Department of Neuroimaging, King's College London, London, United Kingdom.
Department of Physics, Imperial College London, London, United Kingdom.
PLoS Comput Biol. 2020 May 4;16(5):e1007865. doi: 10.1371/journal.pcbi.1007865. eCollection 2020 May.
In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1) it must depend upon a scale-free Lagrangian, and 2) the Lagrangian must change under scale in the same way as the inverse time, [Formula: see text]. Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models-state space models based upon differential equations-that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series.
与空间平移、空间旋转和时间平移的对称性相反，已知的物理定律在尺度变换下并不是普遍不变的。然而，有一种特殊情况，如果作用满足以下两个约束条件，则是尺度不变的：1）它必须依赖于无尺度拉格朗日量，2）拉格朗日量必须以与逆时间相同的方式随尺度变化，[公式：见文本]。我们的贡献在于推导出一个广义拉格朗日量，它以幂级数展开的形式满足这些约束条件。这个广义拉格朗日量为动态因果模型——基于微分方程的状态空间模型——提供了一种规范形式，可以用于在经验数据中区分尺度对称性和无尺度性。我们通过对模拟数据的分析来证明其表面效度，在该分析中，我们展示了如何识别尺度对称性以及如何估计神经元时间序列中的相关守恒量。
