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) for N=100 . Solid curves show the exact finite- N recursion (Eq. ( 3 )); markers show simulations over independently generated graphs. Increasing α narrows the distribution and reduces the typical fan-in.
Figure 2: Mean in-degree ⟨k⟩ versus crowding parameter α for several system sizes. The crowding rule yields logarithmic growth of typical in-degree with N at fixed α (Eq. ( 5 )).
Figure 3: In-degree variance Var(k) versus crowding parameter α for several system sizes. The rigorous result is that Var(k) remains bounded with N at fixed α (Eq. ( 6 )); numerically, in the sparse regime, the saturation value is often close to (2α)−1 .
Figure 4: Variance Var(k) versus system size N for representative crowding strengths, illustrating boundedness and numerical saturation with N at fixed α in the sparse regime.
Figure 5: Out-degree distribution for the crowding ensemble at N=500 and α=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=500 , α=0.77 ).
Figure 7: Maximum-likelihood inference of the crowding parameter α from an observed in-degree histogram. Synthetic experiments show accurate recovery in regimes where Pα(k) is informative.
Figure 8: Heterogeneous mean-field map x(t+1)=Fα(x(t);θ) (Eq. ( 9 )) for representative crowding strengths at θ=0 . The induced degree distribution Pα(k) deforms the macroscopic map and shifts unstable fixed points that govern basin boundaries.
Figure 9: Basin (hitting) probability of reaching the all- +1 state versus initial active fraction for N=100 and θ=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 N while revealing systematic finite-size deviations near the critical initial activity.
Figure 11: Baseline comparison at matched mean degree ⟨k⟩≈3 ( N=500 , θ=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=500 , ⟨k⟩≈3 ) to tie-breaking at θ=0 , changing the threshold to θ=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 j 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 r : later synapses are less likely to be accepted. (c) A small-world interpolation is obtained by rewiring each accepted edge independently with probability ρ to a random source not already connected to the same target (excluding self-edges), preserving each target’s in-degree and hence Pα(k) . The same distance-ordering idea extends to periodic D -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=500 , α=2.66 ). (a) Edge-length distribution on a ring is well approximated by P(d)∝d−1 . (b) Interpolating from spatially ordered graphs ( ρ=0 ) to fully rewired graphs ( ρ=1 ) preserves Pα(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 ) baseline.
Figure 15: Small-world effects on threshold dynamics at fixed degree statistics ( N=500 , α=2.66 , θ=0 ). (a) Basin curves Pr{reach all-active} as a function of initial activity x0 for spatially ordered graphs ( ρ=0 ), a weakly rewired small-world ( ρ=0.05 ), and fully rewired graphs ( ρ=1 ), compared to the degree-only absorbing-Markov prediction. (b) Probability that dynamics does not reach either absorbing state within T=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=24 , N=L2=576 , α=2.66 ). (a) Edge-length distribution under distance-ordered proposals ( ρ=0 ) is well approximated by P(d)∝d−1 . (b) Shortcut rewiring interpolates from the spatially ordered case to a fully rewired graph while preserving Pα(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 ) baseline.
Figure 17: Robustness of the wiring-length distribution to softened distance ordering on a ring ( N=500 , α=2.66 ). Candidate sources are proposed nearest-first with probability 1−β and uniformly at random among remaining candidates with probability β . For small β , the marginal wiring-length distribution remains close to P(d)∝1/d (log-uniform lengths), while for β→1 it approaches the random-order baseline. Dashed line indicates a 1/d reference slope.
Recent breakthroughs in synaptic-resolution network connectomics have revealed that brain circuits feature fine-scale structural connectivity, such as pairs of correlated synaptic couplings known as second-order motifs. Large-scale recordings of neuronal activity in networks containing nonlinear neurons reveal macroscopic heterogeneous population dynamics throughout the brain. These findings rekindle the inquiry into this intriguing question: Can microscale synaptic structures contribute to macroscopic heterogeneous dynamics and computations in ways that canonical brain circuit models cannot? To answer this question, we create random RNNs with various cell types, nonlinear non-negative neural responses, and arbitrary marginal and second-order correlated synaptic statistics. We derive mean-field low-rank equations for P-population networks in which the pre- and postsynaptic neuronal population identities determine the synaptic and motif strengths. Our framework requires 2P latent dynamic variables with P variables describing mean population activity and P variables capturing within-population variability. Theoretical and simulational results demonstrate that chain motifs induce correlations in synaptic variability, enabling microscopic fluctuations to be integrated and influence mesoscopic mean population dynamics. We apply this framework to reverse engineer network connectivity that recapitulates the heterogeneous activity across the population in the mouse primary visual cortex. By bridging the gap between synaptic organization and nonlinear heterogeneous population dynamics, our results offer a principled approach and testable predictions regarding the relationship between fine-scale connectivity, heterogeneous dynamics, and functional computations.
Meiyi Zhang, Jinjian Yu, Louis Tao +1
Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, China · Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing, China · School of Physics, Peking University, China +7
We construct a minimal but biologically plausible spiking neuron model operating in discrete time, combining multiplicative spike-timing-dependent plasticity (WSTDP), divisive normalization of synaptic integration, homeostatic threshold adaptation, and a one-step refractory period. We show that this normalization admits a biologically plausible dendritic implementation in which each binary junction operates using only locally available information. Assembling excitatory-inhibitory pairs of such neurons into a two-dimensional recurrent network and applying periodic localized stimulation, we find that the network spontaneously gives rise to stable, self-propagating wave packets with the properties of dissipative solitons: they maintain a stable spatial profile, propagate at constant speed, and annihilate upon frontal collision. Their emergence requires a geometric asymmetry between excitatory and inhibitory connection radii, and initial inhibitory synapses stronger than excitatory ones. WSTDP engraves the direction of propagation into the synaptic weight profile, so that the network learns by itself to sustain propagation in one direction while suppressing the reverse. When two sources are active simultaneously, the resulting waves annihilate upon collision, defining a semi-persistent boundary whose position encodes the relative phase and frequency of the two sources. These results provide a minimal computational framework for studying the emergence of cortical traveling waves, activity zone delimitation, and spatial memory from local plasticity rules alone.
Ch. Meessen
Independent researcher, computational neuroscience, France
Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. To find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns a distribution over connection weights that is maximally unbiased over unidentifiable components while matching the observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.
Timothy Doyeon Kim, Ulises Pereira-Obilinovic, Yiliu Wang +2
Allen Institute, Seattle, WA, USA · University of Washington, Seattle, WA, USA