Quasi-criticalidade auto-organizada

Fig. 3.

Nested θ- and β/γ-oscillations organize in the form of neuronal avalanches. (A) Definition of neuronal avalanches formed by the nested θ- and β/γ-oscillations. (Top) Threshold detection (broken line) of nLFPs (filled circles) at a single electrode. (Middle) Corresponding time–amplitude raster plot of nLFPs on the MEA (color: nLFP amplitude). (Bottom) Spatiotemporal nLFP clusters occupy successive bins of width Δt avg (dotted rectangles). (B) Average cross-correlation function for nLFPs in vivo at P8 (red) and P13 (black; single experiments). (C) nLFP clusters from nested θ- and β/γ-oscillations organize in the form of neuronal avalanches, i.e., distribute in sizes according to a power law with slope close to α = −1.5 (broken line). Average cluster size distribution in vivo plotted in log–log coordinates for P8 (red open circles; n = 5) and P13 (black; n = 7). (D) Example of two simultaneous burst periods before (black) and after (red) phase-shuffling. (E) The power law in cluster sizes is established for cluster area and cluster intensity (G) in single in vivo experiments and in the average (n = 7; F; cp. also C; all P13), but is destroyed on phase-shuffling of the LFP (open red). (H) Average cluster size distribution in vitro follows a power law with slope α ≅ −1.5 (broken line; n = 15; ≥10 DIV). (Inset) Average nLFP cross-correlation function for single experiment. Published online before print May 22, 2008, doi:10.1073/pnas.0800537105

PNAS May 27, 2008 vol. 105no. 21 7576-7581

Neuronal avalanches organize as nested theta- and beta/gamma-oscillations during development of cortical layer 2/3

  1. Elakkat D. Gireesh and
  2. Dietmar Plenz*

+Author Affiliations

  1. Laboratory of Systems Neuroscience, National Institute of Mental Health, 9000 Rockville Pike, Bethesda, MD 20892
  1. Edited by Nancy J. Kopell, Boston University, Boston, MA, and approved March 27, 2008 (received for review January 18, 2008)


Maturation of the cerebral cortex involves the spontaneous emergence of distinct patterns of neuronal synchronization, which regulate neuronal differentiation, synapse formation, and serve as a substrate for information processing. The intrinsic activity patterns that characterize the maturation of cortical layer 2/3 are poorly understood. By using microelectrode array recordings in vivo and in vitro, we show that this development is marked by the emergence of nested θ- and β/γ-oscillations that require NMDA- and GABAA-mediated synaptic transmission. The oscillations organized as neuronal avalanches, i.e., they were synchronized across cortical sites forming diverse and millisecond-precise spatiotemporal patterns that distributed in sizes according to a power law with a slope of −1.5. The correspondence between nested oscillations and neuronal avalanches required activation of the dopamine D1 receptor. We suggest that the repetitive formation of neuronal avalanches provides an intrinsic template for the selective linking of external inputs to developing superficial layers.

Self-organized (quasi-)criticality: the extremal Feder and Feder model

A simple random-neighbor SOC model that combines properties of the Bak-Sneppen and the relaxation oscillators (slip-stick) models is introduced. The analysis in terms of branching processes is transparent and gives insight about the development of large but finite mean avalanche sizes in dissipative models. In the thermodynamic limit, the distribution of states has a simple analytical form and the mean avalanche size, as a function of the coupling parameter strength, is exactly calculable.
Comments:6 pages, 3 figures
Subjects:Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech)
Cite as:arXiv:cond-mat/9802311v1 [cond-mat.dis-nn]

Self-organization without conservation: true or just apparent scale-invariance?

The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, as forest-fires or earthquake automata, is scrutinized in this paper. By using three different levels of description - (i) a simple mean field, (ii) a more detailed mean-field description in terms of a (self-organized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation-, it is shown on general grounds that non-conserving dynamics does not lead to bona fide criticality. Contrarily to conserving systems, a parameter, which we term "re-charging" rate (e.g. the tree-growth rate in forest-fire models), needs to be fine-tuned in non-conserving systems to obtain criticality. In the infinite size limit, such a fine-tuning of the loading rate is easy to achieve, as it emerges by imposing a second separation of time-scales but, for any finite size, a precise tuning is required to achieve criticality and a coherent finite-size scaling picture. Using the approaches above, we shed light on the common mechanisms by which "apparent criticality" is observed in non-conserving systems, and explain in detail (both qualitatively and quantitatively) the difference with respect to true criticality obtained in conserving systems. We propose to call this self-organized quasi-criticality (SOqC). Some of the reported results are already known and some of them are new. We hope the unified framework presented here helps to elucidate the confusing and contradictory literature in this field. In a second accompanying paper, we shall discuss the implications of the general results obtained here for models of neural avalanches in Neuroscience for which self-organized scale-invariance in the absence of conservation has been claimed.
Comments:40 pages, 7 figures.
Subjects:Statistical Mechanics (cond-mat.stat-mech)
Journal reference:J. Stat. Mech. (2009) P09009
Cite as:arXiv:0905.1799v3 [cond-mat.stat-mech]

Self-organization without conservation: Are neuronal avalanches generically critical?

