[Paper Review] Non-parametric generalized linear model

Matthew Dowling, Yuan Zhao+1 more|arXiv (Cornell University)|Sep 2, 2020

Gaussian Processes and Bayesian Inference18 references4 citations

TL;DR

This paper proposes a nonparametric generalized linear model (NPGLM) that uses sparse variational Gaussian processes to jointly infer temporal filters and hyperparameters in neural spike train analysis, eliminating the need for manual basis function selection. The method achieves superior performance on simulated and real retinal data by automatically learning filter duration and shape with computational efficiency and robust generalization.

ABSTRACT

A fundamental problem in statistical neuroscience is to model how neurons encode information by analyzing electrophysiological recordings. A popular and widely-used approach is to fit the spike trains with an autoregressive point process model. These models are characterized by a set of convolutional temporal filters, whose subsequent analysis can help reveal how neurons encode stimuli, interact with each other, and process information. In practice a sufficiently rich but small ensemble of temporal basis functions needs to be chosen to parameterize the filters. However, obtaining a satisfactory fit often requires burdensome model selection and fine tuning the form of the basis functions and their temporal span. In this paper we propose a nonparametric approach for jointly inferring the filters and hyperparameters using the Gaussian process framework. Our method is computationally efficient taking advantage of the sparse variational approximation while being flexible and rich enough to characterize arbitrary filters in continuous time lag. Moreover, our method automatically learns the temporal span of the filter. For the particular application in neuroscience, we designed priors for stimulus and history filters useful for the spike trains. We compare and validate our method on simulated and real neural spike train data.

Motivation & Objective

• To address the burden of manual basis function selection and hyperparameter tuning in generalized linear models (GLMs) for neural spike train analysis.
• To develop a nonparametric Bayesian approach that jointly infers temporal filters and hyperparameters using Gaussian processes.
• To enable automatic learning of filter temporal span and smoothness without prespecifying basis functions.
• To improve model generalization and robustness, especially in low-sparsity or small-data regimes.
• To provide a computationally efficient alternative to traditional basis-function-based GLMs in neuroscience applications.

Proposed method

• Models temporal filters as sample paths of a Gaussian process (GP) with a mean function and a kernel function, enabling nonparametric inference over continuous time lags.
• Uses a sparse variational approximation with inducing points to reduce computational cost from O(N³) to O(M²N), where M ≪ N.
• Employs a joint variational inference framework to optimize both the GP hyperparameters and the inducing point locations.
• Imposes structured priors on stimulus and history filters tailored for neural spiking data, enhancing interpretability and performance.
• Applies ancestral sampling to generate synthetic spike trains from the fitted model for validation.
• Uses normalized log-likelihood and R² of firing rate to evaluate model fit and generalization on test and repeat stimuli.

Experimental results

Research questions

• RQ1 Can a nonparametric GP-based approach outperform basis-function-based GLMs in modeling neural spike train data?
• RQ2 Can the method automatically infer the temporal extent and shape of neural filters without prespecifying basis functions?
• RQ3 How does NPGLM perform in terms of generalization when data is sparse or noisy?
• RQ4 Does the use of sparse variational inference maintain accuracy while reducing computational cost?
• RQ5 Can the model capture non-smooth or complex filter shapes that standard basis functions fail to represent?

Key findings

• NPGLM consistently outperformed GLM-MLE and GLM-ARD in normalized log-likelihood on test and repeat stimuli across six retinal ganglion cells.
• On average, NPGLM achieved higher normalized log-likelihood and better R² on firing rate predictions, especially on repeat stimuli, indicating superior generalization.
• For neuron 2, NPGLM captured the sharp 10 ms rise in the history filter better than GLM-ARD, despite the challenge of non-smooth behavior.
• The method successfully inferred filters consistent with prior GLM results while avoiding the pitfalls of basis function misselection, such as underfitting or overfitting.
• Even with only 15–25 inducing points, NPGLM produced robust posterior inference and stable predictions, demonstrating computational efficiency.
• Ancestral sampling from NPGLM generated spike trains that closely matched empirical data in raster plots, inter-spike interval distributions, and average firing rates.