A Tractable Approximation to Optimal Point Process Filtering: Application to Neural Encoding
The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitate the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.
This paper describes a tractable method for performing dynamic state estimation from point process observations. The authors derive closed-form expressions for the mean and variance of a Gaussin approximation to the posterior (using assumed density filtering), characterize the optimal placement of tuning curves, and emphasizes the importance of information carried by "silences" to the posterior.The paper is well-written, original, highly novel, and provides a very beautiful extension of previous work on optimal tuning curves for static problems. It is likely to have a substantial impact on the field and should certainly be accepted.Comments:1. The abstract and intro seem slightly misleading in implying that the solution is fully general, when the details presented here seem to require Gaussian tuning curves.(Perhaps I missed it, but do the authors have a solution for dealing with non-Gaussian tuning curves, or a posterior density that is not Gaussian?Even tuning curves given by a Gaussian plus a constant DC baseline rate would seem to create a big problem because one would no longer obtain a closed-form update to the posterior from the spike-related likelihood terms).I still think the work is extremely nice, but a little bit more of a disclaimer about what cases are considered and/or limits on the form of the code might be helpful here. (Alternatively, if they have an idea about how to handle non-zero baseline spike rate, it would helpful to hear it.)2. I'm a little bit confused about one aspec of the Gaussian approximation to the non-spiking terms in the log-likelihood.If we go a long time without seeing any spikes, the true posterior becomes bimodal due to the -f(\theta) term -- intuitively, the probability that the state is far away on one tail or the other becomes increasingly high, the longer we go without spikes.What happens to the approximation in this case - does it simply tend towards uniform?I assume this is an extreme case that the authors regard as relatively unlikely for a reasonable sized population with reasonable marginal spike rate, but would be nice to say a word or two about when the approximation will break down.3.Two refs the authors might consider citing:- Huys, Zemel, Natarajan, & Dayan, NC 2007 - relatively early paper considering the problem of decoding a dynamic posterior distribution from spike trains.- Grabska-Barwinska & Pillow, NIPS 2014 - considered optimal tuning curve problem for uniform case but for different error metrics & constraints. Minor:Fig 2 caption:"dashed green line" - doesn't appear to be green.