http://www.dynamic-connectome.org

http://neuroinformatics.ncl.ac.uk/

@ConnectomeLab

Dr Marcus Kaiser

A Tutorial in Connectome Analysis (I): Topological and Spatial Features of Brain Networks

Professor of NeuroinformaticsSchool of Computing Science /Institute of NeuroscienceNewcastle UniversityUnited Kingdom

2

Outline• What are neural networks?• Introduction to network analysis• How can the fibre tract network

structure be examined?

• Topological network organisation

3

What are neural networks?

4

Axons between neurons

Fibre tracts between brain areas

Links between cortical columns

Levels of connectivity

5

Types of connectivity

• Structural / Anatomical (connection): two regions are connected by a fibre tract

• Functional (correlation): two regions are active at the same time

• Effective (causation): region A modulates activity in region B

Sporns, Chialvo, Kaiser, Hilgetag.Trends in Cognitive Sciences, 2004

A B

Dorsal and ventral visual pathway

Visual system

Cortical networks6

8

Introduction to network analysis

9

Network ScienceRapidly expanding field:Watts & Strogatz, Nature (June 1998)Barabasi & Albert, Science (October 1999)

Modelling of SARS spreading over the airline network(Hufnagel, PNAS, 2004)

Identity and Search in Social Networks(Watts et al., Science, 2002)

The Large-Scale Organization of Metabolic Networks. (Jeong et al., Nature, 2000)

First textbook on brain connectivity (Sporns, ‘Networks of the Brain’, MIT Press, October 2010)

10 Origin of graph theory:Leonhard Euler, 1736

Bridges over the river Pregel in Königsberg (now Kaliningrad)Euler tour: path that visits each edge and returns to the origin

11

Nodes in graphs• Isolated nodes• Degree of a node• Connected graph• Average degree of a graph• Edge density: probability that

any two nodes are connectedd= E___

(N*N-1) /2)

v1

v3

v2v4 v5

Ø Isolated node: v5Ø Degree of a node:

d(v1)=2, d(v4)=1Ø Average degree of a graph:

D = (2+2+2+1+0)/5 = 1.4Ø Edge density

d=4/(5*4/2) = 0.4

12

Examples: edge density

sparse network(density ~ 1%)

dense network(density > 5%)

13

How can the fibre tract network structure be examined?

14

Tract tracing with dyes*

Anterograde: soma → synapseRetrograde: soma ← synapse

* Horseradish peroxidase (HRP) method; fluorescent microspheres; Phaseolus vulgaris-leucoagglutinin (PHA-L) method; Fluoro-Gold; Cholera B-toxin; DiI; tritiated amino acids

PHA-L: Phaseolus vulgaris-leucoagglutinin

15

Diffusion Tensor Imaging (DTI)

16

17

Topological network organisation

18

Archetypes of complex networks

Kaiser (2011) Neuroimage

Note: real complex networks show a combination of these types!

!!A!!Erdös'Rényi!random!!B!!Scale'free!!!!!!!!!!!!!!!C!!Regular!!!!!!!!!!!!!!!!!!!!!!!!D!!Small'world!!!!!!!!!!!!!E!!Modular!!!!!!!!!!!!!!!!!!!!!F!!Hierarchical!

19

Nodes: individuals

Links: social relationship

S. Milgram. Psychology Today (1967)

It’s a small world

20

A Few Good Man

Robert Wagner

Austin Powers

Wild Things

Let’s make it legal

Barry Norton

What Price Glory

Monsieur Verdoux

Kevin Bacon

21

Network propertiesClustering coefficientNeighbours = nodes that are directly connected

local clustering coefficient Clocal= average connectivity between neighboursClocal=1 -> all neighbours are connected

C : global clustering coefficient (average over all nodes)

Characteristic path lengthShortest path between nodes i and j:Lij = minimum number of connections to cross to go from one node to the other node

Characteristic path length L = average of shortest path lengths for all pairs of nodes

A

B

C

D E

Shortest path lengths:A -> C : 2A -> E : 1

A

B

C

E D

F

CA=4/10=0.4

22

Small-world networks

Clustering coefficient is higher than in random networks

(e.g. 40% compared to 15% for the macaque monkey)

Characteristic Path Length is comparable to random networks

Watts & Strogatz, Nature, 1998

23

Modular small-world connectivity

Hilgetag & Kaiser (2004) Neuroinformatics 2: 353

Small-worldNeighbours are well connected; short characteristic path length (~2)

ModularClusters: relatively more connections within the cluster than between clusters

18

Sequential

Kaiser et al. (2010) Frontiers in NeuroinformaticsHilgetag & Kaiser PLoS Comput. Biol. (in preparation)

Hierarchy

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SpatialTemporal Period concatenation in neocortical rhythms

www.frontiersin.org3

turn, lagged behind LII by 41 ± 3 ms (n = 25, Figure 2B and C, P < 0.05 comparing the timing of the two side peaks). The temporal distribution of LV units, with respect to LII units, also showed a similar pattern of con-catenation, with LII spiking alternately following and preceding LV spikes by 22 ± 6 ms and 40 ± 7 ms, respectively (n = 7, Figure 2C, P < 0.05). Separation of deep from superfi cial layers at the level of layer IV abol-ished the population beta1 rhythm (data not shown). This pattern of period concatenation, and the requirement of connectivity between laminae, suggested that microcircuits in LII and LV were still generating spikes in a manner subject to their respective network time constants, but now in a mutually interdependent manner.

