# Estimating within-session changes in firing rate

In the lab meeting on March 23, I presented the following paper:

Learning In Spike Trains: Estimating Within-Session Changes In Firing Rate Using Weighted Interpolation
Greg Jensen, Fabian Munoz, Vincent P Ferrera
bioRxiv (2016).

This paper seeks to develop a method that can track non-stationary dynamics of firing rates. Specifically, it deals with two problems:
1. How to estimate the firing rate accurately from single (or a small number of ) trials?
2. Given a (possibly inhomogeneous) session of consecutive trials, how to subdivide the session into ensembles of trials of similar features?

1. Single-trial rate estimation “ARRIS”

Their method, named ARRIS (Adaptive Rate Regression Involving Steps), uses reversible jump MCMC (RJMCMC) to detect step-like transitions of firing rates.

The RJMCMC (Green 1995) method is a variant of MCMC algorithm which dynamically adjusts the number of parameters (dimensionality of posterior), by adding/removing them as parts of its simulations. RJMCMC was used in a previous method called BARS (Bayesian Adaptive Regression Splines, DiMatteo et al 2001), where they dynamically add/remove knots (spline points).

In this paper’s ARRIS method, they use RJMCMC to determine step functions. The authors emphasize that unlike the previous rate estimation methods which almost always shows attenuation at the transition boundary (and more strongly with smaller samples), the ARRIS method captures sharp transitions successfully even with small samples. The method also works when the transition is smooth, because the MCMC generates an ensemble of steps.

2. Weighted interpolation

On the other hand, the firing rate dynamics varies in a non-random way in many natural applications. It’s nice that one could estimate the firing rates from single trials, but one should also be able to benefit from looking at the next trial in time, which presumably has a similar firing pattern. Based on this idea, the authors consider a weighted ensemble of trials centered on the time of interest, using a tri-cube weight function with a single parameter (the bandwidth). Afterwards, ARRIS could be applied to the weighted ensemble of firing data. Now the problem is: What is the optimal bandwidth of the weight function? In other words, about how many trials should one average over?

Optimal bandwidth determination:
The authors use generalized cross-validation (GCV) to find the optimal bandwidth across trials. Importantly, the across-trial bandwidth is optimized by GCV at each trial $r$ separately. The resulting bandwidth $h(r)$ is itself a good representation of data variability.
More specifically, in order to find the optimal bandwidth $h(r)$ at a given trial $r$, one needs to iterate the following. Specify a fixed bandwidth $h$; Construct a weighted ensemble of spikes with across-trial width $h$ and centered at trial $r$; Calculate a generalized cross-validation score $\textrm{GCV}(h,r)$ by running ARRIS on this weighted ensemble while leaving one time-point out at a time.

Rapid evaluation of the bandwidth:
However, there is a practical difficulty in executing the optimization procedure described above. The most obvious reason is that MCMC is already computationally intensive, and that iterating over many candidate values of $h$ may be too expensive. But more fundamentally, the single-trial estimate of the firing rate would fluctuate, because it depends on a stochastic algorithm (RJMCMC). It means the optimization of $h(r)$ will not be robust, because the GCV score will turn out differently every time it is calculated.

To get around this problem the authors suggest using a kernel density approximation, where the ARRIS estimate is replaced by a box kernel with a fixed bandwidth (*note* this bandwidth is within-trial, not across-trial). This within-trial bandwidth is determined by a simple estimator that depends on spike interval median (rather than being optimized). The approximation is only used to evaluate the optimal bandwidth, then ARRIS can be used to obtain the final estimate at the optimal bandwidth.

Overall, this is a nice paper that provides a readily applicable method for learning non-stationary firing dynamics from small samples. It has two contributions: (1) the step-function-based ARRIS method for single-trial firing rate estimate, and (2) the weighted interpolation framework that involves the determination the optimal bandwidth $h(r)$ which captures data variability.

• For better representation, the authors suggest that the optimal bandwidth $h(r)$ can be smoothed once again with a “smoothing bandwidth $h_{s}$” (which is optimized by another round of GCV).
• There are three different uses of the term “bandwidth” in the paper, which may be confusing: the across-trial bandwidth $h(r)$ for the weighted ensemble construction which is at the core of the paper, the smoothing bandwidth that goes on top of $h(r)$, and finally the within-trial bandwidth that is introduced for the kernel density approximation.