From Self-Organized to Extended Criticality

Abstract

We address the issue of criticality that is attracting the attention of an increasing number of neurophysiologists. Our main purpose is to establish the specific nature of some dynamical processes that although physically different, are usually termed as “critical,” and we focus on those characterized by the cooperative interaction of many units. We notice that the term “criticality” has been adopted to denote both noise-induced phase transitions and Self-Organized Criticality (SOC) with no clear connection with the traditional phase transitions, namely the transformation of a thermodynamic system from one state of matter to another. We notice the recent attractive proposal of extended criticality advocated by Bailly and Longo, which is realized through a wide set of critical points rather than emerging as a singularity from a unique value of the control parameter. We study a set of cooperatively firing neurons and we show that for an extended set of interaction couplings the system exhibits a form of temporal complexity similar to that emerging at criticality from ordinary phase transitions. This extended criticality regime is characterized by three main properties: (i) In the ideal limiting case of infinitely large time period, temporal complexity corresponds to Mittag-Leffler complexity; (ii) For large values of the interaction coupling the periodic nature of the process becomes predominant while maintaining to some extent, in the intermediate time asymptotic region, the signature of complexity; (iii) Focusing our attention on firing neuron avalanches, we find two of the popular SOC properties, namely the power indexes 2 and 1.5 respectively for time length and for the intensity of the avalanches. We derive the main conclusion that SOC emerges from extended criticality, thereby explaining the experimental observation of Plenz and Beggs: avalanches occur in time with surprisingly regularity, in apparent conflict with the temporal complexity of physical critical points.

Cooperation in neural systems: Bridging complexity and periodicity

“…The fast drop in the long-time region close to T P is due to the high sensitivity to periodicity (panel a). It is important to notice that the ML function revealed by the numerical analysis of this article exactly matches the ATA results [29]. This is strong evidence that cooperation generates long-range interactions, thereby making the local-interaction network identical to the ATA network, if a larger K is adopted: Cooperation-induced criticality breaks the local nature of the model [22].…”

“…To prove this important fact, we notice that as an effect of cooperation the neurons tend to fire at the same time [29,39], thereby making the distance between two consecutive firings become larger than 1/G and the intensity of each firing larger than the intensity of a single firing. As a consequence of the fact that the neural network has a finite number N of units, some of these multiple firings cannot be realized, or, let us say, they are not visible.…”

“…It is a rigorously local version of the model of the earlier work of Ref. [29]. Section III illustrates the form of temporal complexity generated by the cooperative interaction between the units of this model.…”

Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency

“…Consequently, these authors proved that, in critical phenomena, spatial and temporal complexity are strictly connected. More recently, emergence of TC was found in some network models at the critical point, such as neural networks or a decision making model, which is basically a 2D version of the Ising model [38,44,45]. This is a strong indication that critical systems are also complex, not only in the topological sense, but also according to TC.…”

“…This means that the dynamics of the noise is approximately Markovian, or even without any memory, and consequently, it is not a direct manifestation of self-organization. Even though, noise can play a fundamental role in complex systems and, in some cases, can also trigger the emergence of complexity itself (see, e.g., [45]). …”