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A new look at state-space models for neural data

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Abstract

State space methods have proven indispensable in neural data analysis. However, common methods for performing inference in state-space models with non-Gaussian observations rely on certain approximations which are not always accurate. Here we review direct optimization methods that avoid these approximations, but that nonetheless retain the computational efficiency of the approximate methods. We discuss a variety of examples, applying these direct optimization techniques to problems in spike train smoothing, stimulus decoding, parameter estimation, and inference of synaptic properties. Along the way, we point out connections to some related standard statistical methods, including spline smoothing and isotonic regression. Finally, we note that the computational methods reviewed here do not in fact depend on the state-space setting at all; instead, the key property we are exploiting involves the bandedness of certain matrices. We close by discussing some applications of this more general point of view, including Markov chain Monte Carlo methods for neural decoding and efficient estimation of spatially-varying firing rates.

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Notes

  1. It is worth noting that other more sophisticated methods such as expectation propagation (Minka 2001; Ypma and Heskes 2003; Yu et al. 2006, 2007; Koyama and Paninski 2009) may be better-equipped to handle these strongly non-Gaussian observation densities p(y t |q t ) (and are, in turn, closely related to the optimization-based methods that are the focus of this paper); however, due to space constraints, we will not discuss these methods at length here.

  2. In practice, the simple Newton iteration does not always increase the objective logp(Q|Y); we have found the standard remedy for this instability (perform a simple backtracking linesearch along the Newton direction \(\hat Q^{(i)} - \delta^{(i)} H ^{-1} \nabla\) to determine a suitable stepsize δ (i) ≤ 1) to be quite effective here.

  3. More precisely, Cunningham et al. (2008) introduce a clever iterative conjugate-gradient (CG) method to compute the MAP path in their model; this method requires O(T logT) time per CG step, with the number of CG steps increasing as a function of the number of observed spikes. (Note, in comparison, that the computation times of the state-space methods reviewed in the current work are insensitive to the number of observed spikes.)

  4. The approach here can be easily generalized to the case that the input noise has a nonzero correlation timescale. For example, if the noise can be modeled as an autoregressive process of order p instead of the white noise process described here, then we simply include the unobserved p-dimensional Markov noise process in our state variable (i.e., our Markov state variable q t will now have dimension p + 2 instead of 2), and then apply the O(T) log-barrier method to this augmented state space.

  5. In some cases, Q may be observed directly on some subset of training data. If this is the case (i.e., direct observations of q t are available together with the observed data Y), then the estimation problem simplifies drastically, since we can often fit the models p(y t |q t ,θ) and p(q t |q t − 1, θ) directly without making use of the more involved latent-variable methods discussed in this section.

  6. It is worth mentioning the work of Cunningham et al. (2008) again here; these authors introduced conjugate gradient methods for optimizing the marginal likelihood in their model. However, their methods require computation time scaling superlinearly with the number of observed spikes (and therefore superlinearly with T, assuming that the number of observed spikes is roughly proportional to T).

  7. An additional technical advantage of the direct optimization approach is worth noting here: to compute the E step via the Kalman filter, we need to specify some initial condition for p(V(0)). When we have no good information about the initial V(0), we can use “diffuse” initial conditions, and set the initial covariance Cov(V(0)) to be large (even infinite) in some or all directions in the n-dimensional V(0)-space. A crude way of handling this is to simply set the initial covariance in these directions to be very large (instead of infinite), though this can lead to numerical instability. A more rigorous approach is to take limits of the update equations as the uncertainty becomes large, and keep separate track of the infinite and non-infinite terms appropriately; see Durbin and Koopman (2001) for details. At any rate, these technical difficulties are avoided in the direct optimization approach, which can handle infinite prior covariance easily (this just corresponds to a zero term in the Hessian of the log-posterior).

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Acknowledgements

We thank J. Pillow for sharing the data used in Figs. 2 and 5, G. Czanner for sharing the data used in Fig. 6, and B. Babadi and Q. Huys for many helpful discussions. LP is supported by NIH grant R01 EY018003, an NSF CAREER award, and a McKnight Scholar award; YA by a Patterson Trust Postdoctoral Fellowship; DGF by the Gulbenkian PhD Program in Computational Biology, Fundacao para a Ciencia e Tecnologia PhD Grant ref. SFRH / BD / 33202 / 2007; SK by NIH grants R01 MH064537, R01 EB005847 and R01 NS050256; JV by NIDCD DC00109.

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

    Liam Paninski, Yashar Ahmadian, Daniel Gil Ferreira, Kamiar Rahnama Rad & Michael Vidne

  2. Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA

    Shinsuke Koyama

  3. Department of Neuroscience, Johns Hopkins University, Baltimore, MD, USA

    Joshua Vogelstein

  4. Department of Statistics, Florida State University, Tallahassee, FL, USA

    Wei Wu

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  1. Liam Paninski
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Correspondence to Liam Paninski.

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Action Editor: Israel Nelken

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Paninski, L., Ahmadian, Y., Ferreira, D.G. et al. A new look at state-space models for neural data. J Comput Neurosci 29, 107–126 (2010). https://doi.org/10.1007/s10827-009-0179-x

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  • DOI: https://doi.org/10.1007/s10827-009-0179-x

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