Consciousness, Graph theory and brain network tsc 2017
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The document explores k-shell decomposition as a method to analyze the hierarchical organization of the human brain, revealing complexities in neural connectivity and the emergence of consciousness. It discusses the significance of graph theory in mapping brain structures, identifying core and peripheral functions within the cortical network, and emphasizes how this modeling aids in understanding data integration and brain functions. Future research aims to further investigate local processors within different shells to deepen our understanding of brain functionality and structure.
Consciousness, Graph theory and brain network tsc 2017
- 1. K-shell decomposition reveals hierarchical cortical organization in human brain Nir Lahav, Baruch Ksherim, Eti Ben Simon, Adi Maron-Katz, Reuven Cohen, Shlomo Havlin Department of physics, Bar-Ilan University, Israel
- 2. There are 100 Billion Neurons in the brain 100,000,000,000,000 connections between the neurons How can we study such high level of complexity? Mathematics allows us to better understand and accurately predict the behavior of the system We need to develop new mathematical tools in order to cope with complexity
- 3. The activity of the network is greater then the sum of the activities of its nodes. Emergence of new properties Consciousness as an emergent phenomenon of the network
- 4. Do we have a Homunculus? The difference between unaware and aware visual perception is correlated with activity of widespread network of cortical regions Amanda Bischoff-Grethe. "Conscious and Unconscious Processing of Nonverbal Predictability in Wernicke’s Area". The Journal of Neuroscience, 2000
- 5. 5 From Dehaene et al, 2001 Experimental results: Wide forward activation due to conscious visual input, but not unconscious.
- 6. Exploring brain structure using graph theory Mapping the brain into a network which consists of Nodes and Edges Node (GM) Edge (WM) Mapping complex systems into a Network Internet Servers network Social network Network Theory
- 7. Defining network properties: node degree -K - the edge amount of a node Node (GM) Edge (WM) K=1 K=2 hub - a node with a significantly higher degree than the average degree Distance between nodes will defined as the shortest path between node i and node j ji ijd NN l )1( 1 Average distance of the network Bassett, D. S. "Small World Brain Networks". The Neuroscientist 12 (2006) dij – distance between node i and node j N – total number of nodes in the network
- 8. Defining network properties: clustering 1 3 2 Your friend’s friend Will also be your friend Large quantity of triangles in the network stering coefficient of the whole network will be: C can be between 0 and 1 i ic n C 1 Newman, M. E. J. "The structure and function of complex networks". SIAM review 45, 167-256 (2003( Local structures of the network
- 9. The brain is assortative, has a small-world organizatio hubs and modules Wang J. et al. 2010 Bassett, D. S. "Small World Brain Networks". The Neuroscientist 12 (2006) Hagmann, P. et al. "Mapping human whole-brain structural networks with diffusion MRI". PLoS ONE 2, e597 (2007) Hagmann, P. et al. "Mapping the structural core of human cerebral cortex". PLoS Biol 6, e159 (2008) Ben Simon et al. “tired and misconnected: a breakdown of brain modularity without sleep”, Human brain mapping (2017)
- 10. Rich-Club Organization of the Human Connectome Van den heuvel and sporns,“Rich-Club Organization of the Human Connectome". Journal of neuroscience (2011)
- 11. focused on the individual degree of the node We want to examine the emergence of global properties in the cortical network We need to go beyond the properties of a specific hub and examine the various structural layers that make up the network Issues with current studies
- 12. • proceed to a more complex analysis called k-shell decomposition in order to build a model of brain topology • Examine the emergence of global properties in the cortical network • Correlate network topology with known brain functions • Reveal hierarchical structure of the human cortical network Study goals
- 13. k-shell decomposition 1 2 4 5 K=1 K=2 1st core=3 1st shell=2 2nd core=0 2nd shell=33 Nucleus=3 nodes 1st Shell=2 nodes
- 14. k-shell decomposition k-shell decomposition not only checks the degree of a node but also the degree of the nodes that connect with that node (the connectivity neighborhood of the node)
