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Complejidad sin Matematicas
Ecología
Biología
Psicologia
Meteorología
MacroEconomíaGeofisica
Dante R. Chialvo Northwestern University. Chicago, IL, USA.
Email: [email protected] www.chialvo.net
Psicologia, Universidad Complutense, Madrid, Mayo 30, 2007.
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Second Latin American School and Conference in Statistical Physics and Interdisciplinary Applications (LASSPIA)
Bento Gonçalves (Brazil), February 5 to 15, 2007
Seminario presentado en:
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Critical Brain Networks
•Why do we need a brain at all?•Why the brain should be critical
Viva la inestabilidad!
Dante R. Chialvo
Physiology, Northwestern University, [email protected]
Reprints: www.chialvo.net
Second Latin American School and Conference in Statistical Physics and Interdisciplinary Applications (LASSPIA)
Bento Gonçalves (Brazil), February 5 to 15, 2007
4D.R. Chialvo “Critical Brain Networks” Physica A 340 (2004)
Brains in DynamicLand
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Plan:
Phenomenology of brain critical dynamics at various scales: frombehavior to a few neurons.
Today:
• Motivation, Why we need a brain?
• Behavior
• Large scale brain dynamics
Tomorrow:
• Spatio-temporal cortical dynamics (Neuronal avalanches)
• Alternative models of biological learning.
Outlook
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Reading
– Articles:» Eguiluz V, Chialvo DR, Cecchi G, Baliki M, AV Apkarian. Scale-free brain functional
networks. Phys. Rev. Letters 92, 018102 (2005).
» Chialvo DR. Critical brain networks. Physica A, 340,4,756-765 (2004).
» Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
» Chialvo DR. Are our senses critical? Nature Phys. 2006
» Copelli M & Kinouchi O. Optimal dynamical range of excitable networks at criticality, Nature Phys. 2006
» Chialvo DR. The brain at the edge (arxiv.org/ q-bio.NC/0610041)
– Review:» Sporns O, Chialvo DR, Kaiser M, and Hilgetag CC. Organization, Development and
Function of Complex Brain Networks. Trends in Cognitive Sciences, 8 (9): 387-(2004).
– Books :» How Nature Works. (Per Bak); Things that think. (Chialvo, 2007?)
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danza
Brain, a collective producing behavior
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Motivation, Why we need a brain
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1. The brain seems a relatively small dynamical system. (composed by a few dozens interacting areas)
2. It performs a small number of behaviors. (also about a few dozens)
3. Even relatively small dynamical systems posed near a 2nd order phase transition, can generate robust and flexible behavior.
(by using the abundance of metastable states at the critical point)
4. The approach pursued here assumes that some of the most fundamental properties of the functioning brain are possible just because often it stays at the border of such instability.
We review the motivation, the arguments and recent results as well as the implications of this view of the functioning brain
Points to be made:
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A few very obvious remarcks
– Almost all interesting macroscopic phenomena, from gravity to Almost all interesting macroscopic phenomena, from gravity to fotosintesis from supeconductivity to muscle contraction are profotosintesis from supeconductivity to muscle contraction are product duct of a underlying collective phenomenaof a underlying collective phenomena
–– Science is often seen as explaining a phenomena at Science is often seen as explaining a phenomena at one levelone level from from fundamental laws at fundamental laws at another levelanother level
–– Biology and neuroscience are not exception, thus we need to explBiology and neuroscience are not exception, thus we need to explain ain behavior (what we see) in terms of the underlying collective (whbehavior (what we see) in terms of the underlying collective (what at often is partially hidden to us)often is partially hidden to us)
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Collectives: A few conflictive demands ...
As a collective As a collective the brain the brain have a few conflictive demandshave a few conflictive demands::
»» “Integrated” but “segregated”. (Sporns, Edelman,Tononi“Integrated” but “segregated”. (Sporns, Edelman,Tononi11; more ; more recently Pietronerorecently Pietronero2,3,42,3,4) )
Q: how different is this from being spontaneously posed at a Q: how different is this from being spontaneously posed at a phase phase transition? transition?
1) O. Sporns, et al, Cerebral Cortex, (2000); 10(2): 127 2) M. De Lucia et al, Physical Review E 71, 016114 (2005)
3) Per Bak, How Nature Works, Oxford Univ. Press, 1997 4) Beggs J. & Plenz D. J. of Neuroscience, 3 23(35):11167 (2003).
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Brains are networks producing behavior ...
“Behavior” (usually bursty, complex, intermittent)
Large scale Small scale
at various scales…
Nowadays we can see these states…
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Why we need a brain ...
