Self-organization without conservation: Are neuronal avalanches generically critical?

Recent experiments on cortical neural networks have revealed the existence of well-defined avalanches of electrical activity. Such avalanches have been claimed to be generically scale-invariant -- i.e. power-law distributed -- with many exciting implications in Neuroscience. Recently, a self-organized model has been proposed by Levina, Herrmann and Geisel to justify such an empirical finding. Given that (i) neural dynamics is dissipative and (ii) there is a loading mechanism "charging" progressively the background synaptic strength, this model/dynamics is very similar in spirit to forest-fire and earthquake models, archetypical examples of non-conserving self-organization, which have been recently shown to lack true criticality. Here we show that cortical neural networks obeying (i) and (ii) are not generically critical; unless parameters are fine tuned, their dynamics is either sub- or super-critical, even if the pseudo-critical region is relatively broad. This conclusion seems to be in agreement with the most recent experimental observations. The main implication of our work is that, if future experimental research on cortical networks were to support that truly critical avalanches are the norm and not the exception, then one should look for more elaborate (adaptive/evolutionary) explanations, beyond simple self-organization, to account for this.Comment: 28 pages, 11 figures, regular pape

On the Finite Size Scaling in Disordered Systems

The critical behavior of a quenched random hypercubic sample of linear size $L$ is considered, within the ``random-$T_{c}$'' field-theoretical mode, by using the renormalization group method. A finite-size scaling behavior is established and analyzed near the upper critical dimension $d=4-\epsilon$ and some universal results are obtained. The problem of self-averaging is clarified for different critical regimes.Comment: 21 pages, 2 figures, submitted to the Physcal Review

Randomly dilute Ising model: A nonperturbative approach

The N-vector cubic model relevant, among others, to the physics of the randomly dilute Ising model is analyzed in arbitrary dimension by means of an exact renormalization-group equation. This study provides a unified picture of its critical physics between two and four dimensions. We give the critical exponents for the three-dimensional randomly dilute Ising model which are in good agreement with experimental and numerical data. The relevance of the cubic anisotropy in the O(N) model is also treated.Comment: 4 pages, published versio

Polymers in long-range-correlated disorder

We study the scaling properties of polymers in a d-dimensional medium with quenched defects that have power law correlations ~r^{-a} for large separations r. This type of disorder is known to be relevant for magnetic phase transitions. We find strong evidence that this is true also for the polymer case. Applying the field-theoretical renormalization group approach we perform calculations both in a double expansion in epsilon=4-d and delta=4-a up to the 1-loop order and secondly in a fixed dimension (d=3) approach up to the 2-loop approximation for different fixed values of the correlation parameter, 2=<a=<3. In the latter case the numerical results need appropriate resummation. We find that the asymptotic behavior of self-avoiding walks in three dimensions and long-range-correlated disorder is governed by a set of separate exponents. In particular, we give estimates for the 'nu' and 'gamma' exponents as well as for the correction-to-scaling exponent 'omega'. The latter exponent is also calculated for the general m-vector model with m=1,2,3.Comment: 13 pages, 5 figure

Critical equation of state of randomly dilute Ising systems

We determine the critical equation of state of three-dimensional randomly dilute Ising systems, i.e. of the random-exchange Ising universality class. We first consider the small-magnetization expansion of the Helmholtz free energy in the high-temperature phase. Then, we apply a systematic approximation scheme of the equation of state in the whole critical regime, that is based on polynomial parametric representations matching the small-magnetization of the Helmholtz free energy and satisfying a global stationarity condition. These results allow us to estimate several universal amplitude ratios, such as the ratio A^+/A^- of the specific-heat amplitudes. Our best estimate A^+/A^-=1.6(3) is in good agreement with experimental results on dilute uniaxial antiferromagnets.Comment: 21 pages, 1 figure, refs adde

Patchiness and Demographic Noise in Three Ecological Examples

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy

Surface critical behavior of random systems at the ordinary transition

We calculate the surface critical exponents of the ordinary transition occuring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation, as well as in $d=4-\epsilon$ dimensions. At $d=4-\epsilon$ we extend, up to the next-to-leading order, the previous first-order results of the $\sqrt{\epsilon}$ expansion by Ohno and Okabe [Phys.Rev.B 46, 5917 (1992)]. In both cases the numerical estimates for surface exponents are computed using Pade approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.Comment: 11 pages, 11 figure

Randomly dilute spin models with cubic symmetry

We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson $\phi^4$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension (d=3) perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.Comment: 23 pages, 1 figure

The three-dimensional randomly dilute Ising model: Monte Carlo results

We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices $L^3$ with $L\le 256$. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining $\nu = 0.683(3)$, $\eta = 0.035(2)$, $\beta = 0.3535(17)$, and $\alpha = -0.049(9)$, in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio $R^+_\xi$ that expresses the universality of the free-energy density per correlation volume. We find $R^+_\xi = 0.2885(15)$.Comment: 34 pages, 7 figs, few correction

Pseudo--epsilon expansion of six--loop renormalization group functions of an anisotropic cubic model

Six-loop massive scheme renormalization group functions of a d=3-dimensional cubic model (J.M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B vol. 61, 15136 (2000)) are reconsidered by means of the pseudo-epsilon expansion. The marginal order parameter components number N_c=2.862(5) as well as critical exponents of the cubic model are obtained. Our estimate N_c<3 leads in particular to the conclusion that all ferromagnetic cubic crystals with three easy axis should undergo a first order phase transition.Comment: 8 page