- 15. k-shell decomposition as a phase transition process Crusts (K<<kmax) Crusts (K<kmax) Crusts (K~kmax) Giant component ~ o(crust) Isolated node ~75% ~25% ~0.5% Topology of Internet sites network
- 16. MRI DTI 998 nodes 14,865 edges 2 3 4 Hagmann, P. et al. "Mapping the structural core of human cerebral cortex". PLoS Biol 6, e159 (2008) Methods Structural networks! n=6 networks
- 17. Nucleus shells (giant component) Nucleus shells (giant component) ~75% ~25% ~0.5% ~18% ~56% ~26% ~80% ~20% ~0.2% Isolated nodes Isolated nodes model of the internet network topology model of the cortical network topology model of an average random cortical network topology Cortical network topology average n=6 One component
- 18. Nucleus shells (giant component) Nucleus shells (giant component) ~18% ~56% ~26% ~80% ~20% Isolated nodes model of the cortical network topology model of a random cortical network topology Cortical network topology One component Not all the highest Hubs are in the Nucleus! Only 50% of the highest hubs are in the nucleus, The rest 50% fall on average in the last 4 shells
- 19. correlation between topology and known brain functions Front FrontBack Back
- 21. Lahav et al, “K-shell decomposition reveals hierarchical cortical organization of the human brain". New journal of physics (2016) Every anatomical region has its own shell “fingerprint”
- 23. The hierarchies have a modular structure Lahav et al, “K-shell decomposition reveals hierarchical cortical organization of the human brain". New journal of physics (2016)
- 24. Nucleus Low hierarchy – low shells, specific functions (e.g. fusiform – face recognition) Data continues to integrate in middle hierarchy – high shells, complex functions (e.g. right DLPFC – hub of executive network) High hierarchy – nucleus, highest amount of data integration (e.g. cingulate cortex , 90% overlap with default network) Model of data integration in the brain
- 25. Nucleus Giant component Most of the local structures are in the Giant component The Nucleus is adding shortcuts and global structures Nucleus 1 3 2 Nucleus Vs. Giant component
- 26. High hierarchy – nucleus, highest amount of data integration One component , spread all over the cortex highly connected within itself has global structures “global interconnected collective”
- 27. Nucleus - “global interconnected collective” has global structures - global function the nucleus might serve as a platform for consciousness to emerge (high correlation with consciousness related regions)
- 28. Nucleus - “global interconnected collective” Both Global Workspace Theory and integrated information theory predict the existence of a Nucleus
- 29. hierarchal structure which mediates data integration To Conclude: k-shell decomposition reveals hierarchal structure of data integration in the human cortex. From local processing to high order global functions and the nucleus is the platform for the emergence of consciousness
- 30. THANK YOU FOR YOUR ATTENTION! Special thanks to: Prof. Shlomo Havlin Dr. Reuven Cohen Dr. Itay Horvitz Kobi Flax Eti Ben Simon
- 31. Giant component has a small-world organization Nucleus Giant component Most of the local structures are in the Giant component The Nucleus is adding shortcuts and global structures 3.06 3.58 2.4 0 0.5 1 1.5 2 2.5 3 3.5 4 Network Giant component Random network Average distance 0.4 0.404 0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Network Giant component Random network Clustering coefficient Nucleus 1 3 2
- 32. Nucleus (high hierarchy)Low hierarchy Data flow (connections amount between the hierarchies) Mediom hierarchy
- 33. בגרעין פעם אף נמצאים שלא מוחיים אזורים ימין בהמיספירה כולם
- 34. Never in Nucleus Areas that never appear in the Nucleus • All from right hemisphere, most of them from temporal lobe • Have specified functions such as localized sensory perception(e.g. the fusiform face area)
- 35. summary • The brain network has an Hierarchical structure Nucleus crust (giant component) ~80% ~20% ~0.2%Isolated nodes • Giant component has a small-world organization • Most of the local structures are in the Giant component • Nucleus is adding shortcuts and global structures model of the brain network topology • The brain network is efficient and very stable
