Attractor computation in continuous-time digital networks

Abstract Attractor dynamics underlie the collective computation of neural networks. Recurrent connections between biological neurons can give rise to stable states towards which network activity relaxes, allowing parallel operation without global coordination while preserving robustness. Implementing physical systems with reliable attractor dynamics is therefore important both for understanding biological intelligence and for developing bio-inspired computing systems. However, parallel physical evolution introduces dynamical challenges. Concurrent evolution of artificial neurons can support rapid collective relaxation, but can also prevent the network from settling. Here, we realise a continuous-time digital Hopfield neural network (CTD-HNN) that combines digitally programmable energy landscapes with state evolution in physical time. Across ferromagnetic and antiferromagnetic Ising models, associative memory tasks, and Max-Cut problems up to 1,000 vertices, we show that this system can be tuned to exhibit stable relaxation, persistent instability, or transient exploration of state space before convergence. By regulating feedback, activation and propagation delays, we control the intrinsic instability as a dynamical degree of freedom, thereby balancing exploration and convergence while preserving the task-defining interactions. This work combines the configurability and precision of digital hardware with continuous-time dynamics, establishing a programmable framework for attractor computation in which energy landscapes are digitally defined but physically explored.