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In analysis of multi-component complex systems, such as neural systems,identifying groups of units that share similar functionality will aidunderstanding of the underlying structures of the system. To find such agrouping, it is useful to evaluate to what extent the units of the system areseparable. Separability or inseparability can be evaluated by quantifying howmuch information would be lost if the system were partitioned into subsystems,and the interactions between the subsystems were hypothetically removed. Asystem of two independent subsystems are completely separable without any lossof information while a system of strongly interacted subsystems cannot beseparated without a large loss of information. Among all the possiblepartitions of a system, the partition that minimizes the loss of information,called the Minimum Information Partition (MIP), can be considered as theoptimal partition for characterizing the underlying structures of the system.Although the MIP would reveal novel characteristics of the neural system, anexhaustive search for the MIP is numerically intractable due to thecombinatorial explosion of possible partitions. Here, we propose acomputationally efficient search to precisely identify the MIP among allpossible partitions by exploiting the submodularity of the measure ofinformation loss. Mutual information is one such submodular information lossfunctions, and is a natural choice for measuring the degree of statisticaldependence between paired sets of random variables. By using mutual informationas a loss function, we show that the search for MIP can be performed in apractical order of computational time for a reasonably large system. We alsodemonstrate that MIP search allows for the detection of underlying globalstructures in a network of nonlinear oscillators.

Fast and exact search for the partition with minimal information loss