A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

翻译：计算神经科学领域日益增长的文献利用梯度下降及其近似学习算法来研究大脑中的突触可塑性。然而，绝大多数研究忽略了其中关键的基础假设：突触变化的距离选择，即突触可塑性的几何结构。梯度下降假设距离是欧几里得的，但存在许多其他可能距离，且并无理由表明生物学必然采用欧几里得几何。本文利用镜像下降提供的理论工具，证明突触权重的分布将取决于突触可塑性的几何结构。我们利用这些结果进一步表明，在多个脑区实验中观察到的对数正态权重分布与标准梯度下降（即欧几里得几何）不一致，而是与非欧几里得距离相符。最后，我们证明通过比较学习前后的突触权重分布，可能实现对不同突触几何结构的实验检验。总体而言，本研究显示，当前假设突触采用欧几里得几何的理论工作范式可能存在偏差，而通过实验确定大脑中突触可塑性的真实几何结构是可行的。