Individual neuronal inputs and outputs, during beta1 population rhythms, revealed the presence of multiple frequency components. The two main principal cell populations [LV intrinsic bursting (n = 7) and LII regular spiking neurons (n = 12)] had unimodal interspike interval distributions corresponding to 2.2 ± 0.4 Hz and 5.9 ± 0.4 Hz, respec-tively (Figure 3A). Two subtypes of superfi cial layer interneurons [fast

spiking (FS) and low threshold spiking (LTS)] also had different spike rates. FS interneurons generated outputs at gamma (39 ± 4 Hz, n = 3) and also beta1 and alpha frequencies (Figure 3A). LTS interneuron output was at beta1 frequency (interspike interval 64 ± 8 ms, n = 5). Excitatory synaptic inputs to these cells were small compared to the control (gamma/beta2 rhythm) condition owing to the presence of NBQX. Modal peaks in the distribution of excitatory potentials were seen at beta1 frequency in all neurons except regular spiking cells, with additional peaks in the theta (6.0 ± 1.6 Hz), alpha (11 ± 2 Hz), beta2 (24 ± 2 Hz) and gamma (42 ± 5 Hz) bands (Figure 3B). A simi-lar pattern of multiple peaks in the spectra for inhibitory postsynaptic potentials was also seen (Figure 3C).

Both EPSPs and particularly IPSPs in LTS cells revealed evidence for period concatenation (Figure 3B and C). Fast synaptic excitation and inhibition occurred in doublets during each fi eld beta1 period, phase locked to the local fi eld potential and with a doublet interval of 27 ± 4 ms and 24 ± 6 ms (for EPSPs and IPSPs, respectively, n = 3). Distribution of

Figure 1. Reduction in excitatory drive to neocortex transforms concurrent gamma (35–45 Hz) and beta2 (22–28 Hz) oscillations into a beta1 (13–18 Hz) rhythm. Spectrograms of local fi eld potential data from LII and LV in rat cortex in vitro. (A) Laminar separation of gamma (LII) and beta2 (LV) frequency population rhythms in the presence of 400 nM kainate. Example traces (1 s epoch) of superfi cial (layer II), and deep (layer V) recordings showing the nature of the two rhythms are shown below. (B) Acute reduction in glutamatergic excitation in neocortex with 2.5 µM NBQX, following 2 h of stable gamma/beta2 oscillations abolished both fi eld potential rhythms, generating persistent beta1 frequency rhythms in both laminae. Example traces (1 s epochs) of fi eld potential beta1 frequency rhythms in layer II and layer V are shown below. Scale bars 100 ms, 50 µV.

Author's personal copy

seen in Fig. 7 (aREMoaWake in N01 and C03, though). This resultmatched well with previous work of Lee et al. who hadinvestigated sleep EEGs of apneic patients [12]. Larger a in deepsleep stages might be understood by decreased spectral power ofalpha rhythms due to slow responses to external stimuli in thosestages [16]. Since the reduced alpha rhythms yielded an increaseof b (see Fig. 6) and a was proportional to b [Eq. (5)], a becamelarger in deep sleep stages.

The group statistics of mean DFA exponents of each sleep stagewere summarized in Table 2, which showed that the narcolepticshad smaller DFA scaling exponents than the controls by 5–10% atREM and WAKE stages. To study statistical significance of thesedifferences, we performed a non-parametric Mann–Whitney

0.1 1

102

103

<v (τ

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REM

0.1 1

102

103 Wake

0.1 1

102

103 N1

0.1 1

102

103

<v (τ

)>

N2

0.1 1

102

103 N3

v ∼ τ1.15 v ∼ τ1.38 v ∼ τ1.14

v ∼ τ1.51v ∼ τ1.22

τ (s) τ (s)

Fig. 5. Typical DFA scaling behaviors of each sleep stage of N01. The scaling exponents were determined in the interval ð0:1 srtr1 sÞ.

1 10f (Hz)

N3

1 10

10−3

10−2

P (f)

N2

1 10

10−3

10−2

N1

1 10

10−3

10−2

Wake

1 10

10−3

10−2

P (f)

REM

β = 1.35 β = 1.45 β = 1.33

β = 1.47

f (Hz)

10−2

10−3 β = 1.96

Fig. 6. Typical power spectra of each sleep stage of N01. Note that the epochs chosen here were the same as those in Fig. 5 and the frequency interval ð1Hzr fr10HzÞused to determine b corresponded to the time interval ð0:1 srtr1 s) of a in Fig. 5.

Table 1Relations between the scaling exponents of the DFA and power spectra at eachsleep stage (rows, as labeled) of subject N01.

