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Modeling of the neural mechanism underlying the terrestrial turning of the salamander

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Abstract

In order to explore the neural mechanism underlying salamander terrestrial turning, an improved biomechanical model is proposed by modifying the forelimb structure of the existing biomechanical model. Based on the proposed improved biomechanical model, a new spinal locomotor network model is constructed which contains the interneuron networks and motoneuron pool. Control methods are also developed for the new model which increase its transient response speed, control the initial swing order of the forelimbs, and generate different walking turning gait and turning on the spot (turning without moving forward). The simulation results show that the biomechanical model controlled by the new spinal locomotor network model can generate different walking turning and turning on the spot, and can control posture and the initial swing order of the forelimbs. Moreover, the transient response speed of the proposed model is very rapid. This paper thus provides a useful tool for exploring the operational mechanism of the spinal circuitry of the salamander. In addition, the research results presented here may inspire the construction of artificial spinal control networks for bionic robots.

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Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under grants 61105110 and 11573011, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under grant 14KJB510004, the Lianyungang “Petrel Project,” the Lianyungang “521” Project, and the Six Talent Peaks Project in Jiangsu Province.

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Authors and Affiliations

  1. School of Electric Engineering, Jiangsu Ocean University, Lianyungang, 222005, China

    Qiang Liu, Jingzhuo Wang, Huizhen Yang & Lu Hong

  2. School of Mechanical and Ocean Engineering, Jiangsu Ocean University, Lianyungang, 222005, China

    Yongshuo Zhang

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  1. Qiang Liu
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  2. Yongshuo Zhang
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  3. Jingzhuo Wang
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  4. Huizhen Yang
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  5. Lu Hong
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Correspondence to Qiang Liu.

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Communicated by Benjamin Lindner.

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Appendices

Appendix 1: Calculation of turning control input for turning gait IV

When the biomechanical model turns left, the turning control inputs applied to the elbow joint and leg-raising oscillators for generating turning gait IV can be calculated by:

$$ \left\{ \begin{array}{l} s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,31) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,32) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,33) \hfill \\ s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,34) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} > F_{T} ), $$
(9)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,31) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,32) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,33) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,34) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} \le F_{T} ), $$
(10)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,31) \hfill \\ s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,32) \hfill \\ s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,33) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,34) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} \le F_{T} ), $$
(11)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,31) \hfill \\ s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,32) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,33) \hfill \\ s_{iT} = k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,34) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} > F_{T} ), $$
(12)

where \( k_{ 3} \) is the control gain of turning gait IV, \( F_{l} \) is the contact force between the left forelimb and the ground, \( F_{r} \) is the contact force between the right forelimb and the ground, and \( F_{T} \) is the contact force threshold between the forelimb and the ground.

When the biomechanical model turns right, the turning control inputs applied to the elbow joint and leg-raising oscillators for generating turning gait IV can be calculated by:

$$ \left\{ \begin{array}{l} s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,32) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,31) \hfill \\ s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,33) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,34) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} > F_{T} ), $$
(13)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,32) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,31) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,34) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,33) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} \le F_{T} ), $$
(14)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,32) \hfill \\ s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,31) \hfill \\ s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,34) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,33) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} \le F_{T} ), $$
(15)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 27,32) \hfill \\ s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 28,31) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 30,34) \hfill \\ s_{iT} = { - }k_{3} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 29,33) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} > F_{T} ). $$
(16)

Appendix 2: Calculation of turning control input for side-stepping applying control signals to the motoneuron pool

When the biomechanical model turns left, the turning control signals applied to the forelimb motoneuron pool for the side-stepping can be calculated by:

$$ \left\{ \begin{array}{l} s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,73) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,74) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,75) \hfill \\ s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} > F_{T} ), $$
(17)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,73) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,74) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,75) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} \le F_{T} ), $$
(18)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,73) \hfill \\ s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,74) \hfill \\ s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,75) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} \le F_{T} ), $$
(19)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,73) \hfill \\ s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,74) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,75) \hfill \\ s_{iT} = k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} > F_{T} ), $$
(20)

where \( k_{ 3} \) is the control gain of the side-stepping applying the turning control signals to the motoneuron pool.

When the biomechanical model turns right, the turning control signals applied to the forelimb motoneuron pool for the side-stepping can be calculated by:

$$ \left\{ \begin{array}{l} s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,74) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,73) \hfill \\ s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,75) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} > F_{T} ), $$
(21)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,74) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,73) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,75) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} \le F_{T} ,F_{r} \le F_{T} ), $$
(22)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,74) \hfill \\ s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,73) \hfill \\ s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,75) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} \le F_{T} ), $$
(23)
$$ \left\{ \begin{array}{l} s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 69,74) \hfill \\ s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 70,73) \hfill \\ s_{iT} = 0,\begin{array}{*{20}l} {} \\ \end{array} (i = 71,75) \hfill \\ s_{iT} = { - }k_{ 4} u,\begin{array}{*{20}l} {} \\ \end{array} (i = 72,76) \hfill \\ \end{array} \right.,\begin{array}{*{20}l} {} \\ \end{array} (F_{l} > F_{T} ,F_{r} > F_{T} ). $$
(24)

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Liu, Q., Zhang, Y., Wang, J. et al. Modeling of the neural mechanism underlying the terrestrial turning of the salamander. Biol Cybern 114, 317–336 (2020). https://doi.org/10.1007/s00422-020-00821-1

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