Two-component feedback loops and deformed mechanics

Abstract

It is shown that a general two-component feedback loop can be viewed as a deformed Hamiltonian system. Some of the implications of using ideas from theoretical physics to study biological processes are discussed.

Deformed Hamiltonian vector fields and Lagrangian fibrations

“…to mirror the formalism of conventional classical mechanics. He showed that this bracket remains invariant under specific representations of canonical transformations, and in [2] these were discussed in the setting of q-deformed groups. Duffin also proved a dissipative version of Liouville's theorem for statistical ensembles on phase space under the assumption that second derivatives ∂ 2 H/∂x i ∂y i are constant.…”

“…He called the dynamics psuedo-Hamiltonian. The same equations have also been shown to include models of very simple biological processes [2]. Properties of these systems are clearly different from those of conservative Hamiltonian systems since, reflecting dissipation, the Hamiltonian H is not an integral of motion…”

“…Then again, perhaps this is not so surprising given that symplectic geometry was developed to accommodate Hamiltonian systems into a geometric setting. Can something similar be achieved for the dynamical systems considered in [1,2]? This is the problem that we attempt to address here.…”

“…Certain dissipative dynamical systems arising in physics are also described using a symplectic viewpoint although not in the setting of differentiable manifolds [1,2]. The current paper grew out of an attempt to put these dynamical systems into the context of symplectic geometry.…”

“…Generally the smooth function H : R 2n → R might also depend on q. Equations of this form were considered a long time ago in physics and used to model dissipative phenomena [1]. There is also a more recent interpretation of these systems in theoretical biology [2]. It is clear that (1) becomes an ordinary Hamiltonian system in the limit q → 1 and so to generalise such dynamical systems to symplectic manifolds we introduce the notion of a deformed Hamiltonian vector field.…”