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Functional Brain Networks - Javier M. Buldù

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COMPLEX NETWORKS: THEORY, METHODS, AND APPLICATIONS (2ND EDITION)
May 16-20, 2016

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Functional Brain Networks - Javier M. Buldù

  1. 1. FUNCTIONAL BRAIN NETWORKS   JAVIER M. BULDÚ UNIVERSIDAD REY JUAN CARLOS (MÓSTOLES) CENTRO DE TECNOLOGÍA BIOMÉDICA (POZUELO) COMPLEJIMAD (MADRID) (… A MINEFIELD!)
  2. 2. OUTLINE ❑ Functional Brain Networks ❑ Measuring Brain Activity ❑Time Series & Network Construction ❑ Network Analysis ❑ Risks & Challenges ❑ Functional Networks are alive ❑ Living in Hillsville ❑ Conclusions ❑ Brain Networks ❑ Anatomical Networks ❑ Functional Networks
  3. 3. (I-III) Brain Networks
  4. 4. COMPLEX SYSTEMS A complex system is composed of interrelated parts which, as a whole, exhibit properties and behaviors that can not be explained analyzing each of the individual parts separately: A neuron A brain
  5. 5. What if we apply network science to the most challenging system we are facing? APPLYING NETWORK SCIENCE TO THE BRAIN
  6. 6. APPLYING NETWORK SCIENCE TO THE BRAIN In brief, (main) types of brain networks From Bullmore & Sporns, Nature Rev. 10, 186 (2009) anatomical functional
  7. 7. M. Rubinov and O. Sporns, NeuroImage 52, 1059–1069 (2010) Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.neuroimage.2009.10.003. References Achard, S., Bullmore, E., 2007. Efficiency and cost of economical brain functional networks. PLoS Comput. Biol. 3, e17. Achard, S., Salvador, R., Whitcher, B., Suckling, J., Bullmore, E., 2006. A resilient, low- frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26, 63–72. Alstott, J., Breakspear, M., Hagmann, P., Cammoun, L., Sporns, O., 2009. Modeling the impact of lesions in the human brain. PLoS Comput. Biol. 5, e1000408. Barabasi, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science 286, 509–512. Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. U. S. A. 101, 3747–3752. Bassett, D.S., Bullmore, E., 2006. Small-world brain networks. Neuroscientist 12, 512–523. Bassett, D.S., Bullmore, E.T., 2009. Human brain networks in health and disease. Curr. Opin. Neurol. 22, 340–347. Bassett, D.S., Meyer-Lindenberg, A., Achard, S., Duke, T., Bullmore, E., 2006. Adaptive reconfiguration of fractal small-world human brain functional networks. Proc. Natl. Acad. Sci. U. S. A. 103, 19518–19523. Bassett, D.S., Bullmore, E., Verchinski, B.A.,Mattay,V.S.,Weinberger, D.R., Meyer-Lindenberg, A., 2008. Hierarchical organization of human cortical networks in health and schizophrenia. J. Neurosci. 28, 9239–9248. Batagelj, V.,M M., Mut Blondel, V.D commun Boccaletti, S Structur Brandes, U., 163–177 Bullmore, E. structur Butts, C.T., 2 Costa, L.d.F. cortical 1, 16. Costa, L.D.F. complex Damoiseaux studies ity. Brain Danon, L., D identific Deuker, L., B D.S., 200 NeuroIm Estrada, E., H Nonline Fagiolo, G., 2 Soft Mat Freeman, L.C 215–239 Table A1 (continued) Measure Binary and undirected definitions W Measures of resilience Degree distribution Cumulative degree distribution of the network (Barabasi and Albert, 1999), P kð Þ = X kVzk p kVð Þ; where p(k′) is the probability of a node having degree k′. C C C Average neighbor degree Average degree of neighbors of node i (Pastor-Satorras et al., 2001), knn;i = P jaN aijkj ki : A B k A k Assortativity coefficient Assortativity coefficient of the network (Newman, 2002), r = l −1 P i;jð ÞaL kikj − l − 1 P i;jð ÞaL 1 2 ki + kj h i2 l−1 P i;jð ÞaL 1 2 k2 i + k2 j − l−1 P i;jð ÞaL 1 2 ki + kj h i2 : W L r D r Other concepts Degree distribution preserving network randomization. Degree-distribution preserving randomization is implemented by iteratively choosing four distinct nodes i1, j1, i2, j2 ∈ N at random, such that links (i1, j1), (i2, j2) ∈ L, while links (i1, j2), (i2, j1) ∉ L. The links are then rewired such that (i1, j2), (i2, j1) ∈ L and (i1, j1), (i2, j2) ∉ L, (Maslov and Sneppen, 2002). “Latticization” (a lattice-like topology) results if an additional constraint is imposed, |i1+j2| + |i2+j1| b |i1+j1| + |i2+j2| (Sporns and Kotter, 2004). T In li p to sc Measure `of network small-worldness. Network small-worldness (Humphries and Gurney, 2008), S = C = Crand L = Lrand ; where C and Crand are the clustering coefficients, and L and Lrand are the characteristic path lengths of the respective tested network and a random network. Small-world networks often have S ≫ 1. W D In All binary and undirected measures are accompanied by their weighted and directed generalizations. G knowledge) are marked with an asterisk (⁎). The Brain Connectivity Toolbox contains Matlab functions Table A1 (continued) Measure Binary and undirected definitions Weighted and directed definitions Modularity Modularity of the network (Newman, 2004b), Q = X uaM euu − X vaM euv !2 # ; where the network is fully subdivided into a set of nonoverlapping modules M, and euv is the proportion of all links that connect nodes in module u with nodes in module v. An equivalent alternative formulation of the modularity (Newman, 2006) is given by Q = 1 l P i;jaN aij − ki kj l δmi ;mj , where mi is the module containing node i, and δmi,mj = 1 if mi = mj, and 0 otherwise. Weighted modularity (Newman, 2004), Qw = 1 lw P i;jaN wij − k w i k w j l w ! δmi ;mj : Directed modularity (Leicht and Newman, 2008), QY = 1 l P i;jaN aij − k out i k in i l ! δmi ;mj : Measures of centrality Closeness centrality Closeness centrality of node i (e.g. Freeman, 1978), L −1 i = n − 1 P jaN;j≠i dij : Weighted closeness centrality, Lw i À Á− 1 = n − 1P jaN; j≠i d w ij . Directed closeness centrality, LY i À Á−1 = n − 1P jaN; j≠i d Y ij . Betweenness centrality Betweenness centrality of node i (e.g., Freeman, 1978), bi = 1 n − 1ð Þ n − 2ð Þ P h;jaN h≠j;h≠i;j≠i; ρ hj ið Þ ρ hj ; where ρhj is the number of shortest paths between h and j, and ρhj (i) is the number of shortest paths between h and j that pass through i. Betweenness centrality is computed equivalently on weighted and directed networks, provided that path lengths are computed on respective weighted or directed paths. Within-module degree z-score Within-module degree z-score of node i (Guimera and Amaral, 2005), zi = ki mið Þ − k mið Þ σk mið Þ ; where mi is the module containing node i, ki (mi) is the within-module degree of i (the number of links between i and all other nodes in mi), and k mið Þ and σk(mi) are the respective mean and standard deviation of the within-module mi degree distribution. Weighted within-module degree z-score, zw i = k w i mið Þ − kw mið Þ σkw mið Þ . Within-module out-degree z-score, zout i = k out i mið Þ − kout mið Þ σ kout mið Þ . Within-module in-degree z-score, zin i = k in i mið Þ − kin mið Þ σ kin mið Þ . Participation coefficient Participation coefficient of node i (Guimera and Amaral, 2005), yi = 1 − X maM ki mð Þ ki 2 ; where M is the set of modules (see modularity), and ki (m) is the number of links between i and all nodes in module m. Weighted participation coefficient, yw i = 1 − P maM kw i mð Þ kw i 2 . Out-degree participation coefficient, yout i = 1 − P maM kout i mð Þ kout i 2 . In-degree participation coefficient, yin i = 1 − P maM kin i ðmÞ kin i 2 . Network motifs Anatomical and functional motifs Jh is the number of occurrences of motif h in all subsets of the network (subnetworks). h is an nh node, lh link, directed connected pattern. h will occur as an anatomical motif in an nh node, lh link subnetwork, if links in the subnetwork match links in h (Milo et al., 2002). h will occur (possibly more than once) as a functional motif in an nh node, lh′ ≥ lh link subnetwork, if at least one combination of lh links in the subnetwork matches links in h (Sporns and Kotter, 2004). (Weighted) intensity of h (Onnela et al., 2005), Ih = P u Π i;jð ÞaLhu wij 1 lh ; where the sum is over all occurrences of h in the network, and L hu is the set of links in the uth occurrence of h. Note that motifs are directed by definition. Motif z-score z-Score of motif h (Milo et al., 2002), zh = Jh − h Jrand;hi σ Jrand;h ; where 〈Jrand,h〉 and σ Jrand,h are the respective mean and standard deviation for the number of occurrences of h in an ensemble of random networks. Intensity z-score of motif h (Onnela et al., 2005), zI h = Ih − hIrand;h i σ Irand;h ; where 〈Irand,h〉 and σ Irand,h are the respective mean and standard deviation for the intensity of h in an ensemble of random networks. Motif fingerprint nh node motif fingerprint of the network (Sporns and Kotter, 2004), Fnh hVð Þ = X iaN Fnh;i hVð Þ = X iaN JhV;i; where h′ is any nh node motif, Fnh,i (h′) is the nh node motif fingerprint for node i, and Jh′,i is the number of occurrences of motif h′ around node i. nh node motif intensity fingerprint of the network, FI nh hVð Þ = P iaNFI nh;i hV À Á = P iaN IhV; i, where h′ is any nh node motif, FI nh,i (h′) is the nh node motif intensity fingerprint for node i, and Ih′,i is the intensity of motif h′ around node i. (continued on next page) Mathematical definitions of complex network measures (see supplementary information for a self-contained version of this table). Measure Binary and undirected definitions Weighted and directed definitions Basic concepts