Regularized Partial Lagged Coherence for Functional Connectivity Analysis in the Presence of Cross-talk Sergul Aydore, Syed Ashrafulla, Richard M. Leahy Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, USA contact: sergulaydore@gmail.com C(z 1 ,z 2 ) , |E[z 1 z * 2 ]| r E h |z 1 | 2 i E h |z 2 | 2 i Background • The rich temporal content of EEG/MEG allow us to study dynamic networks in the brain. • Coherence is a widely used measure that can reveal interactions within the frequency range of interest [1]. • Yet, the limited spatial resolution of EEG/MEG frequently result in cross-talk between these signals introducing instantaneous interactions that make it difficult to reliably detect of networks using coherence. • To overcome this, measures including imaginary coherence (IC) [2], phase lag index (PLI) [3] and lagged coherence (LC) [4] have been proposed. • However, none of these approaches considers the effect of interference from sources at other locations in the brain. We address this problem using a novel measure, partial lagged coherence (PLC) with L1 regularization. Interaction Measures in Presence of Cross-talk Cross-talk Model ··· x 2 x 1 s 1 s 2 a 12 a 21 ? x 2 (t)= a 12 s 1 (t)+ s 2 (t) x 1 (t)= s 1 (t)+ a 21 s 2 (t) {s 1 ,s 2 ,x 1 ,x 2 } ∈ C {a 12 ,a 21 }∈ R Theory E[x 1 x * 2 ]= a 21 E h |s 1 | 2 i + a 12 E h |s 2 | 2 i | {z } Real +E[s 1 s * 2 ]+ a 12 a 21 E[s * 1 s 2 ] | {z } Complex True interaction affects only the complex part. Imaginary Coherence [2]: IC (x 1 ,x 2 )= |I {E[x 1 x * 2 ]}| r E h |x 1 | 2 i E h |x 2 | 2 i Phase Lag Index [3]: PLI (x 1 ,x 2 )= |E[sign (∠x 1 - ∠x 2 )]| Lagged Coherence [4]: LC (x 1 ,x 2 )= |I {E[x 1 x * 2 ]}| r E h |x 1 | 2 i E h |x 2 | 2 i - R{E[x 1 x * 2 ]} 2 Imag Real Aydore 2013, Asilomar Conf. Measures as a function of Cross-talk C(x 1 ,x 2 ) IC (x 1 ,x 2 ) PLI (x 1 ,x 2 ) LC (x 1 ,x 2 ) Only Lagged Coherence is independent of cross-talk! Interference In Addition to Cross-talk Partial Lagged Coherence s k s m x m x k y N -2 y 1 y 2 y 3 ··· ? Regression for Interference Supression ˆ x k (t)= c T k y (t) ˆ x m (t)= c T m y (t) Residuals r k (t)= x k (t) - ˆ x k (t) r m (t)= x m (t) - ˆ x m (t) Partial Lagged Coherence PLC (x k ,x m ; y) = LC (r k ,r m ) Computation of Regression Coefficients ˆ c k = min c k ∈R N -2 n E h   x k - c T k y   2 i + λ k kc k k 1 o tuning parameters interfering signals Mean Squared Error ˆ c m = min c m ∈R N -2 n E h   x m - c T m y   2 i + λ m kc m k 1 o Selection of Tuning Parameters minimum MSE minimum MSE + std λ Results MEG Simulations with Realistic Cross-talk [5] 1 2 3 4 5 6 7 8 Coherence: 0.2 + j0.2 Forward Model + Noise Inverse Model Compute • Lagged Coherence (LC) • Partial Lagged Coherence (PLC) Compute • True Positive Rate • False Positive Rate Receiver Operating Characteristic (ROC) Curves TPR , 1 total connected edges X k∈ { Connected Edges } P(edge(k ) > τ ) FPR , 1 total unconnected edges X k∈ { Unconnected Edges } P(edge(k ) < τ ) P(edge > τ ) , 1 N N X i=1 1 edgeSingleExp(i)>τ P(edge < τ ) , 1 N N X i=1 1 edgeSingleExp(i)<τ Area Under ROC as a function of SNR Conclusions • The main goal of this study was to develop a method to reliably estimate functional connectivity in the presence of cross-talk with interference in EEG and MEG data. • We show LC is invariant to linear mixing when only two signals are present whereas C, IC and PLI change as the degree of mixing changes. • However, this bivariate framework ignores the interference that occurs when additional sources mix into the two signals of interest. • By regressing out reference signals from the interfering regions using real regression coefficients we aimed to improve estimation of true interaction between these two signals. • The resulting method (PLC) uses L1-regularization to control the degree of signal suppression in the regression. Reference 1. S. Aydore, D. Pantazis, Richard M. Leahy, “A note on the phase locking value and its properties” , NeuroImage, vol. 74, no. 1, pp. 231-244, 2013. 2. G. Nolte, U. Bai, L. Wheaton, Z. Mari, S. Vorbach, M. Hallet, “Identifying true brain interaction from EEG data using the imaginary part of coherency”,Clin. Neurophysiol., 115 (2004), pp. 2292–2307. 3. C. J. Stam, G. Nolte, and A. Daffertshofer, “Phase lag index: assessment of functional connectivity from multi channel eeg and meg with diminished bias from common sources,” Human brain mapping, vol. 28, no. 11, pp. 1178–1193, 2007. 4. R. D. Pascual-Marqui, “Coherence and phase synchronization: generalization to pairs of multivariate time series, and removal of zero-lag contributions,” arXiv preprint arXiv:0706.1776, 2007. 5. J.C. Mosher, R.M. Leahy, and P.S. Lewis, “EEG and MEG: Forward solutions for inverse methods,” IEEE Trans. Biomed. Eng., vol. 46, pp. 245-259, 1999. 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