Statistical Models of Spike Trains

One approach for understanding the encoding of information by spike trains is to fit statistical models and then test their goodness of fit. The time-rescaling theorem provides a goodness-of-fit test consistent with the point process nature of spike trains. The interspike intervals (ISIs) are rescaled (as a function of the model's spike probability) to be independent and exponentially distributed if the model is accurate. A Kolmogorov-Smirnov (KS) test between the rescaled ISIs and the exponential distribution is then used to check goodness of fit. This rescaling relies on assumptions of continuously defined time and instantaneous events. However, spikes have finite width, and statistical models of spike trains almost always discretize time into bins. Here we demonstrate that finite temporal resolution of discrete time models prevents their rescaled ISIs from being exponentially distributed. Poor goodness of fit may be erroneously indicated even if the model is exactly correct. We present two adaptations of the time-rescaling theorem to discrete time models. In the first we propose that instead of assuming the rescaled times to be exponential, the reference distribution be estimated through direct simulation by the fitted model. In the second, we prove a discrete time version of the time-rescaling theorem that analytically corrects for the effects of finite resolution. This allows us to define a rescaled time that is exponentially distributed, even at arbitrary temporal discretizations. We demonstrate the efficacy of both techniques by fitting generalized linear models to both simulated spike trains and spike trains recorded experimentally in monkey V1 cortex. Both techniques give nearly identical results, reducing the false-positive rate of the KS test and greatly increasing the reliability of model evaluation based on the time-rescaling theorem.

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An Empirical Model for Reliable Spiking Activity

Neural Computation ◽

10.1162/neco_a_00754 ◽

2015 ◽

Vol 27 (8) ◽

pp. 1609-1623 ◽

Cited By ~ 1

Author(s):

Wanjie Wang ◽

Shreejoy J. Tripathy ◽

Krishnan Padmanabhan ◽

Nathaniel N. Urban ◽

Robert E. Kass

Keyword(s):

Neural Activity ◽

Statistical Models ◽

Generalized Linear Model ◽

Point Processes ◽

Transfer Functions ◽

Spike Trains ◽

Spike Generation ◽

Stimulus Features

Understanding a neuron’s transfer function, which relates a neuron’s inputs to its outputs, is essential for understanding the computational role of single neurons. Recently, statistical models, based on point processes and using generalized linear model (GLM) technology, have been widely applied to predict dynamic neuronal transfer functions. However, the standard version of these models fails to capture important features of neural activity, such as responses to stimuli that elicit highly reliable trial-to-trial spiking. Here, we consider a generalization of the usual GLM that incorporates nonlinearity by modeling reliable and nonreliable spikes as being generated by distinct stimulus features. We develop and apply these models to spike trains from olfactory bulb mitral cells recorded in vitro. We find that spike generation in these neurons is better modeled when reliable and unreliable spikes are considered separately and that this effect is most pronounced for neurons with a large number of both reliable and unreliable spikes.

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Applying the Multivariate Time-Rescaling Theorem to Neural Population Models

Neural Computation ◽

10.1162/neco_a_00126 ◽

2011 ◽

Vol 23 (6) ◽

pp. 1452-1483 ◽

Cited By ~ 9

Author(s):

Felipe Gerhard ◽

Robert Haslinger ◽

Gordon Pipa

Keyword(s):

Statistical Models ◽

Goodness Of Fit ◽

Real Data ◽

Spike Trains ◽

Population Models ◽

Neural Population ◽

Data Sets ◽

Goodness Of Fit Test ◽

Population Activity ◽

Neural Information

Statistical models of neural activity are integral to modern neuroscience. Recently interest has grown in modeling the spiking activity of populations of simultaneously recorded neurons to study the effects of correlations and functional connectivity on neural information processing. However, any statistical model must be validated by an appropriate goodness-of-fit test. Kolmogorov-Smirnov tests based on the time-rescaling theorem have proven to be useful for evaluating point-process-based statistical models of single-neuron spike trains. Here we discuss the extension of the time-rescaling theorem to the multivariate (neural population) case. We show that even in the presence of strong correlations between spike trains, models that neglect couplings between neurons can be erroneously passed by the univariate time-rescaling test. We present the multivariate version of the time-rescaling theorem and provide a practical step-by-step procedure for applying it to testing the sufficiency of neural population models. Using several simple analytically tractable models and more complex simulated and real data sets, we demonstrate that important features of the population activity can be detected only using the multivariate extension of the test.

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