Freezing chaos without synaptic plasticity

TL;DR

The paper addresses chaotic dynamics in high-dimensional random recurrent neural networks and proposes an Onsager reaction term to convert the non-gradient driving force into a gradient form, controlled by a switch $\gamma$ and a kinetic-energy framework. This gradient dynamics reveals three phases—null activity, near-fixed-point behavior with vanishing maximal Lyapunov exponent, and chaos—allowing chaos to be frozen without synaptic plasticity and enabling working-memory-like processing. The approach generalizes to excitatory–inhibitory networks and supports memory and prediction tasks via reservoir-like readouts, with performance comparable to or exceeding that of vanilla non-gradient dynamics at high coupling gain $g$. The work offers a non-plasticity mechanism to stabilize information processing in neural circuits, suggesting new directions for implementing fixed-point memory and robust computation in biologically plausible networks.

Abstract

Chaos is ubiquitous in high-dimensional neural dynamics. A strong chaotic fluctuation may be harmful to information processing. A traditional way to mitigate this issue is to introduce Hebbian plasticity, which can stabilize the dynamics. Here, we introduce another distinct way without synaptic plasticity. An Onsager reaction term due to the feedback of the neuron itself is added to the vanilla recurrent dynamics, making the driving force a gradient form. The original unstable fixed points supporting the chaotic fluctuation can then be approached by further decreasing the kinetic energy of the dynamics. We show that this freezing effect also holds in more biologically realistic networks, such as those composed of excitatory and inhibitory neurons. The gradient dynamics are also useful for computational tasks such as recalling or predicting external time-dependent stimuli.

Figures (9)

• Figure 1: Dynamics-freezing effects of the OR-regulated dynamics. (a) Dynamics with $T=0$, $g=3$, and $N=1000$. $\gamma$ jumps from $0$ to $1$ at the time point $100$. (b) The slowness-measurement $Q$ [between $t=99.99$ and $t'=300$, see (a)] changes with $g$ for both vanilla RNN and OR-regulated dynamics. Five individual trials of experiments are considered for the error bars in the plot.
• Figure 2: The OR-regulated dynamics and the maximal Lyapunov exponents. (a) Dynamics with $T=0$, $g=2$ and $N=1000$. $\gamma$ goes from zero to one and back to zero over time. (b) Maximal Lyapunov exponents versus synaptic gain parameter $g$ for both vanilla and regulated dynamics ($T=0$). The exponents are averaged over five independent estimates (see details in Appendix \ref{['app-a']} with an initial deviation $\delta=10^{-5}$). The inset shows an enlarged view of the intermediate phase (the exponents are close to zero). (c) Dynamics with the same model parameters as (a). $\gamma$ goes from one to zero and back to one over time. (d) The eigenvalue spectrum of the Jacobian matrix at the point marked by x and the point marked by o in (a).
• Figure 3: The dynamics with $\gamma=1$ and the eigenvalue distribution of the Jacobian matrix. (a) Dynamics with $T=0$, $g=2$ and $N=1000$. (b) The eigenvalue spectrum of the Jacobian matrix at the last time point in (a). (c) Dynamics with the same model parameters as (a) but $g=10$. (d) The eigenvalue spectrum of the Jacobian matrix at the last time point in (c).
• Figure 4: Divergence of the driving force in the OR-regulated dynamics. $N=1\,000$, and four different values of $g$ are considered. (a) Evolution of divergence. (b) Distribution of the diagonal of the stability matrix.
• Figure 5: The dynamics statistics for the OR-regulated dynamics and the associated low-dimensional projection. (a) The $\ell_2$ norm of the activity before switching with $g=2$ and $N=1000$. (b) The $\ell_2$ norm of the activity after switching with $g=2$ and $N=1000$. (c) The $\ell_2$ norm of the activity with $\gamma=1$ (gradient dynamics), $g=10$ and $N=1000$. (d) The three-dimensional projection of the neural dynamics where three typical trajectories are shown and the OR term is only on in the later stage of the dynamics. The first three principal components explain about $85.67\%$ of the total variance in the trajectory data. In the data, the number of trajectories $P=100$, and the length is specified by $L=2\,000$, and $N=1\,000$. The cross symbol indicates the starting point of the trajectory. The color bars indicate the flow of time for the dynamics (darker colors mean later stages). The onset of switching (from $\gamma=0$ to $\gamma=1$) is indicated by a change of the color type. The inset shows how the kinetic energy decreases over time for the three trajectories shown in the main plot, and the dashed line indicates the time when the OR term is on.