Network theory offers a rich set of tools for quantifying and comparing networks comprised of nodes and the links between them. We are developing network theory approaches to the analysis of large-scale neural population recordings. In this framework, nodes are individual neurons, and links indicate some form of interaction (e.g. correlation in time) between neurons – creating so-called “functional” networks.
Neuroscience offers formidable challenges to network theory. Typically, its tools are developed or applied to single, static networks; or to the evolution of one network over time. In neuroscience, population recording techniques give us tens to thousands of functional networks from a single study. And all evolve over time as the interactions between neurons changes through development and learning. Consequently, the tools we must develop for studying neuronal networks provide powerful ways of analysing any network.
Current projects include:
Spike-train communities: finding neural ensembles
We recently extended community detection algorithms from network theory to create the state-of-the-art neural activity clustering algorithm, capable of identifying simultaneously the number and composition of neural groups.
Humphries, M. D. (2011) Spike-train communities: finding groups of similar spike trains. Journal of Neuroscience, 31, 2321-2336. http://dx.doi.org/10.1523/JNEUROSCI.2853-10.2011
Bruno, A. M., Frost, W. N. & Humphries, M. D. (2015) Modular Deconstruction Reveals the Dynamical and Physical Building Blocks of a Locomotion Motor Program. Neuron, 86, 304-318. http://dx.doi.org/10.1016/j.neuron.2015.03.005
Code for these algorithms is available: Analysis
Communities in sparse networks
Spectral algorithms based on matrix representations of networks are often used to detect communities, but classic spectral methods based on the adjacency matrix and its variants fail in sparse networks. Unfortunately, many real-world networks are sparse. New spectral methods based on non-backtracking random walks have recently been introduced that successfully detect communities in many sparse networks (Krzkala et al 2013 PNAS). We have improved the robustness of these methods using the so-called "reluctant backtracker" (Singh & Humphries, 2015, Sci Reports).
For us, the problem of comparing multiple networks arose first in a seemingly simple problem: is a given network a "small-world" or not? For a single network, we could make a qualitative decision by computing its path length and clustering coefficient, and compare those to the same measures from an equivalent random network (Watts & Strogatz, 1998, Nature). But what if we have thousands or tens of thousands of networks? To solve this problem, we introduced the quantitative measure of ‘small-world-ness’ (Humphries et al 2006 Proc Biol Sci; Humphries & Gurney, 2008, PLoS One).
Clearly we were not alone in facing this problem. This measure is now a standard part of the network analysis toolkit in imaging studies (Bullmore & Sporns 2009 Nat Rev Neurosci), has been applied across a wide range of network theory studies, critiqued (Hilgetag & Goulas, 2015, Brain Structure & Function), and extended (Telesford et al 2011 Brain Connectivity; Bolanos et al 2013, J Neurosci Methods).