Cell sorting by differential adhesion is one of the basic mechanisms explaining spatial organization of neurons in early stage brain development of fruit flies. The columnar arrangements of neurons determine the large-scale patterns in the fly visual center. Experimental studies indicate that hexagonal configurations regularly appear in the fly compound eye, which is connected to the visual center by photoreceptor axons, while tetragonal configurations can be induced in mutants. We need a mathematical framework to study the mechanisms of such a transition between hexagonal and tetragonal arrangements. Here, we propose a new mathematical model based on macroscopic approximations of agent-based models that produces a similar behavior changing from hexagonal to tetragonal steady configurations when medium-range repulsion and longer-range attraction between individuals are incorporated in previous successful models for cell sorting based on adhesion and volume constraints. We analyze the angular configurations of these patterns based on angle summary statistics and compare between experimental data and parameter fitted ARA (Adhesion-Repulsion-Attraction) models showing that intermediate patterns between hexagonal and tetragonal configuration are common in experimental data as well as in our ARA mathematical model. Our studies indicate an overall qualitative agreement of ARA models in tile patterning and pave the way for their quantitative studies. Our study opens up a new avenue to explore tile pattern transitions, found not only in the column arrangement in the brain, but also in the other related biological processes.

Federated learning is an important framework in modern machine learning that seeks to integrate the training of learning models from multiple users, each user having their own local data set, in a way that is sensitive to data privacy and to communication loss constraints. In clustered federated learning, one assumes an additional unknown group structure among users, and the goal is to train models that are useful for each group, rather than simply training a single global model for all users. In this paper, we propose a novel solution to the problem of clustered federated learning that is inspired by ideas in consensus-based optimization (CBO). Our new CBO-type method is based on a system of interacting particles that is oblivious to group memberships. Our model is motivated by rigorous mathematical reasoning, which includes a mean-field analysis describing the large number of particles limit of our particle system, as well as convergence guarantees for the simultaneous global optimization of general non-convex objective functions (corresponding to the loss functions of each cluster of users) in the mean-field regime. Experimental results demonstrate the efficacy of our FedCBO algorithm compared to other state-of-the-art methods and help validate our methodological and theoretical work.