Construction and analysis of non-Poisson stimulus-response models of neural spiking activity
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.
References (43)
- et al.
A statistical analysis of spontaneous activity of central single neurons
Physiol. Behav.
(1968) - et al.
The hippocampus as a spatial map: preliminary evidence from unit activity in the freely-moving rat
Brain Res.
(1971) - et al.
Spike discharge patterns of spontaneous and continuously stimulated activity in the cochlear nucleus of anesthetized cats
Biophys. J.
(1965) - et al.
Some quantitative methods for the study of spontaneous activity of single neurons
Biophys. J.
(1962) - et al.
Construction and analysis of the inhomogeneous general inverse Gaussian probability model of place cell spiking activity
Soc. Neurosci. Abst.
(2000) Inhomogeneous and modulated gamma processes
Biometrika
(1981)Comment on ‘likelihood analysis of point processes and its applications to seismological data’ by Y Ogata
Bull. Int. Stat. Inst.
(1983)- et al.
Reading a neural code
Science
(1991) - et al.
Statistical analysis of the dark discharge of lateral geniculate neurones
J. Physiol.
(1964) - et al.
Time Series: Forecasting and Control
(1994)
Maximum likelihood analysis of spike trains of interacting nerve cells
Biol. Cyber.
A time-dependent gamma distribution model of spike train activity in hippocampal place cells
Soc. Neurosci. Abstr.
A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells
J. Neurosci.
Pattern and inhibition-dependent invasion of pyramidal cell dendrites by fast spikes in the hippocampus in vivo
Proc. Natl. Acad. Sci.
Statistical Inference
The Inverse Gaussian Distribution: Theory, Methodology and Applications
A point process analysis of the spontaneous activity of anterior semicircular canal units in the anesthetized pigeon
Biol. Cyber.
An Introduction to the Theory of Point Process
Place cell discharge is extremely variable during individual passes of the rat through the firing field
Proc. Natl. Acad. Sci. USA
Principles of spike train analysis
Modeling neural activity using the generalized inverse Gaussian distribution
Biol. Cyber.
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