**A travel guide and a (cognitive) map: a neuroscientific fable from a world of open borders**

Have you ever gone into a different room, only to forget what you were doing? Have you ever been bewildered while travelling somewhere very different, or perhaps while studying or attempting something new? Perhaps you have seen familiar concepts cast in different ways: and perhaps these different perspectives have revealed new insights. Sometimes, those new insights are only apparent after translating the new ideas into more familiar terms; sometimes, we must return to where we started to remember our goal.

These days, I often notice people using spatial metaphors in connection with apparently non-spatial tasks. For instance, āgetting to know the landscapeā of a new field; or, āseeing the way to a solutionā of a problem. Is there more to this than metaphor? My research into the brainās ācognitive mapā suggests that not only is the answer āyesā, but that we can make these metaphors precise, using the new mathematics of category theory.

Donāt let the abstruse name put you off, for category theory is the mathematics of patterns and metaphor: familiar, human stuff! Its fundamental theorem says that we should understand an object by how it relates to others ā how they connect together ā just like we know a place by how to reach it. Each ācategoryā is like a mathematical place, and by formalising notions of connection, we can translate concepts between places. The category of categories turns into something like a travel guide for mathematical sciences.

Compiling this travel guide is still a work in progress. But working in the interdisciplinary world of computational neuroscience, and so often at intellectual sea in othersā fields, I know that such a travel guide will be immensely valuable. Mentors have often said to me that āmodels are metaphorsā, like mathematical fables, each capturing some partial truth. For a system as complex as the brain, there are overwhelmingly many such partial truths, expressed in the particular idiom of each research group. Without a rigorous travel guide, how can we hope to translate these idioms, to piece these partial truths together and understand even just the cognitive map, let alone the whole brain?

You have a cell in your hippocampus ā a seahorse-shaped structure deep in your brain ā that shouts to your other neurons, āI am here!ā But when you go from āhereā to āthereā, this cellās shouts turn to whispers, and another so-called place cell takes up the chorus. Within each environment, each place cell has a preferred location. As in the world, there is a hierarchy of sizes: some place cells prefer big spaces; others small. And when you go from big space to big space, or environment to environment, the smaller-scale cells āre-mapā their preferences to new preferred locations in the new environment.

This cognitive map encodes not just place, however: the hippocampus binds together all the data of your experiences with where and when they took place. This means what you saw, what you smelt, what you felt,… Together, the āplaceā cells represent the topology of your world in the broadest sense: they triangulate the high-dimensional manifold of your experience, as filtered and represented through the rest of your brain. And this explains how just changing rooms can lead to forgetting: after re-mapping, the association between place and task is disrupted.

Numerous mathematical fables exist to explain and predict aspects of these processes. At first, building on data from rodents exploring mazes, the models were explicitly spatial. Subsequent data showed that rodent place cell activity also correlated with reward and goal, and from humans we learnt that the brain represents virtual and abstract environments with similar maps. But the latest models that account for this more abstract processing are not clearly compatible with the earlier spatial ones. Moreover, few models explain how the hippocampus interacts with other brain regions. We need a travel guide to the cognitive map, itself the travel guide in the brain.

This is where my categorical work comes in. Categories describe connection. Neurons are connected to each other; places are connected to each other; categories are connected to each other. By expressing different models categorically, we can use the travel guide to translate concepts and move beyond mere fables, without the disorientation of repeatedly learning new idioms. To illustrate: a deep theorem shows that every category ā every mathematical field ā can be navigated like a space, with the objects of the category turned into places. A logical proof becomes precisely a path from premise to conclusion. So it is no surprise that we use spatial metaphors for abstract problems, or that the brain uses a map for ānon-spatialā environments! Neurons themselves are like scientists, piecing partial truths together, publishing their perspectives. And just as there is a map in the brain, increasingly we have one of it.