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Fast Kalman filtering on quasilinear dendritic trees

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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 = 104 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.

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Notes

  1. Kellems et al. (2009) recently introduced powerful model-reduction methods for decreasing the effective dimensionality of the state vector in large dendritic voltage simulations; these methods are distinct from the techniques we introduce in this paper, but there are a number of interesting connections between these methods which we will describe below.

  2. In some cases it is overly crude to assume that the observations y t are linearly and instantaneously related to the voltage V t ; we will discuss extensions to this basic setup in Section 4 below.

  3. We will assume throughout this paper that the structural anatomy of the dendritic tree (i.e., the neighborhood structure N(x)) is fully known.

  4. For simplicity, we focus exclusively on the forward Kalman filter recursion in this paper, to compute p(V t |Y 1:t ), but note that all of the methods discussed here can be applied to obtain the forward-backward (Kalman smoother) density p(V t |Y 1:T ), for t < T.

  5. It is well-known that the Woodbury formula can be numerically unstable when the observation covariance W is small (i.e., the high-SNR case). Our applications here concern the low-SNR case instead, and therefore we have not had trouble with these numerical instabilities. However, it should be straightforward to derive a low-rank square-root filter (Howard and Jebara 2005; Treebushny and Madsen 2005; Chandrasekar et al. 2008) to improve the numerical stability here, if needed.

  6. We have suppressed the input current i t (x) and noise ε t (x) in the following, for notational simplicity; inclusion of the input current term does not significantly complicate the derivations in this section, since the steady-state covariance C 0 does not depend on i t (x).

  7. We do not have a good theoretical explanation for this fact; this is a purely empirical observation. Indeed, we have found that if the noise scale D varies discontinuously along the tree, then S i tends to be much less sparse, sharply reducing the efficacy of this procedure.

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Acknowledgements

LP is supported by a McKnight Scholar award and an NSF CAREER award. I thank Q. Huys, K. Rahnama Rad, and J. Vogelstein for many helpful conversations.

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  1. Department of Statistics and Center for Theoretical Neuroscience, Columbia University, New York, NY, USA

    Liam Paninski

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  1. Liam Paninski
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Correspondence to Liam Paninski.

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Paninski, L. Fast Kalman filtering on quasilinear dendritic trees. J Comput Neurosci 28, 211–228 (2010). https://doi.org/10.1007/s10827-009-0200-4

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  • DOI: https://doi.org/10.1007/s10827-009-0200-4

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