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WORKSHOP ON DIRECTED FUNCTIONAL CONNECTIVITY ANALYSIS USING WIENERGRANGER CAUSALITY Dr. Steven Bressler Cognitive Neurodynamics Laboratory Center for Complex Systems & Brain Sciences Department of Psychology Florida Atlantic University http://www.ccs.fau.edu/~bressler/ THE PROBLEM To understand the neural basis of cognition To understand the neural basis of cognition requires understanding the influences exerted by neuronal groups in the cortex on one another. Stimulation Ablation Statistical time series analysis DIRECTED FUNCTIONAL CONNECTIVITY Directed Functional Connectivity (DFC) refers to a class of metrics used to infer causal influence (effective connectivity) between neuronal groups. DFC estimates directionally oriented functional relations between neuronal groups. THE CONCEPT OF CAUSALITY David Hume “an object, followed by another, and where all objects similar to the first are followed by objects similar to the second” Bertrand Russell “…to exhibit certain confusions, especially in regard to teleology and determinism, which appear to me to be connected with erroneous notions as to causality" INTERPRETING CAUSALITY Deterministic Causality A change in the activity of one group (A) necessarily produces a change in that of another group (B). P(B|A) = 1 Probabilistic Causality A change in the activity of A contributes an influence that changes the likelihood of a change occurring in the activity of B. P(B|A) > P(B|~A) CAUSALITY IN THE CORTEX [MANNINO & BRESSLER, IN PREP] The complex large-scale anatomical connectivity of the cerebral cortex is characterized by bi-directionality and heavy convergence. The influence from one cortical neuronal group (A) does not necessarily produce a measurable effect in another group (B), and cortical causality is nondeterministic. Group A may change the likelihood of a change in B, so causal influences in the cortex are probabilistic. DFC & CAUSALITY Metrics for DFC should reflect probabilistic causality rather than deterministic causality between neuronal groups. Time series DFC metrics should quantify the degree of predictability of the time series of one group from that of another. Granger causality is a probabilistic concept of causality that is based on predictability. For two simultaneous time series, one series is called ‘Granger causal’ to the other if we can better predict the second series by incorporating knowledge of the first one. STATISTICAL CAUSALITY Norbert Wiener Clive Granger WIENER CAUSALITY Wiener considered 2 functions, f1(α) and f2(α). He proposed a “nonnegative quantity not exceeding 1 which measures the additional effectiveness of the past values of f2(α) in helping to determine the present value of f1(α), and may be considered as a measure C of the causality effect of f2(α) on f1(α).” Norbert Wiener The Theory of Prediction, 1956 GRANGER CAUSALITY “In the early 1960's I was considering a pair of related stochastic processes which were clearly inter-related and I wanted to know if this relationship could be broken down into a pair of one way relationships. It was suggested to me to look at a definition of causality proposed by a very famous mathematician, Norbert Weiner, so I adapted this definition (Wiener 1956) into a practical form and discussed it.” Clive Granger’s personal account in: Seth A. (2007) Granger causality, Scholarpedia 2(7):1667. GRANGER CAUSALITY “Applied economists found the definition understandable and useable and applications of it started to appear.” Clive Granger’s personal account in: Seth A. (2007) Granger causality, Scholarpedia 2(7):1667. “However, several writers stated that ‘of course, this is not real causality, it is only Granger causality.’ Thus, from the beginning, applications used this term to distinguish it from other possible definitions.” UNIVARIATE AUTOREGRESSIVE MODEL xt= [a1xt-1 + a2xt-2 + a3xt-3 + … + amxt-m] + εt where xt is a zero-mean stationary stochastic process, ai are model coefficients, m is the model order, and εt is the residual error. DEFINITION OF GRANGER CAUSALITY [BRESSLER & SETH, NEUROIMAGE, 2011] Granger (1969): Let x1, x2, …, xt and y1, y2, …, yt represent two time series. Compare two autoregressive models: xt = a1xt-1 + … + amxt-m + 1t and restricted xt = b1xt-1 + … + bmxt-m + c1yt-1 + … + cmyt-m + 2t unrestricted Let 1= var(1t) and 2= var(2t). If 2 < 1, then the y time series has a Granger casual influence on the x time series. TIME DOMAIN GRANGER CAUSALITY MEASURE The time domain Granger causality from y to x is: Fy x 1 ln 2 TOTAL INTERDEPENDENCE [DING ET AL, HANDBOOK OF TIME SERIES ANALYSIS, 2006] The total interdependence between time series xt and yt can be decomposed into three components: Fx,y = Fx®y + Fy®x + Fx·y where the 3rd component, called the instantaneous causality, may be due to factors exogenous to the (x, y) system, such as a common driving input. 