Landau–Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization

Abstract

Understanding the origin, nature and functional significance of complex patterns of neural activity, as recorded by diverse electrophysiological and neuroimaging techniques, is a central challenge in Neuroscience. Such patterns include collective oscillations, emerging out of neural synchronization, as well as highly-heterogeneous outbursts of activity interspersed by periods of quiescence, called "neuronal avalanches". Much debate has been generated about the possible scale-invariance or criticality of such avalanches, and its relevance for brain function. Aimed at shedding light onto this, here we analyze the large-scale collective properties of the cortex by using a mesoscopic approach, following the principle of parsimony of Landau-Ginzburg. Our model is similar to that of Wilson-Cowan for neural dynamics but, crucially, including stochasticity and space; synaptic plasticity and inhibition are considered as possible regulatory mechanisms. Detailed analyses uncover a phase diagram including down-states, synchronous, asynchronous, and up-state phases, and reveal that empirical findings for neuronal avalanches are consistently reproduced by tuning our model to the edge of synchronization. This reveals that the putative criticality of cortical dynamics does not correspond to a quiescent-to-active phase transition, as usually assumed in theoretical approaches, but to a synchronization phase transition, at which incipient oscillations and scalefree avalanches coexist. Furthermore, our model also accounts for up and down states as they occur, e.g. during deep sleep. The present approach constitutes a framework to rationalize the possible collective phases and phase transitions of cortical networks in simple terms, thus helping shed light into basic aspects of brain functioning from a very broad perspective.Cortical dynamics | Neuronal avalanches | Criticality | Synaptic plasticity T he cerebral cortex exhibits spontaneous activity even in the absence of any task or external stimuli (1-3). A salient aspect of this, so-called, resting-state dynamics, as revealed by in vivo and in vitro measurements, is that it exhibits outbursts of electrochemical activity, characterized by brief episodes of coherence -during which many neurons fire within a narrow time window-interspaced by periods of relative quiescence, giving rise to collective oscillatory rhythms (4, 5). Shedding light on the origin, nature, and functional meaning of such an intricate dynamics is a fundamental challenge in Neuroscience (6).Upon experimentally enhancing the spatio-temporal resolution of activity recordings, Beggs and Plenz made the remarkable finding that, actually, synchronized outbursts of neural activity could be decomposed into complex spatio-temporal patterns, thereon named "neuronal avalanches" (7). The sizes and durations of such avalanches were reported to be distributed as power-laws, i.e. to be organized in a scale-free way, limited only by network size (7). Furthermore, they obey finite-size scaling (8), a trademark of scale invariance...

The scale-invariant, temporal profile of neuronal avalanches in relation to cortical γ–oscillations

“…Network and biophysical models have demonstrated avalanches to emerge with oscillations at a particular E/I balance or topology; however, the corresponding avalanche profile was not reported 66,75,76 . Our findings are in line with models showing the coexistence of avalanches and neuronal oscillations at a continuous synchronization/desynchronization phase transition 50,68 . These models do not include transmission delays but exhibit exponents expected for a critical branching process.…”

“…For neuronal avalanches, empirical 3 and simulated 69 slopes of α are close to an exponent of 3/2 at Δt = ⟨IEI⟩. These empirical insights are supported within the Landau-Ginzburg theoretical framework of network dynamics 50 . The size exponent of 3/2 is also characteristic for a critical branching process in line with the well-established empirical finding of a critical branching parameter of 1 for neuronal avalanches 3 .…”

“…For neuronal avalanches, empirical3 and simulated69 slopes of α are close to an exponent of 3/2 at Δt = ⟨IEI⟩. These empirical insights are supported within the Landau-Ginzburg theoretical framework of network dynamics50 . The size exponent of 3/2 is also characteristic for a critical branching process in line with the wellestablished empirical finding of a critical branching parameter of 1 for neuronal avalanches 3 .Critical branching processes exhibit a lifetime exponent β = 2 leading to a corresponding χ = 2.Miller et alThe temporal profile of neuronal avalanches 13However, empirically obtained lifetime distributions have been difficult to analyze due to their typically narrow range in values3 .…”

“…with models showing the coexistence of avalanches and neuronal oscillations at a continuous synchronization/desynchronization phase transition 50,68 . These models do not include transmission delays, but exhibit exponents expected for a critical branching process.…”

Criticality between cortical states

“…The model successfully mimics our experimental results, in the sense that it also presents DFA exponents close to one at criticality and avalanche exponents vary continuously near the transition [23]. This supports a scenario in which the transition governing brain dynamics is not between absorbing and active phases, but rather between active and oscillating phases [27][28][29].…”