and measures Basic concepts and notation N is the set of all nodes in the network, and n is the number of nodes. L is the set of all links in the network, and l is number of links. (i, j) is a link between nodes i and j, (i, j ∈ N). aij is the connection status between i and j: aij = 1 when link (i, j) exists (when i and j are neighbors); aij = 0 otherwise (aii = 0 for all i). We compute the number of links as l = ∑i,j∈N aij (to avoid ambiguity with directed links we count each undirected link twice, as aij and as aji). Links (i, j) are associated with connection weights wij. Henceforth, we assume that weights are normalized, such that 0 ≤ wij ≤ 1 for all i and j. lw is the sum of all weights in the network, computed as lw = ∑i,j∈N wij. Directed links (i, j) are ordered from i to j. Consequently, in directed networks aij does not necessarily equal aji. Degree: number of links connected to a node Degree of a node i, ki = X jaN aij: Weighted degree of i, ki w = ∑j∈Nwij. (Directed) out-degree of i, ki out = ∑j∈Naij. (Directed) in-degree of i, ki in = ∑j∈Naji. Shortest path length: a basis for measuring integration Shortest path length (distance), between nodes i and j, dij = X auv agi X j auv; where gi↔j is the shortest path (geodesic) between i and j. Note that dij = ∞ for all disconnected pairs i, j. Shortest weighted path length between i and j, dij w = ∑auv∈gi↔j w f(wuv), where f is a map (e.g., an inverse) from weight to length and gi↔j w is the shortest weighted path between i and j. Shortest directed path length from i to j, dij → = ∑aij∈gi→j aij, where gi→j is the directed shortest path from i to j. Number of triangles: a basis for measuring segregation Number of triangles around a node i, ti = 1 2 X j;haN aijaihajh: (Weighted) geometric mean of triangles around i, tw i = 1 2 P j;haN wijwihwjh À Á1=3 : Number of directed triangles around i, tY i = 1 2 P j;haN aij + aji À Á aih + ahið Þ ajh + ahj À Á . Measures of integration Characteristic path length Characteristic path length of the network (e.g., Watts and Strogatz, 1998), L = 1 n X iaN Li = 1 n X iaN P jaN;j≠i dij n − 1 ; where Li is the average distance between node i and all other nodes. Weighted characteristic path length, Lw = 1 n P iaN P jaN; j≠i d w ij n − 1 . Directed characteristic path length, LY = 1 n P iaN P jaN; j≠i d Y ij n − 1 . Global efficiency Global efficiency of the network (Latora and Marchiori, 2001), E = 1 n X iaN Ei = 1 n X iaN P jaN;j≠i d −1 ij n − 1 ; where Ei is the efficiency of node i. Weighted global efficiency, Ew = 1 n P iaN P jaN; j≠i dw ij − 1 n − 1 . Directed global efficiency, EY = 1 n P iaN P jaN; j≠i dY ij − 1 n − 1 . Measures of segregation Clustering coefficient Clustering coefficient of the network (Watts and Strogatz, 1998), C = 1 n X iaN Ci = 1 n X iaN 2ti ki ki − 1ð Þ ; where Ci is the clustering coefficient of node i (Ci = 0 for ki b 2). Weighted clustering coefficient (Onnela et al., 2005), Cw = 1 n P iaN 2tw i ki ki − 1ð Þ . See Saramaki et al. (2007) for other variants. Directed clustering coefficient (Fagiolo, 2007), CY = 1 n P iaN tY i kout i + kin ið Þ kout i + kin i − 1ð Þ − 2 P jaN aijaji : Transitivity Transitivity of the network (e.g., Newman, 2003), T = P iaN 2ti P iaN ki ki − 1ð Þ : Note that transitivity is not defined for individual nodes. Weighted transitivity⁎, Tw = P iaN 2tw iP iaN ki ki − 1ð Þ . Directed transitivity⁎, TY = P iaN tY i P iaN k out i + k in i À Á k out i + k in i − 1 À Á − v2 P jaN aijaji h i : Local efficiency Local efficiency of the network (Latora and Marchiori, 2001), Eloc = 1 n X iaN Eloc;i = 1 n X iaN P j;haN;j≠i aijaih djh Nið Þ h i−1 ki ki − 1ð Þ ; where Eloc,i is the local efficiency of node i, and djh (Ni) is the length of the shortest path between j and h, that contains only neighbors of i. Weighted local efficiency⁎, Ew loc = 1 2 P iaN P j;haN; j≠i wijwih d w jh Nið Þ½ Š − 1 1 = 3 ki ki − 1ð Þ : Directed local efficiency⁎, EY loc = 1 2n P iaN P j;haN;j≠i aij + ajið Þ aih + ahið Þ d Y jh Nið Þ Â Ã− 1 + d Y hj Nið Þ Â Ã− 1 k out i + k in i À Á k out i + k in i − 1 À Á − 2 P jaN aijaji : … and many more!!
  8. 8. ANATOMICAL BRAIN NETWORKS The connectome is a comprehensive map of neural connections in the brain.The production and study of connectomes, known as connectomics, may range in scale from a detailed map of the full set of neurons and synapses of an organism to a macro scale description of the structural connectivity between all cortical areas and subcortical structures.
  9. 9. Do we have a complete connectome?Yes, we do! C. Elegans: connectivity matrix. (O. Sporns,“The Networks of the Brain”) • C. Elegans, a nematode. • 302 neurons (hermaphrodite), 383 neurones (male). • We now all neurons and connections between them. THE CONNECTOME: A NECESSARY SUBSTRATE
  10. 10. The wiring diagram is an starting point for making hypothesis but… A) … it cannot reveal how neurons behave in real time, nor does it account for the many “mysterious” ways that neurons regulate one another's behavior. B) … the dynamics is quite unpredictable when studying complex movements. C) … we don't have a comprehensive model of how the worm's nervous system actually produces the behaviors. D) … the strength of the synaptic connections changes due to dynamics, also the amount of neurotransmitters… THE CONNECTOME: A NECESSARY SUBSTRATE
  11. 11. Overall structure of PVX, a male-specific interneuron, showing distribution of synapses. (G) Detail of individual synapses. Width of lines indicates synapse size. Edge weights are determined by counting the number of 70- to 90-nm serial sections crossed by individual synapses and summing over all the synapses. onJuly26,2012www.sciencemag.orgDownloadedfrom tween each pair of cells (16). (The resulting struc- tural weight adjacency matrices for the chemical S8 and S9). Individual presynaptic densities varied in size over a 40-fold range, whereas 30-fold range (fig. S2). As a result of the vari- ation in both number of synapses between pairs onJuly26,2012www.sciencemag.orgDownloadedfrom Fig. 1. Specializations of the C. elegans adult male tail for mating. (A) The substeps of mating. (B) Ventral view of the adult male tail showing mating structures with five types of sensilla. (C) Overall structure of the male ner- vous system. (D) Ganglia in the tail containing the neuron cell bodies, con- nected through commissures. Most synaptic connectivity occurs in the preanal ganglion (PAG). DNC, dorsal nerve cord; VNC, ventral nerve cord; DRG, dorsorectal ganglion; LG, lumbar ganglion (left and right); CG, cloacal ganglion (left and right). (E) An example of a male-specific interneuron, PVX, which has a cell body and extensive sensory input in the PAG, and a pr an (d ju ch si po (d m 27 JULY 2012 VOL 337 SCIEN438 each module correlate well with experimental evidence (21–26). Sensory neurons are recurrently connected. Whereas much of the information flow through the network from sensory neurons to end organs— either in monosynaptic pathways or through type Ia interneurons in disynaptic pathways— is feedforward, the 52 sensory neurons are ex- tensively reciprocally and recurrently connected by both chemical and gap junction synapses. Forty-nine percent of the chemical synaptic out- put of sensory neurons is onto other sensory neurons, and this constitutes input to th neurons that is seven times the feedb type Ib and type Ic interneurons. Ninet the 36 ray sensory neurons make auta stituting 6.9% of their input from sen rons. Fifty-eight percent of the gap connectivity of the sensory neurons is sensory neurons. Only on the basis of the recurren tivity of the sensory neurons and the tions to the type Ib interneurons, the n sensory neurons could be partitioned DOI: 10.1126/science.1221762 , 437 (2012);337Science et al.Travis A. Jarrell The Connectome of a Decision-Making Neural Network This copy is for your personal, non-commercial use only. clicking here.ues, clients, or customers by , you can order high-quality copies for yourwish to distribute this article to others here.ng the guidelines can be obtained byssion to republish or repurpose articles or portions of articles ):July 26, 2012ciencemag.org (this information is current as of llowing resources related to this article are available online at ww.sciencemag.org/content/337/6093/437.full.html of this article at: including high-resolution figures, can be found in the onlineed information and services, ww.sciencemag.org/content/suppl/2012/07/25/337.6093.437.DC1.html can be found at:rting Online Material ww.sciencemag.org/content/337/6093/437.full.html#related at: can berelated to this articlef selected additional articles on the Science Web sites ww.sciencemag.org/content/337/6093/437.full.html#ref-list-1 , 16 of which can be accessed free:cites 52 articlesticle ww.sciencemag.org/content/337/6093/437.full.html#related-urls 1 articles hosted by HighWire Press; see:cited byticle has been cience subject collections:ticle appears in the following onJuly26,2012www.sciencemag.orgDownloadedfrom THE CONNECTOME: A NECESSARY SUBSTRATE
  12. 12. (II-III) Functional Brain Networks
  13. 13. BEFORE BEGINNING…. Functional networks are not functional networks! Just a secret* only shared between scientists working on functional networks… * keep the secret, please.
