Stable geometry is inevitable in drifting neural representations

In many brain regions, the stimulus tuning of neurons is stable on a timescale of hours but not on a timescale of weeks, a phenomenon often called ‘representational drift’. For example, in piriform cortex, which is commonly considered primary olfactory cortex, the cells responsive to a given odor are completely uncorrelated with those activated by the same odor two weeks later (Schoonover et al., 2021). This would seem to imply that piriform cortex, like other brain regions whose activity appears to drift, is useless for the retrieval of associative memories learned several weeks prior. However, decoding approaches have demonstrated that stable decoding of drifting representations is possible (Rule et al., 2020). While these previous computational results offer a very plausible resolution to the paradox of how the brain operates with drifting representations, we lack a deep understanding of why this works. Here, we offer a very general mathematical understanding of why stable decoding from drifting representations is possible. We demonstrate that under very weak assumptions, the downstream layer in a two-layer network is guaranteed to act as a tight frame for the representation space of the upstream layer. A tight frame shares many features of an orthogonal basis, including preserving the geometry of relationships between input patterns. Drifting representations that have stable geometry are decodable; thus, the ability to decode from drifting representations is essentially inevitable. Finally, we reconcile these theoretical results with empirical results that appear to show a lack of stable geometry in drifting representations by showing that the discrepancy is due to the number of simultaneously recorded neurons.