A new model of the spinal locomotor networks of a salamander and its properties

Abstract

A salamander is an ideal animal for studying the spinal locomotor network mechanism of vertebrates from an evolutionary perspective since it represents the transition from an aquatic to a terrestrial animal. However, little is known about the spinal locomotor network of a salamander. A spinal locomotor network model is a useful tool for exploring the working mechanism of the spinal networks of salamanders. A new spinal locomotor network model for a salamander is built for a three-dimensional (3D) biomechanical model of the salamander using a novel locomotion-controlled neural network model. Based on recent experimental data on the spinal circuitry and observational results of gaits of vertebrates, we assume that different interneuron sets recruited for mediating the frequency of spinal circuits are also related to the generation of different gaits. The spinal locomotor networks of salamanders are divided into low-frequency networks for walking and high-frequency networks for swimming. Additionally, a new topological structure between the body networks and limb networks is built, which only uses the body networks to coordinate the motion of limbs. There are no direct synaptic connections among limb networks. These techniques differ from existing salamander spinal locomotor network models. A simulation is performed and analyzed to validate the properties of the new spinal locomotor networks of salamanders. The simulation results show that the new spinal locomotor networks can generate a forward walking gait, a backward walking gait, a swimming gait, and a turning gait during swimming and walking. These gaits can be switched smoothly by changing external inputs from the brainstem. These properties are consistent with those of a real salamander. However, it is still difficult for the new spinal locomotor networks to generate highly efficient turning during walking, 3D swimming, nonrhythmic movements, and so on. New experimental data are required for further validation.

Appendix

1.1 A: Detailed description of the biomechanical model of a salamander

For the new biomechanical model of a salamander, built using the ADAMS software, a virtual entity model needs to be built and the constraints added based on Fig. 1. The forces from the water and muscles and the contact forces between the links can then be applied to the virtual entity model of a salamander.

Parameters of virtual entity model The virtual model of a salamander has ten links for the body and eight links for the limbs. Their parameters are shown in Table 4.

Table 4 Parameters of virtual entity model of salamander

Table 5 Parameters of muscle torques of joints

Table 6 Upper bound of inactivity and lower bound of saturation of tonic external inputs for forward walking networks

Table 7 Upper bound of inactivity and lower bound of saturation of tonic external inputs for backward walking networks

Table 8 Upper bound of inactivity and lower bound of saturation of tonic external inputs for swimming networks

Forces due to water Neglecting the fluid forces of limbs, the fluid forces $F_i ,(i=1,\ldots ,10)$ of the links of the body can be calculated as

$$\begin{aligned} F_i =\frac{1}{2}p\rho l_i h_i C_i v_i ^{2},\quad (i=1,\ldots ,10), \end{aligned}$$

(2)

where $\rho$ is the density of the water; $l_i$ is the length of the link i of the body; $h_i$ is the height of the link i of the body; $C_i$ is the drag coefficient of the link i of the body (here $C_i =1$ for all links of the body); $v_i$ is the normal velocity of the link i at the midpoint of the body; and p is the switch variable between walking and swimming (during walking $p=0$ and during swimming).

Torque of muscles A muscle can be simulated using a combination of a spring and a damper (Ekeberg 1993). Using the muscle model as in the reference, the torque acting on the joint of a salamander can be calculated as

$$\begin{aligned} T=\alpha (M_f -M_e )+\beta (M_f +M_e +\xi )\Delta \varphi +\delta \Delta \dot{\varphi }, \end{aligned}$$

(3)

where $M_f$ are the motoneuron activities of the flexor muscles; $M_e$ are the motoneuron activities of the extensor muscles; $\Delta \varphi$ is the difference between the actual angle of the joint and the resting angle; and $\alpha ,\beta$,$\xi$, and $\delta$ indicate respectively the gain, stiffness gain, tonic stiffness, and damping coefficient of the muscles. The parameters of the muscles are shown in Table 5.

Constraint and contact for biomechanical model of salamander Based on the moving range of the shoulder joints and hip joints, constraint blocks surrounding the thighs are added, and the contact between each constraint block and the thighs is added. Based on the moving range of the body joints and the elbow (or knee) joints, the constraint blocks located between the body joints and between the thighs and the shanks, as well as the contact between them, are also added. In addition, the contacts between the biomechanical model of a salamander and the ground are added (see Sect. 2 for further details).

1.2 B: Detailed description of spinal locomotor networks of a salamander

Parameters of spinal locomotor networks During gait generation and gait transition, all parameters of the spinal locomotor networks remain unchanged, except for $\tau _i$ and $\gamma _i$. Both of these are used to control the frequency of the spinal locomotor networks. It is assumed here that the parameters of all oscillators are the same, and all neurons have a rhythmic output under default conditions. Based on Liu and Wang (2018), we choose the parameters of the oscillator to be $\tau _i ={0.5},\gamma _i =0.3,\varepsilon _i =10,\sigma _i =1,a_{ii} =14,b_i =8, \theta _i =0,\bar{{x}}_i ={1}0(i=1,\ldots ,62), a_{ii+1} =-0.8,a_{i+1i} =-0.6(i=1,3,\ldots ,61)$. The excitatory weight between the oscillators of the body is 0.4. The excitatory weight from the oscillators of the body to the oscillators of the limbs is 0.9. The excitatory weight from the oscillators of the limbs to the oscillators of the body is 0.9. The inhibitory weight between the oscillators of the body and the oscillators of the limbs is $-0.9$. The excitatory weight between the oscillators of the limbs is 0.9.

Range of external inputs Based on Liu and Wang (2018), the tonic external inputs of the networks, which make the networks generate rhythmic outputs or become inactive or saturated, are related to the topology of the networks. Since the topology of the networks for forward walking, backward walking, and swimming are different, we calculate their respective tonic external inputs based on Liu and Wang (2018). When only neuron i is inactive or saturated and the other neurons generate rhythmic outputs, the upper bound of inactivity and the lower bound of saturation of the tonic external inputs $s_i$ for the forward walking and backward walking networks are as shown in Tables 6 and 7, respectively. The corresponding results for the swimming networks are shown in Table 8.

Based on Tables 6, 7, and 8, it can be seen that the output of a neuron is inactive, saturated, or rhythmic when the tonic external input is less than the upper bound of inactivity, greater than the lower bound of saturation, or between them, respectively.

Initial states of spinal locomotor networks The initial states are helpful for the transition performance of spinal locomotor networks. Based on the biological properties of a real neuron, the initial states of the spinal locomotor networks are calculated. Since the resting potential of a neuron is lower than the firing threshold, the resting potential $x_{i0} ,(i=1,\ldots ,62)$ is chosen to be −0.1. In addition, during the resting stage, we assume the change rate of the membrane potential of the neuron is $\dot{x}_i =0,(i=1,\ldots ,62)$. Based on Eq. (1), the initial states of the spinal locomotor networks can be calculated as follows:

$$\begin{aligned} x_{i0}^{\prime } =\frac{\varepsilon _i }{b_i }x_{i0} =-0.125,\quad (i=1,\ldots ,62). \end{aligned}$$

(4)