Stochastic neural field theory and the system-size expansion

We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N → ∞) we recover standard activity–based or voltage–based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N ) can be truncated to form a closed system of equations for the first and second order moments. Taking a continuum limit of the moment equations whilst keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the mean–field dynamics; such an analysis is not possible using the system-size expansion since the latter cannot accurately\ud determine exponentially small transitions

Nematic liquid crystals : from Maier-Saupe to a continuum theory

We define a continuum energy functional in terms of the mean-field Maier-Saupe free energy, that describes both spatially homogeneous and inhomogeneous systems. The Maier-Saupe theory defines the main macroscopic variable, the Q-tensor order parameter, in terms of the second moment of a probability distribution function. This definition requires the eigenvalues of Q to be bounded both from below and above. We define a thermotropic bulk potential which blows up whenever the eigenvalues tend to these lower and upper bounds. This is in contrast to the Landau-de Gennes theory which has no such penalization. We study the asymptotics of this bulk potential in different regimes and discuss phase transitions predicted by this model

Ergodic directional switching in mobile insect groups

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including non-monotonic behavior of several order/disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.Comment: Physical Review Focus 26, July 201

Homogenization for advection-diffusion in a perforated domain

The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Carlo algorithm for the computation of the volume growth rate of the sausage is introduced and some numerical results are presented for the Taylor–Green velocity field

Frost heave in compressible soils

We develop a mathematical model of frost heave in compressible soils based on a morphological instability of the ice–soil interface. The theory accounts for heave and soil consolidation,while avoiding the frozen fringe assumption. Using a Lie-B¨acklund transformation an analytical solution to the governing equations is found. Two solidification regimes occur: a compaction regime in which the soil consolidates to accommodate the ice lenses, and a heave regime during which the soil is fully consolidated and heaves. The rate of heave is found to be independent of the rate of freezing, consistent with field and laboratory observations

Modeling networks of spiking neurons as interacting processes with memory of variable length

We consider a new class of non Markovian processes with a countable number of interacting components, both in discrete and continuous time. Each component is represented by a point process indicating if it has a spike or not at a given time. The system evolves as follows. For each component, the rate (in continuous time) or the probability (in discrete time) of having a spike depends on the entire time evolution of the system since the last spike time of the component. In discrete time this class of systems extends in a non trivial way both Spitzer's interacting particle systems, which are Markovian, and Rissanen's stochastic chains with memory of variable length which have finite state space. In continuous time they can be seen as a kind of Rissanen's variable length memory version of the class of self-exciting point processes which are also called "Hawkes processes", however with infinitely many components. These features make this class a good candidate to describe the time evolution of networks of spiking neurons. In this article we present a critical reader's guide to recent papers dealing with this class of models, both in discrete and in continuous time. We briefly sketch results concerning perfect simulation and existence issues, de-correlation between successive interspike intervals, the longtime behavior of finite non-excited systems and propagation of chaos in mean field systems