Learning Discrete Successor Transitions in Continuous Attractor Networks: Emergence, Limits, and Topological Constraints

Continuous attractor networks (CANs) are a well-established class of models for representing low-dimensional continuous variables such as head direction, spatial position, and phase. In canonical spatial domains, transitions along the attractor manifold are driven by continuous displacement signals, such as angular velocity-provided by sensorimotor systems external to the CAN itself. When such signals are not explicitly provided as dedicated displacement inputs, it remains unclear whether attractor-based circuits can reliably acquire recurrent dynamics that support stable state transitions, or whether alternative predictive strategies dominate. In this work, we present an experimental framework for training CANs to perform successor-like transitions between stable attractor states in the absence of externally provided displacement signals. We compare two recurrent topologies, a circular ring and a folded snake manifold, and systematically vary the temporal regime under which stability is evaluated. We find that, under short evaluation windows, networks consistently converge to impulse-driven associative solutions that achieve high apparent accuracy yet lack persistent attractor dynamics. Only when stability is explicitly enforced over extended free-run periods do genuine attractor-based transition dynamics emerge. This suggests that shortcut solutions are the default outcome of local learning in recurrent networks, while attractor dynamics represent a constrained regime rather than a generic result. Furthermore, we demonstrate that topology strictly limits the capacity for learned transitions. While the continuous ring topology achieves perfect stability over long horizons, the folded snake topology hits a geometric limit characterized by failure at manifold discontinuities, which neither curriculum learning nor basal ganglia-inspired gating can fully overcome.