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.

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.

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.