Networks play a central role in understanding the brain. From mapping the anatomy of connections between neurons, to understanding how finely tuned patterns of connections create computation, networks are a central mathematical tool. Further, recent technological advances made possible by the BRAIN Initiative now allow very large-scale reconstructions of the networks of the brain. At the same time, however, a central question remains: even if we know the complete connectivity diagram for a neural system, how can we understand anything about the resulting dynamics? This is a difficult general mathematical problem when systems are nonlinear, and in this sense, the theory is now lagging behind the technology.

This question also has importance for understanding artificial neural networks. Networks such as ChatGPT have captured the world’s attention with their ability to generate coherent, human-like text, and this technology is already having a profound impact on human society. At the same time, however, advances in the underlying theory are now lagging behind the technology: while these networks have an immense potential for practical applications, there are few mathematical approaches for understanding how the precise structure of a trained neural network performs its computation.

New mathematical approaches to neural networks are critically needed at this time, both for understanding the neural systems that make up our brains and to create "explainable AI" (XAI) that will be foundational for ethical applications of this technology for years to come. The Fields Lab for Network Computation, led by Western University mathematics professor Lyle Muller, will bring together experts in discrete mathematics, graph theory, number theory, and physics to spur new work in neural network theory. Collaborations at the lab will be focused on developing new mathematical approaches to neural networks, and applications of these techniques to large-scale network data across science, technology, medicine, and sociology. In this way, the Lab aims to create a hub that can address questions central to human society over the next few years through the study and applications of network theory.

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