Collins Assisi, Mark Stopfer, Maxim Bazhenov (2011) Using the Structure of Inhibitory Networks to Unravel Mechanisms of Spatiotemporal Patterning, DOI 10.1016/j.neuron.2010.12.019 [via]
Initially I was quite excited by this paper. I was a little disappointed that only a qualitative description of the results was provided. The figures were remarkably clear.
The way they constructed graphs with a given k-coloring was just to use bipartite, tripartite ... k-partite graphs to explore 1,2,...,k colorings. This didn't seem all that biologically plausible to me, but they do some controls that suggests that the results will generalize.
Real neural networks probably have connectivity such that the actual number of colors required to color the graph is huge, and that in practice there aren't really as many oscillatory subpopulations as there are colors. They address this with their "multiple coloring" explanation.
They note that as the number of colors increased the orderly activity decreased. It sounds like the timescale of adaptation determines this : you need the k colorings to be able to form a cycle that fits within the natural wavelength of slow inhibitory processes.
I wonder if the sets of neurons with identical color in this paper are related to the potential wells seen in Tkačik & al. In Tkačik & al the wells are stable minima for spontaneous activity. In Assisi & al., if you turn of adaptation single-color clusters are also the stable minima.
I would be interested in seeing a more theoretical analysis using LIF, theta, or rate-based neurons.