Complete synchronizability of chaotic systems: a geometric approach.

Synchronizability of chaotic systems is studied in this contribution. Geometrical tools are used to understand the properties of vector fields in affine systems. The discussion is focused on synchronizability of chaotic systems with equal order. The analysis is based on the synchronous behavior of all states of the master/slave system (complete synchronization). We state sufficient and necessary conditions for complete synchronizability which are based on controllability and observability of nonlinear affine systems. In this sense, the synchronizability is studied for complete synchronization via state feedback control.

Mathematical modeling of dendritic growth in vitro.

The dendritic branching pattern of cultured hippocampal neurons was analyzed to obtain mathematical parameters that fit the time-dependent growth of dendrites under limited extrinsic influence. Cultured neurons were stained with a non-toxic carbocyanine dye (diO) and pyramidal-shaped neurons that were physically separated from one another were analyzed at post-plating days 1, 2, 3, 4, 6 and 7. The geometric branching pattern of the dendrites was analyzed using a mathematical model that incorporates random effects in the form of a Galton-Watson branching process where splitting of one branch is statistically independent of the splitting of all other branches, and deterministic effects in the form of a parameter that measures the extent to which dense patterns (clusters) or sparse patterns (elongated trees) are formed. The geometric branching pattern of the dendrites was analyzed using a mathematical model that incorporates random and deterministic effects. The model parameters were estimated via the method of maximum likelihood. The data suggest that in vitro basal dendrites grow according to a purely random branching process without pronounced dense or sparse patterns, while apical dendrites tend to form elongated trees with fewer secondary bifurcations. This trend is quantified, and it depends on the culture conditions in which the neurons are grown. The quantitative assessment of various influences on dendritic growth patterns are discussed.

Statistical evaluation of dendritic growth models.

A mathematical model (Kliemann, W. 1987. Bull. math. Biol. 49, 135-152.) that predicts the quantitative branching pattern of dendritic tree was evaluated using the apical and basal dendrites of rat hippocampal neurons. The Wald statistic for chi 2-test was developed for the branching pattern of dendritic trees and for the distribution of the maximal order of the tree. Using this statistic, we obtained a reasonable, but not excellent, fit of the mathematical model for the dendritic data. The model's predictability of branching pattern was greatly enhanced by replacing one of the assumptions used for the original method "splitting of branches for all dendritic orders is stochastically independent", with a new assumption "branches are more likely to split in areas where there is already a high density of branches". The modified model delivered an excellent fit for basal dendrites and for the apical dendrites of hippocampal neurons from young rats (30-34 days postpartum). This indicates that for these cells the development of dendritic patterns is the result of a purely random and a systematic component, where the latter one depends on the density of dendritic branches in the brain area considered. For apical dendrites there is a trend towards decreasing pattern predictability with increasing age. This appears to reflect the late arrival of afferents and subsequent synaptogenesis proximal on the apical dendritic tree of hippocampal neurons.