Higher-Order Cumulants Drive Neuronal Activity Patterns, Inducing UP-DOWN States in Neural Populations

A major challenge in neuroscience is to understand the role of the higher-order correlations structure of neuronal populations. The dichotomized Gaussian model (DG) generates spike trains by means of thresholding a multivariate Gaussian random variable. The DG inputs are Gaussian distributed, and thus have no interactions beyond the second order in their inputs; however, they can induce higher-order correlations in the outputs. We propose a combination of analytical and numerical techniques to estimate higher-order, above the second, cumulants of the firing probability distributions. Our findings show that a large amount of pairwise interactions in the inputs can induce the system into two possible regimes, one with low activity (“DOWN state”) and another one with high activity (“UP state”), and the appearance of these states is due to a combination between the third- and fourth-order cumulant. This could be part of a mechanism that would help the neural code to upgrade specific information about the stimuli, motivating us to examine the behavior of the critical fluctuations through the Binder cumulant close to the critical point. We show, using the Binder cumulant, that higher-order correlations in the outputs generate a critical neural system that portrays a second-order phase transition.

Vicsek model of self-propelled particles with hybrid noise

In self-propelled particle models, the interactions are generally subject to some type of noise, which is introduced either during or after the interactions between the particles. Here, self-propelled particles are subject to two different types of noise. If the orientation between them is less than a certain rate [Formula: see text], a vectorial noise is introduced in each particle–particle interaction, otherwise, an angular noise is introduced in the average direction of motion, already calculated, of neighboring particles. We vary the rate [Formula: see text] over a wide range ([Formula: see text]) and study the change in the nature of the phase transition. The product-moment correlation coefficient between the order parameter and the hybrid noise, is calculated. We found that in the combination of the two types of noise, the critical noise decreases with increasing the parameter [Formula: see text], and by analysis of the Binder cumulant we estimate the value of [Formula: see text] as from which begins to appear the effects of the vectorial noise on the phase transition.

Consensus formation in continuous opinion dynamics on quasiperiodic lattices

We studied the Biswas–Chatterjee–Sen (BCS) consensus formation model, also known as the Kinetic Continuous Opinion Dynamics (KCOD) model on quasiperiodic lattices by using Kinetic Monte Carlo simulations and Finite Size Scaling technique. Our results are consistent with a continuous phase transition, controlled by an external noise. We obtained the order parameter [Formula: see text], defined as the averaged opinion, the fourth-order Binder cumulant [Formula: see text] and susceptibility [Formula: see text] as functions of the noise parameter. We estimated the critical noises for Penrose and Ammann–Beenker lattices. We also considered seven-fold and nine-fold quasiperiodic lattices and estimated the respective critical noises as well. Irrespective of rotational and translational long-range order of the lattice, the system falls in the same universality class of the two-dimensional Ising model. Quasiperiodic order is irrelevant and it does not change any critical exponents for BCS model.

Hyperscaling Violation in Ising Spin Glasses

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension D = 6 . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.

Критическая температура трехвершинной модели Поттса на решетке Кагоме

The Potts model on the Kagome lattice is considered. The Monte Carlo method is used to obtain the temperature dependences of the thermodynamic parameters of heat capacity C, order parameter m, and susceptibility χ. The calculations were performed for systems with periodic boundary conditions. Systems with linear dimensions LxL = N, L = 20-90 were considered. Based on the fourth order Binder cumulant method, the critical temperature (Tc) was calculated for the three-vertex Potts model on the Kagome lattice. It is shown that the obtained Tc value within the statistical error is in good agreement with the results obtained using the transfer-matrix and polynomial approximation methods.

Phase transitions and thermodynamic properties of the antiferromagnetic Potts model on a face-centered cubic lattice

Using the Monte Carlo method we investigate the phase transitions and thermodynamic properties of magnetic structures with noncollinear directions of magnetic moments corresponded to antiferromagnetic q=4 Potts model on a face-centered cubic lattice. Monte Carlo simulations are performed on lattices with linear sizes L=20÷44. Thermodynamic parameters: the order parameter mAF, susceptibility χ, internal energy U, and specific heat C are evaluated for all studied systems. By employing the fourth order Binder cumulant method, a first order transition is shown to be occurred in the model.

Magnetic properties and Binder cumulants of a mixed spin-2 and spin-5/2 Ising diamond chain

The mixed spin-2 and spin-5/2 Ising diamond chain is studied using Monte Carlo simulation. The thermal variation of magnetization and magnetic susceptibilities are obtained in diamond chain. The values of transition temperatures of a mixed spin-2 and spin-5/2 Ising diamond chain have been obtained for different sizes of system N, for different crystal field and for different exchange interactions. The Binder cumulant of magnetization [Formula: see text] is investigated. The magnetization with the crystal field and the exchange interaction has been established. The magnetic hysteresis cycle is determined for different values of the crystal field, exchange interactions and temperatures. The system presents the superparamagnetic behavior for a fixed crystal field and temperature value.

Behavioral transitions induced by speed and noise in animal aggregates

In this paper, we used a self-propelled particle model to study the transition between phases of collective behavior observed in animal aggregates. In these systems, transitions occur when individuals shift from one collective state to another. We investigated transitions induced by both the speed and the noise. Statistical quantities that characterize the phase transition driven by noise, such as order parameter, the Binder cumulant and the susceptibility were analyzed, and we used the finite-size scaling theory to estimate the critical exponent ratios [Formula: see text] and [Formula: see text].

Influence of Quenched Nonmagnetic Impurities on Phase Transitions in Two-Dimensional 3-State Antiferromagnetic Potts Model on Triangular Lattice

An influence of quenched nonmagnetic disorder on the phase transitions in the two dimensional antiferromagnetic Potts model with a number of spin state q=3 on a triangular lattice is calculated by the Monte-Carlo method. The systems with linear sizes L=20÷144 at spin concentrations p=1.00, 0.90, 0.80 are studied. By means of the fourth order Binder cumulant method, the inclusion of a quenched disorder as nonmagnetic impurities into a pure antiferromagnetic Potts model is shown to be the cause of the change of the first order phase transition into the second one.