Memory in neural activity: long-range order without criticality

TL;DR

Memory–time non-locality can generate long-range order in cortical dynamics without requiring a critical point. Using a two-timescale mesoscopic model with fast neural activity and slow memory resources, the study demonstrates a robust long-range order (LRO) phase where avalanche distributions follow power laws over a wide range of memory timescales $\tau_D$, independent of criticality. Finite-size scaling and correlation-length analyses show that LRO is an extended phase, not a fine-tuned critical point, challenging the strict criticality hypothesis for brain dynamics. The findings highlight memory as a plausible mechanism for large-scale neural coordination with potential implications for understanding cortical processing and neuromorphic computing.

Abstract

The "criticality hypothesis", based on observed scale-free correlations in neural activity, posits that the brain operates at a critical point of transition between two phases. However, the validity of this hypothesis is still debated. Here, employing a commonly used model of cortical dynamics, we find that a phase of long-range order (LRO) in neural activity may be induced by memory (time non-locality) without invoking criticality. The cortical dynamics model contains fast and slow time scales that govern the neural and resource (memory) dynamics, respectively. When the resource dynamics are sufficiently slow, we observe an LRO phase, which manifests in avalanche size and duration probability distributions that are fit well by power laws. When the slow and fast time scales are no longer sufficiently distinct, LRO is destroyed. Since this LRO phase spans a wide range of parameters, it is robust against perturbations, unlike critical systems.

Figures (16)

• Figure 1: An illustration of our phenomenological model. The bottom image depicts our 2D square lattice network, meant to emulate the cerebral cortex, where neural activities $\rho_{\vec{x}}$ are represented by blue cubes. Each cube has some amount of available resources $R_{\vec{x}}$, represented by the red filling and acting as memory degrees of freedom. The green lines represent diffusive couplings between any activity region and its four nearest-neighbors. The top image zooms in on one region; the orange arrows show the dynamical coupling between $\rho_{\vec{x}}$ and $R_{\vec{x}}$, and the flow fields $F_{\rho}$ and $F_{R}$ can be read from the right hand side of Eqs. \ref{['dynamics_eqns1']} and \ref{['dynamics_eqns2']}, respectively.
• Figure 1: (a, d) The nullclines of the dynamics, for (a) a single neural activity region (from the deterministic, noiseless versions of Eqs. \ref{['dynamics_eqns1']} and \ref{['dynamics_eqns2']}), and (d) the mean field equations, Eq. \ref{['eq:mean']}, both at $\tau_D=8$, $a=1$. The flow field is depicted with gray arrows. For better visualization, the length of each arrow is proportional to the logarithm of the flow field's magnitude. The intersection point is unstable in panel (a), but stable in panel (d), revealing that the existence of the down phase relies on the collective dynamics within the system. (b, e) The dynamics of (b) a single neural activity region and (e) the mean fields, both at $\tau_D=8$, $a=1$. We see a stable spiking oscillation in panel (b), while there is no oscillation in panel (e), corresponding to the down phase. (c)(f) Phase diagrams of (c) a single activity region, and (f) the mean field dynamics, with $\tau_D$ and $a$ as parameters. While no down state exists for an isolated activity region, a narrow stripe of the down phase emerges at low $\tau_D$ in the mean-field dynamics. Additionally, the parameter range of the LRO phase reported in the main text ($a = 1$, $25 \lesssim \tau_D \lesssim 82$) does not lie near any regions of phase transition, ruling out the possibility of criticality or quasi-criticality.
• Figure 1: Dynamics of neural activities $\rho$ on a lattice of size $L^2 = 64^2$, with noise level $\sigma=0.1$. Darker blues correspond to higher activities. (a) Dynamics for $\tau_D = 15$, in the down phase. In this regime, there is no activity except in small bursts due to the presence of noise. (b-d) Dynamics for $\tau_D = 25$, $51$, and $77$, respectively, corresponding to the LRO phase. Here, large-scale, highly correlated waves of activity can be observed. (e) Dynamics for $\tau_D = 88$. When $\tau_D$ is sufficiently large ($\tau_D \gtrsim 82$), the activities remain at high values perpetually since the resource decay timescale is extremely slow.
• Figure 2: A visual depiction of the time interval over which our system evolves. We choose to discretize time into intervals of length $\tau_D$, the memory timescale, each of which is further discretized into intervals of length $1/D$, the neural activity timescale.
• Figure 2: (a) Phase diagram of the collective dynamics, with $\tau_D$ and the noise strength $\sigma$ as variables, on a $64\times 64$ lattice. At small noise levels, all neural activity regions are synchronized, resulting in system-wide avalanches. An appropriate noise level breaks synchronization, and memory effects lead to LRO in the system, characterized by a power-law distribution of avalanches. As the noise level further increases, the system is dominated by noise, leading to a phase of noise-induced disorder. A down state is observed when $\tau_D$ is too small, and an up state is observed when $\tau_D$ is too large, both with short-range correlations. (b) The correlation length, $\xi$, defined as the average distance between two units within the same avalanche (excluding system-wide avalanches), as a function of $\sigma$ and $\tau_D$. An expansive phase of LRO with a large correlation length is observed. (c): The fraction of system-wide avalanches, $\alpha$. In the synchronized state, all avalanches are system-wide, so $\alpha \to 1$. In both the down and up states, avalanche activity is absent, and $\alpha \to 0$. In the LRO phase, avalanches follow power-law distributions, leading to a small but finite value of $\alpha$. As the noise strength $\sigma$ increases, avalanche sizes diminish further, resulting in an even smaller—yet still nonzero—value of $\alpha$.