Scaling conservation laws in neural systems

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 dynamical system described by a Lagrangian that changes under scale in the same way as the inverse time, 1/�. 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.