Modeling and Simulation of Intrafusal Muscle Fiber Using a Multi

; Modeling and simulation of intrafusal muscle ﬁber using a multi-sarcomeric model N. S. Oborin

Modeling and simulation of intrafusal muscle ﬁber using a multi-sarcomeric model

by N. S. Oborin

to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Thursday June 23, 2016 at 9:00 AM.

Student number: 1384945 Supervisor: Dr. Ir. E. de Vlugt

An electronic version of this thesis is available at http://repository.tudelft.nl/.

Abstract

Muscle spindle is an organ of proprioception that plays an important role in neuro-muscular control of the human joints. It is composed of intrafusal fibers, the mechanical properties of which determine the afferent response of the spindle. Intrafusal fibers are not homogenous: they are composed of many sarcomeres, have localized fusimotor innervation and have varying myosin composition throughout their length. Most models of intrafusal fibers do not take these structural considerations into account and model it as a single sarcomere, that way omitting potential emergent behavior that arises from a population of sarcomeres. The effects of sarcomere length inhomogeneity on the behavior of intrafusal fiber is not known. In this study a multi-sarcomeric model is developed and simulated with a varying activation shape along the intrafusal fiber to see whether an emergent behavior in present and how it manifests. Results show that relative activation of contracting sarcomeres has the largest effect. The fiber model with varied activation showed history dependence arising from non-homogenous initial sarcomere length distribution. The model also demonstrated amplitude-dependent behavior under multisine stretches that did not appear in a non-multi-sarcomeric model. In conclusion, it can be stated that multi-sarcomeric models can be beneficial in exploratory studies as they can demonstrate behavior that cannot be described with simplistic models.

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Contents

1 Introduction 1 2 Building the Spindle Model3 2.1 Overview...... 3 2.2 The Model...... 4 2.2.1 Single Half-Sarcomere...... 4 2.2.2 Multiple Sarcomeres...... 5 2.3 Parameter Selection and Model Validation...... 6 2.3.1 Process of Selection of Model Parameters...... 7 2.3.2 Perturbation Selection...... 7 2.3.3 General Criteria for Selecting Parameters...... 7 2.3.4 Parameterization of Response...... 8 2.3.5 Validation of the Model...... 8 2.4 Introduction of Activation Inhomogeneity...... 8 2.4.1 Perturbation Types...... 9 2.4.2 Parametrization of Response...... 10 2.4.3 Data Analysis...... 10 2.5 Multisine Perturbation...... 10 2.5.1 Signal Design...... 10 2.5.2 Data Analysis...... 10 3 Results 11 3.1 Validation and Parameter Selection Results...... 11 3.1.1 Parameter Selection of Extrafusal Fibers...... 11 3.1.2 Validation of Multi-Srarcomere Model...... 11 3.2 Effect of Activation Shape on Isometric Response...... 14 3.2.1 Effects of abg and aoff ...... 14 3.2.2 Effects of abg and aσ ...... 17 3.2.3 Selection of Intrafusal Response...... 17 3.3 Multisine Perturbation...... 20 4 Discussion 23 4.1 Model Derivation...... 23 4.2 Parameter Selection...... 23 4.3 Validation...... 24 4.4 Effects of Activation Shape on Isometric Response...... 25 4.4.1 Effects of Initial Length Inhomogeneity...... 25 4.5 Multisine Perturbation...... 25 4.6 Conclusion...... 26 A Raw Data 27 B Notes on Derivation of Various Parameters 31 B.1 On Selection of cext and kext ...... 31 Bibliography 35

Introduction

A model is a tool to test a scientific hypothesis or to predict and describe a behavior of a system. The tendency in the scientific community is to keep complexity to the minimum. Adopting a more complex model (usually with more parameters) should be justified with a better prediction of an outcome. However, a model which describes behavior in terms of a set of easily interpretable parameters is arguably less complex. For example, a black-box type model would be good at describing behavior of some physiological system, however it would just do that, describe what is already known. On the other hand, a model that mimics the internal structure of the system and its components, gives more insight about the function of the system because now it is apparent how tweaking one parameter, which has a real life counterpart, affects overall behavior. A complex model can potentially solve two problems: a better prediction of physical response and offer a better insight into the function of the system and render it more tractable for study and analysis. The human nervous system is one example of such a complex system. Consisting of 100 billion neural cells, it is studied by numerous disciplines attempting to shed some light on its function and organization. One contributor to this field is the Neuro-Muscular Control (NMC) research group at TU Delft, studying the neural control of movement, in particular the role of proprioceptive feedback. One of the means of studying neural control of movement is through mechanical perturbation of the joint(s) from which the visco-elastic properties can be determined. Visco-elasticity results from passive tissues, muscle mechanics and from the proprioceptive feedback. This way, by measuring joint visco-elasticity, inference can be made on the underlying structures such as proprioceptive feed-back gains that are usually difficult to measure directly. System identification and parameter estimation (SIPE) techniques are employed to facilitate this process. It is therefore vital to have a good model of the system (in this case, a neuromechanical model of the human joint(s)) in order to study the underlying structures and deduce their functions. The organ that is responsible for proprioception and neural feedback is the muscle spindle. A model of a spindle is important for identification of contribution of reflexes to jointś visco-elastic properties. The muscle spindle is a complex sensory organ which is composed of different types of intrafusal fibers (IFs) which are innervated by different afferent and efferent nerves at different locations. Its small size, high degree of inhomogeneity and non-linearity make it a difficult organ to model. There have been a number of attempts of modeling a spindle, but the quality of models could be improved (an overview of models is covered elsewhere [14]). One of the problems of muscle modeling is the fact that there is a lot of experimental data on different scales: from microscopic cross-bridge dynamics to gross muscle behavior; however there is no model which could bridge the gap between different levels of experiments. Furthermore, most of the models are reductionist in nature: the population of sarcomeres is not modeled. This simplification can conceal emergent dynamics which arise because of interaction of sarcomeres in a population. While modeling skeletal muscle fibers as homogenous can be acceptable, doing so for intrafusal fibers which have varying properties and localized innervation can be a hindrance in describing their behavior. Also some models have parameters which are difficult to tie to physiological domain, that way jeopardizing the interpretability of the model. A possible solution to all of the aforementioned problems is a physiologically accurate model that tries to mimic micro and ultra-structure of the muscle. There exist a number of muscle models which can describe muscular behavior, however the Huxley

1 2 1. Introduction model tends to be most physiologically accurate; its parameters can be related to cross-bridge kinetics. Despite predicting a good response [6] for extrafusal fibres (the ones the skeletal muscles are composed of), its effectiveness in describing response of fine IF (the ones which constitute the spindle) is still unknown. The reason is that IFs display great variability in innervation in a very small space [1–3]. Furthermore, each type of IF has different mechanical properties. It is believed that the response of the spindle is mainly determined by localized behavior of the IF [16]. A possible reason for that is that afferent innervation measures strain of the IF in the middle of the fiber, while the efferent innervation creates localized foci of contraction, hence inducing local strain and stress. The matters are complicated by the fact that there are two afferent receptors located in different part of the same IF, and the same IF receives multiple gamma innervations. This makes it critical to consider the distribution of strain throughout the IF. The distribution of strain throughout the IF can be studied by a multi-segmented model (MSM), a fiber model consisting of multiple sarcomeres connected in series. The dawn of multi-segmented models can be traced to the work of [13]. In that work, a “popping-sarcomere" hypothesis was proposed which, as later research shows [18], turned out to be incorrect as there was no sarcomeres “popping" or over-extending to the region of passive force development on the force-length curve. Advancement of microscopic imaging in the last decade provided the tools for examination of the contractile dynamics on a finer level. New research elucidated individual sarcomeric behavior of the muscle fibers and showed inhomogeneity thereof. With that research the idea that sarcomere inhomogeneity (in length or force producing properties) can possibly explain myofibrial mechanics was revitalized. This led to the development of new models which take into account the dynamics of the population of sarcomeres [8, 17, 19]. It is noted that the cause for advent of multi-segmented muscle models was to explain mechanical behavior (residual force enhancement and depression) which cannot be explained by conventional muscle theories. There exist an issue of scalability in the study of muscles, behavior on macro-scale is not the scaled version of the behavior on micro-scale. The force of a single cross-bridge generated in isolated conditions is not the force of the ensemble of cross-bridges, just like the behavior of a single sarcomere is not the behavior of the whole fiber. Huxley model assumes this scalability and therefore cannot predict force enhancement/depression by itself. It is still unknown whether the inhomogeneity of sarcomeres is the sole cause of certain mechanical behavior (residual force enhancement and depression). It is probably a contributing factor along with other mechanisms like activation dependent stiffness of elastic component and cycling cross-bridges [4]. However, while modeling of the residual force enhancement is desirable, for the muscle spindle it is the modeling of variation in sarcomeric length along the IF that seems to be important. The purpose of this paper is to develop a physiologically accurate model and evaluate the benefits, drawbacks and feasibility of multi-sarcomeric models on the modeling of intrafusal fibers and test it by examining how sarcomere inhomogeneity affects the behavior of the intrafusal fiber under stretch. Chapter 2 discusses methods, namely how the IF model was derived. Also a test scenario is established for validation of the model. Chapter 3 deals with results and validation. At first the model is validated for an extrafusal fiber to show that it is feasible, then it is tested to study any emerging behavior of the model in terms of spindle sensitivity. Spindle sensitivity is defined as the relationship between fiber length and afferent output, the latter defined as the strain of the sensory region. The results are discussed in chapter 4. 2