- 36. summary • known FUNCTIONAL networks in Nucleus & crusts - connecting structural data and functional data • k-shell decomposition revels hierarchal structure which mediates data integration in the brain Nucleus (highest processors) related to self processing (Northoff G. et al, 2004) Local processors (hubs) Nodes Giant component Nucleus
- 37. Interconnected collective areas (high hierarchy)
- 38. Distributed interconnected collective areas (high hierarchy)
- 41. Distributed specialized areas - Low hierarchy
- 42. Future work • examine the components of different shells to establish the internal hierarchy of the brain’s local processors • perform k- shell decomposition to The entire brain including sub-cortical structures
- 43. 9.2 3.4 3 2.8 2.8 2.2 0 2 4 6 8 10 12 14 16 precuneus superiorparietalcortex inferiorparietalcortex superiorfrontalcortex superiortemporalcortex cuneus Comparison to previous work Anatomical areas AveragenumberofHubs Anatomical areas with highest number of Hubs average on all networks
- 44. AverageHubsdensity Anatomical areas Anatomical areas according to average Hubs density
- 45. Precuneus paracentral lobule posterior cingulate cortex isthmus of the cingulate cortex superior frontal cortex superior parietal cortex rostral anterior cingulate cortex caudal anterior cingulate cortex precentral gyrus postcentral gyrus inferior parietal cortex cuneus Max core: Comparison to previous work Superior Temporal cortex Frontal pole pericalcarine cortex NOT in Max core: Highest number of Hubs Highest Hubs density No prediction
- 47. small-world index random random sw / / The small-world index is defined as: random random Clustering coefficient : Average distance: For the five individual subjects the average small-world index was: 10.64 (+/- 0.61) high degree of small-world organization Hagmann, P. et al. "Mapping human whole-brain structural networks with diffusion MRI". PLoS ONE 2, e597 (2007)
- 48. Defining network properties: clustering 1 3 2 Your friend’s friend Will also be your friend Large quantity of triangles in the network Number of triangles that involve node i Number of triplets in which node i is in the center ci= Clustering coefficient of the whole network will be: C can be between 0 and 1 i ic n C 1 c3=1/61 2 3 4 5 Newman, M. E. J. "The structure and function of complex networks". SIAM review 45, 167-256 (2003(
- 49. • 6 networks obtained from 5 subjects (network 1 and 2 were obtained from the same subject during two different time points) • Connection matrices were obtained from the networks • Repeated scans were correlated with r = 0.78 • The average correlation between participants is r = 0.65 Hagmann, P. et al. "Mapping human whole-brain structural networks with diffusion MRI". PLoS ONE 2, e597 (2007) Experimental data
- 50. rBSTS rCAC rCMF rCUN rENT rFP rFUS rIP rIT rISTCrLOCC rLOF rLING rMOF rMTrPARCrPARHrPOPErPORBrPTRI rPCALrPSTC rPC rPRECrPCUN rRAC rRMF rSF rSP rSTrSMAR rTP rTT 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 distribution of crusts and core nodes according to right anatomical regions network3 threshold 75% anatomical regions numberofcrust+core lBSTS lCAC lCMF lCUN lENT lFP lFUS lIP lIT lISTC lLOCC lLOF lLING lMOF lMT lPARClPARHlPOPE lPORB lPTRI lPCALlPSTC lPC lPREClPCUN lRAC lRMF lSF lSP lST lSMAR lTP lTT 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 distribution of crusts and core nodes according to left anatomical regions network3 threshold 75% anatomical regions numberofcrust+core 5 10 15 20 5 10 15 20 25
- 51. Max core of network 2 was reveled in degree 19 with 417 nodes and mean degree of 42 Network 2 has 10 Isolated nodes Max core of network 3 was reveled in degree 20 with 151 nodes and mean degree of 46 Network 3 has no Isolated node Max core of network 4 was reveled in degree 19 with 268 nodes and mean degree of 41 Network 4 has 1 Isolated node Max core of network 5 was reveled in degree 19 with 107 nodes and mean degree of 44 Network 5 has no Isolated node Max core of network 6 was reveled in degree 21 with 123 nodes and mean degree of 51 Network 6 has no Isolated node core crust
- 52. correlation between topology and known brain functions 18th crust (out of 19) of network 4 consists of three sub networks: Theory of mind Decision making by language & sensor input motor network FrontBack BackFront Kandel ER, et al. Principles of Neural Science, 4th ed. McGraw-Hill, New York
- 53. Not all the highest Hubs are in the Max core! (k>60) Added nodes crust 17 Hubs (k>60): 5 (from total of 45)
