Comparing Muscle Spindle Afferent Models

Stephanie Hachem George Mason University, Thomas Jefferson High School for Science and Technology

This paper was originally included in the 2020 print publication of the Teknos Science Journal.

Abstract

Proprioception, the internal sense of where body parts are relative to each other, is essential for most activities of daily living. However, modern prosthetics lack this sense, and it is often difficult or impossible to perform tasks that require hand-eye coordination. Thus, upper extremity prosthetics can become a nuisance or burden that amputees frequently abandon at home [1]. This project aimed to create a naturally-functional prosthetic which can provide proprioception by using computational models of muscle spindle afferents and experimental data. The best computational model was then used to predict the voltages that should be provided to specific afferent nerves in a residual limb.

Two muscle spindle models were compared using measured afferent and muscle length data from cats and humans, digitally extracted from figures in ten published articles. Both models implemented the same formulas in either MATLAB or Python, taking muscle length as an input and providing primary and/or secondary afferent output. Comparing the experimentally measured and the predicted afferents, the more recent Python model was found to provide more accurate afferent output.

Keywords: muscle spindle afferents; muscle spindle model; proprioception; proprioceptor; Neural Simulation Tool

Background

Muscle spindles: Proprioception requires the coordinated output of structures contained within our muscles. Muscle spindles are sacks of sensory fibers situated inside capsules that are parallel to muscle fibers in a muscle belly. This arrangement allows the spindles and the muscle to be stretched in tandem. Each spindle holds several such sensory fibers, which fall into three categories: nuclear chain fibers and static nuclear bag fibers that monitor muscle length, and dynamic nuclear bag fibers that primarily monitor the velocity of change in length, as well as monitoring length. Sensory information encoded by these fibers is transmitted by two main types of nerves: primary (Ia) afferent nerves, which are wrapped around all three fiber types to output length and velocity information to the spinal cord and brain, and secondary (II) afferent nerves, which wrap around the nuclear chain and static nuclear bag fibers to output length. Thus, when a muscle is stationary, both the Ia and II afferents are sending signals encoding the muscle's length, but when a muscle is being stretched, Ia output increases drastically to indicate the occurrence and velocity of the stretch while II output increases proportionally to the actual stretch.

Microneurography: Microneurography is a safe and painless technique that involves the insertion of electrode-tipped needles into muscles to monitor nerve activity, including proprioceptor activity. Actual muscle activity is typically recorded through surface EMG, the technique of placing electrodes on the skin to monitor muscle contraction.

Jones, Wessberg, and Vallbo [11] recorded and analyzed microneurographic recordings of proprioceptor response in human wrist-to-wrist movements to determine whether muscle length and bodily movement direction are encoded in afferent output. This study revealed that a more accurate velocity output of spindles occurs in muscles that constantly counteract gravity, while spindles in muscles with a smaller role in counteracting gravity were shown to be equally sensitive to velocity and stretch in accordance with the roles of Ia and II afferents. It was concluded that despite the small amount of data gathered, the wrist joint position could be predicted well from spindle output.

Given the consistency of spindle afferent data across several studies, including the one mentioned above, the possibility of predicting afferent activity arose. In response, several studies were published containing mathematical models that described spindle input based on movement and excitation of the muscle the spindle was situated in. In particular, the muscle spindle model of Mileusnic, Brown, Lan, and Loeb [17] has yielded remarkable accuracy in predicting spindle input based on muscle dynamics. The model was created to closely resemble the physiological elements of spindles and was validated by experimental data for its ability to reproduce experimentally-observed spindle characteristics and behavior under various conditions. This model remains one of the most widely used and accurate spindle afferent output prediction models.

The muscle spindle model by Mileusnic et al. [17] is composed of mathematical elements based off of spindle physiology and exhibits properties observed in spindles. The model was validated using afferent output recorded in cat muscles; models validated over the afferent output of cat muscles have predictive value for the afferent output of human muscles [12]. The types of spindle fibers are represented by similar equations but differ in coefficient values and terms in order to reflect nuclear chain, static nuclear bag, and dynamic nuclear bag fiber afferent behavior. Each fiber's equation takes the input of muscle fascicle length and either static or dynamic muscle spindle motor neuron efferent input. Nuclear chain and static nuclear bag fibers receive static efferent input corresponding to the two fiber types’ afferent output, while dynamic nuclear bag fibers receive dynamic efferent input corresponding to the fiber types’ dynamic afferent output. At the sensory regions of spindle fibers, tension of the spindle fiber's non-sensory regions on either side is converted into afferent output. The sensory regions were modeled with equations defining the properties of an elastic material, or element, based on the material's surroundings. Then, the non-sensory regions on either side of the sensory region were simplified to a single non-sensory region, located in series to the sensory region in the spindle model. This single non-sensory region was modeled with equations defining a spring element in parallel with a contractile element. Properties of elements, outputted by the equations defining elements in the spindle, were input in equations defining mechanical and electrophysical properties of spindles; these properties can predict afferent output. These algorithms were combined with statistical analysis algorithms.

