Polychrony as Chinampas

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.

Automated summary

This paper investigates the flow of signals through linear paths with a nonlinear condition where nodes emit signals based on external stimuli or the coincidence of incoming signals. These nodes form polychrony groups and can lead to cascades, where the number of nodes activated by other nodes exceeds the number of externally activated nodes. The difference between these two numbers is defined as profit. The paper predicts activation conditions for specific vertices and provides an algorithm to efficiently reconstruct cascades, establishing a dictionary between polychrony groups and graph theory. The resulting graphs, called chinampas, can be topologically classified, and enumerations and descriptions of specific chinampas are provided.