q-bio.NCMar 16, 2026

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

Authors: Makoto Fukushima

Organizations: Department of Physics, Waseda University, Tokyo, Japan · Current address: Honda Research Institute Japan, Wako, Saitama, Japan

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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.

Figures & tables

Figure 1: In-degree distributions Pα(k)P_{\alpha}(k) for N=100N=100 . Solid curves show the exact finite- NN recursion (Eq. ( 3 )); markers show simulations over independently generated graphs. Increasing α\alpha narrows the distribution and reduces the typical fan-in.
Figure 2: Mean in-degree ⟨k⟩\langle k\rangle versus crowding parameter α\alpha for several system sizes. The crowding rule yields logarithmic growth of typical in-degree with NN at fixed α\alpha (Eq. ( 5 )).
Figure 3: In-degree variance Var(k)\mathrm{Var}(k) versus crowding parameter α\alpha for several system sizes. The rigorous result is that Var(k)\mathrm{Var}(k) remains bounded with NN at fixed α\alpha (Eq. ( 6 )); numerically, in the sparse regime, the saturation value is often close to (2α)−1(2\alpha)^{-1} .
Figure 4: Variance Var(k)\mathrm{Var}(k) versus system size NN for representative crowding strengths, illustrating boundedness and numerical saturation with NN at fixed α\alpha in the sparse regime.
Figure 5: Out-degree distribution for the crowding ensemble at N=500N=500 and α=0.77\alpha=0.77 , compared to binomial and Poisson approximations with matched mean.
Figure 6: Histogram of Pearson correlations between node in-degree and out-degree (computed across nodes) over an ensemble of crowding graphs ( N=500N=500 , α=0.77\alpha=0.77 ).
Figure 7: Maximum-likelihood inference of the crowding parameter α\alpha from an observed in-degree histogram. Synthetic experiments show accurate recovery in regimes where Pα(k)P_{\alpha}(k) is informative.
Figure 8: Heterogeneous mean-field map x(t+1)=Fα(x(t);θ)x(t+1)=F_{\alpha}(x(t);\theta) (Eq. ( 9 )) for representative crowding strengths at θ=0\theta=0 . The induced degree distribution Pα(k)P_{\alpha}(k) deforms the macroscopic map and shifts unstable fixed points that govern basin boundaries.
Figure 9: Basin (hitting) probability of reaching the all- +1+1 state versus initial active fraction for N=100N=100 and θ=0\theta=0 . Solid curves show the absorbing-Markov committor prediction (Eq. ( 11 )); markers show direct simulations of the threshold dynamics on crowding graphs.
Figure 10: Finite-size scaling of basin curves at matched mean degree across system sizes. The absorbing-Markov closure captures the sharpening of the basin boundary with increasing NN while revealing systematic finite-size deviations near the critical initial activity.
Figure 11: Baseline comparison at matched mean degree ⟨k⟩≈3\langle k\rangle\approx 3 ( N=500N=500 , θ=0\theta=0 ): basin curves for the crowding ensemble, an ER + graph (a directed Erdős–Rényi baseline conditioned to have minimum in-degree 1), and a regular in-degree graph. Vertical dashed lines indicate HMF unstable fixed points for each degree distribution.
Figure 12: Robustness of simulated basin curves in the crowding ensemble ( N=500N=500 , ⟨k⟩≈3\langle k\rangle\approx 3 ) to tie-breaking at θ=0\theta=0 , changing the threshold to θ=1\theta=1 , and asynchronous updates.
Figure 13: Schematic construction of spatially ordered crowding and the controlled small-world interpolation. (a) Nodes are embedded in space (ring shown) and candidates for each target jj are proposed nearest-first, with random tie-breaking inside equal-distance shells. (b) The crowding acceptance rule depends only on the current accepted in-degree rr : later synapses are less likely to be accepted. (c) A small-world interpolation is obtained by rewiring each accepted edge independently with probability ρ\rho to a random source not already connected to the same target (excluding self-edges), preserving each target’s in-degree and hence Pα(k)P_{\alpha}(k) . The same distance-ordering idea extends to periodic DD -dimensional lattices (2D torus schematic).
Figure 14: Spatially ordered crowding yields a power-law wiring-length distribution and small-world structure at fixed degree statistics ( N=500N=500 , α=2.66\alpha=2.66 ). (a) Edge-length distribution on a ring is well approximated by P(d)∝d−1P(d)\propto d^{-1} . (b) Interpolating from spatially ordered graphs ( ρ=0\rho=0 ) to fully rewired graphs ( ρ=1\rho=1 ) preserves Pα(k)P_{\alpha}(k) while reducing clustering. Shown are the mean local clustering coefficient and mean shortest-path length of the symmetrized simple graph (averaged over connected node pairs), normalized by the fully rewired ( ρ=1\rho=1 ) baseline.
Figure 15: Small-world effects on threshold dynamics at fixed degree statistics ( N=500N=500 , α=2.66\alpha=2.66 , θ=0\theta=0 ). (a) Basin curves Pr⁡{reach all-active}\Pr\{\text{reach all-active}\} as a function of initial activity x0x_{0} for spatially ordered graphs ( ρ=0\rho=0 ), a weakly rewired small-world ( ρ=0.05\rho=0.05 ), and fully rewired graphs ( ρ=1\rho=1 ), compared to the degree-only absorbing-Markov prediction. (b) Probability that dynamics does not reach either absorbing state within T=200T=200 synchronous updates, showing that local clustering increases the prevalence of long-lived non-absorbing outcomes near the basin boundary.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 16: Two-dimensional torus embedding of the crowding architecture ( L=24L=24 , N=L2=576N=L^{2}=576 , α=2.66\alpha=2.66 ). (a) Edge-length distribution under distance-ordered proposals ( ρ=0\rho=0 ) is well approximated by P(d)∝d−1P(d)\propto d^{-1} . (b) Shortcut rewiring interpolates from the spatially ordered case to a fully rewired graph while preserving Pα(k)P_{\alpha}(k) ; the plotted clustering coefficient and characteristic path length are the mean local clustering coefficient and mean shortest-path length of the symmetrized simple graph (averaged over connected node pairs), normalized by the fully rewired ( ρ=1\rho=1 ) baseline.
Figure 17: Robustness of the wiring-length distribution to softened distance ordering on a ring ( N=500N=500 , α=2.66\alpha=2.66 ). Candidate sources are proposed nearest-first with probability 1−β1-\beta and uniformly at random among remaining candidates with probability β\beta . For small β\beta , the marginal wiring-length distribution remains close to P(d)∝1/dP(d)\propto 1/d (log-uniform lengths), while for β→1\beta\to 1 it approaches the random-order baseline. Dashed line indicates a 1/d1/d reference slope.
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