Modern neuroscience has evolved into a frontier field that draws on numerous disciplines, resulting in the flourishing of novel conceptual frames primarily inspired by physics and complex systems science. Contributing in this direction, we recently introduced a mathematical framework to describe the spatiotemporal interactions of systems of neurons using lattice field theory, the reference paradigm for theoretical particle physics. In this note, we provide a concise summary of the basics of the theory, aiming to be intuitive to the interdisciplinary neuroscience community. We contextualize our methods, illustrating how to readily connect the parameters of our formulation to experimental variables using well-known renormalization procedures. This synopsis yields the key concepts needed to describe neural networks using lattice physics. Such classes of methods are attention-worthy in an era of blistering improvements in numerical computations, as they can facilitate relating the observation of neural activity to generative models underpinned by physical principles.

Brain–computer interfaces have seen extraordinary surges in developments in recent years, and a significant discrepancy now exists between the abundance of available data and the limited headway made in achieving a unified theoretical framework. This discrepancy becomes particularly pronounced when examining the collective neural activity at the micro and meso scale, where a coherent formalization that adequately describes neural interactions is still lacking. Here, we introduce a mathematical framework to analyze systems of natural neurons and interpret the related empirical observations in terms of lattice field theory, an established paradigm from theoretical particle physics and statistical mechanics. Our methods are tailored to interpret data from chronic neural interfaces, especially spike rasters from measurements of single neuron activity, and generalize the maximum entropy model for neural networks so that the time evolution of the system is also taken into account. This is obtained by bridging particle physics and neuroscience, paving the way for particle physics-inspired models of the neocortex.

An influential theory of increasing returns was proposed by the economist W. B. Arthur in the 1980s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buys the most shared product within this sample. It is known that one of these two companies becomes a monopoly almost surely in the limit of infinite customers. Here we consider a generalization [Dosi, Ermoliev, and Kaniovsky, J. Math. Econom. 23, 1 (1994)] in which the new customer follows the indication of the sample with some probability, and buys the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane, and Sudderth, and it includes several models of physical interest as special cases, such as the Elephant Random Walk, Friedman's urn, and other generalized urn models. We provide a large-deviation analysis of this model at the sample-path level, and we provide a formula that allows us to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sublinear. We also find a nonlinear differential equation for the cumulant-generating function of the market share variable, which can be studied with a suitable perturbation theory.

We describe a microcanonical approach for polymer models that combines atmospheric methods with urn theory. We show that large deviation properties of urn models can provide quite deep mathematical insight by analyzing the random walk range problem in ${\mathbb{Z}}^d$. We also provide a new mean-field theory for the range problem that is exactly solvable by analogy with the Bagchi-Pal urn model.