A fuzzy-genetic model for estimating forces from electromyographical activity of antagonistic muscles due to planar lower arm movements: the effect of nonlinear muscle properties

The aim of this paper is to create a model for mapping the surface electromyogram (EMG) signals to the force that generated by human arm muscles. Because the parameters of each person's muscle are individual, the model of the muscle must have two characteristics: (1) The model must be adjustable for each subject. (2) The relationship between the input and output of model must be affected by the force-length and the force-velocity behaviors are proven through Hill's experiments. Hill's model is a kinematic mechanistic model with three elements, i.e. one contractile component and two nonlinear spring elements.In this research, fuzzy systems are applied to improve the muscle model. The advantages of using fuzzy system are as follows: they are robust to noise, they prove an adjustable nonlinear mapping, and are able to model the uncertainties of the muscle.Three fuzzy coefficients have been added to the relationships of force-length (active and passive) and force-velocity existing in Hill's model. Then, a genetic algorithm (GA) has been used as a biological search method that can adjust the parameters of the model in order to achieve the optimal possible fit.Finally, the accuracy of the fuzzy genetic implementation Hill-based muscle model (FGIHM) is invested as following: the FGIHM results have 12.4% RMS error (in worse case) in comparison to the experimental data recorded from three healthy male subjects. Moreover, the FGIHM active force-length relationship which is the key characteristics of muscles has been compared to virtual muscle (VM) and Zajac muscle model. The sensitivity of the FGIHM has been evaluated by adding a white noise with zero mean to the input and FGIHM has proved to have lower sensitivity to input noise than the traditional Hill's muscle model.     

Transvascular exchange of fluid and plasma proteins

A theoretical model of transvascular exchange of fluid and plasma proteins in the microcirculation is developed based on fundamental laws of the fluid mechanics and on phenomenological transport equations of the irreversible thermodynamics. Intravascular axial changes of the pressure, flow and plasma protein concentration are taken into account as well as axial gradients of vascular permeability. Proper nondimensionalization of the resulting equations leads to the identification of dimensionless parameters which combine the transport characteristics of the endothelial wall and the intravascular flow resistance. In the theory, the dependence of the reflection coefficient on the transport coefficients of the vascular wall and on the plasma protein concentration is established. The model is applied to the cat mesentery and the rat intestinal muscle. The numerical simulations indicate that taking into account vascular protein permeability yields considerable differences in the axial distribution of the plasma protein concentration and transvascular fluxes in comparison with the case of protein impermeability of the endothelial wall. The results show that the maximum of the transvascular fluid and plasma protein movement resides at the site of the small venules while a minimum of the exchange occurs at the site of the midcapillaries.     

Comparison of Caplan's Irreversible Thermodynamic Theory of Muscle Contraction with Chemical Data

Recently Caplan (1) applied the concepts of irreversible thermodynamics and cybernetics to contracting muscle and derived Hill's force-velocity relation. Wilkie and Woledge (2) then compared Caplan's theory to chemical rates inferred from heat data and concluded that the theory was not consistent with the data. Caplan defended his theory in later papers (3, 4) but without any direct experimental verifications. As Wilkie and Woledge (2) point out, the rate of phosphorylcreatine (PC) breakdown during steady states of shortening has not been observed because of technical difficulties. In this paper it is shown that the rate equations may be directly integrated with time to obtain relations among actual quantities instead of rates. The validity of this integration is based on experimental evidence which indicates that certain combinations of the transport coefficients are constant with muscle length. These equations are then directly compared to experimental data of Cain, Infante, and Davies (5) with the following conclusions: (a) The measured variations of ΔPC for isotonic contractions are almost exactly as predicted by Caplan's theory. (b) The value of the chemical rate ratio, ν_m/ν_o, obtained from these data was 3.53 which is close to the value of 3 suggested by Caplan (3). (c) The maximum value of the chemical affinity for PC splitting was found to be 10.6 k cal/mole which is as expected from in vitro measurements (2). Because of the excellent agreement between theory and experiment, we conclude that Caplan's theory definitely warrants further investigation.     

