Decimal Places</th>	Format</th>	Standard Tolerance</th></tr></thead>
1</td>	`0.x</code></td>`	`± 0.2"</code></td></tr>`
2</td>	`0.0x</code></td>`	`± 0.01"</code></td></tr>`
3</td>	`0.00x</code></td>`	`± 0.005"</code></td></tr>`
4</td>	`0.000x</code></td>`	± 0.0005"</code></td></tr> </tbody></table> In general, use equal bilateral tolerance, but there are exceptions to that rule. Remember, dimensions and tolerances communicate intent. Consider the case where you want an eighth-inch hole, but the application permits the hole to be larger more readily than it permits the hole to be smaller. You may want to specify the dimension as something like $\varnothing\ 0.125 + 0.005/-0.001"$. If instead you were to specify the dimension as $\varnothing\ 0.127 \pm .003"$, this might signal to the manufacturer that they would need a special $0.127"$ tool. In general, holes produced by standard drills are over-sized, so a wise machinist might know to use a quarter-inch drill when given a dimension of $\varnothing\ 0.127 \pm .003"$. However, $\varnothing\ 0.125 +0.005/-0.001"$ more clearly communicates your intent: a $0.125"$ hole.</p> Tolerance Analysis</h2> Tolerance analysis is the process of calculating how error might accumulate in dimensions with tolerances. There are two main ways to calculate tolerance: worst-case analysis and statistical analysis.</p> Worst-case analysis is just what it sounds like: you consider each dimension at its largest and smallest acceptable values and you calculate the upper and lower bounds for the combined dimensions. Since most manufacturing processes will produce real dimensions following a normal distribution, it's extremely unlikely that anything close to the worst-case scenario will come true. The power of the worst-case analysis is in its simplicity. It boils down to just adding up the dimensions twice: once for the lower limit dimensions and again for the upper limit dimensions.</p> Statistical tolerance analysis is much more powerful. Instead of assuming each part is at the limits of its acceptable variation, statistical tolerance analysis assumes each dimension is sampled from a statistical distribution. For the sake of simplicity, we'll assume that each dimension is sampled from a normal distribution centered on the nominal dimension and that the tolerance represents three times the standard deviation ($3\sigma$ on either side for a total of $6\sigma$). When we assume the dimensions are normal distributions, we can use the root sum squared (RSS) method to combine their tolerances as standard deviations.</p> Note: Not all manufacturing processes will produce dimensions with a normal distribution. The methods to calculate the tolerance stack-up get more complicated; special software has been developed to deal with these cases. I plan to look into various probabilistic programming packages in Julia (Soss.jl</a>, Turing.jl</a>, and Gen.jl</a>) to see if they can be used for this type of tolerance analysis.</p> Consider the simple case of stacking two parts together.</p> The left, purple part has a dimension of $40 \pm 0.5 mm$.</li> The right, blue part has a dimension of $25 \pm 0.1 mm$.</li> </ul> </p> Worst-Case Tolerance Analysis</h3> Using worst-case analysis, we can calculate the upper and lower bound of the stack. The smallest acceptable dimensions of each part are 39.5 and 24.9 for a total of 64.4. The largest acceptable dimensions of each part are 40.5 and 25.1 for a total of 65.6. The total stack will fall between 64.4 and 65.6—put another way, $64 \pm 0.6 mm$.</p> The worst-case analysis produces a more conservative (larger) tolerance compared with the RSS method.</p> Statistical Tolerance Analysis — RSS Method</h3> Using the RSS method, we sum the standard deviations using the following formula:</p> $$ \sigma_{total}=\sqrt{\sum_{i=1}^{n} \sigma_{i}^{2}}\ = \sqrt{0.5^2 + 0.1^2}\ =\ 0.5099.. $$</p> The nominal dimensions are summed normally to produce a tolerance stack up of $65 \pm 0.51 mm$.</p> These processes are not terribly complicated for simple geometries, but the complexity of these calculations explodes rapidly with more dimensions.