Topological Phases in Nonlinear Oscillatory Systems

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Topology refers to a property of matter that is much desired as a guiding principle to realize robust systems, insensitive to continuous deformations and various sources of noise. A standard example is the quantized Hall conductance discussed in condensed matter physics. Topological phases have been analyzed both in quantum and classical systems, but less in the context of biological applications, although the robustness of biological systems to the inherent stochastic fluctuations is not fully understood. We want to explore whether topological protection mechanisms play also some role in biological systems. We consider three nonlinear systems which can perform autonomous oscillations: (i) a unit composed of a feed-forward and a feed-back loop that amounts to a standard motif in genetic and neural networks; (ii) a unit with hierarchical heteroclinic dynamics, suited to describe transient processes in particular in the brain; and (iii) a modified repressilator, designed to model transient cell dynamics. When dynamical units, belonging to one of these types, are coupled on spatial grids, we choose different couplings, which promise the observation of topological phases. We analyze the expected edge modes, characterize them by topological invariants, and pursue the dependence of topological synchronization as a function of the system parameters, the implemented chirality, the nonlinear interaction strength, and the Hermitian or non-Hermitian properties of the involved effective Hamiltonians. In the context of biological networks, stable clocks, stable biomass transport and stable transition paths in heteroclinic networks will be desirable features when their stability is based on the very efficient topological protection against the various sources of noise.