Efficient spline regression for neural spiking data

Abstract

Point process generalized linear models (GLMs) provide a powerful tool for characterizing the coding properties of neural populations. Spline basis functions are often used in point process GLMs, when the relationship between the spiking and driving signals are nonlinear, but common choices for the structure of these spline bases often lead to loss of statistical power and numerical instability when the signals that influence spiking are bounded above or below. In particular, history dependent spike train models often suffer these issues at times immediately following a previous spike. This can make inferences related to refractoriness and bursting activity more challenging. Here, we propose a modified set of spline basis functions that assumes a flat derivative at the endpoints and show that this limits the uncertainty and numerical issues associated with cardinal splines. We illustrate the application of this modified basis to the problem of simultaneously estimating the place field and history dependent properties of a set of neurons from the CA1 region of rat hippocampus, and compare it with the other commonly used basis functions. We have made code available in MATLAB to implement spike train regression using these modified basis functions.

Integrating Statistical and Machine Learning Approaches to Identify Receptive Field Structure in Neural Populations

“…where N (t) is number of spikes in time interval (0, t] and H(t) is the past spiking history of the neuron or population up to time t. For small ∆, λ(t|H(t))∆ is approximately the probability of observing a single spike in the time interval (t, t + ∆] given the spiking history [15]. A point process neural coding model defines λ(t|H(t)) as a function of a set of covariates influencing spiking.…”

“…Where the g i (•) are a set of basis functions that act on the covariate vector ν(t), and p is the dimension of the model parameter vector θ. GLMs can flexibly capture nonlinear relationships between stochastic signals in a computationally efficient and robust way, and provide powerful tools for assessing goodness-of-fit, and model refinement [1], [11]- [13], [15]. Maximum Likelihood Parameter Estimation Once an encoding model is expressed as a point process GLM with a log link function as in Eq.…”

“…Maximum Likelihood Parameter Estimation Once an encoding model is expressed as a point process GLM with a log link function as in Eq. ( 2), the likelihood surface is guaranteed to be convex, ensuring that there exists a unique maximum [11], [15]. The maximum likelihood estimator for the model parameters can be computed using an iteratively re-weighted least squares (IRLS) procedure [11].…”

“…The maximum likelihood estimator for the model parameters can be computed using an iteratively re-weighted least squares (IRLS) procedure [11]. The IRLS procedure also computes the observed Fisher information matrix, I θ , which allows for the construction of confidence intervals and standard tests of significance of the influence of individual or multiple covariates [11], [12], [15], [40].…”

“…It has been shown that a point process models can be efficiently fit to neural spiking data using a generalized linear model (GLM) framework [11]. Point process GLMs allow researchers to identify significant influences systematically, they provide powerful analysis tools for assessing goodness-of-fit and model refinement, and their parameters are generally interpretable [1], [11]- [15]. However, statistical models are computationally limiting for large scale data analysis due to the fact that the model refinement process is typically performed individually for each neuron.…”