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.

中文翻译：

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理解神经回路的 PSPACE 难度

在神经科学中，了解神经回路功能的一个重要方面是确定回路中哪些神经元（如果有）对神经回路支配的生物行为至关重要。一个类似的问题是确定给定的一小组神经元是否足以显示行为，即使电路中的所有其他神经元都被停用。这样的神经元子集形成了所研究行为的所谓简并回路。实验技术的最新进展为研究人员提供了以非常高的分辨率激活和停用神经元子集的工具，即使在活体动物中也是如此。从此类实验中收集的数据可能具有以下形式：当给定的神经元子集失活时，研究中的行为是否被观察到？当一个神经回路的描述作为输入给出时，这种设置导致确定最小的重要或退化神经元组的算法问题。算法问题需要确定哪些神经元子集应该被扰动（激活/停用），然后使用来自这些扰动的数据来确定最小的生命集或退化集。鉴于大量可能的扰动和神经回路的循环性质，生物学和神经科学文献已经认识到这种方法中组合爆炸的可能性。在本文中，我们证明了寻找最小或最小尺寸退化集以及寻找作为输入的神经回路的重要神经元集的问题实际上是 PSPACE 难的。