Thermodynamic analog of integrate-and-fire neuronal networks by maximum entropy modelling.

Recent results have evidenced that spontaneous brain activity signals are organized in bursts with scale free features and long-range spatio-temporal correlations. These observations have stimulated a theoretical interpretation of results inspired in critical phenomena. In particular, relying on maximum entropy arguments, certain aspects of time-averaged experimental neuronal data have been recently described using Ising-like models, allowing the study of neuronal networks under an analogous thermodynamical framework. This method has been so far applied to a variety of experimental datasets, but never to a biologically inspired neuronal network with short and long-term plasticity. Here, we apply for the first time the Maximum Entropy method to an Integrate-and-fire (IF) model that can be tuned at criticality, offering a controlled setting for a systematic study of criticality and finite-size effects in spontaneous neuronal activity, as opposed to experiments. We consider generalized Ising Hamiltonians whose local magnetic fields and interaction parameters are assigned according to the average activity of single neurons and correlation functions between neurons of the IF networks in the critical state. We show that these Hamiltonians exhibit a spin glass phase for low temperatures, having mostly negative intrinsic fields and a bimodal distribution of interaction constants that tends to become unimodal for larger networks. Results evidence that the magnetization and the response functions exhibit the expected singular behavior near the critical point. Furthermore, we also found that networks with higher percentage of inhibitory neurons lead to Ising-like systems with reduced thermal fluctuations. Finally, considering only neuronal pairs associated with the largest correlation functions allows the study of larger system sizes.

Power spectrum and critical exponents in the 2D stochastic Wilson-Cowan model.

The power spectrum of brain activity is composed by peaks at characteristic frequencies superimposed to a background that decays as a power law of the frequency, [Formula: see text], with an exponent [Formula: see text] close to 1 (pink noise). This exponent is predicted to be connected with the exponent [Formula: see text] related to the scaling of the average size with the duration of avalanches of activity. "Mean field" models of neural dynamics predict exponents [Formula: see text] and [Formula: see text] equal or near 2 at criticality (brown noise), including the simple branching model and the fully-connected stochastic Wilson-Cowan model. We here show that a 2D version of the stochastic Wilson-Cowan model, where neuron connections decay exponentially with the distance, is characterized by exponents [Formula: see text] and [Formula: see text] markedly different from those of mean field, respectively around 1 and 1.3. The exponents [Formula: see text] and [Formula: see text] of avalanche size and duration distributions, equal to 1.5 and 2 in mean field, decrease respectively to [Formula: see text] and [Formula: see text]. This seems to suggest the possibility of a different universality class for the model in finite dimension.

Role of anaxonic local neurons in the crossover to continuously varying exponents for avalanche activity.

Local anaxonic neurons with graded potential release are important ingredients of nervous systems, present in the olfactory bulb system of mammalians and in the human visual system, as well as in arthropods and nematodes. We develop a neuronal network model including both axonic and anaxonic neurons and monitor the activity tuned by the following parameters: the decay length of the graded potential in local neurons, the fraction of local neurons, the largest eigenvalue of the adjacency matrix, and the range of connections of the local neurons. Tuning the fraction of local neurons, we derive the phase diagram including two transition lines: a critical line separating subcritical and supercritical regions, characterized by power-law distributions of avalanche sizes and durations, and a bifurcation line. We find that the overall behavior of the system is controlled by a parameter tuning the relevance of local neuron transmission with respect to the axonal one. The statistical properties of spontaneous activity are affected by local neurons at large fractions and on the condition that the graded potential transmission dominates the axonal one. In this case the scaling properties of spontaneous activity exhibit continuously varying exponents, rather than the mean-field branching model universality class.

Three cooperative mechanisms required for recovery after brain damage.

Stroke is one of the main causes of human disabilities. Experimental observations indicate that several mechanisms are activated during the recovery of functional activity after a stroke. Here we unveil how the brain recovers by explaining the role played by three mechanisms: Plastic adaptation, hyperexcitability and synaptogenesis. We consider two different damages in a neural network: A diffuse damage that simply causes the reduction of the effective system size and a localized damage, a stroke, that strongly alters the spontaneous activity of the system. Recovery mechanisms observed experimentally are implemented both separately and in a combined way. Interestingly, each mechanism contributes to the recovery to a limited extent. Only the combined application of all three together is able to recover the spontaneous activity of the undamaged system. This explains why the brain triggers independent mechanisms, whose cooperation is the fundamental ingredient for the system's recovery.

