Polychrony as Chinampas

TL;DR

This paper models signal cascades in neural-like networks using polychrony groups, introduces the concept of chinampas as graph structures, and provides algorithms for analyzing cascade dynamics and classifications.

Contribution

It introduces the concept of chinampas as graph structures for cascades, linking polychrony groups to graph theory and classifying chinampas topologically.

Findings

Enumerated chinampas with profits zero and one.

Established a correspondence between chinampas and partially ordered sets.

Provided an algorithm for cascade reconstruction.

Abstract

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.