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Analytically tractable model of synaptic crowding explains emergent small-world structure and network dynamics

Makoto Fukushima

Abstract

Neural circuits must balance local connectivity constraints against the need for global integration. Here we introduce a minimal wiring rule motivated by synaptic crowding: as a neuron accumulates incoming connections, each additional synapse becomes progressively harder to form. This single-parameter model admits an exact finite-size solution for the induced in-degree distribution and yields simple scaling laws: mean connectivity grows only logarithmically with network size while variance remains bounded -- consistent with homeostatic regulation of synaptic density. When candidates are encountered in order of spatial proximity, the crowding rule produces a broad, approximately power-law distribution of connection lengths without prescribing any explicit distance-dependent wiring law; combined with shortcut rewiring, this yields networks with small-world characteristics. We further show that the induced degree statistics largely determine attractor basin boundaries in threshold network dynamics, while local clustering primarily modulates the prevalence of long-lived non-absorbing outcomes near these boundaries. The model provides testable predictions linking local developmental constraints to macroscopic network organization and dynamics.

Analytically tractable model of synaptic crowding explains emergent small-world structure and network dynamics

Abstract

Neural circuits must balance local connectivity constraints against the need for global integration. Here we introduce a minimal wiring rule motivated by synaptic crowding: as a neuron accumulates incoming connections, each additional synapse becomes progressively harder to form. This single-parameter model admits an exact finite-size solution for the induced in-degree distribution and yields simple scaling laws: mean connectivity grows only logarithmically with network size while variance remains bounded -- consistent with homeostatic regulation of synaptic density. When candidates are encountered in order of spatial proximity, the crowding rule produces a broad, approximately power-law distribution of connection lengths without prescribing any explicit distance-dependent wiring law; combined with shortcut rewiring, this yields networks with small-world characteristics. We further show that the induced degree statistics largely determine attractor basin boundaries in threshold network dynamics, while local clustering primarily modulates the prevalence of long-lived non-absorbing outcomes near these boundaries. The model provides testable predictions linking local developmental constraints to macroscopic network organization and dynamics.
Paper Structure (36 sections, 1 theorem, 40 equations, 17 figures)

This paper contains 36 sections, 1 theorem, 40 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Testable predictions.
  3. Contributions.
  4. Model
  5. Crowding-based wiring rule
  6. Threshold dynamics
  7. Degree statistics of the crowding ensemble
  8. Exact finite-$N$ recursion and generating function
  9. Mean and variance scaling with system size
  10. Out-degree statistics and in/out correlations
  11. Inferring the crowding parameter from degree data
  12. Heterogeneous mean-field dynamics
  13. Finite-size basin probabilities via an absorbing Markov closure
  14. Spatially ordered crowding and small-world structure
  15. Distance-ordered proposals preserve $P_{\alpha}(k)$
  16. ...and 21 more sections

Key Result

Proposition 1

Consider a spatial embedding in $D$ dimensions and a candidate list ordered by increasing distance from a target node. Under the crowding rule in Eq. eq:accept_prob, the acceptance probability of proposal rank $m$ satisfies $p_m\sim (\alpha m)^{-1}$ for large $m$. If the number of candidates within

Figures (17)

  • Figure 1: In-degree distributions $P_{\alpha}(k)$ for $N=100$. Solid curves show the exact finite-$N$ recursion (Eq. \ref{['eq:Pt_recursion']}); markers show simulations over independently generated graphs. Increasing $\alpha$ narrows the distribution and reduces the typical fan-in.
  • Figure 2: Mean in-degree $\langle k\rangle$ versus crowding parameter $\alpha$ for several system sizes. The crowding rule yields logarithmic growth of typical in-degree with $N$ at fixed $\alpha$ (Eq. \ref{['eq:mean_scaling']}).
  • Figure 3: In-degree variance $\mathrm{Var}(k)$ versus crowding parameter $\alpha$ for several system sizes. The rigorous result is that $\mathrm{Var}(k)$ remains bounded with $N$ at fixed $\alpha$ (Eq. \ref{['eq:var_scaling']}); numerically, in the sparse regime, the saturation value is often close to $(2\alpha)^{-1}$.
  • Figure 4: Variance $\mathrm{Var}(k)$ versus system size $N$ for representative crowding strengths, illustrating boundedness and numerical saturation with $N$ at fixed $\alpha$ in the sparse regime.
  • Figure 5: Out-degree distribution for the crowding ensemble at $N=500$ and $\alpha=0.77$, compared to binomial and Poisson approximations with matched mean.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Proposition 1: Emergent distance kernel from crowding and spatial ordering