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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2103.15265 |
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| _version_ | 1866918303871533056 |
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| author | Dolores-Cuenca, Eric Arciniega-Nevarez, Jose Antonio Nguyen, Anh Zou, Yitong Van Popering, Luke Crock, Nathan Erlebacher, Gordon Mendoza-Cortes, Jose L. |
| author_facet | Dolores-Cuenca, Eric Arciniega-Nevarez, Jose Antonio Nguyen, Anh Zou, Yitong Van Popering, Luke Crock, Nathan Erlebacher, Gordon Mendoza-Cortes, Jose L. |
| contents | In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit.
Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_15265 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Polychrony as Chinampas Dolores-Cuenca, Eric Arciniega-Nevarez, Jose Antonio Nguyen, Anh Zou, Yitong Van Popering, Luke Crock, Nathan Erlebacher, Gordon Mendoza-Cortes, Jose L. Combinatorics Formal Languages and Automata Theory Category Theory Neurons and Cognition 05A10 (Primary), 94C15, 68Q80, 05C30, 37B15, 90C30 (Secondary) F.1.1; I.1.2 In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets. |
| title | Polychrony as Chinampas |
| topic | Combinatorics Formal Languages and Automata Theory Category Theory Neurons and Cognition 05A10 (Primary), 94C15, 68Q80, 05C30, 37B15, 90C30 (Secondary) F.1.1; I.1.2 |
| url | https://arxiv.org/abs/2103.15265 |