Estimating quantities conserved by virtue of scale invariance in timeseries

TL;DR

This paper derives a generalized scale-invariant Lagrangian to identify and estimate conserved quantities related to scale invariance in dynamical systems, demonstrated through simulated and neuronal timeseries data.

Contribution

It introduces a novel power series expansion of a scale-invariant Lagrangian for dynamic models, enabling distinction of scale invariance from scale freeness in empirical data.

Findings

Successfully derived a generalized scale-invariant Lagrangian.

Demonstrated identification of scale invariance in simulated data.

Estimated conserved quantities in neuronal timeseries.

Abstract

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, the action can be invariant under change of scale in the special case of a scale free dynamical system that can be described in terms of a Lagrangian, that itself scales inversely with time. Crucially, this means symmetries under change of scale can exist in dynamical systems under certain constraints. Our contribution lies in the derivation of a generalised scale invariant Lagrangian - in the form of a power series expansion - that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models (i.e., state space models based upon differential equations) that can be used to distinguish scale invariance (scale symmetry) from scale freeness in empirical data. We establish face validity with an analysis of simulated data and then show how scale invariance can be identified - and how the associated conserved quantities can be estimated - in neuronal timeseries.