Continuum Mean-field Model Research Articles

Computational mean-field modeling of confined active fluids

We present a new framework for the efficient simulation of the dynamics of active fluids in complex two- and three-dimensional microfluidic geometries. Focusing on the case of a suspension of microswimmers such as motile bacteria, we adopt a continuum mean-field model based on partial differential equations for the evolution of the concentration, polarization and nematic tensor fields, which are nonlinearly coupled to the Navier-Stokes equations for the fluid flow driven by internal active stresses. A level set method combined with an adaptive mesh refinement scheme on Quad-/Octree grids is used to capture complex domain shapes while refining the solution near boundaries or in the neighborhood of sharp gradients. A hybrid finite volumes/finite differences method is implemented in which the concentration field is treated using finite volumes to ensure mass conservation, while the polarization and nematic alignment fields are treated using a combination of finite differences and finite volumes for enhanced accuracy. The governing equations for these fields are solved along with the Navier-Stokes equations, which are evolved using an unconditionally stable projection solver. We illustrate the versatility and robustness of our method by analyzing spontaneous active flows in various two- and three-dimensional systems. Our results show excellent agreement with previous models and experiments and pave the way for further developments in active microfluidics.

Incorporating Born solvation energy into the three-dimensional Poisson-Nernst-Planck model to study ion selectivity in KcsA K^{+} channels.

Potassium channels are much more permeable to potassium than sodium ions, although potassium ions are larger and both carry the same positive charge. This puzzle cannot be solved based on the traditional Poisson-Nernst-Planck (PNP) theory of electrodiffusion because the PNP model treats all ions as point charges, does not incorporate ion size information, and therefore cannot discriminate potassium from sodium ions. The PNP model can qualitatively capture some macroscopic properties of certain channel systems such as current-voltage characteristics, conductance rectification, and inverse membrane potential. However, the traditional PNP model is a continuum mean-field model and has no or underestimates the discrete ion effects, in particular the ion solvation or self-energy (which can be described by Born model). It is known that the dehydration effect (closely related to ion size) is crucial to selective permeation in potassium channels. Therefore, we incorporated Born solvation energy into the PNP model to account for ion hydration and dehydration effects when passing through inhomogeneous dielectric channel environments. A variational approach was adopted to derive a Born-energy-modified PNP (BPNP) model. The model was applied to study a cylindrical nanopore and a realistic KcsA channel, and three-dimensional finite element simulations were performed. The BPNP model can distinguish different ion species by ion radius and predict selectivity for K^{+} over Na^{+} in KcsA channels. Furthermore, ion current rectification in the KcsA channel was observed by both the PNP and BPNP models. The I-V curve of the BPNP model for the KcsA channel indicated an inward rectifier effect for K^{+} (rectification ratio of ∼3/2) but indicated an outward rectifier effect for Na^{+} (rectification ratio of ∼1/6).

Molecular dynamics simulation of ionic transport at coherent interfaces in fluorite heterostructures

The standard model of enhanced ionic conductivities in solid electrolyte heterostructures follows from a continuum mean-field description of defect distributions that makes no reference to crystalline structure. To examine ionic transport and defect distributions while explicitly accounting for ion-ion correlations and lattice effects, we have performed molecular dynamics simulations of a model coherent fluorite heterostructure without any extrinsic defects, with a difference in standard chemical potentials of mobile fluoride ions between phases induced by an external potential. Increasing the offset in fluoride ion standard chemical potentials across the internal interfaces decreases the activation energies for ionic conductivity and diffusion and strongly enhances fluoride ion mobilities and defect concentrations near the heterostructure interfaces. Non-charge-neutral "space-charge" regions, however, extend only a few atomic spacings from the interface, suggesting a continuum model may be inappropriate. Defect distributions are qualitatively inconsistent with the predictions of the continuum mean-field model, and indicate strong lattice-mediated defect-defect interactions. We identify an atomic-scale "Frenkel polarisation" mechanism for the interfacial enhancement in ionic mobility, where preferentially oriented associated Frenkel pairs form at the interface and promote local ion mobility via concerted diffusion processes.

Dynamics of activity fronts in a continuum mean field model of cortex

The functional organization of cortex appears to be roughly columnar, with the laminar sub-structure of each column organizing its micro-circuitry. These columns tessellate the two-dimensional cortical sheet with high density, e.g., 2,000 cm 2 of human cortex contain 10 5 to 10 6 macrocolumns, comprising about 10 5 neurons each. Continuum mean field models (cMFMs) describe the mean activity of such columns by approximating the cortical sheet as continuous excitable medium [1]. cMFMs can generate rich patterns of emergent spatiotemporal activity [2]. This has been used to understand phenomena from visual hallucinations to the generation of EEG signals. Pattern boundaries are here defined as the interface between low and high states of average neural activity. cMFMs support travelling patterns as well as the formation of intricate structures, as in Fig. 1. Here we derive equations of motion for the pattern boundaries of a simple cMFM, showing that their normal velocities are driven by Biot-Savart-style interactions. The solutions of these exact, but dimensionally reduced, equations for activity fronts are in excellent numerical agreement with those of the full nonlinear integral equation defining the neural field. A linear stability analysis of the dynamics of the interfaces allows us to understand mechanisms of pattern formation arising from instabilities of spots, fronts, and stripes. We further test our results against partial differential equations equivalent to the original integral equation, c.f. [3], and perform numerical simulations on a large spatial grid that represents a sizable cortical sheet. In particular, we clarify how more realistic firing rates (computed with sigmoidal functions instead of Heaviside steps) influence the dynamics of activity fronts.

