Time-delayed mutual information of the phase as a measure of functional connectivity

Abstract

We propose a time-delayed mutual information of the phase for detecting nonlinear synchronization in electrophysiological data such as MEG. Palus already introduced the mutual information as a measure of synchronization. To obtain estimates on small data-sets as reliably as possible, we adopt the numerical implementation as proposed by Kraskov and colleagues. An embedding with a parametric time-delay allows a reconstruction of arbitrary nonstationary connective structures--so-called connectivity patterns--in a wide class of systems such as coupled oscillatory or even purely stochastic driven processes. By using this method we do not need to make any assumptions about coupling directions, delay times, temporal dynamics, nonlinearities or underlying mechanisms. For verifying and refining the methods we generate synthetic data-sets by a mutual amplitude coupled network of Rössler oscillators with an a-priori known connective structure. This network is modified in such a way, that the power-spectrum forms a 1/f power law, which is also observed in electrophysiological recordings. The functional connectivity measure is tested on robustness to additive uncorrelated noise and in discrimination of linear mixed input data. For the latter issue a suitable de-correlation technique is applied. Furthermore, the compatibility to inverse methods for a source reconstruction in MEG such as beamforming techniques is controlled by dedicated dipole simulations. Finally, the method is applied on an experimental MEG recording.

Figure 1. Properties of the synthesized data.

The synthetic data were generated in Eq. (5) by linear amplitude coupled Rössler oscillators. A Linear coupling strength as the input of Eq.(5) for modeling a connective structure. The pattern on the left is given by a Gaussian in the -plane centred at ms and ms. The pattern on the right includes an additional spurious non-zero background activation (right), which is generated by Gaussian filtered Poisson noise and decays for high time-delays. B Simulated time-series with oscillatory (left) and a more stochastic (right) behavior. The maximum of the connectivity is shown via dashed lines: black indicates the driving system and grey the driven system. A comparison of the corresponding power-spectra points out a characteristic (dotted black line) for the system including a spurious background synchronization (dashed black line), which can be also observed in MEG recordings (grey line). C Corresponding mutual phase information for trials. A high connectivity is indicated by a high mutual information. The system featuring a background synchronicity (right) holds a damped, less extended and weaker connectivity in the -plane.

Figure 2. Linear correlated data with additive noise: pre-stimulus based de-correlation.

The underlying bivariate data-set consists of trials with a spectrum, cf. Methods (A modified Rössler network). A of two mixed sources. Each pattern represents a connection directed from (upper) and (lower part). A sliding window ms is applied for the estimation. The dashed line indicates the modeled connectivity, cf. Fig. 1A. Parameter sets the mixing strength as referred to Eq. (7). In B the pattern is de-correlated by subtracting Eq. (8), which is fitted in the pre-stimulus interval from ms to 0 ms. The is computed on a triangular grid with a distance of between neighbored estimates. Significant increased synchronization is indicated by a white dot on the grid using a FDR with .

Figure 3. Applied functions for the de-correlation step.

Shown are three least mean square fits using the Matlab curve fitting toolbox with the following types: A exponential, B Gaussian and C power law function. The functions were fitted on the pre-stimulus interval of a connectivity pattern (cf. Fig. 2A with a mixture of and an additive noise level of 100 RMS).

Figure 4. Linear correlated data with additive noise: ICA based de-correlation.

In contrast to Fig. 2B the bivariate time-series are directly de-correlated with help of an independent component analysis (ICA) using a Matlab based toolbox of . The resulting patterns of the de-mixed sources are shown. The same data-set is used as in Fig. 2A. The computation and statistical validation of are performed analogously to the results of Fig. 2A.

Figure 5. Variation of the total number of trials.

Estimated synchronization for a variate number of trials . A Two uncorrelated and b) two strongly correlated and noisy sources. The same de-correlation step is applied as in Fig. 2B. The same data-set is used as given in Fig. 2. The black dashed line indicates the center of the modeled connectivity.

Figure 6. Qualitative comparison to cross-correlation, phase coherence and phase entropy.

