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.

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 ...

Many complex systems give rise to events that are clustered in space and time, thereby establishing a correlation structure that is governed by power law statistics. In the cortex, such clusters of activity, called "neuronal avalanches," were recently found in local field potentials (LFPs) of spontaneous activity in acute cortex slices, slice cultures, the developing cortex of the anesthetized rat, and premotor and motor cortex of awake monkeys. At present, it is unclear whether neuronal avalanches also exist in the spontaneous LFPs and spike activity in vivo in sensory areas of the mature brain. To address this question, we recorded spontaneous LFPs and extracellular spiking activity with multiple 4 × 4 microelectrode arrays (Michigan Probes) in area 17 of adult cats under anesthesia. A cluster of events was defined as a consecutive sequence of time bins Δt (1-32 ms), each containing at least one LFP event or spike anywhere on the array. LFP cluster sizes consistently distributed according to a power law with a slope largely above -1.5. In two thirds of the corresponding experiments, spike clusters also displayed a power law that displayed a slightly steeper slope of -1.8 and was destroyed by subsampling operations. The power law in spike clusters was accompanied with stronger temporal correlations between spiking activities of neurons that spanned longer time periods compared with spike clusters lacking power law statistics. The results suggest that spontaneous activity of the visual cortex under anesthesia has the properties of neuronal avalanches.

The spread of a virus is an example of a dynamic process occurring on a discrete spatial arrangement. While the mean-field approximation reasonably reproduces the spreading behaviour for topologies where the number of connections per node is either high or strongly fluctuating and for those that show small-world features, it is inaccurate for lattice structured populations. Various improvements upon the ordinary pair approximation based on a number of assumptions concerning the higher-order correlations have been proposed. To find approaches that allow for a derivation of their dynamics remains a great challenge. By representing the population with its connectivity patterns as a homogeneous network, we propose a systematic methodology for the description of the epidemic dynamics that takes into account spatial correlations up to a desired range. The equations that the dynamical correlations are subject to are derived in a straightforward way, and they are solved very efficiently due to their binary character. The method embeds very naturally spatial patterns such as the presence of loops characterizing the square lattice or the tree-like structure ubiquitous in random networks, providing an improved description of the steady state as well as the invasion dynamics.

Supplementing a lattice with long-range connections effectively models small-world networks characterized by a high local and global interconnectedness observed in systems ranging from society to the brain. If the links have a wiring cost associated to their length l, the corresponding distribution q(l) plays a crucial role. Uniform length distributions have received most attention despite indications that q(l) ~ l^{-\alpha} exist, e.g. for integrated circuits, the Internet and cortical networks. While length distributions of this type were previously examined in the context of navigability, we here discuss for such systems the emergence and physical realizability of small-world topology. Our simple argument allows to understand under which condition and at what expense a small world results.Comment: This paper is related to cond-mat/0501420, with a discussion on different possibilities to construct small-world networks with power-law decaying connection length distributions; to appear in Phys. Rev. E, 4 pages, 2 figure

The spreading of an epidemic is determined by the connectiviy patterns which underlie the population. While it has been noted that a virus spreads more easily on a network in which global distances are small, it remains a great challenge to find approaches that unravel the precise role of local interconnectedness. Such topological properties enter very naturally in the framework of our two-timestep description, also providing a novel approach to tract a probabilistic system. The method is elaborated for SIS-type epidemic processes, leading to a quantitative interpretation of the role of loops up to length 4 in the onset of an epidemic.

Abstract. The increased availability of data on real networks has favoured an explosion of activity in the elaboration of models able to reproduce both qualitatively and quantitatively the measured properties. What has been less explored is the reliability of the data, and whether the measurement technique biases them. Here we show that tree-like explorations (similar in principle to traceroute) can indeed change the measured exponents of a scale-free network. PACS

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