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Papers by Christoph Borgers

Hippocampal Microcircuits, 2010

The IMA Volumes in Mathematics and its Applications, 1999

this paper is to describe mathematicalaspects of radiation therapy planning to readers with a bac... more this paper is to describe mathematicalaspects of radiation therapy planning to readers with a background inapplied mathematics.The use of X-rays for cancer therapy began a few days after their discovery.Wilhelm Rontgen announced the discovery of X-rays on December28, 1895, and Emil Grubbe used them for cancer therapy on January 12,1896 [40]. X-rays are still the most common form of radiation used for cancertherapy, but beams of electrons, protons, neutrons, and other particlesare used...

SIAM Journal on Numerical Analysis, 1990

Neural Computation, 2003

In model networks of E-cells and I-cells (excitatory and inhibitory neurons), synchronous rhythmi... more In model networks of E-cells and I-cells (excitatory and inhibitory neurons), synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the E-cells synchronize the I-cells and vice versa. Under ideal conditions -homogeneity in relevant network parameters, and all-to-all connectivity for instance -this mechanism can yield perfect synchronization.

Medical Physics, 1996

Electron beam dose calculations are often based on pencil beam formulas such as the Fermi-Eyges f... more Electron beam dose calculations are often based on pencil beam formulas such as the Fermi-Eyges formula. The Fermi-Eyges formula gives an exact solution of the Fermi equation. The Fermi equation can be derived from a more fundamental mathematical model, the linear Boltzmann equation, in two steps. First, the linear Boltzmann equation is approximated by the Fokker-Planck equation. Second, the Fokker-Planck equation is approximated by the Fermi equation.

Mathematics of Computation, 1989

The Birkhoff equations for the evolution of vortex sheets are regularized in a way proposed by Kr... more The Birkhoff equations for the evolution of vortex sheets are regularized in a way proposed by Krasny. The convergence of numerical approximations to a fixed regularizaron is studied theoretically and numerically. The numerical test problem is a two-dimensional inviscid jet.

Journal of Computational Physics, 2010

Beams of microscopic particles penetrating scattering background matter play an important role in... more Beams of microscopic particles penetrating scattering background matter play an important role in several applications. The parameter choices made here are motivated by the problem of electron-beam cancer therapy planning. Mathematically, a steady particle beam penetrating matter, or a configuration of several such beams, is modeled by a boundary value problem for a Boltzmann equation. Grid-based discretization of such a problem leads to a system of algebraic equations. This system is typically very large because of the large number of independent variables in the Boltzmann equation-six if no dimensionreducing assumptions other than time independence are made. If grid-based methods are to be practical for these problems, it is therefore necessary to develop very fast solvers for the discretized problems. For beams of mono-energetic particles interacting with a passive background, but not with each other, in two space dimensions, the first author proposed such a solver, based on angular domain decomposition, some time ago. Here, we propose and test an angular multigrid algorithm for the same model problem. Our numerical experiments show rapid, grid-independent convergence. For high-resolution calculations, our method is substantially more efficient than the angular domain decomposition method. In addition, unlike angular domain decomposition, the angular multigrid method works well even when the angular diffusion coefficient is fairly large. variables in any kinetic problem (unless the geometry is special), not just in charged-particle transport. However, there are additional difficulties associated specifically with charged-particle transport: The mean free path tends to be small, scattering tends to be very forward-peaked (i.e., particles are typically deflected only very slightly by a single interaction with the background), and particles typically lose very little energy in a single interaction. These properties of charged-particle transport cause difficulties with the accuracy of discretizations and with the efficiency of solution algorithms for the discretized problems [16, Section 3.2], which have lead many in the Medical Physics community to believe that the most efficient way of modeling electron beams may be Monte Carlo simulation. However, based on a rough theoretical complexity estimate presented in [4], we believe that deterministic, grid-based methods could eventually prove to be a very attractive alternative to Monte Carlo simulation, provided that all available tools of numerical computing are brought to bear to develop highly accurate discretizations as well as optimally efficient solution algorithms for the discretized problems. Some algorithm and code development efforts in this direction are in fact underway; see, for instance, .

Inverse Problems, 1999

The possibility of multiple locally optimal dose distributions in radiation treatment planning ha... more The possibility of multiple locally optimal dose distributions in radiation treatment planning has been discussed and documented in the literature. Here we study a different question related to uniqueness: Is it possible for different treatment plans to generate the same dose distribution? For greatly simplified two-dimensional model problems, we show that the answer is 'yes' in regions where two or more beams intersect. In realistic problems, those are of course not the only regions of interest. However, as a result of cancellations in regions of intersection, substantial perturbations of beam profiles in certain directions may still have only small effects on the dose distribution. This is interesting because it offers an opportunity to optimize some other useful property, for instance simplicity, among all treatment plans generating a desired dose distribution with sufficient accuracy. We take a first step beyond our model problems by proving the stability of our results with respect to small perturbations of problem parameters. Since realistic problems differ from our model problems by much more than small perturbations, we plan to present a numerical study of more realistic examples in a sequel to this article.

European Journal of Neuroscience, 2014

Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognit... more Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognitive processing. However, the relationship between the physiological mechanisms producing these rhythms and the functions associated with the rhythms remains mysterious. This article investigates the contributions of rhythms to basic cognitive computations (such as filtering signals by coherence and/or frequency) and to major cognitive functions (such as attention and multi-modal coordination). We offer support to the premise that the physiology underlying brain rhythms plays an essential role in how these rhythms facilitate some cognitive operations. segment of the globus pallidus; GPi, internal segment of the globus pallidus; IB, intrinsically bursting; I-cell, inhibitory fast-spiking cell; IPSP, inhibitory postsynaptic potential; LTS, low-threshold spiking; MSN, medium spiny neuron; NMDA, N-methyl-D-aspartate; PING, pyramidal-interneuronal network gamma; STN, subthalamic nucleus.