Theory of coupled neuronal-synaptic dynamics

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.