Fast Kalman filtering on quasilinear dendritic trees.

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.