The effect of membrane heterogeneity on the predictability of fluxes,with application to the cornea

Many phenomenological treatments of biological membrane transport are based on the assumption that the membrane consists of a single class of passive transport paths; i.e. that the membrane is simple. Simplicity is assumed both in the measurement of the membrane's transport coefficients and in the use of these coefficients to predict membrane fluxes. Such a procedure will in general lead to an error in the prediction of solute flux across parallel arrays. The error depends on the distribution of the reflection coefficients of the parallel paths and upper and lower bounds on it are given in terms of conventionally measured transport coefficients. The frictional representation of electrolyte transport across membranes possessing metabolic pumps is generalized to take this structure effect into account. The flux error resulting from the neglect of membrane heteroreflectivity is essentially the same for nonelectrolytes and electrolytes, irrespective of whether phenomenological or frictional membrane transport properties are used. It is shown that some information about transport structure can be obtained from global measurements made without regard for the organization of pathways across the membrane.The values of the measured transport coefficients of the corneal epithelium and endothelium imply that these cell layers are heteroreflective. Analysis of corneal transport, taking structure effects into account, shows that the corneal thickness may be intrinsically insensitive to tear tonicity, by a mechanism which may be of more general homeostatic significance.     

Correlation of Electrical and Permeability Properties of Ion-Selective Membranes

The linear phenomenological equations giving particle and practical fluxes of a single electrolyte across an ion-selective membrane are stated and interrelated. It is shown that the experimental measurements commonly made in biological and synthetic membrane studies may be used, with minor modification, to obtain the phenomenological transport coefficients and their concentration dependences. It is demonstrated that the electrical properties of a homogeneous membrane may be obtained as functions of the bathing solution concentration by combining fluxes measured under open and short circuit. Attention is paid to the use of radiotracers when measuring ionic fluxes. To obtain all the phenomenological coefficients at least one measurement must be made under a pressure gradient. The experimental difficulties in such measurements are discussed and the merits and demerits of various experiments considered. The problems of measuring potentials and concentrations at the low pressure face of a supported membrane make several mathematically simple approaches experimentally unattractive. The best methods appear to be either the measurement of a succession of “apparent osmotic pressures” under concentration differences sufficiently small that the membrane does not require support or the study of “reverse osmosis”. Sets of equations are given which enable the phenomenological coefficients to be evaluated from convenient experiments. With a stable homogeneous membrane nine coefficients may be obtained thus enabling either the applicability of the reciprocal relations or the applicability of linear theory under the conditions of the experiments to be tested. For a discontinuous system the six independent coefficients may be obtained from experiments in a single membrane cell.     

Variation of muscle stiffness with force at increasing speeds of shortening

Single frog skeletal muscle fibers were attached to a servo motor and force transducer by knotting the tendons to pieces of wire at the fiber insertions. Small amplitude, high frequency sinusoidal length changes were then applied during tetani while fibers contracted both isometrically and isotonically at various constant velocities. The amplitude of the resulting force oscillation provides a relative measure of muscle stiffness. It is shown from an analysis of the transient force responses observed after sudden changes in muscle length applied both at full and reduced overlap and during the rising phase of short tetani that these responses can be explained on the basis of varying numbers of cross bridges attached at the time of the length step. Therefore, the stiffness measured by the high frequency length oscillation method is taken to be directly proportional to the number of cross bridges attached to thin filament sites. It is found that muscle stiffness measured in this way falls with increasing shortening velocity, but not as rapidly as the force. The results suggest that at the maximum velocity of shortening, when the external force is zero, muscle stiffness is still substantial. The findings are interpreted in terms of a specific model for muscle contraction in which the maximum velocity of shortening under zero external load arises when a force balance is attained between attached cross bridges some of which are aiding and others opposing shortening. Other interpretations of these results are also discussed.     

Structural fluctuations in the steady state of muscular contraction.