Pattern recognition with neuronal avalanche dynamics.

Pattern recognition is a fundamental neuronal process which enables a cortical system to interpret visual stimuli. How the brain learns to recognize patterns is, however, an unsolved problem. The frequently employed method of back propagation excels at this task but has been found to be unbiological in many aspects. In this Rapid Communication we achieve pattern recognition tasks in a biologically, fully consistent framework. We consider a neuronal network exhibiting avalanche dynamics, as observed experimentally, and implement negative feedback signals. These are chemical signals, such as dopamine, which mediate synaptic plasticity and sculpt the network to achieve certain tasks. The system is able to distinguish horizontal and vertical lines with high accuracy, as well as to perform well at the more complicated task of handwritten digit recognition. Resulting from the learning mechanism, spatially separate activity regions emerge, as observed in the primary visual cortex using functional magnetic resonance imaging techniques. The results therefore suggest that negative feedback signals offer an explanation for the emergence of distinct activity areas in the visual cortex.

Induced and endogenous acoustic oscillations in granular faults.

The frictional properties of disordered systems are affected by external perturbations. These perturbations usually weaken the system by reducing the macroscopic friction coefficient. This friction reduction is of particular interest in the case of disordered systems composed of granular particles confined between two plates, as this is a simple model of seismic fault. Indeed, in the geophysical context frictional weakening could explain the unexpected weakness of some faults, as well as earthquake remote triggering. In this manuscript, we review recent results concerning the response of confined granular systems to external perturbations, considering the different mechanisms by which the perturbation could weaken a system, the relevance of the frictional reduction to earthquakes, as well as discussing the intriguing scenario whereby the weakening is not monotonic in the perturbation frequency, so that a re-entrant transition is observed, as the system first enters a fluidized state and then returns to a frictional state.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Critical neural networks with short- and long-term plasticity.

In recent years self organized critical neuronal models have provided insights regarding the origin of the experimentally observed avalanching behavior of neuronal systems. It has been shown that dynamical synapses, as a form of short-term plasticity, can cause critical neuronal dynamics. Whereas long-term plasticity, such as Hebbian or activity dependent plasticity, have a crucial role in shaping the network structure and endowing neural systems with learning abilities. In this work we provide a model which combines both plasticity mechanisms, acting on two different time scales. The measured avalanche statistics are compatible with experimental results for both the avalanche size and duration distribution with biologically observed percentages of inhibitory neurons. The time series of neuronal activity exhibits temporal bursts leading to 1/f decay in the power spectrum. The presence of long-term plasticity gives the system the ability to learn binary rules such as xor, providing the foundation of future research on more complicated tasks such as pattern recognition.

Controlled Viscosity in Dense Granular Materials.

We experimentally investigate the fluidization of a granular material subject to mechanical vibrations by monitoring the angular velocity of a vane suspended in the medium and driven by an external motor. On increasing the frequency, we observe a reentrant transition, as a jammed system first enters a fluidized state, where the vane rotates with high constant velocity, and then returns to a frictional state, where the vane velocity is much lower. While the fluidization frequency is material independent, the viscosity recovery frequency shows a clear dependence on the material that we rationalize by relating this frequency to the balance between dissipative and inertial forces in the system. Molecular dynamics simulations well reproduce the experimental data, confirming the suggested theoretical picture.

Synchronized oscillations and acoustic fluidization in confined granular materials.

According to the acoustic fluidization hypothesis, elastic waves at a characteristic frequency form inside seismic faults even in the absence of an external perturbation. These waves are able to generate a normal stress which contrasts the confining pressure and promotes failure. Here, we study the mechanisms responsible for this wave activation via numerical simulations of a granular fault model. We observe the particles belonging to the percolating backbone, which sustains the stress, to perform synchronized oscillations over ellipticlike trajectories in the fault plane. These oscillations occur at the characteristic frequency of acoustic fluidization. As the applied shear stress increases, these oscillations become perpendicular to the fault plane just before the system fails, opposing the confining pressure, consistently with the acoustic fluidization scenario. The same change of orientation can be induced by external perturbations at the acoustic fluidization frequency.

Balance of excitation and inhibition determines 1/f power spectrum in neuronal networks.

