Monte Carlo methods for localization of cones given multielectrode retinal ganglion cell recordings

Abstract

It has recently become possible to identify cone photoreceptors in primate retina from multi-electrode recordings of ganglion cell spiking driven by visual stimuli of sufficiently high spatial resolution. In this paper we present a statistical approach to the problem of identifying the number, locations, and color types of the cones observed in this type of experiment. We develop an adaptive Markov Chain Monte Carlo (MCMC) method that explores the space of cone configurations, using a Linear-Nonlinear-Poisson (LNP) encoding model of ganglion cell spiking output, while analytically integrating out the functional weights between cones and ganglion cells. This method provides information about our posterior certainty about the inferred cone properties, and additionally leads to improvements in both the speed and quality of the inferred cone maps, compared to earlier “greedy” computational approaches.

Fast inference in generalized linear models via expected log-likelihoods

“…We invoke the law of large numbers to approximate the sum over the non-linearity in Eq. (2) by its expectation (Paninski 2004; Field et al 2010; Park and Pillow 2011; Sadeghi et al 2013): $$L\left(\theta \right)=\sum _{n=1}^{N}true(true(x_n^T\theta true)r_n-Gtrue(x_n^T\theta true)true)+\mathrm{const}\left(\theta \right)$$$$\approx true(\sum _{n=1}^N{x}_n^T{r}_{n}true)\theta -N\mathbf{E}true[G\left(x^T\theta \right)true]\equiv \tilde{L}\left(\theta \right),$$ where the expectation is with respect to the distribution of x . The EL trades in the O(KNp) cost of computing the nonlinear sum for the cost of computing E [ G(x T θ)] at K different values of θ , resulting in order O(Kz) cost, where z denotes the cost of computing the expectation E [ G(x T θ)] .…”

“…We can still compute E [ G(xT θ)] approximately in most cases with an appeal to the central limit theorem (Sadeghi et al 2013): we approximate q= x T θ in Eq. (13) as Gaussian, with mean E [ θ T x] = θ T [x] = 0 and variance var (θ T x) θ T Cθ .…”

“…The MELE can therefore be orders of magnitude faster than the MLE here if Np is large, particularly if C has some structure that can be exploited. See Park and Pillow (2011), Sadeghi et al (2013) for further discussion.…”

“…We have already discussed previous work in neuroscience (Paninski 2004; Park and Pillow 2011; Field et al 2010; Sadeghi et al 2013) that exploits the EL approximation in the LNP model. Similar approximations have also been used to simplify the likelihood of GLM models in the context of neural decoding (Rahnama Rad and Paninski 2011).…”

“…The resulting EL approximation can often be computed significantly more quickly than the exact log-likelihood. This approximation has been exploited previously in some special cases (e.g., Gaussian process regression (Sollich and Williams 2005; Rasmussen and Williams 2005) and maximum likelihood estimation of a Poisson regression model (Paninski 2004; Field et al 2010; Park and Pillow 2011; Sadeghi et al 2013)). We generalize the basic idea behind the EL from the specific models where it has been applied previously to all GLMs in canonical form and discuss the associated computational savings.…”