In 1972–1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers of activated and quiescent excitatory and inhibitory neurons, and said nothing about fluctuations and correlations of such activity. However, in 1997 Ohira and Cowan, and then in 2007–2009 Buice and Cowan introduced Markov models of such activity that included fluctuation and correlation effects. Here we show how both models can be used to provide a quantitative account of the population dynamics of neocortical activity.We first describe how the Markov models account for many recent measurements of the resting or spontaneous activity of the neocortex. In particular we show that the power spectrum of large-scale neocortical activity has a Brownian motion baseline, and that the statistical structure of the random bursts of spiking activity found near the resting state indicates that such a state can be represented as a percolation process on a random graph, called directed percolation.Other data indicate that resting cortex exhibits pair correlations between neighboring populations of cells, the amplitudes of which decay slowly with distance, whereas stimulated cortex exhibits pair correlations which decay rapidly with distance. Here we show how the Markov model can account for the behavior of the pair correlations.Finally we show how the 1972–1973 Wilson–Cowan equations can account for recent data which indicates that there are at least two distinct modes of cortical responses to stimuli. In mode 1 a low intensity stimulus triggers a wave that propagates at a velocity of about 0.3 m/s, with an amplitude that decays exponentially. In mode 2 a high intensity stimulus triggers a larger response that remains local and does not propagate to neighboring regions.

Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson–Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson–Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures.Electronic Supplementary MaterialThe online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

Gang violence has plagued the Los Angeles policing district of Hollenbeck for over half a century. With sophisticated models, police may better understand and predict the region's frequent gang crimes. The purpose of this paper is to model Hollenbeck's gang rivalries. A self-exciting point process called a Hawkes process is used to model rivalries over time. While this is shown to fit the data well, an agent based model is presented which is able to accurately simulate gang rivalry crimes not only temporally but also spatially. Finally, we compare random graphs generated by the agent model to existing models developed to incorporate geography into random graphs.

This paper contains an analysis of a simple neural network that exhibits self-organized criticality. Such criticality follows from the combination of a simple neural network with an excitatory feedback loop that generates bistability, in combination with an anti-Hebbian synapse in its input pathway. Using the methods of statistical field theory, we show how one can formulate the stochastic dynamics of such a network as the action of a path integral, which we then investigate using renormalization group methods. The results indicate that the network exhibits hysteresis in switching back and forward between its two stable states, each of which loses its stability at a saddle-node bifurcation. The renormalization group analysis shows that the fluctuations in the neighborhood of such bifurcations have the signature of directed percolation. Thus the network states undergo the neural analog of a phase transition in the universality class of directed percolation. The network replicates precisely the behavior of the original sand-pile model of Bak, Tang & Wiesenfeld.arXiv:1209.3829v1 [q-bio.NC] 18 Sep 2012Self-organized criticality in a neural network

We show that an array of E-patches will self-organize around critical points of the directed percolation phase transition and, when driven by a weak stimulus, will oscillate between UP and DOWN states, each of which generates avalanches consistent with directed percolation. The array therefore exhibits self-organized criticality (SOC) and replicates the behavior of the original sandpile model of [1]. We also show that an array of E∕I patches will also self-organize to a weakly stable node located near the critical point of a directed percolation phase transition, so that fluctuations about the weakly stable node will also follow a power slope with a slope characteristic of directed percolation. We refer to this as self-organized near-criticality (SONC). IntroductionIdeas about criticality in nonequilibrium dynamical systems have been around for at least 50 years or more. Criticality refers to the fact that nonlinear dynamical systems can have local equilibria that are marginally stable, so that small perturbations can drive the system away from the local equilibria toward one of several locally stable equilibria. In physical systems, such marginally stable states manifest in several ways; in particular, if the system is spatially as well as temporally organized, then long-range correlations in both space and time can occur, and the statistics of the accompanying fluctuating activity becomes non-Gaussian, and in fact is self-similar in its structure, and therefore follows a power law. Bak et al.[1] introduced a mechanism whereby such a dynamical system could self-organize to a marginally stable critical point, which they called self-organized criticality. Their paper immediately triggered an avalanche of papers on the topic, not the least of which was a connection with 1∕f or scale-free noise. However, it was not until another paper appeared, by Gil and Sornette [2], which greatly clarified the dynamical prerequisites for achieving SOC, that a real understanding developed of the essential requirements for SOC: (i) an order-parameter equation for a dynamical system with a time-constant o , with stable states separated by a threshold; (ii) a Criticality in Neural Systems, First Edition. Edited by Dietmar Plenz and Ernst Niebur.