Showing papers by "Leslie Greengard published in 2018"

High-density, long-lasting, and multi-region electrophysiological recordings using polymer electrode arrays

Abstract: The brain is a massive neuronal network, organized into anatomically distributed sub-circuits, with functionally relevant activity occurring at timescales ranging from milliseconds to months 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 recording platform that integrates polymer electrodes with a modular stacking headstage design supporting up to 1024 recording channels in freely behaving rats This system can support months-long recordings from hundreds of well-isolated units across multiple brain regions Moreover, these recordings are stable enough to track 25% of single units for over a week We also demonstrate long-lasting, single-unit recordings in songbird This platform enables large-scale electrophysiological interrogation of the fast dynamics and long-timescale evolution of anatomically distributed circuits, and thereby provides a new tool for understanding brain activity

The Anisotropic Truncated Kernel Method for Convolution with Free-Space Green's Functions

Abstract: A common task in computational physics is the convolution of a translation invariant, free-space Green's function with a smooth and compactly supported source density. Fourier methods are natural i...

Decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles

Abstract: We present a new method for the analysis of electromagnetic scattering from homogeneous penetrable bodies. Our approach is based on a reformulation of the governing Maxwell equations in terms of tw...

An adaptive fast gauss transform in two dimensions

Abstract: A variety of problems in computational physics and engineering require the convolution of the heat kernel (a Gaussian) with either discrete sources, densities supported on boundaries, or continuous...

A polymer probe-based system for high density, long-lasting electrophysiological recordings across distributed neuronal circuits

Abstract: The brain is a massively interconnected neuronal network, organized into specialized circuits consisting of large ensembles of neurons distributed across anatomically connected regions. While circuit computations depend upon millisecond timescale interactions, the structure of the underlying networks are remodeled on timescales ranging from seconds to months. Current approaches lack the combination of resolution, spatial coverage, longevity, and stability to measure the detailed dynamics of these networks. Here we describe a large-scale, multisite recording platform that integrates polymer electrodes with a modular stacking headstage design supporting up to 1024 channels of recording in freely-behaving rats. We show that the integrated system can yield months-long recordings from hundreds of well-isolated units across multiple regions. Moreover, the recordings are stable enough to track a substantial fraction of single units for over a week. This platform enables large-scale electrophysiological interrogation of the function and evolution of distributed circuits throughout an animal9s adult life.

Geometry of the Phase Retrieval Problem

Abstract: One of the most powerful approaches to imaging at the nanometer or subnanometer length scale is coherent diffraction imaging using X-ray sources. For amorphous (non-crystalline) samples, the raw data can be interpreted as the modulus of the continuous Fourier transform of the unknown object. Making use of prior information about the sample (such as its support), a natural goal is to recover the phase through computational means, after which the unknown object can be visualized at high resolution. While many algorithms have been proposed for this phase retrieval problem, careful analysis of its well-posedness has received relatively little attention. In this paper, we show that the problem is, in general, not well-posed and describe some of the underlying issues that are responsible for the ill-posedness. We then show how this analysis can be used to develop experimental protocols that lead to better conditioned inverse problems.

Transparent Boundary Conditions for the Time-Dependent Schr\"odinger Equation with a Vector Potential

Abstract: We consider the problem of constructing transparent boundary conditions for the time-dependent Schrodinger equation with a compactly supported binding potential and, if desired, a spatially uniform, time-dependent electromagnetic vector potential. Such conditions prevent nonphysical boundary effects from corrupting a numerical solution in a bounded computational domain. We use ideas from potential theory to build exact nonlocal conditions for arbitrary piecewise-smooth domains. These generalize the standard Dirichlet-to-Neumann and Neumann-to-Dirichlet maps known for the equation in one dimension without a vector potential. When the vector potential is included, the condition becomes non-convolutional in time. For the one-dimensional problem, we propose a simple discretization scheme and a fast algorithm to accelerate the evaluation of the boundary condition.

A new mixed potential representation for the equations of unsteady, incompressible flow

Abstract: We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each individual equation is well-condtioned, and we show that using them in predictor-corrector fashion, combined with spectral deferred correction, leads to high-order accuracy solvers. The fundamental unknowns in the mixed potential representation are densities supported on the boundary of the domain. We refer to one as the vortex source, the other as the pressure source and the coupled system as the combined source integral equation.

Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions

Abstract: We present a hybrid asymptotic/numerical method for the accurate computation of single and double layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called "geometrically-induced stiffness," meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically-induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

On the accurate evaluation of unsteady Stokes layer potentials in moving two-dimensional geometries

Abstract: Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts - a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algorithms. The local part, however, involves complicated singularities in the underlying Green's function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

Integral equation methods for electrostatics, acoustics and electromagnetics in smoothly varying, anisotropic media

Abstract: We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [1] to discretize and apply the relevant integral operators.