Statistical mechanics characterization of neuronal mosaics

Abstract

The spatial distribution of neuronal cells is an important requirement for achieving proper neuronal function in several parts of the nervous system of most animals. For instance, specific distribution of photoreceptors and related neuronal cells, particularly the ganglion cells, in mammal's retina is required in order to properly sample the projected scene. This work presents how two concepts from the areas of statistical mechanics and complex systems, namely the lacunarity and the multiscale entropy (i.e. the entropy calculated over progressively diffused representations of the cell mosaic), have allowed effective characterization of the spatial distribution of retinal cells. * Electronic address: luciano@if.sc.usp.br 1

Multilacunarity as a spatial multiscale multi-mass morphometric of change in the meso-architecture of plant parenchyma tissue

“…The architectural complexity of intercellular air spaces makes single geometrical measurements insufficient to characterize the spatial heterogeneity of the pore space. The inhomogeneity of distribution depends not only on the percentage content of phase, 41,42 but also on how the phase fills the space and the variability in mass content at different spatial scales. The multilacunarity morphometric is a multiscale multi-mass measure of the degree of spatial heterogeneity that provides insights regarding modifications upon the arrangement of cells and voids in plant parenchyma tissue.…”

“…4 Lacunarity, as a second-order statistical measure, quantifies the relationship between neighboring objects/pixels, 5 explicitly characterizes spatial organization, and quantifies the degree of translational invariance. 6,7 It uses multiscale windowing for measuring the scale dependency of heterogeneity, thus characterizing the geometry of deterministic and random sets. 8 It measures how data fill the space, enabling the parsimonious analyses of patterns: aspects of gaps distribution, the presence of structures, homogeneity in gaps distribution, and random or self-similar behavior.…”

Universality of Form: The Case of Retinal Cone Photoreceptor Mosaics

“…which results in µ 2 P 2 6 = 1/2π. For (17) to hold, the probabilities p n s for n / ∈ {5, 6, 7} should be negligible compared to p n s for n ∈ {5, 6, 7}; as a result, the discreteness of n cannot be neglected in this case, since only three ns contribute. The constraint n = 6 implies that p n should sharply peak at n = 6, leading to µ 2 → 0 as p 6 → 1, and thus: µ 2 + p 6 → 1.…”

“…Most two-dimensional cellular networks in nature have an abundance of hexagons, and they likely obey (17) and (18). Low values of p 6 may correspond to amorphous or artificially generated networks.…”

“…To our knowledge, this is the first model capable of predicting the probability of the occurrence of cone cells in various species' eyes by tuning a single parameter. For earlier entropic approaches to study neuronal mosaics, see [17,18].…”