Dimensionality Reduction of Neural spike train data using factor analysis, including a comparative review of linear Gaussian models

Jie Fuhttps://sites.google.com/site/bigaidream/

If we can get the low-dimensional features of the neural signals by doing dimensionality reduction, pattern classification tasks could become easier.

[Key idea] A generalization of PCA, Factor Analysis (FA) handles distinct noise variability in each neuron, making it more appropriate for neural data.

What is PCA? It finds projections in the high-dimensional space, then throwing out the less-informative ones.

PCA finds dimensions with maximum variance of projection. PCA finds dimensions that minimize error of projection. Both lead to a solution based on eigenvector decomposition.

Probabilistic PCA (PPCA)[Key Idea]PCA can also be expressed of as maximum likelihood estimation of a probabilistic continuous latent variable model [pattern recognition and machine learning, Bishop, Section

12.2]. Each data point Xn, can be described by a latent variable Zn in the space Z, which is generative.

Factor Analysis (FA)[Key idea]FA is an extension of PPCA, in which the covariance structure of the noise is less constrained. In PPCA, the covariance matrix corresponding to input noise must be a multiple of the identity

matrix, εI , ε is the global noise level. For limε→0

εI , it is “vanilla” PCA.

For FA, the covariance matrix is diagonal. FA retains distinct noise variance in each dimension.

Equivalence of FA and Mixture Models“Gaussian factor model with k factors is equivalent to some mixture model with k +1 clusters, in the sense that the two models have the same means and covariance.” [Shalizi] “…factor analysis can’t really tell us whether we have k continuous hidden causal variables, or one discrete hidden variable taking k+1 values” [Shalizi]

[KEY] Analysis of Neural DataAssumptions:1. Each neuron is a noisy sensor reflecting temporal evolution of a low-dimensional internal

process.2. Firing rate is sufficient to characterize activity

Challenges:1. Neurons are highly variable: Hence FA is necessary, allowing distinct noise models, especially

to normalize for “loud” vs. “quiet” neurons2. Must smooth binned firing rates: Use a two-step process: first smooth, then fit; Authors

combine two steps into “Gaussian Process Factor Analysis” (GPFA)

Neural Time Series Analysis - Steps

The goal is to determine trajectory from a single trial.

[KEY] Problems with two-step process/traditional approachKernel smoothing technique is ad-hocSame kernel for each neuron implies single timescale, probably erroneously

PCA has no noise model, making it difficult to isolate noise from dataNo interaction between two phases of process

Gaussian Process Factor Analysis

Conclusions and Future WorkGPFA offers a powerful tool with several advantages:– FA allows explicit noise model for each neuron– GPFA extends static model for time-series analysis– Use of parametric GP covariance permits extensiveexploratory modeling

Future Work:– Richer GP neural state evolution– Non-stationary kernels– Non-linear manifolds and point-process likelihood

ReferenceSingle-trial analysis of neural population activity during motor preparationDimensionality Reduction of Neural spike train data using factor analysis, including a comparative review of linear Gaussian models (PPT for CS 545, machine learning)