Abstracts

 

Vivi Rottschäfer:

Title: Modelling drug dynamics in the brain

Abstract: Many drugs need to bind to receptors in the brain to have an effect. In this talk, I will present various models that we developed to study drug distribution into the brain and the central nervous system (CNS). In these models we take the physiology of the CNS and the pharmacokinetic properties of the drugs into account. We developed a compartmental model as well as a spatial model for drug distribution into the brain. We compare the results of the models to results of experiments (in rats). The future aim is to use the compartmental model for prediction of drug concentrations in the brain and their resulting effect in case no experimental data is available for a drug.

 

Anna Shalova:

Title: Porous medium is the message: variational analysis of toy transformers

Abstract: We study an aggregation PDE with competing attractive and repulsive forces on a sphere of arbitrary dimension. In particular, we consider the limit of strongly localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with a porous-medium-type diffusion term. The proof combines variational techniques with elements of harmonic analysis on a sphere. In particular, we characterize the square root of the convolution operator in terms of the spherical harmonics, which allows us to overcome difficulties arising due to the convolution on a sphere being non-commutative. 

Our study is motivated by the toy model of transformers introduced by Geshkovski et al. (2025), in which transformers are treated as an interacting particle system. In the last part of the talk I will show how the attraction-repulsion model is related to transformers and why repulsive layers lead to the diversity of predictions.

This is a joint work with Mark Peletier (arXiv:2512.03185).

 

Daniele Avitabile:

Title: Inferring Parameters and States in Neurobiological Networks

Abstract: The study of cortical dynamics during different states such as decision making, sleep and movement, is an important topic in Neuroscience. Modelling efforts aim to relate the neural rhythms present in cortical recordings to the underlying dynamics responsible for their emergence. We present an effort to characterise the neural activity from the cortex of a mouse during natural sleep, captured through local field potential measurements. Our approach relies on using a discretised Wilson–Cowan Amari neural field model for neural activity, along with a data assimilation method that allows the Bayesian joint estimation of the state and parameters. We demonstrate the feasibility of our approach on synthetic measurements before applying it to a dataset available in literature. Our findings suggest the potential of our approach to characterize the stimulus received by the cortex from other brain regions, while simultaneously inferring a state that aligns with the observed signal.

 

Lara van Vianen:

Title: Travelling fronts between spatially heterogeneous background states: Existence and dynamics

Abstract: We study travelling fronts in a two-component singularly perturbed (FitzHugh-Nagumo-like) reaction-diffusion equation (RDE) with spatially heterogeneous coefficients. The RDE features bistability in terms of two spatially dependent background states. A (travelling) front solution connects  the two motionless background states and has a sharp interface which travels with time varying speed. This renders a comoving frame approach in order to construct travelling fronts practically useless.

To deal with the challenges, we develop new techniques to deal with the spatiotemporal nature of the problem. We manage to prove existence of travelling fronts for a large range of parameters.  In addition, we find a satisfying description of how travelling fronts move. Specifically, we discover that the position of a travelling front can be described in the singular limit by means of a delay differential equation (DDE). Numerically the DDE seems to provide a precise approximation for the position of the front  for small values of the singularly perturbed parameter.

Joint work with Martina Chirilus-Bruckner en Frits Veerman, https://arxiv.org/abs/2507.21797