Networks of living neurons exhibit diverse patterns of activity, including oscillations, synchrony, and waves. Recent work in physics has shown yet another mode of activity in systems composed of many nonlinear units interacting locally. For example, avalanches, earthquakes, and forest fires all propagate in systems organized into a critical state in which event sizes show no characteristic scale and are described by power laws. We hypothesized that a similar mode of activity with complex emergent properties could exist in networks of cortical neurons. We investigated this issue in mature organotypic cultures and acute slices of rat cortex by recording spontaneous local field potentials continuously using a 60 channel multielectrode array. Here, we show that propagation of spontaneous activity in cortical networks is described by equations that govern avalanches. As predicted by theory for a critical branching process, the propagation obeys a power law with an exponent of -3/2 for event sizes, with a branching parameter close to the critical value of 1. Simulations show that a branching parameter at this value optimizes information transmission in feedforward networks, while preventing runaway network excitation. Our findings suggest that "neuronal avalanches" may be a generic property of cortical networks, and represent a mode of activity that differs profoundly from oscillatory, synchronized, or wave-like network states. In the critical state, the network may satisfy the competing demands of information transmission and network stability.

Spontaneous neuronal activity is a ubiquitous feature of cortex. Its spatiotemporal organization reflects past input and modulates future network output. Here we study whether a particular type of spontaneous activity is generated by a network that is optimized for input processing. Neuronal avalanches are a type of spontaneous activity observed in superficial cortical layers in vitro and in vivo with statistical properties expected from a network operating at "criticality." Theory predicts that criticality and, therefore, neuronal avalanches are optimal for input processing, but until now, this has not been tested in experiments. Here, we use cortex slice cultures grown on planar microelectrode arrays to demonstrate that cortical networks that generate neuronal avalanches benefit from a maximized dynamic range, i.e., the ability to respond to the greatest range of stimuli. By changing the ratio of excitation and inhibition in the cultures, we derive a network tuning curve for stimulus processing as a function of distance from criticality in agreement with predictions from our simulations. Our findings suggest that in the cortex, (1) balanced excitation and inhibition establishes criticality, which maximizes the range of inputs that can be processed, and (2) spontaneous activity and input processing are unified in the context of critical phenomena.

Power laws are theoretically interesting probability distributions that are also frequently used to describe empirical data. In recent years, effective statistical methods for fitting power laws have been developed, but appropriate use of these techniques requires significant programming and statistical insight. In order to greatly decrease the barriers to using good statistical methods for fitting power law distributions, we developed the powerlaw Python package. This software package provides easy commands for basic fitting and statistical analysis of distributions. Notably, it also seeks to support a variety of user needs by being exhaustive in the options available to the user. The source code is publicly available and easily extensible.

Spontaneous neuronal activity is an important property of the cerebral cortex but its spatiotemporal organization and dynamical framework remain poorly understood. Studies in reduced systems-tissue cultures, acute slices, and anesthetized ratsshow that spontaneous activity forms characteristic clusters in space and time, called neuronal avalanches. Modeling studies suggest that networks with this property are poised at a critical state that optimizes input processing, information storage, and transfer, but the relevance of avalanches for fully functional cerebral systems has been controversial. Here we show that ongoing cortical synchronization in awake rhesus monkeys carries the signature of neuronal avalanches. Negative LFP deflections (nLFPs) correlate with neuronal spiking and increase in amplitude with increases in local population spike rate and synchrony. These nLFPs form neuronal avalanches that are scale-invariant in space and time and with respect to the threshold of nLFP detection. This dimension, threshold invariance, describes a fractal organization: smaller nLFPs are embedded in clusters of larger ones without destroying the spatial and temporal scale-invariance of the dynamics. These findings suggest an organization of ongoing cortical synchronization that is scale-invariant in its three fundamental dimensions-time, space, and local neuronal group size. Such scale-invariance has ontogenetic and phylogenetic implications because it allows large increases in network capacity without a fundamental reorganization of the system. neuronal synchronization ͉ resting state ͉ rhesus monkey ͉ spontaneous activity ͉ ongoing activity T he cerebral cortex displays spontaneous activity, also known as 'ongoing' or 'resting state' activity, which persists in the absence of sensory stimuli or motor outputs. The ongoing activity is a robust feature of cortical dynamics as it is only modulated to a small extent by stimulus presentation (1), but contributes significantly to the large variability observed in stimulus responses (2-5). In fact, ongoing activity has been found to reflect multiple aspects of neuronal processing. The activity is similar to that observed during stimulus presentation (1, 6-8), incorporates previously acquired information (9), and carries information about the underlying neuronal network [for review see (10)]. Indeed, correlations during resting state activity are altered in disease states such as schizophrenia or chronic pain (11, 12), which raises the question whether there is a general framework that describes the statistics in the spatiotemporal organization of this dynamics.Recently, we found that spontaneous cortical activity in slice cultures, acute slices, and in the anesthetized rat in vivo has a scale-invariant dynamics called neuronal avalanches (13)(14)(15). These spontaneous bursts of synchronized activity occur in clusters of sizes s (where s is the number of active sites in an electrode array) that distribute according to a power law with exponent ␣:where ␣ usually lies between ...

