Dual-Component Eligibility Traces: Structural Necessity of β-Parameterisation in Modular Neuromorphic Systems

This preprint establishes that module-specific β-parameterisation in dual-component eligibility traces is a **strict architectural necessity** — not a tunable hyperparameter — for biologically plausible, modular active inference systems such as the Recurrent Deep Liquid Neural Network (RDLNN). Biological three-factor plasticity rules must simultaneously support two conflicting objectives: - Precise, temporally asymmetric credit assignment via spike-timing-dependent plasticity (STDP), and - Slow, rate-based associative consolidation via Hebbian co-activation. When implemented homogeneously, these mechanisms destructively interfere. We formalise this conflict through a generalized dual-component trace evolution equation: dE₈₉dt = -E₈₉ₑ + K (t) + Aᵢ Aⱼ (with τₑ = 500 ms decay, K (Δt) = precise anti-symmetric STDP kernel, and β controlling co-activation strength) and demonstrate empirically — across identically seeded continuous-time simulations (dt = 1 ms) — that: - **β = 0. 0 is mandatory** for subcortical motor/value-tracking modules (Basal Ganglia, Cerebellum), where precise temporal sign-accuracy (>90% causal credit assignment under delayed reward) and Reinforcement Prediction Error (RPE) integrity must be preserved. Any β > 0 causes catastrophic wash-out of the STDP kernel (p 0 (e. g. , β = 0. 1) ** is required for associative episodic/cortical modules (e. g. , CA3-like Hopfield networks), where slow-wave Hebbian consolidation is essential for robust pattern completion and noise-resistant memory basins (error <5% at 30% input noise only when β = 0. 1; β = 0 fails to converge). These findings directly constrain neuromorphic hardware implementations (e. g. , Intel Loihi 2 or equivalent) by requiring independent, region-specific exposure of the β degree of freedom to segregate pure temporal credit from associative structural consolidation. **Relation to Prior Work** This result is a foundational mechanistic building block for the biologically-constrained modular active inference architecture detailed in the companion preprint: **Biologically-Constrained Modular Active Inference: A Functional Architecture from Trace to Gating** (https: //doi. org/10. 5281/zenodo. insert new DOI once uploaded) It operationalises the module-specific three-factor eligibility paradigm (with β-topology parameters) that enables the macroscopic Variational Free Energy (VFE) gating mechanism to selectively consolidate meaningful plasticity while suppressing background turnover across the full RDLNN system. Both works extend and instantiate concepts from the earlier unified framework: **NeuroForge: A Unified Neural Substrate for Scalable Biological AI** https: //doi. org/10. 5281/zenodo. 17622945 **This paper forms part of a series. The integrated architecture paper is available at https: //doi. org/10. 5281/zenodo. 19186955 **Additional Notes for Zenodo: ** • Preprint (March 2026) – self-contained with mathematical formulation, seeded empirical validation, statistical tests, and explicit hardware implications. • Figure 1 (trace sign-accuracy curves) included in the PDF, demonstrating structural isolation requirement. • Designed as a core mechanistic constraint for neuromorphic porting of modular brain-like systems; complements the full RDLNN integration results. This work mathematically and empirically proves that homogeneous three-factor rules are insufficient for scalable, functionally diverse cognitive architectures — modular segregation of temporal vs. associative plasticity drivers is architecturally mandatory.