Axonal Velocity Distributions in Neural Field Equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Realistic activity propagation for mean field models of human cortex

Inspired by the great density of neurons in cortex (about 106 per macrocolumn), continuum mean field models (CMFMs) treat the cortex as one continuous neural medium. Interactions between neurons thus become flows of activity spreading in this medium. The most popular propagation model [1] is derived by a two-fold ansatz: a pulse of activity will on one hand spread isotropically with a conduction velocity c, and on the other hand its amplitude will decay exponentially with a distance scale σ. This ansatz reflects action potentials traveling with constant speed through axons and that the number of synapses axons form falls roughly exponentially with distance. However, a pulse in a CMFM means a mass of 105 to 106 neurons pulsing together. It is well known that axons can vary greatly in conduction speed, depending on their myelination and diameter. Hence it is natural to expect that such a mass has a broad conduction velocity distribution f(v), rather than a singular one f(v) = δ(v-c). Furthermore, one still has to expand for large wavelengths in order to derive a manageable partial differential equation (PDE) – a damped wave equation as it turns out. If one calculates the velocity distribution of this approximation, one sees that the Dirac delta peak has softened only towards lower velocities, leaving an unnatural distribution with a hard velocity cut-off. We hence propose a new propagation PDE: where Φ is the activity being propagated, S is a local source (e.g., a firing rate function), Nα the total number of connections, c is the conduction velocity and σ the decay parameter, and n > 0 will usually be chosen as an integer.

A Universal Approach to Solvation Modeling

Continuum mean-field models that have been carefully designed to address the various electrostatic and nonelectrostatic interactions that develop between a molecule and a surrounding medium are particularly efficient tools for studying the effects of condensed phases on molecular structure, energetics, properties, spectra, interaction potentials, and dynamics. The SM8 model may be combined with density functional theory or Hartree-Fock theory to describe a solute's electronic structure and its self-consistent-field polarization by a solvent. A key feature is the use of class IV charge models to obtain accurate charge distributions (either in the vapor phase or in solution), even when using small basis sets that are affordable for large systems. A second key feature is that nonelectrostatic effects due to cavity formation, dispersion interactions, and changes in solvent structure are included in terms of empirical atomic surface tensions that depend on geometry but do not require atom-type assignments by the user. Use of an analytic surface area algorithm provides very stable energy gradients that allow geometry optimization in solution. The SM8 continuum model, the culmination of a series of SMx models (x = 1-8), permits the modeling of such diverse media as aqueous and organic solvents, soils, lipid bilayers, and air-water interfaces. In addition to predicting accurate transfer free energies between gaseous and condensed phases or between two different condensed phases, SMx models have been useful for predicting the significant influence of condensed phases on processes associated with a change in molecular charge, including acid/base equilibria and oxidation/reduction processes. In this Account, we provide an overview of the algorithms associated with the computation of free energies of solvation in the SM8 model. We also compare the accuracies of the SM8 model with those of other continuum solvation models. Finally, we highlight applications of the SM8 models to compute ionic solvation free energies, oxidation and reduction potentials, and pK(a) values.

A Variational Principle Governing the Generating Function for Conformations of Flexible Molecules

The generation of a random walk path under the action of an external potential field has been of interest for decades. The motivation derives largely from the prospect of incorporating the nonlocal excluded volume effect through such a potential in characterizing the statistical behavior of a long flexible polymer molecule. In working toward a continuum mean-field model, a central feature is a partial differential equation incorporating the influence of the potential and governing the generating function for the dependence of end to end separation distance of the molecule on its pathlength. The purpose here is to describe an approach in which the differential equation is recast as a global minimization of a functional. The variational approach is illustrated by an application to familiar configurations, the first of which is a molecule attached at one end to a noninteracting plane barrier in the presence of a uniform potential field. As a second illustration, the generating function is sought for a free molecule for the case in which conformations must be consistent with the excluded volume condition. This is accomplished by adapting a local form of the Flory approach to the phenomenon and extracting estimates of the expected end to end separation distance, the entropy and other statistical features of behavior. By means of the variational principle, the problem is recast into a form that admits a direct, noniterative analysis of conformations within the context of the self-consistent field theory.

Diffusion-limited aggregation: A continuum mean field model

Mean field theory is used as a basis for a new approach to analyzing fractal pattern formation by diffusion-limited aggregation. A coarse time scale is introduced to take into account the discrete nature of DLA clusters. A system of equations is derived and solved numerically to determine the fractal dimension and density of a cluster as a function of distance from its center. The results obtained are in good agreement with direct numerical simulations.

Traffic jams and hysteresis in driven one-dimensional systems

The driven underdamped chain of anharmonically interacting atoms in the sinusoidal external potential is studied. It is shown that due to the interatomic interaction the system exhibits hysteresis for any nonzero rate of changing of the dc driving force. Before the transition to the running state the system passes through the traffic-jam inhomogeneous state. The system behavior is explained with the help of two simple models, the discrete lattice-gas model with two states of atoms, and the continuum mean-field model based on the FokkerPlanck equation. @S1063-651X~98!10507-X#