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Qualitative theory of differential equations has been shown to be
successful in many applications. In this talk, I present two dynamical
models designed in terms of ordinary differential equations for
biological oscillatory processes. The first model represents behavior
of protein concentration in a synthetic regulatory network. Modeling
was used for guiding design of the network to achieve oscillatory
behavior. Synchronization of these cellular oscillators across a
population of cells is investigated. I demonstrate the possibility of
both population synchronization and suppression of oscillations
(stationary cluster formation). The approach can be extended on natural
regulatory networks to gain insights into their connectivity and,
therefore, complements biological methods. The second model represents electrical activity of a single neuron: the dopaminergic neuron. The neurons display two functionally distinct modes of activity: low- and high-frequency firing. The high-frequency firing is linked to important behavioral events in vivo. However, it cannot be elicited by standard manipulations in brain slices. Using coupled oscillator model of the neuron, I show a way of combining experimental data and introduce the mechanism of switching dominance for high-frequency firing. According to the mechanism, behavior of the neuron is dominated by its different parts, soma or dendrites under different experimental conditions. I employ the dynamical phenomenon of localization to explain the results. The new theory has allowed formulating important predictions, several of which have been already confirmed experimentally. |