</p> Measurements.jl</h2> using</span> Measurements </span>a = </span>4.5 </span>± </span>0.1</span>; </span>b = </span>3.8 </span>± </span>0.4</span>; </span>@</span>show </span>2</span>a + b </span></code></pre> 2</span>a + b = </span>12.8 </span>± </span>0.45 </span></code></pre> Measurements.jl</a> is a package in Julia that takes care of uncertainty propagation for you. Values with uncertainty can be specified with ±</code>, as we would hope. In this case, the ±</code> symbol is a custom infix operator that creates a new type called a Measurement. Through the unreasonable effectiveness of multiple dispatch, Measurements.jl</a> extends the basic operations of Julia, so all calculations done on a Measurement will look the same as if they were just a float. More details about Measurements.jl</a> can be found in the package documentation. Here is a basic example of how Measurements.jl</a> works:</p> Tolerance Analysis with Measurements.jl</h2> using</span> Measurements </span>dim</span>1 </span>= </span>40 </span>± </span>0.5</span>; </span>dim</span>2 </span>= </span>25 </span>± </span>0.1</span>; </span>@</span>show</span> dim</span>1 </span>+ dim</span>2 </span></code></pre> dim</span>1 </span>+ dim</span>2 </span>= </span>65.0 </span>± </span>0.51 </span></code></pre> Consider our simple example above. The following code is able to replicate the statistical tolerance analysis method without the hassle. There's no special methods to call. The complexities of propagating uncertainty though addition are handled by extending the definition of addition to quantities with uncertainty.</p> Okay, that's too easy. Let's try a problem with more teeth.</p> </p> This example was taken from an example in the documentation for the MITCalc.com</a> tolerance analysis calculator. Imagine a part with two holes. The positions of the two holes are determined by chains of dimensions. We would like to know the center-to-center distance and tolerance between the two holes. Our simple calculation methods above quickly become unwieldy.</p> </p> The MITCalc example suggests first deriving the equations below:</p> $$ X=B-D+(C-E) \cdot \cos (A)-G \cdot \sin (A) $$</p> $$ Y=(C-E) \cdot \sin (A)+G \cdot \cos (A)-F $$</p> $$ Z=\sqrt{X^{2}+Y^{2}} $$</p> Ignore those equations. There's an easier way.</strong> Performing statistical tolerance analysis on this part becomes trivial with Measurements.jl</a>. Simply define each of the dimensions as a vector. For example, dimension $F$ is really a vector with a Measurement for one of its values:</p> $$ F = \begin{bmatrix}0 \44.95 \pm 0.05\end{bmatrix} $$</p> using</span> Pkg; Pkg.add("</span>Measurements</span>") </span>using</span> Measurements, LinearAlgebra </span> </span># Define the degree symbol to convert from degree to radians </span>° = </span>pi </span>/ </span>180</span>; </span> </span># Create a quick function to rotate vectors </span>vec_rotate(θ, vector) = [cos(θ) -sin(θ); sin(θ) cos(θ)] * vector </span> </span>A = (</span>30 </span>± </span>0.2</span>)°; </span>B = [</span>120</span>, </span>0</span>]; </span>C = vec_rotate(A, [</span>120</span>, </span>0</span>]); </span>D = [</span>25</span>, </span>0</span>]; </span>E = vec_rotate(A, [-</span>25</span>, </span>0</span>]); </span>F = [</span>0</span>, </span>44.95 </span>± </span>0.05</span>]; </span>G = vec_rotate(A, [</span>0</span>, </span>44.95 </span>± </span>0.05</span>]); </span> </span>path</span>1 </span>= B + C + G + E; </span>path</span>2 </span>= F + D; </span> </span>Z = path</span>1 </span>- path</span>2</span>; </span> </span>@</span>show</span> norm(Z) </span># The length of Z </span></code></pre> Once all the vectors are defined, determining Z is easy. Z is just a vector derived from all the other dimension vectors. The length of Z can be found by taking its norm. The result is $160.26 \pm 0.24$.</p> norm(Z) = </span>160.26 </span>± </span>0.24 </span></code></pre> Wrapping Up</h2> Smaller Details:</h3> I've only demonstrated doing the RSS statistical tolerance analysis method. I may write a quick function to do worst-case analysis using Measurements.jl</a>—stay tuned.</li> I've converted a unilateral dimension ($45 +0/-0.1$) in the example above into equal bilateral dimensions ($44.95 \pm 0.05$).</li> I've bodged a little convenience function to rotate vectors. I would like to make a more generic function, but it wasn't worth the hassle for this proof of concept. Let me know if you know of a better solution.</li> I've also defined the degree symbol as π/180. I made it a habit to write that at the beginning of all Julia scripts, because it's ridiculously convenient to just write 30°</code> instead of 30π/180</code>.</li> This method is also compatible with Unitful.jl</a>. This means that you could give units to each dimension vector and the calculation proceeds without any additional monkey business. I haven't shown this here, but it does work. Using Unitful.jl</a> may be particularly useful if you're compelled to use mixed imperial and metric dimensions.