Neuronal activity gives rise to behavior, and behavior influences neuronal dynamics, in a closed-loop control system. Is it possible then, to find a relationship between the statistical properties of behavior and neuronal dynamics? Measurements of neuronal activity and behavior have suggested a direct relationship between scale-free neuronal and behavioral dynamics. Yet, these studies captured only local dynamics in brain sub-networks. Here, we investigate the relationship between internal dynamics and output statistics in a mathematical model system where we have access to the dynamics of all network units. We train a recurrent neural network (RNN), initialized in a high-dimensional chaotic state, to sustain behavioral states for durations following a power-law distribution as observed experimentally. Changes in network connectivity due to training affect the internal dynamics of neuronal firings, leading to neuronal avalanche size distributions approximating power-laws over some ranges. Yet, randomizing the changes in network connectivity can leave these power-law features largely unaltered. Specifically, whereas neuronal avalanche duration distributions show some variations between RNNs with trained and randomized decoders, neuronal avalanche size distributions are invariant, in the total population and in output-correlated sub-populations. This is true independent of whether the randomized decoders preserve power-law distributed behavioral dynamics. This demonstrates that a one-to-one correspondence between the considered statistical features of behavior and neuronal dynamics cannot be established and their relationship is non-trivial. Our findings also indicate that statistical properties of the intrinsic dynamics may be preserved, even as the internal state responsible for generating the desired output dynamics is perturbed.
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