Model-based targeted dimensionality reduction for neuronal population data

Abstract

Summarizing high-dimensional data using a small number of parameters is a ubiquitous first step in the analysis of neuronal population activity. Recently developed methods use "targeted" approaches that work by identifying multiple, distinct low-dimensional subspaces of activity that capture the population response to individual experimental task variables, such as the value of a presented stimulus or the behavior of the animal. These methods have gained attention because they decompose total neural activity into what are ostensibly different parts of a neuronal computation. However, existing targeted methods have been developed outside of the confines of probabilistic modeling, making some aspects of the procedures ad hoc, or limited in flexibility or interpretability. Here we propose a new model-based method for targeted dimensionality reduction based on a probabilistic generative model of the population response data. The low-dimensional structure of our model is expressed as a low-rank factorization of a linear regression model. We perform efficient inference using a combination of expectation maximization and direct maximization of the marginal likelihood. We also develop an efficient method for estimating the dimensionality of each subspace. We show that our approach outperforms alternative methods in both mean squared error of the parameter estimates, and in identifying the correct dimensionality of encoding using simulated data. We also show that our method provides more accurate inference of low-dimensional subspaces of activity than a competing algorithm, demixed PCA.

Figure 1:

A: Schematic illustration of low-rank regression model. The n × T response matrix can be decomposed into two response matrices $\left({\mathit{\beta }}_1,{\mathit{\beta }}_2\right),$ each corresponding to one task variable (upper panel). Each response matrix can be factorized into a small number of row and columns vectors, making the population response a linear combination of a small number of common basis functions weighted differently for each neuron. B: Results of simulation study evaluating parameter estimation accuracy for different estimation procedures. Legend indicates the method used. Abscissa indicates the number of trials used for the simulations. Error bars indication 95% confidence intervals over 100 runs. C: Duration of computation for methods and trials used in B.

Figure 2:. Simulation studies.

A: Results of simulation study evaluating performance of Algorithm 1 for dimensionality estimation by means of different parameter estimation procedures. The legend indicates the sample size. Abscissa indicates the error in dimensionality estimate. Ordinate gives the number of estimated subspaces that obtained the corresponding error. Dashed line indicates model-mismatch experiment with Poisson observations and sample size 2000. B: Results of subspace estimation by our MML method compared with dPCA. The MML method out-performs dPCA at all but the highest SNR, where performance is similar.