  14. 14. IT’S A LONG ROAD… FULL OF TROUBLE! Obtaining a functional brain network in three steps: Measuring Brain Activity STEP 1 Time Series Analysis Network Construction Network Analysis STEP 2 STEP 3
  15. 15. OBTAINING FUNCTIONAL BRAIN NETWORKS STEP 1: Measuring Brain Activity Functional MRI (fMRI). The detection of changes in regional brain activity through their effects on blood flow and blood oxygenation (which, in turn, affect magnetic susceptibility and tissue contrast in magnetic resonance images). High spatial resolution (~mm3) but low temporal resolution (~seconds). Electroencephalography (EEG). A technique used to measure neural activity by monitoring electrical signals from the brain, usually through scalp electrodes. EEG has good temporal resolution but relatively poor spatial resolution. Magnetoencephalography (MEG). A method of measuring brain activity by detecting perturbations in the extracranial magnetic field that are generated by the electrical activity of neuronal populations. Like EEG, it has good temporal resolution but relatively poor spatial resolution. It has better resolution than EEG. Others…
  16. 16. OBTAINING FUNCTIONAL BRAIN NETWORKS STEP 1: Measuring Brain Activity Functional MRI (fMRI). The detection of changes in regional brain activity through their effects on blood flow and blood oxygenation (which, in turn, affect magnetic susceptibility and tissue contrast in magnetic resonance images). High spatial resolution (~mm3) but low temporal resolution (~seconds). Typical activation maps.Magnetic Resonance
  17. 17. Example: OBTAINING FUNCTIONAL BRAIN NETWORKS The correlation matrix is calculated and then used to define the network among the highest correlated nodes. Top four images represent snapshots of activity and the three traces correspond to selected voxels from visual (V1), motor (M1) and posterio-parietal (PP) cortices. 147456 voxels. Scale-Free Brain Functional Networks Victor M. Eguı´luz,1 Dante R. Chialvo,2 Guillermo A. Cecchi,3 Marwan Baliki,2 and A. Vania Apkarian2 1 Instituto Mediterra´neo de Estudios Avanzados, IMEDEA (CSIC-UIB), E07122 Palma de Mallorca, Spain 2 Department of Physiology, Northwestern University, Chicago, Illinois, 60611, USA 3 IBM T.J. Watson Research Center, 1101 Kitchawan Rd., Yorktown Heights, New York 10598, USA (Received 13 January 2004; published 6 January 2005) Functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites. Analysis of the resulting networks in different tasks shows that (a) the distribution of functional connections, and the probability of finding a link versus distance are both scale-free, (b) the characteristic path length is small and comparable with those of equivalent random networks, and (c) the clustering coefficient is orders of magnitude larger than those of equivalent random networks. All these properties, typical of scale-free small-world networks, reflect important functional information about brain states. DOI: 10.1103/PhysRevLett.94.018102 PACS numbers: 87.18.Sn, 87.19.La, 89.75.Da, 89.75.Hc Recent work has shown that disparate systems can be described as complex networks, that is, assemblies of nodes and links with nontrivial topological properties, examples of which include technological, biological and social systems [1]. The brain is inherently a dynamic sys- tem, in which the traffic between regions, during behavior or even at rest, creates and reshapes continuously com- plex functional networks of correlated dynamics. An im- portant goal in neuroscience is to understand these spatio- temporal patterns of brain activity. This Letter proposes a method to extract functional networks, as revealed by func- tional magnetic resonance imaging (FMRI) in humans, and analyze them in the context of the current understanding of complex networks (for reviews see [1–3]). Figure 1 shows how underlying functional networks are exposed during any given task. In these experiments, at each time step (typically 400 spaced 2.5 sec.), magnetic resonance brain activity is measured in 36 64 64 brain sites (so-called ‘‘voxels’’ of dimension 3 3:475 3:475 mm3 ). The activity of voxel x at time t is denoted as V x; t . We define that two voxels are functionally con- nected if their temporal correlation exceeds a positive predetermined value rc, regardless of their anatomical connectivity [4,5]. Specifically, we calculate the linear correlation coefficient between any pair of voxels, x1 and x2, as r x1; x2 hV x1; t V x2; t i hV x1; t ihV x2; t i V x1 V x2 ; (1) where 2 V x hV x; t 2i hV x; t i2, and h i repre- sents temporal averages. Figure 2 shows the degree distributions of networks extracted using this method. The data were collected while the subject was opposing fingers one and two during 10 sec, and then resting during 10 sec. We find a skewed distribution of links with a tail approaching a distribution p k k , with around 2. This power law is more evident for networks constructed with higher thresholds rc (more correlated conditions). For decreasing rc, a maxi- mum appears which shifts to the right. Despite changes in parameters, networks remain clearly defined indicating that the main conclusions are robust with respect to the selection of parameters. The small inset in Fig. 2 shows the distribution of links of a network constructed from the randomly shuffled (in time) voxels’ signal. This network displays a Gaussian degree distribution in which the mean and width depend on rc. The largest values of the correla- tion thresholds used to construct the random networks are usually extremely low (rc 0:1) compared to that used to FIG. 1 (color). Methodology used to extract functional net- works from the signals. The correlation matrix is calculated and then used to define the network among the highest correlated nodes. Top four images represent snapshots of activity and the three traces correspond to selected voxels from visual (V1), motor (M1) and posterio-parietal (PP) cortices. PRL 94, 018102 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 14 JANUARY 2005 0031-9007=05=94(1)=018102(4)$23.00 018102-1 © 2005 The American Physical Society
  18. 18. OBTAINING FUNCTIONAL BRAIN NETWORKS Electroencephalography (EEG). A technique used to measure neural activity by monitoring electrical signals from the brain, usually through scalp electrodes. EEG has good temporal resolution (up to kHz) but relatively poor spatial resolution (typically between 19-256 electrodes). There are approximately 10 times more glial cells than neurons. The general shape of neurons is longer and thinner than other cells of the body. They transmit information and communicate with each other using a combination of chemical and electrical signals. Figure 2.1 shows a simplified picture of communica- tion between three neurons. Neuron 2 receives a chemical input from neuron 1, which causes an electrical signal along its length and generates a chemical signal, which is transmitted to neuron 3. The chemical signals are in the form of neurotransmitters, which are released by one neuron and detected by another. The electrical signals are in the form of action potentials, which travel along neuronal axons. While neurons have a variety of shapes and sizes, the most general features are dendrites, which receive input from other neurons, a soma or cell body, and the axon, which conveys electrical information. In the brain, each neuron influences and is influenced by many other neurons. An action potential occurs when inputs at a neuron are summed and the threshold is ex- ceeded. When the action potential reaches the axon terminal of the neuron, it results in the release of neurotransmitter from the presynaptic membrane, which causes post- synaptic potentials (PSP). Two types of PSP’s can be classified. The Excitatory postsynaptic potential (E)PSP, causes a depolarization of the cell due to the arrival of action potentials at a synapse. Neuron 1 (Action potential) Output chemical transmission (Synapse)(Synapse) Dendrites Axon Soma Neuron 2 Neuron 3 Electrical transmission Input chemical transmission Figure 2.1: A simple picture of signal transmission in neurons. Oh my god! A syringe!
  19. 19. OBTAINING FUNCTIONAL BRAIN NETWORKS The sample of human EEG with prominent resting state activity - alpha-rhythm. Left - EEG traces (horizontal - time in seconds; vertical - amplitudes, scale 100uV). Right - power spectra of shown signals (vertical lines - 10 and 20 Hz, scale is linear).Alpha-rhythm consists of sinusoidal-like waves with frequencies in 8–12 Hz range (11 Hz in this case) more prominent in posterior sites.Alpha range is red at power spectrum graph. We are basically recording time series:
  20. 20. OBTAINING FUNCTIONAL BRAIN NETWORKS The samples of main types of artifacts in human EEG. 1 - Electrooculographic artifact caused by the excitation of eyeball's muscles (related to blinking, for example). Big amplitude, slow, positive wave prominent in frontal electrodes. 2 - Electrode's artifact caused by bad contact (and thus bigger impedance) between P3 electrode and skin. 3 - Swallowing artifact. 4 - Common reference electrode's artifact caused by bad contact between reference electrode and skin. Huge wave similar in all channels. Unavoidable artifacts:
  21. 21. OBTAINING FUNCTIONAL BRAIN NETWORKS Magnetoencephalography (MEG). A method of measuring brain activity by detecting perturbations in the extracranial magnetic field that are generated by the electrical activity of neuronal populations. Like EEG, it has good temporal resolution but relatively poor spatial resolution. It has better resolution than EEG. EEG and MEG in brain research +++ ++++++ + + ++++++ ... ... . ... ..... .. .. ..... ...... . ... ..... ... Extracellular current Volume currents Magnetic field lines Electric isopotential lines + ++++ + + + -- - - - - -- c) Propagation + a+ Dendrite Na+ channels in the neuronal membrane open in response to a of the membrane potential. The leading edge of the depolarization y Na+ channels and a wave of depolarization spreads from the ction potentials move in one direction. This is achieved because iod of the Na+ channels. After activation Na+ channels do not ures that the action potential is propagated in only one direction intracellular current and oppositely directed extracellular current rcuit. Within the relatively long time during, which the current ically tens of milliseconds or more), it is reasonable to consider femtoTeslas… yes: 10^(-15)… … Earth doesn’t help (microTeslas)
  22. 22. OBTAINING FUNCTIONAL BRAIN NETWORKS
  23. 23. STEP 1: Measuring Brain Activity Low spatial resolution (we have ~10 11 neurons) Measurements are overlapped In EEG and MEG, we only measure cortical activity Defining the nodes is a complex task Brain is not an isolated system High variability in the results OBTAINING FUNCTIONAL BRAIN NETWORKS
  24. 24. Brain parcellation: what is a node? OBTAINING FUNCTIONAL BRAIN NETWORKS
  25. 25. Source reconstruction: inverse problem Source reconstruction tries to identify the sources of the magnetic field, but …. … every inverse methods makes specific assumptions… … (ideally) performs well if assumptions are met. …
 … but there are no method that performs well in general. OBTAINING FUNCTIONAL BRAIN NETWORKS Inverse methods MNE MCE WMNE Loreta sLORETA eLORETA Laura Electra WROP DICS LCMV-Beamformer Nulling Beamformer FOCUSS Champagne Minimum Entropy Dipole Modeling Multipole Modeling MUSICRAP-MUSIC S-FLEX DCM Different methodologies, the majority are black boxes for the user.
