Avalanches and scaling collapse in the large-N Kuramoto model

Cluster scaling and critical points: A cautionary tale

Many systems in nature are conjectured to exist at a critical point, including the brain and earthquake faults. The primary reason for this conjecture is that the distribution of clusters (avalanches of firing neurons in the brain or regions of slip in earthquake faults) can be described by a power law. Because there are other mechanisms such as 1/f noise that can produce power laws, other criteria that the cluster critical exponents must satisfy can be used to conclude whether or not the observed power-law behavior indicates an underlying critical point rather than an alternate mechanism. We show how a possible misinterpretation of the cluster scaling data can lead one to incorrectly conclude that the measured critical exponents do not satisfy these criteria. Examples of the possible misinterpretation of the data for one-dimensional random site percolation and the one-dimensional Ising model are presented. We stress that the interpretation of a power-law cluster distribution indicating the presence of a critical point is subtle and its misinterpretation might lead to the abandonment of a promising area of research.

Oscillations and collective excitability in a model of stochastic neurons under excitatory and inhibitory coupling

We study a model with excitable neurons modeled as stochastic units with three states, representing quiescence, firing, and refractoriness. The transition rates between quiescence and firing depend exponentially on the number of firing neighbors, whereas all other rates are kept constant. This model class was shown to exhibit collective oscillations (synchronization) if neurons are spiking autonomously, but not if neurons are in the excitable regime. In both cases, neurons were restricted to interact through excitatory coupling. Here we show that a plethora of collective phenomena appear if inhibitory coupling is added. Besides the usual transition between an absorbing and an active phase, the model with excitatory and inhibitory neurons can also undergo reentrant transitions to an oscillatory phase. In the mean-field description, oscillations can emerge through supercritical or subcritical Hopf bifurcations, as well as through infinite period bifurcations. The model has bistability between active and oscillating behavior, as well as collective excitability, a regime where the system can display a peak of global activity when subject to a sufficiently strong perturbation. We employ a variant of the Shinomoto-Kuramoto order parameter to characterize the phase transitions and their system-size dependence.

Identifying epidemic threshold by temporal profile of outbreaks on networks

Identifying epidemic threshold is of great significance in preventing and controlling disease spreading on real-world networks. Previous studies have proposed different theoretical and numerical approaches to determine the epidemic threshold for the susceptible-infected-recovered (SIR) model, but the numerical study of the critical points on networks by utilizing temporal characteristics of epidemic outbreaks is still lacking. Here, we study the temporal profile of epidemic outbreaks, i.e., the average avalanche shapes of a fixed duration. At the critical point, the rescaled average terminating and nonterminating avalanche shapes for different durations collapse onto two universal curves, respectively, while the average number of subsequent events essentially remains constant. We propose two numerical measures to determine the epidemic threshold by analyzing the convergence of the rescaled average nonterminating avalanche shapes for varying durations and the stability of the average number of subsequent events, respectively. Extensive numerical simulations demonstrate that our methods can accurately identify the numerical threshold for the SIR dynamics on synthetic and empirical networks. Compared with traditional numerical measures, our methods are more efficient due to the constriction of observation duration and thus are more applicable to large-scale networks. This work helps one to understand the temporal profile of disease propagation and would promote further studies on the phase transition of epidemic dynamics.

Modeling neuronal avalanches and long-range temporal correlations at the emergence of collective oscillations: Continuously varying exponents mimic M/EEG results

We revisit the CROS ("CRitical OScillations") model which was recently proposed as an attempt to reproduce both scale-invariant neuronal avalanches and long-range temporal correlations. With excitatory and inhibitory stochastic neurons locally connected in a two-dimensional disordered network, the model exhibits a transition where alpha-band oscillations emerge. Precisely at the transition, the fluctuations of the network activity have nontrivial detrended fluctuation analysis (DFA) exponents, and avalanches (defined as supra-threshold activity) have power law distributions of size and duration. We show that, differently from previous results, the exponents governing the distributions of avalanche size and duration are not necessarily those of the mean-field directed percolation universality class (3/2 and 2, respectively). Instead, in a narrow region of parameter space, avalanche exponents obtained via a maximum-likelihood estimator vary continuously and follow a linear relation, in good agreement with results obtained from M/EEG data. In that region, moreover, the values of avalanche and DFA exponents display a spread with positive correlations, reproducing human MEG results.

Hysteresis, neural avalanches, and critical behavior near a first-order transition of a spiking neural network

Many experimental results, both in vivo and in vitro, support the idea that the brain cortex operates near a critical point and at the same time works as a reservoir of precise spatiotemporal patterns. However, the mechanism at the basis of these observations is still not clear. In this paper we introduce a model which combines both these features, showing that scale-free avalanches are the signature of a system posed near the spinodal line of a first-order transition, with many spatiotemporal patterns stored as dynamical metastable attractors. Specifically, we studied a network of leaky integrate-and-fire neurons whose connections are the result of the learning of multiple spatiotemporal dynamical patterns, each with a randomly chosen ordering of the neurons. We found that the network shows a first-order transition between a low-spiking-rate disordered state (down), and a high-rate state characterized by the emergence of collective activity and the replay of one of the stored patterns (up). The transition is characterized by hysteresis, or alternation of up and down states, depending on the lifetime of the metastable states. In both cases, critical features and neural avalanches are observed. Notably, critical phenomena occur at the edge of a discontinuous phase transition, as recently observed in a network of glow lamps.