Exploring human rhythmic gait movement in the role of cerebral cortex signal

Appl. Math. Mech. -Engl. Ed., 32(2), 223 230 (2011) DOI 10.1007/s10483-011-1408-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011 Applied Mathematics and Mechanics (English Edition) Exploring human rhythmic gait movement in the role of cerebral cortex signal Wei DONG ( ), Ru-bin WANG ( ), Zhi-kang ZHANG ( ) (Institute for Cognitive Neurodynamics, School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, P. R. China) (Communicated by Li-qun CHEN) Abstract The rhythmic movement is a spontaneous behavior due to the central pattern generator (CPG). At present, the CPG model only shows the spontaneous behavior, but does not refer to the instruction regulation role of the cerebral cortex. In this paper, a modified model based on the Matsuoka neural oscillator theory is presented to better show the regulation role of the cerebral cortex signal to the CPG neuronal network. The complex interaction between the input signal and other parameters in the CPG network is established, making all parameters of the CPG vary with the input signal. In this way, the effect of the input signal to the CPG network is enhanced so that the CPG network can express the self-regulation movement state instead of being limited to the spontaneous behavior, and thus the regulation role of the cerebral cortex signal can be reflected. Numerical simulation shows that the modified model can generate various movement forms with different modes, frequencies, and interchanges between them. It is revealed in theories that the cerebral cortex signal can regulate the mode and frequency of the gait in the course of the gait movement. Key words central pattern generator (CPG), gait movement, rhythmic movement, cerebral cortex signal, conversion function Chinese Library Classification TP183 2010 Mathematics Subject Classification 92B20 1 Introduction As early as 1911, Brown proved by experiments that the basic pattern of the gait was generated by the spinal cord system, and the movement command from the cerebral cortex was not required [1 2]. Such a spinal system is a central pattern generator (CPG). The concept is the initially form of CPGs. Since the anatomy at that time knew very little about CPGs, the point of Brown was not accepted. In 1961, Wilson clearly proved the existence of the CPG in living organisms (invertebrates). It was found that the nervous system of locusts under the condition of isolated culture could generate a rhythmic output mode similar to that produced as they Received Jul. 26, 2010 / Revised Dec. 31, 2010 Project supported by the National Natural Science Foundation of China (Nos. 10872068 and 10672057) and the Fundamental Research Fund for the Central Universities Corresponding author Ru-bin WANG, Professor, Ph. D., E-mail: rbwang@163.com

224 Wei DONG, Ru-bin WANG, and Zhi-kang ZHANG flied. From then on, the evidence of CPGs in vertebrates gradually emerged, and researchers started simulating CPG models of many lower animals. The rhythmic movements of higher animals and human are also generated and controlled by CPGs. In 1994, Calancie et al. put forward the evidence of the CPG existed in human bodies [3 4]. Modern researches not only confirmed the existence of the CPG that could control human motion, but also, at the same time, proved its stability and adaptability [5 6].Moreover, the CPGs, which can control human motion, cannot only generate a rhythmic mode, but also take on very good adaptability in various environments. In the same time, some researchers modified the mathematical model of the biological CPG to make it relatively more simple. This model is more useful and convenient for engineering applications such as robot control, model and simulation for animal or human movements, and rehabilitation control for paralyzed patients [7]. Some researchers also studied some important properties of CPGs such as the effect of the tonic input and sensory feedback, robustness, stable oscillation, and coupling relationships. Through these studies, we can well understand CPGs, and provide firm foundation for the applications of CPGs in the engineering and medicine rehabilitation. At present, most studies on CPG models are about the spontaneous rhythm movement. The model themselves cannot show the instruction regulation of the cerebral cortex. To better show the regulation role of the signal from the cerebral cortex to the CPG network, in this paper, the relationship of every parameter is rebuilt. It emphasizes the connection of every part of the CPG network so as to show the regulation of the input signal to the CPG network. Theoretically, it realizes the combination of the spontaneous rhythm movement and the rhythm movement regulated by the cerebral cortex. This makes the theoretical model closer to real life. 2 Biology CPG model There are two types of structures of biology CPGs, i.e., chain and network [3]. The chain structure is prevalent in lower aquatic animals. The network CPG exists in the bodies of vertebrates, which is composed with a specific topological structure of a number of oscillators and can control the coordinated movements of animal limbs [4,8]. For higher animals and human, the CPG in bodies is more complicated. The CPG structure existing in a body is more diverse. It is a kind of mutual combination of chain and network structures [9 10]. At present, there are three types of neural oscillator models, which are widely used in the simulations of biological CPGs. The first one is the Matsuoka neural oscillator [11 12],whichembodies the cycle of CPGs. The second one is the oscillator of lampern [13 14], which embodies the nature of the phase coupling of CPGs. The third one is the van der Pol neural oscillator [15 17], which embodies the relative relation among joint angles on the corresponding parts in living organisms when CPGs generate rhythmic movements. Among them, the Matsuoka neural oscillator is the most widely used. For the correlation between the Matsuoka neural oscillator and biological CPGs, refer to [18]. Nowadays, some studies on the theory of rhythmic movements make a series of theoretic foresee on the basis of CPG models. There are different CPG structures in various animals. The structures of the CPG models in different parts of one kind of animals are also different. For the study on human leg rhythmic movements, a leg CPG model must be established, in which the muscle-skeleton structure of legs should be considered. In this paper, the structure of the CPG network model is based on the Matsuoka neural oscillator theory combined with a simplified Hill muscle model, including three couples of neural oscillator units, i.e., leg-hip, knee, and ankle, as shown in Fig. 1. There are two neuron groups in every couple of neural oscillator units, meaning, respectively, two kinds of muscle groups of one joint-flexor and extensor. In this CPG model, the output y i is a state evaluation, and shows the states of muscle. 3 Modified CPG model 3.1 Different representations on gait movement pattern Biology studies show that the CPG network can be coupled with external input signals so

