In this work, we consider a mathematical model describing spontaneous calcium signaling in astrocytes. Based on biologically relevant principles, this model simulates experimentally observed calcium oscillations and can predict the emergence of complicated dynamics. Using analytical and numerical analysis, various attracting sets were found and investigated. Employing bifurcation theory analysis, we examined steady state solutions, bistability, simple and complicated periodic limit cycles and also chaotic attractors. We found that astrocytes possess a variety of complex dynamical modes, including chaos and multistability, that can further provide different modulations of neuronal circuits, enhancing their plasticity and flexibility.

We study the dynamics of multielement neuronal systems taking into account both the direct interaction between the cells via linear coupling and nondiffusive cell-to-cell communication via common environment. For the cells exhibiting individual bursting behavior, we have revealed the dependence of the network activity on its scale. Particularly, we show that small-scale networks demonstrate the inability to maintain complicated oscillations: for a small number of elements in an ensemble, the phenomenon of amplitude death is observed. The existence of threshold network scales and mechanisms causing firing in artificial and real multielement neural networks, as well as their significance for biological applications, are discussed.

Bacterial chemotaxis is one of the most extensively studied adaptive responses in cells. Many bacteria are able to bias their apparently random motion to produce a drift in the direction of the increasing chemoattractant concentration. It has been recognized that the particular motility pattern employed by moving bacteria has a direct impact on the efficiency of chemotaxis. The linear theory of chemotaxis pioneered by de Gennes allows for calculation of the drift velocity in small gradients for bacteria with basic motility patterns. However, recent experimental data on several bacterial species highlighted the motility pattern where the almost straight runs of cells are interspersed with turning events leading to the reorientation of the cell swimming directions with two distinct angles following in strictly alternating order. In this manuscript we generalize the linear theory of chemotaxis to calculate the chemotactic drift speed for the motility pattern of bacteria with two turning angles. By using the experimental data on motility parameters of V. alginolyticus bacteria we can use our theory to relate the efficiency of chemotaxis and the size of bacterial cell body. The results of this work can have a straightforward extension to address most general motility patterns with alternating angles, speeds and durations of runs.