The 1/f-like decay observed in the power spectrum of electro-physiological signals, along with scale-free statistics of the so-called neuronal avalanches, constitutes evidence of criticality in neuronal systems. Recent in vitro studies have shown that avalanche dynamics at criticality corresponds to some specific balance of excitation and inhibition, thus suggesting that this is a basic feature of the critical state of neuronal networks. In particular, a lack of inhibition significantly alters the temporal structure of the spontaneous avalanche activity and leads to an anomalous abundance of large avalanches. Here, we study the relationship between network inhibition and the scaling exponent ß of the power spectral density (PSD) of avalanche activity in a neuronal network model inspired in Self-Organized Criticality. We find that this scaling exponent depends on the percentage of inhibitory synapses and tends to the value ß = 1 for a percentage of about 30%. More specifically, ß is close to 2, namely, Brownian noise, for purely excitatory networks and decreases towards values in the interval [1, 1.4] as the percentage of inhibitory synapses ranges between 20% and 30%, in agreement with experimental findings. These results indicate that the level of inhibition affects the frequency spectrum of resting brain activity and suggest the analysis of the PSD scaling behavior as a possible tool to study pathological conditions.

Synaptic plasticity and neuronal refractory time cause scaling behaviour of neuronal avalanches.

Neuronal avalanches measured in vitro and in vivo in different cortical networks consistently exhibit power law behaviour for the size and duration distributions with exponents typical for a mean field self-organized branching process. These exponents are also recovered in neuronal network simulations implementing various neuronal dynamics on different network topologies. They can therefore be considered a very robust feature of spontaneous neuronal activity. Interestingly, this scaling behaviour is also observed on regular lattices in finite dimensions, which raises the question about the origin of the mean field behavior observed experimentally. In this study we provide an answer to this open question by investigating the effect of activity dependent plasticity in combination with the neuronal refractory time in a neuronal network. Results show that the refractory time hinders backward avalanches forcing a directed propagation. Hebbian plastic adaptation plays the role of sculpting these directed avalanche patterns into the topology of the network slowly changing it into a branched structure where loops are marginal.

Temporal correlations in neuronal avalanche occurrence.

Ongoing cortical activity consists of sequences of synchronized bursts, named neuronal avalanches, whose size and duration are power law distributed. These features have been observed in a variety of systems and conditions, at all spatial scales, supporting scale invariance, universality and therefore criticality. However, the mechanisms leading to burst triggering, as well as the relationship between bursts and quiescence, are still unclear. The analysis of temporal correlations constitutes a major step towards a deeper understanding of burst dynamics. Here, we investigate the relation between avalanche sizes and quiet times, as well as between sizes of consecutive avalanches recorded in cortex slice cultures. We show that quiet times depend on the size of preceding avalanches and, at the same time, influence the size of the following one. Moreover we evidence that sizes of consecutive avalanches are correlated. In particular, we show that an avalanche tends to be larger or smaller than the following one for short or long time separation, respectively. Our analysis represents the first attempt to provide a quantitative estimate of correlations between activity and quiescence in the framework of neuronal avalanches and will help to enlighten the mechanisms underlying spontaneous activity.

Mechanical origin of aftershocks.

Aftershocks are the most striking evidence of earthquake interactions and the physical mechanisms at the origin of their occurrence are still intensively debated. Novel insights stem from recent results on the influence of the faulting style on the aftershock organisation in magnitude and time. Our study shows that the size of the aftershock zone depends on the fault geometry. We find that positive correlations among parameters controlling aftershock occurrence in time, energy and space are a stable feature of seismicity independently of magnitude range and geographic areas. We explain the ensemble of experimental findings by means of a description of the Earth Crust as an heterogeneous elastic medium coupled with a Maxwell viscoelastic asthenosphere. Our results show that heterogeneous stress distribution in an elastic layer combined with a coupling to a viscous flow are sufficient ingredients to describe the physics of aftershock triggering.

Dynamic Weakening by Acoustic Fluidization during Stick-Slip Motion.

The unexpected weakness of some faults has been attributed to the emergence of acoustic waves that promote failure by reducing the confining pressure through a mechanism known as acoustic fluidization, also proposed to explain earthquake remote triggering. Here we validate this mechanism via the numerical investigation of a granular fault model system. We find that the stick-slip dynamics is affected only by perturbations applied at a characteristic frequency corresponding to oscillations normal to the fault, leading to gradual dynamical weakening as failure is approaching. Acoustic waves at the same frequency spontaneously emerge at the onset of failure in the absence of perturbations, supporting the relevance of acoustic fluidization in earthquake triggering.

Non-monotonic dependence of the friction coefficient on heterogeneous stiffness.

