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Measuring information-transfer delays.

Wibral M, Pampu N, Priesemann V, Siebenhühner F, Seiwert H, Lindner M, Lizier JT, Vicente R - PLoS ONE (2013)

Bottom Line: In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another.We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops.While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.

View Article: PubMed Central - PubMed

Affiliation: MEG Unit, Brain Imaging Center, Goethe University, Frankfurt, Germany. wibral@em.uni-frankfurt.de

ABSTRACT
In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another. In the brain, for example, axonal delays between brain areas can amount to several tens of milliseconds, adding an intrinsic component to any timing-based processing of information. Inferring neural interaction delays is thus needed to interpret the information transfer revealed by any analysis of directed interactions across brain structures. However, a robust estimation of interaction delays from neural activity faces several challenges if modeling assumptions on interaction mechanisms are wrong or cannot be made. Here, we propose a robust estimator for neuronal interaction delays rooted in an information-theoretic framework, which allows a model-free exploration of interactions. In particular, we extend transfer entropy to account for delayed source-target interactions, while crucially retaining the conditioning on the embedded target state at the immediately previous time step. We prove that this particular extension is indeed guaranteed to identify interaction delays between two coupled systems and is the only relevant option in keeping with Wiener's principle of causality. We demonstrate the performance of our approach in detecting interaction delays on finite data by numerical simulations of stochastic and deterministic processes, as well as on local field potential recordings. We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops. While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.

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Test case (II).Transfer entropy () values and significance as a function of the assumed delay  for two unidirectionally coupled autoregressive systems. For visualization purposes all values were normalized by the maximal value of the TE between the two systems, i.e. . Red and blue color indicate normalized transfer entropy values and significances for interactions  and , respectively. The nominal interaction delay  used for the generation of the data was 20 sampling units from the process  to . Asterisks indicate those values of  for which the p-value  0.05 once corrected for multiple comparisons. Missing points for  are because the analyses for these ’s failed to pass the shift test (a conservative test in TRENTOOL to detect potential instantaneous cross-talk or shared noise between the two time series, see [42]).
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pone-0055809-g005: Test case (II).Transfer entropy () values and significance as a function of the assumed delay for two unidirectionally coupled autoregressive systems. For visualization purposes all values were normalized by the maximal value of the TE between the two systems, i.e. . Red and blue color indicate normalized transfer entropy values and significances for interactions and , respectively. The nominal interaction delay used for the generation of the data was 20 sampling units from the process to . Asterisks indicate those values of for which the p-value 0.05 once corrected for multiple comparisons. Missing points for are because the analyses for these ’s failed to pass the shift test (a conservative test in TRENTOOL to detect potential instantaneous cross-talk or shared noise between the two time series, see [42]).

Mentions: In test case (II) we investigated two unidirectionally coupled autoregressive (AR) processes where a single interaction delay was present. We investigated as a function of the assumed interaction delay . Figure 5 shows the results of computing and its statistical significance (with a hypothesis of no source-target coupling, see Methods) for the two possible directions of interaction, and . shows a maximal value for units, which matches the nominal value of 20 sampling steps. is statistically significant across a certain interval of delays around the maximum (14 to 23 sampling points) even when corrected for multiple comparisons. This blurring of the statistical significance of the predictive information transfer can be partly explained by memory in the source (via autoregressive terms) meaning the predictive value of the actual directly influential scalar observation of the source is detectable in states of both before and after the actual delay (compare the extension of sources states indicated by shaded boxes in figure 1). An additional factor here is that examination of the source states (instead of scalar observations ) means that full information about the directly influential observation is contained in several source states after. Crucially, the opposite direction () reveals a flat profile with no statistical significance, in correspondence with the absence of a directed interaction from process to .


Measuring information-transfer delays.

Wibral M, Pampu N, Priesemann V, Siebenhühner F, Seiwert H, Lindner M, Lizier JT, Vicente R - PLoS ONE (2013)

Test case (II).Transfer entropy () values and significance as a function of the assumed delay  for two unidirectionally coupled autoregressive systems. For visualization purposes all values were normalized by the maximal value of the TE between the two systems, i.e. . Red and blue color indicate normalized transfer entropy values and significances for interactions  and , respectively. The nominal interaction delay  used for the generation of the data was 20 sampling units from the process  to . Asterisks indicate those values of  for which the p-value  0.05 once corrected for multiple comparisons. Missing points for  are because the analyses for these ’s failed to pass the shift test (a conservative test in TRENTOOL to detect potential instantaneous cross-talk or shared noise between the two time series, see [42]).
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3585400&req=5

pone-0055809-g005: Test case (II).Transfer entropy () values and significance as a function of the assumed delay for two unidirectionally coupled autoregressive systems. For visualization purposes all values were normalized by the maximal value of the TE between the two systems, i.e. . Red and blue color indicate normalized transfer entropy values and significances for interactions and , respectively. The nominal interaction delay used for the generation of the data was 20 sampling units from the process to . Asterisks indicate those values of for which the p-value 0.05 once corrected for multiple comparisons. Missing points for are because the analyses for these ’s failed to pass the shift test (a conservative test in TRENTOOL to detect potential instantaneous cross-talk or shared noise between the two time series, see [42]).
Mentions: In test case (II) we investigated two unidirectionally coupled autoregressive (AR) processes where a single interaction delay was present. We investigated as a function of the assumed interaction delay . Figure 5 shows the results of computing and its statistical significance (with a hypothesis of no source-target coupling, see Methods) for the two possible directions of interaction, and . shows a maximal value for units, which matches the nominal value of 20 sampling steps. is statistically significant across a certain interval of delays around the maximum (14 to 23 sampling points) even when corrected for multiple comparisons. This blurring of the statistical significance of the predictive information transfer can be partly explained by memory in the source (via autoregressive terms) meaning the predictive value of the actual directly influential scalar observation of the source is detectable in states of both before and after the actual delay (compare the extension of sources states indicated by shaded boxes in figure 1). An additional factor here is that examination of the source states (instead of scalar observations ) means that full information about the directly influential observation is contained in several source states after. Crucially, the opposite direction () reveals a flat profile with no statistical significance, in correspondence with the absence of a directed interaction from process to .

Bottom Line: In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another.We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops.While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.

View Article: PubMed Central - PubMed

Affiliation: MEG Unit, Brain Imaging Center, Goethe University, Frankfurt, Germany. wibral@em.uni-frankfurt.de

ABSTRACT
In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another. In the brain, for example, axonal delays between brain areas can amount to several tens of milliseconds, adding an intrinsic component to any timing-based processing of information. Inferring neural interaction delays is thus needed to interpret the information transfer revealed by any analysis of directed interactions across brain structures. However, a robust estimation of interaction delays from neural activity faces several challenges if modeling assumptions on interaction mechanisms are wrong or cannot be made. Here, we propose a robust estimator for neuronal interaction delays rooted in an information-theoretic framework, which allows a model-free exploration of interactions. In particular, we extend transfer entropy to account for delayed source-target interactions, while crucially retaining the conditioning on the embedded target state at the immediately previous time step. We prove that this particular extension is indeed guaranteed to identify interaction delays between two coupled systems and is the only relevant option in keeping with Wiener's principle of causality. We demonstrate the performance of our approach in detecting interaction delays on finite data by numerical simulations of stochastic and deterministic processes, as well as on local field potential recordings. We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops. While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.

Show MeSH
Related in: MedlinePlus