1 Introduction

Life requires movement, and for this reason, the study of neural coding of motion has attracted extensive studies^[1-4]. The human arm is one of the most flexible and complex motion elements of our body. The most direct cause of a conscious or unconscious motion of our arm (such as opening a door and gait movement of the arm during walking) is due to the contraction of muscles, which is regulated by neurons. Therefore, the study of the neural regulation mechanism behind the arm motion is of great significance not only for the design of the rehabilitation of robot arm^[5-6] and the intelligent prosthesis^[7], but also for understanding the mechanisms behind diseases of arm motion. The goal of early research of arm motion was to observe the trajectories. For example, Morasso^[8] found that, when subjects were asked to make point-to-point motion in the free space, they chose unique trajectories with two invariant features. Firstly, the hand pathways in point-to-point motion tended to be straight and smooth. Secondly, the velocity profile of the hand trajectory was bell-shaped. These invariant features give hints about the internal representation of the motion in the central nervous system. To account for these invariant features with the optimization theory is an approach. Flash and Hogan^[9] (1985) simplified the human arm to be like a particle and proposed a minimum hand jerk criterion, implying that the central nervous system planned the point-to-point motion in the free space based on kinematics. By minimizing the criterion, Flash and Hogan^[9] certified that the velocity profile of the hand trajectory was bell-shaped. Based on the idea that trajectory planning should take arm dynamics into account, Uno et al.^[10] (1989) suggested a minimum joint torque change criterion, implying that the central nervous system planned the point-to-point reaching motion in the free space based on the dynamic formulation. By minimizing the criteria, Uno et al.^[10] also certified that the velocity profile of the hand trajectory was bell-shaped. The studies of human arm motion mentioned above are mainly concentrated on the simple motion: point-to-point. In order to study more complex arm motion, Ohta et al.^[11] (2004) designed a crank rotation experiment, and recorded the data of the crank angle, crank angular velocity, and joint angular velocities. Moreover, they achieved the simulation profiles of the crank angle, crank angular velocity, and joint angular velocities by minimizing the torque change criterion, the muscle force change criterion, and the hand force change criterion. Wu and Wang^[12] (2008) proposed a new algorithm for arm motion according to the crank rotation experiment. In this algorithm, sine and cosine functions were used to fit the muscle force. Unfortunately, only the crank angular velocity was achieved. All the studies of arm motion mentioned above are mainly from the perspective of experiment and optimization, and many research results are obtained. However, it is not enough to learn more about arm motion, because they do not cover the neurophysiologic mechanism behind the arm motion. In fact, each motion behavior of our arm is caused by contraction of a group of muscles, and the contraction of muscles is under the regulation of the central nervous system. Thus, the aim of this paper is to explore the neural network model of arm movement from the perspective of neurodynamics^[13].

2 Arm motion model based on central pattern generator (CPG) 2.1 Neural subsystems of arm motion

In the process of human arm motion, neural subsystems such as the cortex^[14], cerebellum^[15], basal ganglia^[16], and the spinal cord^[17] are required. All of these subsystems play different roles in human arm motion. Figure 1 presents the direct connections between alpha motor neurons and arm muscles, and we can find that the alpha motor neurons receive signals from the spinal interneurons and other types of neurons to regulate the arm muscles.

In this paper, we pay attention to the spinal interneurons regardless of other kinds of neurons. In other words, the alpha motor neurons just receive signals from the spinal interneurons to regulate the arm muscle. According to Ref. [18], we know that the spinal interneurons constitute the CPG network. The CPG exists in the vertebrate brain stem and spinal cord^[19]. The studies of motion coding have found that there are many inherent CPG networks in the central nervous system, which produces the basis of neuron function for automatic human behaviors, such as breathing CPG^[20], chewing CPG^[21], leg swing CPG^[22], and arm swing CPG^[23]. The study of CPG has been a recent area of focus^[24-31].