Building the Spindle Model

2.1. Overview This chapter describes a general model of an intrafusal fiber. An intrafusal fiber (IF) is a small myofibril-like fiber that has a length of about 1 cm and a thickness of 20 µm. There are 3 types of intrafusal fibers: bag1, bag2 and chain. They mainly differ in their mechanical properties and how they respond to activation. A spindle is a sensory organ which is composed of these fibers. A spindle usually has 1 bag1 IF, 1 bag2 IF and multiple chain IFs connected in parallel. It is important to note that modeling of an intrafusal fiber is not synonymous with modeling of a muscle spindle. The behavior of the spindle is determined not only by the mechanical behavior of IFs but also by the intricate web of its efferent and afferent innervation. In a spindle, each composing IF can receive non-exclusive innervation (multiple IFs innervated by a same efferent nerve) and/or be innervated by different nerves in different places of the same IF. The output of the spindle or the afferent signal is generated by combination of afferent signals of the composing IFs. The goal of this thesis is to employ a multi-sarcomeric to model a general type of intrafusal fiber rather than a whole spindle. A more complete description of relevent physiology of muscle spindle is given elsewhere [14, Chapter 2] There are models like Huxley and Hill for describing skeletal muscles, however these cannot be readily applied to intrafusal fibers for the following reasons. At first, IFs are much thinner in size (roughly 5 µm in diameter), their thickness consists of one layer of sarcomeres. Their size is comparable to that of myofibrils, which have a different sarcomeric arrangement (less sarcomeres in parallel and also absence of passive connective tissues such as endomissium and perimissium). This means that despite having the same basic functional element (the sarcomere) as extrafusal fibers, different macro structure can result in IFs having different mechanical properties. Second difference lies in the fact that IFs are innervated in a way that produces localized contraction, and the afferent signal is determined by local strain. Finally, the genetic expression of Heavy Myosin Chains, which is responsible for attachment and power stroke of cross-bridges is varied along the length of IFs [15]. A recent trend in muscle research is to take into account modeling of population of sarcomeres [4, 8, 17] to account for emergent behaviors of and possibly explain phenomena such as force enhancement and thixotropy. This modeling approach attempts to shed light onto how organization of low level structures gives rise to the muscle function and provides a good starting point for bridging experimental data from different scales (from actin-myosin interaction to myofilament to sarcomere to myofibril to muscle fiber to fascicle to the whole muscle). For the aforementioned reasons, using a multi-sarcomeric model is a logical choice. Furthermore, to keep a model physiologically accurate a Huxley [21] element is used because it models attachment and detachment of cross-bridge populations. It is noted that a sarcomere can contract asymmetrically and is therefore believed to have 2 degrees of freedom. That is why a basic unit of this model is a 1 DoF half-sarcomere. Two half-sarcomeres attached together would form an equivalent 2 DoF sarcomere. In this chapter a general model of fiber will be described. Then a process of selection of parameters and validation for the fiber will be presented. Finally the simulation scenario for the model will be discussed.

3 4 2. Building the Spindle Model

Figure 2.1: Detailed overview of the half-sarcomere model with all its components.

Figure 2.2: Schematic overview of the whole model containing half-sarcomeres, clamp and the length to force actuator (with kext and cext parameters representing stiffness and damping of the length to force actuator).

2.2. The Model The model is loosely based on the implementation of the multi-segment Huxley model by [17]. This model was also loosely based on MTC Huxley model by [20]. A schematic of a single half-sarcomere is seen in figure 2.1. Whole fiber model is a series of 20 rigidly connected half-sarcomeres 2.2. The three types of intrafusal fibers (bag1, bag2, chain) are modeled with different activation parameters.

2.2.1. Single Half-Sarcomere The single element of the ﬁber is a half-sarcomere. It is discriminated from a sarcomere in the sense that a sarcomere can contract asymmetrically (half of the sarcomere contracts while the other expands) that way a sarcomere has two degrees of freedom while half-sarcomere has only one degree of freedom. The elements of the half-sarcomere (seen in ﬁgure 2.1) are the following:

CE Z ∞ FCE = C × a × N(u) xp(x, t)dx (2.1) −∞

dp(x, t) ∂p ∂p = +u ˙ = f(x)(1 − p(x, t)) − g(x)p(x, t) (2.2) dt ∂t ∂x CE or contractile element describes the dynamics and force generation of the myosin–actin protein interaction. It is modeled as a Huxley element. It generates force based on the length of the element (u), activation (a), scaling factor (C) and distribution of cross-bridge lengths (p). Distribution of cross-bridges governs the amount of force generated (equation 2.1), while the cross-bridge attachment and detachment rates (f(x) and g(x)) govern the change of the p in time (equation 2.2). The force-length relationship of the half-sarcomere is implemented by a function N(u).

Fmax C = R ∞ (2.3) −∞ xpSS(x, t)dx C represents the combined stiffness of crossbridges in a half-sarcomere. Increase in its value can also be interpreted as increase of stiffness of cross-bridges or equivalently as an increase in number of cross-bridges (or thickness of ﬁber). It is the relationship of maximal isometric force (Fmax) to the steady-state average length of cross-bridges [20] as seen in equation 2.3.

Cross-bridge Function Attachment function f(x) is parametrized by parameter f1. Detachment function g(x) is parametrized by 3 parameters g1, g2, g3. The velocity of shortening of contractile element of half-sarcomere (u˙) is translated into shortening of the cross-bridge by a factor h, which is an average length of a cross-bridges. 2.3. Parameter Selection and Model Validation5

SE element FSE = [kSE × w + cSE × w˙ ] × Fmax (2.4)

The actin and myosin are hypothesized to have some elasticity. Although relatively small, this elasticity contributes to the time lag of tension build up of the sarcomere with respect to its stiffness upon activation [12]. The series elastic element is modeled as with a spring and a damper. w represents the length of SE element (w = l − u) while w˙ represents its velocity. The damper is introduced for numerical stability purposes as there is a mass present.

PE element ˙ FPE = fT (l) × φT (l) × Fmax (2.5) ( 0.02l if 0 ≥ l ≤ 1 fT (l) = 0.02l + 1.2(l − 1)3 if l > 1

˙ 2 ˙ φT (l) = arctan(c1l) (2.6) π The parallel element of the model corresponds to the titin protein of the sarcomre. Titin is responsible for passive force generation (due to stretch). The model of titin are varied, in this case a model of [7] was taken and adopted for a half-sarcomere by making it operational on a shorter range. PE element force (2.5) depends on both, length of half-sarcomere (l) and its velocity (l˙).

Masses Since the model is derived from forces, masses are necessary for purposes of numerical stability. They are selected so that they will not contribute significantly to inertial forces in the model (less than 5% influence). The mass of an individual sarcomere is about 10−14 kg. In the model the relationship between applied forces and acceleration is roughly 0.0001 making mass about 10−6 kilograms. Although the masses are significantly heavier, they do not contribute to inertial forces of the model and therefore are acceptable.

2.2.2. Multiple Sarcomeres X ˙ X ˙ Fext = kext(Limposed − li) + cext(Limposed − li) (2.7)

For the multi-sarcomeric model 20 half-sarcomeres were connected in series, sarcomere was clamped at one end and on the opposite end a spring damper element was placed (see 2.2) to act as a transducer and produce force to meet the imposed length requirements. The element has to be signiﬁcantly stiffer than the sarcomeres in order to have no effect on the dynamic response of the model. The stiffness of the element is given with kext and its viscosity with cext as can be seen in equation 2.7. The total force it produces (Fext) is proportional to the difference of the desired length (Limposed) and actual total length ˙ of each sarcomere (li) plus the difference in imposed velocity (Limposed) and the sum of velocities of all ˙ sarcomeres (li).

Activation Modeling ( (S − a)( S − S−1 ) if S ≥ a da/dt = τa τd S−a S < a τd if

The activation is modeled by adjusting a variable a as seen in equation 2.1. The model of calcium dynamics is taken from [20] and it ﬁlters predeﬁned neural stimulation input S to force activation of the sarcomere a (equation 2.2.2). The shape is determined by activation and deactivation time constants τd and τa.

Simulation The model was implemented in Matlab software and simulated using an ODE45 solver. 6 2. Building the Spindle Model

Primary Parameter Selection

Grid of Grid of Parameters Parameters C - h f1 - g3

Simulate Ramp Simulate Ramp Condition Condition

Force Figure A.2 Force Figure A.1 Response Response Bad Result

Parametrization Figure 3.1 Parametrization Figure 3.2 of Response of Response

Data Point Analysis of from [Lombardi90] Model Behavior

Select Bad Fit f1, g3, C, h

Data Points Validation Figure A.3 from [Lombardi90] Validation Velocity Graph Figure 3.3 Simulate Velocity Parametrization Different Ramp Force of Responses Velocities Responses

Figure 2.3: Schematic overview of the parameter selection and model validation.