  26. 26. IT’S A LONG ROAD… FULL OF TROUBLE! Obtaining a functional brain network in three steps: Time Series Analysis Network Construction STEP 2 Measuring Brain Activity STEP 1 Network Analysis STEP 3
  27. 27. STEP 1I: Time Series Analysis Network Construction How to measure coordination between brain regions? Cross-correlation Wavelet coherence Synchronization Likelihood Generalized Synchronization Phase Synchronization Mutual Information Granger Causality Once coordination is evaluated, we construct the functional network. OBTAINING FUNCTIONAL BRAIN NETWORKS * For a review: Pereda et al, Prog. Neurobiol, 77 (2005)
  28. 28. Linear: Evaluate correlation between time series.They are the simplest and, sometimes, good enough. OBTAINING FUNCTIONAL BRAIN NETWORKS Two groups: MCI (21) and Control (21). MEG. Memory task. Classification algorithms: Multi-Layer Perceptron (MLP), Probabilistic Neural Networks (PNN), Decision Tree (DT), y K Nearest Neighbours (KNN).M. Zanin, PhDThesis
  29. 29. OBTAINING FUNCTIONAL BRAIN NETWORKS Non-Linear: Based on a nonlinear function between x(t) and y(t).They also include phase synchronization indexes. 0 50 100 150 time [sec] −5 0 5 10 15 ϕ1,1/2π 0 50 100 150 time [sec] 0 50 100 150 time [sec] yx (a) (b) (c) 0 50 100 150 time [sec] −5 0 5 10 15 ϕ1,1/2π 0 50 100 150 time [sec] 0 50 100 150 time [sec] yx (a) (b) (c) Fig. 7. Stabilograms of a neurological patient for EO (a), EC (b), and A x(t)=f(y(t)) f(y(t))? −5 0 5 10 15 ϕ1,1/2πy (a) (b) 0 50 100 150 −5 0 5 10ϕ1,1/2π 0 50 100 150 yx
  30. 30. OBTAINING FUNCTIONAL BRAIN NETWORKS Spectral: Based on the analysis of the spectrum of the time series. Also include different linear/nonlinear ways of comparing spectra. Spectra of EEG electrodes. M.G. Knyazeva et al., Journal of Neurophysiology
  31. 31. OBTAINING FUNCTIONAL BRAIN NETWORKS And now, what matrix do I analyze? “coordination” matrix adjacency matrix normalized matrix
  32. 32. OBTAINING FUNCTIONAL BRAIN NETWORKS Percentage ThresholdWeighted Basically, you can choose between three options:
  33. 33. OBTAINING FUNCTIONAL BRAIN NETWORKS Binarize the matrix by selecting an adequate threshold: Figure3. Effectsoftaskdifficultyonworkspaceconfigurationofbrainfunctionalnetworksatdifferentfrequencyintervalsoverarangeofnetworkconnectiondensities.A–E,Overallfrequencies (A)andineachfrequencyinterval(B–E),meanbrainnetworkmetricswith95%confidenceinterval(dottedlines)forzero-back(redline),one-back(greenline),andtwo-back(blueline)taskstend toconvergeontheirvaluesinsurrogatenetworks(grayline)asconnectiondensityisincreasedfrom2%to20%ofpossibleedges.Theverticaldottedlinesindicatestheconnectiondensityof10% chosen for ANOVA modeling. Asterisks denote significant difference at p Ͻ 0.05 between two-back and surrogate networks. Kitzbichler et al. • Workspace Configuration of MEG Networks J. Neurosci., June 1, 2011 • 31(22):8259–8270 • 8265 Kitzblicher et al., J. Neurosci. 2011
  34. 34. STEP 1I: Time Series Network Construction It is difficult to evaluate weights and causality in the interactions There is not a unique way of interacting No clear way of defining the network (threshold problem spurious links) Functional networks are not static High variability in the results OBTAINING FUNCTIONAL BRAIN NETWORKS
  35. 35. −5 0 5 10 15 ϕ1,1/2πy (a)(b)(c) OBTAINING FUNCTIONAL BRAIN NETWORKS Example 1: Functional networks are virtual REAL FUNCTIONAL 0 50 100 150 time [sec] −5 0 5 10 15 ϕ1,1/2π 0 50 100 150 time [sec] 0 50 100 150 time [sec] yx (a) (b) (c) 0 50 100 150 time [sec] 0 50 100 150 time [sec] (b) (c) is it correct?
  36. 36. OBTAINING FUNCTIONAL BRAIN NETWORKS 1 4 7 10 13 16 19 40 50 60 70 80 Classificationscore(%) Deleted node 0.775 0.7875 19 16 13 10 7 4 40 50 60 70 80 Classificationscore(%) Number of nodes F3 P8 T7 T8 P7 O2 FZ FP1 F8 T7 FZ O2 P8 T8 F3 F8 P7 FP1 Figure 14: Discarding nodes from the networks. (Top Left) Classification score as a function of the node discarded. The dashed horizontal line represents the best classification score with the complete network (77.5%). (Top Right) Classification score as a function of the number of surviving nodes. (Bottom) The networks of Fig. 5 when only the 9 most important nodes are included. these topological features yielded a rather low score (⇡ 60%). This suggests that these differences were not as important as initially thought, possibly because the analysis (and specifically, its parameters) was not properly tuned. The same data mining techniques were the instrument to increase the significance of the networks, and to point us towards the synchronisation metrics and brain regions most relevant. The upshot is that we end up with higher prognostic capabilities and better understanding of the pathology at hand, EEG (19 electrodes), image recognition task. 40 controls alcoholic individuals. Discarding nodes from the networks. (Top Left) Classification score as a function of the node discarded.The dashed horizontal line represents the best classification score with the complete network (77.5%). (Top Right) Classification score as a function of the number of surviving nodes. (Bottom)The networks with only the 9 most important nodes. (Zanin et al., Phys. Rep. 2016) Example 1I: where to put the threshold?
  37. 37. IT’S A LONG ROAD… FULL OF TROUBLE! Obtaining a functional brain network in three steps: Time Series Analysis Network Construction STEP 2 Measuring Brain Activity STEP 1 Network Analysis STEP 3
  38. 38. Suppose we have the network…. let’s analyze it! a Healthy volunteers b People with schizophrenia 44' 46 37 45 22 45' 39' 39' 22 44 44 44' 8 34' 19 36 36 22 37 9 47 43' 43 39 42'47' 43 21' 20' 21' 19' 10 10' 8' 45' 10 8 9' 39 40 20 19' 19 40' 40 40' 8' 9' 10' 37' 20 12 21 20 36' 37' 43' 46' 47' 47 46 45 Frontal Temporal Parietal Occipital High clustering Low clustering Degree ANALYZING FUNCTIONAL BRAIN NETWORKS
  39. 39. A. Characterize the topology of brain functional networks and its influence in the processes occurring in them. B. Identify differences between healthy brains and those with a certain pathology. C. Develop models in order to explain the changes found in impaired functional networks. Network Analysis: Why? ANALYZING FUNCTIONAL BRAIN NETWORKS
  40. 40. A. Characterize the topology of brain functional networks* and its influence in the processes occurring in them: • Heterogeneous - Crucial nodes. • High clustering - Good local resilience? • Small-world topology - High efficiency in information transmission? • Modularity - Segregation integration of information? • Others: Assortative, degree-degree correlations, rich-clubs, hierarchical structure,… ANALYZING FUNCTIONAL BRAIN NETWORKS * “All generalizations are false, including this one”, MarkTwain (probably…)
  41. 41. Hubs unavoidably appear in functional networks: ❑ Two activities: music and finger tapping ❑ fMRI ❑ 36 x 64 x 64 regione (147456 voxels) ❑ Linear correlation between voxels: ❑ Matrix is thresholded Music Finger tapping FUNCTIONAL NETWORKS ARE HETEROGENEOUS
  42. 42. Highly (functionally) connected nodes: Two different tasks (Eguíluz et al., PRL 2005) FUNCTIONAL NETWORKS ARE HETEROGENEOUS define the functional networks (rc 0:7). Our data were also compared with values from a randomly rewired net- work, where nodes keep their degree by permuting links (i.e., the link connecting nodes i, jis permuted with that connecting nodes k, l) [6] (see below). In this control the degree of each node is maintained but all other correlations (including clustering) are destroyed. To test the generality of these findings the same analysis 10 0 10 1 10 2 10 3 Degree K 10 0 10 1 10 2 10 3 10 4 10 5 )k(stnuoC rc = 0.6 rc = 0.7 rc = 0.8 10 0 10 1 10 2 (mm) 10 -4 10 -3 10 -2 10 -1 10 0 k)(.borP ∆ ∆ 10 0 10 1 10 2 10 3 Degree K 10 0 10 1 10 2 10 3 10 4 Counts(k) rc = 0.5 rc = 0.6 rc = 0.7 700 800 Degree K 0 500 Counts(k) FIG. 2 (color online). Degree distribution for three values of the correlation threshold. The inset depicts the degree distribu- tion for an equivalent randomly connected network. PRL 94, 018102 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 14 JANUARY 2005 fMRI, finger tapping, different thresholds “…scale-free complex networks are known to show resistance to failure, facility of synchronization, and fast signal processing… ”
  43. 43. It is common to observe an exponential cut-off: The degree distribution of all networks at all frequency bands and both behavioral states was best described by a truncated power law, given in the form P(k)∼Ak^(λ−1) e^(k/kc), where A is the coefficient, λ describes the power law, and kc is the exponential parameter. Table 2. Parameters of exponentially truncated power law degree distribution A kc Resting 1 0.8 ± 0.2 1.5 ± 0.4 8 ± 4 2 0.9 ± 0.3 1.6 ± 0.5 5 ± 3 3 0.9 ± 0.3 1.5 ± 0.5 8 ±15 4 0.9 ± 0.3 1.6 ± 0.6 6 ± 4 5 0.8 ± 0.2 1.6 ± 0.4 5 ± 2 6 1.0 ± 0.2 1.4 ± 0.3 8 ± 6 Tapping 1 0.9 ± 0.1 1.4 ± 0.2 9 ± 4 2 0.8 ± 0.2 1.7 ± 0.4 5 ± 1 3 0.8 ± 0.3 1.7 ± 0.5 5 ± 2 4 0.8 ± 0.3 1.7 ± 0.5 6 ± 4 5 0.9 ± 0.2 1.5 ± 0.5 10 ±14 6 1.0 ± 0.1 1.2 ± 0.3 14 ±20 The degree distribution of all networks at all fre- quency bands and both behavioral states was best described by a truncated power law, given in the form P(k) ⇠ Ak 1 ek/kc , where A is the coe cient, describes the power law, and kc is the exponential parameter. These three parameters are given here along with their standard deviation and show a large which reflects the distinctive topological properties (greater density and clustering) of the ␥ network. Spatial Configuration of Scale-Specific Networks. The spatial distri- bution of network hubs was also broadly similar across scales and states (see Fig. 2 and SI Fig. 5). See SI Fig. 6 for average hub distributions across all scales in both states. However, there were striking differences between scales and states in the physical distance between functionally connected network nodes (see Fig. 3). In the resting state, long-range functional connectivity be- tween brain regions was stronger at low frequencies. At higher frequencies (␤, ␥), long-range connectivity was weaker, and most of the edges in the graph represented high-density local con- nections (see Figs. 1E and 4), shown by the increase in charac- teristic length scale of network edges ␨, going from high to low frequency scales; and by the increasing number of connector compared with provincial nodes at low frequencies (see SI Fig. 7 for a schematic and SI Fig. 8 for distributions of provincial and connector hubs in both states and all frequency bands). In the finger-tapping state, long-range functional connections emerged more strongly at high frequencies (␤, ␥), shown by the significant motor task-related increases in characteristic length scale of edges in high-frequency motor networks. It is also represented by the shift from resting-state ␥ networks dominated by provincial hubs (predominantly connected to locally neigh- boring regions of bilateral occipital, parietal, and central cortex) to motor ␥ networks with a larger number of connector hubs in medial premotor and bilateral prefrontal cortex. Some of the new long-range connections engendered by task performance at high frequencies link to topologically pivotal nodes in right medial premotor and prefrontal cortex with high betweenness scores (see Fig. 3; and see SI Fig. 9 for betweenness distributions at all frequencies). This indicates that task performance is associated with reconfiguration of high-frequency networks to favor long-distance connections between prefrontal and premo- Fig. 2. Self-similarity of spatial distribution of highly connected network nodes or ‘‘hubs’’ in the frequency range 2–38 Hz (64). Each column shows the surface distribution of the degree of network nodes in frequency bands ␤ to ␦: red represents nodes with high degree. The last column shows the spatial distribution of degree averaged over these four frequency bands, which emphasizes the similarity of spatial configurations across scales. See SI Fig. 5 for the hub distributions in both states at all frequency bands. Fig. 3. State-related differences in spatial configuration of the highest frequency ␥ network. The top row shows the degree distribution and betweenness scores for the resting state ␥ network; the middle row shows the same maps for the motor ␥ network; the bottom row shows the between-state differences in degree and betweenness. It is clear that motor task performance is associated with emergence of greater connectivity in bilateral prefrontal and premotor nodes, and appearance of topologically pivotal nodes (with high betweenness scores) in medial premotor, right prefrontal, and parietal areas. See SI Fig. 7 for the betweenness distributions in both states at all frequency bands. 19520 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0606005103 Bassett et al. MEG (275 channels), 11 resting and 11 finger tapping. Node degree and betweenness. (Basset et al., PNAS 2006) FUNCTIONAL NETWORKS ARE HETEROGENEOUS
