Fast Kalman filtering on quasilinear dendritic trees
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1
Columbia University, United States
The problem of understanding dendritic computation remains a key open challenge in cellular and computational neuroscience. The major difficulty is in recording physiological signals (especially voltage) with sufficient spatiotemporal resolution on dendritic trees: multiple-electrode recordings from dendrites are quite technically challenging, and provide spatially-incomplete observations, while high-resolution imaging techniques provide more spatially-complete observations, but with significantly lower signal-to-noise. One avenue for extending the reach of these currently available methods is to develop statistical techniques for optimally combining, filtering, and deconvolving these noisy signals. State-space filtering methods are attractive here, since these methods allow us to quite transparently incorporate 1) realistic, spatially-complex multicompartmental models of dendritic dynamics and 2) time-varying, heterogeneous observations (e.g., spatially-scanned multiphoton imaging data) into our filtering equations. The problem is that the time-varying state vector in this problem — which includes, at least, 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 state-space filter methods (e.g., the Kalman filter) require O(N^3) time, and are therefore impractical for applications to large dendritic trees. However, we may take advantage of three special features of the dendritic filtering problem to construct efficient filtering methods. First, dendritic dynamics are governed by a cable equation on a tree, which may be solved using symmetric sparse matrix methods in O(N) time. Second, current methods for imaging dendritic voltage provide low SNR observations, as discussed above. Finally, in typical experiments we record only a few image observations (n < 100 or so coarse pixels) at a time. Taken together, these special features allow us to approximate the Kalman equations in terms of a low-rank perturbation of the steady-state (zero-SNR) solution, which in turn may be obtained in O(N) time using efficient matrix solving methods that exploit the sparse tree structure of the dynamics. The resulting methods provide a very good approximation to the exact Kalman solution, but only require O(N) time and space. In addition, a number of extensions of the basic method are possible: for example, we can incorporate spatially blurred or scanned observations; temporally filtered observations and inhomogenous noise sources on the tree; “quasi-active” resonant membrane dynamics; and even in some cases nonlinear observations of the membrane state. Simulation results using the resulting filter allow us to quantify exactly how much information we can expect to extract about dendritic dynamics from recordings at a given SNR.
Conference:
Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010.
Presentation Type:
Poster Presentation
Topic:
Poster session I
Citation:
Paninski
L
(2010). Fast Kalman filtering on quasilinear dendritic trees.
Front. Neurosci.
Conference Abstract:
Computational and Systems Neuroscience 2010.
doi: 10.3389/conf.fnins.2010.03.00009
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Received:
17 Feb 2010;
Published Online:
17 Feb 2010.
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Correspondence:
Liam Paninski, Columbia University, New York, United States, liam@gatsby.ucl.ac.uk