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Proceedings ArticleDOI

Convex realizations of neural codes in one dimension

06 Nov 2020-Vol. 2277, Iss: 1, pp 150003
TL;DR: In this paper, the authors studied the convex neural codes in one dimension and discussed the possibilities of a convex code to be realized in dimension 1, which is the minimum embedding dimension of a neural code.
Abstract: Neural codes are collective activity of the neurons which are electrically active cells in our brain. In this paper, We study the openness and closeness of the convex neural codes in one dimension. We discuss the possibilities of a convex code to be realized in dimension 1. That is conditions for the minimal embedding dimension of a neural code to be 1.
References
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Journal ArticleDOI
TL;DR: Curto et al. proved that a code has no local obstructions if and only if it contains certain "mandatory" intersections of maximal codewords, and give a new criterion for an intersection of maximalcodewords to be non-mandatory, and prove that it classifies all such non-Mandatory codeword for codes on up to 5 neurons.

72 citations

Journal ArticleDOI
TL;DR: This work provides a complete characterization of local obstructions to convexity and defines max intersection-complete codes, a family guaranteed to have noLocal obstructions, a significant advance in understanding the intrinsic combinatorial properties of convex codes.
Abstract: Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley--Reisner ideals in order to detect violations...

70 citations

Posted Content
TL;DR: In this paper, the authors provide a complete characterization of local obstructions to convexity and define max intersection-complete codes, a family of convex codes with no local obstruction.
Abstract: Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

53 citations

Journal ArticleDOI
TL;DR: It is proved that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed.

17 citations