Michael Hanke

Michael Hanke

For simulating large networks of neurons Hines proposed a method which uses extensively the structure of the arising systems of ordinary differential equations in order to obtain an efficient implementation. The original method requires constant step sizes and produces the solution on a staggered grid. In the present paper a one-step modification of this method is introduced and analyzed with respect to their stability properties. The new method allows for step size control. Local error estimators are constructed. The method has been implemented in matlab and tested using simple Hodgkin-Huxley type models. Comparisons with standard state-of-the-art solvers are provided.

Michael Hanke, Roswitha März

The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and contribute to justify the overdetermined polynomial collocation applied to higher-index differential-algebraic equations. Besides, we discuss several practical aspects concerning higher-order differential-algebraic operators and the associated equations.

Michael Hanke, Marry-Chriz Cabauatan-Villanueva

The simulation of the metabolism in mammalian cells becomes a severe problem if spatial distributions must be taken into account. Especially the cytoplasm has a very complex geometric structure which cannot be handled by standard discretization techniques. In the present paper we propose a homogenization technique for computing effective diffusion constants. This is accomplished by using a two-step strategy. The first step consists of an analytic homogenization from the smallest to an intermediate scale. The homogenization error is estimated by comparing the analytic diffusion constant with a numerical estimate obtained by using real cell geometries. The second step consists of a random homogenization. Since no analytical solution is known to this homogenization problem, a numerical approximation algorithm is proposed. Although rather expensive this algorithm provides a reasonable estimate of the homogenized diffusion constant.

Michael Hanke, Roswitha März

We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations. The numerical experiments show impressive results. In contrast, the theoretical basic concept turns out to be considerably challenging. So far, quite recently convergence proofs for linear problems have been published. In the present paper we come up to a first convergence result for nonlinear problems.

Yaroslav O. Halchenko, Michael Hanke, James V. Haxby et al.

Localizing neuronal activity in the brain, both in time and in space, is a central challenge to advance the understanding of brain function. Because of the inability of any single neuroimaging techniques to cover all aspects at once, there is a growing interest to combine signals from multiple modalities in order to benefit from the advantages of each acquisition method. Due to the complexity and unknown parameterization of any suggested complete model of BOLD response in functional magnetic resonance imaging (fMRI), the development of a reliable ultimate fusion approach remains difficult. But besides the primary goal of superior temporal and spatial resolution, conjoint analysis of data from multiple imaging modalities can alternatively be used to segregate neural information from physiological and acquisition noise. In this paper we suggest a novel methodology which relies on constructing a quantifiable mapping of data from one modality (electroencephalography; EEG) into another (fMRI), called transmodal analysis of neural signals (TRANSfusion). TRANSfusion attempts to map neural data embedded within the EEG signal into its reflection in fMRI data. Assessing the mapping performance on unseen data allows to localize brain areas where a significant portion of the signal could be reliably reconstructed, hence the areas neural activity of which is reflected in both EEG and fMRI data. Consecutive analysis of the learnt model allows to localize areas associated with specific frequency bands of EEG, or areas functionally related (connected or coherent) to any given EEG sensor. We demonstrate the performance of TRANSfusion on artificial and real data from an auditory experiment. We further speculate on possible alternative uses: cross-modal data filtering and EEG-driven interpolation of fMRI signals to obtain arbitrarily high temporal sampling of BOLD.