Analytical Mean-field Approach Research Articles

Intra-ripple frequency accommodation in an inhibitory network model for hippocampal ripple oscillations.

Hippocampal ripple oscillations have been implicated in important cognitive functions such as memory consolidation and planning. Multiple computational models have been proposed to explain the emergence of ripple oscillations, relying either on excitation or inhibition as the main pacemaker. Nevertheless, the generating mechanism of ripples remains unclear. An interesting dynamical feature of experimentally measured ripples, which may advance model selection, is intra-ripple frequency accommodation (IFA): a decay of the instantaneous ripple frequency over the course of a ripple event. So far, only a feedback-based inhibition-first model, which relies on delayed inhibitory synaptic coupling, has been shown to reproduce IFA. Here we use an analytical mean-field approach and numerical simulations of a leaky integrate-and-fire spiking network to explain the mechanism of IFA. We develop a drift-based approximation for the oscillation dynamics of the population rate and the mean membrane potential of interneurons under strong excitatory drive and strong inhibitory coupling. For IFA, the speed at which the excitatory drive changes is critical. We demonstrate that IFA arises due to a speed-dependent hysteresis effect in the dynamics of the mean membrane potential, when the interneurons receive transient, sharp wave-associated excitation. We thus predict that the IFA asymmetry vanishes in the limit of slowly changing drive, but is otherwise a robust feature of the feedback-based inhibition-first ripple model.

Emergence of polarization in a voter model with personalized information

The flourishing of fake news is favored by recommendation algorithms of online social networks which, based on previous users activity, provide content adapted to their preferences and so create filter bubbles. We introduce an analytically tractable voter model with personalized information, in which an external field tends to align the agent opinion with the one she held more frequently in the past. Our model shows a surprisingly rich dynamics despite its simplicity. An analytical mean-field approach, confirmed by numerical simulations, allows us to build a phase diagram and to predict if and how consensus is reached. Remarkably, polarization can be avoided only for weak interaction with the personalized information and if the number of agents is below a threshold. We analytically compute this critical size, which depends on the interaction probability in a strongly non linear way.

Interacting Social Processes on Interconnected Networks.

We propose and study a model for the interplay between two different dynamical processes –one for opinion formation and the other for decision making– on two interconnected networks A and B. The opinion dynamics on network A corresponds to that of the M-model, where the state of each agent can take one of four possible values (S = −2,−1, 1, 2), describing its level of agreement on a given issue. The likelihood to become an extremist (S = ±2) or a moderate (S = ±1) is controlled by a reinforcement parameter r ≥ 0. The decision making dynamics on network B is akin to that of the Abrams-Strogatz model, where agents can be either in favor (S = +1) or against (S = −1) the issue. The probability that an agent changes its state is proportional to the fraction of neighbors that hold the opposite state raised to a power β. Starting from a polarized case scenario in which all agents of network A hold positive orientations while all agents of network B have a negative orientation, we explore the conditions under which one of the dynamics prevails over the other, imposing its initial orientation. We find that, for a given value of β, the two-network system reaches a consensus in the positive state (initial state of network A) when the reinforcement overcomes a crossover value r*(β), while a negative consensus happens for r < r*(β). In the r − β phase space, the system displays a transition at a critical threshold βc, from a coexistence of both orientations for β < βc to a dominance of one orientation for β > βc. We develop an analytical mean-field approach that gives an insight into these regimes and shows that both dynamics are equivalent along the crossover line (r*, β*).

Asymmetric size of ions and orientational ordering of water dipoles in electric double layer model - an analytical mean-field approach

Electric double layer is theoretically described within the mean-field approach by taking into account the orientational ordering of water dipoles and asymmetric size of cations and anions. Analytical expressions for the spatial distribution of ions and water dipoles are derived. The effect of asymmetric ionic size on accumulation of counterions and partial depletion of water molecules near the charged surface, on spatial dependence of relative permittivity and on differential capacitance of electric double layer are presented.

Thermodynamic properties of cubic boron nitride based on an analytic mean field approach

We present a detailed derivation of the analytic expressions for the equation of state (EOS) and internal energy of Morse model solids based on an analytic mean field potential (AMFP) method. The formalism is applied to cubic boron nitride (c-BN). One set of potential parameters are determined by fitting the experimental P-V-T data of c-BN up to 160 GPa at 295 K and 80 GPa in the range 500–900 K. Various physical quantities including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results are consistent with the available experimental data and those calculated by others. These results presented in this paper verify that the AMFP method is a useful approach to consider the anharmonic effect at high temperature. Numerous reasonable predictions and the change trend of the properties for c-BN at extreme conditions have been given.

