The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, and a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.

中文翻译：

---

渲染符合静止动作原理的神经元状态方程

静止作用原理是现代物理学的基石，为研究经典力学中的动力学系统到量子场论提供了强大的框架。然而，尽管计算神经科学严重依赖物理学中的概念，但在这方面是异常的，因为它的主要运动方程与拉格朗日公式不兼容，因此与静止作用原理不兼容。将动态因果建模 (DCM) 神经元状态方程作为计算神经科学中常见的一阶线性微分方程的指导原型，我们表明可以对该方程进行某些修改以使其符合静止动作。具体来说，我们展示了使用复因变量、振荡解和 Hermitian 内在连接矩阵促进了 DCM 神经元状态方程的拉格朗日公式。我们首先通过使用贝叶斯模型反演来证明原理证明，以表明原始模型和修改模型都可以通过直接从各自运动方程生成的计算机数据正确识别。然后，我们通过使用三种不同类型的公开可用的体内神经影像数据集以及开源 MATLAB 代码，为在神经科学中采用改进的模型提供动力，以表明改进的（振荡）模型为其中一些经验提供了更简洁的解释时间序列。我们希望这项工作能够与现有技术相结合，