In this work, we study the theoretical and numerical aspects of the mesoscopic dynamics of neuronal populations. To do so, we adopt a master-equation formulation based on an age-structured Integrate-and-Fire model. The aim of this investigation is to estimate and characterize the pseudo-equilibria and stationary states of the model. To this end, we apply the Scharfetter–Gummel discretization scheme. We obtain simulations showing that the behaviour of the neuronal network varies depending on whether the system is in the inhibitory or the excitatory regime. This is a project in collaboration with María José Cáceres (University of Granada), Alejandro Ramos-Lora (University of Málaga) and Nicolás Torres (Université Côte d'Azur).
Kinetic models for bacterial chemotaxis describe how microorganisms adjust their motion in response to chemical signals. In this talk, we consider kinetic models for bacterial chemotaxis and study their long-time behaviour. We focus on the presence of non-equilibrium effects and on the mechanisms that allow one to establish convergence to equilibrium. These models exhibit a distinctive kinetic mechanism that generates spatial confinement without external potentials. We describe how this mechanism arises and how it influences the asymptotic dynamics. This is joint work with Havva Yoldaş (Delft University of Technology).
The Smoluchowski coagulation equation describes the statistics of growth processes due to pairwise coagulation. We construct a new class of solutions to this equation with increasing mass, called flux solutions. The mass enters the system through the boundary at zero as a flux of dust particles. Flux solutions are expected to approximate the large size behaviour of coagulation equations with source which have many applications in science, including atmospheric science, aerosol science and polymerization. (Based on a joint work with Aleksis Vuoksenmaa - U. Helsinki)
The detailed balance is a property of macroscopic systems that are obtained from an underlying time reversible microscopic model. It states that each elementary process (for instance each chemical reaction) is at equilibrium with its reverse process.
In this talk, I will discuss the role of the lack of detailed balance in a class of stochastic kinetic proofreading models. Kinetic proofreading systems were proposed by Hopfield and Ninio in the 70's in order to explain the low error rate in some biological processes (as immune recognition processes).
The most important property of a kinetic proofreading model is the so-called specificity, i.e. the ability to discriminate between different ligands.
We prove the existence of a critical amount of lack of detailed balance that a class of kinetic proofreading models must have in order to have strong specificity.
We propose an approach to model spatial heterogeneity in SIR-type models for the spread of epidemics via nonlocal aggregation terms. We first consider an SIR model with spatial movements driven by nonlocal aggregation terms, in which the inter-compartment and intra-compartment interaction terms are distinct and modeled through smooth interaction kernels. For the Cauchy problem we provide a full well-posedness theory for $H^1$ solutions. In the second part we discuss on the existence of steady states for these type of models and display a specific example of non-trivial steady states for an SIS model with aggregations, the existence of which is determined by a threshold condition for a suitable "space-dependent" basic reproduction rate. This is a project in collaboration with M. Di Francesco (University of L'Aquila).
Self-organization of active particles drives phase separation and emergent properties in biological systems. In particular, hydrodynamic interactions in populations of microswimmers such as bacteria lead to instabilities and the formation of large-scale patterns. Here, we study how the presence of a director field can lead to collective motion and pattern formation even at low concentrations in a confined population of bacteria. We find that the symmetry breaking and pattern depend on the local confinement geometry, here a rectangular channel or a droplet. In the channel, large-scale plumes and convection flows appear. In the droplet, on the other hand, self-organization can arise from two competing mechanisms, one local, and one global. We then predict very distinct behaviors depending on the hydrodynamic chirality of the swimmers, including a reversible vortex, and oscillating states.