Geometric Modulation of Synaptic Plasticity via MGD Metrics Enables Continual Learning in Spiking Neural Networks

This paper presents BioMGDBrain, a novel, biologically-inspired Spiking Neural Network (SNN) architecture designed to solve the catastrophic forgetting problem in continual learning. The system's mechanisms are strictly grounded in the Generative Mathematics of Dimensions (MGD) framework and validated against neuroscientific principles. Unlike traditional approaches that rely on artificial constraints or global error backpropagation, BioMGDBrain achieves continual learning through local, geometrically-modulated synaptic updates. Core Contributions & Mechanisms: Geometrically Modulated Learning Rate (\eta_{mgd}): Plasticity is dynamically scaled using the Forman-Ricci curvature (\kappa_F) and the effective dimension (D_{avg}) of the network. This acts as a mathematical analogue to dopaminergic novelty signaling, accelerating forward transfer by 3\times compared to standard STDP. Low-Variance Gating & S_{RT} Routing: Task-dedicated neurons are selected by identifying geometrically unspecialised cells (low-variance criterion). New tasks are automatically routed to the cortical area with the maximal residual capacity, measured by the Ryu-Takayanagi entropy proxy (S_{RT}). Priority Sleep Consolidation: An offline sleep phase replays stored experiences ordered by descending S_{RT} priority, mimicking the role of noradrenaline in slow-wave sleep. Local MGD (O(n) Complexity): Global matrix operations (like SVD) are replaced by a per-synapse Forman-Ricci curvature computation based only on immediate neural neighbors, delivering a 47\times speedup and making the system neuromorphic-ready. Biologically Grounded Overlap & Readout: The architecture features a three-modulator system (Dopamine, Acetylcholine, Noradrenaline) to allow task overlap without dedicated hardware. Furthermore, the system is entirely independent of Adam/backpropagation: it utilizes a "BioReadout" module combining the Oja rule, BCM metaplasticity, and a dopamine-gated perceptron for purely online, label-free updates during sleep. Three-Level Temporal Hierarchy: The final architecture (System K) simulates the biological brain's temporal cascade (Hippocampus \to Sensory Cortex \to Deep Neocortex) using two distinct sleep phases for memory crystallization. Hyperparameters for this temporal hierarchy were strictly validated using the OpenEvolve evolutionary coding agent.