Faculty of Electrical Engineering Mechatronics Engineering

Muscle Spindle Modeling - A Tutorial

Professor: Dr. Mehdi Delrobaei

Student Name: Sadaf Yari

January-2019

Muscle Spindle Every day we move around endlessly, walking, exercising, etc. We perform these tasks without thinking about it. In fact, for the human body to make the simplest motion, such as lifting an arm, requires the human brain to perform a dozen calculations and control many complex procedures. Some muscles have to contract while others to expand. The final goal is reached through careful control of the muscles via feedback which provides the brain with information on the current situation of the different parts of the body.

One such feedback mechanism is proprioception which contains information on the current location of body parts and the situation they are in. This is made possible by sensors, called proprioceptors, located in the muscles. Examples of proprioceptors include the muscle spindle and the Golgi tendon organ. The former provides length information of the muscle and the latter detects changes in the muscle stretch. Such information is useful for the brain when attempting to control the motion of the body parts. In this tutorial we will focus on the muscle spindle. First, a detailed anatomical and physiological description of the muscle spindle’s structure and function is given. In this part it is explained where exactly the muscle spindle is located and what it does. Then, a mathematical model, which is currently accepted generally, is discussed.

What is muscle spindle? Muscles are the organs that cause movement in our body. Each motion in the body, wether volunatry or not, is caused by the contraction or release of a muscle. There are three types of muscles in the human body:

1. Skeletal muscles: Movements that are performed consiously, are carried out by these muscles. Most of them are attached to two bones across a joint, so the muscle serves to move parts of those bones close to each other.

2. Smooth muscles: the muscles found inside organs such as the intestines as well as blood vessels and contract to move substances through the organ. Movements casued by this type of muscle is involuntary.

3. Cardiac muscle: this muscle is found only in the heart and has characteristics of both skeletal and smooth muscles.

Figure 1: Types of muscles

Our focus here is on skeletal muscles that provide us the ability to move our body parts voluntarily. The structure of such a muscle can be seen in the figure 3.

Figure 2: The connection of a muscle to bones causing them to move when it is contracted or released

Figure 3: The structure of skeletal muscle

The entire muscle, as well as the individual cells, is wrapped in collagen. Near the end the collagen merges to form the tendons, which attach the muscle to the bone. It is through this connective tissue that the force generated by the individual cells is transmitted to the bone. A group of muscle cells are bundled together by collagen to form a fascicle. Since muscle cells are elongated and cylindrical, each muscle cell is usually called a muscle fiber. In skeletal muscle, the muscle fibers are very large, multinucleated, and up to several millimeters in length. A skeletal muscle is attached to two bones connected by a joint. This system acts as a kinematic chain in which the links are moved through the contraction of the muscle. In order to control this system, the brain needs information on the current positions of the individual links and therefore the length information of the muscle fibers. This information is provided through several feedback sources.

One of the available feedback mechanisms is proprioception. Literally, proprioception means “sense of self” and it is described as our sense of the relative positions of our body parts and the strength of effort being employed in movements. Proprioception is provided by proprioceptors, which are sensors nested in the skeletal muscles and the tendons. The muscle spindle is one type of proprioceptor that provides information about changes in muscle length.

Muscle spindles are small sensory organs that are enclosed within a capsule. They are found throughout the body of a muscle, in parallel with extrafusal fibers. Within a muscle spindle, there are several small, specialized muscle fibers known as intrafusal fibers. Intrafusal fibers have contractile proteins (thick and thin filaments) at either end, with a central region that is devoid of contractile proteins. The central region is wrapped by the sensory dendrites of the muscle spindle afferent. When the muscle lengthens and the muscle spindle is stretched, this opens mechanically-gated ion channels in the sensory dendrites, leading to a receptor potential that triggers action potentials in the muscle spindle afferent.

