Construction and analysis of non-Poisson stimulus-response models of neural spiking activity

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Abstract

A paradigm for constructing and analyzing non-Poisson stimulus-response models of neural spike train activity is presented. Inhomogeneous gamma (IG) and inverse Gaussian (IIG) probability models are constructed by generalizing the derivation of the inhomogeneous Poisson (IP) model from the exponential probability density. The resultant spike train models have Markov dependence. Quantile–quantile (Q–Q) plots and Kolmogorov–Smirnov (K–S) plots are developed based on the rate-rescaling theorem to assess model goodness-of-fit. The analysis also expresses the spike rate function of the neuron directly in terms of its interspike interval (ISI) distribution. The methods are illustrated with an analysis of 34 spike trains from rat CA1 hippocampal pyramidal neurons recorded while the animal executed a behavioral task. The stimulus in these experiments is the animal's position in its environment and the response is the neural spiking activity. For all 34 pyramidal cells, the IG and IIG models gave better fits to the spike trains than the IP. The IG model more accurately described the frequency of longer ISIs, whereas the IIG model gave the best description of the burst frequency, i.e. ISIs≤20 ms. The findings suggest that bursts are a significant component of place cell spiking activity even when position and the background variable, theta phase, are taken into account. Unlike the Poisson model, the spatial and temporal rate maps of the IG and IIG models depend directly on the spiking history of the neurons. These rate maps are more physiologically plausible since the interaction between space and time determines local spiking propensity. While this statistical paradigm is being developed to study information encoding by rat hippocampal neurons, the framework should be applicable to stimulus-response experiments performed in other neural systems.

Introduction

Accurate statistical descriptions of neural spiking activity are crucial for understanding neural information encoding. Simple and inhomogeneous Poisson (IP) processes are the statistical models most frequently used in simulation studies and experimental analyses of neural spike trains (Tuckwell, 1988, Rieke et al., 1997, Gabbiani and Koch, 1998, Reich et al., 1998). Although the Poisson models have provided theoretical predictions and statistical methods for experimental data analysis, spiking activity in many neural systems cannot be completely described by these processes (Gabbiani and Koch, 1998, Shadlen and Newsome, 1998). Statistical analyses of several different neural systems under approximately stationary experimental conditions have described interspike interval (ISI) distributions as unimodal, right skewed probability densities such as the gamma, inverse Gaussian and lognormal (Pfeiffer and Kiang, 1965, Tuckwell, 1988, Iyengar and Liao, 1997, Gabbiani and Koch, 1998). Analytic and simulation studies of elementary stochastic biophysical models of single neurons are consistent with these empirical findings (Tuckwell, 1988, Iyengar and Liao, 1997, Gabbiani and Koch, 1998). Because the exponential probability density is the probability model associated with a simple Poisson counting process, the counting processes associated with these elementary models cannot be Poisson. These elementary ISI models based on unimodal right-skewed probability densities are renewal models, and as such, do not allow for dependence in the spiking pattern of the neurons.

Stimulus-response experiments are widely used neuroscience protocols that require statistical analyses for correct interpretation of experimental data. In these experiments a natural or artificial stimulus is given and the response, the spiking activity of one or a set of neurons, is measured. Examples include motion stimulation of the fly H1 neuron (Bialek et al., 1991), natural sound stimulation of the bullfrog eight nerve (Rieke et al., 1995), position stimulation of of pyramidal (place) cells in the rat hippocampus (O'Keefe and Dostrovsky, 1971), and wind stimulation of the cricket (Miller et al., 1991) and cockroach (Rinberg and Davidowitz, 2000) cercal systems. In addition to the applied stimulus, neural spiking activity can also be modified by background variables not associated with the stimulus, such as the theta rhythm in the case of the hippocampal place cells (O'Keefe and Recce, 1993, Skaggs et al., 1996). A strong appeal of using the IP model to analyze stimulus-response data is that the effects of stimuli and background variables on spiking activity can be simply modeled by making the Poisson rate an explicit function of these variables (Brown et al., 1998b, Zhang et al., 1998). Despite this analytic convenience, goodness-of-fit assessments have shown that the IP model does not completely describe the stochastic structure of spike trains from stimulus-response experiments (Van Steveninck et al., 1997, Brown et al., 1998b, Fenton and Muller, 1998, Reich et al., 1998).

More accurate statistical description of neural spike train activity should result if models that allow dependence in the spike train and include the effects of stimuli and background variables on the spiking activity are considered. Development of these new models also requires appropriate goodness-of-fit procedures to assess model agreement with experimental data.

Here, inhomogeneous neural spike train models are derived by combining a renewal model with a one-to-one transformation function to relate the variable of the renewal probability density to the spike times and the stimulus. The transformation used will generate inhomogeneous models in which the spike trains have Markov dependence. This construction is used to build inhomogeneous gamma (IG) and inverse Gaussian (IIG) probability models and present goodness-of-fit methods using Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), quantile–quantile (Q–Q) plots, and Kolmogorov–Smirnov (K–S) plots, to evaluate model agreement with experimental data. The models and the diagnostic methods are applied to the analysis of spike trains from place cells in the CA1 region of the rat hippocampus collected simultaneously with path data from rats foraging in an open circular environment. In these experiments the stimulus is the animal's position in the environment and the response is the neural spiking activity.

Section snippets

Inhomogeneous interspike interval probability models

The objective is to model the ISI distribution of a neuron as a function of spiking history, stimulus inputs and background variables. This is done by combining a probability model that defines the stochastic structure of the neural spike train with a one-to-one transformation that relates the spike times, the stimulus, and covariates to the random variable of the probability density. Because the construction uses the intensity-rescaling transformation, the history dependence in the spike train

Goodness-of-fit

An essential component of the statistical analysis paradigm is the assessment of goodness-of-fit, i.e. the evaluation of how well the spike train data are described by a given inhomogeneous probability model. Four goodness-of-fit procedures are considered. Two of these are the well-known model comparison statistics, AIC (Box et al., 1994) and BIC (Shwartz, 1978). The other two are the Q–Q plots and K–S plots, derived specifically for point process data analysis.

AIC and BIC give single number

Data analysis: hippocampal place cells

To illustrate the model construction and goodness-of-fit methods developed in Section 3, in this section the IP, IG and IIG models using a Gaussian spatio-temporal intensity function are presented. The model fits are used in two applications: an analysis of burst activity and a study of temporal and spatial rate maps.

Discussion

Our paradigm provides a flexible framework for constructing and analyzing non-Poisson statistical models of neural spike train activity collected in stimulus-response experiments. The paradigm consists of maximum likelihood fitting an inhomogeneous model and measuring of its goodness-of-fit using AIC, BIC, Q–Q plots and K–S plots. The framework is illustrated in an analysis of hippocampal place cells.

The inhomogeneous stimulus-response models are constructed by generalizing the way in which the

Acknowledgements

We thank the anonymous referees for suggestions that helped improve the accuracy and presentation in the manuscript. Support was provided in part by National Institutes of Mental Health grants MH59733 and MH61637, National Science Foundation grant IBN-0081548, DARPA, the Office of Naval Research, and the Center for Learning and Memory at MIT.

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