Network Reconstruction From High-Dimensional Ordinary Differential Equations

Abstract

We consider the task of learning a dynamical system from high-dimensional time-course data. For instance, we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points. We model the dynamical system nonparametrically as a system of additive ordinary differential equations. Most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations. This is known to be challenging and inefficient. We propose a novel approach that does not involve derivative estimation. We show that the proposed method can consistently recover the true network structure even in high dimensions, and we demonstrate empirical improvement over competing approaches. Supplementary materials for this article are available online.

Keywords: Additive model; Group lasso; High dimensionality; Ordinary differential equation; Variable selection consistency.

Figure 1

Performance of network recovery methods on the system of additive ODEs in (26), averaged over 400 simulations. The four curves represent SA-ODE (dashed, red line), NeRDS (dashed, gray line), and GRADE without (solid, red line) and with (solid, gray line) the additional smoothing penalty in (17a) used by NeRDS. Each point on the curves corresponds to average performance for a given sparsity tuning parameter λ_n in (14a) or (17a). The symbols indicate the sparsity tuning parameter λ_n selected using BIC (SA-ODE, red square, and GRADE, red circle and gray circle) or GCV (NeRDS, gray square).

Figure 2

Network recovery on the system of linear ODEs (27), averaged over 200 simulated datasets. The three curves represent GRADE (gray line), Hall and Ma (2014) (blue line), Brunel, Clairon, and d’Alché Buc (2014) (green line).

Figure 3

(a) The graph encoded by a pair of Lotka-Volterra equations as given in (29). Self-edges (solid, gray line) and nonself-edges (dashed, gray line) are shown. (b) Self-edge (solid, gray line) and nonself-edge (dashed, gray line) recovery of GRADE, averaged over 200 simulated datasets. (c) Minimum signals defined in (31), for self-edges, D⁽¹⁾(·) (solid, red line), and nonself-edges, D⁽²⁾(·) (dashed, red line).

Figure 4

Estimated functional connectivities among neuronal populations from the calcium imaging data described in Section 6.2. Each node is positioned near the center of the neuronal population it represents, with jitter added for ease of display. The three red edges are shared between the estimated networks at 1 Hz and 2 Hz; the two blue edges are shared between estimated networks at 2 Hz and 4 Hz; the single green edge is shared between the estimated networks at 1 Hz and 4 Hz. For reference, given two Erdös-Rènyi graphs consisting of 25 nodes and 25 edges, the probability of having three or more shared edges is 0.07, and the probability of having two or more shared edges is 0.26.