Conservation laws by virtue of scale symmetries in neural systems

Authors: Erik D. Fagerholm, Erik D. Fagerholm, W. M. C. Foulkes, Yasir Gallero-Salas, Fritjof Helmchen, Karl J. Friston, Rosalyn J. Moran, Robert Leech

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, 1t. 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.

Author summary

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Considerations of the way in which a dynamical system changes under transformation of scale offer insight into its operational principles. Scale freeness is a paradigm that has been observed in a variety of physical and biological phenomena and describes a situation in which appropriately scaling the space and time coordinates of any evolution of the system yields another possible evolution.

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In the brain, scale freeness has drawn considerable attention, as it has been associated with optimal information transmission capabilities. Scale symmetry describes a special case of scale freeness, in which a system is perfectly unchanged under transformation of scale.

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Noether’s theorem tells us that in a system that possesses such a symmetry, an associated conservation law must also exist. Here we show that scale symmetry can be identified, and the related conserved quantities measured, in both simulations and real-world data.

We achieve this by deriving a generalised equation of motion that leaves the action invariant under spatiotemporal scale transformations and using a modified version of Noether’s theorem to write the associated family of conservation laws. Our contribution allows for the first such statistical characterisation of the quantity that is conserved purely by virtue of scale symmetry.

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