  44. 44. Let’s see what happens at the local level: clustering coefficient Scales 1– 6 denote progressively lower-frequency intervals (Hz). r is the mean inter-regional correlation, and R is the correlation threshold. (N-1 nodes) Lnet and Cnet are the mean path length and clustering coefficient, respectively, of the thresholded network.The lambda︎ and gamma︎ are ratios of brain network path length and clustering coefficient, respectively, to comparable random network metrics. Sigma︎ is a scalar measure of “small-worldness.” (Achard et al., J. Neurosci. 2006) FUNCTIONAL NETWORKS HAVE HIGH CLUSTERING tions between fMRI time series measured in multiple cortical and subcortical human brain regions in normal human volunteers scanned during “rest” or a no-task condition. Analysis of resting- state data has the advantage that it may focus attention on endog- stituted the set of regional mean time series used for wavelet correlation analysis. Wavelet correlation analysis. Wavelet transforms effect a time-scale decomposition that partitions the total energy of a signal over a set of compactly supported basis functions, or little waves, each of which is Table 1. Wavelet scale dependency of functional connectivity and small-world parameters for an entire human brain network Scale Hz r R Lnet Cnet ␭ ␥ ␴ 1 0.23–0.45 0.12 0.13 2.9 0.534 1.28 1.81 1.42 2 0.11–0.23 0.21 0.2 2.6 0.566 1.12 2.14 1.92 3 0.06–0.11 0.39 0.39 2.69 0.555 1.16 2.22 1.91 4 0.03–0.06 0.45 0.44 2.49 0.525 1.08 2.38 2.19 5 0.01–0.03 0.44 0.35 2.4 0.554 1.04 2.39 2.30 6 0.007–0.01 0.41 0.17 2.65 0.515 1.15 2.15 1.88 Scales1–6oftheMODWTdenoteprogressivelylower-frequencyintervals(Hz).risthemeaninter-regionalcorrelation,andRisthecorrelationthreshold.Lnet andCnet arethemeanpathlengthandclusteringcoefficient,respectively,ofthe thresholded network. The ␭ and ␥ are ratios of brain network path length and clustering coefficient, respectively, to comparable random network metrics. The equation ␴ ϭ ␥/␭ is a scalar measure of “small-worldness.” 64 • J. Neurosci., January 4, 2006 • 26(1):63–72 Achard et al. • A Small-World Human Brain Functional Network Behavioral/Systems/Cognitive A Resilient, Low-Frequency, Small-World Human Brain Functional Network with Highly Connected Association Cortical Hubs Sophie Achard,1 Raymond Salvador,1,2 Brandon Whitcher,3 John Suckling,1 and Ed Bullmore1,3 1Brain Mapping Unit and Wolfson Brain Imaging Centre, University of Cambridge, Departments of Psychiatry and Clinical Neurosciences, Addenbrooke’s Hospital, Cambridge CB2 2QQ, United Kingdom, 2Sant Joan de Deu–Serveis de Salut Mental, Sant Boi de Llobregat, Spain, and 3Translational Medicine and Genetics, GlaxoSmithKline, Cambridge CB2 2QQ, United Kingdom Small-world properties have been demonstrated for many complex networks. Here, we applied the discrete wavelet transform to func- tional magnetic resonance imaging (fMRI) time series, acquired from healthy volunteers in the resting state, to estimate frequency- dependent correlation matrices characterizing functional connectivity between 90 cortical and subcortical regions. After thresholding the wavelet correlation matrices to create undirected graphs of brain functional networks, we found a small-world topology of sparse connections most salient in the low-frequency interval 0.03–0.06 Hz. Global mean path length (2.49) was approximately equivalent to a comparable random network, whereas clustering (0.53) was two times greater; similar parameters have been reported for the network of anatomical connections in the macaque cortex. The human functional network was dominated by a neocortical core of highly connected hubs and had an exponentially truncated power law degree distribution. Hubs included recently evolved regions of the heteromodal association cortex, with long-distance connections to other regions, and more cliquishly connected regions of the unimodal association and primary cortices; paralimbic and limbic regions were topologically more peripheral. The network was more resilient to targeted attack on its hubs than a comparable scale-free network, but about equally resilient to random error. We conclude that correlated, low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectiv- ity of the cortex. Because the major hubs of this network are critical for cognition, its slow dynamics could provide a physiological substrate for segregated and distributed information processing. Key words: small world; fMRI; wavelet; functional connectivity; association cortex; neuroimaging Introduction A remarkable variety of social, economic, and biological net- works demonstrate “small-world” properties (Strogatz, 2001), meaning the nodes of the network have greater local interconnec- tivity or cliquishness than a random network, but the minimum path length between any pair of nodes is smaller than would be expected in a regular network or lattice (Watts and Strogatz, 1998). Some small-world networks, such as the worldwide web or the Hollywood network of movie actors, include a small number of “hubs,” nodes with an unusually large number of connections (or large degree) (Baraba´si and Albert, 1999; Baraba´si, 2003). The degree distribution of the worldwide web obeys a power law, implying no meaningful “average” number of links to each site, and for this reason, it has been described as a scale-free network (Baraba´si and Albert, 1999). Many other small-world networks, including Hollywood, have exponential or exponentially trun- cated power law distributions, implying relatively reduced prob- abilities of huge hubs (Amaral et al., 2000; Albert and Baraba´si, 2002). Small-worlds are attractive models for connectivity of nervous systems because the combination of high clustering and short path length confers a capability for both specialized or modular processing in local neighborhoods and distributed or integrated processing over the entire network (Sporns et al., 2004). It has been demonstrated that the neuronal network of Caenorhabditis elegans has small-world topology at a microscopic anatomical scale (Watts and Strogatz, 1998). Anatomical connectivity matri- ces from tract-tracing studies of cat and monkey cortices show small-world properties at a macroscopic scale (Hilgetag et al., 2000). There is evidence also for small-world properties of graphs inferred from functional connectivity matrices measured at a macroscopic (regional) scale in monkey and human neurophys- iological data (Stephan et al., 2000; Stam, 2004; Salvador et al., 2005b) and at a mesoscopic (voxel) scale in human functional magnetic resonance imaging (fMRI) data (Eguı´luz et al., 2005). Here, we used wavelets to decompose the pairwise correla- Received May 17, 2005; revised Oct. 26, 2005; accepted Oct. 27, 2005. This neuroinformatics research was supported by a Human Brain Project grant from the National Institute of Bioengineering and Biomedical Imaging and the National Institute of Mental Health and was conducted in the Medical Research Council (MRC)/Wellcome Trust Behavioral and Clinical Neurosciences Institute (Cambridge, UK). The Wolfson Brain Imaging Centre was supported by an MRC Cooperative Group grant. This work was presented in part at the Brain Connectivity 2005 Workshop at Florida Atlantic University (Boca Raton, FL), April 15–16, 2005. CorrespondenceshouldbeaddressedtoProf.EdBullmore,BrainMappingUnit,UniversityofCambridge,Depart- ment of Psychiatry, Addenbrooke’s Hospital, Cambridge CB2 2QQ, UK. E-mail: etb23@cam.ac.uk. DOI:10.1523/JNEUROSCI.3874-05.2006 Copyright © 2006 Society for Neuroscience 0270-6474/06/260063-10$15.00/0 The Journal of Neuroscience, January 4, 2006 • 26(1):63–72 • 63 rldpropertiesofbrainnetworksasafunctionofcorrelationthreshold.a, shold R is increased, mean degree k monotonically decreases (the net- arselyconnected)atallscalesofthewavelettransform:blacklines,scale en lines, scale 3; dark blue lines, scale 4; light blue lines, scale 5; purple World Human Brain Functional Network J. Neurosci., January 4, 2006 • 26(1):63–72 • 67 fMRI, 5 individuals, resting state, eyes closed. Brain image: Scale 4 Figure2. Small-worldpropertiesofbrainnetworksasafunctionofcorrelationthreshold.a, As the correlation threshold R is increased, mean degree k monotonically decreases (the net-
  45. 45. Small-world everywhere! (I know you already realized) FUNCTIONAL NETWORKS ARE SMALL-WORLD
