A Tucker Decomposition Based Approach for Topographic Functional - PowerPoint PPT Presentation

A Tucker Decomposition Based Approach for Topographic Functional Connectivity State Summarization Arash Mahyari, Selin Aviyente Department of Electrical and Computer Engineering Michigan State University Dec 2015 1 / 24

Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 2 / 24

Introduction Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 3 / 24

Introduction Introduction Higher brain functions depend on the balance between local specialization (functional segregation) and global integration (functional integration) of brain processes (Friston, 2011; Friston, 2001; Le Van Quyen, 2003; Stam, 2005; Tononi et al., 1998). Imaging neuroscience (EEG, MEG, fMRI) has firmly established functional segregation as a principle of brain organization in humans. The integration of segregated areas has proven more difficult to assess. Therefore, there is a need to identify task-related interactions between neuronal populations. Dec 2015 4 / 24

Introduction Functional Connectivity Cognitive control processes are responsible for goal or context representation and maintenance, attention allocation and stimulus-response mapping. In particular, for cognitive control: ▸ Medial prefrontal cortex (mPFC) and lateral prefrontal cortex (lPFC) play an important role. ▸ Synchronization connects anterior cingulate cortex (ACC) and lPFC (Womelsdorf et al. 2014, Current Biology). Impaired cognitive control plays a role in schizophrenia, impulse control and anxiety disorders. Dec 2015 5 / 24

Introduction Dynamic Functional Connectivity Networks Functional connectivity networks transition through quasi-stationary microstates over time (Lehmann et al. 1997). Current Approaches to network state representations: ▸ Sliding window FC analysis (Chang and Glover, 2010) ▸ k-means clustering (Allen et al. 2012) ▸ Principal Component Analysis (Leonardi et al. 2013) Shortcomings: The intrinsic network structure is not preserved: Averaging, Vectorizing. Our solution: Tensors are used to represent and summarize functional connectivity networks. Dec 2015 6 / 24

Time-Frequency Phase Synchrony Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 7 / 24

Time-Frequency Phase Synchrony Functional Connectivity: Phase Synchrony Reduced-interference Rihaczek distribution (RID-Rihaczek): C i ( t , ω ) = ∫ ∫ exp (−( θτ ) 2 ) exp ( j θτ 2 ) A i ( θ, τ ) e − j ( θ t + τω ) d τ d θ. (1) σ �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� Rihaczek kernel Choi-Williams kernel ▸ Ambiguity function: A i ( θ,τ ) = ∫ s i ( u + τ 2 ) s ∗ i ( u − τ 2 ) e j θ u du . The phase distribution: Φ i ( t ,ω ) = arg [ C i ( t ,ω ) ∣ C i ( t ,ω )∣ ] . The phase difference between the two signals can be defined as: ( i , j ) ( t ,ω ) = ∣ Φ k i ( t ,ω ) − Φ k j ( t ,ω )∣ . Φ k Phase locking value (PLV) quantifies the functional integration, as: L PLV ( i , j ) ( t ,ω ) = 1 L ∣ exp ( j Φ k ( i , j ) ( t ,ω ))∣ , 0 ≤ PLV ≤ 1 . ∑ (2) k = 1 Dec 2015 8 / 24

Time-Frequency Phase Synchrony Construction of d-FCNs Functional connectivity matrix: G s , ( i , j ) ( t ) = 1 ∑ PLV s , ( i , j ) ( t ,ω ) , ω b (3) Ω ω = ω a ▸ G ( i , j ) ( t ) ∈ [ 0 , 1 ] , [ ω a ,ω b ] : frequency band of interest, Ω : the number of frequency bins, s: the subject. Dec 2015 9 / 24

Tensor Subspace Analysis Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 10 / 24

Tensor Subspace Analysis Overview of tensors The extension of vectors and matrices to higher dimension is called multiway array, or tensor. X ∈ R m 1 × m 2 × ... × m d is a d -way tensor, where x i 1 , i 2 , i 3 ,..., i d is its ( i 1 , i 2 , i 3 ,..., i d ) th element. Collection of the FC matrices of all subjects, G s ( t ) , forms G( t ) ∈ R N × N × S . Dec 2015 11 / 24

