Estimating effective connectivity in neural networks: comparison of derivative-based and correlation-based methods

Inferring and understanding the underlying connectivity structure of a system solely from the observed activity of its constituent components is a challenge in many areas of science. In neuroscience, techniques for estimating connectivity are paramount when attempting to understand the network structure of neural systems from their recorded activity patterns. To date, no universally accepted method exists for the inference of effective connectivity, which describes how the activity of a neural node mechanistically affects the activity of other nodes. Here, focussing on purely excitatory networks of small to intermediate size and continuous node dynamics, we provide a systematic comparison of different approaches for estimating effective connectivity. Starting with the Hopf neuron model in conjunction with known ground truth structural connectivity, we reconstruct the systems connectivity matrix using a variety of algorithms. We show that, in sparse non-linear networks with delays, combining a lagged-cross-correlation (LCC) approach with a recently published derivative-based covariance analysis method provides the most reliable estimation of the known ground truth connectivity matrix. We outline how the parameters of the Hopf model, including those controlling the bifurcation, noise, and delay distribution, affect this result. We also show that in linear networks, LCC has comparable performance to a method based on transfer entropy, at a drastically lower computational cost. We highlight that LCC works best for small sparse networks, and show how performance decreases in larger and less sparse networks. Applying the method to linear dynamics without time delays, we find that it does not outperform derivative-based methods. We comment on this finding in light of recent theoretical results for such systems. Employing the Hopf model, we then use the estimated structural connectivity matrix as the basis for a forward simulation of the system dynamics, in order to recreate the observed node activity patterns. We show that, under certain conditions, the best method, LCC, results in higher trace-to-trace correlations than derivative-based methods for sparse noise-driven systems. Finally, we apply the LCC method to empirical biological data. Choosing a suitable threshold for binarization, we reconstruct the structural connectivity of a subset of the nervous system of the nematode C. Elegans. We show that the computationally simple LCC method performs better than another recently published, computationally more expensive reservoir computing-based method. We apply different methods to this dataset and find that they all lead to similar performances. Our results show that a comparatively simple method can be used to reliably estimate directed effective connectivity in sparse neural systems in the presence of spatio-temporal delays and noise. We provide concrete suggestions for the estimation of effective connectivity in a scenario common in biological research, where only neuronal activity of a small set of neurons, but not connectivity or single-neuron and synapse dynamics, are known.