To compare the predictive power of spindle input algorithms, Malik, Jabakhanji, and Jones [12] analyzed such research using MATLAB and Simulink software. Specifically, they juxtaposed the ability of six algorithms from the literature to predict actual spindle input results based on kinematics for a set of human arm data from a past study [11]. The three algorithms with best predictability were then used with wrist kinematic data generated through simulating random rotations around the wrist joint axes with a tendon displacement model. The study concluded that muscle spindle models originally made with data from cat hindlegs have predictive power for human muscle spindles.

This study used a spindle model by Vannucci, Falotico, and Laschi [26] that is open source, has been verified and applied in robotic and biological applications, and is accessible through the Neural Simulation Tool [9]. The spindle model’s algorithms are based on simplifications which attempt to minimize loss in prediction accuracy of algorithms presented by Mileusnic et al. [17].

This study also used open source muscle spindle afferent output files from cat hindlegs, measured by Blum et al. [2]. Their results suggested that muscle spindle afferent output is a function of whole-muscle tendon force, not muscle length as previous models have assumed.

Methods

Note that from this point on, the word "afferent(s)" is always used to refer to muscle spindle afferent firing rate, and OFL refers to optimal fiber length.

Required software: MATLAB 2017b version 9.3.0 or above was installed on a Windows 64-bit platform, and the Neural Simulation Tool and dependencies (Python 3) were installed on a Oracle Linux 7 Red Hat Enterprise operating system (this study used Anaconda 3 and Oracle VM VirtualBox version 5.2.12). The muscle spindle module was installed on the VM [27]. IPlotDigitizer version 2.6.8 was installed on Windows [10].

Data sources: Eleven sources of data were simulated with afferent models [2, 5-7, 13-16, 18, 19]. See Table 1 for metadata of each data source.

Blum et al.'s data: Blum et al.'s [3] data was run with two afferent models: Schneider's [24] model and Vannucci et al.'s model.

Schneider's model: Muscle lengths of Blum et al.'s data were converted to a decimal of that muscle’s OFL and written to files, which were fed to Schneider's Simulink model in MATLAB. Blum et al.'s data also provided the time of every afferent action potential; this data was converted to firing rates (to be compared with Vannucci et al.'s model output of firing rates) through averaging the number of action potentials within a 0.0005-second timestep.

Vannucci et al.'s model: Muscle lengths of Blum et al.'s data, with a 0.0005-second timestep, were converted to decimals of OFLs and written to .txt files, which were fed to Vannucci et al.'s model. Vannucci et al.'s model requires the NEST software to run, which requires Linux, so the model was used on a Linux VM (see the Required Software section), and files were shared through folders using VirtualBox Guest Additions. As in Table 1, Blum et al.'s data was recorded only from Ia fibers, so only Ia fiber types were used to simulate the output, and predicted afferents were recorded with a multimeter and spike detector and written to a .dat file. The NEST kernel was reset every time a new simulation was run in order to speed up simulations, and a population size of one Ia/II fiber was used to reduce simulation time and memory needed for output files.

Digitized article data: Experimentally measured muscle lengths and afferent data was obtained by manually and automatically digitizing figures from the literature.

Manual digitization: The software PlotDigitizer version 2.6.8 was used in Windows [10]. A screenshot of each graph, zoomed in to fill the screen, was taken using the software Snipping Tool. The width or height of width/height reference lines (sometimes axes, sometimes a separate line on the side of the graph) whose unit and width/height was known, was measured in pixels with the software Paint, and used to calculate the length/height in units (not pixels) of the x and y axes (if the reference width/height was not the axes). Units were always time (s) for the x-axis, and either afferents (impulses/s, Hz), muscle length (m), or muscle length displacement (m) for the y-axis. These lengths and units were used to calibrate PlotDigitizer; points were then saved to a .csv file. See Appendix A for a detailed explanation of how to manually digitize.