Tensor Subspace Analysis Tucker Decomposition Tucker Decomposition is flexible in representing higher order data, and has orthogonal component matrices. Tucker decomposition is calculated using alternative least square (ALS) method. Tucker decomposition of X ∈ R m 1 × m 2 × ... × m d : X = C × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) ... × d U ( d ) + E , X = ∑ i 1 , i 2 , i 3 ,..., i d C i 1 , i 2 , i 3 ,..., i d ( u ( 1 ) i d ) + E i 1 , i 2 , i 3 ,..., i d , ○ u ( 2 ) ○ u ( 3 ) ○ ... ○ u ( d ) i 1 i 2 i 3 (4) ▸ C ∈ R r 1 × r 2 × ... × r d is the core tensor. ▸ U ( 1 ) ∈ R m 1 × r 1 , U ( 2 ) ∈ R m 2 × r 2 , . . . U ( d ) ∈ R m d × r d . ▸ E ∈ R m 1 × m 2 × ... × m d is the residual. Dec 2015 12 / 24

Tensor Subspace Analysis Tucker Decomposition continued Figure: Tucker decomposition for a 3-way tensor. n–mode product n–mode product is multiplying the tensor unfolded along the n th mode by a matrix. X × n U = U † X ( n ) = ∑ x i 1 , i 2 ,..., i n ,..., i d U j n , i n (5) i n Dec 2015 13 / 24

State Representation Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 14 / 24

State Representation Overview time a) Original nodes . . . . . . connections and event intervals U2 U2 U2 U2 . . . . . . b) Tucker decomposition U3 U3 U3 U3 Core Core Core Core U1 U1 U1 U1 c) Consolidation: . . . . . . ζ ζ ζ ζ define ζ d) Take first . . . slice of ζ . . . d) Collect e) Repeat Tucker f) Summarize slices in an decomposition network using Θ first slice of Θ event interval and consolidation and significant testing Figure: Functional connectivity state summarization algorithm flowchart. Dec 2015 15 / 24

State Representation Subject Summarization Subject Summarization G( t ) ∈ R N × N × S within the time interval t = 1 , 2 ,..., T is fully decomposed using Tucker decomposition: G( t ) = C( t ) × 1 U ( 1 ) ( t ) × 2 U ( 2 ) ( t ) × 3 U ( 3 ) ( t ) . (6) Let’s define: ζ ( t ) = C( t ) × 1 U ( 1 ) ( t ) × 2 U ( 2 ) ( t ) → G( t ) = ζ ( t ) × 3 U ( 3 ) ( t ) . The subtensor θ ( t ) ∈ R N × N captures most of the energy of the activation patterns across subjects at time: S θ ( t ) = ζ i 3 = 1 ( t ) = s , 1 ( t ) G s ( t ) . U ( 3 ) ∑ (7) s = 1 Dec 2015 16 / 24

State Representation Time Summarization Time Summarization θ ( t ) , ∀ t ∈ { 1 , 2 , ⋯ , T } are summarized across time mode to derive the state connectome. The 3-way tensor Θ ∈ R N × N × T is constructed from θ ( t ) , and fully decomposed using Tucker decomposition: U ( 3 ) = ¯ Θ = ϑ × 1 ¯ U ( 1 ) × 2 ¯ U ( 2 ) × 3 ¯ ζ × 3 ¯ U ( 3 ) . (8) The subtensor η = ¯ ζ i 3 = 1 = ∑ T U ( 3 ) t = 1 ¯ t , 1 Θ i 3 = t captures the largest amount of energy across all time steps. Dec 2015 17 / 24

State Representation Time Summarization Significance Testing The significant edges of η is determined through hypothesis testing. A Gaussian distribution for the edge values in η is assumed. This assumption can be validated using Kolmogorov–Smirnov test. z-test is used on the edges of η to Figure: The histogram of the determine the most significant edges. projected tensor edge values η ( i , j ) ∼ N erp ( µ erp ,σ erp ) in the matrix η for ERN. H 0 ∶ η ( i , j ) ∼ N 1 ( µ 1 ≠ µ erp ,σ 1 ≠ σ erp ) H 1 ∶ Dec 2015 18 / 24

Experimental Results Outline Introduction 1 Time-Frequency Phase Synchrony 2 Tensor Subspace Analysis 3 State Representation 4 Subject Summarization Time Summarization Experimental Results 5 Conclusion and Future Work 6 Dec 2015 19 / 24

Experimental Results EEG Data Error-Related Negativity (ERN) occurs 50-100ms after subjects made errors in response to a speeded motor task. Modified Eriksen flanker task for 2 seconds with multiple trials (10-40 error trials per subject). 91 subjects, 63 electrodes collected from undergraduates at the University of Minnesota. Sampling rate: 128 Hz. ERN is dominated by partial phase-locking of intermittent theta band (3-7 Hz) EEG activity between mPFC and lPFC (Cavanagh et al., 2009). Dec 2015 20 / 24

Experimental Results Experimental Results Figure: The most significant edges of the network summarization matrix, η with p = 0 . 95 for: (a) ERN, (b) CRN. Dec 2015 21 / 24