Non-normal Dynamics on Non-reciprocal Networks: Reactivity and Effective Dimensionality in Neural Circuits

TL;DR

This work frames both local and global asymmetries in terms of non-normal dynamics, and shows that their combination leads to emergent collective dynamics, including fluctuation-driven transitions, dimensionality reduction, and novel nonequilibrium steady states.

Abstract

Non-reciprocal interactions are a defining feature of many complex systems, biological, ecological, and technological, often pushing them far from equilibrium and enabling rich dynamical responses. These asymmetries can arise at multiple levels: locally, in the dynamics of individual units, and globally, in the topology of their interactions. In this work, we investigate how these two forms of non-reciprocity interact in networks of neuronal populations. At the local level, each population is modeled by a non-reciprocally coupled set of excitatory and inhibitory neural populations exhibiting transient amplification and reactivity. At the network level, these populations are coupled via directed, asymmetric connections that introduce structural non-normality. Since non-reciprocal interactions generically lead to non-normal linear operators, we frame both local and global asymmetries in terms of non-normal dynamics. Using a modified Wilson-Cowan framework, we analyze how the interplay between these two types of non-normality shapes the system's behavior. We show that their combination leads to emergent collective dynamics, including fluctuation-driven transitions, dimensionality reduction, and novel nonequilibrium steady states. Our results provide a minimal yet flexible framework to understand how multi-scale non-reciprocities govern complex dynamics in neural and other interconnected systems.

Figures (4)

• Figure 1: Inter-population interactions.(a): Schematic representation of the generic non-reciprocal motif. $X$ and $Y$ denote the fractions of active excitatory and inhibitory populations, respectively. Local excitatory couplings are shown in dark green, while inhibitory interactions are indicated in dark red. Light green arrows illustrate inter-population interactions. Two of these connections (represented in light blue) are modulated by the parameter $c$, with $0 \leq c \leq \epsilon$, as defined in Eq. \ref{['Eq.A']}. This modulation allows to interpolate between the homogeneous loop and the feedforward configuration. (b): Schematic representation of the model in the transformed variables ($\Sigma = x+y$, $\Delta = x-y$). In the transformed variables (b) the feedforward configuration includes both within-unit (orange) and between-units (red) couplings, while the cyclic network only includes within-unit (orange) interactions, i.e. the three units are disjoint.
• Figure 2: Bifurcation diagram and scaling of the fixed points.Panels (a--c) Scaling of the fixed points around the bifurcation for $x_1,x_2$, and $x_3$ respectively. The scaling is the same for the decoupled case (purple), for $c = \epsilon/2$ (light blue) and $c = \epsilon$ (orange). (d) Scaling of the fixed points around the bifurcation for the Feedforward case ($c = 0$). Note the change of the slope as expected by theoretical results, depending on the population (see legend). (e) Bifurcation diagram for the basal population $x_1$, obtained via Runge-Kutta integration. Parameters $\alpha=0.1$, $\gamma_{\nu}=0.35$, and $\gamma_{\ell}=0.2$ are used across several setups. The orange line marks the bifurcation point at $\gamma_{\mu} = \gamma_{\mu}^{c,\epsilon} = 0.25$ for the cyclic system. The decoupled diagram, with bifurcation point predicted at $\gamma_{\mu} = \gamma_{\mu}^{c} = 0.45$, is shifted so that $\gamma_{\mu}^{c,\epsilon} = \gamma_{\mu}^{c}$, resulting in a near-perfect overlap of the cyclic and decoupled curves. Several values of $c$ are also evaluated, up to $c=0$ (feedforward setup), with the expected bifurcation point $\gamma_{\mu}^{c,0} = 0.45$.
• Figure 3: Reactivity and comparison of numerical abscissa and non-normality parameters. Left panels. (a,b): Level curves of the numerical abscissa $m$ for different values of $\gamma_\ell$, evaluated at the inactive phase. Colored regions indicate the subspace where the level curves are defined. Straight lines correspond to fixed values of the non-normality parameters ($\zeta = \text{const}$ or $\xi = \text{const}$). Panel (a) shows the cyclic setup, while panel (b) shows the feedforward setup. Increasing $\gamma_\ell$ shifts the $m$ level curves leftward, so that $m$ increases for each pair $(\gamma_\nu,\gamma_\mu)$, demonstrating that all possible values of $m$ can be obtained while keeping the non-normality parameter fixed. Right panels (c--f): Time evolution of $\rho(t)$ under variable reactive conditions. Top row (c,d) corresponds to the decoupled system, with couplings fixed and $\xi$ constant. Bottom row (e,f) corresponds to setups where the non-normality of feedforward and cyclic systems is adjusted to maintain a consistent level across configurations. In this case, only the feedforward system exhibits a significant increase in reactivity. Reactivity increases from left to right with the numerical abscissa $m$.
• Figure 4: Comparison of cyclic and feedforward setups in the stochastic regime. Top panels show the cyclic configuration; bottom panels show the feedforward setup. Comparing the left and right columns, the system shifts from an incoherent regime (b, d) to either a coherent phase in the cyclic case (a) or step-like trajectories in the feedforward case (c). Parameters: (a)$\gamma_{\ell}=1.13$, $\xi=0.004$; (b)$\gamma_{\ell}=0.13$, $\xi=0.004$; (c)$\gamma_{\ell}=1.13$, $\zeta=0.005$; (d)$\gamma_{\ell}=0.07$, $\zeta=0.005$.