Modelling columnarity of pyramidal cells in the human cerebral cortex

Abstract

For modelling the location of pyramidal cells in the human cerebral cortex, we suggest a hierarchical point process in R 3 that exhibits anisotropy in the form of cylinders extending along the z-axis. The model consists first of a generalised shot noise Cox process for the xy-coordinates, providing cylindrical clusters, and next of a Markov random field model for the z-coordinates conditioned on the xy-coordinates, providing either repulsion, aggregation or both within specified areas of interaction. Several cases of these hierarchical point processes are fitted to two pyramidal cell data sets, and of these a final model allowing for both repulsion and attraction between the points seem adequate. We discuss how the final model relates to the so-called minicolumn hypothesis in neuroscience.

Testing goodness of fit for point processes via topological data analysis

“…For the second summand, we can argue similarly as in (14), so that it remains to bound the integral involving m…”

“…As it should serve only to illustrate the application of Theorem 3.2, the present analysis is very limited in scope, and we refer to [14] for a far more encompassing study. For instance, that work considers two datasets and investigates 3D data together with marks for the directions attached to the neurons.…”

Pairwise interaction Markov model for 3D epidermal nerve fibre endings

“…Following Ref. 12, we propose a model for $$X_z$$ conditioned on a realization of $$X_{p}$$ defined by the conditional probability density $$\begin{eqnarray} &&f\left((z_i)_{i=1}^n| (x_i,y_i)_{i=1}^n\right) \propto \gamma ^{s_{B_{r,t}\left((z_i)_{i=1}^n| (x_i,y_i)_{i=1}^n\right)}} \\ &&\times\ {\mathbb {1}}\left({\left\Vert (x_i,y_i,z_i)-(x_j,y_j,z_j)\right\Vert} > h\text{ for } 1\le i < j\le n\right),\nonumber\end{eqnarray}$$where $$h>0$$ denotes the hard core distance, that is the minimum allowed distance between two points, and $$\gamma >0$$ denotes the interaction parameter. If there is no interaction between the points, $$\gamma =1$$.…”

“…11 and then, given the planar coordinates, the third coordinate is modelled by a pairwise interaction Markov random field using cylindrical interaction regions as in Ref. 12. Since the end point patterns cannot be assumed to be isotropic, we use the cylindrical K function estimated in the three coordinate axis directions when describing the 3D spatial structure of the points 13,14 …”

Determinantal shot noise Cox processes

“…So, the distribution of depends only on the intensity, and is a most repulsive DPP in the sense of Lavancier et al (2015); see also Biscio and Lavancier (2016) and Møller and O'Reilly (2021). Christoffersen et al (2021) used this special case of a DSNCP model in a situation where a realization of but not was observed within a bounded region. They estimated with a minimum contrast procedure based on the pair correlation function (pcf) given by (5) in Section 3, where the pcf had to be approximated by numerical methods.…”