Description of spreading dynamics by microscopic network models and macroscopic branching processes can differ due to coalescence

Abstract

Spreading processes are conventionally monitored on a macroscopic level by counting the number of incidences over time. The spreading process can then be modeled either on the microscopic level, assuming an underlying interaction network, or directly on the macroscopic level, assuming that microscopic contributions are negligible. The macroscopic characteristics of both descriptions are commonly assumed to be identical. In this work, we show that these characteristics of microscopic and macroscopic descriptions can be different due to coalescence, i.e., a node being activated at the same time by multiple sources. In particular, we consider a (microscopic) branching network (probabilistic cellular automaton) with annealed connectivity disorder, record the macroscopic activity, and then approximate this activity by a (macroscopic) branching process. In this framework, we analytically calculate the effect of coalescence on the collective dynamics. We show that coalescence leads to a universal non-linear scaling function for the conditional expectation value of successive network activity. This allows us to quantify the difference between the microscopic model parameter and established macroscopic estimates. To overcome this difference, we propose a non-linear estimator that correctly infers the model branching parameter for all system sizes.

A unified picture of neuronal avalanches arises from the understanding of sampling effects

“…With our probability-based update rules, it may happen that tar-460 get neurons are simultaneously activated by multiple sources. 461 This results in so-called coalescence effects that are particularly 462 strong in our model due to the local activity spreading [33]. For 463 instance, naively setting = 1 (with = 300 µm) would result 464 in an effective (measured)̂≈ 0.98, which has considerably 465 different properties.…”

“…All parameters of the model, the sampling and the 61 analysis are closely matched to those known from experiments 62 (see Methods).63Branching dynamics on a local 2D topology 64 In order to evaluate sampling effects, we want to precisely 65 set the underlying dynamics. Therefore, we employ the es-66 tablished branching model, which is well understood analyti-67 cally [9,25,32,33]. Inspired by biological neuronal networks, 68 we simulate the branching dynamics on a dense 2D topology 69 with N = 160 000 neurons where each neuron is connected 70 to ≈ 1000 local neighbors.…”

“…during non-220 zero activity [1,31]. Note that, due to coalescence and drive 221 effects, av can differ from proper [22,33].…”

Tailored ensembles of neural networks optimize sensitivity to stimulus statistics

“…These response measures differ drastically from those of the branching network. The response measures of the branching process differ drastically from those of the branching network, because of coalescence (the simultaneous activation of the same neuron from multiple sources) in the branching network [19]. Coalescence alters the critical non-equilibrium phase transition from subcritical-supercritical in the branching process to absorbing-active in the branching network.…”

“…We hypothesize that the brain combines all strategies for maximized robustness. Internal and external coalescence reduce the effective branching parameter for static connection weights w = m/N [19]. Thereby, macroscopic branching parametersm, estimated from the network rate, differ from the model branching parameter m. For a detailed discussion, we refer to Ref.…”