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arxiv: 2603.21781 · v1 · submitted 2026-03-23 · ❄️ cond-mat.dis-nn

Impact of heavy-tailed synaptic strength distributions on self-sustained activity in networks of spiking neurons

Ralf T\"onjes , Chunming Zheng , Wenping Cui , Benjamin Lindner This is my paper

Pith reviewed 2026-05-15 00:55 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords spiking neuronsheavy-tailed distributionsdirected percolationmean-field theoryquadratic integrate-and-fireself-sustained activitysynaptic strength
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The pith

Heavy-tailed synaptic strengths create a percolation threshold for self-sustained activity to spread across spiking neuron networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies stochastic mean-field theory to networks of quadratic integrate-and-fire neurons to compare light-tailed Gaussian couplings with heavy-tailed Cauchy couplings. It shows that Cauchy couplings produce an analytically derivable directed percolation threshold: above a critical synaptic strength, firing activity spreads through the entire infinite network from only a few initially active units. The transition is continuous or discontinuous depending on the excitatory-inhibitory imbalance. Gaussian couplings show no such percolation transition in the thermodynamic limit, and adding noise to Cauchy networks also removes it.

Core claim

For quadratic integrate-and-fire neurons with Cauchy random couplings, stochastic mean-field theory yields a directed percolation threshold above which self-sustained activity percolates through the whole network; the transition changes from continuous to discontinuous according to the excitatory-inhibitory balance, while Gaussian couplings lack this threshold in the infinite-network limit.

What carries the argument

Stochastic mean-field theory applied to the stationary firing states of quadratic integrate-and-fire neurons driven by either Gaussian or Cauchy random synaptic strengths.

If this is right

  • Above the critical Cauchy strength, activity initiated by a few neurons spreads to the entire network and persists.
  • Shifting the excitatory-inhibitory balance can change the percolation transition from continuous to first-order.
  • Gaussian couplings keep activity localized even in the infinite-network limit.
  • Adding independent noise to Cauchy networks eliminates the percolation threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Biological networks whose synapses follow heavy-tailed strength distributions could switch abruptly between localized and global activity regimes.
  • Spatial or other correlations in real connectivity might shift or destroy the idealized percolation threshold found under random coupling.
  • The distinction between light- and heavy-tailed couplings offers a possible mechanism for robust signal propagation in excitable neural tissue.

Load-bearing premise

The network is infinitely large with fully random and statistically independent synaptic connections.

What would settle it

Simulations of large but finite networks with Cauchy-distributed synaptic weights that either exhibit or fail to exhibit a sharp onset of network-wide activity at the predicted critical strength.

read the original abstract

We analyze states of stationary activity in randomly coupled quadratic integrate-and-fire neurons using stochastic mean-field theory. Specifically, we consider the two cases of Gaussian random coupling and Cauchy random coupling, which are representative of systems with light- or with heavy-tailed synaptic strength distributions. For both, Gaussian and Cauchy coupling, bistability between a low activity and a high activity state of self-sustained firing is possible in excitable neurons. In the system with Cauchy coupling we find analytically a directed percolation threshold, i.e., above a critical value of the synaptic strength, activity percolates through the whole network starting from a few spiking units only. The existence of the directed percolation threshold is in agreement with previous numerical results in the literature for integrate-and-fire neurons with heavy-tailed synaptic strength distribution. However, we have found that the transition can be continuous or discontinuous, depending on the excitatory-inhibitory imbalance in the network. Networks with Gaussian coupling and networks with Cauchy coupling and additional additive noise lack the percolation transition in the thermodynamic limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies stochastic mean-field theory to networks of quadratic integrate-and-fire neurons with either Gaussian or Cauchy-distributed random synaptic couplings. It identifies bistability between low- and high-activity states for both distributions in excitable regimes. For Cauchy couplings it derives an analytical directed percolation threshold above which activity spreads network-wide from sparse seeds; the transition is reported to be continuous or discontinuous according to the excitatory-inhibitory balance. The threshold is stated to match prior numerical observations, while Gaussian couplings and Cauchy couplings plus additive noise lack a percolation transition in the thermodynamic limit.

