Identification of Functional Information Subgraphs in Complex Networks

We present a general information theoretic approach for identifying functional subgraphs in complex networks. We show that the uncertainty in a variable can be written as a sum of information quantities, where each term is generated by successively conditioning mutual informations on new measured variables in a way analogous to a discrete differential calculus. The analogy to a Taylor series suggests efficient optimization algorithms for determining the state of a target variable in terms of functional groups of other nodes. We apply this methodology to electrophysiological recordings of cortical neuronal networks grown in vitro. Each cell’s firing is generally explained by the activity of a few neurons. We identify these neuronal subgraphs in terms of their redundant or synergetic character and reconstruct neuronal circuits that account for the state of target cells.

Figure 1

(a) Neuronal culture over a microelectrode array (white circles; $\mathrm{\text{bar}}=40\mu \mathrm{m}$). (b) Detail of a spike time series. The box shows network state 011101100 (bottom to top).

Figure 2

Joint entropy of neuron 46 and a set of other neurons of size $k$. The next neuron measured is chosen by maximizing the variation to various orders; the neuron numbers appear for the exact curve. Inset: Histogram of entropy fraction for each neuron remaining after all possible measurements.

Figure 3

(a) Sorted global information deficit or excess of a multiplet, relative to the sum of the pairwise mutual informations: $R_k^S$. (b) Values of each term in the expansion in Eq. (1) vs $k$ for 36 000 randomly sampled variable combinations. White to blue: 0–0.5%; red to yellow: 0.5–100%.

Figure 4

(a) Frequency of redundant (red) and synergetic (blue) cores versus size $k$. (b) Purely redundant (red) and purely synergetic (blue) circuits relative to neuron 46. Neurons and groups with the most information about 46 are closest to the center; c.f. Fig. 2. Arcs identify neurons that participate in multiple functional groups.