Sleep stages aa aestb bc

REM 1.15 1.18 1.35WAKE 1.38 1.23 1.45N1 1.14 1.17 1.33N2 1.22 1.24 1.47N3 1.51 1.48 1.96

a See Fig. 5.b aest ¼ ðbþ1Þ=2 [Eq. (5)].c See Fig. 6.

J.W. Kim et al. / Computers in Biology and Medicine 40 (2010) 831–838 835

f [Hz]

Author's personal copy

seen in Fig. 7 (aREMoaWake in N01 and C03, though). This resultmatched well with previous work of Lee et al. who hadinvestigated sleep EEGs of apneic patients [12]. Larger a in deepsleep stages might be understood by decreased spectral power ofalpha rhythms due to slow responses to external stimuli in thosestages [16]. Since the reduced alpha rhythms yielded an increaseof b (see Fig. 6) and a was proportional to b [Eq. (5)], a becamelarger in deep sleep stages.

The group statistics of mean DFA exponents of each sleep stagewere summarized in Table 2, which showed that the narcolepticshad smaller DFA scaling exponents than the controls by 5–10% atREM and WAKE stages. To study statistical significance of thesedifferences, we performed a non-parametric Mann–Whitney

0.1 1

102

103

<v (τ

)>

REM

0.1 1

102

103 Wake

0.1 1

102

103 N1

0.1 1

102

103

<v (τ

)>

N2

0.1 1

102

103 N3

v ∼ τ1.15 v ∼ τ1.38 v ∼ τ1.14

v ∼ τ1.51v ∼ τ1.22

τ (s) τ (s)

Fig. 5. Typical DFA scaling behaviors of each sleep stage of N01. The scaling exponents were determined in the interval ð0:1 srtr1 sÞ.

1 10f (Hz)

N3

1 10

10−3

10−2

P (f)

N2

1 10

10−3

10−2

N1

1 10

10−3

10−2

Wake

1 10

10−3

10−2

P (f)

REM

β = 1.35 β = 1.45 β = 1.33

β = 1.47

f (Hz)

10−2

10−3 β = 1.96

Fig. 6. Typical power spectra of each sleep stage of N01. Note that the epochs chosen here were the same as those in Fig. 5 and the frequency interval ð1Hzr fr10HzÞused to determine b corresponded to the time interval ð0:1 srtr1 s) of a in Fig. 5.

Table 1Relations between the scaling exponents of the DFA and power spectra at eachsleep stage (rows, as labeled) of subject N01.

Sleep stages aa aestb bc

REM 1.15 1.18 1.35WAKE 1.38 1.23 1.45N1 1.14 1.17 1.33N2 1.22 1.24 1.47N3 1.51 1.48 1.96

a See Fig. 5.b aest ¼ ðbþ1Þ=2 [Eq. (5)].c See Fig. 6.

J.W. Kim et al. / Computers in Biology and Medicine 40 (2010) 831–838 835

f [Hz]

25

Summary

1. Types of connections:- Structural- Functional- Effective

2. Finding structural fibre tract connectivity:- Diffusion tensor imaging- Tract tracing

3. Topological properties:- multiple clusters/ modularity- small-world: path lengths and local neighbourhood clustering

26 Further readings

Jeff Hawkins with Sandra Blakeslee. On Intelligence. Henry Holt and Company, 2004

Olaf Sporns. Networks of the Brain. MIT Press, 2010

Duncan J. Watts. Six Degrees: The Science of a Connected Age. Norton & Company, 2004

Sporns, Chialvo, Kaiser, Hilgetag. Trends in Cognitive Sciences (September 2004) www.dynamic-connectome.org

27 Practicaluse Matlab or Octave

• Measures for brain connectivity structure and development(including data for the macaque and cat): http://www.dynamic-connectome.orghttp://www.dynamic-connectome.org/t/tutorial/honey.mat

• Brain Connectivity Toolbox: http://www.brain-connectivity-toolbox.net/

• Connectome Viewer: http://www.connectomeviewer.org/

28 Matlab analysis - topology%% Network features using adjacency matrixmatrix%% see networks under the resources link at %% http://www.dynamic-connectome.org/%% for example, cat55.mat or mac95.mat

% how many nodes are there?N = length(matrix)

% how many edges are there (i.e. non-zero matrix elements)?E = nnz(matrix)

% what is the edge density (likelihood that any two nodes are connected?d = E / (N * (N-1))

% are there any loops (connections from node to itself)?min(min(matrix)) % any negative value out there?trace(matrix) % any non-zero diagonal elements (aka self-loops)

29 Matlab analysis – spatial organisation% network with 3D coordinates in variable pos e.g. using% http://www.dynamic-connectome.org/t/tutorial/honey.mat

%% visualize networkspy(matrix) % binary viewpcolor(matrix) % view of values for weighted networks

hist(nonzeros(matrix))unique(nonzeros(matrix))

%% Spatial Network visualisation% view from topsubplot(1,3,1); gplot(pos(:, [1,2])); axis equal

% view from sidesubplot(1,3,2); gplot(pos(:, [1,3])); axis equal

% view from backsubplot(1,3,3); gplot(pos(:, [2,3])); axis equal