SOC neunoral networks

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This document summarizes a presentation on self-organized criticality and neuronal avalanches in networks with dynamical synapses. It discusses how synaptic depression can tune networks to a critical state in a non-conservative model. The model uses a probabilistic cellular automata on a random graph with synaptic depression, which slowly loads the system. Mean-field theory and simulations show the branching ratio converges to unity, making the system conservative on average. Analysis of neuronal interspike interval distributions in different regimes is proposed to further understand the model's behavior.

Self-Organized Criticality and Neuronal
Avalanches in Networks
with Dynamical Synapses
Ariadne Costa, Mauro Copelli,
Osame Kinouchi
First Workshop of Young Researchers of NeuroMat, São Paulo, May 2015
University of São Paulo (USP), Ribeirão Preto, Brazil
Federal University of Pernambuco (UFPE), Recife, Brazil

Size distribution of neuronal avalanches
P(s) = c s-3/2

Quick historical background of SOC
• Microscopically conservative models
with true SOC
(Bak, Tang & Wiesenfeld PRL 87, Jensen
SOC 98)
• Non-conservative models with after-
avalanche (hard) loading, presenting
pseudo-SOC
(Drossel PRL 92, Kinouchi & Prado, PRE 99)
• Non-conservative models with intra-
avalanche (soft) loading, presenting
SOqC (self-organized quase-criticality)
(Levina, Herrmann & Geisel Nat Phys 07,
Bonachela et al JSTAT 09, 10)

Self-Organized Criticality and Neuronal
Avalanches in Networks
with Dynamical Synapses
• Microscopically non-conservative model
with “soft” loading, but
• Probabilistic cellular automata
(larger system sizes & easier mean field)
• Finite connectivity (random graph)
• Stationary regime: conservative on average
• Fluctuations vanish when N → ∞
Here:

Random network of excitable cellular automata
Random graph with K outcoming synapses
Kinouchi & Copelli, Nat. Phys. 06
Control parameter:
Branching ratio:
Order parameter:
Mean firing rate:
Mean-field equation: (I)

Directed-percolation phase transition at
Kinouchi & Copelli, Nat. Phys. 06
Chialvo, Nat. Phys. 06

Directed-percolation phase transition at
Kinouchi & Copelli, Nat. Phys. 2006
Chialvo, Nat. Phys. 2006
Can synaptic
depression tune
If so, is it SOC?

Synaptic Depression
Levina et al, Nat. Phys. 07 (LHG)
- Integrate and fire neurons (deterministic)
- all-to-all coupling
Synaptic depression: slow load
Here
- Cellular automata (probabilistic)
- random graph
Synaptic depression: slow load

Synaptic Depression: two variants
Quenched
Annealed
(random neighbor)

Results
• Analytical (mean-field theory)
• Simulations

Mean-field theory
Mean-field equation (I)
Mean-field equation (II)
Kinouchi & Copelli, Nat. Phys. 06

Mean-field theory
Mean-field equation (I)
Mean-field equation (II)

Branching ratio converges to unity
System becomes conservative, on
average!
Is on average good enough?

Our model
Fluctuations decrease with system size
Bonachela et al., JSTAT 10
LHG model

Pruessner-Jensen model:
- Similar parameter dependence
- Different cutoff exponent (Bonachela & Muñoz, JSTAT 09)

Conservative
Bonachela & Muñoz, JSTAT 09
Non-Conservative

Bonachela & Muñoz, JSTAT 09
Conservative

Conclusions
Costa, Copelli and Kinouchi – Can dynamical synapses produce true
Self-organized criticality?
accepted for publication in Journal of Statistical Mechanics (2015)

Perspectives
• Study of neuronal
interspike interval (ISI)
probability distribution p(t)
in the subcritical, critical
and supercritical regime

PLoS Computational Biology
Power-Law Inter-Spike Interval Distributions
Infer a Conditional Maximization of Entropy in
Cortical Neurons
•Yasuhiro Tsubo ,
•Yoshikazu Isomura,
•Tomoki Fukai
•Published: April 12, 2012
•DOI: 10.1371/journal.pcbi.1002461
Study of neuronal interspike
interval (ISI) probability
distribution p(t) in the
subcritical, critical and
supercritical regime
Perspectives
ISI distributions (log-log graphs)

Interspike Intervals (ISI) distributions
seems to have long (power laws) tails
Collaboration with Lézio Soares Bueno Júnior started here!

Thank you
Come visit us in Ribeirão Preto!

Scaling with system size
Mean-field equation (II)
A similar scaling was found for the Pruessner-
Jensen model (Bonachela & Muñoz, JSTAT 09)

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SOC neunoral networks

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