Dynamical properties of neuronal systems with complex network structure

An important question in neuroscience is how the structure and dynamics of a neuronal network relate to each other. We approach this problem by modeling the spiking activity of large-scale neuronal networks that exhibit several complex network properties. Our main focus lies on the relevance of two particular attributes for the dynamics, namely structural heterogeneity and degree correlations. Although these are fundamental properties of many brain networks, their influence on the system’s activity is not yet well understood. As a central result, we introduce a novel mean-field method that makes it possible to calculate the average activity of heterogeneous, degree-correlated neuronal networks without having to simulate each neuron explicitly. The method is based on grouping neurons with equal number of inputs (their in-degree) into populations and describing the dynamics of these populations in terms of reduced firing rate equations. We find that the connectivity structure is sufficiently captured by a reduced matrix that contains only the coupling between the populations. This matrix can be calculated analytically for some generic random networks, which allows for an efficient analysis of the system’s dynamics. With the mean-field method and numerical simulations we demonstrate that assortative degree correlations enhance the network’s ability to sustain activity for low external excitation, thus making it more sensitive to small input signals. This effect is reminiscent of an increased structural robustness of assortative networks that is well-known in network sciences. Therefore, we additionally examine the structural robustness of correlated networks with a simplified percolation model, additionally taking into account the fact that neuronal networks among other real-world systems are subject to metric constraints: Spatially close neurons are usually more likely to connect to each other than distant neurons. The main result of this examination is that for weak metric constraints assortativity generally makes the network more robust, whereas for strong metric constraints assortativity can strengthen or weaken the network, depending on its local structure. Therefore, we conclude that degree correlations and metric constraints strongly interplay and that they should not be regarded as independent features of real-world networks.