Mycelium-Mediated Reservoir Computing: A Mathematical Framework for Bio-Hybrid AI Systems with Self-Repairing Substrates

Fungal mycelium networks exhibit electrical spiking behavior, memristive dynamics, and self-repairing topology that collectively satisfy the requirements for physical reservoir computing. This work originated from the author's submission to the DARPA Expedited Research Implementation Series (ERIS), Submission 4-25-0358: "Biological AI Interfaces: Mycelium as a Scalable Sensor Grid," which was reviewed, deemed compliant, and advanced to assessment but was non-fundable at the time of evaluation due to insufficient experimental maturation of the underlying substrate technology. Since that submission, three independent lines of experimental evidence have substantially validated the core premises: (i) Boolean gate implementation via nonlinear signal transformation in mycelium composites [1,2], (ii) memristive switching at frequencies up to 5.85 kHz with 90 ± 1% accuracy in dehydrated shiitake mycelium [3], and (iii) confirmation of small-world network topology with high clustering coefficients in modeled mycelial growth suitable for reservoir computing [4]. We present a unified mathematical framework formalizing the mycelium network as a weighted directed graph G = (V, E, w) whose edges carry time-varying conductances governed by a fractional-order memristive state equation. We derive the echo state property conditions for mycelial reservoirs, prove that the self-repair dynamics of hyphal regrowth preserve reservoir memory capacity under edge deletion, and establish information-theoretic bounds on the computational capacity of a mycelium substrate as a function of network degree distribution and spike propagation velocity. We propose a complete bio-hybrid architecture coupling a living mycelium reservoir with a trainable silicon readout layer and identify three experimentally testable predictions.