A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation.

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. We formulate, test, and disseminate a direct (i.e.

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation.

Background: When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling.

Objective: We formulate, test, and disseminate a direct ( non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~200,000 or more arbitrarily selected observation points within approximately 5 sec, with the solution time itself of 3 sec.

Method: The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods - an FMM-based LU decomposition.

Robust ab initio solution of the cryo-EM reconstruction problem at low resolution with small data sets.

Single particle cryo-electron microscopy has become a critical tool in structural biology over the last decade, able to achieve atomic scale resolution in three dimensional models from hundreds of thousands of (noisy) two-dimensional projection views of particles frozen at unknown orientations. This is accomplished by using a suite of software tools to (i) identify particles in large micrographs, (ii) obtain low-resolution reconstructions, (iii) refine those low-resolution structures, and (iv) finally match the obtained electron scattering density to the constituent atoms that make up the macromolecule or macromolecular complex of interest. Here, we focus on the second stage of the reconstruction pipeline: obtaining a low resolution model from picked particle images.

Forced and spontaneous symmetry breaking in cell polarization.

How does breaking the symmetry of an equation alter the symmetry of its solutions? Here, we systematically examine how reducing underlying symmetries from spherical to axisymmetric influences the dynamics of an archetypal model of cell polarization, a key process of biological spatial self-organization. Cell polarization is characterized by nonlinear and non-local dynamics, but we overcome the theory challenges these traits pose by introducing a broadly applicable numerical scheme allowing us to efficiently study continuum models in a wide range of geometries. Guided by numerical results, we discover a dynamical hierarchy of timescales that allows us to reduce relaxation to a purely geometric problem of area-preserving geodesic curvature flow.

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators.

A fast spectral method for electrostatics in doubly periodic slit channels.

We develop a fast method for computing the electrostatic energy and forces for a collection of charges in doubly periodic slabs with jumps in the dielectric permittivity at the slab boundaries. Our method achieves spectral accuracy by using Ewald splitting to replace the original Poisson equation for nearly singular sources with a smooth far-field Poisson equation, combined with a localized near-field correction. Unlike existing spectral Ewald methods, which make use of the Fourier transform in the aperiodic direction, we recast the problem as a two-point boundary value problem in the aperiodic direction for each transverse Fourier mode for which exact analytic boundary conditions are available.

High-Density, Long-Lasting, and Multi-region Electrophysiological Recordings Using Polymer Electrode Arrays.

The brain is a massive neuronal network, organized into anatomically distributed sub-circuits, with functionally relevant activity occurring at timescales ranging from milliseconds to years. Current methods to monitor neural activity, however, lack the necessary conjunction of anatomical spatial coverage, temporal resolution, and long-term stability to measure this distributed activity. Here we introduce a large-scale, multi-site, extracellular recording platform that integrates polymer electrodes with a modular stacking headstage design supporting up to 1,024 recording channels in freely behaving rats.

A Fully Automated Approach to Spike Sorting.

Understanding the detailed dynamics of neuronal networks will require the simultaneous measurement of spike trains from hundreds of neurons (or more). Currently, approaches to extracting spike times and labels from raw data are time consuming, lack standardization, and involve manual intervention, making it difficult to maintain data provenance and assess the quality of scientific results. Here, we describe an automated clustering approach and associated software package that addresses these problems and provides novel cluster quality metrics.

A Fast Summation Method for Oscillatory Lattice Sums.

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge.

Validation of neural spike sorting algorithms without ground-truth information.

Background: The throughput of electrophysiological recording is growing rapidly, allowing thousands of simultaneous channels, and there is a growing variety of spike sorting algorithms designed to extract neural firing events from such data. This creates an urgent need for standardized, automatic evaluation of the quality of neural units output by such algorithms.

New Method: We introduce a suite of validation metrics that assess the credibility of a given automatic spike sorting algorithm applied to a given dataset.

Fast Direct Methods for Gaussian Processes.

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n ×n covariance matrix, C, as well as the evaluation of its determinant, det(C).

A fast solver for multi-particle scattering in a layered medium.

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory.

Visualizing skin effects in conductors with MRI: (7)Li MRI experiments and calculations.

While experiments on metals have been performed since the early days of NMR (and DNP), the use of bulk metal is normally avoided. Instead, often powders have been used in combination with low fields, so that skin depth effects could be neglected. Another complicating factor of acquiring NMR spectra or MRI images of bulk metal is the strong signal dependence on the orientation between the sample and the radio frequency (rf) coil, leading to non-intuitive image distortions and inaccurate quantification.

A mesh-free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators.

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape.

A mathematical tool for exploring the dynamics of biological networks.

We have developed a mathematical approach to the study of dynamical biological networks, based on combining large-scale numerical simulation with nonlinear "dimensionality reduction" methods. Our work was motivated by an interest in the complex organization of the signaling cascade centered on the neuronal phosphoprotein DARPP-32 (dopamine- and cAMP-regulated phosphoprotein of molecular weight 32,000). Our approach has allowed us to detect robust features of the system in the presence of noise.

Plasmon-assisted chemical vapor deposition.

We introduce a new chemical vapor deposition (CVD) process that can be used to selectively deposit materials of many different types. The technique makes use of the plasmon resonance in nanoscale metal structures to produce the local heating necessary to initiate deposition when illuminated by a focused low-power laser. We demonstrate the technique, which we refer to as plasmon-assisted CVD (PACVD), by patterning the spatial deposition of PbO and TiO(2) on glass substrates coated with a dispersion of 23 nm gold particles.

Sensitivity analysis of photonic crystal fiber.

Photonic crystal fibers are well-known to offer a number of unusual properties, including supercontinuum generation, large mode-areas and controllable dispersion behavior. Their manufacturability would be enhanced by a more detailed understanding of how small perturbations in the fiber's geometric structure cause variations in the fiber's fundamental modes. In this paper, we demonstrate that such sensitivity analysis is feasible using highly accurate boundary integral techniques.

Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory.

We present a new integral equation method for calculating the electromagnetic modes of photonic crystal fiber (PCF) waveguides. Our formulation can easily handle PCFs with arbitrary hole geometries and irregular hole distributions, enabling optical component manufacturers to optimize hole designs as well as assess the effect of manufacturing defects. The method produces accurate results for both the real and imaginary parts of the propagation constants, which we validated through extensive convergence analysis and by comparison with previously published results.

Fast algorithms for classical physics.

Some of the recently developed fast summation methods that have arisen in scientific computing are described. These methods require an amount of work proportional to N or N log N to evaluate all pairwise interactions in an ensemble of N particles. Traditional methods, by contrast, require an amount of work proportional to N(2).