Synaptic plasticity alters the nature of chaos transition in neural networks

TL;DR

This work addresses how learning-induced synaptic plasticity reshapes chaotic transitions in high-dimensional recurrent networks. It develops a neuron-synapse coupled quasi-potential framework combined with a canonical-ensemble replica calculation in the zero-speed limit, deriving order parameters and saddle-point equations for Hebbian, feedback, and homeostatic plasticities. A key finding is that strong Hebbian plasticity can induce a discontinuous chaos transition at a smaller synaptic gain $g$ than in non-plastic networks (the baseline being $g_c=1$ for $k=0$), while feedback and homeostatic rules preserve the transition’s location and type but modulate chaotic fluctuations; these predictions are supported by Lyapunov-exponent analyses and numerical simulations. The results illuminate how different plasticity mechanisms influence computation and memory in recurrent networks and point to future work on time-scale separation and potential relevance to neural dysfunctions.

Abstract

In realistic neural circuits, both neurons and synapses are coupled in dynamics with separate time scales. The circuit functions are intimately related to these coupled dynamics. However, it remains challenging to understand the intrinsic properties of the coupled dynamics. Here, we develop the neuron-synapse coupled quasi-potential method to demonstrate how learning induces the qualitative change in macroscopic behaviors of recurrent neural networks. We find that under the Hebbian learning, a large Hebbian strength will alter the nature of the chaos transition, from a continuous type to a discontinuous type, where the onset of chaos requires a smaller synaptic gain compared to the non-plastic counterpart network. In addition, our theory predicts that under feedback and homeostatic learning, the location and type of chaos transition are retained, and only the chaotic fluctuation is adjusted. Our theoretical calculations are supported by numerical simulations.

Figures (7)

• Figure 1: Phase diagram and order parameters with varying $g$ under Hebbian plasticity of different plasticity strengths. (a) The phase diagram is divided into two regions: the pink-colored area represents the fixed-point region, while the blue-colored area represents the chaotic region. The boundary is determined by equating free energies of two phases. (b,d) Plots of order parameters $q$ and $r$ against $g$ for different positive values of $k$. In (c), the plot shows the derivative of $q$ with respect to $g$. The dashed lines in (b) indicate the point where a first-order phase transition occurs, and in (c), the dashed line marks a sharp increase of the order parameter $q$. The threshold for the sharp slope is set to $4.0$. (e-f) Plots of order parameters $q$ and $r$ against $g$ for different negative values of $k$. The vertical lines in (e,f) indicate the phase transition point. Results are the averages over five independent runs of the SDE solver (see Appendix \ref{['app-b']}).
• Figure 2: The profile of the order parameters $q$ and $r$ with respect to the gain parameter $g$ and the strength of feedback learning $\delta\in\{0.0, 0.5, 1.0\}$. The vertical lines in (a,b) indicate the phase transition point. Five independent runs of the SDE solver are considered.
• Figure 3: The profile of the order parameters with respect to the gain parameter $g$ and the target firing rate of homeostatic learning $r_{\operatorname{tg}}\in\{-0.8, 0.0, 0.8\}$. The learning strength $k$ is set to $0.5$; we also present the simulation results for $r_{\text{tg}} = 0$ in networks of $N=1\,000$ ($\tau=0$). The simulation results are averages over the last $500$ time steps of the simulated dynamics. The vertical lines in (a,b) indicate the phase transition point. Five independent runs are used to obtain the averaged data points.
• Figure 4: Neural dynamics under different plasticity rules (a network of $1\,000$ neurons, five of which are randomly selected and shown). Three different background colors distinguish different learning types. (a, d, g, h, i) Neural dynamics under Hebbian learning. In (a), the time constant $\tau=1.5$; in (d, g, h, i), $\tau=0$. In the main plot, parameters are $(g,k)=(1.2, 0.5)$, and in the inset plot parameters are $(g,k)=(0.5, 0.5)$. (b, e) Neural dynamics under feedback learning. In (b), $\tau=1.5$; in (e), $\tau=0$. In the main plot, parameters are $(g,\delta)=(1.2, 0.6)$, and in the inset parameters are $(g,\delta)=(0.5, 0.6)$. (c, f) Neural dynamics under homeostatic learning. In (c), $\tau=1.5$; in (f), $\tau=0$. In the main plot, parameters are $(g,k)=(1.2, 0.5)$, and $r_{\rm tg} = 0.6$, and in the inset parameters are $(g,k)=(0.5, 0.5)$, and $r_{\rm tg} = 0.6$.
• Figure 5: Maximal Lyapunov exponent $\lambda_{\max }$ as a function of plasticity strength $g$ for different types of synaptic plasticity, computed by orbit separation method with N = 1000 and an initial perturbation of magnitude $\delta_0=10^{-5}$. The exponents are averaged over five independent estimates. (a) Hebbian learning. The connected symbols indicate a chaos transition boundary (continuous or discontinuous, depending on the value of $k$). (b) Feedback learning. (c) Homeostatic learning ($k=0.5$). Insets display typical examples of dynamics trajectries.