Weighted Network Analysis for Groups:

Transcription

1 Weighted Network Analysis for Groups: Separating Differences in Cost from Differences in Topology Cedric E. Ginestet Department of Neuroimaging, King s College London Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

2 Connectivity Data Subject-specific Correlation Matrices For the i th subject in the j th condition: R ij. AAL Cortical Regions AAL Cortical Regions Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

3 Connectivity Data Experimental Paradigm J conditions (columns), and n subjects (rows). R 11 R 12 R 1J R n1 R nj Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

4 Part I N-back Task on Working Memory Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

5 N-back Paradigm Figure: N-back task. There are here four levels of difficulties from 0-back to 3-back. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

6 Experimental Paradigm Ginestet et al., Neuroimage, i. 43 (incl. 21 females) healthy controls. ii. Mean age of years (sd = 13.17). iii. 12 randomised blocks lasting each 31 seconds. iv. 186 T2 -weighted EPI volumes on 1.5T scanner. v. TE=40ms, TR=2s, flip angle 90. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

7 Experimental Paradigm Ginestet et al., Neuroimage, i. 43 (incl. 21 females) healthy controls. ii. Mean age of years (sd = 13.17). iii. 12 randomised blocks lasting each 31 seconds. iv. 186 T2 -weighted EPI volumes on 1.5T scanner. v. TE=40ms, TR=2s, flip angle 90. Subject-specific Weighted Networks i. Anatomical Automatic Labeling (AAL) Parcellation. ii. Regional Mean time series. iii. Maximal Overlap Discrete Wavelet Transform (MODWT). iv. Scale 4 Wavelet Coefficient: ( Hz interval). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

8 Wavelet Decomposition + Concatenation W Concatenated Volumes W Concatenated Volumes Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

9 Wavelet Decomposition + Concatenation Concatenation Only Concatenation Only Concatenation Only Density Density Density Differences in Correlations (0 back to 1 back) Differences in Correlations (0 back to 2 back) Differences in Correlations (0 back to 3 back) Wavelet Concatenated Wavelet Concatenated Wavelet Concatenated Density Density Density Differences in Correlations (0 back to 1 back) Differences in Correlations (0 back to 2 back) Differences in Correlations (0 back to 3 back) Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

10 Differences in Cost/Density Main Effect of N-back Experimental Factor? 0-back 1-back 2-back 3-back Figure: Heatmaps corresponding to subject-specific correlation matrices for the four N-back conditions. (Ginestet et al., Neuroimage, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

11 Part II Statistical Parametric Networks (SPNs) Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

12 Statistical Parametric Networks (SPNs) R 11 R 12 R 1J R n1 R nj Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

13 Mass-univariate Approaches to Network Inference Previous Approaches i. Achard et al. (Jal of Neuroscience, 2006). ii. He et al. (PLoS one, 2009). iii. Kramer et al. (Phys. Rev. E., 2009). Method i. Z-test on Fisher-transformed correlation coefficients. ii. Parametric/Non-parametric significance testing. iii. Control for multiple comparison (False Discovery Rate). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

14 Cost/density Decreases with Cognitive Load Sagittal SPN j 0-back 1-back 2-back 3-back Figure: Mean Statistical Parametric Networks (SPN j ), based on wavelet coefficients in the Hz frequency band. The locations of the nodes correspond to the stereotaxic centroids of the cortical regions (Ginestet et al., Neuroimage, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

15 Task-related Physiological Variability Sagittal SPN j 0-back 1-back 2-back 3-back i. Could N-back connectivity differences be solely explained by task-correlated physiological variability, such as breathing? ii. As breathing accelerates with task difficulty, its frequency 0.03Hz. iii. See Birn et al. (HBM, 2008), and Birn et al. (Neuroimage, 2009). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

16 Connectivity Strength Predicts Task Performance (a) Penalized RT (b) Weighted Cost prt(ms) K(G) back 1 back 2 back 3 back 0 back 1 back 2 back 3 back Figure: Boxplots of (a) penalized reaction time and (b) weighted cost. Regression of prt on subject-specific weighted cost (K W (G ij ) for the i th subject under the j th condition) after controlling for the N-back factor was found to be significant (p <.001) (Ginestet et al., Neuroimage, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

17 Differential SPNs R 11 R 12 R 1J F-test for all e E(G), v V (G): r e i = X e i βe + Z e i be i + ɛ e i ; y v i = X v i βv + Z v i bv i + ɛ v i. R n1 R nj Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

18 Differential SPNs L R Figure: Differential SPN. Sagittal section of the negative differential SPN, which represents the significantly lost edges, due to the N-back experimental factor. The presence of an edge is determined by the thresholding of p-values at.01, uncorrected (Ginestet et al., Neuroimage, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

19 Part III Differences in Topology vs. Differences in Density Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

20 Differences in Topology vs. Differences in Density Regular Random Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

21 Differences in Topology vs. Differences in Density Regular Random Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

22 Classical Measures of Topology Efficiencies (Latora et al., 2001) For any unweighted graph G = (V, E), connected or disconnected, E(G) := 1 N V (N V 1) N V N V i=1 j i d 1 ij, (1) where d ij is the length of the shortest path between vertices i and j in G. Global and Local Efficiencies E Glo (G) := E(G), and E Loc (G) := 1 N V N V i=1 E(G i ), (2) where G i is the subgraph of G that includes all the neighbors of the i th node. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

