Statistical models of spike trains
Book chapter in Stochastic Methods in Neuroscience,
Oxford University Press, ed. Laing, C. and Lord, G., 2008.
Spiking neurons make inviting targets for analytical methods based on
stochastic processes: spike trains carry information in their temporal
patterning, yet they are often highly irregular across time and across
experimental replications. The bulk of this volume is devoted to
mathematical and biophysical models useful in understanding
neurophysiological processes. In this chapter we consider
statistical models for analyzing spike train data.
Strictly speaking, what we would call a statistical model for spike
trains is simply a probabilistic description of the sequence of
spikes. But it is somewhat misleading to ignore the data-analytical
context of these models. In particular, we want to make use of these
probabilistic tools for the purpose of scientific inference.
The leap from simple descriptive uses of probability to inferential
applications is worth emphasizing for two reasons. First, this leap
was one of the great conceptual advances in science, taking roughly
two hundred years. It was not until the late 1700s that there emerged
any clear notion of inductive (or what we would now call statistical)
reasoning; it was not until the first half of the twentieth century
that modern methods began to be developed systematically; and it was
only in the second half of the twentieth century that these methods
became well understood in terms of both theory and practice. Second,
the focus on inference changes the way one goes about the modeling
process. It is this change in perspective we want to highlight here,
and we will do so by discussing one of the most important models in
neuroscience, the stochastic integrate-and-fire (IF) model for spike
trains.
The stochastic IF model has a long history (Gerstein and Mandelbrot,
1964; Stein, 1965; Knight, 1972; Burkitt, 2006): it is the simplest
dynamical model that captures the basic properties of neurons,
including the temporal integration of noisy subthreshold inputs, all-
or-none spiking, and refractoriness. Of course, the IF model is a
caricature of true neural dynamics (see, e.g., (Ermentrout and Kopell,
1986; Brunel and Latham, 2003; Izhikevich, 2007) for more elaborate
models) but, as demonstrated in this book and others (Ricciardi, 1977;
Tuckwell, 1989; Gerstner and Kistler, 2002), it has provided much
insight into the behavior of single neurons and neural populations.
In this chapter we explore some of the key statistical questions that
arise when we use this model to perform inference with real neuronal
spike train data. How can we efficiently fit the model to spike train
data? Once we have estimated the model parameters, what can the model
tell us about the encoding properties of the observed neuron? We also
briefly consider some more general approaches to statistical modeling
of spike train data.
We begin, in section 1, by discussing three distinct useful ways of
approaching the IF model, via the language of stochastic (diffusion)
processes, hidden Markov models, and point processes,
respectively. Each of these viewpoints comes equipped with its own
specialized analytical tools, and the power of the IF model is most
evident when all of these tools may be brought to bear
simultaneously. We discuss three applications of these methods in
section 2, and then close in 3 by indicating the scope of the general
point process framework of which the IF model is a part, and the
possibilities for solving some key outstanding data-analytic problems
in systems neuroscience.
Reprint | Related work
on generalized linear models | Liam Paninski's research page