Over the past decade, neural representations have been studied from the lens of low-dimensional subspaces defined by the co-activation of neurons. However, this view has overlooked other forms of covarying structure in neural activity, including i) condition-specific high-dimensional neural sequences, and ii) representations that change over time due to learning or drift. In this talk, I will present a new framework that extends the classic view towards additional types of covariability that are not constrained to a fixed, low-dimensional subspace. In addition, I will present sliceTCA, a new tensor decomposition that captures and demixes these different types of covariability to reveal task-relevant structure in neural activity. Finally, I will close with some thoughts regarding the circuit mechanisms that could generate mixed covariability. Together this work points to a need to consider new possibilities for how neural populations encode sensory, cognitive, and behavioral variables beyond neural subspaces.

Remembering events in the past is crucial to intelligent behaviour. Flexible memory retrieval, beyond simple recall, requires a model of how events relate to one another. Two key brain systems are implicated in this process: the hippocampal episodic memory (EM) system and the prefrontal working memory (WM) system. While an understanding of the hippocampal system, from computation to algorithm and representation, is emerging, less is understood about how the prefrontal WM system can give rise to flexible computations beyond simple memory retrieval, and even less is understood about how the two systems relate to each other. Here we develop a mathematical theory relating the algorithms and representations of EM and WM by showing a duality between storing memories in synapses versus neural activity. In doing so, we develop a formal theory of the algorithm and representation of prefrontal WM as structured, and controllable, neural subspaces (termed activity slots). By building models using this formalism, we elucidate the differences, similarities, and trade-offs between the hippocampal and prefrontal algorithms. Lastly, we show that several prefrontal representations in tasks ranging from list learning to cue dependent recall are unified as controllable activity slots. Our results unify frontal and temporal representations of memory, and offer a new basis for understanding the prefrontal representation of WM