2.3. Parameter Selection and Model Validation Before a model can be used to test various hypotheses, a set of correct parameters needs to be selected and a model needs to be validated to demonstrate that it is capable of producing required behavior. The overview of this step is given in the figure 2.3. The approach to building a model towards intrafusal fibers is in two steps. At first parameters for generalized fiber-model (homogenous activation distribution throughout sarcomeres) are selected. The model is validated by comparing its force responses (to different velocity ramp perturbations) with force responses from the literature under similar perturbation conditions. This validation step will demonstrate whether the model can capture behavior of a real system via force response, how well the model performs under dynamic conditions and where it diverges from reality. Due to the large amount of parameters and high computational times, selection of parameters by means of a brute-force search is not a viable option. Instead, the parameters which contribute to the behavior of the system are tweaked, while the rest are held constant and are either taken from the literature or set to realistic values which would make the system behave in a desired way. Furthermore, parameters which affect the function of CE are tweaked: C, f1, g3, h, while all others are held constant. The parameters have to be determined visually, so that the response of the fiber matched the one of the literature. C and h are tweaked first, then f1 and g3 are varied. 2.3. Parameter Selection and Model Validation7

2.3.1. Process of Selection of Model Parameters The parameters are initially seeded with values from literature and similar models as a starting points. Then the parameters are separated into two groups. First groups is varied to see the effects on the behavior but they are not used to change behavior. Second group of parameters (C, h, f1 and g3) are varied to a greater degree and are used to fine tune the behavior of the model. The effects of the second group on the model response will be analyzed (figure 2.3). The subset of parameters which will be optimized (C, f1, g3, h) are grouped into 2 subgroups. For first subgroup C and h will be varied while f1 and g3 will be held constant. For a second group, f1 and g3 will be varied. The two groups are separated because f1 and g3 affect the cross-bridge dynamics, while C and h scale the relative force of CE, hence 2 groups are somewhat orthogonal. There will be 8 values for each variable producing 128 combinations in total (64 for each group). The parameters are selected based on the force response of the model to a ramp stretch.

2.3.2. Perturbation Selection The “blueprint” for force response, to which the model response will be compared is taken from experiments of Lombardi and Piazzesi [11]. In that research a muscle ﬁber was perturbed with length perturbations of different velocities (ramps) and the force was measured. After that, the force response was parametrised with two values and a comparison was made to how these values vary with increase of ramp velocity. The reason for selection of this particular study is that it tests muscle ﬁbers individually rather than a whole muscle with tendons. For parameter selection process the model will be perturbed with length ramps of similar magnitude (0.1L0) and velocity of 0.9L0/s. The velocity should be high enough to evoke the biphasic nature of the force response. It is noted that in the experimental initial conditions are set such that popping would not be a problem, at the plateau part of the force-length relationship of the half-sarcomere (1.03µm). In the model same initial conditions will be implemented.

2.3.3. General Criteria for Selecting Parameters

Figure 2.4: Generic force response of a muscle ﬁber to a ramp perturbation

The model parameters are selected by how well they replicate shape of the experimental force response. The force response can at ﬁrst be analyzed visually to discard obviously faulty responses with popping. A quantitative analysis is then performed by extracting numerical value of the force response at critical points in time. 8 2. Building the Spindle Model

2.3.4. Parameterization of Response Force responses described in previous section are complex in nature and need to be parametrized to preform quantitative analysis and compare with existing literature. There is a number of ways to characterize the force response to the ramp length perturbation (slope of SRS, duration of SRS, peak force, etc.) but the general idea is to extract critical points and preform some type of a transformation on them. In this case, at higher speeds, the force response is biphasic (as seen in ﬁgure 2.4). The two phases occur during rising part of the ramp: the short stiffness part and steady state part. They can be described by 3 parameters. Initial force before the beginning of the perturbation is assumed to be a point of reference with respect to which other points are scaled. The 3 coordinates of 2 points on the force-time curve which matter and can possibly vary with variation in length and speed are: time of the SRS region, force at the end of SRS region and force at the end of the stretch. The other criteria (such as slope of the SRS and force overshoot) can be derived from these points. It is also possible to take into account the shape of the transition and the steady-state force at the ramp plateau after stretch, but that would complicate matters and will not be done. Lombardi and Piazzesi [11] characterizes force response to a ramp strain with two values, maximum force and force of the steady state region of the response: Fmax/F0 and FSS/F0. Same type of charac- terization will be used in this study. Relative nature of the ratios (Fmax/FSS) will also be looked at for simpliﬁcation of analysis. If this ratio is less then 1 then the force response is monotonically increasing, if it is more than 1 then the biphasic nature becomes more pronounced and the force at the end of the SRS region (until tmax) is greater than the force at the steady state phase making a transient force peak as seen in Figure 2.4. The two values, Fmax/F0 and FSS/F0, will be plotted against varied parameters (C, f1, g3, h) for analysis. Based on these graphs a decision will be made which parameters to use. The point after which the FSS and Fmax stop being the same value (in other words the Fmax/FSS is greater than 1) is referred to as bifurcation point.

2.3.5. Validation of the Model Once it is known how the variation of parameters affects the response and a set of parameters is selected, the model will be tested with different ramp velocities (ﬁgure 2.3, Validation box). This step is not used for estimation of parameters but rather to demonstrate the degree of convergence to the experimental data, under a new set of experimental conditions. The experiments done by Lombardi and Piazzesi [11] perturbs frog muscle ﬁber by a ramp stretch at varied speeds (0.02 to 1.34 L0/s) to show how they affect force values. Same speed variation will be imposed on the model to compare results.

2.4. Introduction of Activation Inhomogeneity Intrafusal fibers, unlike extrafusal fibers, exhibit localized contraction when activated. Under isometric activation, sensory region of the intrafusal fiber stretches while the region with efferent innervation contracts. The cause for the inhomogeneity of contraction is hypothesized to stem from a spatial variation of activation over the fiber length. Since there is no direct data on the concentration of calcium ions in the intrafusal fiber during activation, it is assumed that activation is solely (as opposed to other factors like different CE properties in the sensory region) responsible for contraction inhomogeneity and sarcomere contraction data will be used to find the right shape of the activation curve. This data is taken from the experiments of [2]. Generally speaking, the fiber model can be turned into generalized intrafusal fiber model by introducing activation inhomogeneity. The activation distribution curve is modeled as a sum of 2 normal distribution curves, at two different locations (see figure 2.6). The shape is varied by varying the background activation (abackground), distance between the foci (aoffset) and the standard deviation (or the width of the curves aσ). The effects of activation shape on the dynamic response of the intrafusal fiber regions is then analyzed (overview in figure 2.5). The desired shape parameters are then selected to match the behavior of the intrafusal bag1 and chain fiber, based on the criteria in table 2.1. It is possible that a few iterations with refinement of parameters are necessary to pinpoint the right set of parameters. 2.4. Introduction of Activation Inhomogeneity9

Simulation of Variation of Activation

Initial Grid of Grid of Grid of Sarcomere Parameters Parameters Parameters Distribution asig - aoff abg - aoff asig - abg

Exclude from Simulate Isometric Condition Analysis

Insignificant Sarcomere Length Response Effect

Parametrization of Response

Magnitude of Elongation in Fig 3.4 Fig 3.6 Sensory and Focal Zones Velocity of Elongation in Fig 3.5 Fig 3.7 Sensory and Focal Zones Processed Parameters Results

Shortening Select Activation asig, abg, aoff Data from Curve for Parameters for Boyd intrafusal fiber Bag2, Chain

Figure 2.5: Schematic overview of introduction of activation variation into the model

Figure 2.6: The distribution of activation value across half-sarcomeres and corresponding parameters which deﬁne the shape of activation

2.4.1. Perturbation Types The activation distribution is imposed under isometric conditions. The three activation parameters determine the shape of activation distribution among the sarcomeres. Background activation (abackground) sets the overall tone of all sarcomeres and how much are sarcomeres activated in relation to the foci of contraction. The foci sarcomeres (represented by two peaks in Figure 2.6) will be constrained to a maximum activation value of 1. The other parameter (aσ or equivalently astd) determines standard deviation of the compounding Gaussian distribution. The smaller the value the more localized the contraction of the activation. Finally, aoffset, determines how close is the peak of contraction to the equatorial region of intrafusal ﬁber. Initial conditions (initial distribution of the sarcomere lengths before the onset of isometric shortening) 10 2. Building the Spindle Model

Quantity value for bag1 value for bag2/chain Sarcomere shortening at foci of contraction [%] 7 25 Amplitude of stretch of sensory zone ISO [%] 2-8 12 - 30 Time to maximum [s] 0.5-1.2 0.2 -0.8 Velocity of stretch [%/s] 5-10 25-40

Table 2.1: Overview of physiological strain parameters of intrafusal ﬁbers; taken from [2] will also be varied. There will be simulations with homogenous initial distribution of sarcomeres lengths (all sarcomeres are the same length) and with non-homogenous initial length distribution. The nonhomogenous length distribution is attained after one isometric perturbation of a homogenous distribution. This will help elucidate whether and how initial variation of sarcomeres affects shortening behavior and shed light into the history dependence behavior of the multi-sarcomeric model.

2.4.2. Parametrization of Response The sarcomere length distribution is analyzed in two regions: the sensory region (designated by the sarcomeres 9 - 12) which lengthens and the two focal regions (sarcomeres 5,6,15,16) which shorten. The raw length response as a result of isometric activation is parametrized with 2 values in order to simplify analysis. The strain of a region and the rate of strain (or velocity of elongation) of a region. Strain is the percentage increase of the region length from before onset of isometric activation, until a steady state is reached. Velocity of elongation is the maximum speed of shortening.

