Theoretical Neuroscience of Self‐Organized Criticality: From Formal Approaches to Realistic Models

Abstract

Self-organized criticality (SOC) is a common phenomenon in nature [1] and became a fascinating research subject for neuroscience when critical avalanches were predicted theoretically and observed experimentally to occur in networks of neurons [2][3][4]. Models of SOC in neural network evolve from early sandpile-like models [5, 6] on a lattice to large-scale realistically connected networks [7,8]. In this chapter we will present a model of criticality in the brain which identifies a number of contributing factors such as short-term synaptic dynamics and homeostatic plasticity in the synapses. This model is but one approach among an ever-increasing number of other studies on the subject. We were aiming for a family of models that are simple enough to permit an analytical solution but can still account for several biological principles that are known to be relevant here.As we try to identify the components of a neuronal network that can make it self-organized critical or, at least, bring it closer to a critical state, we have to be aware that criticality (in the sense of the existence of full-fledged power law event distributions) does not exist in finite systems. This emphasizes the importance of analytical treatments for infinite systems, while a theory of finite-size effects is also required to relate precisely to biological experiments. Such a theory exists only for the simplest abstract models (see Section 20.2), which forces us to apply a combination of analytical and numerical work on models including dynamic synapses (Section 20.3) and homeostatic adaptation (Section 20.4). The Eurich Model of Criticality in Neural NetworksIn 2002, Eurich et al. [2] presented a model of a globally coupled neural network that exhibits critical avalanches. In contrast to earlier work on criticality [5,6,[9][10][11], here an explicit analytical derivation of the distribution of sizes of neural avalanches was given not only for the idealization of very large networks but Criticality in Neural Systems, First Edition. Edited by Dietmar Plenz and Ernst Niebur.

Subsampling scaling

“…Thus we need to deal with a combination of both: subsampling effects as a result of incomplete data acquisition and finite-size effects inherited from the full system. To disentangle influences from system size and system dynamics, finite size scaling (FSS) has been introduced 32 33 . It allows to infer the behaviour of an infinite system from a set of finite systems.…”

Criticality in the brain: A synthesis of neurobiology, models and cognition

“…the pH, temperature, etc). Simple models of criticality classically rely upon fine-tuning of system parameters to a critical value (Levina et al, 2014). What is the basis for a robustness that apparently eschews the need for such a balancing act?…”

“…Several models of criticality in the brain incorporate such slow processes (Marković and Gros, 2014). A considerable body of research has focused upon the role of various forms of synaptic plasticity, including simple activity-dependent up-and down-regulation (de Arcangelis, 2008), activitydependent synaptic plasticity (de Arcangelis et al, 2006), synaptic potentiation (Stepp et al, 2015), short-term synaptic depression through depletion of synaptic vesicles (Bonachela et al, 2010;Levina et al, 2014;Mihalas et al, 2014;Millman et al, 2010), Hebbian (Van Kessenich et al, 2016) and anti-Hebbian synaptic plasticity (Cowan et al, 2014;Magnasco et al, 2009), and spike-time dependent plasticity (de Andrade Costa et al, 2015;Rubinov et al, 2011). As with physical systems, the (relatively) slow synaptic plasticity serves to broaden the critical point to a broad, stable region.…”

Mechanisms of Self-Organized Quasicriticality in Neuronal Network Models

“…From this, they conclude that the relation between criticality and learning is more complex, and it is not obvious if criticality optimizes learning. Levina et al [86] studied the combined effect of LHG synapses, homeostatic branching parameter W h , and STDP:…”