Quantitative relations among causality measures with applications to pulse-output nonlinear network reconstruction

Abstract

The causal connectivity of a network is often inferred to understand the network function. It is arguably acknowledged that the inferred causal connectivity relies on the causality measure one applies, and it may differ from the networks underlying structural connectivity. However, the interpretation of causal connectivity remains to be fully clarified, in particular, how causal connectivity depends on causality measures and how causal connectivity relates to structural connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., neural networks with spike output, and address the above issues based on four intensively utilized causality measures, i.e., time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy. We theoretically show how these causality measures are related to one another when applied to pulse signals. Taking the simulated Hodgkin-Huxley neural network and the real mouse brain network as two illustrative examples, we further verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network structural connectivity, therefore establishing a direct link between the causal and structural connectivity. We stress that the structural connectivity of networks can be reconstructed pairwise without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for pulse-output network reconstruction. Significance StatementInferring network connectivity is a key challenge in many diverse scientific fields. We investigate networks with pulse signal as measured output and solve the above reverse-engineering issue by establishing a direct link between the networks causal connectivity and structural connectivity. Here, the causal connectivity can be inferred by any one of the four causality measures, i.e., time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy. We analytically reveal the relationship among these four measures and show that they are equally effective to fully reconstruct the network connectivity pairwise. Our work provides a practical framework to reconstruct the structural connectivity in general pulse-output nonlinear networks or subnetworks.