- 54. 0 20 40 60 80 100 120 2 2.5 3 3.5 4 4.5 Scattering of mean distances for each vertex vs. vertex degree for network 2 vertex degree meandistanceofthevertex Scattering of mean distances for each vertex vs. vertex degree Scattering of mean distances vs. degree (of all vertexes with same degree) Mean distance of each vertex according to the vertex Degree, network 2 Mean distance of each vertex according to the vertex Degree Mean distance of all vertices with the same Degree
- 55. 0 5 10 15 20 25 רכזות כמות 2 3 4 5 6 רשת מספר השונות ברשתות 60 דרגה מעל רכזות כמות ימין בהמיספרה 60 דרגה מעל רכזות כמות שמאל בהמיספרה 60 דרגה מעל רכזות כמות
- 56. rBSTS rCAC rCMF rCUN rENT rFP rFUS rIP rIT rISTC rLOCC rLOF rLING rMOF rMT rPARC rPARH rPOPE rPORB rPTRI rPCAL rPSTC rPC rPREC rPCUN rRAC rRMF rSF rSP rST rSMAR rTP rTT 0 10 20 30 40 50 Number of nodes in each right anotmical region. mean = 15.0909, std = 11.0916 Anatomical region Numberofnodes lBSTS lCAC lCMF lCUN lENT lFP lFUS lIP lIT lISTC lLOCC lLOF lLING lMOF lMT lPARC lPARH lPOPE lPORB lPTRI lPCAL lPSTC lPC lPREC lPCUN lRAC lRMF lSF lSP lST lSMAR lTP lTT 0 10 20 30 40 50 Number of nodes in each left anotmical region. mean = 15.1515, std = 10.7997 Anatomical region Numberofnodes
- 57. 0 5 10 15 20 25 1 1.5 2 2.5 3 3.5 4 4.5 5 mean distance of crust of degree k. last k is crusts+core network 6 degree of crust meandistance 0 5 10 15 20 25 0 200 400 600 800 1000 number of nodes in crust of degree k. last k is crusts+core network 6 number of isolate nodes: 0 degree of crust numberofnodes mean distance of k crust mean distance of biggest cluster in k crust mean distance of 2nd biggest cluster in k crust size of k crust size of biggest cluster in k crust size of 2nd biggest cluster in k crust X10 S - mean size of finite component P - node probability to connect to a component P∞ - node probability to connect the giant component Percolation process
- 61. Diffusion Tensor ImagingDTI = Axons Myelin ~2m Axons Myelin ~2m Brain imaging techniques DTI : imaging white matter by water diffusion measurement
- 62. D// , λ1 D^ , λ2 D^ , λ3 Fractional Anisotropy Anisotropy Map Directionality Map Basser P.J., Pierpaoli C. Magn. Reson. Med. 39, 928-934 (1998). FA=
- 63. Results 0 20 40 60 80 100 120 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 nodes density vs. degree (averaged over 6 networks). The average degree is 29.1887. the Standard deviation is 15.0205 degree nodesdensity(averagedover6networks) 1 2 3 4 5 6 Average poisson Gaussian Nodesdensity Nodes density according to Degree average degree is 29.18 standard deviation: 15
- 64. 10 0 10 1 10 2 10 -4 10 -3 10 -2 10 -1 lof of degree logofnodesdensity nodes density vs. degree (averaged over 6 networks). The average degree is 29 the Standard deviation is 15 0 20 40 60 80 100 120 10 -3 10 -2 10 -1 degree logofnodesdensity nodes density vs. degree. The average degree is 29.18 the Standard deviation is 15.02 Nodes density according to Degree average degree is 29.18 standard deviation: 15 Nodesdensity Degree Average Degree density (all networks) Gaussian distribution Average Degree density (all networks) No linear dependence between the Degrees and the Nodes density (logarithmic scale)
- 65. 20 30 40 50 60 70 80 90 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Clustering coefficients as function of degree network 4 Degree Meanofclusteringcoefficients 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 Logarithm of clustering coefficients as function of Log(Degree) network 4 log (Degree) Log(meanofclusteringcoefficients) y = - 0.69*x + 0.61 Mean Clustering coefficient =0.4y = - 0.00012*x + 0.05 Clustering coefficient as function of Degree (k=20 cutoff), network 4
- 66. 20 30 40 50 60 70 80 90 0.03 0.035 0.04 0.045 0.05 0.055 Clustering coefficients as function of degree RANDOM network 4 Degree Meanofclusteringcoefficients 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -1.5 -1.45 -1.4 -1.35 -1.3 -1.25 Logarithm of clustering coefficients as function of log (degree) RANDOM network 4 Log (Degree) Log(meanofclusteringcoefficients) y = - 0.017*x - 1.3 data 1 data 2 linear Clustering coefficient as function of Degree, randomnetwork 4 Mean Clustering coefficient =0.043
- 67. 0 5 10 15 20 25 0 200 400 600 800 1000 numberofnodes in crust ofdegree k. last k is crusts+core network 6 numberofisolate nodes: 0 degree ofcrust numberofnodes size ofk crust size ofbiggest clusterin k crust size of2nd biggest clusterin k crust k-shell decomposition as a phase transition process Number of nodes in k- core = 123 Number of isolate nodes= 0
