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Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential starting point is basal cognition, which gives an abstract representation of a range of organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner, mimicking the way these organisms function. A key ingredient in our model, not found in previous basal cognition models, is that we explicitly model oscillations in the number of particles (i.e. the nutrients, chemical signals or similar, which make up the biological system) and the flow of these particles within the modelled organisms. Using this approach, our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We further demonstrate that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. In the context of these findings, we discuss connections between our model and basal cognition in biological systems and slime moulds, in particular, how oscillations might contribute to self-organized problem-solving by these organisms.

While football analytics has changed the way teams and analysts assess performance, there remains a communication gap between machine learning practice and how coaching staff talk about football. Coaches and practitioners require actionable insights, which are not always provided by models. To bridge this gap, we show how to build wordalisations (a novel approach that leverages large language models) for shots in football. Specifically, we first build an expected goals model using logistic regression. We then use the coefficients of this regression model to write sentences describing how factors (such as distance, angle and defensive pressure) contribute to the model's prediction. Finally, we use large language models to give an engaging description of the shot. We describe our approach in a model card and provide an interactive open-source application describing shots in recent tournaments. We discuss how shot wordalisations might aid communication in coaching and football commentary, and give a further example of how the same approach can be applied to other actions in football.

The way in which a social network is generated, in terms of how individuals attach to each other, determines the properties of the resulting network. Here, we study an intuitively appealing ‘friend of a friend’ model, where a network is formed by each newly added individual attaching first to a randomly chosen target and then to  randomly chosen friends of the target, each with probability ⁠. We revisit the master equation of the expected degree distribution for this model, providing an exact solution for the case when nq allows for attachment to all of the chosen target’s friends [a case previously studied by Bhat et al. (2016, Phys. Rev. E, 94, 062302)], and demonstrating why such a solution is hard to obtain when nq is fixed [a case previously studied by Levens et al. (2022, R. Soc. Open Sci., 9, 221200)]. In the case where attachment to all friends is allowed, we also show that when ⁠, the expected degree distribution of the model is stationary as the network size tends to infinity. We go on to look at the clustering behaviour and the triangle count, focusing on the cases where nq is fixed.

Despite convincing arguments by mathematicians, philosophers, sociologists and machine learning practitioners to the contrary, there remains a widespread notion amongst many members of the general public (and some practitioners) that mathematics is neutral, that it is free from human values. One reason why this notion persists is that we lack clear-cut examples that demonstrate how mathematics and values are intertwined. In this paper, we offer one such example. In particular, we show that when sharing four biscuits between three people, several possible mathematical and ethical frameworks can be used. We demonstrate that different solutions-hiding one biscuit, arbitrarily sharing the extra biscuit, randomizing allocation, dividing the extra biscuit into three parts, and successively dividing it into smaller and smaller parts-involve different mathematical methods and evoke different human values. We discuss the construction of quantum biscuit splitting devices and the use of machine learning to divide biscuits. We argue that the multitude of different mathematically-correct solutions to this problem (each with its own ethical justification) might influence the values held by practicing mathematicians. The example we propose here has been used in teaching to help students understand why mathematics cannot be cleanly separated from human values.

There is a greater need in today's society, to understand and critically discuss how the limited resources of our planet are allocated. Often, mathematical models are used in connection with resource allocation problems, and a common view is that mathematics in itself is neutral. In this article, we challenge this view of mathematics as a neutral practice through an analysis of possible solutions to a sharing task. The tasks come from a research project aiming to study how mathematics can support ethical reasoning and ethical arguments can support different mathematical solutions when sharing a resource. In ethical reasoning, three components are addressed: Information, Coherence, and Engagement. We show that ethical reasoning is part of mathematical reasoning in all the solutions to the task, independent of whether the dividend is treated as indivisible or divisible.

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities - (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data - inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomenon can be modelled in an infinite number of different ways, through the adoption of a pluralistic approach, where we view a system from multiple, different points of view. We explain this pluralistic approach using fish locomotion as a case study and illustrate some of the pitfalls - universalism, creating models of models, etc. - that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling.

While mathematical models, in particular self-propelled particle models, capture many properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the use of intermittent locomotion, often referred to as burst-and-glide, by many species. In this paper, we propose a model of social burst-and-glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We first show that our model can capture the motion of a single fish swimming down a channel. Extending to a two-fish model, where visual stimulus of a neighbour affects the internal burst or glide state of the fish, we observe a rich set of dynamics found in many species. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and tit-for-tat turn taking. Moreover, unlike previous studies where a randomness is required for leadership switching to occur, we show that this can instead be the result of deterministic interactions. We give several empirically testable predictions for how bursting fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.

One of the best-known models in network science is preferential attachment. In this model, the probability of attaching to a node depends on the degree of all nodes in the population, and thus depends on global information. In many biological, physical and social systems, however, interactions between individuals depend only on local information. Here, we investigate a truly local model of network formation-based on the idea of a friend of a friend-with the following rule: individuals choose one node at random and link to it with probability p, then they choose a neighbour of that node and link with probability q. Our model produces power-laws with empirical exponents ranging from 1.5 upwards and clustering coefficients ranging from 0 up to 0.5 (consistent with many real networks). For small p and q = 1, the model produces super-hub networks, and we prove that for p = 0 and q = 1, the proportion of non-hubs tends to 1 as the network grows. We show that power-law degree distributions, small world clustering and super-hub networks are all outcomes of this, more general, yet conceptually simple model.

This book introduces machine learning for readers with some background in basic linear algebra, statistics, probability, and programming. In a coherent statistical framework it covers a selection of supervised machine learning methods, from the most fundamental (k-NN, decision trees, linear and logistic regression) to more advanced methods (deep neural networks, support vector machines, Gaussian processes, random forests and boosting), plus commonly-used unsupervised methods (generative modeling, k-means, PCA, autoencoders and generative adversarial networks). Careful explanations and pseudo-code are presented for all methods. The authors maintain a focus on the fundamentals by drawing connections between methods and discussing general concepts such as loss functions, maximum likelihood, the bias-variance decomposition, ensemble averaging, kernels and the Bayesian approach along with generally useful tools such as regularization, cross validation, evaluation metrics and optimization methods. The final chapters offer practical advice for solving real-world supervised machine learning problems and on ethical aspects of modern machine learning