Variational analysis of sensory feedback mechanisms in powerstroke–recovery systems

Biological Cybernetics https://doi.org/10.1007/s00422-024-00996-x ORIGINAL ARTICLE Variational analysis of sensory feedback mechanisms in powerstroke–recovery systems Zhuojun Yu1 · Peter J. Thomas2 Received: 29 March 2024 / Accepted: 21 August 2024 © The Author(s) 2024 Abstract Although the raison d’etre of the brain is the survival of the body, there are relatively few theoretical studies of closed-loop rhythmic motor control systems. In this paper we provide a unified framework, based on variational analysis, for investigating the dual goals of performance and robustness in powerstroke–recovery systems. To demonstrate our variational method, we augment two previously published closed-loop motor control models by equipping each model with a performance measure based on the rate of progress of the system relative to a spatially extended external substrate—such as a long strip of seaweed for a feeding task, or progress relative to the ground for a locomotor task. The sensitivity measure quantifies the ability of the system to maintain performance in response to external perturbations, such as an applied load. Motivated by a search for optimal design principles for feedback control achieving the complementary requirements of efficiency and robustness, we discuss the performance–sensitivity patterns of the systems featuring different sensory feedback architectures. In a paradigmatic half-center oscillator-motor system, we observe that the excitation–inhibition property of feedback mechanisms determines the sensitivity pattern while the activation–inactivation property determines the performance pattern. Moreover, we show that the nonlinearity of the sigmoid activation of feedback signals allows the existence of optimal combinations of performance and sensitivity. In a detailed hindlimb locomotor system, we find that a force-dependent feedback can simultaneously optimize both performance and robustness, while length-dependent feedback variations result in significant performance-versus-sensitivity tradeoffs. Thus, this work provides an analytical framework for studying feedback control of oscillations in nonlinear dynamical systems, leading to several insights that have the potential to inform the design of control or rehabilitation systems. Keywords Sensory feedback · Closed-loop control · Central pattern generator · Power stroke · Robustness · Efficiency 1 Introduction Physiological systems underlying vital behaviors such as breathing, walking, crawling, and feeding, must generate Communicated by Benjamin Lindner. B Zhuojun Yu Peter J. Thomas 1 Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA 2 Department of Mathematics, Applied Mathematics, and Statistics, Department of Biology, Department of Electrical, Control and Systems Engineering, Case Western Reserve University, Cleveland, OH 44106, USA motor rhythms that are not only efficient, but also robust against changes in operating conditions. Although central neural circuits have been shown to be capable of producing rhythmic motor outputs in isolation from the periphery (Brown 1911, 1914; Harris-Warrick and Cohen 1985; Pearson 1985; Smith et al. 1991), the role of sensory feedback should not be underestimated. Sensory feedback can play a crucial role in stabilizing motor activity in response to unexpected conditions. For example, modeling work suggests that walking movements can be stably restored after spinal cord injury by enhancing the strengths of the afferent feedback pathways to the spinal central pattern generator (CPG) (Markin et al. 2010; Spardy et al. 2011). Feedback control can also improve the performance and efficiency of movements. For instance, in a model of feeding motor patterns in the marine mollusk Aplysia californica, seaweed intake can 123 Biological Cybernetics be increased by strengthening the gain of sensory feedback to a specific motor neural pool (Wang et al. 2022). We are interested in understanding how sensory feedback contributes to control and stabilization within a specific class of rhythmic motor behaviors, namely, behaviors in which an animal (or robot) repeatedly engages and disengages with the outside world (see Fig. 1, top). We refer to the phase of the motion during which the animal is in contact with an external substrate as the power stroke, and the component during which the animal is disengaged as the recovery phase. The decomposition of a repetitive movement into powerstroke and recovery applies naturally to many motor control systems, including locomotion (Jahn and Votta 1972) and swallowing (Shaw et al. 2015); a similar dynamical structure also appears in mechanical stick–slip systems (Galvanetto and Bishop 1999) as well as abstract two-stroke relaxation oscillators (Jelbart and Wechselberger 2020). In the motor control context, when the animal is in contact with an external substrate or load opposing the motion, we say the animal makes “progress" (food is consumed, distance is traveled, oxygen is absorbed) relative to the outside world. During the recovery phase, the animal disconnects from the external component, and repositions relative to the substrate in order to prepare for the next power stroke. Consider, for example, the ingestive behavior of Aplysia (Shaw et al. 2015; Lyttle et al. 2017; Wang et al. 2022). When the animal’s grasper is closed on a stipe of seaweed, it drags the food into the buccal cavity; meanwhile, the food applies a mechanical load on the grasper. Then the grasper opens, releasing its grip on the food. The grasper moves in the absence of the force exerted by the seaweed and returns to the original position to begin the next swallowing cycle. In this paper, we present a novel analysis of feedback control for powerstroke–recovery systems. To quantitatively evaluate the behavior of a system controlled by different feedback mechanisms, we measure the sensitivity (or robustness) and performance (or efficiency) (see Fig. 1, bottom). The complementary objectives of sensitivity and performance have been studied in a variety of motor control systems, from both empirical and theoretical perspectives (Lee and Tomizuka 1996; Yao et al. 1997; Ronsse et al. 2008; Hutter et al. 2014; Lyttle et al. 2017; Sharbafi et al. 2020; Mo et al. 2023). There are a myriad of ways to interpret performance and robustness used by engineers, biologists, neuroscientists, and applied mathematicians. Here we define the performance of a powerstroke–recovery system to be the total progress divided by the period of the rhythm (i.e., the average rate of progress), and the sensitivity to be the ability of the system to maintain performance in response to some specific external perturbation, such as an increased mechanical resistance while pulling on a load, or increased slope while walking. That is, we take the sensitivity to be the derivative of the performance with respect to the external perturb (...truncated)