2.2 Single neuron calculation model of CPG

The Japanese researcher Matsuoka^[32] proposed a single neuron calculation model of the CPG network in 1985, which is presented as follows:

(1)

(2)

(3)

(4)

where x is the membrane potential of the neuron body, r is the tonic input, y is the output of the neuron, and θ is the threshold. T₁ is the time constant, and it specifies the rise time when a step input is given. υ is the variable that represents the degree of fatigue or adaptation in the neuron. β is the parameter that determines the steady-state firing rate for a constant input, and T₂ is the time constant that specifies the time lag of the adaptation effect.

2.3 CPG network model based on muscle anatomy

We build our CPG network based on the muscle anatomy. Considering the fact that the arm contains numerous muscles, we choose the upper arm as our study object. According to anatomy, the upper arm mainly consists of six muscles (see Fig. 2).

In Fig. 2, f₁ is the posterior deltoid, f₂ is the pectoralis major, f₃ is the lateral head of triceps brachii, f₄ is the brachialis, f₅ is the long head of triceps, and f₆ is the biceps brachii. f₁, f₂, f₃, and f₄ are monarticular muscles, and f₅ and f₆ are biarticular muscles.

In our paper, a new kind of CPG network, consisting of six neurons, is proposed to regulate the contraction of six muscles of the upper arm. The new kind of CPG network is presented as follows:

(5)

(6)

(7)

(8)

where ω_ji=0 when i=j, which is a weight of connection from the jth neuron to the ith neuron. The output y_i (i=1, 2, …, 6) of the CPG network regulates the contraction of the ith muscle. The contraction of muscle generates the torque to drive the motion of arm. From Fig. 2, the contractions of f₁, f₂, f₅, f₆ muscles generate the torque of the shoulder τ₁, and the contractions of f₃, f₄, f₅, f₆ muscles generate the torque of the elbow τ₂. According to Ref. [33], we propose a new relationship between the output y_i (i=1, 2, …, 6) and τ₁ and τ₂, which is presented by

(9)

where d_ij (i=1, 2, …, 6, j=1, 2) is the conversion factor, k_i (i=1, 2, …, 6) is the gain coefficient, and it can be understood as amplification of the alpha neuron for the neural output y_i (i=1, 2, …, 6).

2.4 Arm motion model based on CPG

The arm moves in a unique trajectory under the torques τ₁ and τ₂. In order to study the trajectory of the arm, we combine our model with the crank-rotation task (see Fig. 3) designed by Ohta et al.^[11].

In the task, the wrist joint of each subject (there are five subjects in the task) is fixed by a cast in order to exclude the kinematic redundancy of the arm, and they sit in a chair with a harness attached to their trunk in order to constrain the shoulder joint and grasp the handle of the crank lever with their right hand. The subjects are asked to make one clockwise circular movement at their own pace, as comfortable as possible in about 2 s.

The simplified model of the crank-rotation task is shown in Fig. 4. In the experiment of Ref. [11], the arm is modeled as a planar two-link manipulator with lengths l₁ and l₂, centers of mass at l_g1 and l_g2, masses m₁ and m₂, and moments of inertia I₁ and I₂. The dynamic equation of a planar two-link manipulator is presented by

(10)

where M(q) is the inertia matrix of the arm,

(11)

is the generalized vector of centrifugal and Coriolis forces,

(12)

and B is the joint viscosity matrix,

(13)

τ₁ and τ₂ are the shoulder torque and the elbow torque, respectively. τ₁ and τ₂ are derived from Eq. (9). J(q) is the Jacobian matrix, and F is the hand contact force. From what have been discussed above, we establish an arm motion control model based on the CPG, which is presented by

(14)

3 Results of simulation and analysis

In our model, the outputs of the CPG network are converted into torques by Eq. (9), and the motion of arm is derived. In other words, the outputs of the CPG network decide the trajectory of the motion of arm. We adjust the parameters of the CPG network according to the following two principles, which are found during our simulation: (ⅰ) The system tonic input r mainly affects the amplitude of the output, which means that the increase or decrease in r leads to small change in the output frequency. (ⅱ) The output frequency is positively correlated to the parameter β, and is negatively correlated to the rise time constant T₁ and the adaptation time constant T₂. Finally, a set of parameters are chosen as follows:

(15)

(16)

The neural responses of the CPG network based on the parameters as shown in Eq. (16) are achieved by MATLAB in Fig. 5(a). By comparing the neural responses y_i (i=1, 2, …, 6) of the CPG network with the electromyography (EMG) signals in Fig. 5(b), it can be found that the firing states of the neurons are similar to the EMG signals. In this paper, the neural responses y_i are converted into the torque by Eq. (9). The parameters of Eq. (9), including the conversion factor d_ij (i=1, 2, …, 6, j=1, 2) and the gain coefficient k_i (i=1, 2, …, 6) are adjusted according to Refs. [11] and [12]. After achieving the torque, we solve the differential equation (10) with the MATLAB ODE45 function and achieve the numerical solution of joint angles q₁ and q₂ (see Fig. 6). The parameters of the differential equation (10) are shown as follows^[11]:

(17)

Ohta et al.^[11] did not record the experimental results of q₁ and q₂ in his crank-rotation task. The joint angle profiles shown in Fig. 6 are achieved by simulation. In order to observe the trend of joint angle more clearly, we obtain the first derivative of q₁ and q₂ with MATLAB, which are the joint angular velocities and in Fig. 7(a). Comparing Fig. 7(a) with Fig. 7(b), we find that the simulation profile of and the experimental profile of do not have good consistency during 0.8 s to 1.2 s. However, the overall trend are basically the same for both and .

According to Fig. 4, the relationship between the crank angle and the joint angle is

(18)

where x₀ = −0.185. Combining it with the numerical solution of joint angles q₁ and q₂ in Fig. 6, the simulation profile of crank angle θ is obtained as shown in Fig. 8(a).

The comparison between the simulation and experimental data of crank angle is presented in Figs. 8(a) and 8(b). The simulation results of crank angle consistently match the experimental data. In order to observe the trend of crank angle more clearly, we obtain the first derivative of θ with MATLAB, which is the crank angular velocity , as shown in Fig. 9(a). From Fig. 9(b), we can find that the crank angular velocity exhibits an obvious upward trend in the experimental profile (see the small circle in Fig. 9(b)), which is not shown in the simulation profile. However, the whole trends of by simulation and experiment are basically the same.

4 Conclusions

The human arm is one of the most flexible and complex motion elements of our body. The most direct cause of a conscious or unconscious movement of the arm is due to the contraction of muscles, which is regulated by neurons. Therefore, the study of the neural mechanism behind the arm motion is of great significance. First, this paper analyzes the current research of arm motion, and then establishes an arm motion control model based on the CPG network from the perspective of physiology. Further, we obtain the neural responses of the CPG network, the joint angle and the hand angle of the arm motion model with MATLAB. In order to verify the effectiveness of the model, we refer to the crank rotation experiment designed by Ohta et al.^[11]. At last, the simulation results are compared with the experimental results of the crank-rotation task. The comparison results show that

(ⅰ) The neural responses of the CPG network are similar to the EMG signals recorded in the crank-rotation task.

(ⅱ) The simulation profile of joint angular velocity and the experimental profile do not have a good consistency during 0.8 s to 1.2 s, but the overall trends are basically the same for both joint angular velocities.

(ⅲ) The crank angular velocity has an obvious upward trend in the experimental profile, although it is not shown in the simulation profile, the simulation results of crank angular velocity match the experimental data mostly.

(ⅳ) The study in Ref. [12] proposed a new algorithm for arm motion according to the crank-rotation experiment. In the algorithm, sine and cosine functions were used to generate torques of the shoulder and the arm, and it did not explain the source of the torque clearly. We establish our model from the perspective of neurodynamics, and as a result, the source of the torque can be explained more clearly compared with the study in Ref. [12], i.e., the outputs of the neurons of the CPG network regulate the contractions of muscles, and the contractions of muscles generate the torque. Moreover, we achieve more simulation results including the joint angle and the joint angular velocity. In summary, the arm motion control model based on the CPG network is reasonable and effective.