A Multivariate Time-Frequency Based Phase Synchrony Measure for Quantifying Functional Connectivity in the Brain

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1 A Multivariate Time-Frequency Based Phase Synchrony Measure for Quantifying Functional Connectivity in the Brain Dr. Ali Yener Mutlu Department of Electrical and Electronics Engineering, Izmir Katip Celebi University

2 Introduction: Human Brain The Big Picture A Highly Complex Network Number of neurons range from 80 to 120 billions Responsible for: Thought, action Memory, cognition Vital life functions Fig. 1: The Big Picture How Is it Connected and Organized? Structural or anatomical Connectivity Functional connectivity The focus of our work A Recent Project: Human Connectome Build a network map of the brain Facilitate research into brain diseases

3 Fig. 2: Functional Connectivity vs Anatomical Connectivity [1] [1] E. Bullmore and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems, Nature Reviews Neuroscience, vol. 10, no. 3, pp , Functional Connectivity Statistical dependencies across brain regions Why is it important? Understanding cognitive and executive processes Memory, attention Addressing diseases such as schizophrenia EEG, fmri Generated by nonlinear and non-stationary neural processes How to quantify?

4 Existing Measures to Quantify Functional Connectivity Reciprocal interactions are the key features of functional connectivity Linear Measures: Cross-correlation Spectral Coherence Granger Causality Nonlinear Measures: Mutual Information Solution? Use Phase Synchronization [2] Large-scale synchrony during cognition and perception Studies on animals: Visual Stimuli Limited to stationary processes and linear dependencies Reliable estimation requires a large amount of data [2] M. Le Van Quyen, J. Foucher, J. P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie, and F. J. Varela, Comparison of hilbert transform and wavelet methods for the analysis of neuronal synchrony, Journal of Neuroscience Methods, vol. 111, no. 2, pp , 2001.

5 Phase Synchrony The temporal adjustment of the phases of two oscillators Advantages: A widely studied nonlinear measure Does not rely on a parametric model A better indicator of dependency than amplitude dependent measures First step: Extract the time-varying phase of each signal Existing methods to phase estimation: Hilbert Transform: Requires filtering Short-time Fourier Transform: Resolution trade-off due to window function Continuous Wavelet Transform Non-uniform time-frequency resolution Fig. 3: Phase Synchrony Between Two Sinusoidal Signals

6 A Time-Frequency Based Approach to Phase and Phase Synchrony Estimation

7 Cohen s Class of Represent signal energy across time and frequency Choose a complex TFD: Rihaczek TFD Advantages: Uniformly high time-frequency resolution Better localization for phase modulated signals Time-varying phase Problem: Cross-terms appear for multicomponent signals Biased energy and phase estimates Time-Frequency Distributions

8 Solution: Reduced-Interference Rihaczek Distribution Use an appropriate kernel to filter out cross-terms: RID-Rihaczek Distribution [3]: Choi-Wiliams (CW), Born-Jordan or binomial kernels can be used How to quantify phase synchrony? Find the phase difference: Phase Locking Value (PLV): A measure of how the relative phase is distributed over the unit circle Based on circular variance PLV is always between 0 and 1 [3] S. Aviyente and A. Mutlu, A Time-Frequency Based Approach to Phase and Phase Synchrony Estimation, IEEE Transactions on Signal Processing, vol. 59, no. 7, pp , July, 2011.

9 Simulation Results: Evaluation of Time-Frequency Resolution We consider:, where for 128 time points Fig. 4: RID-Rihaczek (left) and Wavelet (right) based phase synchrony in the time-frequency plane for two chirp signals with constant phase shift

10 Simulation Results: Robustness of RID-TFPS to Noise RID-TFPS and Wavelet- TFPS are compared in terms of robustness to noise Consider: where are independent white Gaussian processes SNR db 64 time points, 200 trials, 200 simulations Fig. 5: Comparison of noise robustness: Performances of the RID-TFPS and Wavelet-TFPS Ideal synchrony value is equal to one RID-TFPS is significantly larger at all SNR values for, especially for

11 Kuramoto Model: Simulation Results: Tracking Real Synchrony within a Network Describes phase dynamics of a large network of oscillators [4] Lorentzian distribution is used: Phase transition to a synchronized state at 16 sinusoidal oscillators SNR = 10 db, and rad/s Fig. 6 Mean RID-TFPS and mean Wavelet-TFPS (PLV, averaged over all possible pairs of 16 oscillators) as a function of the coupling strength K, where the number of trials, L= 200 [4] Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in International symposium on mathematical problems in theoretical physics, Lecture notes in Physics, vol. 39, 1975, pp

12 A Potential Application to fmri Fig. 7 Error-response vs Correct-response ERN (Error-related negativity): A brain potential response Occurs and peaks within 100 ms after error is committed Related to brain s self-monitoring ability Reduced ERN when functional connectivity is impaired Largest synchrony at central electrodes (FCz) Fig. 8 RID-Rihaczek based synchrony for the error-correct difference Results from 92 subjects, 62 electrodes : Hypothesis: Larger phase synchrony values compared to correct responses at central electrodes Significantly larger at 5% (Wilcoxon signedrank test) near the primary motor areas Indicator of interactions between Anterior Cingulate Cortex and motor areas during error processing