Exploring human rhythmic gait movement in the role of cerebral cortex signal 225 Fig. 1 Theoretic foundation of CPG model as to transfer the output pattern. Therefore, changing the input excitation signals can adjust the output behavior of the network, and then can realize the gait conversion [19].Thesimulation of the present CPG model only shows a single mode in the gait movement. It cannot show the interchange between different movement modes. That is, it cannot show the instruction regulation role of cerebral cortex signals to rhythm movements. Other modes of the output of rhythmic movements are also introduced. The simulation and experiment on CPG models by Zhang et al. have shown two basis states of movements on legs, i.e., walking (see Fig. 2) and treadmill (see Fig. 3). The above two simulation results both are simulated by resetting different parameters to show different gait representations in the human leg model based on the inhibition neural network. For details, refer to [3 4]. However, these two kinds of gait representations are presented independently. There is no relationship between them. Thereby, we try to show the relationship of different gait representations, which presents that different gait modes can interchange. 3.2 Revised model for the interchange relationship of different gait movements In real life, gait movements are various, e.g., walking, running, treadmill and so on. How to show these gait movement modes by CPG network models? From the above, we know that different movement modes with different modes and frequencies can be obtained by changing every parameter of the CPG network. However, because of the complex relationship between theories and facts, at present, the corresponding relation about a certain rhythm movement and a special CPG simulation mode cannot be exactly explained. Here, we want to show that the output of CPGs has different modes, of which every one can interchange with each other. We believe that this phenomenon is in line with the fact, and it realizes the interchange between the gait modes, which is the combination of theories and facts. The CPG neural network is an important network structure showing rhythm movements of legs, which is not directly controlled by the cerebral and is a spontaneous behavior. Original CPG models use the tonic input sig-

226 Wei DONG, Ru-bin WANG, and Zhi-kang ZHANG nal as stimulus, which shows spontaneous rhythm movements. In real life, different movement modes are regulated by the cerebral. When the cerebral cortex gives out a special instruction signal, the network structure of CPGs makes this kind of movements go on rhythmically. We try to show the interchange between different modes, that is, considering the regulation role of the cerebral cortex in the CPG network. In most simulations, the sinusoid is used as the instruction signal of the cerebral cortex [19]. Fig. 2 Walking Fig. 3 Treadmill In the modified CPG model, the equations are as follows: K f τx i & = X i βν i + K t r + ω i,j y i + Le, i, j =1, 2,L,12, i j Tν i & = ν i + y i, { r0, r = r 0 + r brain, ( 2π ) r brain = A sin T t, t [t start,t end ], τ = bt r r brain dt + τ 0, T = c r brain dt + τ 0, T r y i = f(x i )=max(x i, 0), (1)