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Because the world around us Because the world around us -- in which brains have to survivein which brains have to survive--more often looks like thismore often looks like this
Subcritical SuperCriticalCritical
not like that!not like that!
Why do we need a brain at all?Why do we need a brain at all?
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T<TC T>TCT~TC
CriticalCriticalSubCriticalSubCritical SuperCriticalSuperCriticalCr
itica
l Tem
pera
ture
Criti
cal T
empe
ratu
re
Snapshots of Snapshots of spins states spins states in a model in a model system system ((IsingIsing))
Long Range Correlations Only at the Critical state!Long Range Correlations Only at the Critical state!
What is special about being critical? What is special about being critical? Recall FerromagneticRecall Ferromagnetic--paramagnetic Phaseparamagnetic Phase--TransitionTransition
Ok, even if the physical world is plenty of critical Ok, even if the physical world is plenty of critical stuff but... Why the brain should be Critical?stuff but... Why the brain should be Critical?
Why do we need a brain at all?Why do we need a brain at all?
••In a subIn a sub--critical world everything would be simple and uniform critical world everything would be simple and uniform -- there there would be nothing to learn. would be nothing to learn.
••In a supercritical world, everything would be changing all the tIn a supercritical world, everything would be changing all the time ime -- it it would be impossible to learn. would be impossible to learn.
The brain is necessary to navigate in a complex, critical world The brain is necessary to navigate in a complex, critical world ..
A brain not only have to remember, but also to forget and adapt.A brain not only have to remember, but also to forget and adapt.
••In a subIn a sub--critical brain memories would be frozen. critical brain memories would be frozen.
••In a supercritical brain, patterns change all the time so no lonIn a supercritical brain, patterns change all the time so no long term g term memory would be possible.memory would be possible.
To be highly To be highly susceptiblesusceptible, the brain itself has to be in the in, the brain itself has to be in the in--between between critical state.critical state.
What one should be able to observe?What one should be able to observe?
Brain dynamics can be usefully described in similar terms as Brain dynamics can be usefully described in similar terms as thermodynamic systems at the thermodynamic systems at the criticalcritical point including:point including:
At largescaleAt largescale11::•• Cortical Long range correlations in space and time (scaleCortical Long range correlations in space and time (scale--free)free)•• Intrinsically defined antiIntrinsically defined anti--correlated resting state networks.correlated resting state networks.
At smaller scaleAt smaller scale22::•• “Neuronal avalanches” is the normal homeostatic state of “Neuronal avalanches” is the normal homeostatic state of neocortical neocortical circuits. ( “corticalcircuits. ( “cortical--quakes” ).quakes” ).
At behavioral level:At behavioral level:••Adaptive behavior as “Adaptive behavior as “burstybursty” and apparently unstable, (always at ” and apparently unstable, (always at the “edge of failing”, “rising the bar effect”)the “edge of failing”, “rising the bar effect”)
1Eguiluz V, et al Phys. Rev. Letters (2005); Chialvo DR. Physica A, (2004). 2Beggs J. & Plenz D, J. Neuroscience (2003).
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Behavior:
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Behavior: is there an average rate for animal motion?
Chialvo et al, Submitted(2007) Data from Toni Diez Noguera, Trini Cambras (Univ. Barcelona)
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More behavior, better resolution same scaling law
Chialvo et al, Submitted(2007)
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Easy problem # 1:Easy problem # 1:
Write a toy dynamical model for the scaling behavior seen in animal and humans data.
Price: plane tickets plus a week expenses in Chicago
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Large Scale brain dynamics
How to look at brain networks?
23Jefferson High
Networks (A whole other lecture)
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Networks (A whole other lecture)
Graph
node
edge
C: Cluster coefficientLp: Pathlength
C = “how well connected” at the local level (number of triangles/total possible)
Lp = “how well connected” at the global level
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Small-WorldLattice Scale-FreeRandom
Pathlength
Clustering
Long Short Short Short
Large Large Small Small
≥
≥ ≥
> ≥
Networks (A whole other lecture)
Degree Dist. OneScale > Uniform > Uniform > ScaleFree
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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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Why we care to know this for the case of the brain?
Pospone the answer until the end
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How toHow to extract extract functional brain networks? functional brain networks?
( )( ) ( ) ( ) 222 ,, txVtxVxV −=σ
( ) ( ) ( ) ( ) ( )( )( ) ( )( )21
212121
,,,,,
xVxVtxVtxVtxVtxV
xxrσσ−
=
(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 “airports”Indicate “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).
