Fast Kalman filtering on quasilinear dendritic trees

Abstract

Optimal filtering of noisy voltage signals on dendritic trees is a key problem in computational cellular neuroscience. However, the state variable in this problem -the vector of voltages at every compartment -is very high-dimensional: realistic multicompartmental models often have on the order of N = 10 4 compartments. Standard implementations of the Kalman filter require O(N 3 ) time and O(N 2 ) space, and are therefore impractical. Here we take advantage of three special features of the dendritic filtering problem to construct an efficient filter: (1) dendritic dynamics are governed by a cable equation on a tree, which may be solved using sparse matrix methods in O(N ) time; and current methods for observing dendritic voltage (2) provide low SNR observations and (3) only image a relatively small number of compartments at a time. The idea is to approximate the Kalman equations in terms of a low-rank perturbation of the steady-state (zero-SNR) solution, which may be obtained in O(N ) time using methods that exploit the sparse tree structure of dendritic dynamics. The resulting methods give a very good approximation to the exact Kalman solution, but only require O(N ) time and space. We illustrate the method with applications to real and simulated dendritic branching structures, and describe how to extend the techniques to incorporate spatially subsampled, temporally filtered, and nonlinearly transformed observations.

Optimal experimental design for sampling voltage on dendritic trees in the low-SNR regime

“…(1) corresponds to a row of B t , along with the corresponding elements of the covariance matrix W t . Note that we will only consider temporally uncorrelated noise here, though generalizations are possible (Paninski, 2010). We define A using a backward Euler (implicit) discretization for the cable equation, as discussed in (Paninski, 2010): in the noiseless case,…”

“…In addition, competitions such as the Digital Reconstruction of Axonal and Dendritic Morphology (DIADEM) Challenge 1 continue to encourage research in this area. Now that the model has been specified, we may derive an efficient smoothed backward Kalman recursion using methods similar to those employed by (Paninski, 2010) for the forward Kalman recursion. We assume that the forward mean µ f t = E(V t |Y 1:t ) and covariance C f t = Cov(V t |Y 1:t ) (where Y 1:t denotes all of the observed data {ys} up to time t) have already been computed by such methods.…”

“…See (Paninski, 2010) for details on calculating C f t and efficiently multiplying by C 0 and C −1 0 without having to represent these large matrices explicitly. In short, we can take advantage of the fact that the matrix K that implements the backward Euler propagation is symmetric and tree-tridiagonal (also known as "Hines" form (Hines, 1984)): all offdiagonal elements Kxw are zero unless the compartments x and w are nearest neighbors on the dendritic tree.…”

“…For dendritic trees which have on the order of N ∼ 10 4 compartments, the standard implementations of the Kalman filter-smoother are impractical because they require O(N 3 ) time and O(N 2 ) space. We therefore extend the results of (Paninski, 2010) to approximately calculate the filter-smoother estimator covariance matrix as a low rank perturbation to the steady-state (zero-SNR) solution in O(N ) time and space. Using these computed covariance matrices we can easily calculate the expected mean-squared error, the mutual information, and other design metrics.…”

“…Statistical techniques offer a path to partially offset these difficulties by providing methods to de-noise the data and infer the voltage in unobserved compartments while also maximizing the amount of information that can be extracted from the data that can be gathered with currently available methods. As discussed in previous work (Huys and Paninski, 2009;Paninski, 2010), state-space filtering methods such as the Kalman filter are an attractive choice for addressing these concerns because they allow us to explicitly incorporate the known biophysics of dendritic dynamics, along with the known noise properties of the measurement device.…”