Estimating quantities conserved by virtue of scale invariance in timeseries

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.

💡 Research Summary

The paper tackles the problem of detecting scale‑invariant dynamics in time‑series data, a task that is fundamentally different from the well‑known symmetries of spatial translation, rotation, and temporal translation. The authors begin by recalling that a physical system is said to possess a symmetry if a transformation leaves the mathematical form of its governing equations unchanged. While most fundamental laws are invariant under the three classic symmetries, they are generally not invariant under a change of scale. The authors argue that a scale‑free dynamical system can nevertheless exhibit scale symmetry provided its Lagrangian scales inversely with time.

Mathematically, they consider a Lagrangian (L(q,\dot q,t)) with explicit time dependence. Under a scale transformation (q\rightarrow F q) and (t\rightarrow F^{G} t) the action (S=\int L,dt) remains invariant if the Lagrangian transforms as (L\rightarrow F^{-G}L). This condition leads to a constraint on the exponents of any monomial term in a power‑series expansion of the Lagrangian. The authors therefore propose a generalized, scale‑invariant Lagrangian written as a triple sum
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