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Complejidad Dia 5

Ecología

Biología

Psicolo

gia

Meteorología

MacroEconomíaGeofisica

UBA, Junio 12, 2012.

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Vimos que:

La física tiene leyes simples mientras que la naturaleza es Compleja.La Complejidad en la naturaleza refleja la tendencia de los sistemas con muchos componentes de evolucionar hacia el estado crítico.Revisamos ejemplos de complejidad emergiendo de la interaccion de muchos grados de libertad no lineales (Modelos de trafico de Nagel, pila de arena de BTW, Bak-Sneppen macroevolucion, game of life,ect)

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Hoy: 1)Evidencia empirica de Avalanchas Neuronales2) 2 modelos (de muchos) de avalanchas neuronales

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Brain is a collective that produces behavior usefull to survive

Emphasizes the study of

Information processing

via

“connectivity”

Instead we foccus on emergent dynamicsand “collectivity”

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•Neuronal avalanches are peculiar dynamics occurring in the cortex (probably like an “action potential traveling trough cortex*”...)

Critical at Small scale

*But not a “traveling wave”

WM

V/VI

WM

V/VI

II/III

early 2 weeks

Rat cortex development

Rat brain PD 1-2

cut

CPu

Cx

Cpu1 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).

7Beggs 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).

9Beggs 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 Local field potential

600 m

400 m

200 m

Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).

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All Models of neuronal avalanches foccus on explaining (or not) the origin of this:

avalanche sizes distribution

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s i z e : 2 6 0 V 4 e l e c . a v a l t i + 1a v a l t i

W a i t i n g t i m e TQ u i e t t i m e QL i f e t i m e D t

14elec. (n)

AA B

C01

6 0

8 0 0 T i m e ( s ) e x t e r n a l d r i v i n g

n o d r i v i n g

1

6 0

+ 7 5 °- 7 5 °

elec. (n)

elec. (n)

Here we discuss spatiotemporal properties of cortical avalanchesrelevant to decide which of the model is closer.

1st Point: Is there a separation of time scales and if so how long last?

We look at spontaneous as well as slowly driven cortical cultures

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Driven-NonDriven:At relatively short time scales size fluctuations are identical

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Separation of time scales: the (fast) time scale of the avalanches is not perturbed by the (slow) drive.

3/2 3/2

Driven-NonDriven: Size distributions are identical

Non driven driven

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Mean time to the next avalanche is independent of the time elapsed since the last avalanche.

o The slow driving changes the statistic only for relatively long times.

Driven-NonDriven:

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Omori Law: Number of avalanches as a function of time from a given avalanche

foreshocksaftershocks

mainshock

Driven-NonDriven:

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Bath Law ? Mean avalanche size after an avalanche of relatively very large size

Shuffled Experiment

Each color corresponds to a different experiment.Red circles are averages of the six experiments.

mainshock…

NonDriven:

main shock

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1st Preliminary conclusion:

There is time scales separation and the limit of the scaling is around the (fast) time scale of about 10-1 sec.

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Second Point: Is the avalanche size distribution stationary? Yes

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Six different experiments (211512 avalanches) overimposed, identical stationary

Second Point: Is the avalanche size distribution stationary? Yes

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Avalanche’ rate scaling with space

Third Point: What about Space?

23(415254 avalanches SI cortex of a freely moving rat druing 4587

sec.)

4rd point: what about in vivo?

“Scaling in the Recurrence of Neuronal Avalanches in vivo”, Ribeiro T, Ribeiro S, Nicolellis M, Chialvo DR,

Copelli M, 2009)

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In vivo

4 animals exhibiting the same scaling during awake state

“Scaling in the Recurrence of Neuronal Avalanches in vivo”, Ribeiro T, Ribeiro S, Nicolellis M, Chialvo DR, Copelli M, (TB

submitted 2009)

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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.

Nada es realmente nuevo!

1941 McCulloch Chemical Neuronography...

Illinois Neuropsychiatric Institute (Chicago).

Recording cortical activity after local Strychninization

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new? 1941 McCulloch Chemical Neuronography...

Adjacency matrix of cortico-cortical “functional” connectivity, after

McCulloch (1940)

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new? 1941 McCulloch Chemical Neuronography...

Network analysis of 1941 Chemical Neuronography

Chimpanzee’ Degree and Link Length distribution(calculated from McCullock ,1941 data)

• Non-homogeneous degree• Similar scaling

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2 (de los muchos)modelos de avalanchas neuronales

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Cultured cerebral cortex

Spatiotemporal patterns that often

repeats

Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).

Neuronal avalanches:

30Beggs J. & Plenz D, Neuronal Avalanches in Neocortical Circuits J. of Neuroscience, 3 23 (2003).

Cultured cerebral cortex

Neuronal avalanches:

Branching process with ~ 1

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Simulations with N=105 sites, K=10 and n=10 states. a, Probability density function of local branching ratio and (inset) connectivity ('degree') distribution. b, Instantaneous density of active sites for subcritical (black), critical (red) and supercritical (blue) branching parameters as functions of time (three different runs for each case). c,d, Instantaneous density (of all sites; upper panels) and raster plot (of 103 randomly chosen sites; lower panels) in response to a square pulse of stimulus (r=0.5 ms -1 for 100 mst300 ms, null otherwise) for critical (c) and supercritical (d) branching parameters.

Optimal dynamical range of excitable networks at criticalityKinouchi & Copelli; Nature Physics 2, 348 - 351 (2006)

pmax=2*rho/K

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The points represent simulation results with N=105 sites, K=10, n=5 states and T=103 ms, whereas the lines correspond to the mean-field model described in the text. a, Response curves (mean firing rate versus stimulus rate) from =0 to 2 (in intervals of 0.2). The line segments are power laws F=rm with m=1 (subcritical) and m=1/2 (critical). Inset: spontaneous activity F0 versus branching ratio . b, The same as in a, but with a linear vertical scale. c, Response curve for =1.2 and relevant parameters for calculating the dynamic range . d, Dynamic range versus branching ratio is optimized at the critical point =1.

Optimal dynamical range of excitable networks at criticalityKinouchi & Copelli; Nature Physics 2, 348 - 351 (2006)

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Networks constructed with branching ratios close to one maintain, on average, the input activity (green, followed by yellow and red), thus optimizing the dynamic range. Instead, supercritical networks explode with activity, whereas subcritical ones are unable to sustain any input pattern. Psychophysics: Are our senses critical? Dante R. Chialvo, Nature Physics 2, 301 - 302 (2006)

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“Learning as a phenomenon occurring in a critical state” De Arcangelis & Herrmann. PNAS, 2010.

Actividad: Nodes redistribuyen “cargas”

Plasticidad: -if the output neuron is in the correct state we keep the value of synaptic strengths.-if response is wrong, we modify the strengths of those synapses involved in the information propagation by alfa ∕ dk, where dk is the chemical distanceof the presynaptic neuron from the output neuron.

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“Learning as a phenomenon occurring in a critical state” De Arcangelis & Herrmann. PNAS, 2010.

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“Learning as a phenomenon occurring in a critical state” De Arcangelis & Herrmann. PNAS, 2010.

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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.

uno mas:

“Synapsis dinamicas”