Recent studies of the intensity fluctuation spectra of coherent light scattered from striated muscle have demonstrated the existence of large scale fluctuations in position and polarizability at the level of the myofibrillar sarcomere and its major structural subunits during the steady state of contraction. The existence of these fluctuations implies a fluctuating driving force. Various possible fluctuating motions of the thick and thin filaments, A and I bands, and entire sarcomeres are described. The magnitude of the fluctuating forces associated with the making and breaking of cross bridges is estimated. A mechanical model is proposed for coupling structural elements of a single sarcomere to one another and for coupling myofibrillar sarcomeres to one another. It is shown that the fluctuating force generated by the spontaneous making and breaking of cross bridges in conjunction with the model accounts for some of the features of the observed intensity fluctuation spectra.     

Caldesmon inhibits formation of strongly bound myosin cross-bridges and activates an ability of weakly bound cross-bridges to transform actin monomers to the off-conformation

The effect of caldesmon and its actin-binding C-terminal 35 kDa fragment on conformational alterations of actin in a muscle fiber at relaxation, rigor and at simulation of strong and weak binding of myosin heads to actin was studied by polarizational fluorimetry technique. The strong and weak binding forms were mimicked during binding of F-actin of ghost muscle fibers to myosin subfragment-1 modified with NEM (NEM-S1) or pPDM (pPDM-S1), respectively. As a test for alterations in actin conformation, changes in orientation and mobility of a fluorescent probe, TRITC-phalloidin, bound specifically to F-actin were used. The results obtained have shown that during transition of the muscle fiber from the relaxed state into the rigor and during binding of actin filaments to NEM-S1, changes of polarization parameters take place, which are characteristic of formation between actin and myosin of the strong binding and of transformation of actin subunits from the "turned-off" (inactive) to the "turned-on" (active) conformation. Binding of pPDM-S1 to actin and relaxation of the muscle fiber are accompanied, on the contrary, by the changes of orientation and of the fluorescent probe mobility, which are typical of formation of the weak ("non-force-producing") form of actin-myosin binding and of transformation of actin subunits from the active conformation into the inactive one. Caldesmon and its C-terminal fragment markedly inhibit formation of the strong binding at rigor and activate transition of actin monomers to the switched off conformation at relaxation of muscle fiber. In parallel experiments, these regulatory proteins have been shown to inhibit an active force developed at the transition of a muscle fiber from relaxation to rigor. Besides, caldesmon and its fragment decrease the rate of actin filament sliding over myosin in an in vitro motility assay. Caldesmon is suggested to regulate the smooth muscle contraction in an allosterical manner. The alterations in actin conformation inhibit formation of strong binding of myosin cross bridges to actin and activate the ability of weakly bound cross bridges to switch actin monomers from the "on" to the "off" conformation.     

Frictional properties of multisubunit structures

The translational drag, rotational drag, and intrinsic viscosity of spherical multisubunit structures have been calculated analytically using the Felderhof–Deutch theory of polymer frictional properties. The structures considered were hollow shells, spheres with uniform subunit density, and spheres covered with a subunit layer of different density. Changes in the transport coefficients resulting from the random removal of subunits and from the variation of subunit size are calculated. For the case of the shell, the results agree with the numerical computations of Bloomfield, Dalton, and Van Holde [Biopolymers 5 , 135, 149 (1967)].     

Pre-power stroke cross bridges contribute to force during stretch of skeletal muscle myofibrils

When activated skeletal muscle is stretched, force increases in two phases. This study tested the hypothesis that the increase in stretch force during the first phase is produced by pre-power stroke cross bridges. Myofibrils were activated in sarcomere lengths (SLs) between 2.2 and 2.5 microm, and stretched by approximately 5-15 per cent SL. When stretch was performed at 1 microms-1SL-1, the transition between the two phases occurred at a critical stretch (SLc) of 8.4+/-0.85 nm half-sarcomere (hs)-1 and the force (critical force; Fc) was 1.62+/-0.24 times the isometric force (n=23). At stretches performed at a similar velocity (1 microms-1SL-1), 2,3-butanedione monoxime (BDM; 1 mM) that biases cross bridges into pre-power stroke states decreased the isometric force to 21.45+/-9.22 per cent, but increased the relative Fc to 2.35+/-0.34 times the isometric force and increased the SLc to 14.6+/-0.6 nm hs-1 (n=23), suggesting that pre-power stroke cross bridges are largely responsible for stretch forces.     