Figure 4: Muscle spindle

Figure 5: Muscle spindle structure

Modelling In this section we give a, as much as possible, simple model of the spindle which consists of mathematical elements that are closely related to the anatomical parts of the spindle and show the same physiological properties. We begin by giving a general description of the spindle model. We then continue by examining the individual blocks. Next the model for an intrafusal fiber is considered and the underlying equations are explained in detail. For this purpose, first, the fusimotor activation is addressed. Secondly the mechanics of stretch within the intrafusal fiber is analyzed. Finally, sensory transduction from stretch to afferent endings is formulized. Ultimately, we describe the afferent firing model that deals with nonlinear summation between the intrafusal fibers’ transduction regions.

The spindle model Three types of specialized fibers, known as intrafusal fibers, constitute the muscle spindle. These are the long nuclear bag 1 and bag 2 fibers and the shorter chain fibers which are in parallel with the extrafusal fibers (the fibers of the muscle itself). A typical spindle consists of one bag 1, one bag 2, and about 4 to 11 chain fibers. Using these fibers, the spindle provides feedback in the form of nerve impulses, a sequence of electrical potential spikes, to the central nervous system.

Figure 3: Intrafusal and extarfusal muscles incorporation with motor neuron

Figure 4: Three types of intrafusal fibers

The dynamic range of the nerve impulses or action potentials fired from the nerve to activate the muscle is limited. Nevertheless, the spindle should be capable of sensing and accurately encoding the length and velocity over a wide range of kinematic conditions. In order to be able to accomplish this, the sensitivity to length or velocity has to be shifted by means of specialized fusimotor efferents (γ motorneurons). These efferents can be divided into dynamic fusimotor efferent and the static fusimotor efferent, which signal the transitions in phase and the intended movement of the motor unit to the CNS respectively. The bag 1 fiber is primarily sensitive to the velocity of the spindle, i.e. its rate of stretching; hence it has dynamic fusimotor efferent endings located on it. The bag 2 and chain fibers are modulated via static fusimotor efferents and contribute mainly to length sensitivity. The feedback to the CNS containing length and velocity information from the muscle is collected via the afferent endings which are nerve fibers wound around the intrafusal fibers. The primary afferent signal is provided through group Ia fibers which have their endings attached to the region in the middle of all three types of the intrafusal fibers. The secondary endings, located a little further from the middle of only the bag 2 and chain fibers, generate the secondary afferent signal which corresponds primarily to information on the length of the muscle.

Now, we are going to construct a model of the spindle based on the anatomical and physiological descriptions explained above. The model is composed of three intrafusal fiber models corresponding to a bag 1, bag 2 and a chain fiber. At its input, the system receives the fascicle length (L) and two fusimotor inputs, the static and the dynamic fusimotor, in the form of potential spikes. From these inputs the model generates two output signals which are to be fed to the CNS via the primary and the secondary afferents.

Figure5: schematics of the overall spindle model

Figure6: spindle model consists of 3 intrafusal fiber models; it receives 3 inputs (fascicle

length, in terms of optimal length L0, and static and dynamic fusimotor drives) to produce primary and secondary afferent firing

The three intrafusal models are similar in the constituent blocks but have different coefficients to account for the differences in their physiological behaviors. Each intrafusal fiber model responds to two inputs: the fascicle length (L; in units of L0, which represents the optimal muscle fascicle length) and the relevant fusimotor drive (in the case of bag1 fiber it is dynamic fusimotor drive (γdynamic), whereas in the case of bag2 it is chain static fusimotor drive (γstatic)). The outputs of the intrafusal fiber models are combined to produce the final outputs of the model i.e. the primary (Ia) and secondary (II) afferent activity. In the coming sections we will first explain the intrafusal fiber model in more detail. We do so by first considering the fusimotor activation. Next the mechanics of stretch within the intrafusal fiber is analyzed. Finally it is clarified how stretch information is converted to signals for the afferent endings. In the second major section it is explained how the intrafusal fibers’ outputs are nonlinearly summed to yield the afferent firing model.

