揭示稀疏神经网络中抑制性时变差异如何塑造临界动力学，提升工作记忆性能。
Random matrix theory of sparse neuronal networks with heterogeneous timescales
- 构建基于随机矩阵的理论模型，模拟训练后网络的雅可比矩阵特征。
- 发现网络在临界边缘处的谱特性与抑制性核心-兴奋性外围结构相关。
- 适用于研究神经动力学、工作记忆机制及类脑计算系统的读者。
训练包含兴奋性（E）和抑制性（I）单元的递归神经网络进行工作记忆计算时，会减缓并多样化抑制性时变，从而提升任务表现，其原因被归因于涌现的近临界稳定平衡态。然而，训练后网络特性与其对理想动力学景观的塑造作用之间的联系尚不明确。本文研究了这些平衡态附近的雅可比矩阵，发现其为稀疏、非厄米、分块矩形矩阵，受异质突触衰减速率和激活函数增益影响。我们提出一个能忠实捕捉训练后雅可比矩阵谱特性的随机矩阵系综，源于训练后观测到的抑制性核心-兴奋性外围网络结构（修剪后的E权重，广泛分布的I权重）。通过统计场论方法建立该系综的解析理论：采用福约多罗夫与米尔林风格的超对称处理，结合厄米化核密度表示，精确推导出谱边的解析描述。该理论将雅可比矩阵的统计参数（稀疏度、权重方差、E/I比例，以及时变和增益的分布）与平衡态的近临界特征关联起来，这些特征对鲁棒的工作记忆计算至关重要。
原文摘要 · Abstract (English)
Training recurrent neuronal networks consisting of excitatory (E) and inhibitory (I) units with additive noise for working memory computation slows and diversifies inhibitory timescales, leading to improved task performance that is attributed to emergent marginally stable equilibria [PNAS 122 (2025) e2316745122]. Yet the link between trained network characteristics and their roles in shaping desirable dynamical landscapes remains unexplored. Here, we investigate the Jacobian matrices describing the dynamics near these equilibria and show that they are sparse, non-Hermitian rectangular-block matrices modified by heterogeneous synaptic decay timescales and activation-function gains. We specify a random matrix ensemble that faithfully captures the spectra of trained Jacobian matrices, arising from the inhibitory core - excitatory periphery network motif (pruned E weights, broadly distributed I weights) observed post-training. An analytic theory of this ensemble is developed using statistical field theory methods: a Hermitized resolvent representation of the spectral density processed with a supersymmetry-based treatment in the style of Fyodorov and Mirlin. In this manner, an analytic description of the spectral edge is obtained, relating statistical parameters of the Jacobians (sparsity, weight variances, E/I ratio, and the distributions of timescales and gains) to near-critical features of the equilibria essential for robust working memory computation.
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