Complejidad Dia 8
Geofisica
MacroEconomía
Biologí
a
Ecología
Psicologia
Meteorología
UBA, Junio 26, 2012.
1
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 transition
2da 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 degree, centrality and modularity.
Graph
clustering-coefficient
characteristic path length
connectivity degree
centrality
modularity
Network topologies: regular, random, small-world, scalefree and modular networks.
the “small-world” phenomenon
–1011 neurons
–104 synapses per neuron
–On average two neurons are only 2
~ 3 “synapses” apart
• 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?
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How to extract functional brain networks?
fMRI
(I)
(III)
(II)
 2 V x   V x, t 2  V x, t 
2
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
My brain’s network (finger tapping)
Indicate “airports”
Nodes spatial location
Undirected Degree (k)
Colors indicate the number of links (or “degree”) of each node.
yellow=1, green 2, red=3, blue=4, etc
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
Group statistics
fM RI-results
rc
N
C
L
<k>
g
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
Brain
networks
are smallword
“Small-world”

C >> Crand

L ~ Lrand
Previous related results
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
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
Brain’s degree distribution (i.e., how many links each node have)
Scale-free
k
-g
with g ~ 2
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
Average Degree Distribution
n=22 from 7 subjects
Few but very
well connected
brain sites
g =2
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
Average Links Length Distribution
Probability of finding a
link between two
nodes separated by a
distance x < D
Voxel length
“~ Brain radius”
k(D) ~ 1/x2
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
Something that bother us: Degree vs Clustering
Recall that clustering
estimates the proportion of
nodes forming “triangles”.
Clustering relatively independent of
connectivity
Assortative
From Eguiluz et al, Phys. Rev. Letters (2005).
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fMRI
A node tends to
be either an inhub or an outhub
(Directed links)
in-hub vs und.
out-hub vs und.
few “airports”
From Cecchi et al, BME (2007).
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fMRI
(Directed links)
From Cecchi et al, BME (2007).
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(Directed links)
lattice
brain
random
Assortativ
e?
From Cecchi et al, BME (2007).
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Networks are scale free across different tasks
Finger tapping vs. Music
•Different tasks
•Different networks
•Similar scaling
From Eguiluz et al, Phys. Rev. Letters (2005).
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”
22
~
1
Chialvo et al., “Beyond feeling: chronic pain hurts
the brain disrupting the default-mode network
dynamics” J.Neuroscience (2008)
23
What is special about being critical?
Recall the Ferromagnetic-paramagnetic Phase-Transition
Snapshots of
spins states
in a model
system
(Ising)
Critical Point
disorder
T
C
T<T
T~T
T>T
C
C
C
Subcritical
Critical
SuperCritica
l Ising
Snapshots of spins states in the
model.
Long range correlations
emerges at the critical point
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Critical Ising networks ~ brain networks
SubCritical
Critical
SuperCritical
Positive correlated net
Ising
Brain
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/ condmat/0811.3721)
EJ S <i,j> Si Sj – B Sk Sk
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Critical Ising networks ~ brain networks
Brains
Ising
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Critical Ising networks ~ brain networks
Negative correlated networks
SubCritical
Critical
SuperCritical
Ising
Brain
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).
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Assortativity
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Critical Ising networks ~ brain networks
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Resting-state networks. (functionally linked resting-state networks during rest
identified using different methods (e.g. seed, ICA or clustering)
Laspia 2007
Easy problem # 2:
Define a (reasonable) heuristic order parameter
for the large scale brain dynamics seen in the
fMRI experiments
Price: A year postdoct salary in Chicago
(renewable)
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Related results
32
33
1/x2 replicated independently with fMRI
Average Links Length Distribution agrees with recent results (in resting
condition)
interhemispheric
PC(D) ~ 1/x2
intrahemispheric
Functional connectivity vs.
anatomical distance.
(
Symmetric interhemispheric)
From Salvador et al, (Cerebral
Cortex, 2005.)
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C/Crandom = 2.08
L/Lrandom = 1.09
EEG
1
3
2
4
C
threshold
Synchronization I
L
Graph
Path length is related to cognitive score
Control subjects
Clustering
Path
Length
cognitive score
Alzheimer patients
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Clauset, Newman & Moore
Algorithm*
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