Article

Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking

Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA 02129, USA.
Neural Computation (Impact Factor: 1.69). 10/2010; 22(10):2477-506. DOI: 10.1162/NECO_a_00015
Source: PubMed

ABSTRACT 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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    • "In our previous publications on the MIMO signal model (Song et al., 2006, 2007, 2009a,b, 2011, 2013; Song and Berger, 2010), the model goodness-of-fit are validated with a Kolmogorov-Smirnov (KS) test based on the time-rescaling theorem (Brown et al., 2002; Haslinger et al., 2010). This KS test is a powerful tool that allows the firing probability intensity function predicted by the MIMO model to be directly validated with the actual output spike train, and the model goodness-of-fit to be quantified statistically with confidence bounds. "
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    • "Application of the timerescaling theorem to sampled data produces some artifacts within KS plots since the actual spike times could have occurred anywhere within the finite-sized time bins. These artifacts are addressed within nSTAT using the discrete time rescaling theorem (Haslinger et al., 2010). Independence of the rescaled spike times can be assessed by plotting u i s+1 vs. u i s (Truccolo et al., 2005). "
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    • "Prior work on modeling of spike trains (Barbieri et al., 2001; Brown et al., 2002; Truccolo et al., 2005; Haslinger et al., 2010) and LFPs (Banerjee et al., 2010) to decode task-relevant events has argued that goodness of fit of models to experimental recordings is needed for model validation. We have validated unified spike-field models in simulated data where the independent variables (amplitude scaling factors/latency lags) are known, as well as in experimental recordings. "
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