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Back to the Continuous Attractor
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Shows that continuous attractors are structurally unstable but their bifurcations and approximations retain a persistent attractive slow manifold, so approximate continuous attractors remain functionally useful for analog memory, supported by theory (Fenichel persistent manifold) and task-trained RNN experiments.
Problem
Continuous attractors, the standard model of analog working memory, are structurally unstable: infinitesimal perturbations of the dynamics destroy the continuum of fixed points, raising doubts about their biological relevance.
Approach
Applies Fenichel's persistent manifold theorem and fast-slow decomposition to show that bifurcations and approximations of continuous attractors leave an attractive slow invariant manifold with the same topology; the uniform norm of the slow flow bounds memory error, so such systems act as approximate continuous attractors.
Key results
- Trained RNNs (10 per architecture and hidden size 64/128/256) develop slow invariant manifolds with a gap between the first two eigenvalues, consistent with approximate continuous attractors.
- Memory error over the trained duration is bounded by the uniform norm of the slow flow on the manifold, and average asymptotic error is roughly inversely proportional to the number of fixed points.
- In a 6-unit ring attractor, random parametric perturbations collapse the continuum into between 2 and 12 isolated fixed points, with the fixed-point count depending on perturbation direction.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Perturb the bounded line attractor with random parameter changes | Bounded line attractor · Bounded line attractor perturbations | probability of one stable fixed point: \frac{3}{4} | Mixed |
| Perturb the bounded line attractor with random parameter changes | Bounded line attractor · Bounded line attractor perturbations | probability of three fixed points: \frac{1}{4} | Mixed |
| Apply parametric perturbations | Piecewise-linear ring attractor, N = 6 · Central-complex ring attractor | number of isolated fixed points: continuum -> between 2 and 12 isolated fixed points | Mixed |
| Replace the continuous ring attractor with a limit-cycle approximation | Ring attractor memory system · Circular-variable memory | memory error over time | Hurt |
| Use a fixed-point ring manifold instead of a limit-cycle approximation | Ring attractor memory system · Circular-variable memory | asymptotic memory error | Mixed |
| Train RNNs with state noise and use the resulting slow-manifold dynamics for memory | Task-trained RNNs · Angular velocity integration task | finite-time memory error: T_{1} = 256 time steps | Helped |
Surprising findings
- Continuous attractors bifurcate under almost any perturbation, yet the resulting systems retain a ghost invariant ring manifold with nearly the same topology.
- The number of fixed points stays constant over a range of perturbation sizes and depends mainly on perturbation direction.
Limitations
- Numerical slow-manifold identification relies on time-scale separation and may pick up transient, only forward-invariant sets.
- Linear readout restricts the possible solutions found by RNNs; nonlinear readouts may be needed for alternatives such as oscillator-based solutions.
Experimental setup
Datasets: simulated head direction integration task simulated memory-guided saccade task
Benchmarks: angular velocity integration task memory-guided saccade task
Metrics: normalized MSE average accumulated angular error memory capacity eigenvalue spectrum gap
Builds on: Fenichel's persistent manifold theorem continuous attractor network models geometric singular perturbation theory
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