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The PSPACE-hardness of understanding neural circuits

In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a subset of neurons forms what is called a degenerate circuit for the behavior being studied. Recent advances in experimental techniques have provided researchers with tools to activate and deactivate subsets of neurons with a very high resolution, even in living animals. The data collected from such experiments may be of the following form: when a given subset of neurons is deactivated, is the behavior under study observed? This setting leads to the algorithmic question of determining the minimal vital or degenerate sets of neurons when one is given as input a description of the neural circuit. The algorithmic problem entails both figuring out which subsets of neurons should be perturbed (activated/deactivated), and then using the data from those perturbations to determine the minimal vital or degenerate sets. Given the large number of possible perturbations, and the recurrent nature of neural circuits, the possibility of a combinatorial explosion in such an approach has been recognized in the biology and the neuroscience literature. In this paper, we prove that the problems of finding minimal or minimum-size degenerate sets, and of finding the set of vital neurons, of a neural circuit given as input, are in fact PSPACE-hard. More importantly, we prove our hardness results by showing that a simpler problem, that of simulating such neural circuits, is itself PSPACE-hard.