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).