Automatic digitization: Because manual digitization is time-consuming, Open Source Computer Vision Library (openCV) [4] and Python 3 were used with the Spyder IDE [21] to automatically digitize files. OpenCV converted images to grayscale and recorded the x-y coordinates of each pixel whose greyscale value was greater than some threshold; thresholds were adjusted for some figures. The length between the left-most x-coordinate and the image's left edge was subtracted from all x-coordinates to align the data to the starting time. The lengths or heights (in pixels and in units) of reference lines were used to calculate scale factors, which were used to convert aligned coordinates into units, and these converted and aligned coordinates were written to .txt files. See Appendix A for a detailed explanation of how to automatically digitize.

Vannucci et al.'s model’s digitized article data: Muscle lengths were converted to decimals of OFLs and written to .txt files, which were fed to Vannucci et al.'s model. Since many of the digitized graphs do not have constant timesteps for the data, the NEST Simulate function's timestep argument was calculated for each timestep. Ia or II fibers (both with a population of one fiber) were used for each graph depending on which type of fiber (Ia or II) was experimentally measured in the article. OFLs, which were assumed to be a muscle's resting length, were either taken from the article, taken from other articles, or were estimated (see next paragraph). Decimals of OFL were then input to Vannucci et al.'s model. The same procedures for running Vannucci et al.’s model on a Linux VM, as outlined previously, were followed.

Estimating OFLs of digitized article data: The OFLs of muscles whose muscle lengths and afferents were measured by Marasco et al. [13], Prochazka & Gorassini [19], and Edin & Vallbo [7] could not be found in the article or literature, so the lengths were estimated as follows: for graphs that did not provide units, the limits (max and min y-coordinates) of the muscle length graph were assumed to be the resting length (that is, the OFL) and the muscle's limit of stretch, respectively. The muscle was assumed to have a range of stretch that is from 100% of the OFL to 108% of the OFL, since those are the limits of Vannucci et al.'s model's input OFLs in that model's example file, and the limits of that model's input OFLs before the simulation is killed by the Linux Out of Memory killer. Therefore, the muscle's OFL was calculated given that this range of stretch (limit of stretch y coordinate - OFL y coordinate, where OFL y coordinate is 0) is 8% of the muscle's OFL.

Results

Color legend: For Figures 1, 2, 3, 4, 10, and 12, the experimentally-measured OFL are yellow, the experimentally-measured afferents are red, Vannucci et al.'s predicted afferents are blue, Schneider's predicted afferents are green, and (only in the plots of Blum et al.'s data) the averages of Blum et. al's experimentally-measured afferents and Schneider's predicted afferents are orange.

General remarks: For Blum et al.'s data, experimentally-measured afferents generally form horizontal lines, are far above Schneider's predicted afferents, and are aligned with or slightly above Vannucci et al.'s predicted afferents. For digitized article data, experimentally-measured afferents are far above Schneider's predicted afferents, and generally aligned with or below Vannucci et al.'s predicted afferents. Examples of graphs of Blum et al.'s data are in Figure 1, and examples of graphs of digitized article data are in Figure 2.

Using both models for more accurate predictions of experimentally-measured afferents for Blum et al.'s data: In graphs of Blum et al.'s data, it was observed that Vannucci et al.'s predicted afferents sometimes fell between the experimentally-measured afferents and Schneider's predicted afferents, and thus perhaps the average of the experimentally-measured afferents and Schneider's predicted afferents would be a good approximation of Vannucci et al.'s predicted afferents. Then,

("~=" means "approximately equals")

Vannucci et al.'s predicted afferents ~= average of Schneider's predicted afferents and experimentally-measured afferents = (Schneider's predicted afferents + experimentally-measured afferents) * .5

2 * Vannucci et al.'s predicted afferents ~= Schneider's predicted afferents + experimentally-measured afferents

2 * Vannucci et al.'s predicted afferents - Schneider's predicted afferents ~= experimentally-measured afferents

In this way, Vannucci et al.'s predicted afferents and Schneider's predicted afferents could be used to approximate experimentally-measured afferents. However, the average of Schneider's predicted afferents and experimentally-measured afferents was not a good approximation of Vannucci et al.'s predicted afferents, and thus those graphs were not included here.