The repertoire of neural activity patterns that a cortical network can produce constrains the ability of the network to transfer and process information. Here, we measured activity patterns obtained from multisite local field potential recordings in cortex cultures, urethaneanesthetized rats, and awake macaque monkeys. First, we quantified the information capacity of the pattern repertoire of ongoing and stimulus-evoked activity using Shannon entropy. Next, we quantified the efficacy of information transmission between stimulus and response using mutual information. By systematically changing the ratio of excitation/inhibition (E/I) in vitro and in a network model, we discovered that both information capacity and information transmission are maximized at a particular intermediate E/I, at which ongoing activity emerges as neuronal avalanches. Next, we used our in vitro and model results to correctly predict in vivo information capacity and interactions between neuronal groups during ongoing activity. Close agreement between our experiments and model suggest that neuronal avalanches and peak information capacity arise because of criticality and are general properties of cortical networks with balanced E/I.

A major goal of neuroscience is to elucidate mechanisms of cortical information processing and storage. Previous work from our laboratory (Beggs and Plenz, 2003) revealed that propagation of local field potentials (LFPs) in cortical circuits could be described by the same equations that govern avalanches. Whereas modeling studies suggested that these "neuronal avalanches" were optimal for information transmission, it was not clear what role they could play in information storage. Work from numerous other laboratories has shown that cortical structures can generate reproducible spatiotemporal patterns of activity that could be used as a substrate for memory. Here, we show that although neuronal avalanches lasted only a few milliseconds, their spatiotemporal patterns were also stable and significantly repeatable even many hours later. To investigate these issues, we cultured coronal slices of rat cortex for 4 weeks on 60-channel microelectrode arrays and recorded spontaneous extracellular LFPs continuously for 10 hr. Using correlation-based clustering and a global contrast function, we found that each cortical culture spontaneously produced 4736 Ϯ 2769 (mean Ϯ SD) neuronal avalanches per hour that clustered into 30 Ϯ 14 statistically significant families of spatiotemporal patterns. In 10 hr of recording, over 98% of the mutual information shared by these avalanche patterns were retained. Additionally, jittering analysis revealed that the correlations between avalanches were temporally precise to within Ϯ4 msec. The long-term stability, diversity, and temporal precision of these avalanches indicate that they fulfill many of the requirements expected of a substrate for memory and suggest that they play a central role in both information transmission and storage within cortical networks.

Rapidly growing empirical evidence supports the hypothesis that the cortex operates near criticality. Although the confirmation of this hypothesis would mark a significant advance in fundamental understanding of cortical physiology, a natural question arises: What functional benefits are endowed to cortical circuits that operate at criticality? In this review, we first describe an introductory-level thought experiment to provide the reader with an intuitive understanding of criticality. Second, we discuss some practical approaches for investigating criticality. Finally, we review quantitative evidence that three functional properties of the cortex are optimized at criticality: 1) dynamic range, 2) information transmission, and 3) information capacity. We focus on recently reported experimental evidence and briefly discuss the theory and history of these ideas.

The subthalamic nucleus of the basal ganglia (STN) is important for normal movement as well as in movement disorders. Lesioning or deep-brain stimulation of the STN can alleviate resting tremor in Parkinson's disease. The STN and its target nuclei display synchronized oscillatory burst discharge at low frequencies, some of which correlate with tremor, but the mechanism underlying this synchronized bursting is unknown. Here we show that the excitatory STN and inhibitory, external globus pallidus (GPe) form a feedback system that engages in synchronized bursting. In mature organotypic cortex-striatum-STN-GPe cultures, neurons in the STN and GPe spontaneously produce synchronized oscillating bursts at 0.4, 0.8 and 1.8 Hz. Pallidal lesion abolishes this bursting, whereas cortical lesion favours bursting at 0.8 Hz. Pallidal bursts, although weaker than STN bursts, were required for synchronized oscillatory burst generation by recruitment of subthalmic rebound excitation. We propose that the STN and GPe constitute a central pacemaker modulated by striatal inhibition of GPe neurons. This pacemaker could be responsible for synchronized oscillatory activity in the normal and pathological basal ganglia.