The complexity of the frictional dynamics at the microscopic scale makes difficult to identify all of its controlling parameters. Indeed, experiments on sheared elastic bodies have shown that the static friction coefficient depends on loading conditions, the real area of contact along the interfaces and the confining pressure. Here we show, by means of numerical simulations of a 2D Burridge-Knopoff model with a simple local friction law, that the macroscopic friction coefficient depends non-monotonically on the bulk elasticity of the system. This occurs because elastic constants control the geometrical features of the rupture fronts during the stick-slip dynamics, leading to four different ordering regimes characterized by different orientations of the rupture fronts with respect to the external shear direction. We rationalize these results by means of an energetic balance argument.

Modelling the influence of photospheric turbulence on solar flare statistics.

Solar flares stem from the reconnection of twisted magnetic field lines in the solar photosphere. The energy and waiting time distributions of these events follow complex patterns that have been carefully considered in the past and that bear some resemblance with earthquakes and stockmarkets. Here we explore in detail the tangling motion of interacting flux tubes anchored in the plasma and the energy ejections resulting when they recombine. The mechanism for energy accumulation and release in the flow is reminiscent of self-organized criticality. From this model, we suggest the origin for two important and widely studied properties of solar flare statistics, including the time-energy correlations. We first propose that the scale-free energy distribution of solar flares is largely due to the twist exerted by the vorticity of the turbulent photosphere. Second, the long-range temporal and time-energy correlations appear to arise from the tube-tube interactions. The agreement with satellite measurements is encouraging.

Brain modularity controls the critical behavior of spontaneous activity.

The human brain exhibits a complex structure made of scale-free highly connected modules loosely interconnected by weaker links to form a small-world network. These features appear in healthy patients whereas neurological diseases often modify this structure. An important open question concerns the role of brain modularity in sustaining the critical behaviour of spontaneous activity. Here we analyse the neuronal activity of a model, successful in reproducing on non-modular networks the scaling behaviour observed in experimental data, on a modular network implementing the main statistical features measured in human brain. We show that on a modular network, regardless the strength of the synaptic connections or the modular size and number, activity is never fully scale-free. Neuronal avalanches can invade different modules which results in an activity depression, hindering further avalanche propagation. Critical behaviour is solely recovered if inter-module connections are added, modifying the modular into a more random structure.

Spatial organization of foreshocks as a tool to forecast large earthquakes.

An increase in the number of smaller magnitude events, retrospectively named foreshocks, is often observed before large earthquakes. We show that the linear density probability of earthquakes occurring before and after small or intermediate mainshocks displays a symmetrical behavior, indicating that the size of the area fractured during the mainshock is encoded in the foreshock spatial organization. This observation can be used to discriminate spatial clustering due to foreshocks from the one induced by aftershocks and is implemented in an alarm-based model to forecast m > 6 earthquakes. A retrospective study of the last 19 years Southern California catalog shows that the daily occurrence probability presents isolated peaks closely located in time and space to the epicenters of five of the six m > 6 earthquakes. We find daily probabilities as high as 25% (in cells of size 0.04 × 0.04deg(2)), with significant probability gains with respect to standard models.

Balance between excitation and inhibition controls the temporal organization of neuronal avalanches.

Neuronal avalanches, measured in vitro and in vivo, exhibit a robust critical behavior. Their temporal organization hides the presence of correlations. Here we present experimental measurements of the waiting time distribution between successive avalanches in the rat cortex in vitro. This exhibits a nonmonotonic behavior not usually found in other natural processes. Numerical simulations provide evidence that this behavior is a consequence of the alternation between states of high and low activity, named up and down states, leading to a balance between excitation and inhibition controlled by a single parameter. During these periods, both the single neuron state and the network excitability level, keeping memory of past activity, are tuned by homeostatic mechanisms.

Multiple-time scaling and universal behavior of the earthquake interevent time distribution.

The interevent time distribution characterizes the temporal occurrence in seismic catalogs. Universal scaling properties of this distribution have been evidenced for entire catalogs and seismic sequences. Recently, these universal features have been questioned and some criticisms have been raised. We investigate the existence of universal scaling properties by analyzing a Californian catalog and by means of numerical simulations of an epidemic-type model. We show that the interevent time distribution exhibits a universal behavior over the entire temporal range if four characteristic times are taken into account. The above analysis allows us to identify the scaling form leading to universal behavior and explains the observed deviations. Furthermore, it provides a tool to identify the dependence on the mainshock magnitude of the c parameter that fixes the onset of the power law decay in the Omori law.