Global Sensitivity Analysis of High-Dimensional Neuroscience Models: An Example of Neurovascular Coupling

Abstract

The complexity and size of state-of-the-art cell models have significantly increased in part due to the requirement that these models possess complex cellular functions which are thought—but not necessarily proven—to be important. Modern cell models often involve hundreds of parameters; the values of these parameters come, more often than not, from animal experiments whose relationship to the human physiology is weak with very little information on the errors in these measurements. The concomitant uncertainties in parameter values result in uncertainties in the model outputs or quantities of interest (QoIs). Global sensitivity analysis (GSA) aims at apportioning to individual parameters (or sets of parameters) their relative contribution to output uncertainty thereby introducing a measure of influence or importance of said parameters. New GSA approaches are required to deal with increased model size and complexity; a three-stage methodology consisting of screening (dimension reduction), surrogate modeling, and computing Sobol’ indices, is presented. The methodology is used to analyze a physiologically validated numerical model of neurovascular coupling which possess 160 uncertain parameters. The sensitivity analysis investigates three quantities of interest, the average value of $\hbox {K}^{+}$ in the extracellular space, the average volumetric flow rate through the perfusing vessel, and the minimum value of the actin/myosin complex in the smooth muscle cell. GSA provides a measure of the influence of each parameter, for each of the three QoIs, giving insight into areas of possible physiological dysfunction and areas of further investigation.

Appendix

Determining the coefficients $c_{\varvec{\alpha }}$ in the Polynomial Chaos surrogate (8) is challenging. Ideally, one would solve the least-squares problem

$$\begin{aligned} \min \sum \limits _{k=1}^M \left( g(\varvec{\theta }^k)-\sum _{\varvec{\alpha }} c_{\varvec{\alpha }} \psi _{\varvec{\alpha }}(\hat{\varvec{\theta }}) \right) ^2 \end{aligned}$$

(18)

to determine the coefficients. This approach is not currently feasible for the problems considered in this article. If there are, for instance, 18 input variables ($\theta _{j_i}$’s), then a 3rd-degree polynomial has 1330 unknown coefficients and a 4th-degree polynomial has 7315 unknown coefficients. With less than 1000 sample points, as in our case, (18) will admit infinitely many solutions which interpolate the data but will yield poor approximations of the QoI. Rather, we seek an approximate solution of (18) for which most of the coefficients are exactly 0. This may be achieved by introducing a penalty term and solving

$$\begin{aligned} \min \sum \limits _{k=1}^M \left( g(\varvec{\theta }^k)-\sum _{\varvec{\alpha }} c_{\varvec{\alpha }} \psi _{\varvec{\alpha }}(\hat{\varvec{\theta }}) \right) ^2 + \lambda \sum _{\varvec{\alpha }} \vert c_{\varvec{\alpha }} \vert \end{aligned}$$

(19)

instead of (18). Adding the sum of absolute values of the coefficients encourages a sparse solution, i.e., one with many 0 coefficients. However, it comes at the cost of making the objective function non-differentiable and hence (19) requires a more sophisticated optimization approach in comparison with (18). A plurality of well-documented methods exist for solving (19). In this article, we use Least Angle Regression (LAR) (Efron et al. 2004) with its implementation in Marelli and Sudret (2014), and a maximum polynomial degree of 5.

Because the basis function of the Polynomial Chaos surrogate is orthogonal with respect to the PDF $p_{\hat{\varvec{\theta }}}$, the variance and conditional expectation in (11) may be computed analytically as a function of the coefficients. Hence, the total Sobol’ indices of the Polynomial Chaos surrogate are given in closed form as a function of the coefficients.