Thermodynamic properties of solid FCC C84 based on an analytic mean field approach

The analytic mean field potential (AMFP) approach is applied to the poly-exponential model solid. The analytic expressions for the Helmholtz free energy, internal energy and equation of state (EOS) are derived. The formalism for the case of the double-exponential (DE) model is applied to fcc C84. One set of potential parameters are determined by fitting the experimental compression data of C84 up to 9.24 GPa at ambient temperature (297 K). The equilibrium distance and well depth for C60, C70 and C84 molecules are plotted. The thermophysical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available of C84. Basing the results of our calculations, we may also predict the behaviors of C84 at extreme conditions.

Analytic equation of state and thermodynamic properties of diamond based on an analytic mean field approach

The analytic mean field potential (AMFP) method is applied to diamond. The analytic expressions for the Helmholtz free energy, internal energy and equation of state (EOS) are derived. The formalism for the case of the Morse potential is used in this work. The six potential parameters are determined by fitting the compression experimental data of diamond at ambient temperature. The thermophysical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available for diamond. These results presented in this paper verify that the AMFP method is a useful approach to consider the anharmonic effects at high temperature. Numerous reasonable predictions and the change trend of the properties for diamond at extreme conditions have been given.

The dune size distribution and scaling relations of barchan dune fields

Barchan dunes emerge as a collective phenomena involving the generation of thousands of them in so called barchan dune fields. By measuring the size and position of dunes in Moroccan barchan dune fields, we find that these dunes tend to distribute uniformly in space and follow an unique size distribution function. We introduce an analytical mean-field approach to show that this empirical size distribution emerges from the interplay of dune collisions and sand flux balance, the two simplest mechanisms for size selection. The analytical model also predicts a scaling relation between the fundamental macroscopic properties characterizing a dune field, namely the inter-dune spacing and the first and second moments of the dune size distribution.

Analytic Equation of State and Thermodynamic Properties for α-, β-, and γ-Si3N4Based on Analytic Mean Field Approach

Analytic Equation of State and Thermodynamic Properties for α-, β-, and γ-Si₃N₄Based on Analytic Mean Field Approach

Analytic equation of state for solids with multi-exponential potential based on analytic mean field potential approach and applied to the epsilon phase of solid oxygen

The thermodynamic properties of the ε phase of solid oxygen are studied by using the analytic mean field approach (AMFP). Analytic expressions for the Helmholtz free energy, internal energy and equation of state of solid oxygen have been derived based on the multi-exponential potential. The formulism for the case of double-exponential (DE) model is applied to the ε phase of solid oxygen. Its four potential parameters are determined through fitting the experimental compression data of the ε phase of solid oxygen. Numerical results of the pressure dependence of the volume calculated by using the AMFP are in good agreement with the original experimental data. This suggests that the AMFP is a useful approach to study the thermodynamic properties of the ε phase of solid oxygen. Furthermore, we predict the variation of the volume, lattice parameters and intermolecular distances with pressure, and some thermodynamic quantities versus volume, at several higher temperatures.

An analytic equation of state based on analytic mean-field potential approach and application to three phases of metallic Zirconium

Abstract The analytic mean-field approach (AMFP) was applied to study the thermodynamic properties of Zirconium (Zr). The analytic expressions for the Helmholtz free energy, internal energy and equation of state have been derived. The formalism for the case of the Morse potential is used in this work. The four potential parameters are determined by fitting the molar volume of the three phases of Zr. The calculated molar volume of α, β and ω Zr are in fairly good agreement with the available experimental data. The results presented in this paper verify that the AMFP is a useful approach to study the thermodynamic properties of Zr. Furthermore, we predict the variation of the relationship of free energy and internal energy versus the molar volume at various temperatures and the dependence of the bulk modulus, the thermal expansion coefficient and the heat capacity on temperature at zero pressure of α, β and ω Zr.