Average RMS error with Blum et al.'s data: The average root mean squared (RMS) error (across all Blum et al. data graphs) by which Vannucci et al.'s predicted afferents fail to match the experimentally-measured afferents is 57.844 impulses/(s*points). The average RMS error (across all Blum et al. data graphs) by which Schneider's predicted afferents fail to match the experimentally-measured afferents is 86.120 impulses/(s*points). The average RMS error (across all Blum et al. data graphs) by which Blum et. al's experimentally-measured afferents and Schneider's predicted afferents fail to match Vannucci et al.'s predicted afferents in Blum et al.'s data is 52.687 impulses/(s*points). This error is less than the former two RMS errors (57.844 impulses/(s*points) and 86.120 impulses/(s*points)), and this error should be the same as the error between 2*Vannucci et al.'s predicted afferents - Schneider's predicted afferents and experimentally-measured afferents, so this error being lower than the former two RMS errors shows that, for Blum et al.'s data, using Vannucci et al.'s predicted afferents and Schneider's predicted afferents together predicts experimentally-measured afferents better than using either alone. See Table 2 for RMS errors for all Blum data graphs.

Average RMS error with digitized article data: The average RMS error (across all digitized article data graphs) by which Vannucci et al.'s predicted afferents fail to match the experimentally-measured afferents is 32.941 impulses/(s*points). The average RMS error (across all digitized article data graphs) by which Schneider's predicted afferents fail to match the experimentally-measured afferents is 33.205 impulses/(s*points). See Table 3 for RMS errors for all digitized article graphs.

Similarities in average RMS error between Blum et al.'s data and digitized article data: With both Blum et al.'s data and digitized article data, the average RMS error by which Vannucci et al.'s predicted afferents fail is more than the average RMS error by which Schneider's predicted afferents fail. While the shape of Schneider's predicted afferents is often similar to experimentally-measured afferents, the height is typically off, which can cause a greater RMS error than having the shape be off. The shape of Vannucci et al.'s predicted afferents is less similar to experimentally-measured afferents than the shape of Schneider's predicted afferents, but the height of Vannucci et al.'s predicted afferents is often closer to that of experimentally-measured afferents; this is likely the cause of the lower RMS error of Vannucci et al.'s data.

Differences in average RMS error between Blum et al.'s data and digitized article data: The average RMS error by which both models fail with digitized article data is less than the average RMS error by which both models fail with Blum et al.'s data. This could be because the experimentally-measured muscle lengths and afferents of some digitized article data graphs are in humans or rats (see Table 1), while the experimentally-measured muscle lengths and afferents of Blum et al.'s data are in cats. However, Malik, Jabakhanji, and Jones [12] demonstrated that muscle spindle models originally made based on cat experiments have strong predictive value for modeling human muscle spindle afferents. Moreover, of the ten articles from which data was digitized, six have data from measurements taken in cats.

Conclusion

I obtained experimentally -measured muscle lengths and corresponding experimentally -measured muscle spindle afferents from online article supplemental information or from digitizing data in graphs of online articles, and compared experimentally -measured muscle spindle afferents with afferents predicted from Schneider’s model and Vannucci et al.’s model, when providing those models the input of the experimentally -measured muscle lengths. With both Blum et al.'s data and digitized article data, the average RMS error by which Vannucci et al.'s predicted afferents fail is more than the average RMS error by which Schneider's predicted afferents fail. This is likely because the height of Vannucci et al.’s predicted afferents is more accurate than that of Schneider’s predicted afferents, although the shape of Schneider’s predicted afferents is more accurate than that of Vannucci et al.’s predicted afferents, by observation. The error between 2*Vannucci et al.'s predicted afferents - Schneider's predicted afferents and experimentally -measured afferents is less than the error between Vannucci et al.’s predicted afferents and Blum et al.’s experimentally -measured afferents, and is less than the error between Schneider’s predicted afferents and Blum et al.’s experimentally -measured afferents, and therefore using Vannucci et al.'s predicted afferents and Schneider's predicted afferents together predicts experimentally -measured afferents better than using either alone.

References

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[17] Mileusnic, M. P., Brown, I. E., Lan, N., & Loeb, G. E. (2006). Mathematical models of proprioceptors. I. Control and transduction in the muscle spindle. Journal of Neurophysiology, 96(4), 1772-1788. doi:10.1152/jn.00868.2005

[18] Nichols, T. R., & Cope, T. C. (2004). Cross-bridge mechanisms underlying the history-dependent properties of muscle spindles and stretch reflexes. Canadian Journal of Physiology and Pharmacology, 82(8-9), 569-576.