2.4.3. Data Analysis After the sensory elongation is extracted it would become possible to analyze the effects of the spatial distribution of activation and initial sarcomere distribution on sensory behavior.

2.5. Multisine Perturbation The model will be perturbed with multisine signal to see how the sensitivity of the sensory region vary with frequency and amplitude and bandwidth of the signal. The effects of the ﬁber activation and amount of sarcomeres on frequency response will also be studied. The amplitude of perturbation is taken from Cathers et al. [5], and is converted from wrist angle degrees into L0 units. The paper presents two types of frequency compositions of the signal, however for sake of simplicity only a wide band signal will be used.

2.5.1. Signal Design The signal is generated from a speciﬁed power spectrum by means of a MATLAB function. The spectrum has two regions: a high power region and a low power one, and is separated by a sub-frequency. The sub-frequency of a signal determines the bandwidth of the signal and will be varied. The whole power region spans to a maximum frequency which was set to 40Hz and is constant throughout all trials. The reason for having lower power in the frequencies higher than sub-frequency is to excite potential nonlinear dynamics. The minimal frequency and frequency resolution is 0.25Hz and taking into account 4 band ﬁltering, effective observation time would be 16 seconds.

2.5.2. Data Analysis The significant results will be fitted with a PID model to see how parameters vary with bandwidth and amplitude. The model will be fitted to the data by minimizing error vector using a lsqnonlin MATLAB function: Hfiber e = Cxy lg( ) (2.8) Hmodel where Cxy, Hfiber and Hmodel are coherence, fiber transfer function and the PID model transfer function respectively. 3

Results

In this chapter the results of parameter estimation, validation and simulation will be presented. The ﬁrst part, presents results associated with parameter selection and validation of the model. Second part is the results of simulation of activation shapes and frequency response.

3.1. Validation and Parameter Selection Results 3.1.1. Parameter Selection of Extrafusal Fibers The raw force response under a ramp stretch (as described in section 2.3) is presented in appendixA. The extracted feature points Fmax/F0 and FSS/F0 are displayed in the graphs below.

Effects of C and h on Response The raw data for force response due to variation of cross-bridge stiffness C and the cross-bridge step h can be seen in appendix A.2. The parametrized responses (as described in section 2.3.4) are shown in Figure 3.1 and they will be the focus of analysis. From ﬁgure 3.1 it is seen that as C increases, force ratios are generally increasing, but Fmax/FSS is decreasing. The bifurcation point (where Fmax/FSS is more than one) occurs for low values of C and h. The increase in C causes increase of cross-bridge stiffness and as a result the cross-bridge dynamics which are responsible for the pronounced biphasic nature of the force response are not recruited. For the same reason, high values of h do not produce bifurcation in the response: the cycling of cross-bridges is signiﬁcantly faster than elongation of CE and transient response is not apparent. For low values of h the transient force overshoot emerges and biphasic nature of the response is emphasized. The desired ratio of Fmax/FSS is 1.2 to match the data of Lombardi and Piazzesi [11], it could be attained with low values of C and h. At low values of C and h, however the problem of popping and sarcomere instability occurs, which is avoided.

Effects of f1 and g3 on Response From ﬁgure 3.2 it is seen that increasing f1, while keeping g3 constant, results in increase of peak and steady state force; meanwhile their ratio is decreasing and the clear biphasic transition disappears. This occurs because for relatively high values of f1 the cross-bridges are not broken fast enough and the CE continues to generate force as it is elongated. At higher absolute magnitudes of parameters the point of bifurcation occurs at a lower ratio of f1 to g3. From the Figure 3.2 it is seen that f1 sets the slope of the Fmax/FSS curves. Like in the case of C and h, the lower values of f1 and g3 are desired to attain the correct Fmax/FSS ratio. However, that poses a problem with possible popping.

3.1.2. Validation of Multi-Srarcomere Model Multi sarcomere model is validated by the plot of force characteristics (Fmax/F0 and FSS/F0) against velocities as seen in ﬁgure 3.3. It can be seen that bifurcation occurs roughly in the same point and the fact that relative magnitudes match. The popping however occurs at high velocities, meaning that model diverges from experimental results and becomes not valid.

11 12 3. Results

C:0.0199 h:−−− C:0.0298 h:−−− C:0.0398 h:−−− C:0.0497 h:−−− C:0.0696 h:−−− C:0.0796 h:−−− C:0.1591 h:−−− 2.9 F /F max 0 2.8 F /F SS 0 F /F 2.7 max SS

2.6

2.5

2.4

2.3

2.2

2.1

1.9

1.8

Sarco Force Ratio [−] 1.7

1.6

1.5

1.4

1.3

1.2

1.1

0.9 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 50 100 150 H value H value H value H value H value H value H value

C:−−− h:150.0 C:−−− h:60.0 C:−−− h:30.0 C:−−− h:20.0 C:−−− h:15.0 2.2 F /F max 0 F /F 2.1 SS 0 F /F max SS

1.9

1.8

1.7

1.6

1.5 Sarco Force Ratio [−] 1.4

1.3

1.2

1.1

0.9 0 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.2 C value C value C value C value C value Figure 3.1: (top panel) Force ratios vs h for different C. Derived from ﬁgure A.2. (bottom panel) Force ratios vs C for different h values. Derived from ﬁgure A.2 3.1. Validation and Parameter Selection Results 13

f :12.50 g :−− r:−− f :31.25 g :−− r:−− f :62.50 g :−− r:−− f :93.75 g :−− r:−− f :125.00 g :−− r:−− f :162.50 g :−− r:−− f :200.00 g :−− r:−− 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3 F /F max 0 F /F 2.85 SS 0 F /F max SS 2.7

2.55

2.4

2.25

2.1

1.95

1.8

1.65 Sarco Force Ratio [−]

1.5

1.35

1.2

1.05

0.9

0.75

0.6 25 125 225 325 425 25 125 225 325 425 25 125 225 325 425 25 125 225 325 425 25 125 225 325 425 25 125 225 325 425 25 125 225 325 425 g values g values g values g values g values g values g values 3 3 3 3 3 3 3

f :−−− g :20.00 r:−−−f :−−− g :50.00 r:−−−f :−−− g :100.00 r:−−−f :−−− g :150.00 r:−−−f :−−− g :200.00 r:−−−f :−−− g :240.00 r:−−−f :−−− g :320.00 r:−−−f :−−− g :440.00 r:−−− 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3 F /F max 0 F /F 2.85 SS 0 F /F max SS 2.7

2.55

2.4

2.25

2.1

1.95

1.8

Sarco Force Ratio [−] 1.65

1.5

1.35

1.2

1.05

0.9

0.75

25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 25 75 125 175 225 f value f value f value f value f value f value f value f value 1 1 1 1 1 1 1 1 Figure 3.2: (top panel) Force ratios vs g3 for different f1. Derived from ﬁgure A.1. (bottom panel) Force ratios vs f1 for different g3. Derived from ﬁgure A.1 14 3. Results

0 F /F Simulated Curve SS 0 F Lombardi90 fig2 Data SS F /F Simulated Curve max 0 F Lombardi90 fig2 Data max

0.2

0.4

/s] 0.6 0

0.8 Velocity of stretch [L

1.2

1.4 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 F/F 0 Figure 3.3: Force ratios vs velocity of perturbation. Simulated data is derived from ﬁgureA. Experimental data is taken from [11]

3.2. Effect of Activation Shape on Isometric Response In this section the effect of activation shape on sarcomere length distribution are presented. The sarcomere length distribution is parametrized with 2 parameters (see section 2.4.2) for every fiber region: elongation and rate of strain. The effect on these 2 elongation parameters with activation shape parameters is displayed in Figures 3.4- 3.7. The effects of initial sarcomere distribution (before onset of isometric activation) are also presented in the same graphs. Since the activation shape parameters (abg, aoff and aσ) are varied in pairs, there are 2 figures for each pair of parameters for the sake of clarity, as that way the data trends become more visible. In this analysis the relative activation of the foci of contraction to the rest of the fiber (abg) has the largest effect on the behavior, therefore the other two shape parameters are varied with respect to it.

