Abstract
An isotropic dynamical system is one that looks the same in every direction, i.e., if we imagine standing somewhere within an isotropic system, we would not be able to differentiate between different lines of sight. Conversely, anisotropy is a measure of the extent to which a system deviates from perfect isotropy, with larger values indicating greater discrepancies between the structure of the system along its axes. Here, we derive the form of a generalised scalable (mechanically similar) discretized field theoretic Lagrangian that allows for levels of anisotropy to be directly estimated via timeseries of arbitrary dimensionality. We generate synthetic data for both isotropic and anisotropic systems and, by using Bayesian model inversion and reduction, show that we can discriminate between the two datasets – thereby demonstrating proof of principle. We then apply this methodology to murine calcium imaging data collected in rest and task states, showing that anisotropy can be estimated directly from different brain states and cortical regions in an empirical in vivo biological setting. We hope that this theoretical foundation, together with the methodology and publicly available MATLAB code, will provide an accessible way for researchers to obtain new insight into the structural organization of neural systems in terms of how scalable neural regions grow – both ontogenetically during the development of an individual organism, as well as phylogenetically across species.
Highlights
Two of the main concepts upon which computational neuroscience models are based are those of the ‘particle’ (Sears, 1964) and the ‘field’ (McMullin, 2002) – both terms that are inherited from theoretical physics.Action Editor: Abraham Zvi SnyderCollege London, London, United Kingdom1.1 Particle theoretic modelsIn the particle theoretic approach we treat every node within a neural system as a zero-dimensional element – a so-called ‘particle’ that evolves in time
We use murine data collected in both rest and task states to map levels of anisotropy across different cortical regions directly via the in vivo timeseries. These wide-field calcium imaging data were collected across the left hemisphere of mouse cortex expressing GCaMP6f in layer 2/3 excitatory neurons (Fagerholm et al, 2021; Gallero-Salas et al, 2021; Gilad et al, 2018), with cortical areas aligned to the Allen Mouse Common Coordinate Framework (Mouse & Coordinate, 2016), We suggest that the presented methodology could be valuable in future large-scale studies of neural systems, in which the quantification of region-wise anisotropy may shed light on how neural systems grow both ontogenetically within the lifespan of an individual animal, as well as phylogenetically across species (Buzsaki et al, 2013)
Following model inversion and reduction, we demonstrate proof of principle by showing that there is higher evidence for the ground truth isotropic data having been created with the isotropic model (Fig. 1C) and for the ground truth anisotropic data having been created with the anisotropic model (Fig. 1D)
Summary
In the particle theoretic approach we treat every node within a neural system as a zero-dimensional (point-like) element – a so-called ‘particle’ that evolves in time. Particle theoretic frameworks yield experimental advantages for neuroimaging modalities such as electroencephalography (EEG), in which there are usually very few measurement locations. Particle theoretic frameworks have computational and statistical advantages for neuroimaging analyses due to associated dimensionality reduction – an attribute that becomes increasingly important for largescale recordings of neural systems (Izhikevich & Edelman, 2008). This computational expediency comes at the cost of losing the spatial information contained in a continuum description
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