Memory-induced long-range order drag

TL;DR

This work asks whether memory-induced long-range order (MILRO) can be transmitted from a memory-rich base layer to downstream memory-free layers via local feedforward couplings. It develops an analytical framework where slow memory with frequency $\gamma$ creates effective long-range intra-layer correlations in the base, and shows that sufficiently strong inter-layer coupling $J^{\perp}$ drags this order into deeper layers, producing LRO across the stack. Numerical simulations confirm that dragged layers display identical scale-free avalanche statistics and finite-size scaling with $D=2$, validating the drag mechanism beyond a single layer. The results point to practical guidelines for layered neuromorphic architectures and offer a potential explanation for sustained inter-laminar cortical correlations through local connectivity.

Abstract

Recent research has shown that memory, in the form of slow degrees of freedom, can induce a phase of long-range order (LRO) in locally-coupled fast degrees of freedom, producing power-law distributions of avalanches. In fact, such memory-induced LRO (MILRO) arises in a wide range of physical systems. Here, we show that MILRO can be transferred to coupled systems that have no memory of their own. As an example, we consider a stack of layers of spins with local feedforward couplings: only the first layer contains memory, while downstream layers are memory-free and locally interacting. Analytical arguments and simulations reveal that MILRO can indeed drag across the layers, enabling downstream layers to sustain intra-layer LRO despite having neither memory nor long-range interactions. This establishes a simple, yet generic mechanism for propagating collective activity through media without fine tuning to criticality, with testable implications for neuromorphic systems and laminar information flow in the brain cortex.

Figures (10)

• Figure 1: Schematic of the memory-induced LRO drag phenomenon in our system. In all layers, continuously relaxed spins interact via traditional spin-glass interactions. In the base $k = 0$ layer, spins also experience memory interactions, where the memory $x_{ij}$ is an auxiliary dynamical field which depends on the state of neighboring spins $s_i^{(0)}$ and $s_j^{(0)}$ and their mutual interaction $J_{ij}^{(0)}$. In deeper layers, memory does not appear, but instead, unidirectional, inter-layer interactions exist which enable the memory-induced LRO in layer 0 to be "dragged" through deeper layers.
• Figure 2: Schematic of two layers of continuously relaxed spins alongside avalanche size $s$ probability distributions $P(s)$. We notice that, in both layers, scale-free avalanche distributions exist which collapse well under finite-size scaling (shown in the insets). Furthermore, the scaling exponents in these distributions are nearly identical. This suggests that the collective, LRO state is effectively transferred from the base layer $k=0$ to the $k=1$ layer due to the inter-layer interactions. Both layers are simulated over 100 instances for $T=200$ simulation time (a.u.), with $\gamma = 0.2$ and $J^\perp = 3.5$.
• Figure 3: Depth-resolved phase structure and representative avalanche statistics. (a) Avalanche size distributions $P(s)$ for $\{\gamma, J^\perp\} = \{0.045, 5.0\}, \{0.045, 3.1\}, \{2.2, 3.1\}$ (stars correspond to locations in the phase diagram in panel b). Dashed lines indicate power-law guides with exponents $\gamma\simeq 1.90$ and $1.92$, respectively. (b) Phase diagrams in the $\{\gamma, J^\perp\}$ plane at depths $k=0, 1, 2, 5, 10$. Colors denote long-range order, crossover, and short-range correlations. Stars mark the parameter points used in (a). All points in the parameter space are simulated over 25 instances for $T=200$ simulation time (a.u.) (with initial transient dynamics removed) on a $64 \times 64$ lattice.
• Figure 4: Transient build-up of long-range order across layers. Time evolution of the in-plane correlation length $\xi_k(t)$ for the driving layer ($k=0$) and ten downstream layers ($k=1-10$) at $\{\gamma, J^\perp\} = \{0.045, 5.0\}$ on a $64\times 64$ lattice with $J=1$, simulated over 100 instances for 4096 time steps, where each time step $\Delta t=0.017$. All layers exhibit a rapid rise over $\sim 10^2$ steps followed by a plateau. Mild overshoots and subsequent relaxation reflect the exclusion of system-wide events from $\xi$.
• Figure 5: The flipping time $\Delta t$ over which a single spin $s_i^{(k)}$ flips, in both a) the base layer $k = 0$ and b) other layers $k \geq 1$. This timescale varies depending on the nature of the relevant spin-glass interactions. a) When $J_4 = 0$, $\Delta t = \sqrt{3/g \gamma \delta^*}$. When $J_4 < 0$, the flipping time decays as $\Delta t \sim 1/\gamma$, until $\gamma \sim O(1)$. When $J_4 > 0$, the flipping time remains constant at values near $1$ for $J_4 = 2J$ and $1/2$ for $J_4 = 4J$, until $\gamma \sim O(1)$. b) $\Delta t$ decays monotonically when both $J_4 > 0$ and $J_1^\perp > 0$. Divergences appear at $J^\perp = 2J$ and $J^\perp = 4J$ when the inter-layer and intra-layer spin-glass interactions are in opposition, and in these cases, flips can only occur within a particular range of $J^\perp$. When $J_4 < 0$ and $J_1^\perp < 0$, no flip will ever occur, so these curves cannot be plotted. Parameters chosen: $g = 2$, $\delta^* \equiv (\delta - 1/2) = 1/4$, $J = 1$.