3.2.1. Effects of abg and aoff Elongation From the figure 3.4 it is visible that elongation of sensory region decreases almost linearly with abg. abg has the most effect on lengthening. aoff seems to play no role, except for high values, where the foci of contraction are symmetrically close to the sensory region. At these values the elongation of the sensory zone becomes small and does not change with variation of abg. The elongation of sensory region of the fiber is smaller than the elongation of the focal region because of the fact that other part of the fiber elongates as the fiber shortens. Different initial conditions seem to have largest effect on the elongation, with homogeneous length distribution (dashed line) producing less strain relative to varied initial distribution. 3.2. Effect of Activation Shape on Isometric Response 15

0.05 a bg

0.10 a bg 0.2 0.15 a bg 0.20 a bg 0.05 a 0.25 a bg bg 0.10 a 0.30 a bg bg 0.15 a 0.35 a 0.20 a bg bg 0.1 0.25 a bg 0.40 a 0.30 a bg bg 0.35 a bg 0.40 a bg bg ] 0

0 Elongation [L

−0.1

0.40 a bg 0.40 a 0.35 a bg bg 0.30 a 0.35 a −0.2 bg bg 0.25 a 0.30 a bg bg 0.20 a bg 0.25 a 0.15 a bg bg 0.10 a 0.20 a bg bg 0.05 a 0.15 a bg bg 0.10 a bg −0.3 0.05 a bg 3.5 4 4.5 5 5.5 6 6.5 7 7.5 a values off

4.50 a off 5.50 a 4.00 aoff 5.00 aoff 6.00 aoff off 0.2 4.50 a 5.50 aoff 6.50 a off off 5.00 a off 4.00 a 6.00 aoff off

6.50 a 0.1 off

7.00 a off

] 7.50 a 0 7.50 a off

0 off

Elongation [L −0.1

−0.2 6.50 a off 5.50 a off 4.50 a 6.00 aoff 5.00 aoff 7.00 aoff 7.50 aoff 4.00 aoff 4.50 a off 5.50 aoff −0.3 off 7.50 a 6.50 aoff 5.006.00 aoff 4.00off a 7.00 aoff off

−0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a values bg Figure 3.4: (top panel) Elongation (blue line) of sensory zones and shortening (red line) of foci of contraction vs aoff values and variation of abg, for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). (bottom panel) Elongation (blue line) of sensory zones and shortening (red line) of foci of contraction vs abg values and variation of aoff , for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). 16 3. Results

0.6

0.10 act 0.4 0.10 act

0.15 act 0.15 act

0.20 act 0.2 0.20 act 0.25 act 0.30 act 0.25 act 0.35 act 0.30 act 0.40 act 0.35 act /s]

0 0.40 act 0 Speed [L 0.40 act −0.2 0.35 act 0.30 act 0.40 act 0.35 act 0.25 act 0.30 act 0.25 act 0.20 act −0.4 0.20 act

0.15 act 0.15 act

−0.6

0.10 act 0.10 act

−0.8 3.5 4 4.5 5 5.5 6 6.5 7 7.5 a values off

4.50 a off 5.50 a 5.00 aoff 1 off 4.00 a off

6.00 a off

6.50 a off 5.00 a 0.5 off 5.50 a off 6.00 a off 4.50 a off 4.00 a off 6.50 a off 7.00 a /s] off 0 0 7.00 a off

7.50 a off 7.50 a Speed [L off

−0.5

5.00 a off 5.50 a off 6.50 a off 7.50 a −1 off 7.50 a 6.00 a 6.50 aoff off off 5.50 a 4.50 a 4.50 aoff off off 4.00 a 7.00 a off off 6.00 a off 4.00 a 5.00 a off off 7.00 a −1.5 off 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a values bg Figure 3.5: (top panel) Velocity of elongation (blue line) of sensory zones and velocity of shortening (red line) of foci of contraction vs aoff values and variation of abg, for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). (bottom panel) Velocity of elongation (blue line) of sensory zones and velocity of shortening (red line) of foci of contraction vs abg values and variation of aoff ,for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). 3.2. Effect of Activation Shape on Isometric Response 17

Quantity value for bag2/chain Background activation (abg) [-] 0.15 Standard deviation of activation (asig) [-] 1.6 Focii of activation (aoff ) [sarc number] 5.5, 15.5

Table 3.1: Selected activation shape parameters to describe isometric strain and velocity of the intrafusal ﬁber.

The initial distribution of sarcomeres has little effect on the shortening of the sarcomeres.

Velocity Velocity of elongation, like elongation itself, is predominately dependent on abg rather than aoff (see Figure 3.5. The velocity, however, is not linearly related to abg, but seems to be inversely related. For higher values of abg the velocity seems to reach a steady state. As aoff becomes larger moving closer to sensory region the velocity of elongation becomes smaller. Homogeneous distribution of sarcomeres at the start of activation tends to result in lower contraction speeds of the sensory region while having little effect on the contraction speed of the focal region.

3.2.2. Effects of abg and aσ Elongation abg like in the case of aoff variation seems to linearly affect elongation magnitude (Figure 3.6), but this case there is more variation as aσ is introduced. aσ is responsible for how smooth the activation is transitioned between activation foci and the rest of the ﬁber. The elongation increases as aσ becomes larger also the spread of elongation values and the sensitivity to abg becomes greater. This happens roughly up to the value of aσ = 1.6 after which the activation curve becomes too smooth and further increase in aσ results in both decrease of elongation. Initial conditions play a role, as there is a signiﬁcant smaller magnitude of both sensory and focal strain.

Velocity From the Figure 3.7 the effects of abg on velocity are similar to the case of variation of aoff . aσ produces a signiﬁcant increase in velocity for low values of abg and has little effect for high values of abg. There is a point where the co-recruitment of neighboring sarcomeres (increase of abg) helps with the elongation speed of sensory region. Once that point is crossed, the parts of sensory region become activated and that induces shortening and results in decrease of lengthening of sensory region. At aσ =1.4 the velocity of lengthening reaches maximum value and maximum sensitivity to abg. It is also interesting to note that aft that point the velocities of shortening and lengthening become symmetrical (as opposed to the case of low aσ where shortening velocity was higher in magnitude than lengthening) meaning that all strain is taken up by the sensory zone and does not propagate to other parts of the ﬁber.

3.2.3. Selection of Intrafusal Response The bag1 and chain ﬁbers are described by the the following parameters of the abg, aoff and aσ. These parameters are listed in table 3.1. 18 3. Results

0.4

0.3

0.2

0.05 a bg 0.10 a bg 0.15 a bg 0.1 0.20 a 0.25 a bg 0.30 a bg 0.35 a bg 0.05 a bg bg 0.40 a 0.10 a bg 0.15 a bg ] 0.20 a bg 0 0.25 a bg 0.30 a bg 0.35 a bg 0.40 a bg 0 bg Elongation [L

0.40 a 0.35 a bg 0.30 a bg −0.1 bg 0.40 a 0.25 a 0.35 a bg 0.20 a bg bg bg 0.30 a 0.15 a 0.25 a bg bg 0.20 a bg 0.10 a bg bg 0.15 a bg 0.05 a bg 0.10 a bg

0.05 a −0.2 bg

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−0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 a values sig

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1.60 a 1.40 aoff 1.80 aoff 0.3 off 1.20 a 2.00 aoff off

1.60 a 1.00 a off off 1.80 a off 1.40 a off 0.80 a 0.2 2.00 a off off 1.20 a off 1.00 a off 0.60 a off

0.80 a 0.1 off

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0 Elongation [L

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0.60 a off

−0.2 0.60 a 2.00 a off off 1.80 a off 1.60 a off 2.00 a 0.80 a off off 1.00 a 1.80 a off off 1.60 a 1.40 a off off 0.80 a 1.20 a off −0.3 off 1.40 a off 1.20 a off

1.00 a off

−0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a values bg

Figure 3.6: (top panel) Velocity of elongation (blue line) of sensory zones and velocity of shortening (red line) of foci of contraction vs aσ values and variation of abg, for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). (bottom panel) Elongation (blue line) of sensory zones and shortening (red line) of foci of contraction vs abg values and variation of aσ, for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). 3.2. Effect of Activation Shape on Isometric Response 19

0.6

0.4

0.10 a bg 0.10 a bg 0.15 a bg 0.15 a bg 0.20 a 0.20 a bg 0.2 bg 0.25 a bg 0.25 a 0.30 a bg bg 0.30 a 0.35 a bg bg 0.40 a bg 0.35 a bg /s]

0 0.40 a bg 0 0.40 a bg 0.40 a bg 0.35 a Speed [L bg 0.35 a bg 0.30 a 0.30 a bg bg 0.25 a 0.25 a bg bg −0.2 0.20 a bg 0.20 a bg 0.15 a bg 0.15 a bg

−0.4

0.10 a bg 0.10 a bg

−0.6

−0.8 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aσ values

1.5

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1.00 a σ

1.60 a 1 σ

1.40 a σ 0.80 a σ

1.60 a σ 1.20 a σ 0.60 a σ 1.80 a 0.5 σ 1.00 a σ

0.80 a 1.80 a σ σ 2.00 a σ 2.00 a 0.60 a σ σ /s] 0

0 Speed [L

2.00 a σ

−0.5 2.00 a σ

1.80 a σ 0.60 a σ 1.60 a σ 1.80 a σ 0.80 a 0.60 a −1 σ σ

0.80 a σ 1.40 a σ 1.20 a 1.60 a σ σ 1.00 a 1.00 a σ σ

1.20 a σ 1.40 a σ −1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a values bg

Figure 3.7: (top panel) Velocity of elongation (blue line) of sensory zones and velocity of shortening (red line) of foci of contraction vs aσ values and variation of abg, for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line). (bottom panel) Velocity of elongation (blue line) of sensory zones and velocity of shortening (red line) of foci of contraction vs abg values and variation of aσ,for homogenous initial length conditions (dashed line) and nonhomogenous initial length conditions (solid line) 20 3. Results

x 10 −4 0.25 0.055 2 5 Hz 5 Hz 5 Hz 10 Hz 10 Hz 10 Hz 20 Hz 20 Hz 20 Hz 20 Hz Max Activation 20 Hz Max Activation 20 Hz Max Activation 0.2 0.05 1

0.15 0.045 0

0.1 0.04 −1 I value P value D value 0.05 0.035 −2

0 0.03 −3

−0.05 0.025 −4

−0.1 0.02 −5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Amplitude of Sinusoid [deg] Amplitude of Sinusoid [deg] Amplitude of Sinusoid [deg]

Figure 3.8: Plot of PID model parameters against the magnitude of perturbation as fitted in the figure 3.9 (legend specifies different bandwidths).

