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Measuring Brain Connectivity

Zhefu Chen MATH-553

Dec 9th ,2013

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OUTLINE

• Introduction and Background • Preliminaries and Example • A Real World Example—Macaque Neocortex • Conclusion and further research

INTRODUCTION AND BACKGROUND

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• Graph is connected– if every pair of vertices in graph belongs to a path[1]. • Connectivity of graph can be quantified by examining lengths of each of

these paths[8]. • What is brain connectivity? Brain connectivity refers to a pattern of anatomical links of statistical dependencies or of causal interactions between distinct units within a nervous system[2]. • Brain connectivity can be modeled as a graph where:

Brain Connectivity Graph

Neural elements Vertices

Synapses and axonal pathways Edges

INTRODUCTION AND BACKGROUND

Main accomplishment:

we studied four parameters of characteristic path length, global efficiency, local efficiency, and the clustering coefficient, and how these four parameters can be used to analyze brain connectivity[8].

Concepts of four parameters:

Characteristic path length L: average distance between two vertices,

𝐿 =1

𝑛∗ 𝑛−1 𝑑(𝑖, 𝑗)𝑖≠𝑗 [8].

Global efficiency 𝑬𝒈𝒍𝒐𝒃[3]: 𝐸𝑔𝑙𝑜𝑏 =1

𝑛∗ 𝑛−1

1

𝑑(𝑖,𝑗)𝑖≠𝑗

Local efficiency 𝑬𝒍𝒐𝒄: average of the global efficiencies over the

subgraphs 𝐺𝑖, 𝐸𝑙𝑜𝑐 =1

𝑛 𝐸(𝐺𝑖)𝑖∈𝐺 [8]

Clustering coefficient 𝑪: 𝐶 =1

𝑛 𝐶𝑖𝑖 [8] where 𝐶𝑖 is the subgraph

induced by the neighbors of i.

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PRELIMINARIES AND EXAMPLE In this chapter, we elaborately explain these four parameters with a simple example.

• 𝐿 =1

𝑛∗ 𝑛−1 𝑑(𝑖, 𝑗)𝑖≠𝑗 , where 𝑑(𝑖, 𝑗) is shortest path from 𝑖 to 𝑗.

• 𝐸𝑔𝑙𝑜𝑏 =1

𝑛∗ 𝑛−1

1

𝑑(𝑖,𝑗)𝑖≠𝑗

• 𝐸𝑙𝑜𝑐 =1

𝑛 𝐸(𝐺𝑖)𝑖∈𝐺 , where 𝐺𝑖 is the subgraphs of G which are induced by

the neighbors of i.

• 𝐶 =1

𝑛 𝐶𝑖𝑖 , where 𝐶𝑖 is the subgraph induced by the neighbors of i.

• 𝐶𝑖: local clustering coefficient[4] is :the ratio of the number of actual edges there are between neighbors to the number of potential edges there are between neighbors (all possible edges between the vertices including blue edges).

𝐶𝑖 =𝑒𝑥𝑖𝑠𝑡𝑒𝑑 𝑛𝑒𝑖𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠

𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙𝑙𝑦 𝑛𝑒𝑖𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠=2 {𝑒𝑗𝑘:𝑣𝑗,𝑣𝑘∈𝑁𝑖,𝑒𝑗𝑘∈𝐸 |

𝑘𝑖∗(𝑘𝑖−1)

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PRELIMINARIES AND EXAMPLE

Exercise 1: Let G be the following graph, determine L(G), 𝐸𝑔𝑙𝑜𝑏 𝐺 , 𝐶 𝐺 , 𝑎𝑛𝑑 𝐸𝑙𝑜𝑐 𝐺 :

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A

B

C D

E

𝐿 𝐺 : starting from vertex A, the shortest distance from A to other vertices are:{ 1,2,2,1 } ; vertex B, the shortest distance from B to other vertices are: {1,1,1 ,1}; vertex C, the shortest distance from C to other vertices are:{ 2,1,1,1 } ; vertex D, the shortest distance from D to other vertices are:{ 2,1,1,1 } ; vertex E, the shortest distance from E to other vertices are :{1,1,1,1}; The order of G is n=5, therefore:

