神经网络中从函数预测结构的几何框架
Geometric framework to predict structure from function in neural networks
摘要：
生物和人工网络中的神经计算依赖于许多输入的非线性求和。神经元之间突触权重的结构连接矩阵是整个网络功能的关键决定因素，但神经网络结构和功能之间的定量联系是复杂和微妙的。例如，许多网络可以产生类似的功能反应，同一个网络可以根据上下文的不同而发挥不同的功能。是否需要某些突触连接模式来生成特定的网络级计算在很大程度上是未知的。在这里，我们介绍了一个几何框架，用于识别阈值线性神经元的递归网络中的稳态响应所需的突触连接。假设指定的响应模式的数量不超过输入突触的数量，我们分析计算所有前馈和递归连接矩阵的解决方案空间，可以从网络输入生成指定的响应。考虑噪声的一般化进一步揭示，随着允许误差的增加，解空间几何形状可以经历拓扑转变，这可以提供对神经科学和机器学习的洞察。我们最终使用这种几何特征来获得保证神经元之间的非零突触的确定性条件。因此，我们的理论框架可以应用于神经活动数据，使严格的解剖预测，一般遵循的模型架构。
英文摘要：
Neural computation in biological and artificial networks relies on the nonlinear summation of many inputs. The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function, but quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of threshold-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate the solution space of all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. A generalization accounting for noise further reveals that the solution space geometry can undergo topological transitions as the allowed error increases, which could provide insight into both neuroscience and machine learning. We ultimately use this geometric characterization to derive certainty conditions guaranteeing a non-zero synapse between neurons. Our theoretical framework could thus be applied to neural activity data to make rigorous anatomical predictions that follow generally from the model architecture.
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