Highlights d The geometry of abstraction supports generalization d Hippocampal and PFC representations are simultaneously abstract and high dimensional d Multiple task-relevant variables are represented in an abstract format d Representations in simulated neural networks are similar to recorded ones

normale supérieure * These authors contributed equally, † co-senior authorsThe curse of dimensionality plagues models of reinforcement learning and decision-making. The process of abstraction solves this by constructing abstract variables describing features shared by different specific instances, reducing dimensionality and enabling generalization in novel situations. Here we characterized neural representations in monkeys performing a task where a hidden variable described the temporal statistics of stimulus-response-outcome mappings. Abstraction was defined operationally using the generalization performance of neural decoders across task conditions not used for training. This type of generalization requires a particular geometric format of neural representations. Neural ensembles in dorsolateral pre-frontal cortex, anterior cingulate cortex and hippocampus, and in simulated neural networks, simultaneously represented multiple hidden and explicit variables in a format reflecting abstraction. Task events engaging cognitive operations modulated this format. These findings elucidate how the brain and artificial systems represent abstract variables, variables critical for generalization that in turn confers cognitive flexibility.When encountering a new situation, the ability to determine right away what to think, feel, or do is a hallmark example of cognitive, emotional and behavioral flexibility. This ability relies on the fact that the world is structured. In other words, new situations in the world often share features with previously experienced ones. These shared features may correspond to hidden variables not directly observable in the environment, as well as to explicit variables. If past and future situations can be described by a small number of variables, then generalization in novel situations could be achieved by a reduced number of observations. In this scenario, the identification of few relevant variables enables generalization and can overcome the 'curse of dimensionality', obviat-1 ing the need to enumerate and observe all possible combinations of values of all features appearing in the environment. This suggests that a process of dimensionality reduction can identify a compact set of relevant variables which exhaustively describe the environment. These variables -that correspond to features shared by multiple examples (instances) -can be represented in the brain as abstract variables, or concepts.An account of how the brain may represent abstract variables has remained elusive, in part because we lack a precise definition of what constitutes an abstract variable. Motivated by the central role of generalization for defining abstraction, we developed analytic methods for defining when a variable is represented in an abstract format by determining the geometric properties of neural representations that support generalization. More specifically, we operationally defined a neural representation of a variable as being in an abstract format (an "abstract variable") as a representation for which a linear ...