23 Efficiencies are Monotonic Increasing with Density (a) Global Efficiency (b) Lobal Efficiency E (Glo) back 1 back 2 back 3 back E (Loc) back 1 back 2 back 3 back Cost Cost Figure: Efficiencies under the four conditions of the N-back task, with density-equivalent random (red) and regular (blue) networks, for each condition. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

24 Integrating over Densities Cost-integrated Topological Metrics Given a weighted graph G = (V, E, W) and a topological metric T ( ), T p (G) := k Ω K T (γ(g, k))p(k), (3) where γ(g, k) thresholds G and returns an unweighted graph with density/cost k. Treating Cost/Density as a Random Variable Here, the number of edges in G, denoted k, is given distribution p(k), defined over { ( )} NV Ω k := 1,...,, (4) 2 with N V := V(G). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

25 Prior Distribution over Graph Densities Beta Binomial Distribution p(k=k) 0e+00 2e 04 4e 04 6e n=ne a=b=1 a=b=2 a=b=3 a=b=4 a=b=5 Ne Figure: Symmetric versions of the Beta-binomial distribution for different choices of parameters, with N E = 4005 (Ginestet et al., PLoS one, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

26 Integrating over Cost/Density Proposition (Ginestet et al., PLoS one, 2011) Let a weighted undirected graph G = (V, E, W). For any monotonic function h( ) acting elementwise on a real-valued matrix, W, corresponding to the weight set W, and any topological metric T, the cost-integrated version of that metric, denoted T p, satisfies T p (W) = T p (h(w)). (5) Proof. Since h( ) is applied elementwise to W, we have R ij (h(w)) = 1 2 N V N V I{h(w ij ) h(w uv )} = R ij (W). (6) u=1 v u Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

27 Topological Differences does not Predict Performance (a) Global Efficiency (b) Local Efficiency E (Glo) E (Loc) back 1 back 2 back 3 back 0 back 1 back 2 back 3 back Figure: Boxplots of subject-specific cost-integrated global and local efficiencies in panels (a) and (b), respectively, where G ij denotes the functional network for the i th subject in the j th condition (Ginestet et al., Neuroimage, 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

28 Part IV Weighted Metrics for Weighted Networks? Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

29 Weighted Topological Metrics Weighted Global Efficiency As introduced by Latora et al. (2001), E W (G) := 1 N V (N V 1) where G is a weighted graph, G = (V, E, W). N V N V i=1 j i 1 dij W. (7) Weighted Shortest Path The weighted shortest path d W ij is defined as d W ij := min P ij P ij (G) w uv W(P ij ) w 1 uv, (8) Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

30 Integrating over Cutoff Proposition (Ginestet et al., PLoS one, 2011) For any weighted graph G = (V, E, W), whose weight set is denoted by W(G), if we have min w ij 1 max w ij W(G) 2 w ij, (9) w ij W(G) then E W (G) = K W (G). (10) Proof. Assume that dij W w 1 ij for at least one edge (i, j), and then show that this contradicts the hypothesis. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

31 Modularity & Edge Density A Number of Modules B Number of Modules Networks Random Regular Random Rewirings Number of Edges Figure: Topological randomness and number of edges predict number of modules. (A) Relationship between the number of random rewirings of a regular lattice and the number of modules in such a network. (B) Relationship between the number of edges in a network and its number of modules (Bassett et al., PNAS 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

32 Modularity & Edge Density C N E = 100 N E = 600 N E = 1100 N E = 1600 N E = 2100 D N E = 100 N E = 600 N E = 1100 N E = 1600 N E = 2100 Figure: Topological randomness and number of edges predict number of modules. Modular structures of regular (C) and random (D) networks for different number of edges, N E (Bassett et al., PNAS 2011). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

33 Part V Some Conclusions. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

34 Summary Main Messages 1 Thresholding: Discrete mathematics on Continuous (Real-valued) data. 2 What matters when comparing weighted networks: i. Weighted cost/density (e.g. mean correlation). ii. Cost-integrated topological metrics. iii. Problem does not vanish with weighted metrics. 3 Cost-integration approximated using Monte Carlo sampling scheme. 4 R package for cost-integration: NetworkAnalysis on CRAN. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

35 Summary Main Messages 1 Thresholding: Discrete mathematics on Continuous (Real-valued) data. 2 What matters when comparing weighted networks: i. Weighted cost/density (e.g. mean correlation). ii. Cost-integrated topological metrics. iii. Problem does not vanish with weighted metrics. 3 Cost-integration approximated using Monte Carlo sampling scheme. 4 R package for cost-integration: NetworkAnalysis on CRAN. Future Work 1 Replicate these findings in other MRI cognitive tasks. 2 Weighted network analysis in neuropharmacological studies. Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

36 Activity vs. Connectivity Sepulcre et al. (PLoS CB, 2010). Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35

37 Collaborators & Funding Agencies Collaborators 1 Andy Simmons, Mick Brammer, Andre Marquand, Vincent Giampietro, Orla Doyle, Jonny O Muircheartaigh, Owen G. O Daly (King s College London) 2 Arnaud Fournel (Lyon, France) 3 Ed Bullmore (Cambridge, UK) 4 Tom Nichols (Warwick, UK) 5 Randy Buckner (Harvard, MA) 6 Dani Bassett (UCLA, CA) Funding Agencies Cedric E. Ginestet (KCL) Weighted Network Analysis 24th January / 35