Systematic, cross-cortex variation in neuron numbers in rodents and primates.

Uniformity, local variability, and systematic variation in neuron numbers per unit of cortical surface area across species and cortical areas have been claimed to characterize the isocortex. Resolving these claims has been difficult, because species, techniques, and cortical areas vary across studies. We present a stereological assessment of neuron numbers in layers II-IV and V-VI per unit of cortical surface area across the isocortex in rodents (hamster, Mesocricetus auratus; agouti, Dasyprocta azarae; paca, Cuniculus paca) and primates (owl monkey, Aotus trivigratus; tamarin, Saguinus midas; capuchin, Cebus apella); these chosen to vary systematically in cortical size. The contributions of species, cortical areas, and techniques (stereology, "isotropic fractionator") to neuron estimates were assessed. Neurons per unit of cortical surface area increase across the rostro-caudal (RC) axis in primates (varying by a factor of 1.64-2.13 across the rostral and caudal poles) but less in rodents (varying by a factor of 1.15-1.54). Layer II-IV neurons account for most of this variation. When integrated into the context of species variation, and this RC gradient in neuron numbers, conflicts between studies can be accounted for. The RC variation in isocortical neurons in adulthood mirrors the gradients in neurogenesis duration in development.

Modeling local and cross-species neuron number variations in the cerebral cortex as arising from a common mechanism.

A massive increase in the number of neurons in the cerebral cortex, driving its size to increase by five orders of magnitude, is a key feature of mammalian evolution. Not only are there systematic variations in cerebral cortical architecture across species, but also across spatial axes within a given cortex. In this article we present a computational model that accounts for both types of variation as arising from the same developmental mechanism. The model employs empirically measured parameters from over a dozen species to demonstrate that changes to the kinetics of neurogenesis (the cell-cycle rate, the progenitor death rate, and the "quit rate," i.e., the ratio of terminal cell divisions) are sufficient to explain the great diversity in the number of cortical neurons across mammals. Moreover, spatiotemporal gradients in those same parameters in the embryonic cortex can account for cortex-wide, graded variations in the mature neural architecture. Consistent with emerging anatomical data in several species, the model predicts (i) a greater complement of neurons per cortical column in the later-developing, posterior regions of intermediate and large cortices, (ii) that the extent of variation across a cortex increases with cortex size, reaching fivefold or greater in primates, and (iii) that when the number of neurons per cortical column increases, whether across species or within a given cortex, it is the later-developing superficial layers of the cortex which accommodate those additional neurons. We posit that these graded features of the cortex have computational and functional significance, and so must be subject to evolutionary selection.

Systematic, balancing gradients in neuron density and number across the primate isocortex.

The cellular and areal organization of the cerebral cortex impacts how it processes and integrates information. How that organization emerges and how best to characterize it has been debated for over a century. Here we demonstrate and describe in the isocortices of seven primate species a pronounced anterior-to-posterior gradient in the density of neurons and in the number of neurons under a unit area of the cortical surface. Our findings assert that the cellular architecture of the primate isocortex is neither arranged uniformly nor into discrete patches with an arbitrary spatial arrangement. Rather, it exhibits striking systematic variation. We conjecture that these gradients, which establish the basic landscape that richer areal and cellular structure is built upon, result from developmental patterns of cortical neurogenesis which are conserved across species. Moreover, we propose a functional consequence: that the gradient in neurons per unit of cortical area fosters the integration and dimensional reduction of information along its ascent through sensory areas and toward frontal cortex.

Network structure implied by initial axon outgrowth in rodent cortex: empirical measurement and models.

The developmental mechanisms by which the network organization of the adult cortex is established are incompletely understood. Here we report on empirical data on the development of connections in hamster isocortex and use these data to parameterize a network model of early cortical connectivity. Using anterograde tracers at a series of postnatal ages, we investigate the growth of connections in the early cortical sheet and systematically map initial axon extension from sites in anterior (motor), middle (somatosensory) and posterior (visual) cortex. As a general rule, developing axons extend from all sites to cover relatively large portions of the cortical field that include multiple cortical areas. From all sites, outgrowth is anisotropic, covering a greater distance along the medial/lateral axis than along the anterior/posterior axis. These observations are summarized as 2-dimensional probability distributions of axon terminal sites over the cortical sheet. Our network model consists of nodes, representing parcels of cortex, embedded in 2-dimensional space. Network nodes are connected via directed edges, representing axons, drawn according to the empirically derived anisotropic probability distribution. The networks generated are described by a number of graph theoretic measurements including graph efficiency, node betweenness centrality and average shortest path length. To determine if connectional anisotropy helps reduce the total volume occupied by axons, we define and measure a simple metric for the extra volume required by axons crossing. We investigate the impact of different levels of anisotropy on network structure and volume. The empirically observed level of anisotropy suggests a good trade-off between volume reduction and maintenance of both network efficiency and robustness. Future work will test the model's predictions for connectivity in larger cortices to gain insight into how the regulation of axonal outgrowth may have evolved to achieve efficient and economical connectivity in larger brains.

Seed size strongly affects cascades on random networks.

The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions-these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low- z transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.