From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems

The collective movements of unicellular organisms such as bacteria or amoeboid (crawling) cells are often modeled by partial differential equations (PDEs) that describe the time evolution of cell density. In particular, chemotaxis equations have been used to model the movement towards various kinds of extracellular cues. Well-developed analytical and numerical methods for analyzing the time-dependent and time-independent properties of solutions make this approach attractive. However, these models are often based on phenomenological descriptions of cell fluxes with no direct correspondence to individual cell processes such signal transduction and cell movement. This leads to the question of how to justify these macroscopic PDEs from microscopic descriptions of cells, and how to relate the macroscopic quantities in these PDEs to individual-level parameters. Here we summarize recent progress on this question in the context of bacterial and amoeboid chemotaxis, and formulate several open problems

Mathematical models for cell migration: A non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'

Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods

On a hyperbolic-parabolic chemotaxis model on networks for medical tissue engineering applications

Negli ultimi decenni l’analisi matematica si è rivelata uno strumento efficace per descrivere fenomeni di tipo biologico. Uno dei più importanti è la *chemiotassi*, ovvero il meccanismo secondo il quale organismi uni o multicellulari modificano il loro movimento in risposta alla variazione di concentrazione di una certa sostanza chimica nell’ambiente. Nella presente tesi viene proposto un modello per lo studio del movimento dei fibroblasti sotto l’influenza di chemiotassi. Tali cellule sono le principali responsabili del processo di chiusura di una ferita. Quando la pelle è soggetta a un grave trauma spesso non riesce a ricostruirsi da sola, per questo motivo vengono inseriti dei ponteggi artificiali, in modo da poter agevolare il trasporto dei fibroblasti e dunque accellerare il processo di ricostruzione della pelle. Il modello proposto è un modello di tipo iperbolico-parabolico su reti, dove la rete mima il ponteggio artificiale. A partire dalla derivazione del modello, condizioni ai nodi e al bordo vengono discusse. Inoltre alcuni risultati analitici vengono riportati

Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter