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Moritz Gruber

Publications and source records attributed to Moritz Gruber.

6 recordsLinked to original sources
arXiv2021

Conductance-based Dynamic Causal Modeling: A mathematical review of its application to cross-power spectral densities

Dynamic Causal Modeling (DCM) is a Bayesian framework for inferring on hidden (latent) neuronal states, based on measurements of brain activity. Since its introduction in 2003 for functional magnetic resonance imaging data, DCM has been extended to electrophysiological data, and several variants have been developed. Their biophysically motivated formulations make these models promising candidates for providing a mechanistic understanding of human brain dynamics, both in health and disease. However, due to their complexity and reliance on concepts from several fields, fully understanding the mathematical and conceptual basis behind certain variants of DCM can be challenging. At the same time, a solid theoretical knowledge of the models is crucial to avoid pitfalls in the application of these models and interpretation of their results. In this paper, we focus on one of the most advanced formulations of DCM, i.e. conductance-based DCM for cross-spectral densities, whose components are described across multiple technical papers. The aim of the present article is to provide an accessible exposition of the mathematical background, together with an illustration of the model's behavior. To this end, we include step-by-step derivations of the model equations, point to important aspects in the software implementation of those models, and use simulations to provide an intuitive understanding of the type of responses that can be generated and the role that specific parameters play in the model. Furthermore, all code utilized for our simulations is made publicly available alongside the manuscript to allow readers an easy hands-on experience with conductance-based DCM.

q-bio.QM↗
arXiv2019

Non-Euclidean isoperimetric inequalities for nilpotent Lie groups

This article treats isoperimetric inequalities for integral currents in the setting of stratified nilpotent Lie groups equipped with left-invariant Riemannian metrics. We prove that for each such group there is a dimension in which no Euclidean isoperimetric inequality is admitted, while in all smaller dimensions strictly Euclidean isoperimetric inequalities are satisfied.

math.MG↗
arXiv2018

Higher divergence for nilpotent groups

The higher divergence of a metric space describes its isoperimetric behaviour at infinity. It is closely related to the higher-dimensional Dehn functions, but has more requirements to the fillings. We prove that these additional requirements do not have an essential impact for many nilpotent Lie groups. As a corollary, we obtain the higher divergence of the Heisenberg groups in all dimensions.

math.MG↗
arXiv2017

Filling Invariants of Stratified Nilpotent Lie Groups

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of the Euclidean space and in the high dimensions slower than the filling functions of the Euclidean space. We do this by developing a purely algebraic condition on the Lie algebra of a stratified nilpotent Lie group. Further, we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the Euclidean space. Further we bound the higher divergence functions of stratified nilpotent Lie groups.

math.DG↗
arXiv2015

Higher Divergence Functions for Heisenberg Groups

We prove a Filling Theorem for the Heisenberg Groups $H^{2n+1}$: For a given $k$-cycle $a$ we construct a $(k+1)$-chain $b$ (the filling) with boundary $\partial b=a$ and controlled volume. For this filling $b$ we prove a uniform bound on the distance of points in $b$ to its boundary $a$. Using this we compute the higher divergence functions for the Heisenberg Groups $H^{2n+1}$. Further we generalise these results to the Jet-Groups $J^m(\mathbb R^n)$ for dimension less or equal $n$ .

math.DG↗