Peter Bastian

Electrodiffusion Models of Neurons and Extracellular Space Using the Poisson-Nernst-Planck Equations—Numerical Simulation of the Intra- and Extracellular Potential for an Axon Model

In neurophysiology, extracellular signals-as measured by local field potentials (LFP) or electroencephalography-are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Planck equations of electrodiffusion. The propagation of an action potential along the axonal membrane is investigated by means of numerical simulations. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. We focus on the evolution of the extracellular electric potential. A characteristic up-down-up LFP waveform in the far-field is found. Close to the membrane, the potential shows a more intricate shape. A comparison with the widely used line source approximation reveals similarities and demonstrates the strong influence of membrane currents. However, the electrodiffusion model shows another signal component stemming directly from the intracellular electric field, called the action potential echo. Depending on the neuronal configuration, this might have a significant effect on the LFP. In these situations, electrodiffusion models should be used for quantitative comparisons with experimental data.

Fast extraction of neuron morphologies from large-scale SBFSEM image stacks

Neuron morphology is frequently used to classify cell-types in the mammalian cortex. Apart from the shape of the soma and the axonal projections, morphological classification is largely defined by the dendrites of a neuron and their subcellular compartments, referred to as dendritic spines. The dimensions of a neuron’s dendritic compartment, including its spines, is also a major determinant of the passive and active electrical excitability of dendrites. Furthermore, the dimensions of dendritic branches and spines change during postnatal development and, possibly, following some types of neuronal activity patterns, changes depending on the activity of a neuron. Due to their small size, accurate quantitation of spine number and structure is difficult to achieve (Larkman, J Comp Neurol 306:332, 1991). Here we follow an analysis approach using high-resolution EM techniques. Serial block-face scanning electron microscopy (SBFSEM) enables automated imaging of large specimen volumes at high resolution. The large data sets generated by this technique make manual reconstruction of neuronal structure laborious. Here we present NeuroStruct, a reconstruction environment developed for fast and automated analysis of large SBFSEM data sets containing individual stained neurons using optimized algorithms for CPU and GPU hardware. NeuroStruct is based on 3D operators and integrates image information from image stacks of individual neurons filled with biocytin and stained with osmium tetroxide. The focus of the presented work is the reconstruction of dendritic branches with detailed representation of spines. NeuroStruct delivers both a 3D surface model of the reconstructed structures and a 1D geometrical model corresponding to the skeleton of the reconstructed structures. Both representations are a prerequisite for analysis of morphological characteristics and simulation signalling within a neuron that capture the influence of spines.

Parallel implementation of the recursive Green's function method

A parallel algorithm for the implementation of the recursive Greens function technique, which is extensively applied in the coherent scattering formalism, is developed. The algorithm performs a domain decomposition of the scattering region among the processors participating in the computation and calculates the Schurs complement block in the form of distributed blocks among the processors. If the method is applied recursively, thereby eliminating the processors cyclically, it is possible to arrive at a Schurs complement block of small size and compute the desired block of the Greens function matrix directly. The numerical complexity due to the longitudinal dimension of the scatterer scales linearly with the number of processors, though, the computational cost due to the processors cyclic reduction establishes a bottleneck to achieve efficiency 100%. The proposed algorithm is accompanied by a performance analysis for two numerical benchmarks, in which the dominant sources of computational load and parallel overhead as well as their competitive role in the efficiency of the algorithm will be demonstrated.

An Unfitted Discontinuous Galerkin method for pore-scale simulations of solute transport

Abstract: For the simulation of transport processes in porous media effective parameters for the physical processes on the target scale are required. Numerical upscaling, as well as multiscale approaches can help where experiments are not possible, or hard to conduct. In 2009, Bastian and Engwer proposed an Unfitted Discontinuous Galerkin (UDG) method for solving PDEs in complex domains, e.g. on the pore scale. We apply this method to a parabolic test problem. Convergence studies show the expected second order convergence. As an application example solute transport in a porous medium at the pore scale is simulated. Macroscopic breakthrough curves are computed using direct simulations. The method allows finite element meshes which are significantly coarser then those required by standard conforming finite element approaches. Thus it is possible to obtain reliable numerical results for macroscopic parameter already for a relatively coarse grid.

A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier–Stokes equations

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on

Validity limits for the van Genuchten–Mualem model and implications for parameter estimation and numerical simulation

H(text{div})

Couplex benchmark computations obtained with the software toolbox UG

reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.