The intrafusal fiber model As mentioned earlier, the intrafusal fiber models are the same in structure but to account for the fibers’ different physiologies the weights of the model parameters in the calculations varies (see Table 1). Each intrafusal fiber model is divided into a sensory and polar region. The sensory region has afferent endings wound around it. Stretch of this region causes distortion in the afferent endings, resulting in increase in the rate of action potential firing. This region is modeled with a spring whose stretch is proportional to the afferent firing. The rest of the intrafusal fiber on either side of the sensory region is called the polar region, constituting essentially a striated muscle fiber innervated by fusimotor endings; for simplicity, the two polar regions of the biological spindle are combined into one polar region for the model. The polar region is modeled by a passive spring in parallel with a contractile element which further consists of an active force generator and a damping element. The contractile element was designed to represent the properties of the spindle that change in response to fusimotor activation.

Figure7: Intrafusal fiber model. All intrafusal fibers consist of polar and sensory zones with qualitatively identical mechanical components. For each fiber model the stretch in the sensory and polar regions is calculated to determine its contribution to the firing of each afferent. Model parameters for each intrafusal fiber type is provided in Table 1

Table 1: Spindle model parameters

Parameter Parameter Definition Bag1 Bag2 Chain KSR Sensory region spring constant [FU/L0] 10.4649 10.4649 10.4649 KPR Polar region spring constant [FU/L0] 0.1500 0.1500 0.1500 M Intrafusal fiber mass [FU/(L0/s2)] 0.0002 0.0002 0.0002

B0 Passive damping coefficient [FU/(L0/s)] 0.0605 0.0822 0.0822

β1 Coef. of damping due to dyn. fusimotor input [FU/(L0/s)] 0.2592 * β2 Coef. of damping due to stat. fusimotor input [FU/(L0/s)] -0.0460 -0.0690

Г1 Coef. of force generation due to dyn. fusimotor input [FU] 0.0289

* Г2 Coef. of force generation due to stat. fusimotor input [FU] 0.0639 0.0954

CL Coef. of asymmetry in F–V curve during lengthening 1 1 1

Cs Coef. of asymmetry in F–V curve during shortening 0.4200 0.4200 0.4200 X Percentage of the secondary afferent on sensory region 0.7** 0.7** SR LN Sensory region threshold length (L0) 0.0423 0.0423 0.0423 PR ** ** LN Polar region threshold length (L0) 0.89 0.89 G Term relating the sensory region’s stretch to afferent firing 20000 10000(7250) 10000(7250)

a Nonlinear velocity dependence power constant 0.3 0.3 0.3 R Fascicle length below which force production is zero (L0) 0.46 0.46 0.46 SR L0 Sensory region rest length (L0) 0.04 0.04 0.04 PR L0 Polar region rest length (L0) 0.76 0.76 0.76

Lsecondary Secondary afferent rest length (L0) 0.04 0.04 τ Low-pass filter time constant (see APPENDIX 1) (s) 0.149 0.205 freq Constant relating the fusimotor frequency to activation 60 60 90 p Power constant relating the fusimotor frequency to 2 2 2 activation FU is force unit. Values in FUs are arbitrary because they can be scaled by a constant without a change in the model’s behavior. *Chain fiber values that needed to be scaled from bag2 values because of different fusimotor saturation frequencies for two fibers (bag2 values × 1.5). Values in parentheses are values used to model the secondary afferent. ** Parameters that need to be adjusted to capture the variability in secondary afferent response across different spindles. PR Note that the parameter values below the LN parameter were extracted directly from the experimental literature.

FUSIMOTOR ACTIVATION Fusimotor activation is done by the application of action potentials. Hence the fusimotor inputs of the model are in the form of a sequence of potential spikes with a specific frequency (γdynamic or γstatic in pulses per second). The frequency of these action potentials has to be converted to an activation level (fdynamic or fstatic, defined within range 0 to 1) in order to be applied to the mechanical building blocks in the model. To do this an equation similar to the Hill equation in biochemistry, which is used to describe the fraction of a macromolecule saturated by ligand as a function of the ligand concentration, is utilized in order to account for the saturation effects that take place at high fusimotor stimulation frequencies.

In addition, the model incorporates the different temporal properties of intrafusal fiber responses that were measured previously for individual intrafusal fibers in response to step changes in fusimotor activation. These different temporal properties are thought to arise from differences in the spread of activation in twitch muscle fibers that propagate action potentials (including chain fibers) versus tonic muscle fibers where synaptic depolarization spreads electrotonically (including bag fibers), as well as being related to differences in calcium kinematics. To model these differences, low-pass filters between the fusimotor inputs and the equivalent activation levels were introduced for the two relatively slow bag intrafusal fibers.

