bibkey: fernandez2018revisiting
authors: Bastien Fernandez
year: 2018
title: "Revisiting asymptotic periodicity in networks of degrade-and-fire oscillators"
doi: 10.1140/epjst/e2018-800068-8
url: "https://arxiv.org/abs/1809.03228v1"
claim: "For networks of degrade-and-fire oscillators the paper improves the conditions for asymptotic periodicity under exhaustive firing, analyses the unidirectional 3-cell chain, shows that the simplest non-exhaustive periodic orbit of the epsilon > 1/2 transition graph cannot exist, and states the belief that the same impossibility holds for every other periodic trajectory with non-exhaustive firing in that graph."
strata_touched:
  - D5/S3/StatisticalMechanics/DegradeFireNonExhaustiveOrbitRefutation
license: citation-only
triage: anchor

Fernandez, Revisiting asymptotic periodicity in networks of degrade-and-fire oscillators

B. Fernandez, Eur. Phys. J. Spec. Top. 227, 1267–1279 (2018), DOI 10.1140/epjst/e2018-800068-8; arXiv:1809.03228v1. Quotations are from the arXiv v1 TeX source DynamicsDFNetworksRevised.tex; macros are expanded.

Expression levels $x\in [0,1{]}^N$ evolve under a threshold $\eta \in (0,1)$ and a stochastic non-negative matrix $W$ with $w_{ii}>\eta$ (l. 46–64): “If $W{x}_i(t)>\eta$, then the expression level $x_i(t)$ degrades at constant speed $-1$, unless $x_i(t)=0$ (in which case, it remains steady)”; “If $W{x}_i(t)\le \eta$, a firing takes place and resets the expression level to the value 1.” A firing sequence is exhaustive when “each of its length $N$ segment must be a permutation of $\{1,\dots ,N\}$” (l. 68). Section 4 takes $N=3$, $W$ with rows $(1-ϵ,0,ϵ)$, $(ϵ,1-ϵ,0)$, $(0,ϵ,1-ϵ)$ and $ϵ\in (0,1-\eta )$, restricted to trajectories whose coordinates are always distinct (l. 239–255). The Concluding Remarks state (l. 373):

For instance, one can check that a (minimal) periodic orbit with the simplest non-exhaustive firing sequence of the right transition graph in Fig. 1 cannot exist (see analysis below). We believe that similar impossibility prevails for any other periodic trajectory with non-exhaustive firing in this graph.

Verified locator

• DOI: https://doi.org/10.1140/epjst/e2018-800068-8
• Source: https://arxiv.org/abs/1809.03228v1