2.2
2.1
2.0
γ
8.9x10-4
3.7x10-4
4.3x10-4
Crand
6.04.126.0.1548910.8
5.36.2912.90.13171740.7
3.913.4111.40.14315030.6
Lrand<k>LCNrc
“Small-world”C >> Crand
L ~ Lrand
17.48
9.85
7.68
<k>
.
.
.
.
0.273
0.318
0.025
Crand
1.4
1.5
2.1
Lrand
1.870.5465Cat Cortex2
1.770.5532Macaque VC2
2.650.28282C. Elegans1
LCNNetwork
fM RI-results
Previous related results
Brain networks Brain networks are smallare small--wordword
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fMRI
Brain’s degree distribution (i.e., how many links each node haveBrain’s degree distribution (i.e., how many links each node have))
Scale-free
k-γ with γ ~ 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).
γ =2Few 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 < Δ
k(Δ) ~ 1/x2
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fMRI
Something that bother us: Degree Something that bother us: Degree vsvs ClusteringClustering
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. MusicFinger tapping vs. Music
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, Submitted(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 (inAverage Links Length Distribution agrees with recent results (in resting resting condition)condition)
Functional connectivity vs. Functional connectivity vs. anatomical distance.anatomical distance.
( Symmetric ( Symmetric interhemisphericinterhemispheric))
From Salvador et al, (Cerebral From Salvador et al, (Cerebral Cortex, 2005.)Cortex, 2005.)
PC(Δ) ~ 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*
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Easy problem # 2:Easy problem # 2:
Define a (reasonable) heuristic order parameter Define a (reasonable) heuristic order parameter for the large scale brain dynamics seen in the for the large scale brain dynamics seen in the fMRI fMRI experimentsexperiments
Price: A year Price: A year postdoct postdoct salary in Chicagosalary in Chicago(renewable)(renewable)
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Is anything
ever new?
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Warren McCulloch Gerhardt von Bonin
Percival Bailey
J. Neurophysiology, 1941. J. G. Dusser de Barenne, Garol and McCulloch FUNCTIONAL ORGANIZATION OF SENSORY AND ADJACENT CORTEX OF THE MONKEY.
Ever new? 1940 McCulloch Chemical Neuronography...
Illinois Neuropsychiatric Institute (Chicago).
Recording cortical activity after local Strychninization
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Ever new? 1940 McCulloch Chemical Neuronography...
Adjacency matrix of cortico-cortical “functional” connectivity, after McCulloch (1940)
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Ever new? 1940 McCulloch Chemical Neuronography...
Network analysis of 1940 Chemical Neuronography
Chimpanzee’ Degree and Link Length distribution(calculated from McCullock ,1940 data)
• Non-homogeneous degree• Similar scaling
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Saturday 6:30 PM
Futebol, Futbol, Soccer, Calcio, φυτβολ!!!
Sign up with Secretary and show up at the lobby at 6:30
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Day two: Small Scale
•Neuronal Avalanches are a normal state of the cortex.•Learning is critical
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Reading
– Articles:» Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3
23 (2003).
» J. Beggs and D. Plenz. Neuronal Avalanches Are Diverse and Precise Activity Patterns That Are Stable for Many Hours in Cortical Slice Cultures. J Neurosci, 24(22):5216-5229, 2004.
» C. W. Eurich, M. Herrmann, and U. Ernst. Finite-size effects of avalanche dynamics. Phys. Rev. E, 66, 2002.
» Plenz D & Thiagarajan T.C. The organizing principles of neuronal avalanches: cell assemblies in the cortex?. Trends in Neurosciences (in press, 2007)
» Chialvo DR and Bak P.(1999) Learning from mistakes. Neuroscience 90 (4) 1137–1148.
» Bak P and Chialvo DR (2001) Adaptive learning by extremal dynamics and negative feedback. Physical Review E .63(3) 1912-1924.
WM
V/VI
WM
V/VI
II/III
early 2 weeks
Rat cortex development
Rat brain PD 1-2
cut
CPu
Cx
Cpu 1 mm
White matter
Culture for up to 6 weeks
Cultured cerebral cortex
Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
60Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
Cultured cerebral cortex
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Cultured cerebral cortex
Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
62Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
Cultured cerebral cortex Neuronal avalanches
Power law exponent of 3/2 :Array size
n = 15
n = 30
n = 60
Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
Power law exponent of 3/2 :Electrode distance
# electrodes Σ Local field potential
600 μm
400 μm
200 μm
Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
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Cultured cerebral cortex
Spatiotemporalpatterns that often
repeats
Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
Neuronal avalanches:
66Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).