The following equations were used to convert the actual fusimotor frequency (dynamic or static) to an equivalent activation level (fdynamic or fstatic) for three types of intrafusal fibers:

푃 훾푑푦푛푎푚𝑖푐 푝 −푓푑푦푛푎푚𝑖푐 훾푃 −푓푒푟푞 푑푓푑푦푛푎푚𝑖푐 푑푦푛푎푚𝑖푐 푏푎푔1 = for bag1 fiber, which saturates at about 100 pps 푑푡 휏

푃 훾푠푡푎푡𝑖푐 푝 −푓푠푡푎푡𝑖푐 훾푃 −푓푒푟푞 푑푓푠푡푎푡𝑖푐 푠푡푎푡𝑖푐 푏푎푔2 = for bag2 fiber, which saturates at about 100 pps (1) 푑푡 휏

푃 훾푠푡푎푡𝑖푐 푓푠푡푎푡푖푐 = 푃 푝 for chain fiber, which saturates at about 150 pps 훾푠푡푎푡𝑖푐+푓푒푟푞푐ℎ푎𝑖푛

MECHANICS OF SENSORY AND POLAR REGIONS In the spindle model, the sensory region of each intrafusal fiber is represented as an elastic element with spring constant KSR. The tension within this region is equal to

푆푅 푃푅 푆푅 푇 = 퐾 × [(퐿 − 퐿 ) − 퐿0 ] (2)

PR SR where L is fascicle length, L is polar region length, and L0 is the unloaded sensory region length, all PR in units of L0. The polar region is modeled as a spring with a spring constant K and a parallel contractile element that consists of an active force generator and a damping element, both of whose properties are modulated by fusimotor input. The tension within this region is equal to

푎 푃푅 푃푅 푃푅̇ ̈ 푃푅̇ 푃푅 푃푅 푃푅 푇 = 푀 × 퐿 + 훽 × 퐶 × (퐿 − 푅) × 푠푖푔푛(퐿 ) × 푎푏푠(퐿 ) + 퐾 × (퐿 − 퐿0 ) + Г (3)

Where

훽 = 훽0 + 훽1 × 푓푑푦푛푎푚푖푐 + 훽2 × 푓푠푡푎푡푖푐 (4)

And

Г = Г1 × 푓푑푦푛푎푚푖푐 + Г2 × 푓푠푡푎푡푖푐 (5)

M refers to the intrafusal fiber mass required for computational stability in a series-elastic system with PR velocity-dependent contractility. L0 is the polar region’s rest length. β represents the polar region’s damping term; increases in β result in increases in the velocity sensitivity of the primary afferent, which plays an important role in modulating the spindle’s behavior during dynamic fusimotor stimulation of bag1 fiber (β1) (the only intrafusal fiber receiving dynamic fusimotor drive).

By contrast, static fusimotor activation produces a small decrease in β (note that β2 is negative). Γ is defined as the active-state force generator term; increases in Γ result in an increase in the bias activity of the dependent afferent. Static fusimotor input causes a sustained, strong contraction within the bag2 and chain polar regions (Γ2 * fstatic), producing a stretch in the sensory region and a bias in the afferent activity.

Dynamic fusimotor input produces a similar but much weaker effect (Γ2 * fdynamic). The model incorporates the experimentally observed nonlinear velocity dependency of the spindle’s afferent response and was modeled with the velocity power term ( 퐿̇푃푅 )a. C is a constant describing the experimentally observed asymmetric effect of velocity on force production during lengthening and shortening. Although this asymmetry has been observed directly only in extrafusal striated muscle, we have assumed its existence also in the case of the intrafusal fiber’s contractile polar region. C was set to unity during polar region lengthening (C = CL = 1) and to CS during shortening (C = CS). Finally, the model incorporates the length dependency of the force–velocity relationship (term LPR - R), where an increase in fascicle length results in increased slope of the force–velocity relationship for slow to moderate velocities. This effect, observed in extrafusal fibers, is believed to result from the influence of myofilament lattice spacing on cross-bridge kinetics. Under most physiological conditions the sarcomere length of the intrafusal fiber’s polar region tends to follow that of extrafusal fibers, so the extrafusal fiber measurements were used to estimate the length (R) of the polar region of intrafusal fibers for this effect (assuming that length changes in sensory region are minor comparing to those in the polar region).

