Efficient computation of the maximum a posteriori path and parameter estimation in integrate-and-fire and more general state-space models

Abstract

A number of important data analysis problems in neuroscience can be solved using state-space models. In this article, we describe fast methods for computing the exact maximum a posteriori (MAP) path of the hidden state variable in these models, given spike train observations. If the state transition density is log-concave and the observation model satisfies certain standard assumptions, then the optimization problem is strictly concave and can be solved rapidly with Newton–Raphson methods, because the Hessian of the loglikelihood is block tridiagonal. We can further exploit this block-tridiagonal structure to develop efficient parameter estimation methods for these models. We describe applications of this approach to neural decoding problems, with a focus on the classic integrate-and-fire model as a key example.

Appendices

Appendix A: Point process filter and smoother

A simple version of the point process filter approximates the filtered distribution Eq. (26) to a Gaussian centered by its mode (Brown et al. 1998). Let x _i|i and V _i|i be the (approximate) mode and covariance matrix for the filtered distribution Eq. (26), and x _{i|i − 1} and V _{i|i − 1} be the mode and covariance matrix for the predictive distribution Eq. (27) at time i. Let l(x _i ) = log{ p(y _i |x _i ) p(x _i |y _{1:i − 1}) }. The filtered distribution is then approximated to a Gaussian whose mean and covariance are $x_{i|i} = \arg\max_{x_i}l(x_i)$ and $V_{i|i} = -[\nabla\nabla_{x_i} l(x_{i|i})]^{-1}$, respectively. When the state-transition density is linear Gaussian, $p(x_i|x_{i-1}) = \mathcal{N}(F_i x_{i-1}, Q_i)$, the predictive distribution Eq. (27) is also Gaussian, whose mean and covariance are computed as

$$ \begin{array}{rcl} x_{i|i-1} &=& F_i x_{i-1|i-1}, \\ \end{array} $$

(67)

$$ \begin{array}{rcl} V_{i|i-1} &=& F_i V_{i-1|i-1}F_i^T + Q_i. \end{array} $$

(68)

Since the filtered and predictive distributions are Gaussian, the smoothing distribution Eq. (28) is also Gaussian, which can be computed by the standard Kalman smoother (Smith and Brown 2003). Let x _i|N and V _i|N be the mean and covariance the smoothing distribution at time i. The recursive smoothing equation corresponding to Eq. (28) is given by

$$ \begin{array}{rcl} x_{i|N} &=& x_{i|i} + V_{i|i} F_i V_{i+1|i}^{-1}(x_{i+1|N}-x_{i+1|i}), \\ \end{array} $$

(69)

$$ \begin{array}{rcl} V_{i|N} &=& V_{i|i} + V_{i|i} F_i V_{i+1|i}^{-1}(V_{i+1|N}-V_{i+1|i}) V_{i|i}^{-1} F_i^T V_{i+1|i}.\\ \end{array} $$

(70)

There are now several versions of the point process filter depending on the choice of the mean and variance of the approximate filtered distribution. In Eden et al. (2004), the filtered distribution at each time step i is approximated to a Gaussian by expanding its logarithm in a Taylor series about x _{i|i − 1} up to the second-order term, which results in a simpler algorithm. Koyama et al. (unpublished manuscript) proposed a more sophisticated method by utilizing the fully exponential Laplace approximation (Tierney et al. 1989), which achieves second-order accuracy in approximating the posterior expectation.

For the leaky IF model with hard-threshold, the standard Taylor-series-based recursions (Brown et al. 1998) do not apply (due to the discontinuity of log p(y _i |x _i )), and therefore we have not included comparisons to the point-process smoother in Figs. 4–6. However, it is worth noting that in this case the filtered distribution Eq. (26) can be approximated recursively as a truncated Gaussian defined on ( − ∞ , x _th ], and hence the approximate mean and variance can be obtained analytically; we found that this moment-matching method behaves similarly to the EP method (this is unsurprising, since EP is also based on a moment-matching procedure; data not shown).

Appendix B: Gaussian quadrature in EP algorithm

The expectation of a function of x _i , f(x _i ), with respect to p(x _i |y _1:N ) in Eq. (38) is expressed as

$$ \begin{array}{rcl} E_i[f(x_i)] &=& \int\int f(x_i) p(x_{i-1},x_i|y_{1:N}) dx_{i-1}dx_i \\ &\propto& \int f(x_i)p(y_i|x_i)\beta_i(x_i) \\&& \times \bigg[ \int \alpha_{i-1}(x_{i-1})p(x_i|x_{i-1})dx_{i-1} \bigg]dx_i \\ &\equiv& \int f(x_i) p(y_i|x_i)\beta_i(x_i) g(x_i)dx_i, \end{array} $$

(71)

where

$$ g(x_i) = \int \alpha_{i-1}(x_{i-1})p(x_i|x_{i-1})dx_{i-1} $$

(72)

is Gaussian since α _i − 1(x _i − 1) and p(x _i |x _i − 1) are also Gaussian. By introducing the Laplace approximation, $p_L(x_i)\equiv \mathcal{N}(m, v) \approx p(y_i|x_i)\beta_i(x_i) g(x_i)$, as a proposal distribution, the expectation can be expressed as

$$ \begin{array}{rcl} E_i[f(x_i)] &\propto& \int p_L(x_i) \bigg[\frac{f(x_i) p(y_i|x_i)\beta_i(x_i) g(x_i)}{p_L(x_i)}\bigg]dx_i \\ &\equiv& \int p_L(x_i)F(x_i)dx_i. \end{array} $$

(73)

After a linear change of variable, $x_i=\sqrt{v}u+m$, we have the standard form of the Gauss-Hermite quadrature,

$$ \begin{array}{rcl} E_i[f(x_i)] &\propto& \int e^{-\frac{u^2}{2}} F(\sqrt{v}u+m) du \\ &\approx& \sum\limits_{l=1}^nw_l F(\sqrt{v}u_l+m), \end{array} $$

(74)

where the weights w _l and evaluation points u _l are chosen according to a quadrature rule. The advantages of this method is that it requires only an inner product once the weights and evaluation points are calculated. (These only have to be computed once.) The expectation of f(x _i − 1) with respect to p(x _i − 1|y _1:N ) in Eq. (39) can be computed in the same way.

For the leaky IF model with hard-threshold, the observation model is given by the step-function, and thus the integral in Eq. (73) becomes

$$ \int_{-\infty}^{\infty}p(y_i=0|x_i)\ldots dx_i = \int_{-\infty}^{x_{th}}\ldots dx_i. $$

(75)

As a result, the expectation Eq. (73) is reduced to the integral over a truncated Gaussian, which can be computed analytically.