Exploring human rhythmic gait movement in the role of cerebral cortex signal 227 where K f and K t mean, respectively, the sensory gain and the tonic input gain (in simulation, K f,k t =1),i and j mean the 1st, 2nd,, 12th neurons, X i mean the two states of neural groups in the ith joint-flexor and extensor, ν i is the adaptability in the process of muscle recovery, ω ij is the interconnection weight between the 12 neuronal groups of both legs, τ is a status constant, and T is a fitness constant. The output y i is a state evaluation, 1 shows the activation, and 0 shows inactivation. As for the parameters in details, please refer to [19]. From the aspect of simulation, sine functions are one of the simplest periodic functions. Any other function can be converted as a combination of sine functions with different frequencies. The complexity of brain still is unknown. Thus, we take sine functions as the simulation signal of the cerebral cortex. It has a certain representativeness and significance. In (1), the formulas with stars are the modifications on the parameters in the original equations. They enhance the relationship of every parameter. We take r brain as a sine function so that the simulation result can better show the interrelation between the input signal and the CPG network. The simulation result of the modified CPG model is closer to the fact in biology. In original CPG models, there is no relationship between the input signal and other parameters, and only constant input signals are used to simulate the spontaneous rhythm movement. There is a close relationship between the modified input signal and the CPG network. By use of the sinusoid input signal to simulate the instruction of the cerebral, the interchange of modes and frequencies on the gait movement can be realized. By the modified CPG model, the variability of the gait movement can be better shown, which makes the model closer to the fact. 4 Results and discussion Under the condition of rhythmic movements, the movement control system centering as the spinal cord can spontaneously generate neural signals depending on a lower neural system through the self-oscillation network. This kind of neural signals can be set as internal stimulus, which can be seen as constant input signals in the simulation. When the rhythm of the gait movement changes, brain will give the instruction for changing the rhythm of the movements to limbs. The input signals controlling the limb movements from the cerebral cortex mostly take as the form of sine waves [19]. So, in the simulation, we take constant signals as the stimulus when self-oscillation, and take the sine wave as the instruction stimulus of the cerebral cortex. When the cerebral cortex sends out instantaneous instruction signals, the CPG output modes occur to change so as to realize the regulation function of the cerebral cortex. The introduction on the simulation results is as follows. 4.1 Interchange between different modes Here, we only give two examples, i.e., two kinds of different CPG output modes. The interchange between these two modes is shown in Fig. 4. According to three kinds of different states of sine waves, i.e., three types of different regulation instructions from the cerebral cortex, we can obtain three interchange relationships on the CPG output mode as follows: Condition I When the sine wave meaning the instruction signal from the cerebral cortex only is the state of the positive semi-period, the CPG output mode can transform from mode ItomodeII(seeFig.4). Condition II When the sine wave meaning the instruction signal from the cerebral cortex only is the state of the negative semi-period, the CPG output mode can transform from mode II to mode I (see Fig. 4). Condition III When the sine wave meaning the instruction signal from the cerebral cortex is the state of the whole period, the CPG output mode cannot transform. 4.2 Interchange between different frequencies in the same mode As for the regulation signal from the cerebral cortex r brain, it not only can make different CPG modes interchange, but also can make different frequencies in the same CPG mode interchange. There is a close relationship between the cycle T r of the regulation signal from the

228 Wei DONG, Ru-bin WANG, and Zhi-kang ZHANG Fig. 4 Interchange between modes cerebral cortex r brain, the time parameter τ, andt in the CPG network, which can adjust the effect parameters b and c. Then, the interchange between different frequencies in the same mode can be realized in the simulation. Accordingly, there are three conditions as follows: Condition I When the sine wave meaning the instruction signal from the cerebral cortex only is the state of the positive semi-period, the CPG output frequency in the same mode can transform from fast to slow (see Fig. 5). Condition II When the sine wave meaning the instruction signal from the cerebral cortex only is the state of the negative semi-period, the CPG output frequency in the same mode can transform from slow to fast (see Fig. 5). Condition III When the sine wave meaning the instruction signal from the cerebral cortex is the state of the whole period, the CPG output frequency in the same mode cannot transform. In our modified model, it is shown that there are two types of different transformations, i.e., mode interchange and frequency interchange. The parameters in the model have the following properties: (i) Under the precondition that the instruction signal from the cerebral cortex does not change, the values of the parameters b and c directly affect the interchange types. When b increases gradually, the interchange types can transform from the frequency interchange to the mode interchange, e.g., the frequency interchange b = 0.01 (see Fig. 5) to the mode interchange b = 0.1 (see Fig. 4). However, the parameter b cannot increase illimitably, when b reaches a

Exploring human rhythmic gait movement in the role of cerebral cortex signal 229 Fig. 5 Interchange between frequencies in the same mode certain value, the rhythm property of the CPG output vanishes. The parameter c affects the shift degree of the CPG output mode between before and after the transform. The bigger the value of the parameter c is, the more obvious the shift degree is. (ii) The parameter τ affects mostly the type of the CPG output modes. When τ is small, thetypeisclosetomodeli;whenτ isbig,thetypeisclosetomodeii. (iii) The parameter T affects the initial frequency of the CPG output. The bigger the parameter T is, the faster the initial frequency is. 5 Conclusions CPGs are a kind of lower neural center systems, which can control the rhythmic movement of the creature. In the course of rhythmic movements, it is not directly controlled by the signal from the cerebral cortex. For the human, the regulation role of the cerebral cortex is unassailable, and the regulation function of the cerebral cortex to rhythmic movements exists necessarily. Original CPG models only show spontaneous behaviors of CPGs. In this paper, a function of the relationship of the parameters within an original CPG model is established, which makes the relationship between the input signal and every parameter closer. The parameters in the CPG model change with the change of the input signal [19]. Theoretically, the modified model can show the effect of the cerebral cortex to the CPG network, and can better

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