Nonlinear demixed component analysis for neural population data as a low-rank kernel regression problem

Here I introduce an extension to demixed principal component analysis (dPCA), a linear dimensionality reduction technique for analyzing the activity of neural populations, to the case of nonlinear components. This extension, kernel demixed principal component analysis (kdPCA), relies on kernel least-squares regression techniques, and it resembles kernel-based extensions to standard principal component analysis and canonical correlation analysis. kdPCA includes dPCA as a special case when the kernel is linear. I present simulated examples of high-dimensional neural activity generated from low-dimensional trajectories and compare the results of kdPCA to dPCA. These simulations demonstrate that neurally relevant nonlinearities - such as stimulus-dependent gain and rotations - impede the ability of dPCA to demix neural activity corresponding to experimental parameters. However, kdPCA can still recover interpretable components from such data. Additionally, I apply kdPCA to a neural population previously analyzed by dPCA from rat orbitofrontal cortex during an odor classification task in recovering decision-related activity. The components recovered by kdPCA achieve better generalization and demixing performance compared to dPCA by accounting for a nonlinear interaction between stimulus and decision in the neural activity. In conclusion, simple nonlinear interactions inhibit the ability of linear dimensionality reduction techniques to recover interpretable demixed components in neural data, but this problem can be tackled by nonlinear dimensionality reduction approaches like kdPCA.