Dynamically selected steady states and criticality in non-reciprocal networks

TL;DR

We study how non-reciprocal interactions in a fully connected neural-rate network drive dynamical selection of steady states near stability, bridging spin-glass physics and dynamical systems. Using a path-integral dynamic mean-field approach, we derive self-consistent equations for order parameters $M$, $q$, and two-time correlators, and analyze fixed-point and non-fixed-point states across uncorrelated ($\gamma=0$) and correlated ($\gamma\neq0$) couplings, with and without noise $\sigma$. We identify two types of criticality: type-I fixed-point transitions to ferromagnetic order and type-II transitions to spin-glass-like states; in the spin-glass region, non-reciprocity induces chaos, with the system dynamically selecting states on the separatrix and exhibiting aging-like, non-time-translation-invariant dynamics. Adding noise or strong signal collapses the multiplicity of steady states, shrinking the chaotic region and recovering single-boundary behavior, akin to transitions to paramagnetic or ferromagnetic phases. The work thus links chaotic non-equilibrium phases to equilibrium fRSB spin-glass structure and outlines a framework for exploring non-reciprocal disordered systems beyond reciprocal coupling paradigms.

Abstract

Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric. Our results are relevant to advance the knowledge of disordered systems beyond the paradigm of reciprocal couplings, and to develop an interface between statistical physics of equilibrium spin-glasses and dynamical systems theory.

Figures (14)

• Figure 1: The left panel illustrates a fully-connected network with $N$ nodes and coupling strengths $J_{ij}$, drawn randomly and independently from a Gaussian distribution with mean $J_0/N$ and variance $J^2/N$. The central and right panels sketch typical trajectories, $x_i(t)$, obtained from a numerical simulation of Eq. \ref{['eq:dynamics-def']} for a network instance (with $\gamma=0$). The trajectories of a few nodes (solid grey lines) are plotted together with the mean $\hat{M}(t)=N^{-1}\sum_i x_i(t)$ (blue line) and variance $\hat{q}(t)=N^{-1}\sum_i x_i^2(t)$ (red line). For type-I phase transitions (central panel), the trajectories converge to steady-state values (on the same timescale) while for type-II (right panel) trajectories are highly irregular or chaotic. The insets show the distribution of eigenvalues (in the complex plane) for each case: for type-I transitions only one eigenvalue, the outlier, has crossed the instability line ${\rm Re}[\lambda]=1/g_c$ (shown by the dashed line), while for type-II, a section of the bulk has crossed the instability line (without a gap), giving rise to a more complex dynamics.
• Figure 2: Phase diagram of the model for uncorrelated ($\gamma = 0$) and noiseless dynamics ($\sigma^2 = 0$). Panels (A, B): Heat map of the time-averaged mean activity $\hat{M}$ (panel (A)) and mean-squared activity (or equal-time correlator) $\hat{C}(0)$ (panel (B)), obtained from simulations as a function of the control parameters $J_0/J$ (as color coded) and $1 / g J$. The white lines represent the theoretically predicted critical curves (from stability analyses of fixed-point solutions) separating the paramagnetic (P), ferromagnetic (F) and spin-glass (SG) phases. Panels (C, D): Symbols show $\hat{M}$, from panel (A), and $\hat{C}(0)$, from panel (B), obtained from simulations, versus $1/gJ$, at three different values of $J_0/J$, corresponding to the three dashed vertical lines on (A) and (B). Solid lines show the theoretically predicted asymptotic behaviour around the critical point such that $M \sim (g-g_c)^{\alpha}$ (defined only for the ferromagnetic transition) and $q \sim (g-g_c)^{\beta}$. (Note that $M$ and $q$ denote, respectively, the values of the mean and the mean-squared activity obtained from the theory ---by averaging over initial conditions and network ensemble---, assuming a fixed-point solution, whereas $\hat{M}$ and $\hat{C}(0)$ denote their numerical counterparts obtained from numerical simulations by averaging over different realizations.) The inset illustrates the effect of increasing the system size ($N = 100, 1000, 10000$).
• Figure 3: Analysis of the motion in the effective potential $V(C|C(0),M)$ for the uncorrelated ($\gamma = 0$) and noiseless ($\sigma^2 = 0$) case. Panels (A, D): potential $V(C|C(0),M)$ as a function of $C$ for $M\!\neq\! 0$ (ferromagnetic phase, with $J_0/J = 1.5$) and $M\!=\!0$ (spin-glass phase, with $J_0/J = 0.5$), respectively. Each curve is obtained for a different value of $C(0)$ (as shown in the legend with a color code) and is plotted in the range $C\in[-C(0),C(0)]$. Note that only in the SG phase the potential may exhibit a shape with either one or two wells. The heat maps in panels (B, C, E, F) describe the probability distribution function (PDF) $P(C_0)$ (as defined in Eq.\ref{['eq:PDF_Cs']}) of finding a given value of $C(0)$ for the ferromagnetic case (B, C) and the SG phase (E, F), for different values of $1/gJ$ (obtained from $S=1000$ simulations of the microscopic dynamics, averaged over a time window $t_m=2000$, for system size $N=100$ (B, E) and $N=1000$ (C, F)). Each simulation corresponds to a different realization of the random initial condition and quenched disorder. Stars show the values of $1/gJ$ and $C(0)$ used to plot panels (A, D). In panels (B, C), the solid line ("q-line") describes the solution $q$ obtained from the fixed-point solution, revealing that for large $N$, the PDF becomes peaked around this value, so that $C(0)=q$. On the other hand, in panels (E, F), the PDF becomes more and more peaked (as the system size is enlarged from $N=100$ to $N=1000$) around a value $C_0^\star \neq q$, corresponding to the separatrix point (lying in the green curve) at which the potential verifies $V(C(0) | C(0), 0) = 0$ ($C(0) = 0.48$ in (D)). Note that such a green line is in between the q-line (yellow) and the threshold curve $C_{\rm th}$ (red) as described in the main text, so that in particular $C(0) \neq q$ in the SG phase.
• Figure 4: Illustration of how fluctuations in the initial condition translate into changes in the shape of the potential (upper). Small fluctuations have a large impact when the initial condition varies around the separatrix value $C^*_0$ (lower).
• Figure 5: Single instances of the dynamics for uncorrelated ($\gamma = 0$) and noiseless dynamics ($\sigma^2 = 0$). The (upper panels) show the time-dependent mean-squared activity $q(t_0+\tau)=N^{-1}\sum_i x_i^2(t_0+\tau)$ (gray line in the upper panels) plotted together with its time average $\hat{C}_s(0)$ and (lower panels) $\hat{C}_s(\tau)$. Curves are obtained from simulations with system size $N=100$ and $1/gJ=0.58$ (i.e., within the SG phase). Observe that the trajectories reveal either chaotic behavior (A) or periodic motion (B, C, D, E). The yellow and the red dashed lines show $q$ and $C_{\rm th}$, respectively while the green dashed line --in between the previous two--- marks the separatrix $C_0^\star$. In (A) the mean-squared activity fluctuates randomly above and below the separatrix line; in (B, C) the mean-squared activity remains at only one side of the separatrix; in (D, E) the mean-squared activity fluctuates about the separatrix in sync with $\hat{C}_s(\tau)$.