Theory of coupled neuronal-synaptic dynamics

TL;DR

This work addresses how joint neuronal and synaptic dynamics shape computation in recurrent networks. It develops a plastic random-network model with Hebbian and anti-Hebbian updates and analyzes it with dynamical mean-field theory to map a rich phase diagram, including chaotic and fixed-point regimes. A key finding is the emergence of a two-band, synapse-dominated spectrum that slows or destabilizes activity, and the discovery of freezable chaos, where halting plasticity can store and retrieve a neuronal state as memory. The results connect synaptic dynamics to working-memory mechanisms and offer a quantitative framework for testing how plasticity influences computation in neural circuits and related artificial systems.

Abstract

In neural circuits, synaptic strengths influence neuronal activity by shaping network dynamics, and neuronal activity influences synaptic strengths through activity-dependent plasticity. Motivated by this fact, we study a recurrent-network model in which neuronal units and synaptic couplings are interacting dynamic variables, with couplings subject to Hebbian modification with decay around quenched random strengths. Rather than assigning a specific role to the plasticity, we use dynamical mean-field theory and other techniques to systematically characterize the neuronal-synaptic dynamics, revealing a rich phase diagram. Adding Hebbian plasticity slows activity in chaotic networks and can induce chaos in otherwise quiescent networks. Anti-Hebbian plasticity quickens activity and produces an oscillatory component. Analysis of the Jacobian shows that Hebbian and anti-Hebbian plasticity push locally unstable modes toward the real and imaginary axes, explaining these behaviors. Both random-matrix and Lyapunov analysis show that strong Hebbian plasticity segregates network timescales into two bands with a slow, synapse-dominated band driving the dynamics, suggesting a flipped view of the network as synapses connected by neurons. For increasing strength, Hebbian plasticity initially raises the complexity of the dynamics, measured by the maximum Lyapunov exponent and attractor dimension, but then decreases these metrics, likely due to the proliferation of stable fixed points. We compute the marginally stable spectra of such fixed points as well as their number, showing exponential growth with network size. In chaotic states with strong Hebbian plasticity, a stable fixed point of neuronal dynamics is destabilized by synaptic dynamics, allowing any neuronal state to be stored as a stable fixed point by halting the plasticity. This phase of freezable chaos offers a new mechanism for working memory.

Figures (9)

• Figure 1: (a) Dynamics of a pair of neurons (top panel) and of the synapses through which they are reciprocally coupled (bottom panel). Synapses fluctuate about quenched random strengths (dashed lines) in response to pre- and postsynaptic activity according to a Hebbian rule. (b) Left: phase diagram of the plastic network for $p = 2.5$. Right: example neuronal traces $x_i(t)$ from simulations of each phase-diagram region, with parameters given by the location of the associated square marker.
• Figure 2: Chaotic states with ${g > 1}$. (a) $C(\tau)$ from the DMFT (solid lines) and in simulations (dashed lines) for $g = 2$, $p = 2.5$, and various values of $k$ (indicated by the lower color bar). (b) Dynamic timescale $\tau^*$ [Eq. (\ref{['eq:dynamic-timescale-def']})] as a function of $k$ for various values of $g$. Dotted line indicates $p$. (c) Power spectra (normalized such that $S(0) = 1$) for the autocovariance functions shown in (a). For anti-Hebbian ($k < 0$) power spectra, triangular markers indicate an oscillatory frequency computed from the zero-crossings of $C(\tau)$.
• Figure 3: Chaotic states for ${g < 1}$. (a) $C(\tau)$ from the DMFT (lines) and in simulations for $g = 0.9$, $p=2.5$, and various values of $k$. (b) Left: median log-lifetime of transient activity, before collapsing to the trivial fixed point, as a function of $N$ for $g = 0.9$ and values of $k$ from (a). Right: histograms of the log-lifetime of transient chaos, corresponding to stars in the left plot. Simulations were terminated at time $T_\text{sim} = 10^7$. (c) Curves $y(s)$ for solutions of the model of stern2014dynamics for various values of $g$. Dashed horizontal lines correspond to different values of $k$, intersecting $y(s)$ at self-consistent solutions of Eq. (\ref{['eq:slow-single-site-picture']}).
• Figure 4: Spectra of the Jacobian for $p = 2.5$ and various values of $g$ (running horizontally) and $k$ (running vertically). Lines: predicted boundary curves from random matrix theory and DMFT. Dots: spectra measured in simulations of chaotic plastic networks. Modes are colored by $f_{\bm{a}}(\lambda)$, the weight on the synaptic part of the corresponding eigenvector of the reduced Jacobian [Eq. (\ref{['eq:weight']})]. The red dot at $\lambda = -1/p$ indicates a delta function of $N^2 - N$ synaptic modes. For anti-Hebbian ($k < 0$) spectra, triangular markers indicate an oscillatory frequency computed from the zero-crossings of $C(\tau)$.
• Figure 5: (a) Maximum Lyapunov exponent $\lambda_\text{max}$, computed by a perturbation method, throughout $(g, k)$ parameter space with $p = 2.5$, $N=4000$. White: quiescence. Hatched: convergence to nonzero fixed points. (b) Histograms of Lyapunov spectra, computed using tangent-vector propagation with $N = 900$, for $p = 2.5$ and various values of $g$ (running horizontally) and $k$ (running vertically). Black outline for $k = 0$ histograms: spectra of nonplastic network. The red triangle marks $-1/p$, where there are $\mathcal{O}(N^2)$ exponents in the full spectrum. (c) $\lambda_\text{max}$ as a function of $k$ for various values of $g$. Solid lines: estimate from tangent-vector propagation. Dashed lines: estimate from perturbation method. (d) Attractor dimension divided by $N$ as a function of $k$ for various value of $g$.