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How does one quantify dynamic complexity in large stochastic networks? While measures of integrated information serve as a good start to address these issues, all existing versions of the measure have been plagued with normalization ambiguities and combinatorial explosions which has hindered applications to large-scale networks. In this paper, we propose a new version of integrated information which resolves all these problems and brings us a step closer to addressing complexity in large biological networks. We also show that our measure is the only one which accounts for the total integrated information of a network. We apply this measure to prototypical networks and interestingly find the existence of complexity resonances in the solutions, which suggests a new way of looking at the informational spectrum of complex dynamical systems. Finally, as a proof of principle, we compute how much information is integrated by the anatomical connectivity network of the human cerebral cortex.

How does our nervous system successfully acquire feedback control strategies in spite of a wide spectrum of response dynamics from different musculo-skeletal systems? The cerebellum is a crucial brain structure in enabling precise motor control in animals. Recent advances suggest that synaptic plasticity of cerebellar Purkinje cells involves molecular mechanisms that mimic the dynamics of the efferent motor system that they control allowing them to match the timing of their learning rule to behavior. Counter-Factual Predictive Control (CFPC) is a cerebellum-based feed-forward control scheme that exploits that principle for acquiring anticipatory actions. CFPC extends the classical Widrow-Hoff/Least Mean Squares by inserting a forward model of the downstream closed-loop system in its learning rule. Here we apply that same insight to the problem of learning the gains of a feedback controller. To that end, we frame a Model-Reference Adaptive Control (MRAC) problem and derive an adaptive control scheme treating the gains of a feedback controller as if they were the weights of an adaptive linear unit. Our results demonstrate that rather than being exclusively confined to cerebellar learning, the approach of controlling plasticity with a forward model of the subsystem controlled, an approach that we term as Model-Enhanced Least Mean Squares (ME-LMS), can provide a solution to wide set of adaptive control problems.