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UBA, Junio 26, 2012.

Martes 26:

1era parte More on preprocessing of fmri images

2da Parte Redes, desde Eguiluz a Tagliazuchi.

Jueves 28 1era parte anomalous scaling and phase transition2da parte Modeling

“Our brain is a network. A very efficient network to be precise. It is a network of a large number of different brain regions that each have their own task and function, but who are continuously sharing information with each other. As such, they form a complex integrative network in which information is continuously processed and transported between structurally and functionally linked brain regions: the brain network”

Where is the router?

conventional task-related fMRI

Resting state fMRI

From: Exploring the brain network: A review on resting-state fMRI functional connectivity. Martijn P. van den Heuvel, Hilleke E. Hulshoff Pol. European Neuropsychopharmacology, 20 (2010) 519–534

modeling the brain as a functional network with connections between regions that are functionally linked

Graph clustering-coefficient

characteristic path length

connectivity degreecentrality modularity

Graph, clustering-coefficient, characteristic path length, connectivity degree, centrality and modularity.

Network topologies: regular, random, small-world, scale-free and modular networks.

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the “small-world” phenomenon

• Connectivity is sparse (i.e., 104 / 1011 )• Most connections are local (high clustering

coefficient)• The distance between any two network nodes is still

relatively small: how is possible?

–1011 neurons

–104 synapses per neuron

–On average two neurons are only 2 ~ 3 “synapses” apart

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How toHow to extract extract functional brain networks? functional brain networks?

222 ,, txVtxVxV

(I)

(II)

(III)

From Eguiluz et al, Phys. Rev. Letters (2005).

fMRIfMRI

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fMRI

My brain’s network (finger tapping)My brain’s network (finger tapping)

Undirected Degree (k)

Nodes spatial locationNodes spatial location

Colors indicate the number of links (or “degree”) of each node. yellow=1, green 2, red=3, blue=4, etc

Indicate Indicate “airports”“airports”

From Eguiluz et al, Phys. Rev. Letters (2005).

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fMRI

Group statisticsGroup statistics

From Eguiluz et al, Phys. Rev. Letters (2005).

rc N C L <k> Crand Lrand

0.6 31503 0.14 11.4 13.41

2.0 4.3x10-

4

3.9

0.7 17174 0.13 12.9 6.29 2.1 3.7x10-

4

5.3

0.8 4891 0.15 6. 4.12 2.2 8.9x10-

4

6.0

“Small-world”

C >> Crand

L ~

Lrand Network N C L <k> . Crand Lrand

C. Elegans1

282 0.28 2.65 7.68 . 0.025 2.1

Macaque VC2

32 0.55 1.77 9.85 . 0.318 1.5

Cat Cortex2

65 0.54 1.87 17.48

. 0.273 1.4

fM RI-results

Previous related results

Brain Brain networks networks are small-are small-wordword

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fMRI

Brain’s degree distribution (i.e., how many links each node Brain’s degree distribution (i.e., how many links each node have)have)

Scale-free

k- with ~ 2

From Eguiluz et al, Phys. Rev. Letters (2005).

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fMRI

Average Degree DistributionAverage Degree Distribution

From Eguiluz et al, Phys. Rev. Letters (2005).

=2

Few but very well connected brain sites

n=22 from 7 subjects

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fMRI

Average Links Length Distribution Average Links Length Distribution

From Eguiluz et al, Phys. Rev. Letters (2005).

Voxel length“~ Brain radius”

Probability of finding a link between two nodes separated by a distance x <

k() ~ 1/x2

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fMRI

Something that bother us: Degree vs ClusteringSomething that bother us: Degree vs Clustering

From Eguiluz et al, Phys. Rev. Letters (2005).

Recall that clustering estimates the proportion of nodes forming “triangles”.

Clustering relatively independent of connectivity

Assortative

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fMRI (Directed links)

A node tends to be either an in-hub or an out-hub

few “airports”

in-hub vs und. out-hub vs und.

From Cecchi et al, BME (2007).

