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

"in the table \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_p$$\end{document} s p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_p$$\end{document} j p and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {j}}_p$$\end{document} j p denote the saltation matrix the jump matrix and the time-reversed jump matrix at some boundary crossing point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{x}_p=\textbf{x}(t_p)$$\end{document} x p = x ( t p ) respectively \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_p^-=\lim _{\textbf{x}\rightarrow \textbf{x}_p^-}f(\textbf{x})$$\end{document} f p - = lim x -> x p - f ( x ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_p^+=\lim _{\textbf{x}\rightarrow \textbf{x}_p^+}f(\textbf{x})$$\end{document} f p + = lim x -> x p + f ( x ) denote the vector fields of the nonsmooth system just before and just after the crossing at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{x}_p$$\end{document} x p i denotes the identity matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_p$$\end{document} n p denotes the unit normal vector of the crossing boundary at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{x}_p$$\end{document} x p simulation codes written in matlab are available at https://github com/yangyang-wang/aplysiamodel ."

"This work was supported in part by National Institutes of Health BRAIN Initiative grant RF1 NS118606-01 to HJC and PJT, by NSF grant DMS-2052109 to PJT, by NSF grant IOS-1754869 to HJC, by NIH/NIDA R01DA057767 to YW, as part of the NSF/NIH/DOE /ANR/BMBF/BSF/NICT/AEI/ISCIII Collaborative Research in Computational Neuroscience Program and by NSF grant DBI 2015317 to HJC, as part of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks for Neuroscience Program. This work was supported in part by the Oberlin College Department of Mathematics. We thank Zhuojun Yu for providing a critical reading of the manuscript. We thank the anonymous reviewer for suggesting an alternative way of deriving the infinitesimal shape response curve, which is now included in the “Appendix C.”"