Topological and geometrical signatures of computation in rate, spiking, and oscillatory neural reservoirs
Oleg V. Maslennikov · Chaos An Interdisciplinary Journal of Nonlinear Science · 2025
The computational efficiency of a neural substrate is shaped by the geometry and topology of its state-space manifold. We propose and test a "computational matching" principle: efficiency is maximized when the intrinsic geometry of a reservoir's representation aligns with the latent structure of the task. We investigate this on a phase-coherent burst detection task, comparing three physically distinct reservoir types: rate-based (tanh-neurons), spiking (leaky integrate and fire), and oscillatory (Kuramoto). We find that oscillatory reservoirs exhibit the highest neural efficiency, requiring approximately 4 times fewer neurons than rate-based models and 7.5 times fewer than spiking models with a static readout to reach 95% accuracy. Using persistent homology on final-state representations, we uncover the underlying mechanism. Both oscillatory and rate-based reservoirs generate manifolds with a non-trivial one-dimensional cycle (β1>0), reflecting the task's circular structure. However, the oscillatory reservoir's cycle is geometrically "straight," enabling nearly perfect linear decoding of the input phase [mean absolute angular error (MAE) ≈0.02°], whereas the rate-based cycle is distorted (MAE≈8°). Spiking reservoirs only reveal robust cyclic topology and accurate phase decoding (MAE≈0°) when states are integrated over a dynamic time window. Our findings suggest that computational efficiency in reservoirs is predicted not by topology alone, but by the geometric alignment of the state-space representation, offering a design heuristic for task-specialized neuromorphic systems.