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Non-Monotonicity of Closed Convexity in Neural Codes


Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes – open convex and closed convex codes – are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserves open convexity; but here we show that this property fails to hold for closed convex codes. Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also present an example of a code that is neither open convex nor closed convex.

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The authors thank Aaron Chen, Chad Giusti, Kaitlyn Phillipson, and Thomas Yahl for insightful comments and suggestions, and thank Nida Obatake for editorial suggestions. The authors are also grateful to four referees for detailed comments which improved this work. BG, SM, and AS are grateful to Ramona Heuing and Martina Juhnke-Kubitzke for finding a crucial error in a prior version of this work. BG and SM initiated this research in the 2019 REU in the Department of Mathematics at Texas A&M University, supported by the NSF (DMS-1757872). AS was supported by the NSF (DMS-1752672). RAJ was supported by the NSF (DGE-1761124).

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Correspondence to Anne Shiu.

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Dedicated to Bernd Sturmfels on the occasion of his 60th birthday.

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Gambacini, B., Jeffs, R.A., Macdonald, S. et al. Non-Monotonicity of Closed Convexity in Neural Codes. Vietnam J. Math. 50, 359–373 (2022).

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  • Neural code
  • Place cell
  • Convex
  • Simplicial complex

Mathematics Subject Classification (2010)

  • 52A20
  • 52C99
  • 32F27