UCLA

In this dissertation, I study two types of networks: biopolymer filament networks (part I) and biological neural networks (part II).

First, I focus on the most common structural element of the filament networks: bundles of parallel stiff filaments held together by smaller molecules (cross links). The experiment and numerical simulations show that such bundles can have localized regions of high curvature that are long-lived metastable states (kinks). I suggest the mechanism of kink stabilization as a topological defect, with three possible defect types: a difference in trapped length of the filament segments between two cross-links (loop); the braiding of the filaments in the bundle; and the dislocation where the endpoint of a filament occurs within the bundle. I show that the pairs of defects are produced under compressive loading. Loops then get separated, while braids remain coupled. The braid separation requires cycles of compression and tension.

Second, I explore the force propagation in the whole network from the point-like source. Numerical simulations show a particular pattern of stress propagation. In particular, the force decays exponentially, which I can capture using an analytical model in a particular parameter region.

Third, I study the statistical properties of the filament in the network modeling the rest of the network as a quenched random potential. I compare the average end-to-end distance in this model with one obtained numerically validating it.

Finally, I switch to neural networks and explore the preB\"{o}tzinger complex which produces rhythm that times inspiration in mammals. We explore the initiation of the activity by external stimulation of a small subset of neurons. In the leaky integrate-and-fire neuron model, I observe that only a small subset of network patterns can produce results resembling experimental data. Using the firing-rate model with dendritic adaptation, we observe that if the system quits the oscillatory phase, it gets separated into two subnetworks: highly firing and low firing one. This separation occurs even for the all-to-all coupled networks thus being a spontaneous symmetry breaking. For the arbitrary network, simplified ("on-off") version of neuronal dynamics the separation is exactly controlled by the topological feature of the network - k-cores.