- 68. 0 2 4 6 8 10 12 14 16 18 20 22 0 100 200 300 400 500 600 700 800 900 1000 number of nodes in crust of degree k. last k is crusts+core network 6 degree of crust numberofnodes size of k crust size of biggest cluster in k crust size of 2nd biggest cluster in k crust X10 Number of nodes in k- core = 123 Number of isolate nodes= 0 2nd biggest cluster has 21 nodes in K=14 2nd biggest cluster has 4 nodes in K=15
- 69. 0 2 4 6 8 10 12 14 16 18 20 0 200 400 600 800 1000 numberofnodes incrustofdegreek.lastk is crusts+corerandomnetwork 6numberofisolatenodes:17 degreeofcrust numberofnodes sizeofk crust sizeofbiggestclusterink crust sizeof2ndbiggestclusterink crust Nodes in k core185 K core degree: 18 isolate nodes17 Random network 2nd biggest cluster has 9 nodes in K=13 2nd biggest cluster has 7 nodes in K=14
- 70. 0 5 10 15 20 25 1 1.5 2 2.5 3 3.5 4 4.5 5 mean distance ofcrust ofdegree k. last k is crusts+core network 6 degree ofcrust meandistance mean distance ofk crust mean distance ofbiggest cluster in k crust mean distance of2nd biggest cluster in k crust Mean distance of k core= 1.7893 Mean distance of whole network= 3.09 2nd biggest cluster has mean distance of 2.77 in K=14 2nd biggest cluster has mean distance of 1 in K=15
- 71. 0 2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 meandistanceofcrustofdegreek.lastk is crusts+corerandomnetwork 6 degreeofcrust meandistance meandistanceofk crust meandistanceofbiggestclusterink crust meandistanceof2ndbiggestclusterink crust Mean distance of k core: 2.18 Mean distance: 2.44 Random network 2nd biggest cluster has mean distance of 2.83 in K=14 2nd biggest cluster has mean distance of 2.09 in K=15
- 72. correlation between topology and known brain functions Front FrontBack Back Four major known FUNCTIONAL networks in the nucleus (form one component): Sensorimotor network and motor planningintegrating (precentral gyrus, postcentral gyrus, paracentral lobule) Default network (Precuneus, posterior cingulate cortex, isthmus of the cingulate cortex) Executive network (rostral & caudal anterior cingulate cortex, superior frontal cortex) High order visual areas (inferior parietal cortex, cuneus, superior parietal cortex)
- 73. k-shell decomposition reveals hierarchal structure which mediates data integration in the brain V1 V2 V3/MT Parietal’s High order visual areas Dorsal stream 2nd Crust below nucleus 2nd Crust below nucleus 2nd Crust below nucleus & Nucleus Nucleus & 2nd Crust below nucleus Dorsal stream (Where) Ventral stream (what) V2/ V1 V3/MT Parietal’s High order visual areas Nucleus Nucleus (highest processors) Local processors (hubs) Nodes
- 75. Nucleus (high hierarchy)Low hierarchy Data flow (connections amount between the hierarchies) Middle hierarchy
- 76. Never in Nucleus Always in Nucleus
- 77. Distance is the shortest path between two nodes (the distance between the two red nodes is 2) node edge hub hub Degree = 4 Hierarchical structure Clustering coefficient
- 78. Nucleus crust (giant component) Nucleus Crust (giant component) ~75% ~25% ~0.5% ~18% ~56% ~42% ~80% ~20% ~0.2% Isolated nodes Isolated nodes model of the internet network topology model of the brain network topology model of a random brain network topology Brain network topology average n=6 One component
- 79. 3.06 3.58 2.4 0 0.5 1 1.5 2 2.5 3 3.5 4 Network Giant component Random network Average distance 0.4 0.404 0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Network Giant component Random network Clustering coefficient
- 81. 9.2 3.4 3 2.8 2.8 2.2 0 2 4 6 8 10 12 14 16 precuneus superiorparietalcortex inferiorparietalcortex superiorfrontalcortex superiortemporalcortex cuneusAnatomical areas AveragenumberofHubs Anatomical areas with highest number of Hubs average on all networks for high hubs: k>60 always in nucleus Highest hierarchy Highest hierarchy Highest hierarchy left s.f. always in nucleus Left i.p. always in nucleus Highest hierarchy Right i.p. never in nucleus Middle hierarchy Left s.t. Highest hierarchy Right s.t. never in nucleus Middle hierarchy Highest hierarchy
- 82. AverageHubsdensity Anatomical areas Anatomical areas according to average Hubs density for high hubs: k>60 always in nucleus Highest hierarchy Highest hierarchy Left LOCC Highest hierarchy Right LOCC Middle hierarchy Middle hierarchy Highest hierarchy Highest hierarchy