  46. 46. 30 individuals (17 young/13 old). Functional network during resting state (fMRI). Modularity optimization through a greedy algorithm. 5% of the total number of links is considered. 5 functional modules are detected. Meunier et al., Front. Neuroinformatics 3:37 (2009). Anatomical or functional, in any case they are modular: FUNCTIONAL NETWORKS ARE MODULAR 719D. Meunier et al. / NeuroImage 44 (2009) 715–723 D. Meunier et al. / NeuroImage 4 Age-related changes in modular organization of human brain functional networks David Meunier a,b , Sophie Achard a,b,c , Alexa Morcom d , Ed Bullmore a,b, ⁎ a Brain Mapping Unit, University of Cambridge, Cambridge, UK b Behavioural and Clinical Neurosciences Institute, University of Cambridge, Cambridge, UK c Grenoble Image Parole Signal Automatique, Centre National de la Recherche Scientifique, Grenoble, France d Centre for Cognitive and Neural Systems, University of Edinburgh, Edinburgh, UK a b s t r a c ta r t i c l e i n f o Article history: Received 21 May 2008 Revised 4 September 2008 Accepted 30 September 2008 Available online 5 November 2008 Keywords: Aging Complex networks Functional MRI Modularity Graph theory allows us to quantify any complex system, e.g., in social sciences, biology or technology, that can be abstractly described as a set of nodes and links. Here we derived human brain functional networks from fMRI measurements of endogenous, low frequency, correlated oscillations in 90 cortical and subcortical regions for two groups of healthy (young and older) participants. We investigated the modular structure of these networks and tested the hypothesis that normal brain aging might be associated with changes in modularity of sparse networks. Newman's modularity metric was maximised and topological roles were assigned to brain regions depending on their specific contributions to intra- and inter-modular connectivity. Both young and older brain networks demonstrated significantly non-random modularity. The young brain network was decomposed into 3 major modules: central and posterior modules, which comprised mainly nodes with few inter-modular connections, and a dorsal fronto-cingulo-parietal module, which comprised mainly nodes with extensive inter-modular connections. The mean network in the older group also included posterior, superior central and dorsal fronto-striato-thalamic modules but the number of intermodular connections to frontal modular regions was significantly reduced, whereas the number of connector nodes in posterior and central modules was increased. Crown Copyright © 2008 Published by Elsevier Inc. All rights reserved. Introduction Modularity is a word with many meanings in neuroscience (Fodor, 1983; Zeki and Bartels, 1998; Redies and Puelles, 2001; Callebaut and Rasskin-Gutman, 2005). Here we are concerned with the topological organization of whole human brain functional networks and the partitioning of these networks into a set of modules, each module being defined by dense internal or intra-modular connectivity and relatively sparse external or inter-modular connectivity (Newman and Girvan, 2004); see Fig. 1. This pattern of complex network organiza- tion, also sometimes described as a community structure, is wide- spread in biochemical, social and infrastructural networks (Guimerà et al., 2005). A key advantage of modular organization, which may explain its ubiquity in diverse systems, is that it favours evolutionary and developmental optimization of multiple or changing selection criteria (Redies and Puelles, 2001; Slotine and Lohmiller, 2001; Kashtan and Alon, 2005; Pan and Sinha, 2007): a modular network can evolve or grow one module at a time, without risking loss of function in other modules. Mathematical tools have recently been developed to quantify the modularity of any network that can be abstractly described as a set of nodes and links (Newman and Girvan, 2004; Newman, 2004a; Danon et al., 2005; Newman, 2006). Once the modules have been identified, this information can be further used to refine the definition of the topological role of any particular node. For example, the global air transportation network has a modular organization (Guimerà and Amaral, 2005b), broadly conforming to geopolitical constraints, which informed the assignment of distinct roles to the component nodes (cities) based on the ratio of intra- and inter-modular links (flights) connecting each node to the rest of the network. Thus a highly- connected city, like London, with many long-haul flights to other modules (different continents), was designated a connector hub; whereas a regionally important city, like Barcelona, with relatively few long-haul flights outside Europe and North Africa, was designated a provincial hub. Here we extend the analysis of modularity and topological roles in functional brain networks, using tools drawn from the literature on physics of complex networks (Newman and Girvan, 2004; Guimerà and Amaral, 2005b) that have not been previously applied to analysis of human functional neuroimaging data. However, we note that there have been several prior studies using conceptually related multi- variate or graph theoretical methods to explore the clustered or modular organization of mammalian cortex. Young (1992) applied non-metric multidimensional scaling (MDS) to anatomical connectiv- ity matrices to demonstrate dorsal and ventral “streams” of inter- regional connectivity, and a predominance of local neighbourhood NeuroImage 44 (2009) 715–723 ⁎ Corresponding author. Brain Mapping Unit, University of Cambridge, Department of Psychiatry, Addenbrooke's Hospital, Cambridge CB2 2QQ, UK. Fax: +44 1223 336581. E-mail address: etb23@cam.ac.uk (E. Bullmore). 1053-8119/$ – see front matter. Crown Copyright © 2008 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2008.09.062 Contents lists available at ScienceDirect NeuroImage journal homepage: www.elsevier.com/locate/ynimg 5 modules were detected in the young individuals. 5 modules were detected in the young individuals.
  47. 47. Representation in the topological space. Left:Young adults (18-33); Right: Older adults (62-73) The modular organization changes with age: FUNCTIONAL NETWORKS ARE MODULAR
  48. 48. Assortativity in functional brain networks: More (functionally) connected regions are prone to be connected between them. (finger tapping) music finger tapping FUNCTIONAL NETWORKS ARE ASSORTATIVE
  49. 49. B. Identify differences between healthy brains and those with a certain pathology: • Identify differences with respect to a control group. • Evaluate the effects of a certain disease in the functional network. • Quantify evolution towards “impaired” topologies. • Evaluate the loss of segregation/integration in the functional networks. • Quantify the increase of energy expenses. ANALYZING FUNCTIONAL BRAIN NETWORKS
  50. 50. Self-portrait of William Utermohlen (american painter (1993-2007)). In 1995 (62 years old) he began to suffer problems with memory and writing. FROM A HEALTHY TO AN IMPAIRED FUNCTIONAL NETWORK
  51. 51. Fig. 5 Mean PLI averaged over all pairs of MEG sensors for Alzheimer’s disease patients and controls in six frequency bands. Error bars are SDs. The mean PLI was significantly lower in Alzheimer’s disease patients compared to controls in the lower alpha band (two-tailed t-test, P50.022) and the beta band (two-tailed t-test, P = 0.036). Fig. 4 Average weighted graphs of Alzheimer’s disease patients and controls in six frequency bands. The value of the PLI for all individual pairs of MEG sensors is indicated in colour (blue: low PLI; red: high PLI). Fig. 3 Damage modelling procedure. The mean PLI of a control subject network is lowered by randomly weakening edges in the network, until it reaches the same value as in a Alzheimer’s disease patient network. The effect of this damage is then examined by comparing the network characteristics of the damaged network to the Alzheimer’s disease patient net- work characteristics. RF = Random Failure, TA = Targeted Attack, Cw = mean weighted clustering coefficient, Lw = mean weighted path length. byguestonApril7,2011brain.oxfordjournals.orgnloadedfrom The non-parametric Mann–Whitney U-test for independent samples revealed that Cw was lower in Alzheimer’s disease subjects in the 8–10 Hz band (U = 89.5; P = 0.022), but not in network in the Alzheimer’s disease group. Please note that, by definition, the average PLI of both models is the same as the average PLI of the Alzheimer’s disease data. Further analysis of the model data compared with the real data is shown in Fig. 8. For the Random Failure model the ^Cw was not different from the control data, and significantly higher than ^Cw of the Alzheimer’s disease group (Mann–Whitney U-test, U = 76.5; P = 0.007). In contrast, ^Cw of the Targeted Attack model was not significantly different from the Alzheimer’s disease group, but significantly lower than ^Cw of the control group (U = 87.0; P = 0.018). The weighted path length ^Lw showed a decreasing trend going from controls to Random Failure, Targeted Failure and controls (Fig. 8, right panel). ^Lw of both models did not differ significantly from control data. Correlation with MMSE No significant correlations between MMSE and PLI or network measures were found in the Alzheimer’s disease patient group (Fig. 9). When correlation with MMSE was analysed for all subjects (Alzheimer’s disease and control) put together in one group, we found significant effects between MMSE and mean PLI in the beta band (Spearman’s r = 0.570, P = 0.001) and between MMSE and ^Cw in the lower alpha band (Spearman’s r = 0.475, P = 0.008). Table 1 Results of weighted graph analysis for Alzheimer’s disease patients and controls in six frequency bands Cw Lw ^Cw ^Lw Alzheimer’s disease Control Alzheimer’s disease Control Alzheimer’s disease Control Alzheimer’s disease Control 0.5–4 Hz 0.12 (0.10–0.32) 0.12 (0.10–0.17) 4.05 (1.69–4.40) 3.92 (2.89–4.59) 1.04 (1.03–1.12) 1.04 (1.02–1.11) 1.09 (1.06–1.33) 1.08 (1.05–1.34) 4–8 Hz 0.11 (0.09–0.20) 0.10 (0.09–0.15) 4.23 (2.48–4.99) 4.44 (3.22–5.01) 1.05 (1.03–1.17) 1.04 (1.03–1.13) 1.14 (1.04–1.41) 1.15 (1.05–1.43) 8–10 Hz 0.15 (0.12–0.21) 0.17 (0.13–0.29) 3.27 (2.25–3.76) 2.69 (1.80–3.73) 1.04 (1.02–1.12) 1.07 (1.04–1.13) 1.08 (1.05–1.32) 1.19 (1.07–1.30) 10–13 Hz 0.12 (0.11–0.14) 0.13 (0.11–0.22) 3.83 (3.28–4.36) 3.72 (2.36–4.30) 1.04 (1.03–1.10) 1.04 (1.03–1.21) 1.10 (1.05–1.35) 1.12 (1.04–1.45) 13–30 Hz 0.06 (0.05–0.06) 0.06 (0.05–0.08) 7.97 (6.44–9.24) 7.61 (5.18–9.35) 1.04 (1.02–1.07) 1.04 (1.03–1.16) 1.11 (1.05–1.50) 1.12 (1.04–1.50) 30–45 Hz 0.05 (0.05–0.09) 0.05 (0.05–0.08) 8.70 (5.17–9.07) 8.54 (6.06–9.14) 1.02 (1.02–1.07) 1.02 (1.02–1.07) 1.09 (1.06–1.33) 1.04 (1.02–1.30) Values are medians, with range printed between parentheses. Cw = mean weighted clustering coefficient; Lw = mean weighted path length; ^Cw = mean normalized average weighted clustering coefficient (see Materials and Methods section), ^Lw = mean normalized average weighted path length. Significant differences between Alzheimer’s disease and controls with non-parametric testing (Mann–Whitney U-test, P50.05) are given in bold. Fig. 6 Schematic illustration of significant differences in long distance (indicated by arrows) and short distance (indicated by filled squares) PLI in the 8–10 Hz and 13–30 Hz band. Alzheimer’s disease patients had lower left sided fronto- temporal, fronto-parietal, temporo-occipital and parieto- occipital PLI in the 8–10 Hz band. Local left frontal and tem- poral, and right parietal PLI were also decreased in Alzheimer’s disease patients (A). For the 13–30 Hz band, Alzheimer’s dis- ease patients had lower inter hemispheric frontal, right fronto- parietal and bilateral frontal PLI (B). byguestonApril7,2011brain.oxfordjournals.orgDownloadedfrom BRAINA JOURNAL OF NEUROLOGY Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer’s disease C. J. Stam,1 W. de Haan,2 A. Daffertshofer,3 B. F. Jones,4 I. Manshanden,1 A. M. van Cappellen van Walsum,5,6 T. Montez,7 J. P. A. Verbunt,1,8 J. C. de Munck,8 B. W. van Dijk,1,8 H. W. Berendse2 and P. Scheltens2 1 Department of Clinical Neurophysiology and MEG, Amsterdam, The Netherlands 2 Department of Neurology, Alzheimer Center, VU University Medical Center, Amsterdam, The Netherlands 3 Research Institute MOVE, VU University, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands 4 Dementia Research Centre, Institute of Neurology, UCL, London, UK 5 Department of Anatomy, Radboud University Nijmegen Medical Centre, Nijmegen, The Netherlands 6 Institute of Technical Medicine, University of Twente, Enschede, The