Transport properties of oligomeric subunit structures

The translational friction coefficients, rotational friction coefficient, and intrinsic viscosity of rigid regular structures composed of up to eight identical spherical subunits have been accurately calculated. The aim of this calculation is to interpret the hydrodynamic properties of oligomeric subunit proteins. To avoid the well-known failure of the theory in the evaluation of rotational coefficients and intrinsic viscosities, each subunit is hydrodynamically modeled as a polyhedral array of smaller spheres. The analysis of several alternatives suggests that a cubic array is the best choice. The reliability of this strategy is checked by comparison of the calculated values for all the transport properties of a sphere and the translational friction coefficients of a dimer with their exact values. Finally, the hydrodynamic properties of a number of subunit structures with varying number of subunits and different geometries are tabulated.     

Phenomenological coefficients and general proof of reciprocal relation for a certain class of sliding filament models of muscle contraction.

General expressions for the near-equilibrium phenomenological coefficients of any sliding filament model of muscle contraction, in the class of models considered by Hill, are derived using a matrix method which is ordinarily used in the solution of a system of inhomogeneous first order differential equations. With these general expressions, the Onsager reciprocal relation is easily verified. This provides a check on the general validity of Hill's formalism. Furthermore, these expressions are suitable for numerical evaluation even for rather complicated models and are therefore useful for the calculation of the efficiency near or at equilibrium. The analysis is also applicable to other coupling systems, such as active transport across membranes.     

Theory of muscle contraction II. Isotonic contraction

The theory of muscle contraction developed in Part I is extended to non-isometric cases. The basic feature of the approach is the strong viscous coupling of the movement of the counterionic (K⁺) layer with the movement of I-filaments. The surface conductance of the K⁺ layer governs the flux of H⁺ along the I-filaments which in turns regulates the rate of ATP hydrolysis. The energy output of the muscle becomes the function of its mechanical activity. By assuming linear dependence of the K⁺ layer's surface conductance on the velocity of shortening Hill's equation has been derived. With a set of reasonably chosen values of the basic parameters of the theory the values of Hill's constants have been computed. The theory has been also shown to provide the observed dependence of the isometric tension on the degree of the myofilamental overlap.     

Contractile Characteristics of Mimosa pudica L

The force-velocity characteristics of the primary pulvinus of Mimosa pudica have been determined using a new polytonic measurement technique. The contractile characteristics were determined from a modified form of Hill's equation (Hill, A. V. 1938. Proc. Roy. Soc. London B126: 136-195) describing the physiological contractile behavior of animal muscle. The values of the resulting Hill's constants were found to be remarkably similar to those of intact animal muscle and reconstituted contractile collagen.     

A Phenomenological Theory of Muscular Contraction: II. Generalized Length Variations

In part I of this series, the theory of irreversible thermodynamics was applied to the sliding filament model to obtain rate equations for a contracting muscle at the in situ length l_o. In this paper we extend the theory to include length variations derived from the sliding filament model of contracting muscle using the work of Gordon, Huxley, and Julian (1). Accepting the validity of Hill's forcevelocity relation (2) at the in situ length, we show that Hill's equation is valid for any length provided that the values of the parameters, a, b, and V_m vary with length as derived herein. The predicted variation with length of the velocity for a lightly loaded isotonic contraction is shown to agree well with that measured by Gordon, Huxley, and Julian (1). Chemical rates are derived as functions of length using parameters that can be obtained experimentally.     

Comparative evaluation of quantum theory of nerve excitation

Deliberate evaluation of the quantum theory of nerve excitation is made by comparing it with Hill's theory in fitting the experimental data on threshold-frequency relation, optimum frequency (v₀) for nerve excitation and strength-duration relation. Decrease of v₀ and increase of all the time constants (Hill's λ andk, Wei'sT ₂ and spike durationw) with decreasing temperature are interpreted on the basis of the dipole relaxation timeT ₂ but inexplicable from Hill's theory or any other existing theory. The closeness ofk,T ₂ andw values is explained. A variety of experimental results obtained by others is discussed. Finally, a comparison is made between the Hodgkin-Huxley equations and the quantum theory. Most of the facts (electrical and non-electrical) tend to support the thesis that nerve excitation is a macroscopic expression of quantum transitions of dipoles between energy states.     