3.3. Multisine Perturbation The response of sensory zones (middle 4 sarcomeres in a 20 sarcomere model and 2nd sarcomere in 3 sarcomere model) of intrafusal fiber to multisine perturbation are seen in figure 3.9. Dashed lines represent the best fit for PID model.

Bandwidth Effects Increasing the bandwidth of the signal from 5Hz to 20Hz (top two panels and middle left panel in ﬁgure 3.9) does not introduce any noticeable non-linearities. The effect for bandwidth of 5Hz is inconclusive due to low coherence.

Magnitude effects The increase of magnitude decreases the sensitivity (the P gain decreases in ﬁgure 3.8), except for the case with 3 sarcomeres (ﬁgure 3.9 bottom panel). At higher frequencies the phase lag increases with the magnitude.

Frequency effects As frequency increases the sensitivity stays the same, there is also a phase delay present.

Activation effects Submaximal activation (left plot in middle panel in ﬁgure 3.9) of the fusimotor regions (see 2.4.2) tends to have higher variability in sensitivities with increase of magnitudes while maximum activation (right plot in middle panel) tends to increase sensitivity of sensory region. It is noted that for small amplitude of perturbation the sensitivity did not change with activation.

Sarcomere number effects A 3 sarcomeric fiber (bottom panel in figure 3.9) has overall higher gain than 20 sarcomeric fiber (middle panel) and the variability of response due to change in magnitude are not present. The coherence is higher, and there is negligible phase lag; the effects of activation are not present as in the case of 20 sarcomeres. Sensory Sarcs: 9:12 Sensory Sarcs: 9:12 10 −1

Figure 3.9: Plot of multisine simulation

Magnitude [−] results with different bandwidths, activa-

10 −2 tions, and number 10 0 10 1 10 0 10 1 0.9 of sarcomeres in 0.8 0.7 the system. Dashed 0.6 lines of same color 0.5 0.4 represent best ﬁt Coherence [−] 0.3 0.7 deg 0.7 deg 0.2 1.3 deg 1.3 deg of the PID model. 2.5 deg 2.5 deg 0.1 3.5 deg 3.5 deg (Top panel) Graph of

80 10 0 10 1 10 0 10 1 limited bandwidth of 60 40 5Hz (left graph) and 20 10Hz (right graph). 0 −20 (Middle Panel) Graph Phase [deg] −40 of full 20 Hz band- −60 −80 with and variation 10 0 10 1 10 0 10 1 of lengths, for small Freq. [Hz] Freq. [Hz]

Sensory Sarcs: 9:12 Sensory Sarcs: 9:12 activation of peaks −1 10 (sarcomeres [5,6, 15, 16]) (left graph) and full activation of peaks (sarcomeres

Magnitude [−] 5,6,15,16) (right graph). (Bottom 10 −2 10 0 10 1 10 0 10 1 Panel) Same as 0.9 0.8 middle panel, but 0.7 0.6 the system is com- 0.5 prised from only 3 0.4 Coherence [−] 0.3 0.7 deg 0.7 deg sarcomeres instead 0.2 1.3 deg 1.3 deg 2.5 deg 2.5 deg of 20. The legend 0.1 3.5 deg 3.5 deg

0 1 0 1 speciﬁes magnitude 80 10 10 10 10 60 of length input in 40 terms of equivalent 20 0 wrist angle as used in −20 Phase [deg] Cathers et al. [5] −40 −60 −80 10 0 10 1 10 0 10 1 Freq. [Hz] Freq. [Hz]

Sensory Sarcs: 2 Sensory Sarcs: 2 10 0 Magnitude [−]

10 −1 10 0 10 1 10 0 10 1 0.9 0.8 0.7 0.6 0.5 0.4 Coherence [−] 0.3 0.7 deg 0.7 deg 0.2 1.3 deg 1.3 deg 2.5 deg 2.5 deg 0.1 3.5 deg 3.5 deg

80 10 0 10 1 10 0 10 1 60 40 20 0 −20 Phase [deg] −40 −60 −80 10 0 10 1 10 0 10 1 Freq. [Hz] Freq. [Hz]

Discussion

The goal of the study was to develop a physiologically accurate model and evaluate the benefits, drawbacks and feasibility of a multi-sarcomeric model on the modeling of intrafusal fibers and test it by examining how sarcomere inhomogeneity affects the behavior of the intrafusal fiber. Issues related to the model derivation, parametrization and validation are discussed first. Results show that the sarcomere inhomogeneity, expressed as a variation of activation and initial length, affected the strain and velocity of lengthening of the sensory zone once activated under isometric conditions. The history dependence was also demonstrated in the model. Under multisine perturbation, the fiber behavior varied with the activation and amplitude and it showed a dependence on the number of sarcomeres.

4.1. Model Derivation The model of intrafusal fiber was made physiologically interpretable and accurate with the idea that it would facilitate extensibility and interpretability of the results. The variation of fusimotor innervation (activation) along the length of the fiber led to adaptation of multi-sarcomeric model, that would allow to implement that variation and see how it affects behavior. The model was first constructed as a model of myofibril with homogenous activation, then it was made specific to intrafusal fiber by varying activation. Stoecker et al. [17] model was used as a basis for this model because he had a Huxley type contractile element (CE) and discrete amount of sarcomeres. He analyzed the effects of variance of force capacity between sarcomeres (C in this model) under isometric shortening on the transient velocity of shortening. The results are similar to this study where increase in variation of force capacity (done via variation of activation) results in an increased lengthening and shortening velocities (as seen in figure 3.5). The difference between this study and Stoecker et al. model is that little masses were introduced to translate force into acceleration. The introduction of masses, which made the model much heavier than in reality, limited maximal elonation velocity at which the contribution of masses to inertial forces would still be negligible. For computational reasons it was not possible to decrease the size of the masses further. However it was not possible to construct the model without them. For the simulation of isometric strain distribution the effects of masses are insignificant. For sinusoidal and ramp the masses can affect the sarcomeres closest to the force transducer.

4.2. Parameter Selection The parameter selection step of the model development proved to be difficult due to presence of many parameters, high computational times and the issues with sarcomere instability. The parameters which affected the behavior of CE (f1, g3, h, C) were varied (others were taken from the literature and stayed fixed) and their effect on the force response was studied (figure 3.1 and figure 3.2). The shape of force response was the criteria for the parameter selection as the mechanical properties of the CE element, specifically the bi-phasic nature of force response was the desired behavior to be captured. The CE was focused on as it is the nonlinear element that could give rise to new behavior of the whole fiber from

23 24 4. Discussion interaction with other components; it is also an element that is mostly responsive for viscosity which makes it the crucial part of the model. From figure 3.2 it is seen that the bi-phasic force characteristic can be attained with a high Fmax/FSS value but it would be at the edge of stability envelope (continuing to change the parameters in the direction of increasing Fmax/FSS results in popping). Depending on the shape of the parallel elastic (PE) element’s force-length relationship in relation to CE, the half-sarcomere can be unstable: the series of sarcomeres would pop one by one. The PE element was offset to make force-length relationship of a half-sarcomere monotonically increasing that way ensuring sarcomere stability. This is rationalized by the fact that sarcomeres in majority of experiments are stable but the mechanism of stability are not yet understood and there is no way in researched literature to reconcile a descending limb of the sarcomere force-length relationship with stability. Model’s stability can potentially be improved by making the force-length relationship curve of a sarcomere stem from the distribution of cross-bridges rather than explicitly prescribing it. This would also be more accurate from physiological point of view. The descending limb of the force-length relationship was derived from the isometric shortening experiments, as the length of the sarcomere increases, the total force produced goes down. This is done to account for the decrease in overlap of myosin and actin. The issue with the decreasing force-length limb is that it affects the whole population of cross-bridges regardless of their length. The problem arises for large velocities when the population is overstretched: the cross-bridges are overextended and in the process of breaking but under normal circumstances they generate large force (because of the large elongation). With the prescribed force-length relationship the force would decrease as the lengthening is increased. In other words the decreasing force-length limb counteracts the transient force generation. By contrast, if f1 would decrease with length there would be no new cross-bridges formed (as in the physiological case) but the existing over-stretched cross-bridges would not experience a decrease in their ability to generate transient force as they are attached. As a result there would be more transient force generation which could potentially affect sarcomere stability. The main limitation during parametrization step was the high computation time of the model that resulted from its complexity. Because of that, not all parameters could be varied and their effects studied. Second issue is that the force response to a ramp stretch that was used as the criteria for the parameter selection was not from intrafusal fiber but from an extrafusal one. This captures the general mechanical properties of the intrafusal fiber but not the nuanced behavior that can arise from variation of CE rate constants from sarcomere to sarcomere. Incorporation of these types of variations would not be possible due to the further increase of computing time.

