Browsing by Subject "Singularity theory"
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Item Open Access Homeostasis-Bifurcation Singularities and Identifiability of Feedforward Networks(2020) Duncan, WilliamThis dissertation addresses two aspects of dynamical systems arising from biological networks: homeostasis-bifurcation and identifiability.
Homeostasis occurs when a biological quantity does not change very much as a parameter is varied over a wide interval. Local bifurcation occurs when the multiplicity or stability of equilibria changes at a point. Both phenomena can occur simultaneously and as the result of a single mechanism. We show that this is the case in the feedback inhibition network motif. In addition we prove that longer feedback inhibition networks are less stable. Towards understanding interactions between homeostasis and bifurcations, we define a new type of singularity, the homeostasis-bifurcation point. Using singularity theory, the behavior of dynamical systems with homeostasis-bifurcation points is characterized. In particular, we show that multiple homeostatic plateaus separated by hysteretic switches and homeostatic limit cycle periods and amplitudes are common when these singularities occur.
Identifiability asks whether it is possible to infer model parameters from measurements. We characterize the structural identifiability properties for feedforward networks with linear reaction rate kinetics. Interestingly, the set of reaction rates corresponding to the edges of the graph are identifiable, but the assignment of rates to edges is not; Permutations of the reaction rates leads to the same measurements. We show how the identifiability results for linear kinetics can be extended to Michaelis-Menten kinetics using asymptotics.
Item Open Access Invariance of Knot Lattice Homology and Homotopy(2022) Niemi-Colvin, Seppo MatthewLinks of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice homology is an invariant of the smooth knot type of a generalized algebraic knot in a rational homology sphere. In that case, knot lattice homology can be realized as the cellular homology of a doubly-filtered homotopy type, which is itself invariant. Along the way, we show that the topological link type of a generalized algebraic link determines the nested singularity type.