Netherlands 7 Institute of Biophysics and Biomedical Engineering, Faculty of Sciences, University of Lisbon, Portugal 8 Department of Physics and Medical Technology, VU University Medical Center, Amsterdam, The Netherlands Correspondence to: Willem de Haan, Department of Neurology, Alzheimer Center, VU University Medical Center, PO Box 7057, 1007 MB Amsterdam, the Netherlands E-mail: w.dehaan@vumc.nl In this study we examined changes in the large-scale structure of resting-state brain networks in patients with Alzheimer’s disease compared with non-demented controls, using concepts from graph theory. Magneto-encephalograms (MEG) were recorded in 18 Alzheimer’s disease patients and 18 non-demented control subjects in a no-task, eyes-closed condition. For the main frequency bands, synchronization between all pairs of MEG channels was assessed using a phase lag index (PLI, a synchronization measure insensitive to volume conduction). PLI-weighted connectivity networks were calculated, and char- acterized by a mean clustering coefficient and path length. Alzheimer’s disease patients showed a decrease of mean PLI in the lower alpha and beta band. In the lower alpha band, the clustering coefficient and path length were both decreased in Alzheimer’s disease patients. Network changes in the lower alpha band were better explained by a ‘Targeted Attack’ model than by a ‘Random Failure’ model. Thus, Alzheimer’s disease patients display a loss of resting-state functional connectivity in lower alpha and beta bands even when a measure insensitive to volume conduction effects is used. Moreover, the large-scale structure of lower alpha band functional networks in Alzheimer’s disease is more random. The modelling results suggest that highly connected neural network ‘hubs’ may be especially at risk in Alzheimer’s disease. Keywords: Alzheimer’s disease; functional connectivity; MEG; synchronization; small-world networks Abbreviations: EEG = electro-encephalography; MEG = Magneto-encephalography; MMSE = mini mental state examination; PLI = phase lag index; SL = synchronization likelihood doi:10.1093/brain/awn262 Brain 2009: 132; 213–224 | 213 Received May 5, 2008. Revised September 12, 2008. Accepted September 18, 2008. Advance Access publication October 24, 2008 ß The Author (2008). Published by Oxford University Press on behalf of the Guarantors of Brain. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org byguestonApril7,2011brain.oxfordjournals.orgDownloadedfrom MEG, resting state. 18 AD patients and 18 controls. (Phase Lag Index).Weighted Clustering coefficient (Cw) and shortest path (Lw). Only one frequency band showed statistically significant differences (pval0.05) FROM A HEALTHY TO AN IMPAIRED FUNCTIONAL NETWORK
  52. 52. Some general features of different brain diseases: ❑ Alzheimer’s Disease: ❑ The overall synchronization of the network is decreased. ❑The average path length increases (probably as a consequence of the reduction of the synchronization). ❑The clustering coefficient is significantly reduced (the network evolves to random topologies). ❑ Mild Cognitive Impairment: ❑ The average synchronization increases. ❑The network becomes more random. ❑ Network outreach increases as a consequence of an unbalanced increase of the synchronization in the long-range connections. FROM A HEALTHY TO AN IMPAIRED FUNCTIONAL NETWORK
  53. 53. ❑ Schizophrenia: ❑ The small-world properties of the network are impaired (specially at low-frequency bands). ❑ Clustering and average path length are shifted to random configurations. ❑The hierarchical configuration of the network is also affected. ❑ Epilepsia: ❑ Synchronization increases during the epileptic episodes. ❑ As a consequence, clustering coefficient increases and average path length decreases. ❑ Changes are more significant at delta, theta and alpha bands. FROM A HEALTHY TO AN IMPAIRED FUNCTIONAL NETWORK
  54. 54. A brain disorder in which thinking abilities are mildly impaired. Individuals with MCI are able to function in everyday activities but have difficulty with memory, trouble remembering the names of people they met recently, the flow of a conversation, and a tendency to misplace things. Every year, around 10% of MCI patients develop Alzheimer’s disease. We perform magnetoencephalograms (MEG) to a group of 19 MCI's patients and 19 control subjects during a memory task. By means of the synchronization likelihood (SL) we quantify the interaction between the 148 channels of the MEG system and we obtained a weighted connectivity matrix between cortical areas. ❑ What is Mild Cognitive Impairment (MCI)? ❑ The experiment EXAMPLE 1: MILD COGNITIVE IMPAIRMENT J.M. Buldú, R. Bajo, F. Maestú et al., Reorganization of Functional Networks in Mild Cognitive Impairment, PLoS ONE 6(5): e19584 (2011)
  55. 55. Topological analysis of the functional networks of both groups (Control and MCI): EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  56. 56. Differences between the MCI and Control groups: ❑ Global Parameters: ❑The network strength K increases (+15.9%) ❑ Network outreach increases (+23.4%) (and more than the increase in K) ❑The network modularity decreases (-13.5%) ❑ Normalized Parameters: ❑ Normalized clustering decreases (-13.6%): CCONTROL =1.76 CMCI =1.52 ❑ Normalized outreach increases (+6.7%): OCONTROL =0.63 OMCI =0.67 CAUTION! The functional network is becoming random ^ ^ ^ ^ EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  57. 57. ❑ Intra-lobe synchronization: ❑The intra-lobe synchronization increases ❑The inter-lobe synchronization increases (more than the intra-lobe sync.) ❑ Modularity decreases CAUTION! The segregated operation of the brain is decreasing In-strengthOut-strengthModularity Differences between the MCI and Control groups, Inter-lobe connections: EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  58. 58. Within module degree Participation coefficient From macroscopic (network) to microscopic (node) analysis: EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  59. 59. Δ Δ Nodes increase their participation From macroscopic (network) to microscopic (node) analysis: EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  60. 60. MCI diagnostic must be done by analysing longitudinal recordings EXAMPLE 1: MILD COGNITIVE IMPAIRMENT Caution, GIGO is around...
  61. 61. Caution, GIGO is around... “Lies, damned lies and statistics” From :The Evolution of Adult Height in Europe: A Brief Note* Jaume Garcia and Climent Quintana-Domeque I’m Swedish! EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  62. 62. ? Randomness Networkstrength Control Alzheimer M.C.I. We n e e d l o n g i t u d i n a l experiments in order to understand the emergence of MCI The evolution of MCI to Alzheimer is still unknown … despite there are some clues ❑ High Synchronization ❑ Low clustering ❑ Higher outreach ❑ Low modularity ❑ Higher Randomness ❑ Low Synchronization ❑ Low clustering ❑ Higher Randomness Some conclusions: EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
  63. 63. Another good candidate:Trauma recovering therapy Accident Head Trauma Cognitive Therapy MEG recording (after the accident) MEG recording (9-14 months of therapy) Comparison between both networks N.P. Castellanos, I. Leyva, J.M. Buldú, et al., “Principles of recovery from traumatic brain injury: reorganization of functional networks, Neuroimage, 55, 1189-1199 (2011). EXAMPLE 1I: TRAUMATIC BRAIN INJURY
  64. 64. Band δ [1-4 Hz] Band α [8-13 Hz] Network changes: ❑ The delta band is overconnected ❑ The alfa band is underconnected ❑ The cognitive therapy shifts network parameters towards control values Black bars:After the TBI Grey bars:After the therapy EXAMPLE 1I: TRAUMATIC BRAIN INJURY Another good candidate:Trauma recovering therapy
  65. 65. C. Develop models in order to explain the changes found in impaired functional networks: • Identify what are the rules that determine the network distortion. ANALYZING FUNCTIONAL BRAIN NETWORKS
  66. 66. Two specific applications of network modeling: ❑ Mild Cognitive Impairment ❑Traumatic Brain Injury EVOLUTIONARY NETWORK MODELS
  67. 67. 1) We select a link randomly. 2) We change the weight of the link according to a certain function: w'ij=wij [1+λ+η] ξ(dij) 3) We normalize and recalculate the network parameters. 4) We go back to step 1. w'ij= modified link weight wij = previous link weight λ=degradation rate (λ 0) η= noise term ξ(dij)= length dependence function dij= link length EVOLUTIONARY NETWORK MODELS: MCI Develop models in order to explain the changes found in impaired functional networks:
  68. 68. Mild Cognitive Impairment: Real data versus evolutionary models Real data Models EVOLUTIONARY NETWORK MODELS: MCI
  69. 69. Healthy brain Impaired brain Length dependent Length independent EVOLUTIONARY NETWORK MODELS: MCI Develop models in order to explain the changes found in impaired functional networks:
  70. 70. The goal of this model is enhancing those links with higher initial weights. This leads to an increase of the relative difference between higher and lower weights along the evolution. Modeling network recovery inTraumatic Brain Injury: Contrasting model (T+): Unifying model (T-): The global average strength of the matrix decreases and, in addition, the relative differences between link weights are reduced at each time step. EVOLUTIONARY NETWORK MODELS: TBI
  71. 71. Post (after therapy) *Pre (before therapy) Control (healthy subject) Alpha band Contrasting model Unifying model EVOLUTIONARY NETWORK MODELS: TBI
  72. 72. We are accumulating errors from the previous two steps Functional networks are not static High variability in the results (Functional networks do not evaluate function) But… above all… STEP III: Network Analysis ANALYZING FUNCTIONAL BRAIN NETWORKS
  73. 73. … NETWORK MEASURES ARE COMMONLY MISINTERPRETED…. … SINCE WE NORMALLY FORGET THAT WE ARE ANALYZINGTHE BRAIN! ANALYZING FUNCTIONAL BRAIN NETWORKS
  74. 74. (III-III)
  75. 75. THE WHOLE PROCESS IS A MINELAND 2.4 The Brain as a Complex Network 39 0MROW *MPXIVMRK1IXVMGW 7XEXMWXMGW (ITIRHIRGMIW2SHIW Brain activity Recorded signals Connectivity Matrix Graphs Topological properties Neuromarkers Healthy vs. Diseased Rest vs. Task Figure 2.5: The general framework of brain networks. Clockwise guideline. Nodes can be regarded as sensor or electrodes recording the electromagnetic signals of the brain, which may contain dependencies based on correlation or causality. These interdependencies, or link weights, lead to a weighted connectivity matrix, which is the mathematical representation of a network. This
  76. 76. HOW CAN I COMPARE NETWORKS BETWEEN THEM? Anatomical network (Hagmann et aI., 2008) and functional network (Honey et aI., 2009) of the same group of individuals. 998 regions of interest (ROIs).The anatomical matrix is positive while the functional one has both positive/negative values. RH: right hemisphere, LH: left hemisphere. Anatomical Network (DTI) Functional Network (fMRI)
  77. 77. A SOCIAL ANALOGY: Facebook: Four views of the same Facebook network. Respectively: friendship network, profiles visited, unidirectional c o m m u n i c a t i o n a n d bidirectional communication. Same network, different levels of information. D. Easley J. Kleinberg, Networks, crowds and markets. HOW CAN I COMPARE NETWORKS BETWEEN THEM?