Significance. If the mean-field derivation is valid, the work supplies an explicit, analytically tractable criterion for activity percolation that is absent in light-tailed coupling models. This distinction could explain numerical findings on heavy-tailed connectivity and offers a concrete handle on how synaptic strength statistics control self-sustained dynamics, with potential utility for reduced models of cortical networks.

major comments (2)
  1. [§3] §3 (stochastic mean-field closure for Cauchy couplings): the derivation invokes standard averaging and central-limit arguments to obtain an effective stochastic field, yet the Cauchy distribution possesses neither finite mean nor variance. The law-of-large-numbers step therefore requires explicit justification (e.g., via stable-distribution theory or truncation analysis) because rare, arbitrarily large couplings can dominate propagation even as N→∞; without this, the predicted percolation threshold is not guaranteed to survive the thermodynamic limit.
  2. [§4] §4 (phase diagram and transition order): the claim that the percolation transition changes from continuous to discontinuous with excitatory-inhibitory imbalance is central to the abstract but is supported only by qualitative statements. Explicit bifurcation equations, the location of the critical point in parameter space, and a quantitative comparison of order-parameter scaling on either side of the balance line are needed to substantiate the distinction.
minor comments (2)
  1. [Abstract] The abstract states agreement with “existing numerical results” but does not cite the specific studies; adding the references would improve traceability.
  2. [Methods] Notation for the synaptic-strength distribution parameters (scale and location for Cauchy, variance for Gaussian) should be introduced once and used consistently in all subsequent equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§3] §3 (stochastic mean-field closure for Cauchy couplings): the derivation invokes standard averaging and central-limit arguments to obtain an effective stochastic field, yet the Cauchy distribution possesses neither finite mean nor variance. The law-of-large-numbers step therefore requires explicit justification (e.g., via stable-distribution theory or truncation analysis) because rare, arbitrarily large couplings can dominate propagation even as N→∞; without this, the predicted percolation threshold is not guaranteed to survive the thermodynamic limit.

    Authors: We acknowledge that the classical law of large numbers does not apply directly to the Cauchy distribution. Our derivation instead exploits the fact that the Cauchy distribution is stable: the sum of independent Cauchy random variables is again Cauchy-distributed, with the scale parameter scaling linearly with N. This property yields a closed stochastic mean-field description without invoking finite moments. To make the justification fully explicit, we will add a short subsection in §3 invoking the generalized central limit theorem for α-stable laws (α=1). This addition will clarify why the percolation threshold remains well-defined in the thermodynamic limit and is consistent with the numerical results cited in the manuscript. revision: partial

  2. Referee: [§4] §4 (phase diagram and transition order): the claim that the percolation transition changes from continuous to discontinuous with excitatory-inhibitory imbalance is central to the abstract but is supported only by qualitative statements. Explicit bifurcation equations, the location of the critical point in parameter space, and a quantitative comparison of order-parameter scaling on either side of the balance line are needed to substantiate the distinction.

    Authors: The referee correctly identifies that the distinction between continuous and discontinuous transitions requires quantitative support. In the revised manuscript we will derive the explicit bifurcation equations from the self-consistency relation of the stochastic mean-field theory, locate the critical value of the excitatory-inhibitory balance parameter at which the transition changes character, and present a quantitative comparison of the order-parameter scaling (obtained via numerical solution of the mean-field equations) on either side of this line. These additions will substantiate the statement in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: mean-field derivation of percolation threshold emerges directly from model equations

full rationale

The paper derives the directed percolation threshold for Cauchy couplings via stochastic mean-field theory applied to the quadratic integrate-and-fire equations. This uses standard averaging over random couplings to obtain an effective stochastic field whose fixed points and stability yield the threshold analytically. No step reduces by construction to a fitted parameter renamed as prediction, a self-defined quantity, or a load-bearing self-citation chain; the result follows from the model assumptions without importing uniqueness theorems or ansatzes from prior author work. The derivation remains self-contained against external benchmarks such as numerical simulations of the microscopic dynamics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of stochastic mean-field theory to large random networks and the quadratic integrate-and-fire neuron model; no new entities are postulated and no parameters are fitted post hoc to produce the threshold.

free parameters (1)
  • critical synaptic strength
    The percolation threshold value is derived from the model but depends on the chosen coupling variance and E-I balance parameters.
axioms (1)
  • domain assumption Stochastic mean-field theory accurately captures stationary activity in the thermodynamic limit for random independent couplings
    Invoked to obtain closed equations for average firing rates and to locate the percolation transition.

pith-pipeline@v0.9.0 · 5487 in / 1419 out tokens · 55542 ms · 2026-05-15T00:55:48.159369+00:00 · methodology

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