Likelihood Methods for Point Processes with Refractoriness

Abstract

Likelihood-based encoding models founded on point-processes have received significant attention in the literature because of their ability to significantly improve neural decoding in Brain-Machine Interface applications. We propose an approximation to the likelihood of a point-process model of neurons which holds under assumptions about the continuous time process that are physiologically reasonable for neural spike trains: the presence of a refractory period, the predictability of the conditional intensity function, and its integrability. These are properties which apply to a large class of pointprocesses arising in applications other than neuroscience. The proposed approach has several advantages over state-of-the-art conventional ones. In particular, one can use 1 standard fitting procedures for generalized linear models (GLMs) based on iterativelyreweighted least-squares (IRWLS) while improving the accuracy of the approximation to the likelihood and reducing bias. As a result, the proposed approach can use a larger bin size to achieve the same accuracy as conventional approaches would with a smaller bin size. This is particularly important when analyzing neural data with high mean and instantaneous firing rates. We demonstrate these claims on simulated and real neural spiking activity. By allowing a substantive increase in the required bin size, our algorithm has the potential of lowering the barrier to the use of point-process methods in an increasing number of neural engineering applications.

Fast maximum likelihood estimation using continuous-time neural point process models

“…Here n is the number of time-steps in the observation duration. In Citi et al (2014), computational efficiency is gained be replacing a Riemann approximation with an improved, but discrete, approximation to an integral appropriate for restricted but often realistic models of neural spiking activity.…”

On Quadrature Methods for Refractory Point Process Likelihoods

“…For point processes with refractory effects, in which the CIF jumps discontinuously to zero following a spike, (Citi, Ba, Brown & Barbieri, 2014) recently developed an improved such discretization. Here we show that an alternative approach, in which we apply standard quadrature methods directly to the original continuous time integral, leads to significant further improvements beyond those offered by the approach of (Citi et al, 2014), with minimal additional computational cost 1 .…”

“…To begin, it is useful to discuss both the standard discretization approach and also the more refined method of (Citi et al, 2014). Both of these approaches begin by discretizing the observed continuous-time spike train process {t i } into a binary sequence ∆N j , with a one in each bin (indexed by j) containing a spike time t i , and a zero otherwise.…”

“…Notice that the only difference between these two approaches is in the second term; as discussed in (Citi et al, 2014), the latter approximation is more accurate because it accounts for the loss of intensity due to refractoriness: roughly speaking, on average, spikes are in the center of bins, and if bins are small then the CIF over the second half should be zero. See (Citi et al, 2014) for full details; we use the abbreviation "DR," for discrete Riemann, for each of these approaches, with DR1 corresponding to the standard method and DR2 for the approach of (Citi et al, 2014).…”

“…See (Citi et al, 2014) for full details; we use the abbreviation "DR," for discrete Riemann, for each of these approaches, with DR1 corresponding to the standard method and DR2 for the approach of (Citi et al, 2014). Our approach avoids the discretization of the observed spike train into a binary sequence ∆N j , and instead works with the original continuous-time loglikelihood more directly.…”