A Unmixed case with and without additive uncorrelated white noise, B strong correlated sources with and high additive uncorrelated white noise with . denotes the absolute value of the cross-correlation of the amplitude. is the phase coherence in time-domain, cf. Eq. (9) with , the phase synchronization based on the Shannon-entropy, cf. Eq. (10) with , and the mutual information of the phase, cf. Eq. (4). The same data-set with trials is used as shown in Fig. 2A. The connectivity patterns are estimated with a moving time window ms and de-correlated analogously to Fig. 2B. The temporal coordinate of the underlying connectivity is indicated by a black dashed line. Because , and are estimated in every bin, the significant increased correlation is indicated by a white area (FDR with ).

Figure 7

A Connectivity of the input data. Modified Rössler system with a spectrum is used as time-course of three sequentially chained dipoles, cf. Table 1 and 2. trials were simulated with an additive uncorrelated sensor noise of RMS. Brain noise is adapted by uncorrelated, randomly located and orientated dipoles in the grey matter. B Connectivity of the reconstructed source activation with a conventional LCMV beamforming method. C Reconstructed connectivity by applying a weighted LCMV in a preprocessing step: data in sensor space is filtered by a linear weighted moving average in order to suppress thermal noise of the MEG device. The smoothed signals are mapped onto the cortex and the functional connectivity among the reconstructed source activation is estimated. In B and C patterns are unprocessed (left) and processed by an de-correlation on pattern level (right) as suggested in the Results (Pattern de-correlation).

Figure 8. Influence of the additive noise level 10%, 15% and 20% RMS on the reconstruction of the connectivity.

A Connectivity of the input data ( trials of a Rössler system), cf. Table 1 and 2. B Connectivity of the reconstructed source activation with a conventional LCMV beamforming. C Reconstructed connectivity by applying a WMA filter on sensor data in a preprocessing step (WMA+LCMV). In B and C the patterns are unprocessed (left) and de-correlated (right) as suggested in the Resutls (Pattern de-correlation). Significant improvements in the connectivity reconstruction compared to the patterns of the conventional beamformer without de-correlation are marked by white solid circles.

Figure 9. Dipole simulation and reconstruction with Rössler oscillations as source activation with a variable number of (left), (center) trials (right).

The data is simulated with additive uncorrelated sensor noise of 20% RMS and brain noise generated by uncorrelated, randomly located and orientated dipoles in the grey matter. A Connectivity of the connected Rössler oscillators indexed with and , cf. Table 1 and 2. B Connectivity of the reconstructed source activation with a conventional LCMV beamforming approach. C Reconstructed connectivity by applying a WMA filter to reduce thermal noise captured by the MEG device on sensor data (WMA+LCMV). All patterns in b) and c) are processed by removing exponentially decaying correlations as described in the Results (Pattern de-correlation).

Figure 10. Data from the MEG study of in processing of olfactorily conditioned faces.

Shown is for 3–150 Hz in terms of the functional connectivity between a temporal and frontal cortical area during a passive viewing of faces with neutral expression. The interval –0 ms was taken to estimate a statistical threshold with a FDR ratio of . Significant changes are marked by white dots. The patterns were de-correlated with . The rows are given by a Negative conditioning with and a Neutral control with . In the contrast Negative-Neutral the processing of faces is compensated by subtracting the Neutral baseline. The columns denote connectivity before conditioning (Pre), after conditioning (Post) and the contrast Post-Pre, which shows the change in connectivity through the conditioning process. Cluster locations are calculated by the center of gravity and marked by cross hairs. Numerical values of the location are listed in Table 3.

Figure 11. Impact of different types of de-correlations and baselines on a) the connectivity pattern Post Negative and b) the contrast Post Negative - Pre Negative - (Post Neutral - Pre Neutral).

Four types of de-correlation were implemented: None without any de-correlation, Exponential with , Gaussian with and Power law with . Data from the interval –0 ms provides a baseline for the FDR threshold in Pre-Stimulus. As an alternative approach Surrogate data was estimated by destroying correlation in phases. Significant higher connectivities are marked by white dots.