Abstract
Self-organized bistability (SOB) is the counterpart of 'self-organized criticality' (SOC), for systems tuning themselves to the edge of bistability of a discontinuous phase transition, rather than to the critical point of a continuous one. The equations defining the mathematical theory of SOB turn out to bear strong resemblance to a (Landau-Ginzburg) theory recently proposed to analyze the dynamics of the cerebral cortex. This theory describes the neuronal activity of coupled mesoscopic patches of cortex, homeostatically regulated by short-term synaptic plasticity. The theory for cortex dynamics entails, however, some significant differences with respect to SOB, including the lack of a (bulk) conservation law, the absence of a perfect separation of timescales and, the fact that in the former, but not in the second, there is a parameter that controls the overall system state (in blatant contrast with the very idea of self-organization). Here, we scrutinize --by employing a combination of analytical and computational tools-- the analogies and differences between both theories and explore whether in some limit SOB can play an important role to explain the emergence of scale-invariant neuronal avalanches observed empirically in the cortex. We conclude that, actually, in the limit of infinitely slow synaptic-dynamics, the two theories become identical, but the timescales required for the self-organization mechanism to be effective do not seem to be biologically plausible. We discuss the key differences between self-organization mechanisms with/without conservation and with/without infinitely separated timescales. In particular, we introduce the concept of 'self-organized collective oscillations' and scrutinize the implications of our findings in neuroscience, shedding new light into the problems of scale invariance and oscillations in cortical dynamics.
Highlights
The theory of self-organized criticality (SOC) explains how systems can become self-organized to the edge of a continuous phase transition, i.e., to the vicinity of a critical point without the apparent need of parameter fine tuning [1,2,3,4,5]
As discussed in detail in Appendix B, the limit cycle is created via a homoclinic bifurcation at ξ = a and destroyed via a Hopf bifurcation at a value of ξ that depends on the timescales
From a theoretical point of view, our analyses reveal that different regulatory mechanisms for the control parameter—i.e., different equations for the “energy” field, with different meanings and possibly with diverse features such as conserved dynamics or not, or with or without a diffusion term—can be considered in the context of self-organization to the neighborhood of a discontinuous transition
Summary
The theory of self-organized criticality (SOC) explains how systems can become self-organized to the edge of a continuous phase transition, i.e., to the vicinity of a critical point without the apparent need of parameter fine tuning [1,2,3,4,5]. Our research group has recently proposed a physiologically motivated mesoscopic (Landau-Ginzburg) theory, designed to shed light on the large-scale dynamical features of cortical activity [35] The outcome of such an approach is that the relevant phase transition for cortical dynamics is a synchronization phase transition (which occurs in concomitance with scaleinvariant avalanches) with no self-organization to such a transition: parameters need to be fine tuned to observe it. We pose the following question: can self-organization to the very edge of a discontinuous transition with scale-free avalanches be possibly observed in the (Landau-Ginzburg) model? Can the SOB theory be modified to reproduce the phenomenology of the Landau-Ginzburg equation? Answering these questions will pave the way to a deeper understanding of self-organization mechanisms and their relevance in neuroscience
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