Biol. Cybern. 73, 37 47 (1995)

9 Springer:Verlag 1995

Control of a one-link arm by burst signal generators Jaywoo Kim, Hooshang Hemami Department of Electrical Engineering, Ohio State University, 2015 Neil Ave., Columbus, OH 43210-1272, USA Received: 30 May 1994/Accepted in revised form: 13 January 1995

Abstract. The focus of this paper is the study of stability and point-to-point movement of a one-link arm. The sagittal arm has two musculotendon actuators, two neural oscillators that generate burst signals as motoneuron inputs, and spindles and Golgi tendon organs for extrinsic reflex feedbacks. It is shown that coactivation leads to intrinsic position and velocity feedback, and that the tendons introduce intrinsic force and rate of force feedback. In addition, the integrating effects of the tendons are studied when the actuator is constructed from a large number of identical fibers that are excited by alpha signals whose arrival times at the fiber are randomly distributed. Each of the musculotendon actuators receives two input signals - a burst signal analogous to alpha inputs and a conventional analogue signal that represents fusimotor input to the spindles. The process of combining burst signals and conventional analogue signals is studied. Simulation results show that the movement of the system with burst signals as inputs has overshoot and speed similar to the system with analogue signals.

1 Introduction The objective of this paper is to introduce a simple musculoskeletal model that is qualitatively a better approximation to physiological systems than many previous models. This model accepts input signals analogous to alpha and gamma inputs, has extrinsic sensory feedback with transmission delays, and qualitatively imitates certain behaviors of the natural system under both muscular and alpha-gamma coactivation. Philosophically speaking, the study of such intermediate models leads to a better understanding of the central nervous system in the long run. These simulation models, given more detail and complexity, can better mimic the structure, morphology, and behavior of natural systems. In this paper the major emphasis lies on a more physiological form of alpha excitation to the actuators, Correspondence to: J. Kim

namely burst signal inputs. The sensory spindle and Golgi tendon organ feedback signals are left in their original analogue form for the time being. For this reason, a comparison of the behavior of the system to be studied is undertaken under two kinds of inputs: analogue continuous inputs and pulse-code type of burst signals. A second point of emphasis involves the stabilizing and force-integrating properties of tendons. The two actuators are assumed to be composed of fibers that are excited by alpha signals whose arrival times at the fiber are random. Hence, the tendon has an averaging effect in assimilating and smoothing the eventual total force the actuator exerts. The complex electrical activity called bursting has been observed during the recording of transmembrane potential in various nerve, muscle, and secretory cells (Lee 1993; Morris and Lecar 1981). For example, the fl-cells of isolated pancreatic islets respond with periodic bursting in the presence of a certain level of glucose, and this activity is known to be correlated with the release of insulin. In particular, the ratio of the active phase duration time to the overall period is proportional to the insulin release rate of the fl-cell. Another example involves a barnacle giant muscle fiber. Several clamp studies of the barnacle muscle (Morris and Lecar 1981) show continuous or damped burst voltage behavior in response to various levels of current clamping. It is known that the character of the burst oscillation is influenced by Ca 2+ inside the muscle fiber as well as the current strength (Morris and Lecar 1981). Furthermore, burst signals have been observed in controlling locomotion of lower vertebrates (Grillner et al. 1988), respiratory patterns of mammals, and mastication of higher vertebrates (Rossignol et al. 1988). Atwater et al. (1980) proposed a biophysical mechanism for bursting electrical activity based on voltagegated potassium and calcium channels and a calciumactivated potassium channel. Since then, several researchers (Chay and Keizer 1985; Sherman et al. 1991) have studied mathematical models of bursting signal generation. Chay and Keizer (1985) developed a minimal mathematical model for the burst pattern of fl-cell oscillations by modifying the Hodgkin-Huxley kinetics for the