  78. 78. fMRI in (A) resting state and (B) during a memory task. Functional relations between the most active nodes. Node description: rMTL, right medial temporal lobe; IMTL, left medial temporal lobe; dmPFC, dorsomedial prefrontal cortex; vmPFC, ventro medial prefrontal cortex; rTC, right temporal cortex; lTC, left temporal cortex; rIPL, right inferior parietal lobe; lIPL, left inferior parietal lobe. Fransson et al., Neuroimage (2008). Functional networks adopt different configurations depending on the task you are carrying out: PROBLEM: FUNCTIONAL NETWORKS CHANGE
  79. 79. Functional network (fMRI) for groups of different ages.. In the picture, nodes are grouped following a spring algorithm.The frontal region is depicted in blue. We can observe how it segregates with the maturity of the functional network. Fair et al. PLoS Comp. Bio.(2009). They also change with age: PROBLEM: FUNCTIONAL NETWORKS CHANGE
  80. 80. The underlying anatomical network influences the dynamics but, in turn, the dynamics influences the anatomical network. For example, hebbian learning reinforces connexions between regions that are usually coordinated. Sporns, The networks of the Brain. Functional networks do not evolve…. they co-evolve! determina afecta evolución topológica afecta dinámica neuronal topología estado determina PROBLEM: TOPOLOGY AND DYNAMICS ARE INTERRELATED
  81. 81. FUNCTIONAL BRAIN NETWORKS: RISKS CHALLENGES When projecting the brain activity into a network, we are loosing a lot of information… … and we may forget what is behind…
  82. 82. How to interpret the results of the network analysis? FUNCTIONAL BRAIN NETWORKS: RISKS CHALLENGES
  83. 83. “…the analysis reported here looks at the synchronizability from different perspective and considers the synchronization properties of the brain networks rather than looking for a synchronous pattern in the original EEG signal…” EXAMPLE 1: Synchronizability M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011)1184 compared to other bands), where the synchroniz Fig. 7. Measure of synchronizability of brain functional netw index, i.e., the eigenratio (the largest eigenvalue of the Laplac patients and normal controls. Other designations are as Fig. M. Jalili, M.G. Kny1184 compared to other bands), where the synchronization properties of the SZ networks were worse than those of controls. Ref. [43]), decreased cortic increased cell packing dens neurons [40], suggest the d dysconnection hypothesis Fig. 7. Measure of synchronizability of brain functional networks in SZ patients compared to normal controls. The graphs index, i.e., the eigenratio (the largest eigenvalue of the Laplacian matrix of the connection graph divided by its second sma patients and normal controls. Other designations are as Fig. 3. M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–11184 Synchronizability parameter for the control and patient (schizophrenia) group in the alpha band. Fig. 2. Whole-head difference maps of nod in Eq. (4)) for delta, theta, alpha, beta, and with strength values significantly higher in gray regions. Fig. 3. Functional segregation and integra worldness index as a function of the thres brain functional networks were based on EEG-based functional networks in schizophrenia Mahdi Jalili a,n , Maria G. Knyazeva b,c a Department of Computer Engineering, Sharif University of Technology, Tehran, Iran b Department of Clinical Neuroscience, Centre Hospitalier Universitaire Vaudois (CHUV), and University of Lausanne, Lausanne, Switzerland c Department of Radiology, Centre Hospitalier Universitaire Vaudois and University of Lausanne, Switzerland a r t i c l e i n f o Keywords: EEG Schizophrenia Functional connectivity Graph theory Unpartial cross-correlation Partial cross-correlation a b s t r a c t Schizophrenia is often considered as a dysconnection syndrome in which, abnormal interactions between large-scale functional brain networks result in cognitive and perceptual deficits. In this article we apply the graph theoretic measures to brain functional networks based on the resting EEGs of fourteen schizophrenic patients in comparison with those of fourteen matched control subjects. The networks were extracted from common-average-referenced EEG time-series through partial and unpartial cross-correla- tion methods. Unpartial correlation detects functional connectivity based on direct and/or indirect links, while partial correlation allows one to ignore indirect links. We quantified the network properties with the graph metrics, including mall-worldness, vulnerability, modularity, assortativity, and synchronizability. The schizophrenic patients showed method-specific and frequency-specific changes especially pro- nounced for modularity, assortativity, and synchronizability measures. However, the differences between schizophrenia patients and normal controls in terms of graph theory metrics were stronger for the unpartial correlation method. 2011 Elsevier Ltd. All rights reserved. 1. Introduction Techniques from graph theory are increasingly being applied to model the functional and/or structural networks of the brain [1,2]. The brain networks can be studied at different levels ranging from micro-scale containing a number of interconnected neurons to macro-scale containing distributed brain regions. To construct the large-scale networks, signals recorded from the brain via methods such as electroencephalography (EEG), magnetocephalography (MEG), functional magnetic resonance imaging (fMRI), or diffusion tensor imaging (DTI), are used [3–7]. Often, binary (directed or undirected) adjacency matrices are analyzed [1,2], where binary links represent the presence or absence of a connection. The first step in analyzing brain networks is to extract its structure from the time-series. Possible methods are cross-correlation, coherence, and synchronization likelihood [3–6]. The next step is to represent it in a number of biologically meaningful measures. To this end, measures such as characteristic path length, efficiency of connec- tions, clustering coefficient, modularity, node degree and central- free properties [9,10]. Graph theoretical analysis on anatomical and functional networks of the brain have revealed its economical small-world structure characterized by high clustering (transitiv- ity) and a short characteristic path length [11]. The brain func- tional networks are cost-efficient in the sense that they provide efficient parallel processing for low connection cost [12]. Brain disorders influence the anatomical and functional brain networks. Brain wirings may show abnormal patterns in schizophrenia (SZ). SZ symptoms affect the patients by manifesting as auditory hallucinations, paranoid or bizarre delusions and/or disorganized speech and thinking in the context of significant social and/or occupational dysfunction. About 1% of the population worldwide suffers from different forms of SZ [13]. Additionally, another 3% of the population has SZ-type personality disorders. SZ is the fourth leading cause of disability in the developed counties for people at the age of 15–44. Schizophrenic patients show the abnormal patterns of brain connectivity. MRI-based studies on a large group of SZ patients revealed the reduced hierarchy of multimodal networks and Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cbm Computers in Biology and Medicine Computers in Biology and Medicine 41 (2011) 1178–1186 FUNCTIONAL BRAIN NETWORKS: RISKS CHALLENGES
  84. 84. EXAMPLE 1: Synchronizability 60 120 180 time [s] es during a seizure. Vertical broken lines d Tend of the seizure. ͑a͒ Exemplary time e cluster coefficient C͑w͒ of the epileptic arger than the corresponding value of a ure onset and attains a maximum devia- the seizure. Already prior to seizure end, 15 25 λ max (w) 0.2 0.6 1 b λ min (w) 20 60 100 c S(w) 0 60 0.2 0.4 0.6 0.8 dε(w) Evolving functional network properties and synchronizability during human epileptic seizures Kaspar A. Schindler,1,2,a͒ Stephan Bialonski,1,3 Marie-Therese Horstmann,1,3,4 Christian E. Elger,1 and Klaus Lehnertz1,3,4,b͒ 1 Department of Epileptology, University of Bonn, Sigmund-Freud-Strasse 25, 53105 Bonn, Germany 2 Department of Neurology, Inselspital, Bern University Hospital and University of Bern, Switzerland 3 Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany 4 Interdisciplinary Center for Complex Systems, University of Bonn, Römerstrasse 164, 53117 Bonn, Germany ͑Received 21 May 2008; accepted 10 July 2008; published online 15 August 2008͒ We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with different anatomical onset locations by statistical and spectral properties of functionally defined networks. We observe a concave-like temporal evolution of characteristic path length and cluster coefficient indicative of a movement from a more random toward a more regular and then back toward a more random functional topology. Surprisingly, synchronizability was significantly de- creased during the seizure state but increased already prior to seizure end. Our findings underline the high relevance of studying complex systems from the viewpoint of complex networks, which may help to gain deeper insights into the complicated dynamics underlying epileptic seizures. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2966112͔ Epilepsy represents one of the most common neurological disorders, second only to stroke. Patients live with a con- siderable risk to sustain serious or even fatal injury dur- ing seizures. In order to develop more efficient therapies, the pathophysiology underlying epileptic seizures should be better understood. In human epilepsy, however, the exact mechanisms underlying seizure termination are still as uncertain as are mechanisms underlying seizure initia- tion and spreading. There is now growing evidence that an improved understanding of seizure dynamics can be achieved when considering epileptic seizures as network phenomena. By applying graph-theoretical concepts, we analyzed seizures on the EEG from a large patient group and observed that a global increase of neuronal synchro- nization prior to seizure end may be promoted by the underlying functional topology of epileptic brain dynam- ics. This may be considered as an emergent self- regulatory mechanism for seizure termination, providing clues as to how to efficiently control seizure networks. techniques.11–13 Then two nodes are connected by an edge, or direct path, if the strength of their interaction increases above some threshold. Among other structural ͑or statistical͒ pa- rameters, the average shortest path length L and the cluster coefficient C are important characteristics of a graph.1,14 L is the average fewest number of steps it takes to get from each node to every other, and is thus an emergent property of a graph indicating how compactly its nodes are interconnected. C is the average probability that any pair of nodes is linked to a third common node by a single edge, and thus describes the tendency of its nodes to form local clusters. High values of both L and C are found in regular graphs, in which neigh- boring nodes are always interconnected yet it takes many steps to get from one node to the majority of other nodes, which are not close neighbors. At the other extreme, if the nodes are instead interconnected completely at random, both L and C will be low. Recently, the emergence of collective dynamics in com- CHAOS 18, 033119 ͑2008͒ Evolving synchronizability during an epileptic seizure. The synchronizability parameter increases, thus being the network LESS synchronizable. FUNCTIONAL BRAIN NETWORKS: RISKS CHALLENGES “…we observed a concave-like temporal evolution, with highest values of S ︎i.e., lowest synchronizability︎ in the middle of the seizure, followed by a decline ︎i.e., an increasing synchronizability︎…” “…while the aforementioned interpretation WOULD indicate that the transient evolution in graph properties is an active process of the brain to abort a seizure, our findings could also be understood as a passive consequence of the seizure itself.”

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