Criticality hypothesis in the brain: from neutral theory to self-organisation and synchronisation

Many biological systems show spatial and temporal long range correlations, high sensitivity to stimuli and the ability to adapt to perform a wide variety of complex tasks. This observation suggested the hypothesis that they might be operating close to the critical point of some phase transition, that would confer them an optimal trade-off between robustness (resilience to external perturbations, a property of ordered phases) and flexibility (responsiveness to external stimuli, a property of disordered phases). This hypothesis was proposed also for neural networks, in particular after an experiment of 2003, where rodent neocortex electrical activity was measured with unprecedented spatial and temporal resolution, allowing to recognize clusters of activity and paving the way to the definition of neuronal avalanches, which, under physiological conditions, show power-law distributed sizes and durations. The debate on the hypothesis that scale-invariant neural avalanches are a symptom of criticality is fully on-going and in this PhD thesis it is tackled in detail, from a simple and unifying perspective. We evidence that some of the most celebrated neurophysiologically-inspired models reproduce empirically measured avalanche distributions, not as a consequence of the alleged criticality toward which the system self-organizes, but as a result of a neutral principle: each cascade of events has the same probability to shrink or expand as the other coexisting ones. Moreover, we highlight that the experimental protocol for the measure of avalanches hides an approximation on the evaluation of the causality of registered events, which must be taken into account. Neuronal signals are composed by the alternation of high and low activity periods named “Up and Down states”. This bistable behavior suggests to analize the possibility that a system self-organizes to a first-order †"rather that second-order†" phase transition. Following the footprints of the study of Self-Organised Criticality (SOC), we develop a general theory for the auto-organisation to a phase coexistence point, showing aspects of scale-invariance in combination with anomalous events, which involve the whole system. This new paradigm is candidate to find realisations in many fields. Often, within the debate on the criticality hypothesis in the brain, the phases between which the putative transition occurs are not completely clear. The common reference to SOC and Branching Process implies an absorbing/active phase transition, but such states would not have a clear counterpart in any eventual pathological conditions. A parsimonious approach a la Landau-Ginzburg allows us to create a neat physiologically-inspired theory, with a reduced number of parameters, that shows a synchronization phase transition †"more plausible from a biologic point of view†" at which (following the experimental protocol) we measure neuronal avalanches that reproduce many aspects of the experimentally measured ones. Moreover we analyse in detail a mechanism for the generation of scale-invariant neuronal avalanches proposed in literature as an alternative to criticality and based on the “balance” between excitation and inhibition. The role played by the various ingredients (deterministic dynamics, demographic noise and thermal noise) is clarified, making use of numerical experiments as well as analytic techniques. Lastly we present an analytic super-universal approach (including at least 4 different universality classes) for the calculation of avalanche distribution exponents in systems with absorbing states.