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fMRI (Directed links)

From Cecchi et al, BME (2007).

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random

lattice

brain

(Directed links)

Assortative?

From Cecchi et al, BME (2007).

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Finger tapping vs. Finger tapping vs. MusicMusic

From Eguiluz et al, Phys. Rev. Letters (2005).

•Different tasksDifferent tasks•Different networksDifferent networks•Similar scalingSimilar scaling

Networks are scale free across different tasksNetworks are scale free across different tasks

And during “resting state” And during “resting state” =>=>

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Summary until now:

The large scale brain network extracted from correlations seems to be scale-free and small word

But what about dynamics?

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Even in resting state, each positively correlated clique have a negatively correlated contrapart

Areas coloured redish have significant positive correlation with seed regions and are significantly anticorrelated with regions coloured blueish

(Fox et al , PNAS, 102, 2005)

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Each positively correlated clique have a negatively

correlated contrapart

Healthy Controls

Chronic Pain Patients

Chialvo et al. 2007, “Beyond feeling: chronic pain hurts the brain disrupting the default-mode network dynamics”

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Chialvo et al., “Beyond feeling: chronic pain hurts the brain disrupting the default-mode network dynamics” J.Neuroscience (2008)

~ 1

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Snapshots Snapshots of spins of spins states in a states in a model model system system (Ising)(Ising)

What is special about being critical? What is special about being critical? Recall the Ferromagnetic-paramagnetic Phase-TransitionRecall the Ferromagnetic-paramagnetic Phase-Transition

Snapshots of spins states in the Ising model. Long range

correlations emerges at the critical point

Subcritical

SuperCritical

Critical

T<TC

T>TC

T~TC

TC

Critical Pointorder

disorder

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Critical Ising networks ~ brain networks

Ising

Brain

EJ<i,j> Si Sj – B k Sk

Only local positive interactions Chialvo DR, Balenzuela P, Fraiman D. The brain:

What is critical about it? (arXiv.org/ cond-mat/0804.0032)

Fraiman D, Balenzuela P, Foss J. Chialvo DR, Ising like dynamics in large-scale brain networks. (arXiv.org/ cond-mat/0811.3721)

Positive correlated networks

SubCritical

Critical

SuperCritical

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Critical Ising networks ~ brain networks

Brains

Ising

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Critical Ising networks ~ brain networks

Negative correlations with fat tails similar to the brain data appear in the Ising data, despite the absence of negative “structural” interactions (i.e. no “inhibitory” connectivity).

Ising

Brain

Negative correlated networksSubCritical

SuperCriticalCritical

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Assortativity

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Critical Ising networks ~ brain networks

Resting-state networks. (functionally linked resting-state networks during rest identified using different methods (e.g. seed, ICA or clustering)

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Easy problem # 2:Easy problem # 2:

Define a (reasonable) heuristic order Define a (reasonable) heuristic order parameter for the large scale brain parameter for the large scale brain dynamics seen in the fMRI experimentsdynamics seen in the fMRI experiments

Price: A year postdoct salary in ChicagoPrice: A year postdoct salary in Chicago(renewable)(renewable)

20072007

Laspia 2007 Laspia 2007

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Related results

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1/x2 replicated independently with fMRI

Average Links Length Distribution agrees with recent results (in Average Links Length Distribution agrees with recent results (in resting condition)resting condition)

Functional connectivity vs. Functional connectivity vs. anatomical distance.anatomical distance.

( Symmetric ( Symmetric interhemispheric)interhemispheric)

From Salvador et al, From Salvador et al, (Cerebral Cortex, 2005.)(Cerebral Cortex, 2005.)

PC()~1/x2

interhemisphericinterhemispheric

intrahemisphericintrahemispheric

C/Crandom = 2.08

L/Lrandom = 1.09

C

Lthreshold

1 2

3 4

EEG

GraphSynchronization I

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Alzheimer patients

Path length is related to cognitive score

Control subjects

cognitive score

Clustering Path Length

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Clauset, Newman & MooreAlgorithm*