Linking causal and structural connectivity in nonlinear networks

Accessing structural connectivity information is crucial for understanding intricate functions of complex networks. In situations where direct measurement of structural connectivity is limited, like in cortical neuronal networks, causal inference tools such as transfer entropy are employed to reconstruct structural connectivity from network activity data. However, the causal connectivity inferred may not always align with the network's underlying structural connectivity, leaving the interpretation of inferred causal connectivity open for further clarification. Specifically, the relationship between causal and structural connectivity needs more exploration. In this context, our focus is on nonlinear pulse-coupled networks, e.g., spiking neural networks, and propose a new framework, pairwise time-delayed transfer entropy (PTD-TE), to reconstruct the structural connectivity of several simulated nonlinear pulse-coupled networks. Our findings show orders of magnitude differences in the PTD-TE values between connected and unconnected pairs of nodes, which allows for the reconstruction of structural connectivity. We delve deeper to understand the mechanism that enables this accurate reconstruction and establish that the PTD-TE value between two nodes is proportionate to the square of their coupling strength. Importantly, we highlight that the structural connectivity can be reconstructed pairwise without the need for global information from all other nodes in a network. This approach helps bypass the "curse of dimensionality" problem commonly encountered with transfer entropy. Our study thus offers a practicable and effective method to infer the structural connectivity of nonlinear pulse-coupled systems.