Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization

Project Information

Nonlocal-CPD

Grant agreement ID: 883363

DOI

10.3030/883363

EC signature date 12 May 2020

Start date 1 October 2020

End date 30 September 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 2 478 560,00

EU contribution

€ 2 478 560,00

Coordinated by

THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
United Kingdom

Periodic Reporting for period 3 - Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization)

Reporting period: 2023-10-01 to 2025-03-31

Summary of the context and overall objectives of the project

Our project focuses on the development of new mathematical tools to analyse theoretical, numerical and modelling aspects of novel applications of nonlinear nonlocal aggregation-diffusion equations in Mathematical Biology and in classical problems of kinetic theory. Among the numerous areas of applications of kinetic modelling in Mathematical Biology, we will concentrate on phenomena identified, at the modelling stage, as systems involving a large number of "individuals" showing "collective behaviour" and how to obtain "averaged" information from them. Individuals behavior can be modelled via stochastic/deterministic ODEs from which one obtains mesoscopic/macroscopic descriptions based on mean-field PDEs leading to continuum mechanics, hydrodynamic and/or kinetic systems. Understanding the interplay between the interaction behaviour (nonlocal, nonlinear), the diffusion (nonlinear), the transport phenomena, and the synchronization is our main mathematical goal.

The proposed research is centred on developing tools underpinning the analysis of long time asymptotics, phase transitions, stability of patterns, consensus and clustering, and qualitative properties of these models. On the other hand, designing numerical schemes to accurately solve these models is key not only to understand theoretical issues but also crucial in applications. We will focus on the important case of the Landau equation with applications in weakly nonlinear plasmas by means of the gradient flow techniques. On the other hand, we showcase our tools in patterns and consensus by focusing on zebra fish patterning formation, as example of spontaneous self-organisation processes in developmental biology, and grid cells for navigation in mammals, as prototype for the synchronization of neural networks. This project connects with other areas of current interest in science and technology such as agent-based models in engineering: global optimization, clustering, and social sciences.

Our precise objectives include 3 main research avenues: equilibration patterns for repulsive-attractive potentials and nonlinear aggregation/diffusions: gradient Flows, blow-up profiles, phase transitions and functional inequalities; the (Vlasov-Maxwell)-Landau equation in plasma physics; pattern formation in tissue growth and synchronization of neuronal networks in mathematical biology. We refer to the figure attached for the numerical finding of hexagonal patterns in synchronization of neural networks by Fokker-Planck models.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Despite the slow starting in October 2020 due to the pandemic hitting the first hirings, I was able to make a contract to one PDRA in the first year where we focused in Strands A and B. Strand C in its theoretical part started with a PhD student during the first year of the project, while the numerical part started in September 2021 with the arrival of another PDRA. Strands D and E were started in early 2021 and took a boost with the incorporation of other 2 PDRAs. The rest of PDRAs started between October 2021 and January 2022 except one that started in September 2022. The project has taken full cruise speed since mid 2021 and all strands in the project have been started and developed.

We have obtained major results in the concentration of solutions for aggregation-diffusion equations, objectives related to Strand B showing that concentration happens in the fast diffusion regime for certain range of the homogeneity of the interaction potential and even for given confining potentials both at the level of the variational problems as well as the associated gradient flow equations. We have also obtained almost sharp conditions on the interaction potential leading to diffusive behavior asymptotically in time.

Concerning properties of minimizers of the interaction energy, objectives of Strand A, we have obtained quite sharp conditions leading to radial symmetry or fractal behavior of their minimizers. These ideas have also been applied to the case of anisotropic potentials. Numerical approaches have been developed to tackle with these difficulties proving their convergence and their efficiency by using optimal transportation techniques.

The gradient flow approach to the Landau equation for plasma physics has been fully developed from the theoretical viewpoint obtaining well-posedness results of the approximated Landau equations and making the grazing collision limit rigorous from this perspective. This has led the ground to develop semidiscrete deterministic schemes for the inhomogeneous Landau equation with the right physical properties whose implementation has been accomplished in early 2023 and writing-up is underway as one of the main objectives of Strand C. A PhD student finalized in May 2022 where the main theoretical ground works on the gradient flow structure of Landau were laid.

Strand D is less developed since the PDRA in this research line started in January 2022. We have excellent results for cell-cell adhesion models applied to zebra-fish skin patterning that are being writing-up a this moment. We have also advanced a lot in the modelling of micro glia cells migration in brain development and neural crest formation in mammal embryos.

Finally, we have finished the analysis of the linear instability of homogeneous states of neural field networks in computational neuroscience. This was the starting brick to analyse more complicated attractor network models used in practise. We have also achieved one of the main objectives of Strand E in which we analyse the bifurcation diagram of these attractor network dynamics leading to the formation of hexagonal standing waves as observed in experiments.

In short, the project is developing all strands of research in the proposal at cruise speed, all hirings have been working very successfully and we have obtained major results in all the topics of research. We already substituted some of the PDRAs in May and October 2022 since two of them left having obtained tenure-track or permanent positions in other institutions. Two more hirings are under way in September 2023 again to replace and advance in the different lines of research due to PDRAs that left the project for better career opportunities.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We are at the edge of knowledge in the research topic of aggregation-diffusion equations and their applications in mathematical biology and plasma physics. This is substantiated with the invitation as plenary speaker of the PI of the project to the ICIAM 2023, the most important international venue for applied mathematics at a conference organized every 4 years. We are advancing equally well in all strands of the project connecting with researchers of the applied fields in neuroscience and tissue growth and in plasma physics. We expect the results of this project to be fully achieved by the end of the project in March 2026. Some of main goals have already been achieved as specified in the major achievements section. The PI has been invited as plenary speaker of the SIAM APDE conference, the ENUMATH23 conference, the SIMAI23 conference and many other events in Oberwolfach, Mittag-Leffler, INI, CIRM institutes to showcase his research. All PDRAs of the project have also been invited to conferences of the specific topics of research of highest level. This is also another landmark of the prestige of the research developed.

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