Journal: Scientific Reports

Article Title: An optimality principle for locomotor central pattern generators

doi: 10.1038/s41598-021-91714-1

Figure Lengend Snippet: Dynamic walking model controlled by CPG controller with feedback. ( A ) Pendulum-like legs are controlled by motor commands for hip torques \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{1}$$\end{document} T 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}$$\end{document} T 2 , with sensory feedback of leg angle and ground contact “GC” relayed back to controller. ( B ) Controller produces alternating motor commands versus time, which drive ( C ) leg movement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uptheta }$$\end{document} θ . Sensory measurements of leg angle and ground contact in turn drive the CPG. ( D ) Resulting motion is a nominal periodic gait (termed a “limit cycle”) plotted in state space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\theta }}$$\end{document} θ ˙ versus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uptheta }$$\end{document} θ . ( E ) Discrete perturbation to the limit cycle can cause model to fall.

Article Snippet: The ratio between the noise levels determines the optimal linear quadratic estimator gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ (Matlab function “lqe”).

Techniques:

Journal: Scientific Reports

Article Title: An optimality principle for locomotor central pattern generators

doi: 10.1038/s41598-021-91714-1

Figure Lengend Snippet: Locomotion control circuit interpreted in two representations: ( A ) Neural central pattern generator with mutually inhibiting half-center oscillators, and as ( B ) state estimator with feedback control. Each half-center has a primary neuron with two states ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u$$\end{document} u and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} v , respectively), an auxiliary neuron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} c for registering ground contact, and an alpha motoneuron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} α driving leg torque commands. Inputs include a tonic descending drive, and afferent sensory data with gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} L . State estimator acts as second-order internal model of leg dynamics to estimate leg states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }$$\end{document} θ ^ (hat symbol denotes estimate) and ground contact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{{\text{GC}}}}$$\end{document} GC ^ , which drive state-based command \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} T . The estimator dynamics and estimator parameters including sensory feedback \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} L , and thus the corresponding neural connections and weights, are designed for minimum mean-square estimation error. Leg dynamics have nonlinear terms (see “ ” section) of small magnitude (thin grayed lines).

Article Snippet: The ratio between the noise levels determines the optimal linear quadratic estimator gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ (Matlab function “lqe”).

Techniques: Control

Journal: Scientific Reports

Article Title: An optimality principle for locomotor central pattern generators

doi: 10.1038/s41598-021-91714-1

Figure Lengend Snippet: State estimation accuracy and walking performance under noisy conditions, as a function of sensory feedback gain. The theoretically optimal sensory feedback gain (normalized gain of 1) yielded best performance, in terms of mechanical cost of transport (mCOT), step length variability, mean time between falls (MBTF), and state estimator error. Normalized sensory feedback gain varies between extremes of pure feedforward (to the left) and pure feedback (to the right), with 1 corresponding to theoretically predicted optimum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ . Formally, normalized gain is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| L \right|/\left| {L_{{{\text{lqe}}}}^{*} } \right|,$$\end{document} L / L lqe ∗ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \cdot \right|$$\end{document} · denotes matrix norm. Vertical arrow indicates best performance (minimum for all measures except maximum for MTBF). For all gains, model was simulated with a fixed combination of process and sensor noise as input to multiple trials, yielding ensemble average measures. Each data point is an average of 20 trials of 100 steps each, and errorbar indicates standard deviation of the trials. Mechanical cost of transport (mCOT) was defined as positive work divided by body weight and distance travelled, and step variability as root-mean-square (RMS) variability of step length. Falling takes time and dissipates mechanical energy, and so mCOT was computed both including and excluding losses from falls (work, time, distance).

Article Snippet: The ratio between the noise levels determines the optimal linear quadratic estimator gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ (Matlab function “lqe”).

Techniques: Standard Deviation

Journal: Scientific Reports

Article Title: An optimality principle for locomotor central pattern generators

doi: 10.1038/s41598-021-91714-1

Figure Lengend Snippet: Theoretically optimal sensory feedback gains increase with greater process noise. Effect of three conditions of increasing process noise (L low, M medium, H high) on walking performance as a function of sensory feedback gain. The theoretically optimal gains (vertical lines) led to best performance, as quantified by mechanical cost of transport (mCOT, including falls), step length variability, mean time between falls (MBTF), and state estimator error. (An exception was step length variability, which had a broad and indistinct minimum.) The predicted optimal sensory feedback gains for each noise condition are indicated with vertical lines. Arrows indicate best performance for each measure for each noise condition. Performance is plotted with normalized sensory feedback gain ranging between extremes of pure feedforward (to the left) and pure feedback (to the right), with 1 corresponding to theoretical optimum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ of the previous testing condition (Fig. ). The process noise covariance was set to multiples of the previous reference values: 0.36 for L, 1.15 for M, and 2.06 for H. Sensor noise covariance was set to 1.15 of previous value.

Article Snippet: The ratio between the noise levels determines the optimal linear quadratic estimator gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ (Matlab function “lqe”).

Techniques:

Journal: Scientific Reports

Article Title: An optimality principle for locomotor central pattern generators

doi: 10.1038/s41598-021-91714-1

Figure Lengend Snippet: Emergence of fictive locomotion from CPG model. ( A ) Block diagram of intact control loop, where sensory measurements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{y}}$$\end{document} y and estimation error \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} e are fed into internal model. Motor command \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} T drives the legs and (through efference copy) the internal model of legs. ( B ) Two models of fictive locomotion, starting with the intact system but with sensory feedback removed in two ways. Error feedback refers to sensors that receive efferent copy as inhibitory drive (e.g., some muscle spindles). Removal of error (dashed line) results in sustained fictive rhythm, due to feedback between internal model and state-based command. Measurement feedback refers to other, more direct sensors of limb state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{x}}$$\end{document} x . Removal of such feedback can also produce sustained rhythm from internal model of legs and sensors, interacting with state-based command. ( C ) Simulated motor spike trains show how fictive locomotion can resemble intact. Measurement feedback case produces slower and weaker rhythm than error feedback.

Article Snippet: The ratio between the noise levels determines the optimal linear quadratic estimator gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{{{\text{lqe}}}}^{*}$$\end{document} L lqe ∗ (Matlab function “lqe”).

Techniques: Blocking Assay, Control