INTRAFUSAL FIBER CONTRIBUTIONS TO AFFERENT FIRING Firing in this context means the transmission of an action potential by a neuron. Action potentials are the main means of cell-to-cell communication and occur when the electric potential of a specific part on the axon rises and falls very quickly. They are generated by special types of voltage-gated ion channels embedded in a cell’s plasma membrane. These channels are shut when the membrane potential is near the (negative) resting potential of the cell, but they rapidly begin to open if the membrane potential increases to a precisely defined threshold voltage, depolarizing the transmembrane potential.

More specifically, when the channels open, Na+ ions move into the axon of the neuron cell, causing depolarization. This marks the beginning of the action potential. This depolarization travels through the axon by the consequent depolarization of adjacent segments. Repolarization occurs when the K+ channels open and K+ ions move out of the axon, creating a change in polarity between the outside of the cell and the inside. In the afferents attached to the intrafusal fibers, stretching of the afferent endings causes the ion channels to open which results in depolarization. Hence to obtain the afferent firing rate, the stretch in the afferent endings has to be evaluated first. In our spindle model, the stretches in the sensory and polar regions are independently calculated for each intrafusal fiber to determine their potential contributions to afferent firing. Because tensions in equations 2 and 3 are the same, the sensory region’s equation for tension (equation 2) can be rearranged to express polar region length (LPR) in terms of tension (T) and fascicle length (L). This polar region length is then substituted into equation 3 to obtain a second- order differential equation of tension in terms of fascicle length.

푎 퐾푆푅 푇̇ 푇̇ 푇 푇 푇̈ = × [퐶 × 훽 × 푠푖푛푔 (퐿̇ − ) × 푎푏푠 (퐿̇ − ) × (퐿 − 퐿푆푅 − − 푅) + 퐾푃푅 × (퐿 − 퐿푆푅 − − 퐿푃푅) + 푀 퐾푆푅 퐾푆푅 0 퐾푆푅 0 퐾푆푅 0 푀 × 퐿 +̈ Г − 푇] (6)

Because the primary afferent endings are located on the sensory regions of all three intrafusal fibers, the stretch in the sensory region of each intrafusal fiber is calculated (T/KSR). Once the afferent endings SR are stretched passed a certain sensory region length (sensory region threshold length: LN ) the ion channels open and depolarization/impulse generation takes place. The stretch above the threshold length is scaled by a constant G (the term that relates the stretch of the intrafusal fiber’s sensory region to primary afferent firing) to obtain the intrafusal fiber’s contribution to the activity of the primary afferent (before nonlinear intrafusal fiber firing summation between bag1 and combined bag2 and chain fiber models)

푇 퐴푓푓푒푟푒푛푡_ = 퐺 × [ − (퐿푆푅 − 퐿푆푅)] (7) 푃표푡푒푛푡푖푎푙푃푟𝑖푚푎푟푦 퐾푆푅 푁 0

Contrary to the primary afferent endings, the secondary afferent transduction zones are located more eccentrically, straddling both sensory and polar regions of bag2 and chain intrafusal fibers. Therefore, action potential generation reflects the stretch in both sensory and polar regions

퐿 퐿 푠푒푐표푛푑푎푟푦 푇 ( 푆푅 푆푅) ( ) 푠푒푐표푛푑푎푟푦 푇 푆푅 퐴푓푓푒푟푒푛푡푃표푡푒푛푡푖푎푙푠푒푐표푛푑푎푟푦 = 퐺 × [ 푆푅 × [ 푆푅 − 퐿푁 − 퐿0 ] + 1 − 푋 × 푃푅 × (퐿 − 푆푅 − 퐿0 − 퐿0 퐾 퐿0 퐾 푃푅 퐿푁 )] (8)