38 squid giant axon. Although the mathematical model generates accurate burst signals, it is shown that it is not robust in terms of gain changes and parameter variations. Simpler neural oscillator models have been proposed for temporal sequence learning and visual recognition (Bay and Hemami 1993; Malsberg and Buhmann 1992; Rowat and Selverston 1993; Wang et al. 1990). The advantage of these neural models is that the behavior of the oscillator is robust and controllable. Many analytical studies on the control of muscle length and tension in human movement (Dinneen and Hemami 1993; Gielen and Houk 1987; Houk and Rymer 1981) have been based on continuous neural input rather than burst signals. In this study, the agonist-antagonist muscle pair is excited by a pair of burst signal generators. The low-pass filter function of the muscle is also considered (Agarwal and Gottlieb 1982, 1984, 1985; Baratta and Solomonow 1990; Bahill 1981). The low-pass filtering is in the transition between the pulse step and the active-phase tensions ( Bahill 1981). This filtering is due to both spreading in time and rate-limiting factors. The spreading is due to variations between cells in synchronization, synaptic transmission delays, motoneuronal firing frequency acceleration, neuronal conduction velocity, depolarization, spread of activity in the sarcoplasmic reticular formation, and acceleration of actomyosin cross-bridges. The rate-limiting process includes the synaptic transmissions, the release and reuptake of Ca 2+ and its modification of the actomyosin fibers. It is known that most of the low-pass filtering is probably due to the Ca 2+ activation process in the muscle fiber. A clue for a physiological value of the time constant of the low-pass filter can be taken from Ruders work on the frog with a Ca 2 + sensitive bioluminescent protein which emitted light in the presence of Ca 2+ (Taylor et al. 1975). The tendon also performs low-pass filtering with a cut-off frequency related to the stiffness constant of the tendon (Zajac 1989). Many muscle physiologists, including Hill, recognized that the determination of the properties of the contractile apparatus of muscle tissue requires the elimination of the series elasticity which cannot be acquired from the muscle itself, but from coactivation of muscle and tendon (Zajac 1989). Our aim, however, is not to implement more accurate physiological models, but to consider a practically reasonable model in terms of computation cost and generation of human-like movement. In Sect. 2, the system model is presented. Sect. 3 is devoted to the burst signal generator. In Sect. 4, the stability of the system is studied by considering the behavior of the linearized system in the vicinity of an equilibrium point. There, point-to-point movement is also discussed. Numerical examples and simulations of the single-fiber actuator case are presented in Sect. 5 in order to study the nonlinear behavior of the system, point-to-point movement, and functional behavior of the tendon. Simulation of the multifiber actuators with random delays in the arrival times of alpha signals to the fibers are presented in Sect. 6. The paper concludes with Sect. 7.

2 The system model

The system consists of a one-link arm hinged to an inertial reference frame and acted upon by two musclelike actuators a flexor and an extensor. The flexor rotates the arm counter-clockwise and the extensor rotates the arm clockwise. The two states of the arm are 0 and 0 (Figs. 1, 2). 2.1 Arm dynamics

The equation of the arm shown in Fig. 1 is JO - mgdsin O = a(F1

-

(1)

F2)

where m is the mass of the system, d is the distance of the center of gravity from the hinge, J is the moment of inertia of the arm about the hinge, a is the moment arm of the actuator, and 9 is the constant of gravity. F 1 and F 2 a r e the extensor and flexor forces at the end of the tendon, respectively. The moment arm, a, in general, is a function of the state, i.e., angle 0. However, in order to make the analysis easier, it is assumed to be constant in this paper. 2.2 Excitation dynamics and low-pass filtering

The flexor and the extensor actuators consist of either single or multiple fibers. The fibers are assumed to be identical: they have the same viscous constant and passive elastic constant. Also, the fibers are assumed to be attached to a solid plate at each end, which assures that they all have same length. The force of the actuator, F, is the total tension generated by the fibers and is given by

O i

Fig. 1. The arm model with extensor force, F1, and flexor force, Fz. The actuator consists of either single or multiple numbers of parallel contractile fibers

39

~

l~h?

.

,F

l?

6

Mu~le

__o _

~

~

I th'

Muscle

I

Fig. 2. Block diagram of the overall system with external inputs ll and 12 and outputs 0 and 0

I I,,.~ Feedback

I ~ . ~

(2)

can, therefore, be represented by a linear time-invariant filter whose unit impulse response is p(t), and whose frequency response is the Fourier transform of p(t), i.e., the characteristic function of the random variable z. As an example, if a uniform pdf is assumed:

(3)

p(t)=~

(4)

Then the transfer function

(Gielen and Houk 1987; Ong et al. 1990): N

F,= j=~ V i, i = 1 , 2 ~-I--

Vj = (krj + cAff) z + bAL~u(- Ai~, ), i=1,2

j=I...N

rj = ri(t + z j) Air=it-iS;

Ai~ = i~ - i8

(5)

where superscript c refers to contractile fiber, it is the length of the extensor contractile fiber, J~ is the length of the flexor contractile fiber, J8 is the initial length of these fibers, and AJ? (i = 1, 2) is the difference in length from its initial length, rj is the firing rate input to the fiber with different delay. N, b, c, and k are total number of the fiber in each actuator, viscous constant, passive elastic constant, and amplification factor, respectively. Vj is the tension of the fiber. The alpha burst signals are assumed to reach the corresponding muscle fiber with some delay z (Bahill 1981). These delays can be implemented by a random variable that has a well-defined probability density function (pdf) over a finite time interval Tb. Let this pdf be defined by p(z) over the range 0