Shape versus Timing: Linear Responses of a Limit Cycle with Hard Boundaries under Instantaneous and Static Perturbation

Variational and phase response analysis for limit cycles with hard boundaries, with applications to neuromechanical control problems

“…However, since the perturbed limit cycle has a different period T ε and hence a different perturbed time τ ε due to sustained perturbations, the forward variational equation which neglects such changes in timing fails to give a valid comparison between the perturbed and unperturbed trajectories for times on the order of a full cycle or longer (see Figure 2C and D). Hence, we adopt a new tool developed in Wang et al (2021), the infinitesimal shape response curve (ISRC), which incorporates both the shape and timing aspects and captures a more accurate first-order approximation to the change in shape of the limit cycle under a parameteric perturbation.…”

“…Infinitesimal Shape Response Curve (ISRC) Suppose the rescaled perturbed time can be written as Wang et al (2021) denote the linear shift in the periodic orbit, γ 1 (t) in (3.10), as the ISRC and show it satisfies the following variational equation:…”

“…Thus, in order to compare perturbed and unperturbed motions with greater accuracy, in the remainder of the paper (cf. §2.2 and following) we show how to extend the local-in-time variational analysis to a global analysis by applying the infinitesimal shape response curve (ISRC) analysis and local timing response curve (LTRC) analysis developed in Wang et al (2021). We review these methods in §3.…”

“…Such temporary slowing down of neural variables allows the muscles, which evolve on slower timescales, to "catch up"; hence the seaweed can be swallowed and ingested successfully. Following the terminology from (Wang et al, 2021), we call attracting periodic trajectories that experience sliding motions limit cycles with sliding components (LC-SCs).…”

“…This makes the study of robustness even more challenging.To understand how a motor system produces robust behaviors in a variable environment, we consider a neuromechanical model of motor patterns in the feeding apparatus of the marine mollusk Aplysia californica Lyttle et al, 2017). We established in (Wang et al, 2021) the tools for studying combined shape and timing responses of limit cycle systems under sustained perturbations and here…”