𝐿 𝐺 =1

𝑛 ∗ 𝑛 − 1 𝑑 𝑖, 𝑗 =

1

5 ∗ 4 6 + 4 + 5 + 5 + 4

𝑖≠𝑗

=24

20= 1.2

𝐸𝑔𝑙𝑜𝑏 =1

𝑛 ∗ 𝑛 − 1 1

𝑑(𝑖, 𝑗)𝑖≠𝑗

We have all the distance 𝑑 𝑖, 𝑗 from 𝐿 𝐺 , let D be the

set of distance,

𝐷 = 1,2,2,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,

𝐸𝑔𝑙𝑜𝑏 =1

5 ∗ 4 1

1+1

2+1

2+1

1…+1

1=18

20

= 0.9

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PRELIMINARIES AND EXAMPLE A

B

C D

E

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PRELIMINARIES AND EXAMPLE

A

B

C D

E

G

• 𝐸𝑙𝑜𝑐 =1

𝑛 𝐸(𝐺𝑖)𝑖∈𝐺 , where 𝐺𝑖 is the subgraphs of G which

are induced by the neighbors of i. • In this graph we have 5 induced graph are listed below, we

compute their globe efficiency and sum them up to get the local efficiency value of graph G.

• 𝐸𝑙𝑜𝑐 =1

𝑛=5[𝐸𝑔𝑙𝑜𝑏 𝐺𝐴 +

𝐸𝑔𝑙𝑜𝑏 𝐺𝐵 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐶 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐷 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐸 ]

B

C D

E

𝐺𝐴, induced graph by

neighbor of vertex A

A

B

C D

E

𝐺𝐵, induced graph by

neighbor of vertex B

A

B

C D

E

𝐺𝐶 , induced graph by

neighbor of vertex C

A

B

C D

E

𝐺𝐷, induced graph by

neighbor of vertex D

A

B

C D

E

𝐺𝐸, induced graph by

neighbor of vertex E

A

Computing 𝐸𝑙𝑜𝑐 continued’

𝐸𝑙𝑜𝑐 =1

𝑛=5[𝐸𝑔𝑙𝑜𝑏 𝐺𝐴 + 𝐸𝑔𝑙𝑜𝑏 𝐺𝐵 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐶 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐷 +𝐸𝑔𝑙𝑜𝑏 𝐺𝐸 ]

=1

5

1

1212 + 10 + 11 + 11 + 10 =

54

60= 0.9

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PRELIMINARIES AND EXAMPLE

B

C D

E

𝐺𝐴, induced graph by

neighbor of vertex A

A

B

C D

E

𝐺𝐵, induced graph by

neighbor of vertex B

A

B

C D

E

𝐺𝐶 , induced graph by

neighbor of vertex C

A

B

C D

E

𝐺𝐷, induced graph by

neighbor of vertex D

A

B

C D

E

𝐺𝐸, induced graph by

neighbor of vertex E

A

• 𝐶 =1

𝑛 𝐶𝑖𝑖 , where 𝐶𝑖 is the subgraph induced by the neighbors of i.

• 𝐶𝑖 =𝑒𝑥𝑖𝑠𝑡𝑒𝑑 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠

𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙𝑙𝑦 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠=2 {𝑒𝑗𝑘:𝑣𝑗,𝑣𝑘∈𝑁𝑖,𝑒𝑗𝑘∈𝐸 |

𝑘𝑖∗(𝑘𝑖−1)

• For computing 𝐶𝑖, numerator is number of existed edges in neighbors of i, denominator is number of edges in that complete graph of neighbors of i.

For example, at vertex A, it has neighbors of B and E , existed edges between

B and E is 1, and number of edges in a complete 𝐾2 graph is 1. Therefore

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PRELIMINARIES AND EXAMPLE

𝐶𝐴 =1

1= 1 𝐶𝐵 =

4

6=2

3

𝐶𝐶 =3

3= 1 𝐶𝐷 =

3

3= 1

𝐶𝐸 =4

6=2

3

Potentially neihgborhood edges

Existed neighborhood

Computing 𝐶 𝐺 continued’

𝐶 𝐺 =1

𝑛 = 5𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶 + 𝐶𝐷 + 𝐶𝐸

=1

51 +2

3+ 1 + 1 +

2

3=13

15=0.8667

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PRELIMINARIES AND EXAMPLE

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

For a real world example, we study a macaque neocortex case. The network consists 47 nodes are represented macaque brain ([5]) and 505 edges are presented directed anatomical connection.

