Limit-Cycle Proliferation Under Parametric Delayed Feedback in a Conductance-Based Neuron: Bifurcation Landscape, Orbit Catalog, and Capacity Analysis.

We show that a single Hodgkin-Huxley (HH) neuron with Pyragas-type delayed feedback control (DFC) can store multiple symbols as stable periodic orbits, where the specific orbit is selected by tuning the DFC gain K and time delay τ. Sweeping the (K,τ) parameter plane at fixed bias current Ibias = 10.0 μA/cm2 reveals 207 orbit types across 12 topological categories, with inter-spike interval (ISI) means from 5.9 to 56.9 ms. We establish: (i) a write protocol that reliably locks orbits with 13.9 ms median settling time; (ii) a novel Pattern-Oriented Limit-cycle Decoder (POLD) that reads orbits at 100% accuracy from only five observed ISIs (1200 trials across 12 orbits; Wilson 95% CI: 99.7-100%); (iii) a complete single-symbol write-read-erase (W-R-E) cycle with 100% read accuracy, 92% erase verification, and no decay over hold durations up to 50 s; and (iv) a fully validated 12-symbol memory capacity with a read-discriminable upper bound of 67 symbols (11.2× over rate coding; write viability confirmed only for the conservative 12-symbol subset). Reliable orbit addressing needs delay precision of ±2%, which constitutes a write-precision specification and not a fundamental capacity limit. These findings show that parametric delayed feedback is a viable mechanism for limit-cycle-based information storage in conductance-based spiking neurons. The biological interpretation is analogical, not direct: the ±2% delay-precision requirement exceeds what has been demonstrated for biological autaptic variability, and the orbit-coded memory framing is best understood as a computational proof-of-principle aimed at neuromorphic engineering, not as a claim about biological working memory.