# BIOMECHANICS OF MUSCLES

### Abstract

INTRODUCTION A basic one degree of freedom neuromuscular assemblage, the mechanoeffector unit (MU) whose structure and properties are defined and its dynamical response to small perturbations around a quiescent working point are considered so that linear approximations to its functional relationships may be used. This dynamical response is simulated numerically using publlshed data on the characteristics of muscles and sensors and its capacIty to meet requirements of (a) precision (b) stability (c) speed of res~onse and (d) insensitivity to external loadlng IS evaluated and checked with experimental known data The role of the spinal cord in providmg filtering circuitry to improve the mechanoeffector assemblage design and to achieve for it a configuration which fits weil the known overall characteristics of its natural counterparts.' METHODS The diagram of figure 1 illustrates the structure of the MU control assemblage Two types of muscle fibres are considered, namely, the extrafusal fibres (EFF) which constitue the muscle contractile element and the intrafusal fibres (IFF) which are part of the muscle spindie sensorial structure. Two types os sensors are considered: the muscle spindie (MS) and the Golgi tendon organ (TO) F1, F2, F3, F4 are neural correcting filters located at spinal cord level and !:!. is the propagation delay in the afferent nerve links and which account for both the afferent and efferent delays The small signal variables are the as folIows: muscle (u) and fusimotor (v) central commands, muscle (a) and fusimotor (y) efferent commands, the afferent signals from the spindie (cr) and trom the tendon organ (r), the correspondlng filtered signals (4)) and (1jI) and the length changes (or strains) of the EFF (xm ), of the external load (x.,) and of the overall MU assemblage (Xo = Xm = -xe); Yo is the force (or stress) developed by the MU The feedback relationships for the sensorial outputs are those which have been found to be most suitable for the optimization of the control assemblage. Actually, the feedback arrangements have reasonable experimental support and both have, separately or together, been considerated by other authors (Stein, 1974): the length feedback from the MS is negative as a result of the differential assemblage of this sensor and the force feedback from the TO is positive Feedback loops involving higher CNS areas are not considered and so the only propagation delay to be taken into account is that which occurs in the afferent pathways The Commands Synthetizer represents the cerebral cortex areas which are responsible for generating the commands that control the MU. The fact that we consider only small incremental variations arount quiescent working points allows lhe linearization of the func ional relationships between the different elements of the MU assemblage and the use of the Laplace Transform formalism. The transfer functions 52 of the EFF and of the TO and the MS are, respectively: Yo(s) = C(s) -a(s) + Yg(s) _Xo(s) 1(S) = AT(s) _Yo (s) and o(s) = As(s) _Xo (s) + 8s(s) _y (s) (1 ) where s is the Laplace variable, C(s) is the complex frequency-to-force conversion factor, Yg(s) is the museie stiffness and AT(s), As(s) and 8s(s) are transfer funtions relating .(5) to Yo (s) and o(s) to Xo (s) and to y (s). The controlled variables. namely, the output length (xo), force (Yo) and elastance (Zo) are expressed, in terms of the corresponding Laplace transforms, by the equations (2), (3) and (4): X() (s) :: G(s) [ u(s) +Bs "*(s). ~s) 1+l{)(S). Ye(s) 1 + G(s).l---(s) (2) c•(s) yo(s):: • [u(5)+Bs ••(S).v(s)]1 -C•(s). AT(S) . Yg(s) +C' (5).As(s) 1-C.(s)AT(5) .X()(5) (3) 1 1-C•(5).AT(S) (4) zQ(s) :: Y (5) + Yg(s) 1+ G(s) H (5) e where: As"(s) = e ",.\5 F3 (s) _As (s) 85 '(s)= e-\S F3 (s) _8s(s) AT" (s) =e•.'\ s F4 (s) . AT(s) 85 "(s) =F1 (5). 8s(5) Cs"(s) = F2 (5) . C(5) GC '(5) (s):: Ye(s) + Y(5) g H(s) = As'(S) -AT' (5). Y.(5) (5) e ".\ is the delay transfer funclion and S Y.(s) is the externaiload stiffness. RESULTS AND DISCUSSION The parameters and relationships which charaeterize the muscle fibres properties and the mechanoreceptors parameters ware obtained from the literature;. Figure 2 represents the Nyquist Diagram for to the MU. This diagram is the complex plane plot of the product G(s) H(s), the MU open loop transfer function whieh occurs in the denominator of equation (2), when s=j (J) and the angular frequency (J) varies fram zero to infinite This diagram pravides a powerful tool for the study of the system behaviour at low and high frequencies. Diagram 1 refers to the situation when there is no filtering and corresponds to a position servomechanism of type zero. Le., with no pole et the origin end therefore with a finite position error. Diagram 2 represents the same when the filters mate the transfer function of the corresponding link, i.e., F1 = k1. 85. 1, F2=k2. CS• 1, F3= k3. AT .1, F4 = k4. As •1 with the k's adjusted adequately_ For this situation, which corresponds to remove the frequencydependent parte of the sensors and effector characteri5tics, the dynamical behaviour of the MU is optimal and demonstrates the capacity of the nervous system to compensate for the biological material characteristics. This solution brings out clearly the role played by the two types of sensors. The MS controlling lhe MU length error end the TO controlling its ourtput (Zo) which is easily broughlto small value found experimentally. The position error is minimized and the speed of response increased by the high transfer gains made possible by Ihis filtering scheme and the various components of the MU performance become weil consistent with those found experimentally. REFERENCES Stein, R 8 (1974) Physiological Rev., 54(1), 215,243
Section

Keynote-Lectures

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