Topological Origin of the Diversity of Timescales in Recurrent Neural Circuits

TL;DR

The results provide a mechanistic link between connectome topology, neural dynamics, and computation, identifying hubs in partially symmetric networks as a natural substrate for multiplexed processing across timescales.

Abstract

Structural and functional heterogeneity are hallmarks of cortical circuits, from broad degree distributions in the mouse connectome to diverse intrinsic neuronal timescales. Yet a mechanistic link between connectivity heterogeneity and functional diversity is lacking. To bridge this gap, we introduce a random recurrent network in which connectivity is generated by a configuration model with tunable degree heterogeneity and synaptic weights exhibiting varying levels of correlation. Using generating-functional methods, we derive a heterogeneous dynamical mean-field theory (hDMFT) with degree-conditioned stochastic dynamics. The theory shows that the interaction of partial symmetry in the weights and degree heterogeneity induces a non-Markovian memory term in the form of an emergent self-coupling whose strength scales with degree and produces a broad distribution of activity timescales. We obtain analytic stability criteria demonstrating that degree heterogeneity lowers the critical gain and localizes unstable modes onto hubs. The resulting rich dynamical landscape includes silent, chaotic, and multistable regimes, which we uncover via spectral, replica, and Lyapunov exponent analyses. We highlight the computational benefits of the observed timescale heterogeneity by revealing that, under an external input drive featuring a broadband spectrum, slow hub neurons act as integrators, demixing slow input components. Finally, instantiating the model with the empirically measured topology from the MICrONS cubic-millimeter mouse connectome explains the broad range of single-neuron timescales and their positive correlation with in-degree observed in resting-state recordings. Our results provide a mechanistic link between connectome topology, neural dynamics, and computation, identifying hubs in partially symmetric networks as a natural substrate for multiplexed processing across timescales.

Figures (17)