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Figure 1. Large-scale anatomical connection matrix of macaque neocortex[6]

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

We download CIJ matrix from the Brain Connectivity Tool-box[7], it is a adjacent matrix, the figure below is a clearer view of this adjacency matrix.

13 Figure 2. The adjacency matrix for CIJ [8]

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

Computing four parameters: 𝐿 𝐶𝐼𝐽 , 𝐸𝑔𝑙𝑜𝑏 𝐶𝐼𝐽 , 𝐸𝑙𝑜𝑐 𝐶𝐼𝐽 , 𝐶 𝐶𝐼𝐽

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𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 𝑳(𝑪𝑰𝑱): In Matlab, we have function to obtain shortest path between two vertices: DA=sparse(CIJ)

[dist]=graphallshortestpaths(DA);

By applying this code, we can get a matrix with shortest paths, and figure 3[8] on the right is a clearer view of distance matrix:

Figure 3 distance matrix

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX 𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 𝑳(𝑪𝑰𝑱):

Computing L(CIJ) is simply summing up all value of entries in the distance matrix then divide by n*(n-1),n is number of vertices in graph. We examine this by applying algorithm to previous example.

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A

B

C D

E

Figure 4 Matlab output for L(G)

The result is the same value with our manually computing result, thus we apply this algorithm to CIJ matrix. Value of L(CIJ) =2.0541.

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 𝑬𝒈𝒍𝒐𝒃 𝑪𝑰𝑱 :

𝐸𝑔𝑙𝑜𝑏 =1

𝑛 ∗ 𝑛 − 1 1

𝑑 𝑖, 𝑗𝑖≠𝑗

Considering there will be zeros value in the distance matrix which are values in diagonal entries, we use condition statement in Matlab to

avoid sum up zero value in 1

𝑑 𝑖,𝑗:

if dist(i,j)~=0

Eglob=Eglob+1/dist(i,j);

end

Therefore, we have 𝑬𝒈𝒍𝒐𝒃 𝑪𝑰𝑱 = 𝟎. 𝟓𝟕𝟏𝟒.

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A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 𝑬𝒍𝒐𝒄 𝑪𝑰𝑱 :

𝐸𝑙𝑜𝑐 =1

𝑛 𝐸(𝐺𝑖)𝑖∈𝐺 , we need to compute global efficiency of induced

graph of each vertex.

We use previous example to illustrate our algorithm here: in graph G, for computing 𝐸 𝐺𝐴 , we need to get its adjacency matrix and then compute its distance. The steps are:

A. For a vertex, delete row and column of that vertex in original matrix to get a new adjacency matrix of induced graph.

B. Compute distance matrix for 𝐺𝐴 and calculate 𝐸𝑔𝑙𝑜𝑏 𝐺𝐴

C. Keep doing step B until we finish all 𝐸𝑔𝑙𝑜𝑏 𝐺𝑖

D. Compute 𝐸𝑙𝑜𝑐 =1

𝑛 𝐸𝑔𝑙𝑜𝑏 𝐺𝑖

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A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

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𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 𝑬𝒍𝒐𝒄 𝑪𝑰𝑱 :

Previous example for examining algorithm of 𝑬𝒍𝒐𝒄: A

B

C D

E

Induced graph of neighborhood of A

Distance matrix of 𝐺𝐴

0111

1011

1101

1110

Adjacency matrix of G New adjacency matrix of 𝐺𝐴

01001

10111

01011

01101

11110

0111

1011

1101

1110

We can compute 𝐸 𝐺𝐴 according to distance matrix of induced graph of neighborhood of A . Thus, 𝑬𝒍𝒐𝒄 𝑪𝑰𝑱 = 𝟎. 𝟓𝟕𝟎𝟑

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

Computing 𝑪 𝑮 :

• 𝐶 =1

𝑛 𝐶𝑖𝑖 , where 𝐶𝑖 is the subgraph induced by the neighbors of i.

