T1. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. Information transfer correctly drops when wavelet coefficients are selectively shuffled at scale 4 at the source (S0, left column). The corresponding reception of information at the target (T1) is shown on the right, where a drop for shuffled wavelet coefficients is observed for the frequency band receiving the information in this simulation (i.e. scale 1). The temporal surrogate analysis using surrogates constructed by permuting blocks of samples in the time-domain is shown in the bottom row. (C) SOSO analysis. Blue bars display the distribution of distances between the median of the surrogate data distribution with a shuffled source (compare Fig 5B) when also the target is shuffled. The red line indicates the median of the distribution of
. The black line indicates the original distance between the median of the surrogate data distribution with a shuffled source and the mTE value computed on the original data. If this latter value is found in the upper rejection interval of the distribution of
, there is significant direct information transfer from the source to the target frequency band under investigation. (Left Panel) No information transfer remains when the source sending scale and the target receiving scale are simultaneously shuffled and no drop of mTE can be seen (the distribution
approaches 0); the original drop in mTE is significantly larger. (Right panel) Information transfer remains when an unrelated target frequency band is shuffled.
(black bar), median of the
distribution (red dotted bar).
Table 1.
Results of TE analysis.
Table 2.
Results of spectral TE analysis.
Fig 6.
Comparison of spectrally resolved transfer entropy to spectral Granger causality for within-band transfer (example 3).
(A) Left, a source S0, is unidirectionally coupled, at scale j = 1 (frequency band 30 − 60 Hz), with a target T1. Right, power spectra of S0 and T1. (B) Spectrally resolved TE. Information transfer, correctly, drops when wavelet coefficients are selectively shuffled at scale 1 (frequency band 30-60 Hz) at the source site (left panel). At the target site the drop of wavelet coefficients at scale 1 exhibits the frequency entering the target linearly. (right panel). (C) Parametric spectral Granger analysis. First panel, a significant source was identified with peak at 45 Hz, (Granger causality estimates blue line). The 95% significance level obtained by permutation is indicated by a black dashed line).
Fig 7.
Nonparametric spectral Granger causality in a system with cross-frequency information transfer (CFIT, example 2).
(A) Nonparametric spectral Granger causality. No significant source could be found in either directions. Granger causality estimates (blue line) and the 95% significance level obtained by permutation (black dashed line). (B) Spectrally resolved information transfer of the same system, in contrast, reveals the CFIT.
Fig 8.
Spectrally resolved transfer entropy for example 4.
(A) Left, a system with bidirectionally coupled nodes: y0 and y1. The process y0 is linearly coupled with y1 at scale j = 4 (frequency band 4-8 Hz) and the process y1 is linearly coupled with y0 at scale j = 1 (frequency band 30-60 Hz). Right, power spectral of y0 and y1. (B) Spectrally resolved Transfer Entropy for source y0 and target y1. See Fig 4 for display conventions. (Left panel) Information transfer, drops when wavelet coefficients are selectively shuffled at scale 4 (frequency band 4-8 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is also observed at scale 4 (frequency band 30-60 Hz). A significant drops is also observed at scale 1, in relation to the autonomous oscillations of the target. (C) Spectrally resolved Transfer Entropy for source y1 and target y0. See Fig 4 for display conventions. (Left panel) Information transfer, drops when wavelet coefficients are selectively shuffled at scale 1 (frequency band 30-60 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is also observed at scale 1 (frequency band 30-60 Hz). A significant drops is also observed at scale 4, in relation to the autonomous oscillations of the target. (D) SOSO analysis for source y0 and target y1 (E) SOSO analysis for source y1 and target y0. Both, within- and cross-frequency information transfer is detected. (For plotting conventions see Fig 5).
Fig 9.
Spectrally resolved transfer entropy for example 5.
(A) Top, a source S0, but not S2, is unidirectionally coupled, at scale j = 5 (frequency band 4-8 Hz), with a target T1 at scale j = 1 (frequency band 60-120 Hz). Bottom, power spectral of S0, S2 and T1. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. (Left panel) Information transfer, correctly, drops when wavelet coefficients are selectively shuffled at scale 5 (frequency band 4-8 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is observed at scale 1 (frequency band 60-120 Hz), which contained the simulated target frequency.
Fig 10.
Spectrally resolved transfer entropy for example 6.
(A) Top, a source S1 is unidirectionally coupled, at scale j = 5 (frequency band 8-16 Hz), with a target T0. Bottom, power spectra of S1 and T0. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. (Left panel) Information transfer, correctly, drops when wavelet coefficients are selectively shuffled at scale 5 (frequency band 8-16 Hz) on the source site. (Right panel) No significant drop is present at the target site.
Fig 11.
Spectrally resolved transfer entropy for example 7.