Analytic Equation of State and Thermodynamic Properties of Solid FCC C61D2 Based on an Analytic Mean Field Approach

Analytic expressions for the equation of state and internal energy of poly-exponential solids are derived based on the analytic mean field potential (AMFP) method. The formalism is applied to fcc C61D2. Two sets of potential parameters are determined by fitting the experimental compression data of C61D2 up to 1 GPa at 343 and 307 K, respectively. The difference between the two sets of parameters is small. Whereas the difference between the potential of the C61D2 molecules and that of the C60 molecules is fairly prominent, the conclusion is different from that in the literature, and the reason is unclear at present. The thermo-physical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available for solid C61D2.

Memory Capacity for Sequences in a Recurrent Network with Biological Constraints

The CA3 region of the hippocampus is a recurrent neural network that is essential for the storage and replay of sequences of patterns that represent behavioral events. Here we present a theoretical framework to calculate a sparsely connected network's capacity to store such sequences. As in CA3, only a limited subset of neurons in the network is active at any one time, pattern retrieval is subject to error, and the resources for plasticity are limited. Our analysis combines an analytical mean field approach, stochastic dynamics, and cellular simulations of a time-discrete McCulloch-Pitts network with binary synapses. To maximize the number of sequences that can be stored in the network, we concurrently optimize the number of active neurons, that is, pattern size, and the firing threshold. We find that for one-step associations (i.e., minimal sequences), the optimal pattern size is inversely proportional to the mean connectivity c, whereas the optimal firing threshold is independent of the connectivity. If the number of synapses per neuron is fixed, the maximum number P of stored sequences in a sufficiently large, nonmodular network is independent of its number N of cells. On the other hand, if the number of synapses scales as the network size to the power of 3/2, the number of sequences P is proportional to N. In other words, sequential memory is scalable. Further-more, we find that there is an optimal ratio r between silent and nonsilent synapses at which the storage capacity α = P/[c (1 +r)N] assumes a maximum. For long sequences, the capacity of sequential memory is about one order of magnitude below the capacity for minimal sequences, but otherwise behaves similar to the case of minimal sequences. In a biologically inspired scenario, the information content per synapse is far below theoretical optimality, suggesting that the brain trades off error tolerance against information content in encoding sequential memories.

Memory Capacity for Sequences in a Recurrent Network with Biological Constraints

The CA3 region of the hippocampus is a recurrent neural network that is essential for the storage and replay of sequences of patterns that represent behavioral events. Here we present a theoretical framework to calculate a sparsely connected network's capacity to store such sequences. As in CA3, only a limited subset of neurons in the network is active at any one time, pattern retrieval is subject to error, and the resources for plasticity are limited. Our analysis combines an analytical mean field approach, stochastic dynamics, and cellular simulations of a time-discrete McCulloch-Pitts network with binary synapses. To maximize the number of sequences that can be stored in the network, we concurrently optimize the number of active neurons, that is, pattern size, and the firing threshold. We find that for one-step associations (i.e., minimal sequences), the optimal pattern size is inversely proportional to the mean connectivity c, whereas the optimal firing threshold is independent of the connectivity. If the number of synapses per neuron is fixed, the maximum number P of stored sequences in a sufficiently large, nonmodular network is independent of its number N of cells. On the other hand, if the number of synapses scales as the network size to the power of 3/2, the number of sequences P is proportional to N. In other words, sequential memory is scalable. Furthermore, we find that there is an optimal ratio r between silent and nonsilent synapses at which the storage capacity alpha = P//[c(1 + r)N] assumes a maximum. For long sequences, the capacity of sequential memory is about one order of magnitude below the capacity for minimal sequences, but otherwise behaves similar to the case of minimal sequences. In a biologically inspired scenario, the information content per synapse is far below theoretical optimality, suggesting that the brain trades off error tolerance against information content in encoding sequential memories.

Analytic equation of state for FCC C60 solid based on analytic mean-field potential approach

The analytic mean-field approach (AMFP) was applied to the FCC C60 solid. For the intermolecular forces the Girifalco potential has been utilized. The analytic expressions for the Helmholtz free energy, internal energy and equation of state have been derived. The numerical results of thermodynamic quantities are compared with the molecular dynamic (MD) simulations and the unsymmetrized self-consistent field approach (CUSF) in the literature. It is shown that our AMFP results are in good agreement with the MD data both at low and high temperatures. The results of CUSF are in accordance with the AMFP at low temperature, but at high temperature the difference becomes prominent. Especially the AMFP predicted that the FCC C60 solid is stable upto 2202 K, the spinodal temperature, in good agreement with 2320 K from the MD simulation. However, the CUST just gives 1916 K, a temperature evidently lower than the MD data. The AMFP qualifies as a useful approach that can reasonably consider the anharmonic effects at high temperature.

