Neural Systems Under Change of Scale.

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

doi:10.3389/fncom.2021.643148

Abstract

We derive a theoretical construct that allows for the characterisation of both scalable and scale free systems within the dynamic causal modelling (DCM) framework. We define a dynamical system to be “scalable” if the same equation of motion continues to apply as the system changes in size. As an example of such a system, we simulate planetary orbits varying in size and show that our proposed methodology can be used to recover Kepler’s third law from the timeseries. In contrast, a “scale free” system is one in which there is no characteristic length scale, meaning that images of such a system are statistically unchanged at different levels of magnification. As an example of such a system, we use calcium imaging collected in murine cortex and show that the dynamical critical exponent, as defined in renormalization group theory, can be estimated in an empirical biological setting. We find that a task-relevant region of the cortex is associated with higher dynamical critical exponents in task vs. spontaneous states and vice versa for a task-irrelevant region.

Highlights

Scalable Dynamical SystemsLet us consider a dynamical system that is evolving in time and generating a certain series of states
When we speak of varying the size of a planetary orbit, we do not allude to the size of the planet itself, but rather to the spatial coordinate of its centre of mass as it orbits its host star
Results are presented for n = 3 mice, with analyses performed separately within two regions of interest (ROIs) (Figure 2C)

Summary

Introduction

Scalable Dynamical SystemsLet us consider a dynamical system that is evolving in time and generating a certain series of states. When we speak of varying the size of a planetary orbit, we do not allude to the size of the planet itself, but rather to the spatial coordinate of its centre of mass as it orbits its host star. Upon making such a transformation, we refer to the system as being “scalable” if the equations of motion describing both its scaled and unscaled versions are identical in form – a system that Landau referred to as possessing “mechanical similarity” (Landau and Lifshitz, 1976).

Methods

Results

Conclusion