(A) Top, a source S0 is unidirectionally coupled, at multiple scales: j = 3 (frequency band 8-16 Hz), j = 4 (frequency band 4-8 Hz) and j = 5 (frequency band 2-4 Hz), with a target T1 at scale j = 1 (frequency band 31-63 Hz). Bottom, power spectral of S0 and T1. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. (Left panel) Information transfer, correctly, drops when wavelet coefficients are selectively shuffled at scale 3 (frequency band 8-16 Hz), 4 (frequency band 4-8 Hz), 5 (frequency band 2-4 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is observed at scale 1 (frequency band 31-63 Hz).
Fig 12.
Spectrally resolved transfer entropy for example 8.
(A) Top, a source S0 is unidirectionally coupled, at scales j = 5 (frequency band 4-8 Hz), with a target T1 at multiple scales: j = 1 (frequency band 63-125 Hz), j = 2 (frequency band 31-63 Hz) and j = 3 (frequency band 16-31 Hz). Bottom, power spectrum of S0 and T1. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. (Left panel) Information transfer, drops when wavelet coefficients are selectively shuffled at scale 5 (frequency band 4-8 Hz) and 6 (frequency band 2-4 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is observed at scale 1 (frequency band 63-125 Hz), scale 2 (frequency band 31-63 Hz) and scale 3 (frequency band 16-31 Hz).
Fig 13.
Spectrally resolved transfer entropy for example 9.
(A) Top, a source S0 is unidirectionally coupled, with target T1. Multiple scales (j = 1, 2, 3, 5) of S0 are coupled with a single target scale 2 and at the same time a single source scale 4 of S0 is coupled with multiple target scales (j = 1, 2, 3, 4, 5). Bottom, power spectrum of S0 and T1. (B) Spectrally resolved Transfer Entropy. See Fig 4 for display conventions. (Left panel) Information transfer drops when wavelet coefficients are selectively shuffled at scale 4 (frequency band 8-16 Hz) and 5 (frequency band 4-8 Hz) on the source site. The corresponding reception of information at the target is shown on the right panel, where a drop for shuffled wavelet coefficients is observed at scale 1 (frequency band 63-125 Hz), scale 2 (frequency band 31-63 Hz) and scale 3 (frequency band 16-31 Hz). (C) SOSO application to redundant information flow. See Fig 5C, for display conventions. Information transfer remains when the source scale 4 and the target scale 2 are simultaneously shuffled, ruling out a direct information transfer between these two frequency bands.
Fig 14.
Spectrally resolved information transfer between MEG sources when preparing to detect faces.
(A) Spectrally resolved information transfer between aIT as a source and FFA as a target in the condition where subjects are trying to detect target faces. aIT sends information mainly at 75-150Hz (left column), whereas FFA receives information at high frequencies (75-150Hz and above) as well as low frequencies (9-19Hz and 5-9Hz) (right column). See Fig 4 for display conventions. (B) Analyses of cross-frequency information transfer between specific source frequency in aIT and multiple target-frequencies in FFA (second panel on the right side). All four scales (2, 3, 6, 7) at the target side showed a significant direct information transfer from the source at scale 3. (black bar), median of the
distribution (red dotted bar).
Fig 15.
Spectrally resolved information transfer for the ferret.
See Fig 4 for display conventions for B and D and Fig 5 for C and D. (A) Schematic location of recording sites on the ferret brain (from [39]). (B) Spectrally resolved information transfer from PFC to V1 in the ferret. (Left panel) Information transfer, drops at scale 7, 8, and 9 on the source site (PFC), when the wavelet coefficients are shuffled. (Right panel) A significant drop is observed at scale 2 at the target site (V1). mTEtot original (black dotted bar), (red dotted bar). Scale 1 is not shown since LFP were low passed at 300 Hz. (C) Analyses of cross-frequency information transfer between specific source- and target-frequencies in PFC and V1 of the ferret. (Left Panel) No information transfer is present when the source sending scale and the target receiving scale are simultaneously shuffled and no drop of mTE can be seen (the distribution
approaches 0). (Right panel) Information transfer is still present when unrelated target frequency band is shuffled.
(black bar), median of the
distribution (red dotted bar). (D) Spectrally resolved information transfer from V1 to PFC. (Left panel) Information transfer, drops at scale 2, 3, and 4 on the source site (V1), when the wavelet coefficients are shuffled. (Right panel) A significant drop is observed at scale 2 at the target site (PFC). mTEtot original (black dotted bar),
(red dotted bar). Scale 1 is not shown since LFP were low passed at 300 Hz. (E) SOSO application to source V1 and target PFC of ferret.(Left Panel) No information transfer is present when the source sending scale and the target receiving scale are simultaneously shuffled and no drop of mTE can be seen (the distribution
is centered on 0). This means that there is indeed a direct transfer of information from source scale 3 to target scale 2 (Right panel) Information transfer into target scale 2 is still present when the source scale 4 is shuffled, meaning information does not flow from source scale 4 to target scale 2.
(black bar), median of the
distribution (red dotted bar).
Table 3.
Results of TE analysis neural data.
Table 4.
Results of spectral TE analysis.