13 Methods for Quantifying Multivariate Phase Synchronization

14 Problem Statement Phase Synchrony: Bivariate Relationship between two signals Imposes a limitation to multivariate analysis of EEG Requires working in the computationally costly space of signal pairs of N signals Not sufficient to reveal indirect relationships Goal: Extend bivariate phase synchrony to multivariate case Quantify synchrony within a group of signals Obtain a global understanding of functional brain connectivity

15 Multivariate Multivariate Phase Synchrony: Captures global synchronization patterns Characterizes the dynamics of a network Existing Approaches: Phase Synchronization Compute the whole set of pairwise synchrony values Indirect way of computing global synchrony Graph theoretical methods using cluster analysis: E.g., spectral clustering using correlation matrix Partial phase synchrony Mean field approach [5] [5] C. Allefeld and J. Kurths, An approach to multivariate phase synchronization analysis and its application to eventrelated potentials, International journal of bifurcation and chaos, vol. 14, no. 2, pp , 2004.

16 Within Group Multivariate Synchrony For a network,, S-estimator is defined as [6]: where s are the N normalized eigenvalues Quantifies the group synchrony Exploits the Eigenvalue spectrum of the synchrony matrix Complement to the entropy of normalized Eigenvalues If the network is completely synchronized, i.e.,, then and indicates perfect within group multivariate synchrony [6] K. Oshima, C. Carmeli, and M. Hasler, State Change Detection Using Multivariate Synchronization Measure from Physiological Signals, J Signal Process, vol. 10, no. 4, pp , 2006.

17 Hyperspherical Phase Synchrony for Quantifying Multivariate Functional Brain Connectivity

18 Hyperspherical Phase Synchrony Bivariate Synchrony: Based on the circular variance is mapped onto a unit circle: 1- sphere Multivariate Synchrony: N signals Extend to N-1 angular coordinates Form direction vectors in an N- dimensional hyperspherical coordinate system The set of coordinates in N+1 dimensional space,, defines N-sphere: where, r is radius. Fig. 9 Line crossings indicate the direction vectors sampled from a 2-sphere

19 Hyperspherical Phase Synchrony The set of N Cartesian coordinates on a unit N-1 sphere is defined as: Forms a direction vector : Hyperspherical phase synchrony (HPS) : Equivalent to PLV when two signals exist A direct and novel method Offers low computational complexity,, compared to S-estimator,

20 Simulation Results: Consider highly synchronized sinusoidal signals with constant phase differences Robustness of HPS to Noise SNR from -30 to 30 db 200 trials, 1024 time points Fig. 10 Performances of the hyperspherical phase synchrony and S-estimator in estimating the true multivariate phase synchrony within a group of oscillators consisting of highly synchronized sinusoidal signals having constant phase differences for different SNR values, bars indicate the standard deviations.

21 Application to EEG Data Data: 92 subjects, 62 electrodes ERN interval, theta band (4-8 Hz) Hypothesis: Increased synchrony between frontal-central vs parietalcentral electrodes Fig. 11 The mean HPS values computed over all subjects at each time and frequency point within the ERN interval and theta band for the two electrode groups. Fig. 12: EEG System Results: Significantly larger mean HPS values within ERN window and theta band for frontal-central group for all subjects (p<0.01) Medial prefrontal cortex (mpfc) and lateral prefrontal cortex (lpfc) are synchronized during error-related cognitive control processes Consistent with the previous work and the hypotheses in neuroscience [6]

22 Comparison with S-estimator Fig. 13 The mean S-values computed over all subjects at each time and frequency point within the ERN interval and theta band for the two electrode groups. S-values for frontal-central group are significantly larger with p<0.05 HPS values for frontal-central group are significantly larger with p<0.01 For each subject, larger mean HPS values compared to S-values for the frontal-central group HPS performs better in discriminating between the multivariate synchronization of the two groups

23 Comparison with S-estimator: ROC Curve Fig. 14 ROC curves for HPS and S-estimator For each subject: A true positive (detection) is determined when the mean multivariate synchrony for the frontal-central electrodes is larger than the threshold A false alarm is defined when the mean multivariate synchronization for the parietal-central group is larger than the threshold From the figure, HPS performs better than the S-estimator in detecting frontal-central multivariate synchronization

24 Conclusions and Further Directions Bivariate Phase Synchrony A new phase estimation method based on the RID- Rihaczek distribution More robust to noise, better time-frequency resolution, better at detecting synchrony Further Directions: Explore different complex TFDs Multivariate Phase Synchrony S-estimator: Within Group Synchrony Hyperspherical Phase Synchrony:A novel and direct method Revealed the frontalcentral electrode interactions Further Directions: Discover underlying synchronization clusters

25 Thank You for Listening!

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