Direct Estimation of Inhomogeneous Markov Interval Models of Spike Trains

Abstract

A necessary ingredient for a quantitative theory of neural coding is appropriate "spike kinematics": a precise description of spike trains. While summarizing experiments by complete spike time collections is clearly inefficient and probably unnecessary, the most common probabilistic model used in neurophysiology, the inhomogeneous Poisson process, often seems too crude. Recently a more general model, the inhomogeneous Markov interval model (Berry & Meister, 1998 ; Kass & Ventura, 2001 ), was considered, which takes into account both the current experimental time and the time from the last spike. Several techniques were proposed to estimate the parameters of these models from data. Here we propose a direct method of estimation that is easy to implement, fast, and conceptually simple. The method is illustrated with an analysis of sample data from the cat's superior colliculus.

Applying the Multivariate Time-Rescaling Theorem to Neural Population Models

“…Inhomogeneous Poisson processes (Olson et al, 2000; Kass & Ventura, 2001) as well as stationary renewal processes for the interspike interval distributions (typically involving gamma (Brown et al, 2003; Maimon & Assad, 2009), log-normal (Barbieri et al, 2001) or inverse Gaussian distributions (Iyengar & Liao, 1997)) can be described using the conditional intensity. Moreover, inhomogeneous Markov interval processes (IMI) are a sub-class of conditional intensity models (Johnson, 1996; Kass & Ventura, 2001; Brown et al, 2002; Muller et al, 2007; Wojcik et al, 2009). The conditional intensity can also be non-parametrically estimated (Berry & Meister, 1998; Jacobs et al, 2009).…”

“…In the case of deterministic models, their predictions can be compared to the observed data set using spike train metrics (e. g. the G coincidence factor (Kistler et al, 1997; Jolivet et al, 2008)). In the case of stochastic models evaluations have been made using receiver-operating characteristics (Truccolo et al, 2010) and, more extensively, the (univariate) time-rescaling theorem (see e. g. Ogata (1988); Barbieri et al (2001); Smith & Brown (2003); Brown et al (2003); Rigat et al (2006); Koyama & Kass (2008); Wojcik et al (2009); Shimokawa & Shinomoto (2009)). It should be noted that both the classical and the multivariate extension of the time-rescaling theorem can be used as a relative measure by ranking models according to the absolute value of KS statistics.…”

“…In practice, one generally hypothesizes a parametric form (which may be a function of stimuli, the neurons’ own spiking history and so on) and fits it to the recorded spikes, generally using maximum-likelihood estimation3. This approach has become standard for simultaneously quantifying the influence of multiple external covariates (such as stimuli) and also the neuron’s own spike history upon the output spikes (Brillinger, 1988; Chornoboy et al, 1988; Kass & Ventura, 2001; Brown et al, 2003; Paninski, 2003; Truccolo et al, 2005; Okatan et al, 2005; Kass et al, 2005; Rigat et al, 2006; Pillow, 2007; Stevenson et al, 2008; Pillow et al, 2008; Czanner et al, 2008; Wojcik et al, 2009). …”

Skellam process with resetting: a neural spike train model

“…However, biological phenomena such as neural refractoriness—the period after each spike during which a second spike cannot be initiated—and burst spiking activity—a sudden and short‐term high frequency spike generation—violate the basic assumptions of Poisson process. Other approaches include integrate‐and‐fire models , generalized linear model based approaches , filtering and general renewal processes . Although the likelihood method is commonly used with each of these approaches, , Bayesian methods have also been widely employed .…”

Rescaling, thinning or complementing? On goodness-of-fit procedures for point process models and Generalized Linear Models

“…Specifically, Generalized Linear Models (GLMs) are built on this representation. Such discretized models of time series have mostly been seen as an approximation to continuous point processes and hence, the time-rescaling theorem was also applied to such models [4,5,6,7,8].…”