• 𝐶𝑖 =𝑒𝑥𝑖𝑠𝑡𝑒𝑑 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠

𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙𝑙𝑦 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑒𝑑𝑔𝑒𝑠=2 {𝑒𝑗𝑘:𝑣𝑗,𝑣𝑘∈𝑁𝑖,𝑒𝑗𝑘∈𝐸 |

𝑘𝑖∗(𝑘𝑖−1)

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For computing each 𝐶𝑖 , we still do row and column operations on adjacency matrix of graph: A. Check the value of entries of row and column where vertex 𝑉𝑖 belongs to; B. If value of entry 𝑎𝑖𝑗 is 0, substitute value 0 to row 𝑖 and column 𝑗.

C. Keep doing step B until we finish checking all entries. D. Sum up all value of entries in this new adjacency matrix for 𝐶𝑖

E. Applying 𝐶𝑖=2 {𝑒𝑗𝑘:𝑣𝑗,𝑣𝑘∈𝑁𝑖,𝑒𝑗𝑘∈𝐸 |

𝑘𝑖∗(𝑘𝑖−1) where numerator is summation value of

new adjacency matrix, 𝑘𝑖 is degree of 𝑉𝑖

A REAL WORLD EXAMPLE—MACAQUE NEOCORTEX

Computing 𝑪 𝑮 :

Previous example for examining algorithm of C(G):

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Adjacency matrix of G New matrix for existed edges for neighbor vertices of vertex A

E B

. A

𝐶𝐴 =1+1

2∗(2−1)=1

Therefore, 𝑪 𝑪𝑰𝑱 =𝟏

𝒏 𝑪 𝑪𝑰𝑱𝒊 = 𝟎. 𝟔𝟎𝟗𝟖

CONCLUSION

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CIJ 𝑲𝒏

𝐿(𝐺) 2.0541 1

𝐸𝑔𝑙𝑜𝑏(𝐺) 0.5714 1

𝐸𝑙𝑜𝑐(𝐺) 0.5703 1

𝐶(𝐺) 0.6098 1

Comparison of results with graph CIJ and a complete graph

Our conclusion for these four parameters are: For characteristic path length, the less edges in the graph, the larger value of 𝐿 𝐺 will be. For global efficiency, local efficiency and clustering coefficienct, the less edges in the graph, the smaller value of these parameters will be.

CONCLUSION AND FURTHER RESEARCH

For considering other random graphs with same edges at this brain net work, the results of four parameters follows properties of our conclusion. However, when we generate larger graphs with same edges, it will have chance to generate a disconnected graph in our Matlab programming. For further research, one of improvements will be always generating a connected graph with certain edges.

In addition, our project is subjected to a unweigthed graph, in the further research we can consider a weighted graph or even a weighted direct graph for these parameters.

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REFERENCES

[1] Douglas B. West, Introduction of Graph Theory 2nd Edition, Pearson Education, 2002

[2] Brain Connectivity http://www.scholarpedia.org/article/Brain_connectivity

[3] V. Latora and M. Marchiori, Efficient Behavior of Small-World Networks, Physical Review Letters E, Vol. 87, No. 19, (2001).

[4] local clustering coefficient: http://www.learner.org/courses/mathilluminated/interactives/network/

http://en.wikipedia.org/wiki/Clustering_coefficient

[5] C.J. Honey, R.Kötter, M. Breakspear, and O. Sporns, Network structure of cerebral cortex shapes functional connectivity on multiple time scales, PNAS vol. 104 no. 24 10240-10245, (2007).

[6] D.J. Felleman, and D.C. Van Essen, Cereb Cortex 1:1—47 (1991).

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[7] https://sites.google.com/site/bctnet/datasets File: macaque47.mat.

[8] Nathan D. Cahill, Joy Lind, Darren A. Narayan, Measuring Brain Connectivity.

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REFERENCES

THANK YOU

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