Robust variability of grid cell properties within individual grid modules enhances encoding of local space.

Although grid cells are one of the most well studied functional classes of neurons in the mammalian brain, the assumption that there is a single grid orientation and spacing per grid module has not been carefully tested. We investigate and analyze a recent large-scale recording of medial entorhinal cortex to characterize the presence and degree of heterogeneity of grid properties within individual modules. We find evidence for small, but robust, variability and hypothesize that this property of the grid code could enhance the ability of encoding local spatial information. Performing analysis on synthetic populations of grid cells, where we have complete control over the amount heterogeneity in grid properties, we demonstrate that variability, of a similar magnitude to the analyzed data, leads to significantly decreased decoding error, even when restricted to activity from a single module. Our results highlight how the heterogeneity of the neural response properties may benefit coding and opens new directions for theoretical and experimental analysis of grid cells.

Long-term transverse imaging of the hippocampus with glass microperiscopes.

The hippocampus consists of a stereotyped neuronal circuit repeated along the septal-temporal axis. This transverse circuit contains distinct subfields with stereotyped connectivity that support crucial cognitive processes, including episodic and spatial memory. However, comprehensive measurements across the transverse hippocampal circuit in vivo are intractable with existing techniques. Here, we developed an approach for two-photon imaging of the transverse hippocampal plane in awake mice via implanted glass microperiscopes, allowing optical access to the major hippocampal subfields and to the dendritic arbor of pyramidal neurons. Using this approach, we tracked dendritic morphological dynamics on CA1 apical dendrites and characterized spine turnover. We then used calcium imaging to quantify the prevalence of place and speed cells across subfields. Finally, we measured the anatomical distribution of spatial information, finding a non-uniform distribution of spatial selectivity along the DG-to-CA1 axis. This approach extends the existing toolbox for structural and functional measurements of hippocampal circuitry.

On Koopman mode decomposition and tensor component analysis.

Koopman mode decomposition and tensor component analysis [also known as CANDECOMP (canonical decomposition)/PARAFAC (parallel factorization)] are two popular approaches of decomposing high dimensional datasets into modes that capture the most relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by different scientific communities and are formulated in distinct mathematical languages. We examine the two together and show that, under certain conditions on the data, the theoretical decomposition given by the tensor component analysis is the same as that given by Koopman mode decomposition. This provides a "bridge" with which the two communities should be able to more effectively communicate. Our work provides new possibilities for algorithmic approaches to Koopman mode decomposition and tensor component analysis and offers a principled way in which to compare the two methods. Additionally, it builds upon a growing body of work showing that dynamical systems theory and Koopman operator theory, in particular, can be useful for problems that have historically made use of optimization theory.

Renormalization group as a Koopman operator.

Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents η and δ, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant directions.