Conservation laws by virtue of scale symmetries in neural systems

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, 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, 1 = t. 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.

A Complex Systems Perspective on Neuroimaging Studies of Behavior and Its Disorders

“…Therefore, local oscillations in the gamma frequency are the binding force that generates rapid metastable dynamics, moving between synchronization and de-synchronization, which spread across the cortex. These elementary computational motifs have been observed across various brain regions, as well as across scales ( Turkheimer and others 2015 ), reaching the long-range synchrony of neural activity ( Agrawal and others 2019 ; Fagerholm and others 2020 ; Meshulam and others 2019 ). This supports the view of the brain as exhibiting SOC ( Cabral and others 2011 ; Cabral and others 2014 ; Roberts and others 2019 ; Tognoli and Kelso 2014 ), with a distinctive fractal signature both in its structure and function ( Expert and others 2011 ; Squarcina and others 2015 ; Turkheimer and others 2015 ).…”

The brain and its time: intrinsic neural timescales are key for input processing

“…Despite different temporal (and spatial) scales across body, brain, and environments, they are nevertheless connected through scale-free self-similarity in their shape or form. Just like the smaller Russian doll is contained within the next larger one (same shape, different size), the brain and its temporo-spatial organization nest in a scale-free self-similar way within the much larger environment 132 . Given such temporo-spatial self-similarity between brain and environment, we may better focus on “what our head’s inside of” rather than searching for “what inside our heads” 86 .…”

Estimating anisotropy directly via neural timeseries

“…In the third section, we use murine wide-field calcium imaging data [6][7][8] collected in both rest and task states to map levels of anisotropy across different cortical regions directly via the in vivo timeseries.…”

“…All murine calcium imaging data were collected as previously reported [6][7][8]. As with the synthetic data, we perform Bayesian model inversion to obtain posterior estimates for the parameter quantifying the extent to which the time series for each pixel deviate from isotropy at = 0.…”