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

TL;DR

This paper analyzes how neuromechanical models of motor systems, specifically in Aplysia, achieve robustness against perturbations by examining shape and timing responses of limit cycles, revealing nonlinear biomechanical resilience and delayed sensory feedback effects.

Contribution

It introduces a combined shape and timing response analysis for limit cycle systems with hard boundaries, applied to a neuromechanical model of Aplysia's feeding behavior, highlighting mechanisms of robustness.

Findings

Nonlinear biomechanical properties increase immediate load resistance.

Sensory feedback delays influence timing adjustments in motor responses.

Robustness is achieved through shifts in neural activation timing.

Abstract

Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness is difficult. In many rhythmic systems, robustness against perturbations involves response of both the shape and the timing of the trajectory. 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 \textit{Aplysia californica} \citep{shaw2015,lyttle2017}. We established in \citep{WGCT2021} the tools for studying combined shape and timing responses of limit cycle systems under sustained perturbations and here apply them to study robustness of the neuromechanical model against increased mechanical load during swallowing. Interestingly, we discover that nonlinear biomechanical properties confer resilience by immediately increasing resistance to applied loads. In contrast, the effect of changed sensory feedback signal is significantly delayed by the firing rates' hard boundary properties. Our analysis suggests that sensory feedback contributes to robustness in swallowing primarily by shifting the timing of neural activation involved in the power stroke of the motor cycle (retraction). This effect enables the system to generate stronger retractor muscle forces to compensate for the increased load, and hence achieve strong robustness. The approaches that we are applying to understanding a neuromechanical model in \textit{Aplysia}, and the results that we have obtained, are likely to provide insights into the function of other motor systems that encounter changing mechanical loads and hard boundaries, both due to mechanical and neuronal firing properties.