P051 Dynamical systems principles underly the ubiquity of neural data manifolds
Isabel M. Cornacchia*1, Arthur Pellegrino*1,2, Angus Chadwick1
1 Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, UK 2Gatsby Computational Neuroscience Unit, School of Life Sciences, University College London, UK
The manifold hypothesis posits that low-dimensional geometry is prevalent in high-dimensional data. In neuroscience, such data emerge from complex interactions between neurons, most naturally described as dynamical systems. While these models offer mechanistic descriptions of the processes generating the data, the geometric perspective remains largely empirical, relying on dimensionality reduction methods to extract manifolds from data. The link between the dynamic and geometric views on neural systems therefore remains an open question. Here, we argue that modelling neural manifolds in a differential geometric framework naturally provides this link, offering insights into the structure of neural activity across tasks and brain regions.
Methods In this work, we argue that many manifolds observed in high-dimensional neural systems emerge naturally from the structure of their underlying dynamics. We provide a mathematical framework to characterise the conditions for a dynamical system to be manifold-generating. Using the framework, we verify in datasets that such conditions are often met in neural systems. Next, to investigate the relationship between the dynamics and geometry of neural population activity, we apply this framework to jointly infer both the manifold and the dynamics on it directly from large-scale neural recordings. Results In recordings of macaque motor and premotor cortex during a reach task [1], we uncover a manifold with behaviourally-relevant geometry: neural trajectories on the inferred manifold closely resemble the hand movement of the animal, without a need to explicitly decode the behaviour. Furthermore, from 2-photon imaging of mouse visual cortex during a visual discrimination task [2], we show that neurons tracked over one month of learning have a stable curved manifold shape, despite the neural dynamics changing. In these two example datasets, we show that considering the curvature of neural manifolds and dynamics on them allows to extract more behaviourally relevant neural representations and to probe for their change over learning (Fig. 1). Discussion Overall, our framework offers a formal mathematical link between the geometric and dynamical perspectives on population activity, and provides a generative model to uncover task manifolds from experimental data. We use this framework to highlight how behavioural and stimulus variables are naturally encoded on curved manifolds, and how this encoding evolves over learning. This lays the mathematical groundwork for systematically modelling neural manifolds in the language of differential geometry, which can be reused across tasks and brain regions. Overall, bridging geometry and dynamics is a key step towards a unified view of neural population activity which can be used to generate and test hypotheses about neural computations in the brain.
Figure 1. a. The framework (MDDS) jointly fits the manifold and dynamics to data. b. Reach task. c. Inferred manifold and trajectories within it. d. Visual task. e. Neural representation of the angle over time. f. Variance explained by a model trained on pre-learning and (top): tested on pre-learning (bottom): tested on post-learning while refitting components, either separately or in combination. Acknowledgements