Cultured cerebral cortex
Neuronal avalanches:
Branching process with σ ~ 1
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Pre-Post Avalanche Correlations
Experimental Model
Cultured cerebral cortex
Neuronal avalanches:
No good models yet!
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Levina et al, NIPS (2005)
Based on C. W. Eurich, M. Herrmann, and U. Ernst. Finite-size effects of avalanche dynamics. Phys. Rev. E, 66, 2002.
Power-law exponent for a range of connection strength in two models.
Probability distributions of avalanche sizes P(L;N; ). (a) in the subcritical, (b) the critical, and (c) supra-critical regime.
“One of the best in town”
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Learning
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Learning is never smooth
What Is the Problem?
The current emphasis is in …
• To understand how billions of neurons learn, remember and forget on a self-organized way.
• To find a relationship between neuronal long-term potentiation, (so called “LTP”) of synapses and memory.
Steps of LongSteps of Long--term term PotentiationPotentiation
1. Rapid stimulation of neurons depolarizes them.
2. Their NMDA receptors open, Ca2+ ions flows into the cell and bind to calmodulin.
3. This activates calcium-calmodulin-dependent kinase II (CaMKII).
4. CaMKII phosphorylates AMPA receptors making them more permeable to the inflow of Na+ ions (i.e., increasing the neuron’ sensitivity to future stimulation.
5. The number of AMPA receptors at the synapse also increases.
6. Increased gene expression (i.e., protein synthesis - perhaps of AMPA receptors) and additional synapses form.
Biology is concerned with “Long-Term Potentiation”
If A and B succeed together to fire the neuron (often enough) synapse B will be reinforced
What Is Wrong With “LTP”?First of all:There is no evidence* linking memory LTP
Furthermore:• It is a process purely local (lacking any global
coupling).• It implies a positive feedback (“addictive”).• It needs multiple trials (“rehearsal”).
Finally: Network components are not constant, often
neurons are replaced (even in adults).
*(non-circumstantial)
How difficult would be for a neuronal network to learn
The idea was not to invent another “learning algorithm” but to play with the simplest, still biologically realistic, one.
• Chialvo and Bak, Neuroscience (1999)
• Bak and Chialvo, Phys. Rev. E (2001).
• Wakeling J. Physica A, 2003)
• Wakeling and Bak, Phys.Rev. E (2001).
Self-organized Learning: Toy Model
1) Neuron “I*” fires
2) Neuron “j*” with largest W*(j*,I*) fires
and son onneuron with largest W*(k*,j*) fires…
3) If firing leads to success:Do nothingDo nothing
otherwiseotherwisedecrease W* by ΔΔ
That is allThat is all
How It Works on a Simple Task
Connect one (or more) input neurons with a given output neuron.
Chialvo and Bak, Neuroscience (1999)
A simple gizmo
a)left <->right
b)10% “blind”
c)10% “stroke”
d)40% “stroke”
Chialvo and Bak, Neuroscience (1999)
How performance scales with “brain” size
More neurons ->faster learning.
It makes sense!The only model where larger is better
Chialvo and Bak, Neuroscience (1999)
How It Scales With Problem Size (on the Parity Problem)
• A) Mean error vs Time for various problem’ sizes (i.e., N=2m bit strings)
• B) Rescaled Mean error (with k=1.4)
Chialvo and Bak, Neuroscience (1999)
Order-Disorder Transition
Learning time is optimized for ζ > 1
Order-Disorder Transition
At ζ = 1 the network is critical
Synaptic landscape remains rough
• Elimination of the least-fit connections
• Activity propagates through the best-fit ones
• At all times the synaptic landscape is rough
Fast re-learning
Chialvo and Bak, Neuroscience (1999)
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If you make a mistake, next do something different
H. Ohta, Y.P. Gunji / Neural Networks 19 (2006) 1106–1119
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By “inhibiting” the past states
H. Ohta, Y.P. Gunji / Neural Networks 19 (2006) 1106–1119
87H. Ohta, Y.P. Gunji / Neural Networks 19 (2006) 1106–1119
So you can learn new thing without deleting the old ones
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Easy problem # 3:Easy problem # 3:
Study how the input statistics (critical, subcritical, boring, surprissing etc) shapes and affects the final network configuration.(I.e., it does matter how it is the world outside?)
Price: plane tickets plus a week expenses in Chicago
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Brains are critical
DRC & Per Bak (Brookhaven N. Lab. 1991)
“Per, for me the brain is critical”…
“Yes, for me too Dante!”
