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 .
Many studies of neural activity in behaving animals aim to discover interpretable low-dimensional structure in large-scale neural population recordings. One approach to this problem is demixed principal component analysis (dPCA), a supervised linear dimensionality reduction technique to find components that depend on particular experimental parameters. Here, I introduce kernel dPCA (kdPCA) as a nonlinear extension of dPCA by applying kernel least-squares regression to the demixing problem. I consider simulated examples of neural populations with low-dimensional activity to compare the components recovered from dPCA and kdPCA . These simulations demonstrate that neurally relevant nonlinearities , such as stimulus-dependent gain and rotation, interfere with linear demixing of neural activity into components that represent to individual experimental parameters. However, kdPCA can still recover interpretable components in these examples. Finally, I demonstrate kdPCA using two examples of neural populations recorded during perceptual decision-making tasks .
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Many studies of neural activity in behaving animals aim to discover interpretable low-dimensional structure in large-scale neural population recordings. One approach to this problem is
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for analyzing the activity of neural populations, to the case of nonlinear components . This extension, kernel demixed principal component analysis
to find components that depend on particular experimental parameters. Here, I introduce kernel dPCA
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as a nonlinear extension of dPCA by applying
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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
to the demixing problem. I consider
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high-dimensional neural activity generated from
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trajectories and compare the results of kdPCA to dPCA
activity to compare the components recovered from dPCA and kdPCA
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rotations - impede the ability of dPCA to demix neural activity corresponding to
rotation, interfere with linear demixing of neural activity into components that represent to individual
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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
in these examples. Finally, I demonstrate kdPCA using two examples of neural populations recorded during perceptual decision-making tasks