4.3. Validation The extrafusal model is validated by different velocity stretches and as a result the bi-phasic force response follows the same trends as demonstrated by Lombardi and Piazzesi [11] (seen in figure 3.3). This behavior was not explicitly prescribed in the model but rather it emerged from the cross-bridge dynamics. This validation demonstrates that a model can exhibit a velocity dependent force nonlinearity which can introduce interesting dynamics in the further simulations with varied activation throughout the fiber. The model agrees with experimental data up to the velocity of 1.2L0/s (figure 3.3). This velocity is sufficient for the isometric shortening, as during these inputs local velocity never reaches high magnitudes. For multisine perturbations the composing sinusoids have a velocity higher than that, but since the magnitude of perturbation is significantly lower, it is acceptable. At velocities higher than 1.2L0/s the divergence from the experimental data occurs due to popping. The overstretching occurred only in the sarcomere closest to the length actuator. According to the myofibril experiments of Telley et al. [18], sarcomeres near the clamped ends did in fact exhibit significantly larger length, but they did not pop. In contrast, Telley et al. [18] had a significantly lower stretch velocity of 0.2L0/s for a myofibril of 52 half-sarcomeres. The cause of divergence with the Lombardi and Piazzesi [11] data was the fact that with popping a transient force decrease was produced. As stated earlier, due to the lack of mechanical data for the myofibrils, muscle fibers data of Lombardi and Piazzesi [11] was used instead. There is difference in ultra-structure as, unlike the myofibril, the muscle fiber has a number sarcomeres connected in parallel, allowing for multiple pathways of stress distribution. Sarcomere popping was also demonstrated in other multi-sarcomeric models like these of Campbell and Campbell [4], Denoth et al. [7], Givli [8], especially under fast stretches. The popping could be an inherent property of myofibril stretch at high velocities but there are no contrary or supportive evidence to that was found in literature. 4.4. Effects of Activation Shape on Isometric Response 25

4.4. Effects of Activation Shape on Isometric Response The introduction of sarcomere inhomogeneity by means of activation variation resulted in shortening of the sensory zone to a varying degree. The main parameter (abg) which was responsible for the degree of variation between largest and smallest activation in a distribution showed a direct effect on the sensory zone elongation and rate of stretch (figures 3.4, 3.5, 3.6, 3.7 bottom panel). The same trends were observed in the study of Boyd [2] where increase of relative activation of the focal region resulted in increase of elongation of the sensory region. The other parameters: aoff and aσ which are responsible for the offset and the width of the focal region did not affect the result to the same degree. Offsetting of focal zone from the sensory region tended to cause a decrease in lengthening velocity and elongation (figures 3.4 and 3.5 top panel). Increasing spread of activation amongst the sarcomeres, as opposed to making it localized, resulted in increase of lengthening velocity and strain of sensory zone up to a point, after which the activation became too ‘smoothed out’ resulting in lack of variation and hence decrease of the sensory zone strain (figures 3.6 and 3.7 top panel)

4.4.1. Effects of Initial Length Inhomogeneity Initial sarcomere length variability has an effect on the strain behavior. Sarcomeres with same initial length before the onset of activation (dashed lines on figures 3.4- 3.7), resulted in less velocity and elongation of sensory zone as opposed to sarcomeres with varied initial length distribution (solid line on the same figures). As the non-uniform sarcomere distribution resulted from a previous isometric activation, this demonstrates history dependent behavior of intrafusal fiber. It is noted that having correct isometric activation dynamics of intrafusal fiber does not necessary equate to having a correct sensory zone response dynamics under imposed strain, because the later can involve velocities higher than these experienced during isometric activation.

4.5. Multisine Perturbation The sarcomere inhomogeneity and multi-sarcomeric model affected the behavior of intrafusal fiber under multisine perturbation. From the results (section 3.3) it becomes apparent that the nonlinear behavior emerges from the number sarcomeres. In case of 3 sarcomeres, the fiber response did not vary under variation of activation, frequency and magnitude of perturbation (figure 3.9 bottom panel). A 20 sarcomere fiber, on the other hand, showed a varied response (figure 3.9 middle panel). In both cases, bandwidth effects were not observed. In the study of Cathers et al. [5] a wrist was perturbed and muscle activation measured. It demonstrated that smaller stretches result in more sensitivity than larger ones. However, the smallest amplitude response increased with frequency and largest stayed constant, in this model the former stayed constant while the later decreased (figure 3.9 middle panel). It is possible, since Cathers et al. [5] tested the whole system (muscle, golgi tendon organs, spindles and neural connections in-between) opposed to the stimulation of just an individual intrafusal fiber, his results will differ. Like Cathers et al. [5] in this study there is a similar trend of decreasing phase with increase of amplitude and the difference in both cases is roughly 20deg. However this behavior occurs at higher frequencies than in the experiments. The explanation for the effects of activation (figure 3.9 middle panel, left vs right figure) arises from the fact that focal sarcomeres became stiffer and more strain was transferred to the sensory region of the intrafusal fiber. In literature, for larger amplitudes, a spindle with dynamic stimulation tends to have higher sensitivity than a passive one [9]. There is no data for individual intrafusal fibers, but it can be inferred that individual fiber response will demonstrate an increase in sensitivity with increase of activation also. The benefit of the intrafusal fiber model is that it shows variation of sensitivity with amplitude of perturbation while model was not specifically trained on sinusoidal input. This shows that structural bottom approach has some merit. The model did not model transduction from strain to afferent signal which can introduce frequency dependent dynamics (like the case of [10]). A further area of study can focus on how different frequencies of vibration propagate throughout the myofibril or intrafusal fiber. By attaching a force sensors to both ends of the fiber and looking at time delay and force dissipation, some light can be shed on the kinetics of cross-bridges and effects of myosin isoform variation. 26 4. Discussion

4.6. Conclusion The developed multi-sarcomeric model of intrafusal fiber demonstrated variation of strain and velocity of the sensory zone with variation of activation under isometric conditions with relative activation of contracting sarcomeres having the largest effect. The fiber model with varied activation showed history dependence arising from non-homogenous initial sarcomere length distribution. The activation shape also had an effect on the sensitivity to the amplitude of the multisine stretches. The amount of sarcomeres in the model played a role in the emergent behavior and this supports the notion that sarcomere inhomogeneity can affect behavior or, on more general note, that the structure gives rise to function. Ideally, the benefits of a physiologically accurate models are that the parameters could be related to physiology and it would be easier to translate and incorporate scientific research. These benefits, however, come at a price: this approach resulted in many parameters, very large computation times and as a result, difficulty in selecting parameter values. Furthermore, sarcomere stability of a fiber can be a potential issue for large velocities and certain parameter values. It is possible the modeling of CE and/or PE in this study do not take into account dynamics which are critical to stability. These drawbacks make the model only feasible for exploratory studies rather than descriptive ones. Further research, adhering to same idea of modeling structure, can address the problems of stability by modeling sarcomere to a finer level of detail. A

Raw Data

27 28 A. Raw Data 0.35 0.25 :50.00 r:5.00 :320.00 r:0.78 :20.00 r:12.50 :100.00 r:2.50 :150.00 r:1.67 :200.00 r:1.25 :240.00 r:1.04 :440.00 r:0.57 3 3 3 3 3 3 3 3 time [s] 0.15 :250.00 g :250.00 g :250.00 g :250.00 g :250.00 g :250.00 g :250.00 g :250.00 g 1 f 1 1 1 1 1 1 1 0.05 f f f f f f f 0.35 0.25 :50.00 r:4.00 :320.00 r:0.63 :20.00 r:10.00 :100.00 r:2.00 :150.00 r:1.33 :200.00 r:1.00 :240.00 r:0.83 :440.00 r:0.45 3 3 3 3 3 3 3 3 time [s] 0.15 :200.00 g :200.00 g :200.00 g :200.00 g :200.00 g :200.00 g :200.00 g :200.00 g 1 f 1 1 1 1 1 1 1 0.05 f f f f f f f 0.35 0.25 :20.00 r:8.13 :50.00 r:3.25 :100.00 r:1.63 :150.00 r:1.08 :200.00 r:0.81 :240.00 r:0.68 :440.00 r:0.37 :320.00 r:0.51 3 3 3 3 3 3 3 3 time [s] 0.15 :162.50 g :162.50 g :162.50 g :162.50 g :162.50 g :162.50 g :162.50 g :162.50 g 1 1 1 1 1 1 1 1 f f 0.05 f f f f f f 0.35 0.25 :20.00 r:6.25 :50.00 r:2.50 :320.00 r:0.39 :100.00 r:1.25 :150.00 r:0.83 :200.00 r:0.63 :240.00 r:0.52 :440.00 r:0.28 3 3 3 3 3 3 3 3 time [s] 0.15 :125.00 g :125.00 g :125.00 g :125.00 g :125.00 g :125.00 g :125.00 g :125.00 g 1 1 1 f f 1 1 1 1 1 0.05 f f f f f f 0.35 0.25 :20.00 r:4.69 :50.00 r:1.88 :100.00 r:0.94 :150.00 r:0.63 :200.00 r:0.47 :240.00 r:0.39 :320.00 r:0.29 :440.00 r:0.21 3 3 3 3 3 3 3 3 time [s] 0.15 :93.75 g :93.75 g :93.75 g :93.75 g :93.75 g :93.75 g :93.75 g :93.75 g 1 1 f f 1 1 1 1 1 1 f f f f f f 0.05 0.35 0.25 :20.00 r:3.13 :50.00 r:1.25 :100.00 r:0.63 :150.00 r:0.42 :200.00 r:0.31 :240.00 r:0.26 :320.00 r:0.20 :440.00 r:0.14 3 3 3 3 3 3 3 3 time [s] 0.15 :62.50 g :62.50 g :62.50 g :62.50 g :62.50 g :62.50 g :62.50 g :62.50 g 1 1 f f 1 1 1 1 1 1 f f f f f f 0.05 0.35 0.25 :20.00 r:1.56 :50.00 r:0.63 :100.00 r:0.31 :150.00 r:0.21 :200.00 r:0.16 :240.00 r:0.13 :320.00 r:0.10 :440.00 r:0.07 3 3 3 3 3 3 3 3 time [s] 0.15 :31.25 g :31.25 g :31.25 g :31.25 g :31.25 g :31.25 g :31.25 g :31.25 g 1 1 1 1 1 1 1 1 f f f f f f f f 0.05 0.35 0.25 :20.00 r:0.63 :50.00 r:0.25 :100.00 r:0.13 :150.00 r:0.08 :200.00 r:0.06 :240.00 r:0.05 :320.00 r:0.04 :440.00 r:0.03 3 3 3 3 3 3 3 3 time [s] 0.15 :12.50 g :12.50 g :12.50 g :12.50 g :12.50 g :12.50 g :12.50 g :12.50 g 1 1 f f 1 1 1 1 1 1 f f f f f f 0.05