Mean field theory of the coherent to random-phase state transition in three-wave interactions

The crossover of three-wave interactions from the coherent monochromatic limit to the wide bandwidth random-phase limit is investigated as the bandwidth of the waves is varied in a system exhibiting nonlinear three-wave oscillations. A recently observed sudden transition between the coherent and incoherent interaction is confirmed. As the bandwidth is increased from the monochromatic limit, it is found that the coherence of the interaction decreases slowly. At the transition point of the interaction the coherence then falls abruptly and nonlinear oscillations cease. An analytic mean-field approach is used to model the transition. Below the transition point, each frequency component of the wave spectra oscillates at its own individual frequency about the aggregate quasicoherent oscillation of the wave as a whole. It is when the frequency component at the spectral edge cannot sustain such oscillations that the system switches to random phase behavior. The analytic model’s estimate of the transition point and other interaction properties agrees semiquantitatively with numerical solutions of the full equations.

Dynamics of chainlike molecules on surfaces

We consider the diffusion and spreading of chainlike molecules on solid surfaces. We first show that the steep spherical cap shape density profiles, observed in some submonolayer experiments on spreading polymer films, imply that the collective diffusion coefficient ${D}_{C}(\ensuremath{\theta})$ must be an increasing function of the surface coverage $\ensuremath{\theta}$ for small and intermediate coverages. Through simulations of a discrete model of interacting chainlike molecules, we demonstrate that this is caused by an entropy-induced repulsive interaction. Excellent agreement is found between experimental and numerically obtained density profiles in this case, demonstrating that steep submonolayer film edges naturally arise due to the diffusive properties of chainlike molecules. When the entropic repulsion dominates over interchain attractions, ${D}_{C}(\ensuremath{\theta})$ first increases as a function of $\ensuremath{\theta}$ but then eventually approaches zero for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\theta}}1$. The maximum value of ${D}_{C}(\ensuremath{\theta})$ decreases for increasing attractive interactions, leading to density profiles that are in between spherical cap and Gaussian shapes. We also develop an analytic mean-field approach to explain the diffusive behavior of chainlike molecules. The thermodynamic factor in ${D}_{C}(\ensuremath{\theta})$ is evaluated using effective free-energy arguments and the chain mobility is calculated numerically using the recently developed dynamic mean-field theory. Good agreement is obtained between theory and simulations.

Finite-size effects for phase segregation in a two-dimensional asymmetric exclusion model with two species

We investigate the stationary states of a two-dimensional lattice gas model with exclusion, in the presence of an external field. The lattice is populated by equal numbers of positively and negatively charged particles. An analytical mean-field approach and Monte Carlo simulations give strong evidence of the fact that at any finite density the only relevant stationary state of the system in the thermodynamic limit is inhomogeneous, consisting of a strip of particles transverse to the field. In the inhomogeneous phase, the density profiles and the current measured by Monte Carlo simulations are closely related to those found in mean field. The same is true for the finite-size behavior of the system.

Kinetic roughening phase transition in surface growth: Numerical study and mean-field approach.

Numerical simulations are reported on a ballistic deposition model that interpolates between the weak- and strong-coupling limits of the stochastic Burgers equation introduced by M. Kardar, G. Parisi, and Y. C. Zhang [Phys. Rev. Lett. 56, 889 (1986)]. The model, which is built on a d-dimensional cubic lattice, considers a random deposition of two kinds of particles, with concentrations c and 1-c, that stick to the deposit or may slide on it by no more than one lattice spacing. Evidence is given for the existence of a phase transition, when varying c, in d=3 and 4, but not in d=2, in agreement with an analytical prediction due to Halpin-Healy [Phys. Rev. Lett. 62, 442 (1989)]. The transition is investigated by analyzing, as a function of c, the scaling exponents of the surface thickness as well as the compactness of the bulk of the deposit. An analytical mean-field approach is proposed that describes most of the features of the phase transition.