Memory decay and loss of criticality in quorum percolation

Renaud Renault, Pascal Monceau, and Samuel Bottani

Phys. Rev. E 88, 062134 – Published 20 December 2013

Abstract

In this paper, we present the effects of memory decay on a bootstrap percolation model applied to random directed graphs (quorum percolation). The addition of decay was motivated by its natural occurrence in physical systems previously described by percolation theory, such as cultured neuronal networks, where decay originates from ionic leakage through the membrane of neurons and/or synaptic depression. Surprisingly, this feature alone appears to change the critical behavior of the percolation transition, where discontinuities are replaced by steep but finite slopes. Using different numerical approaches, we show evidence for this qualitative change even for very small decay values. In experiments where the steepest slopes can not be resolved and still appear as discontinuities, decay produces nonetheless a quantitative difference on the location of the apparent critical point. We discuss how this shift impacts network connectivity previously estimated without considering decay. In addition to this particular example, we believe that other percolation models are worth reinvestigating, taking into account similar sorts of memory decay.

Received 30 May 2013

DOI:https://doi.org/10.1103/PhysRevE.88.062134

Authors & Affiliations

Renaud Renault¹, Pascal Monceau^1,2, and Samuel Bottani^1,*

¹MSC (Université Paris-Diderot, CNRS-UMR 7057), 5 Rue Thomas Mann, 75013 Paris, France
²Université d'Evry, France

^*samuel.bottani@univ-paris-diderot.fr

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Vol. 88, Iss. 6 — December 2013

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Images

Figure 1
Illustration of quorum percolation and decay. (a) Successive frames taken during QP showing how the inactive neurons (in light gray) become active (in orange) as soon as their potentials reach $m$ . In this particular example, $m = 3$ . (b) Relationship between $f$ , $ϕ$ , and $m$ in the classical QP model, for a Gaussian “in-connectivity” with $\bar{k} = 50$ and $σ = 10$ . $f$ is the initial active fraction (externally activated) and $ϕ$ is the fraction of active neurons at the end of the cascade. $m$ defines the number of active “in-neighbors” necessary for a neuron to become active if it was not part of $f$ . For $m \leq m_{c}$ , we observe a first order transition in $ϕ$ . The height of the jump at the discontinuity is called $g (m)$ (see text). (c) Schematic representation of QP with decay. Random disintegration of signals (decay) between steps (glowing links) changes the final state of the network although the initial configuration is the same as in (a).Reuse & Permissions
Figure 2
Representation of a mean field network. (a) Shows the different $p_{k_{e q}, s}$ that constitute the network at a given time step. Note that the actual sizes of the active and inactive fractions are independent from the number of squares that constitute them. (b) Pictorial description of Eq. (4) showing how it is applied to two different $p_{k_{e q}, s}$ displayed as green filled squares. Their values at step $t + 1$ are computed from their previous state and those of a subset of $p_{k_{e q}, s} (t)$ , displayed as gray hatched squares. Neurons “inside” those squares which lose through decay and receive from incoming neighbors the right amount of signals will become equivalent to the neurons in the associated green square at the end of the iteration. For one of the hatched squares, outlined in yellow, these two antagonist effects are represented by a first red arrow for the decay and the associated decrease of $k_{e q}$ [see Fig. 2c], and a second outlined blue arrow for the reception of new signals. (c) Equivalence between neurons. The two neurons in the middle have the same probability to become active in the next step.Reuse & Permissions
Figure 3
Comparison of explicit and mean field simulations. Lines are computed using the mean field algorithm while points of the same color correspond to 10 different equivalent explicit simulations. As $N$ increases from (a) 1000 to (b) $100 000$ , explicit simulations converge to the mean field behavior. $\bar{k} = 50$ , $σ = 10$ , $d = 0.1$ .Reuse & Permissions
Figure 4
Apparent approximate size of the discontinuity $g_{ε}$ as a function of $d$ for (a) $ε = 10^{- 3}$ and (b) $ε = 10^{- 10}$ . The connectivity $p_{k}$ follows a Gaussian distribution of parameters $\bar{k} = 25$ and $σ = 5$ . $ν = 10^{- 10}$ .Reuse & Permissions
Figure 5
Apparent size of the discontinuity $g_{ε} (2)$ as a function of $ε$ , in a network where $\bar{k} = 25$ , $σ = 5$ , and $d = 1$ (immediate decay). What is interpreted as a very large discontinuous transition at larger $ε$ vanishes completely when $ε \approx 10^{- 40}$ . For each point, $ν = ε / 10$ . High accuracy computations were done with matlab and the VPA package.Reuse & Permissions

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