X represents the percentage of the secondary afferent located on the sensory region, which can vary among spindles. The stretch within the part of the secondary afferent that is located on the sensory region SR is obtained by multiplying X by sensory region stretch above the sensory region threshold length (LN ) and normalizing it by the ratio of the secondary afferent rest length (Lsecondary) and sensory region rest SR length (L0 ). The stretch within the part of the secondary afferent located on the polar region is similarly obtained. The polar region length at and above which secondary afferent sensory endings are stretched PR is defined as the polar region threshold length (LN ). Once the stretches of the secondary afferent portions that are located on both sensory and polar regions are obtained, they are summed together and multiplied by G to obtain the intrafusal fiber contribution to secondary afferent firing.

Afferent firing model To produce the final outputs, the outputs of the individual fiber model blocks are combined as follows:

The secondary afferent output is simply the sum of the outputs of bag 2 and chain intrafusal fiber models. However, the primary afferent output is more complex due to an experimentally observed effect called partial occlusion. This occurs during simultaneous static and dynamic fusimotor stimulation which produces an afferent response that is greater than the larger of the individual responses (during either static or dynamic fusimotor stimulation) but smaller than their sum. Though the mechanism responsible is yet to be fully know, it has been well described experimentally and thus, it is incorporated in the model. To obtain the primary afferent, a nonlinear summation is carried out between the bag 1 and the combined bag 2 plus chain intrafusal fiber outputs, inorder to account for the partial occlusion. More specifically, the driving potentials produced by bag1 and combined bag2 and chain intrafusal fibers are compared and the larger of the two plus a fraction (S) of the smaller are summed to obtain the primary afferent firing.

Model verification The figures below shows experimental results on the primary afferent output and compares it with the model’s response to similar inputs during ramp-and-hold and triangular stretches. Several conditions are tested including situations in which the no fusimotor activation is present. It is evident that the model closely agrees the experimental data.

Figure8 : Spindle model performance during 6-mm whole muscle ramp stretches. Primary afferent response at 3 different velocities (whole muscle stretches at 5, 30, and 70 mm/s; fascicle stretches at 0.11, 0.66, and 1.55L0/s; fascicle length changes 0.95–1.08L0) were performed in the absence of fusimotor stimulation (A, B, C), during constant dynamic fusimotor stimulation at 70 pps (D, E, F), and during constant static fusimotor stimulation at 70 pps (G, H, I). Solid thin lines represent model output; experimental data are shown as dots _. Secondary afferent responses at 3 different velocities (whole muscle stretches at 5, 30, and 50 mm/s; fascicle stretches at 0.11, 0.66, and 1.12L0/s) are compared with the secondary afferent activity from 2 different muscle spindles (shown as+, . ) originating from the same experimental preparation (J, K, L).

Figure9: Model’s ability to capture primary afferent activity during triangular stretches. Whole muscle (8 mm) stretches (fascicle length changes 0.90–1.08L0) at 8 mm/s (fascicle stretches at 0.18L0/s) were performed during constant dynamic (A, B, C) or static (D, E, F) fusimotor stimulations at 3 different frequencies (35, 70, and 200 pps). Solid thin lines represent model output; experimental data are shown as dots.

Summary Muscle spindles are proprioceptors located in the muscle. Each spindle provides the brain with information on the length of a single fascicle and its rate of change. A typical muscle spindle consists of three types of special fibers called intrafusal fibers, which lie in parallel with the fibers of the muscle. These fibers combine to generate static and dynamic information of the muscle length. This information is fed back to the brain via the primary and the secondary afferents. The mathematical model of the muscle spindle contains three separate intrafusal fiber models. The outputs of these are summed non- linearly to produce the output action potentials.

When modelling the intrafusal fibers, one has to distinguish between a sensory and a polar region. The former has afferent endings wound around it and produces the action potentials that are transmitted to the brain. The latter has fusimotor endings which are used to control the spindle’s sensitivity. Two fusimotor drives are provided to the model as inputs. The other input is the fascicle length.

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