[19] Prochazka, A., & Gorassini, M. (1998). Ensemble firing of muscle afferents recorded during normal locomotion in cats. The Journal of Physiology, 507(1), 293-304.

[20] Puljak, L. (n.d.). Extracting data from figures using software. Retrieved from https://training.cochrane.org/sites/training.cochrane.org/files/public/uploads/resources/downloadable_resources/2016_11_webinar_Puljak-extracting-data-from-figures.pdf

[21] Raybaut, P. & Cordoba, C. (2009). Spyder: The Scientific Python Development Environment, version 3.3.3, with Python 3.7.3, Qt 5.9.6, PyQt5 5.9.2, and Windows 10 . [Computer software]. Retrieved from https://github.com/spyder-ide/spyder.

[22] Rosenthal, N. P., McKean, T. A., Roberts, W. J., & Terzuolo, C. A. (1970). Frequency analysis of stretch reflex and its main subsystems in triceps surae muscles of the cat. Journal of Neurophysiology, 33(6), 713-749.

[23] Saul, K. R., Hu, X., Goehler, C. M., Vidt, M. E., Daly, M., Velisar, A., & Murray, W. M. (2015). Benchmarking of dynamic simulation predictions in two software platforms using an upper limb musculoskeletal model. Computer Methods in Biomechanics and Biomedical Engineering, 18(13), 1445-1458. doi:10.1080/10255842.2014.916698

[24] Schneider, O. (2013). An Interactive Simulation of The Muscle Spindle. Retrieved from https://github.com/oschneid/RTMuscleSpindle.

[25] Spector, S. A., Gardiner, P. F., Zernicke, R. F., Roy, R. R., & Edgerton, V. R. (1980). Muscle architecture and force-velocity characteristics of cat soleus and medial gastrocnemius: implications for motor control. Journal of Neurophysiology, 44(5), 951-960. Retrieved from https://pdfs.semanticscholar.org/b19d/1a307c801870cc09773d67083e28648b983e.pdf.

[26] Vannucci, L., Falotico, E., & Laschi, C. (2017). Proprioceptive Feedback through a Neuromorphic Muscle Spindle Model. Frontiers in Neuroscience, 11, 341. doi:10.3389/fnins.2017.00341

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Appendix A: How to manually digitize data in figures of scientific articles

Puljak's (n.d.) instructions on slides 13-28 (inclusive) explain this process well [20].

I added to that explanation with Figures 5 and 6, which show how to manually digitize a figure. The steps below explain each step (green numbers) of Figures 5 and 6.

Steps

(Note that the figure being digitized in this example is digitized article 2, figure 1.)

1. Open the file in PlotDigitizer. Click "Digitize." Click on the left-most end of the x-axis.

2. Type the x-axis minimum. Click "Okay."

3. Click on the right-most end of the x-axis.

4. Type the x-axis maximum. Click "Okay." (The x-axis maximum was calculated by finding the number of pixels in a line with a width or height of known units which are n'ot pixels. For the image shown, I determined that the 1-second measuring bar in this figure (digitized article 2, figure 1) is 68 pixels, and the x-axis length in the figure being digitized is 300 pixels, so (300 pixels)(1 s/68 pixels)=4.412 s, which is the x-axis maximum given 0 is the x-axis minimum.)

5. Click on the left-most end of the y-axis.

6. Type the y-axis minimum. Click "Okay."

7. Click on the right-most end of the y-axis.

8. Type the y-axis maximum. Click "Okay."

9. Type the x-axis units. Click "OK."

10. Type the y-axis units. Click "OK."

11. Click "Zoom: In" until zoomed in as much as possible (700%).

12. Click on each point of interest in the graph.

13. Finish clicking on each point of interest in the graph. Click "Done."

14. Click "File" in the pop-up containing the (x,y) coordinates of all the digitized points.

15. Click "Save As..." in the drop down from the pop-up. Save the file.

Appendix B: How to automatically digitize data in figures of scientific articles

Get a scale. All images used were .png format, and were screenshots (taken with Microsoft Snipping Tool) of .pdf files of research papers (viewed with Google Chrome). Open the image in Microsoft Paint, use the Select Tool, and measure the length (height or width, depending on how the measuring line is aligned) (in pixels) of some line in the image which indicates how many graph units the line is. This line may be the axes (as in purple number 1, 2, 3, and 4 of Figure 7), or may be a separate bar on the side (as in purple number 5 and 6 of Figure 7). Axes may apply to several graphs (as in all graphs of Figure 7), including the graph of interest, so you may have to screenshot several unrelated graphs so that your screenshot includes the axes.