1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3 1.1 0.9 0.7 0.5 0.3

N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco N] [ Force Sarco µ µ µ µ µ µ µ µ

Figure A.1: The force response of extrafusal ﬁber under f1 and g3 variation 29 0.4 0.3 0.2 time [s] 0.1 C:0.1591 h: 5.0 C:0.1591 h: 7.5 C:0.1591 h:60.0 C:0.1591 h:30.0 C:0.1591 h:20.0 C:0.1591 h:15.0 C:0.1591 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0796 h: 7.5 C:0.0796 h: 5.0 C:0.0796 h:60.0 C:0.0796 h:30.0 C:0.0796 h:20.0 C:0.0796 h:15.0 C:0.0796 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0696 h: 5.0 C:0.0696 h: 7.5 C:0.0696 h:60.0 C:0.0696 h:30.0 C:0.0696 h:20.0 C:0.0696 h:15.0 C:0.0696 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0497 h: 5.0 C:0.0497 h: 7.5 C:0.0497 h:60.0 C:0.0497 h:30.0 C:0.0497 h:20.0 C:0.0497 h:15.0 C:0.0497 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0398 h: 7.5 C:0.0398 h: 5.0 C:0.0398 h:60.0 C:0.0398 h:30.0 C:0.0398 h:20.0 C:0.0398 h:15.0 C:0.0398 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0298 h: 5.0 C:0.0298 h: 7.5 C:0.0298 h:60.0 C:0.0298 h:30.0 C:0.0298 h:20.0 C:0.0298 h:15.0 C:0.0298 h:150.0 0 0.4 0.3 0.2 time [s] 0.1 C:0.0199 h: 7.5 C:0.0199 h: 5.0 C:0.0199 h:60.0 C:0.0199 h:30.0 C:0.0199 h:20.0 C:0.0199 h:15.0 C:0.0199 h:150.0 0 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5

N] [ Force Sarc N] [ Force Sarc N] [ Force Sarc N] [ Force Sarc N] [ Force Sarc N] [ Force Sarc N] [ Force Sarc µ µ µ µ µ µ µ

Figure A.2: The force response of extrafusal ﬁber under C and H variation 30 A. Raw Data 0.4 0.3 0.2 time [s] Vel:1.40 L_0/s Vel:0.20 L_0/s Vel:0.60 L_0/s Vel:1.00 L_0/s 0.1 0 0.4 0.3 0.2 time [s] Vel:0.15 L_0/s Vel:0.50 L_0/s Vel:0.90 L_0/s Vel:1.30 L_0/s 0.1 0 0.4 0.3 0.2 time [s] Vel:0.10 L_0/s Vel:0.40 L_0/s Vel:0.80 L_0/s Vel:1.20 L_0/s 0.1 0 0.4 0.3 0.2 time [s] Vel:1.10 L_0/s Vel:0.05 L_0/s Vel:0.30 L_0/s Vel:0.70 L_0/s 0.1 0 1 1 1 1

1.2 0.8 0.6 0.4 0.2 1.2 0.8 0.6 0.4 0.2 1.4 1.2 0.8 0.6 0.4 0.2 1.4 1.2 0.8 0.6 0.4 0.2

Sarcomere Force [uN] Force Sarcomere [uN] Force Sarcomere [uN] Force Sarcomere [uN] Force Sarcomere

Figure A.3: The force response and elongation of extrafusal ﬁber under different velocity ramps, from 0.1914 L0\s to 1.34 L0\s at 0.1L0 ∆L. B

Notes on Derivation of Various Parameters

B.1. On Selection of cext and kext The stiffness of the force transducer which is modeled as a spring-damper in parallel is determined by the length error of the system. The length error is the determining factor for stiffness . To avoid force jerks (as seen for the low kext values in ﬁgure B.2), error in length was considered. It is noted that real life transducers do not have instantaneous response. Hence a degree of length error is acceptable. kext is selected based on the error of the the transducer element (4 values different by an order of magnitude), then cext selected such that the system would be critically damped, under-damped and over-damped.

31 32 B. Notes on Derivation of Various Parameters 0.4 0.3 : 270000.00 : 270000.00 : 270000.00 : 270000.00 : 270000.00 ext ext ext ext ext 0.2 time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k 0.1 ext ext ext ext ext c c c c c 0 0.4 0.3 : 54000.00 : 54000.00 : 54000.00 : 54000.00 : 54000.00 ext ext ext ext ext 0.2 time [s] : 0.046 k : 0.147 k : 0.465 k : 1.470 k : 3.286 k 0.1 ext ext ext ext ext c c c c c 0 0.4 0.3 : 5400.00 : 5400.00 : 5400.00 : 5400.00 : 5400.00 ext ext ext ext ext 0.2 time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k 0.1 ext ext ext ext ext c c c c c 0 0.4 0.3 : 540.00 : 540.00 : 540.00 : 540.00 : 540.00 ext ext ext ext ext 0.2 time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k ext ext ext ext ext 0.1 c c c c c 0 0.4 0.3 : 54.00 : 54.00 : 54.00 : 54.00 : 54.00 ext ext ext ext ext 0.2 time [s] : 0.046 k : 0.147 k : 0.465 k : 1.470 k : 3.286 k ext ext ext ext ext 0.1 c c c c c 0 0 0 0 0 0

−1 −2 −3 −4 −5 −6 −7 −1 −2 −3 −4 −5 −6 −7 −1 −2 −3 −4 −5 −6 −7 −1 −2 −3 −4 −5 −6 −7 −1 −2 −3 −4 −5 −6 −7

ext 10 ext 10 ext 10 ext 10 ext 10

)[−] (L Log )[−] (L Log )[−] (L Log )[−] (L Log )[−] (L Log

Figure B.1: Plot of log of length error (length of the force transducer) for different cext and kext values. B.1. On Selection of cext and kext 33 0.3 0.25 : 270000.00 : 270000.00 : 270000.00 : 270000.00 : 270000.00 0.2 ext ext ext ext ext time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k ext ext ext ext ext 0.15 c c c c c 0.1 0.3 0.25 : 54000.00 : 54000.00 : 54000.00 : 54000.00 : 54000.00 0.2 ext ext ext ext ext time [s] : 0.046 k : 0.147 k : 0.465 k : 1.470 k : 3.286 k ext ext ext ext ext c c c c c 0.15 0.1 0.3 0.25 : 5400.00 : 5400.00 : 5400.00 : 5400.00 : 5400.00 ext ext ext ext ext 0.2 time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k ext ext ext ext ext c c c c c 0.15 0.1 0.3 0.25 : 540.00 : 540.00 : 540.00 : 540.00 : 540.00 ext ext ext ext ext 0.2 time [s] : 3.286 k : 0.046 k : 0.147 k : 0.465 k : 1.470 k ext ext ext ext ext c c c c c 0.15 0.1 0.3 0.25 : 54.00 : 54.00 : 54.00 : 54.00 : 54.00 ext ext ext ext ext 0.2 time [s] : 0.046 k : 0.147 k : 0.465 k : 1.470 k : 3.286 k ext ext ext ext ext c c c c c 0.15 0.1 1 1 1 1 1

1.1 0.9 0.8 0.7 0.6 0.5 1.1 0.9 0.8 0.7 0.6 0.5 1.2 1.1 0.9 0.8 0.7 0.6 0.5 1.2 1.1 0.9 0.8 0.7 0.6 0.5 1.2 1.1 0.9 0.8 0.7 0.6 0.5

N] [ sarcomeres the of Force N] [ sarcomeres the of Force N] [ sarcomeres the of Force N] [ sarcomeres the of Force N] [ sarcomeres the of Force µ µ µ µ µ

Figure B.2: Plot of the force response to a ramp perturbation for different cext and kext values.

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