Neural Dynamics and the Geometry of Population Activity Abigail A. Russo Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2019 © 2019 Abigail A. Russo All Rights Reserved Abstract Neural Dynamics and the Geometry of Population Activity Abigail A. Russo A growing body of research indicates that much of the brain’s computation is invisible from the activity of individual neurons, but instead instantiated via population-level dynamics. According to this ‘dynamical systems hypothesis’, population-level neural activity evolves according to underlying dynamics that are shaped by network connectivity. While these dynamics are not directly observable in empirical data, they can be inferred by studying the structure of population trajectories. Quantification of this structure, the ‘trajectory geometry’, can then guide thinking on the underlying computation. Alternatively, modeling neural populations as dynamical systems can predict trajectory geometries appropriate for particular tasks. This approach of characterizing and interpreting trajectory geometry is providing new insights in many cortical areas, including regions involved in motor control and areas that mediate cognitive processes such as decision-making. In this thesis, I advance the characterization of population structure by introducing hypothesis-guided metrics for the quantification of trajectory geometry. These metrics, trajectory tangling in primary motor cortex and trajectory divergence in the Supplementary Motor Area, abstract away from task- specific solutions and toward underlying computations and network constraints that drive trajectory geometry. Primate motor cortex (M1) projects to spinal interneurons and motoneurons, suggesting that motor cortex activity may be dominated by muscle-like commands. Observations during reaching lend support to this view, but evidence remains ambiguous and much debated. To provide a different perspective, we employed a novel behavioral paradigm that facilitates comparison between time- evolving neural and muscle activity. We found that single motor cortex neurons displayed many muscle-like properties, but the structure of population activity was not muscle-like. Unlike muscle activity, neural activity was structured to avoid ‘trajectory tangling’: moments where similar activity patterns led to dissimilar future patterns. Avoidance of trajectory tangling was present across tasks and species. Network models revealed a potential reason for this consistent feature: low trajectory tangling confers noise robustness. We were able to predict motor cortex activity from muscle activity by leveraging the hypothesis that muscle-like commands are embedded in additional structure that yields low trajectory tangling. The Supplementary Motor Area (SMA) has been implicated in many higher-order aspects of motor control. Previous studies have demonstrated that SMA might track motor context. We propose that this computation necessitates that neural activity avoids ‘trajectory divergence’: moments where two similar neural states become dissimilar in the future. Indeed, we found that population activity in SMA, but not in M1, reliably avoided trajectory divergence, resulting in fundamentally different geometries: cyclical in M1 and helix-like in SMA. Analogous structure emerged in artificial networks trained without versus with context-related inputs. These findings reveal that the geometries of population activity in SMA and M1 are fundamentally different, with direct implications regarding what computations can be performed by each area. The characterization and statistical analysis of trajectory geometry promises to advance our understanding of neural network function by providing interpretable, cohesive explanations for observed population structure. Commonality between individuals and networks can be uncovered and more generic, task-invariant, fundamental aspects of neural response can be explored. Table of Contents List of Figures ........................................................................................................................................... v Acknowledgements ................................................................................................................................ viii Chapter 1 Introduction................................................................................................................. 1 Overview of dissertation ........................................................................................................................... 3 Characterizing population structure .......................................................................................................... 4 Visualizing population activity ............................................................................................................................. 5 Identifying computationally-relevant trajectory structure .................................................................................... 7 Characterizing neural covariance ......................................................................................................................... 9 Predicting and interpreting population structure ..................................................................................... 12 Studying population structure in artificial networks .......................................................................................... 13 Metrics of geometric properties .......................................................................................................................... 15 Chapter 2 Motor cortex embeds muscle-like commands in an untangled population response ........................................................................................................................................ 18 Introduction ............................................................................................................................................. 19 Results ..................................................................................................................................................... 23 Task and behavior ............................................................................................................................................... 23 Single-neuron responses ..................................................................................................................................... 26 Non-muscle-like signals dominate the neural population response ................................................................... 28 Potential explanations and caveats ..................................................................................................................... 29 Smooth dynamics predict low trajectory tangling .............................................................................................. 32 Neural- versus muscle-trajectory tangling .......................................................................................................... 36 Tangling across tasks, species, and areas ........................................................................................................... 38 Noise-robust networks display low tangling ...................................................................................................... 39 i Hypothesis-based prediction of neural responses ............................................................................................... 42 Alternative predictions ....................................................................................................................................... 45 Signals introduced by optimization yield incidental correlations ...................................................................... 46 Muscle-like signals are embedded in trajectories with low tangling.................................................................. 48 Tangling in sulcal motor cortex .......................................................................................................................... 49 Discussion ............................................................................................................................................... 51 Are the dominant signals in motor cortex representational or computational? .................................................. 51 Differences and commonalities across tasks ...................................................................................................... 52 Tangling across areas ......................................................................................................................................... 54 Methods ................................................................................................................................................... 55 Experimental apparatus ...................................................................................................................................... 55 Task .................................................................................................................................................................... 56 Neural recordings during cycling ....................................................................................................................... 57 EMG recordings ................................................................................................................................................