The brain exhibits complex spatio-temporal patterns of activity. This phenomenon is governed by an interplay between the internal neural dynamics of cortical areas and their connectivity. Uncovering this complex relationship has raised much interest, both for theory and the interpretation of experimental data (e.g., fMRI recordings) using dynamical models. Here we focus on the so-called inverse problem: the inference of network parameters in a cortical model to reproduce empirically observed activity. Although it has received a lot of interest, recovering directed connectivity for large networks has been rather unsuccessful so far. The present study specifically addresses this point for a noise-diffusion network model. We develop a Lyapunov optimization that iteratively tunes the network connectivity in order to reproduce second-order moments of the node activity, or functional connectivity. We show theoretically and numerically that the use of covariances with both zero and non-zero time shifts is the key to infer directed connectivity. The first main theoretical finding is that an accurate estimation of the underlying network connectivity requires that the time shift for covariances is matched with the time constant of the dynamical system. In addition to the network connectivity, we also adjust the intrinsic noise received by each network node. The framework is applied to experimental fMRI data recorded for subjects at rest. Diffusion-weighted MRI data provide an estimate of anatomical connections, which is incorporated to constrain the cortical model. The empirical covariance structure is reproduced faithfully, especially its temporal component (i.e., time-shifted covariances) in addition to the spatial component that is usually the focus of studies. We find that the cortical interactions, referred to as effective connectivity, in the tuned model are not reciprocal. In particular, hubs are either receptors or feeders: they do not exhibit both strong incoming and outgoing connections. Our results sets a quantitative ground to explore the propagation of activity in the cortex.

MG and GD acknowledge funding from the FP7 FET ICT Flagship Human Brain Project (604102). RMB acknowledges funding from the Ramon y Cajal Spanish Award (RYC-2010-05952), the Marie Curie FP7-PEOPLE-2010-IRG (PIRG08-GA-2010-276795), and the Programa Estatal de Investigacion Cientifica y Tecnica de Excelencia (PSI2013-44811-P). PR acknowledges funding from the James S. McDonnell Foundation(JSMF22002082), the German Ministry of Education and Research (01GQ0971), and the Max-Planck-Gesellschaft (Minerva Program). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.