Language-model-based agents operating over extended interaction horizons face persistent challenges in preserving temporally grounded information and maintaining behavioral consistency across sessions, a failure mode we term "soul erosion." We present BMAM (Brain-inspired Multi-Agent Memory), a general-purpose memory architecture that models agent memory as a set of functionally specialized subsystems rather than a single unstructured store. Inspired by cognitive memory systems, BMAM decomposes memory into episodic, semantic, salience-aware, and control-oriented components that operate at complementary time scales, organised as a six-phase memory lifecycle. To support long-horizon reasoning, BMAM organises episodic memories along explicit timelines and retrieves evidence by fusing multiple complementary signals. Experiments on the LoCoMo benchmark show that BMAM achieves 78.45% accuracy, outperforming seven memory-augmented baselines. Pairwise ablations reveal super-additive synergy between brain-region components rather than redundant stacking, and a Soul Portability Test demonstrates 87.5% identity-integrity across full memory export, clear, and restore. A targeted refinement of the temporal-trigger heuristics raises LongMemEval multi-session accuracy from 45.2% to 56.4%, validating the architectural decomposition behind BMAM.Code is available at https://github.com/innovation64/BMAM.