Estimating small-world topology of neural networks from multi-electrode recordings
Since the seminal work of Watts in the late 90s , graph-theoretic analyses have been performed on many complex dynamic networks, including brain structures . Most studies have focused on functional connectivity defined between whole brain regions, using imaging techniques such as fMRI, EEG or MEG. Only very few studies have attempted to look at the structure of neural networks on the level of individual neurons [3,4]. To the best of our knowledge, these studies have only considered undirected connectivity networks and have derived connectivity based on estimates on small subsets or even pairs of neurons from the recorded networks.Here, we investigate scale-free and small-world properties of neuronal networks, based on multi-electrode recordings from the awake monkey on a larger data set than in previous approaches. We estimate effective, i.e. causal, interactions by fitting Generalized Linear Models on the neural responses to natural stimulation. The resulting connectivity matrix is directed and a link between two neurons represents a causal influence from one neuron to the other, given the observation of all other neurons from the population. We use this connectivity matrix to estimate scale-free and small-world properties of the network samples. For this, the quantity proposed by Humphries et al. (2008) for quantifying small-world-ness is generalized to directed networks . We find that the networks under consideration lack scale-free behavior, but show a small, but significant small-world structure.Finally, we show that the experimental design of multi-electrode recordings typically enforces a particular network structure that can have a considerate impact on how the small-world structure of the network should be evaluated. Random graphs that take the geometry of the experiment into account can serve as a more refined null model than the homogeneous random graphs that are usually proposed as reference models to evaluate small-world properties. References  Watts, D. J., Strogatz, S. H., 1998. Collective dynamics of 'small-world' networks. Nature 393 (6684), 440-442.  Bullmore, E., & Sporns, O. (2009). Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3), 186–198.  Bettencourt, L. M. A., Stephens, G. J., Ham, M. I., Gross, G. W., 2007. Functional structure of cortical neuronal networks grown in vitro. Physical Review E 75 (2), 021915+.  Yu, S., Huang, D., Singer, W., Nikolic, D., 2008. A small world of neuronal synchrony. Cereb. Cortex, bhn047+.  Humphries, M. D., Gurney, K., April 2008. Network 'small-world-ness': a quantitative method for determining canonical network equivalence. PloS one 3 (4), e0002051+.