4 POSSIBLE RELATIONS BETWEEN 2 TIME SERIES Unidirectional GC only from x to y Unidirectional GC only from y to x Bidirectional GC between x and y Independence of x and y BIVARIATE VOXELWISE GC FROM FMRI [BRESSLER ET AL, J NEUROSCI, 2008] Granger Causality was computed as an F-statistic for every voxel pair of a pair of Regions of Interest (ROIs). F _value = RSSr - RSSur / m ( RSS T = # observations = ~700 m = model order = 1 Distributions of F-statistics in top-down and bottom-up directions for ROIs RaIPS & RV3A. ) RSSur / T - 2m -1 The fraction of voxel pairs having significant GC from one ROI to another was greater in topdown than bottom-up direction. 60 ROI pairs from 8 ROIs in each cerebral hemisphere of 6 subjects were analyzed. VoxelRandomized Control By subject By ROI pair TrialRandomized Control MULTIVARIATE VOXELWISE GC WITH LASSO[TANG ET AL, PLOS COMP BIOL, 2012] The MVAR Model: m Zt = å Bk Zt-k + Et k=1 The b-coefficients represent GC: The Least Absolute Shrinkage and Selection Operator (LASSO) makes model estimation feasible when too few observations are available to otherwise estimate the model coefficients. LASSO GC VS. CORRELATION LASSO GC shows directional asymmetry between ROIs by use of summary statistic W. Connectivity between FEF and VP is more sparse for LASSO GC (A) than for Correlation (B), indicating greater spatial specificity. CONDITIONAL GC OF DAN & VOC IN VISUAL SPATIAL ATTENTION Positive W Negative W Modified method after Garg et al 2011, FARM using fMRI SPECTRAL GRANGER CAUSALITY[DING ET AL, HANDBOOK OF TIME SERIES ANALYSIS, 2006] MVAR Model: Xt = A1Xt-1 + The + AmXt-m + Εt Spectral Matrix is defined as: S( f ) = <X (f ) X (f )*> = H(f ) H*(f ) where * denotes matrix transposition & complex conjugation; is the covariance matrix of Et ; and 1 m is the transfer function of the 2 i k f H(f ) Ak e k 0 system. SPECTRAL GRANGER CAUSALITY[DING ET AL, HANDBOOK OF TIME SERIES ANALYSIS, 2006] Geweke (1982) found a spectral representation of the time domain Granger causality: () Iy®x f = ln () Sxx f () () * H xx f S2H xx f where Sxx(f) is the spectral power of x, Hxx(f) is an element of the transfer matrix H(f)=A-1(f), and * denotes matrix transposition & complex conjugation. X(f)=H(f)E(f); X(f)=A-1(f), where the first p component of A is: a jk (f ) = 1- åa jk e-2p ikf k=1 Geweke showed that: Fy x 1 1 ln 2 2 I f df yx SPECTRAL GRANGER CAUSALITY OF MACAQUE LFPS Coefficient matrices are obtained by solving the multivariate YuleWalker equations (of size mp2), using the Levinson, Wiggens, Robinson algorithm, as implemented by Morf et al. (1978). Repeated trials are treated as realizations of a stationary stochastic process. The model order is determined by parametric testing. SPECTRAL GRANGER CAUSALITY OF MACAQUE LFPS Spectral GC is bidirectional in the frequency domain. It shows predictability in 2 directions as a function of frequency. Coherence between PAR1 and PAR2 GC from PAR1 to PAR2 (black) GC from PAR2 to PAR1 (orange) Model order = 15 SENSORIMOTOR GC IN MOTOR CONTROL [BROVELLI ET AL, PNAS, 2004] GC pattern consistent with: • Anatomical connectivity • MacKay functional loop • Clinical deafferentation evidence EXTRASTRIATE-V1 GC IN VISUAL EXPECTATION [BRESSLER ET AL, STAT MED, 2007] GC pattern consistent with: • Descending anatomical connectivity • Visual anticipation BIVARIATE VERTEXWISE SPECTRAL GC FROM MEG[MAHALINGAM ET AL, CNS, 2008] WORKING MEMORY GC IN FRONTOPARIETAL SYSTEM[SALAZAR ET AL, SCIENCE, 2012] GC spectra from PPC to PFC (red) and from PFC to PPC (blue) during late delay period Directional GC difference (solid). Significance level from surrogate difference distribution (dashed) PPCPFC GC > PFCPPC GC CONCLUSIONS Granger Causality is a useful tool to investigate influences between neural groups. It can be applied to multivariate time series of different neural signal types. When applied to fMRI BOLD data, it can be useful to non-invasively study inter-regional interactions in humans under normal physiological conditions. When applied to LFP data, it can be useful to study inter-regional interactions with high spatial, temporal, and frequency precision.