• Figure 1: Nontrivial topology of neural circuits in mouse visual cortex microns2025functional. (a) Spatial distribution of neurons within the MICrONS Cubic Millimeter volume. Colored markers indicate the subset of proofread neurons analyzed in this study, highlighted according to visual cortical area: primary visual cortex (V1), anterolateral visual area (AL), and rostrolateral visual area (RL). Gray points show the surrounding neuronal population within the same volume. (b) Corresponding binary adjacency matrix of synaptic connectivity among the $N=1965$ proofread neurons shown in (a). (c) In-degree distribution $p(k)$ of the neurons together with a lognormal fit (red curve; shape parameter $\sigma=0.69$, scale parameter $\mu=3.83$), indicating a heavy-tailed connectivity profile. (d) Pairwise connectivity between neurons compared to an Erdős--Rényi null model with connection probability matching empirical observation ($p=2.67\%$). Bars report the observed counts of non-connected, unidirectional, and reciprocal pairwise connections (95.25%, 4.17%, and 0.58% of all neuron pairs, respectively) normalized by their expectation under the null model (horizontal gray line at 1). Reciprocal connections are strongly overrepresented, while unidirectional connections are less with respect to the the null model (Monte Carlo test with $10000$ random networks, $p<10^{-4}$). Quantifying reciprocity by the Pearson correlation between opposite directed edges gives $r = 0.197$ (see Appendix \ref{['app:microns']}).
• Figure 2: Effects of heterogeneous degree distribution at $\gamma=0$ (asymmetric case). (a) Phase diagram for $\gamma = 0$ with a lognormal degree distribution. The solid black line corresponds to the theoretical prediction, Eq. (\ref{['eq:crit_cond_LN_gamma0']}), for the critical gain, separating the trivial fixed-point phase from the chaotic phase. Insets show examples of dynamics in the two regions. (b) Example of a synthetic degree distribution for a network with $N = 2000$ units, drawn from a lognormal distribution with $\mu = 3$ and $\sigma = 1$. The solid line represents the theoretical degree distribution, and the dots correspond to degrees obtained through the configuration model sampling procedure. (c) Numerical validation of the critical gain prediction for a network with a lognormal degree distribution ($\mu = 3$, $\sigma = 1$) and no cross-correlations ($\gamma = 0$). The parameter $f$ quantifies the temporal volatility of neuronal activities at stationarity and is evaluated over $20$ network realizations with $1000$ neurons each. (d) Eigenvalue spectrum of the Jacobian matrix computed at the trivial fixed point for a lognormal degree distribution with $\mu = 3$ and $\sigma = 1$ for a network realization with $N = 2000$ nodes and $g = 0.8$. Scatter points are colored based on the degree score defined in Eq. (\ref{['eq:degree_score']}). (e) Radial density distribution of the eigenvalue disk for networks with $N = 2000$ units and Poisson ($K = 100$, $g = 0.8$) or lognormal ($\mu = 3$, $\sigma = 1$, $g = 0.8$) degree distributions.
• Figure 3: Phase diagram with heterogeneous and reciprocal connections. (a) Phase diagram for a network with a lognormal connectivity distribution, $P(k) = \mathcal{LN}(\mu = 3, \sigma = 1)$, showing phase boundaries. Insets (i)–(iv) illustrate representative dynamical regimes within different regions of the phase diagram. Histograms display the stationary activity distributions of neurons, while trajectories are colored according to the degree percentile (red for high-degree neurons, blue for low-degree neurons). High-degree neurons exhibit larger activity fluctuations and, in some regimes, bistability in their stationary distributions. b) Numerical validation of the critical condition obtained by solving Eqs. (\ref{['eq:crit_cond_silent']}) together with (\ref{['eq:chi']}), shown as the black solid line. The blue error bars are obtained by sampling $100$ interaction matrices with $N = 4000$ neurons and interpolating the values of $g$ for which the largest real eigenvalue of the stability matrix crosses zero girko1985circular. The heatmap represents the largest Lyapunov exponent computed using the Jacobian method engelken2023lyapunov with $N = 3000$ and a simulation time of $t_{\mathrm{sim}} = 3000$. c) Replica distance (blue) and LLE (black) of networks with $g = 3$, $\mu = 3$, $\sigma = 1$, and varying $\gamma$. The dashed-dotted line indicates the interpolated transition between ergodicity and replica symmetry breaking at $\gamma \simeq 0.6$ (color-coded shades as in panel b.
• Figure 4: Emergence of heterogeneous timescales. (a) Distribution of neuronal timescales for poisson and lognormal degree distributions at gain $g = 3$ and symmetry parameter $\gamma = 0.4$. The neuronal timescale $\tau$ is defined as the half-width at half-maximum (HWHM) of each neuron’s autocorrelation function. Parameters of the degree distributions are reported in the legend. (b) Coefficient of variation of neuronal timescales, $\mathrm{CV}(\tau)$, as a function of the symmetry parameter $\gamma$ and the coefficient of variation of the lognormal degree distribution $\mathrm{CV}(k)=\sqrt{e^{\sigma^2}-1}$. Increasing heterogeneity in the degree distribution and larger symmetry both lead to a progressive broadening of the timescale distribution, indicating enhanced heterogeneity in neuronal dynamics. For each degree distribution, dynamics are simulated over 20 independent network realizations with $N = 4000$.
• Figure 5: Network response to broadband input. (a) Population-averaged power spectral density (PSD) of the network activity under a broadband periodic drive composed of the superposition of $M=11$ sinusoidal components with different frequencies (vertical dashed lines; input amplitude $A=0.5$). Red and blue curves show the PSD averaged over nodes over nodes in a network with $\gamma=0.3$, degree distribution $k \in \{100,1000\}$ and probabilities $P_k=\{0.9,0.1\}$, respectively. Circles indicate the PSD amplitudes evaluated at the driven frequencies. (b) Modulation index ($\mathrm{MI}$) between the high- and low-degree PSD responses as a function of the driving frequency for different values of the symmetry parameter $\gamma$ (color-coded; $\gamma=0.0,\,0.1,\,0.3,\,0.4,\,0.5$). Increasing $\gamma$ enhances the frequency selectivity of the degree-dependent response, making $\mathrm{MI}$ increasingly concentrated at low frequencies, consistent with a stronger dominance of high-degree nodes for the slow input components. Network parameters: $g=3.0$, $N=4000$.