## Fast Kalman filtering on quasilinear dendritic trees

Journal of Computational Neuroscience 28: 211-28.

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: typical realistic
multicompartmental models 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.

Preprint | low_rank_speckle.mp4 (5MB)
| low_rank_horiz.mp4
(5MB) | sample
code | Liam
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