Information processing in dendrites II. Information theoretic complexity

In the companion paper, we established a rationale for exploring the general principles of dendritic processing using a class of Boolean functions-the Multi-Cube Units (MCUs). Here, we use this approach to further characterise dendritic processing using ideas from information theory and studies in complexity. The starting point is a novel decomposition of a Boolean function's total mutual information (between input variables and the output). Each component of the decomposition is a mutual information measure with respect to a single input, conditioned on a subset of the remaining inputs. We call this decomposition the information spectrum and conceive of it as a re-representation of the function in the information domain. Furthermore, the information spectrum of a Boolean function may be assigned a complexity value using the approximate entropy introduced by Pincus (Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA, 88, 2297-2301). Using Monte Carlo methods, we provide evidence that the information spectral complexity of MCUs is larger than that of any other class of Boolean function. We explain this phenomenon in terms of information flow through the 2-stage MCU architecture. Under our modelling assumptions, the implication for biological neural processing is that dendrites implement functions that have maximal information spectral complexity with respect to the class of multivariate functions from which they are drawn.