Remove lines. Remove any axes, labels, titles, legends, graph lines, or anything except the data points to be digitized. See Figures 8 and 9 for examples of removing guiding dots and lines which aren't data points, but are the same color of the data points.

Run the program. Run digitize.py, which searches for the color with the blue-green-red (BGR) lower and upper bounds specified in the code, and writes each point of that color to a comma-separated values (.csv) file. See Figures 10, 11, and 12 for examples of data points of a certain color extracted from a graph which has curves of various colors.

Figures

Figure 1. Examples of graphs of Blum et al.'s data. The experimentally-measured OFL (m/m; unitless) are yellow, the experimentally-measured afferents ((impulses/s)/(impulses/s); unitless) are red, Vannucci et al.'s predicted afferents ((impulses/s)/(impulses/s); unitless) are blue, Schneider's predicted afferents ((impulses/s)/(impulses/s); unitless) are green, and (only in the plots of Blum et al.'s data) the averages of Blum et. al's experimentally-measured afferents ((impulses/s)/(impulses/s); unitless) and Schneider's predicted afferents ((impulses/s)/(impulses/s); unitless) are orange. All y-values are normalized and therefore unitless.

Figure 2. Examples of graphs of digitized article data. The experimentally-measured OFL (m/m; unitless) are yellow, the experimentally-measured afferents ((impulses/s)/(impulses/s); unitless) are red, Vannucci et al.'s predicted afferents ((impulses/s)/(impulses/s); unitless) are blue, Schneider's predicted afferents ((impulses/s)/(impulses/s); unitless) are green.

Figure 3. For Blum et al.’s data, the graphs with the minimum and maximum RMS errors by which Vannucci et al.'s predicted afferents fail to match the experimentally-measured afferents, the graphs with the minimum and maximum RMS errors by which Schneider's predicted afferents fail to match the experimentally-measured afferents, and the graphs with the minimum and maximum RMS errors by which Blum et. al's experimentally-measured afferents and Schneider's predicted afferents fail to match Vannucci et al.'s predicted afferents.

Figure 4. For digitized article data, the graphs with the minimum and maximum RMS errors by which Vannucci et al.'s predicted afferents fail to match the experimentally-measured afferents, and the graphs with the minimum and maximum RMS error by which Schneider's predicted afferents fail to match the experimentally-measured afferents.

Figure 5. A walkthrough on manual digitization. Steps 1-8 (green numbers) explain the figure.

Figure 6. The continuation of a walkthrough on manual digitization. Steps 9-15 (green numbers) explain the figure.

Figure 7. Examples of some line (circled in purple) in the image which indicates how many graph units the line is. This line may be the axes (as in purple number 1, 2, 3, and 4), or may be a separate bar on the side (as in purple number 5 and 6). Examples of graphs where x or y axes apply to multiple graphs, not just the graph closest to the axes.

Purple number 1. Data from article 1, figure 4, muscleShortening

2. Data from article 6, figure 6, Sartorius and Posterior hamstrings

3. Data from article 9, figure 2.

4. Data from article 4, figure 2.

5. Data from article 5, figure 4.

6. Data from article 7, figure 7.

Figure 8. Examples of removing guiding dots which aren't data points, but are the same color of the data points. The figure is from digitized data article 8, figure 4.

Figure 9. Examples of removing guiding lines which aren't data points, but are the same color of the data points. The figure is from digitized data article 6, figure 6, extensor digitorum longus Ia afferents.

Figure 10. Examples of data points of a certain color extracted from a graph which has curves of various colors.

Figure 11. Article 1, figure 4, muscleLengthening graph, where a threshold level of 50 captures the grey line, but some of that line is covered by other colors, producing an incorrect graph. Therefore, although the threshold level of 150 captures multiple lines, I used that threshold level in order to get a graph with y values for each x value, for smooth simulation with Vannucci et al.'s and Schneider's models.

Figure 12. An example of obtaining graphs with holes since colors cover other colors when extracting data points of a certain color from a graph which has curves of various colors.