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

Daniel Brownell

TL;DR

This work probes whether continuous attractor networks (CANs) can autonomously acquire stable successor-like transitions without external displacement inputs. By comparing Ring and folded Snake manifolds across a three-phase curriculum and an automated hyperparameter search, the authors demonstrate that short-horizon training yields impulse-driven shortcut solutions that mimic correct transitions but lack sustained attractor dynamics, while long-horizon, attractor-enforced evaluation reveals true persistent transitions, contingent on topology. The Ring topology supports robust long-horizon transitions, whereas the Snake topology exhibits a geometric limit at manifold discontinuities that timing control alone cannot overcome. The study highlights a crucial distinction between predictive success and attractor-consistent dynamics, emphasizing the need for evaluation designs that measure late-trial stability and suggesting architectural directions (e.g., non-local connectivity) to bridge topological gaps. These findings inform how CAN-like circuits could serve as stable internal simulators and outline practical constraints on learning endogenous transition dynamics.

Abstract

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.

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

TL;DR

This work probes whether continuous attractor networks (CANs) can autonomously acquire stable successor-like transitions without external displacement inputs. By comparing Ring and folded Snake manifolds across a three-phase curriculum and an automated hyperparameter search, the authors demonstrate that short-horizon training yields impulse-driven shortcut solutions that mimic correct transitions but lack sustained attractor dynamics, while long-horizon, attractor-enforced evaluation reveals true persistent transitions, contingent on topology. The Ring topology supports robust long-horizon transitions, whereas the Snake topology exhibits a geometric limit at manifold discontinuities that timing control alone cannot overcome. The study highlights a crucial distinction between predictive success and attractor-consistent dynamics, emphasizing the need for evaluation designs that measure late-trial stability and suggesting architectural directions (e.g., non-local connectivity) to bridge topological gaps. These findings inform how CAN-like circuits could serve as stable internal simulators and outline practical constraints on learning endogenous transition dynamics.

Abstract

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.
Paper Structure (16 sections, 3 figures, 1 table)

This paper contains 16 sections, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Overview
  3. Continuous Attractor Dynamics
  4. Successor Transitions as Learned Displacements
  5. Gating and Control
  6. Training Curriculum and Evaluation Tasks (Phases A--C)
  7. Automated Hyperparameter Search (Megatune Phase B/C)
  8. Results
  9. Associative Shortcut Dynamics as the Default Regime
  10. Regime Selection via Composite Constraints
  11. The Geometric Limit
  12. Control vs. Geometry
  13. Discussion
  14. Relationship to oscillatory-interference displacement mechanisms.
  15. Implications for architectural extensions.
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Topology Comparisons. Force-directed graphs (top) and weight matrices (bottom) revealing the structural differences between Ring and Snake manifolds. Note the continuous connectivity in the Ring matrix versus the hard boundary cutoffs in the Snake matrix.
  • Figure 2: Transition stability traces across horizons and controllers. Cosine similarity to the target prototype during free-run dynamics after a successor transition. Short-horizon evaluation ($t=40$) permits impulse-dominated transients, while long-horizon evaluation ($t=120$) reveals robust Ring stability and persistent Snake drift that BG gating does not eliminate.
  • Figure 3: Impulse success versus geometric failure in Phase C. Under short-horizon evaluation ($t=40$), both Ring and Snake achieve high apparent accuracy, despite the Snake relying on transient impulse dynamics. Under long-horizon attractor-enforced evaluation ($t=120$), the Ring remains perfectly stable, while the Snake exhibits systematic errors at topological discontinuities (notably $9 \to 0$), revealing a structural geometric limit.