The off rate of Ca(2+) from troponin C is regulated by force-generating cross bridges in skeletal muscle.

The effects of dissociation of force-generating cross bridges on intracellular Ca(2+), pCa-force, and pCa-ATPase relationships were investigated in mouse skeletal muscle. Mechanical length perturbations were used to dissociate force-generating cross bridges in either intact or skinned fibers. In intact muscle, an impulse stretch or release, a continuous length vibration, a nonoverlap stretch, or an unloaded shortening during a twitch caused a transient increase in intracellular Ca(2+) compared with that in isometric controls and resulted in deactivation of the muscle. In skinned fibers, sinusoidal length vibrations shifted pCa-force and pCa-actomyosin ATPase rate relationships to higher Ca(2+) concentrations and caused actomyosin ATPase rate to decrease at submaximal Ca(2+) and increase at maximal Ca(2+) activation. These results suggest that dissociation of force-generating cross bridges during a twitch causes the off rate of Ca(2+) from troponin C to increase (a decrease in the Ca(2+) affinity of troponin C), thus decreasing the Ca(2+) sensitivity and resulting in the deactivation of the muscle. The results also suggest that the Fenn effect only exists at maximal but not submaximal force-activating Ca(2+) concentrations.     

Relationship between force and stiffness in muscle fibers after stretch.

The purpose of this study was to evaluate the relationship between force and stiffness after stretch of activated fibers, while simultaneously changing contractility by interfering with the cross-bridge kinetics and muscle activation. Single fibers dissected from lumbrical muscles of frogs were placed at a length 20% longer than the plateau of the force-length relationship, activated, and stretched by 5 and 10% of fiber length (speed: 40% fiber length/s). Experiments were conducted with maximal and submaximal stimulation in Ringer solution and with the addition of 2 and 5 mM of the myosin inhibitor 2,3-butanedione monoxime (BDM) to the solution. The steady-state force after stretch of an activated fiber was higher than the isometric force produced at the corresponding length in all conditions investigated. Lowering the frequency of stimulation decreased the force and stiffness during isometric contractions, but it did not change force enhancement and stiffness enhancement after stretch. Administration of BDM decreased the force and stiffness during isometric contractions, but it increased the force enhancement and stiffness enhancement after stretch. The relationship between force enhancement and stiffness suggests that the increase in force after stretch may be caused by an increase in the proportion of cross bridges attached to actin. Because BDM places cross bridges in a weakly bound, pre-powerstroke state, our results further suggest that force enhancement is partially associated with a recruitment of weakly bound cross bridges into a strongly bound state.     

Mathematical modelling of corneal swelling

This paper presents a differential model of the corneal transport system capable of modelling thickness changes in response to osmotic perturbations applied to either limiting membrane. The work is directed towards understanding corneal behaviour in vivo. The model considers the coupled viscous flows within the corneal stroma and across the epithelial and endothelial membranes. The flows within the stroma are established based on transport theory in porous media, while the flows across the membranes are described using the phenomenological equations of irreversible thermodynamics. The ability of the numerical model to reproduce corneal thickness changes in response to endothelial perturbations was tested against available experimental data. The sensitivity of the model to changes in stromal and membrane transport coefficients was examined.     

Permeability of human granulocytes to dimethyl sulfoxide

The permeability of the membrane of human granulocytes to the permeating solute dimethyl sulfoxide (DMSO) was studied using the Onsager form of the phenomenological equations derived from the theory of irreversible thermodynamics. Changes in cellular volume were monitored with an electronic particle counter as samples of that population were introduced into hypertonic osmotica. Temperature and concentration sensitivity analyses of the permeability coefficients were carried out. It is shown that the introduction of the Onsager formalism allows further insight into the observed transport phenomena. It was found that DMSO may affect the water permeability properties of the membrane for that population of cells.