Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

The role of sparsity in inverse problems for networks with nonlinear dynamics

Pages: 1291 – 1311



Victor J. Barranca (Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania, U.S.A.)

Gregor Kovačič (Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, New York, U.S.A.)

Douglas Zhou (School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)


Sparsity is a fundamental characteristic of numerous biological, social, and technological networks. Network connectivity frequently demonstrates sparsity on multiple spatial scales and network inputs may also possess sparse representations in appropriate domains. In this work, we address the role of sparsity for solving inverse problems in networks with nonlinear and time-evolving dynamics. In the context of pulse-coupled integrate-and-fire networks, we demonstrate that nonlinear network dynamics imparts a compressive coding of both network connectivity and inputs provided they possess a sparse structure. This work thereby formulates an efficient sparsity-based framework for solving several classes of inverse problems in nonlinear network dynamics. Driving the network with a small ensemble of random inputs, we derive a mean-field set of underdetermined linear systems relating the network inputs to the corresponding activity of the nodes via the feed-forward connectivity matrix. In reconstructing the network connections, we utilize compressive sensing theory, which facilitates the recovery of sparse solutions to such underdetermined linear systems. Using the reconstructed connectivity matrix, we are capable of accurately recovering detailed network inputs, which may vary in time, distinct from the random input ensemble. This framework underlines the central role of sparsity in information transmission through network dynamics, providing new insight into the structure-function relationship for high-dimensional networks with nonlinear dynamics. Considering the reconstruction of structural connectivity in large networks is a significant and challenging problem in the study of complex networks, we hypothesize that analogous reconstruction methods taking advantage of sparsity may facilitate key advances in the field.


sparsity, neuronal networks, nonlinear dynamics, compressive sensing, network reconstruction, inverse problems

2010 Mathematics Subject Classification

82B05, 92C20, 92C42, 94A08

This work is supported by NSF DMS-1812478 (V.J.B.), a Swarthmore Faculty Research Support Grant (V.J.B.), NSFC-11671259, NSFC-11722107, NSFC-91630208, SJTU-UM Collaborative Research Program (D.Z.), and the Student Innovation Center at Shanghai Jiao Tong University.

Received 30 October 2018

Accepted 4 May 2019

Published 6 December 2019