Recent approaches have been restricted to two broad classes (Boccaletti et al., 2006). The first class examines complex dynamics, albeit in relatively simple networks. These networks (often consisting of a few [<10] neurons) (Shilnikov et al., 2008) are designed to resemble the characteristic building blocks of networks known as motifs that recur with considerable frequency in neuroanatomical TSA HDAC cell line data sets (Milo et al., 2002, Sporns et al., 2004 and Sporns and
Kötter, 2004). Another commonly studied example of the same class is a large network with simple connectivity (all-to-all, or nearest neighbor) that can also demonstrate a rich repertoire of dynamical patterns (Assisi et al., 2005). A second class of studies examines simple dynamics in networks with arbitrarily complex topologies (Arenas et al., 2008). These studies have been largely limited to the analysis of the stability of completely synchronized states (Pecora and Carroll, 1998) and have been applied to a variety of systems (see Boccaletti et al., 2006 for a review). Our approach is a departure from these two classes. Here we establish a relationship between a structural property of the network, its colorings, and the dynamics it constrains. Furthermore, we show that the description of network dynamics
based on its coloring comes with a fortuitous benefit. It helps us define a low-dimensional space in which the seemingly high-dimensional dynamics of networks of excitatory and inhibitory neurons reliably forms a series of orthogonally propagating waves, a predictable MLN0128 order and simple pattern where synchronous ensembles of excitatory projecting cells are successively recruited. A “coloring” of a network (or graph) is an assignment of colors to the nodes of the graph so that nodes (such as neurons) that are directly connected to each other are assigned Rolziracetam different colors (see Supplemental Information for a detailed example). Graph coloring problems first emerged as interesting mathematical curiosities (Biggs et al.,
1986 and Kubale, 2004), but have since been applied to resolving scheduling conflicts, reinterpreting the Sudoku puzzle (Herzberg and Murty, 2007) and, most prominently, coloring maps (see Appel and Haken, 1989 for a proof of the famous four-color theorem; Appel and Haken, 1977 and Biggs et al., 1986 describe the colorful history). We found that antagonistic interactions in a network of inhibitory neurons can be usefully related to its coloring. To illustrate, consider the simplest possible inhibitory network, one consisting of two reciprocally connected inhibitory interneurons. Since these neurons are directly connected to each other, by definition, we must assign a different color to each neuron. A general property of such a network is that neurons inhibiting each other via fast GABAergic synapses tend to spike asynchronously.