Recent experiments on cortical neural networks have revealed the existence of well-defined avalanches of electrical activity. Such avalanches have been claimed to be generically scale-invariant -- i.e. power-law distributed -- with many exciting implications in Neuroscience. Recently, a self-organized model has been proposed by Levina, Herrmann and Geisel to justify such an empirical finding. Given that (i) neural dynamics is dissipative and (ii) there is a loading mechanism "charging" progressively the background synaptic strength, this model/dynamics is very similar in spirit to forest-fire and earthquake models, archetypical examples of non-conserving self-organization, which have been recently shown to lack true criticality. Here we show that cortical neural networks obeying (i) and (ii) are not generically critical; unless parameters are fine tuned, their dynamics is either sub- or super-critical, even if the pseudo-critical region is relatively broad. This conclusion seems to be in agreement with the most recent experimental observations. The main implication of our work is that, if future experimental research on cortical networks were to support that truly critical avalanches are the norm and not the exception, then one should look for more elaborate (adaptive/evolutionary) explanations, beyond simple self-organization, to account for this.
Comments:28 pages, 11 figures, regular paper
Subjects:Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Computational Physics (physics.comp-ph); Neurons and Cognition (q-bio.NC)
Cite as:arXiv:1001.3256v1 [cond-mat.dis-nn]


Generic aspects of complexity in brain imaging data and other biological systems
Purchase the full-text article

Ed Bullmorea, Corresponding Author Contact Information, E-mail The Corresponding Author, Anna Barnesa, Danielle S. Bassetta, b, c, Alex Fornitoa, d, Manfred Kitzbichlera, David Meuniera and John Sucklinga

aBehavioural and Clinical Neurosciences Institute, University of Cambridge, Department of Psychiatry, Addenbrooke's, Hospital, Cambridge, UK

bBiological Soft Systems Sector, Department of Physics, University of Cambridge, Cambridge, UK

cGenes Cognition and Psychosis Program, Clinical Brain Disorders Branch, National Institute of Mental Health, NIH, Bethesda, USA

dMelbourne Neuropsychiatry Centre, Department of Psychiatry, University of Melbourne, Parkville, Australia

Received 13 January 2009;
revised 3 May 2009;
accepted 8 May 2009.
Available online 19 May 2009.


A key challenge for systems neuroscience is the question of how to understand the complex network organization of the brain on the basis of neuroimaging data. Similar challenges exist in other specialist areas of systems biology because complex networks emerging from the interactions between multiple non-trivially interacting agents are found quite ubiquitously in nature, from protein interactomes to ecosystems. We suggest that one way forward for analysis of brain networks will be to quantify aspects of their organization which are likely to be generic properties of a broader class of biological systems. In this introductory review article we will highlight four important aspects of complex systems in general: fractality or scale-invariance; criticality; small-world and related topological attributes; and modularity. For each concept we will provide an accessible introduction, an illustrative data-based example of how it can be used to investigate aspects of brain organization in neuroimaging experiments, and a brief review of how this concept has been applied and developed in other fields of biomedical and physical science. The aim is to provide a didactic, focussed and user-friendly introduction to the concepts of complexity science for neuroscientists and neuroimagers.

V Pasquale, P Massobrio, LL Bologna, M Chiappalone, … - Neuroscience, 2008 - Elsevier
... experimental model for studying the universal mechanisms governing the formation and
conservation of neuronal ... In our work, while trying to understand the phenomenon of
self-organization in dissociated ... In this study, no attempt was made to discriminate and sort the ...
Citado por 19 - Artigos relacionados - Todas as 4 versões

JD Halley, DA Winkler - Biosystems, 2008 - Elsevier
... Propulsive Actin Networks A.3. Example 3. Neuronal Avalanches References. ... is a need for new
theoretical frameworks that explain how natural selection and self-organization interact ([Kauffman ...
As originally defined, the sand pile model seems to involve no tuneable parameters ...
Citado por 6 - Artigos relacionados - Todas as 4 versões

upd.edu.ph [PDF]DE Juanico, C Monterola, C Saloma - New Journal of Physics, 2007 - iop.org
... The non-conservative transfer rule is repeated until no more excited agents remain—when the ...
by using the definition ε = 1–α–β and then taking β = 0 without loss of ... q* = q c , which indicates
that the non-conservative system evolves towards its critical state by self-organization. ...
Citado por 4 - Artigos relacionados - Todas as 5 versões

Self-organising mechanism of neuronal avalanche criticality

Juanico, Dr D.E. (2007) Self-organising mechanism of neuronal avalanche criticality. [Preprint]

Full text available as:



A self-organising model is proposed to explain the criticality in cortical networks deduced from recent observations of neuronal avalanches. Prevailing understanding of self-organised criticality (SOC) dictates that conservation of energy is essential to its emergence. Neuronal networks however are inherently non-conservative as demonstrated by microelectrode recordings. The model presented here shows that SOC can arise in non-conservative systems as well, if driven internally. Evidence suggests that synaptic background activity provides the internal drive for non-conservative cortical networks to achieve and maintain a critical state. SOC is robust to any degree $\eta \in (0,1]$ of background activity when the network size $N$ is large enough such that $\eta N\sim 10^3$. For small networks, a strong background leads to epileptiform activity, consistent with neurophysiological knowledge about epilepsy.

Role of membrane potential fluctuations to the criticality of neuronal avalanche activity

Juanico, Dr Dranreb Earl (2007) Role of membrane potential fluctuations to the criticality of neuronal avalanche activity.[Preprint]

Full text available as:



Experimental evidence for self-organised criticality (SOC) in non-conservative systems has recently been found in studies of rat cortical slices. The size distribution of observed neuronal avalanches has been attested to obey $3/2$ power-law scaling. A mean-field sandpile model of a noisy neuronal system is proposed to refute the irreconcilability between non-conservation and criticality put forward by longstanding SOC hypotheses. The model predicts that neuronal networks achieve and maintain criticality despite non-conservation due to the presence of background activity originating from membrane potential fluctuations within individual neurons. Furthermore, small networks are demonstrated to tip towards epileptiform activity when background activity is strong. This finding ties in redundancy, an intriguing feature of brain networks, to robustness of SOC behaviour.


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