Neural codes serve as a language for neurons in the brain. Open (or closed) convex codes, which arise from the pattern of intersections of collections of open (or closed) convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly understood. Here we find that a code that can be realized by a collection of open convex sets may or may not be realizable by closed convex sets, and vice versa, establishing that open convex and closed convex codes are distinct classes. We establish a non-degeneracy condition that guarantees that the corresponding code is both open convex and closed convex. We also prove that max intersection-complete codes (i.e., codes that contain all intersections of maximal codewords) are both open convex and closed convex, and provide an upper bound for their minimal embedding dimension. Finally, we show that the addition of non-maximal codewords to an open convex code preserves convexity.

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关于开和闭凸码

神经代码充当大脑中神经元的语言。开（或闭）凸代码，由欧几里德空间中开（或闭）凸集集合的交集模式产生，与神经科学特别相关。然而，并非每个代码都是开凸或闭凸的，并且代码的组合特性决定其由此类集合实现的特性仍然知之甚少。在这里，我们发现可以由一组开凸集实现的代码可能会或可能不会被闭凸集实现，反之亦然，确定开凸代码和闭凸代码是不同的类。我们建立了一个非简并条件，保证相应的代码既是开凸又是闭凸。我们还证明了最大交集完全码（即 包含最大码字的所有交集的代码都是开凸和闭凸的，并为其最小嵌入维度提供了上限。最后，我们展示了向开放凸码添加非最大码字保留了凸性。