Identification of Criticality in Neuronal Avalanches: II. A Theoretical and Empirical Investigation of the Driven Case

Abstract

The observation of apparent power laws in neuronal systems has led to the suggestion that the brain is at, or close to, a critical state and may be a self-organised critical system. Within the framework of self-organised criticality a separation of timescales is thought to be crucial for the observation of power-law dynamics and computational models are often constructed with this property. However, this is not necessarily a characteristic of physiological neural networks—external input does not only occur when the network is at rest/a steady state. In this paper we study a simple neuronal network model driven by a continuous external input (i.e. the model does not have an explicit separation of timescales from seeding the system only when in the quiescent state) and analytically tuned to operate in the region of a critical state (it reaches the critical regime exactly in the absence of input—the case studied in the companion paper to this article). The system displays avalanche dynamics in the form of cascades of neuronal firing separated by periods of silence. We observe partial scale-free behaviour in the distribution of avalanche size for low levels of external input. We analytically derive the distributions of waiting times and investigate their temporal behaviour in relation to different levels of external input, showing that the system’s dynamics can exhibit partial long-range temporal correlations. We further show that as the system approaches the critical state by two alternative ‘routes’, different markers of criticality (partial scale-free behaviour and long-range temporal correlations) are displayed. This suggests that signatures of criticality exhibited by a particular system in close proximity to a critical state are dependent on the region in parameter space at which the system (currently) resides.

Single-cell membrane potential fluctuations evince network scale-freeness and quasicriticality

“…For continuous time-series such as the Vm recording, one selects a threshold and a baseline. We defined a neuronal avalanche based on the positive threshold crossing followed by a negative threshold crossing of the Vm time series (Poil et al, 2012;Hartley et al, 2014;Larremore et al, 2014;Karimipanah et al, 2017b). We quantified each neuronal avalanche by (i) its size A, i.e., the area between the curve and the baseline, and (ii) its duration , i.e., the time between threshold crossings ( Figure 2B).…”

Spike avalanches in vivo suggest a driven, slightly subcritical brain state

“…An important property of SOC systems, which is potentially absent in neural activity, is the separation of time scales (STS) (Bak et al, 1987; Drossel and Schwabl, 1992; Clar et al, 1996; Dickman et al, 2000; Pruessner, 2012; Hartley et al, 2013) whereby pauses between avalanches last much longer than the avalanches proper. For example, forest fires last for a much shorter time than it takes to regrow the forest.…”

“…Likewise, in the classical sandpile model, scale-free avalanche distributions are observed only if the grains are dropped at a low enough rate (Vespignani and Zapperi, 1997, 1998). This low rate of external input, called drive, is a necessary condition for the long pauses and hence for SOC (Bak et al, 1987; Drossel and Schwabl, 1992; Clar et al, 1996; Dickman et al, 2000; Pruessner, 2012; Hartley et al, 2013). …”

Aluminum?s Role in CNS-immune System Interactions leading to Neurological Disorders

“…Many researchers have long sought data to show that the brain operates at a critical state to benefit from the maximal dynamic range of processing, fidelity of information transmission, coherence between multiple "nested" biosemiotic levels [19] and information capacity [74][75][76][77][78][79][80]. A very appropriate marker of criticality may prove to be the percolation transition of interphase water at neurolemmal membranes, e.g.…”

“…In 2003, Beggs and Plenz [75] showed that in vitro propagation of spontaneous activity in cortical networks obeys a power law, and is described by equations that govern avalanches [62]. They proposed that these so-called "neuronal avalanches" may represent new modes of neuronal network activity [74][75][76], which differ profoundly from oscillatory, synchronized or wave-like network states. They further proposed that in the critical state, the branching network may satisfy competing demands of information transmission and network stability [75].…”