</li> </ul> Future Work</h3> There is still a lot that could be done. I've considered creating a package to perform tolerance analysis, but for now I'm happy with using Measurements.jl</a>. More refined statistical tolerance analysis processes could probably still use Julia, but I believe I'll need to use one of the probabilistic programming packages (Soss.jl</a>, Turing.jl</a>, and Gen.jl</a>) to perform uncertainty propagation with arbitrary distributions. Here are some features that could be part of a Julia package specifically for tolerances:</p> Supporting all of the tolerance types above, including a way to parse a dimension written as a string.</li> Support for multiple different tolerance analysis methods, including worst-case, RSS, Monte Carlo simulations, and others.</li> Support for units (a must!).</li> Integration with Plots.jl</a> (this would be interesting).</li> The ability to specify standard tolerances (e.g., ANSI or ISO hole tolerance standards)</li> </ul> If you end up using this, let me know! I'd be really curious to hear what you think.</p> Assistive Bottle Opener 2020-01-17T00:00:00+00:00 Biomedical engineering was a natural choice for me. It combines my knack for engineering with my desire to help people. Biomedical engineers are more or less guaranteed to work on projects intended to help people. The spark of altruism at the beginning of a project can slowly fade as a project gets longer. This project came out of a desire to work on a smaller project where I could see that spark through from beginning to end. Inspiration for this project comes from two places. First, I stumbled upon an article</a> discussing the cost savings from 3D printing assistive devices. Second, I stumbled across MakersMakingChange.com</a> where such devices are posted.</p> Starting Point</h2> </p> I can't claim credit for this design. I believe I downloaded this design from MyMiniFactory</a>, though there appears to be several versions of this opener floating around on the web. Armed with my Prusa i3 MK3</a>, I printed this design as a starting point for my prototyping.</p> This version works well for bottles within a small range of cap diameters. There were many bottle caps in my kitchen with smaller caps than this opener could accommodate.</p> I also wanted to improve the aesthetics of this device if I could. The hard surface transitions and utilitarian design doesn't make for an inviting object to hold. I liked the curved profile of the handles. The outside of the curve finds the center of my palm nicely making for a secure grip.</p> Prototype 1 - Complete Failure</h2> </p> My next thought was to add a conical taper to the cap-gripping portion. I increased the height of the cap-gripping section to have a gently tapering cone. As I was designing this, I suspected that the opener would simply slide upward as the grips were closed. This was indeed the case.</p> If there was enough friction and the taper angle of the cone were shallow enough, I'm pretty sure this design would work. I wanted this design to be printable in a variety of different materials.</p> Prototype 2 - Progress</h2> </p> With this new prototype I added three different sections of different diameters. This opened most of the caps that I tested. I added a taper between each section of different diameters. This taper helps with 3D printing. The added benefit is that if the cap is slightly larger than one of the sections, a small amount of downward pressure will expand the opener to allow the cap to fit.</p> There were a few issues that I wanted to improve:</p> The handles could be a bit longer to get the right amount of leverage and to improve the grip comfort.</li> The force required to close the opener is higher than I would like. This is mostly because of how thick the plastic is around the cap-gripping portion. This section acts like a large live hinge.</li> The gaps between the three different diameter sections were too large. More diameters would work better.</li> </ul> Prototype 3 - Ready for Testing</h2> </p> I added several more sections of different diameters to the cap-gripping section. The handles were lengthened and material was taken away from around the cap-gripper. This prototype works wonderfully. There are only a few bottles that it doesn't work on. If a bottle has a small cap and the neck of the bottle is much wider, the bottle opener can run into the bottle's neck before the cap gets to the right section of the cap-gripper.</p> Next steps are to give it to a family member that has trouble opening bottles to get their feedback. Stay tuned.</p> Other Ideas - Jamming</h2> I would like to try a design based on a jamming principle next. That is, when the bottle opener is torqued against the cap, the applied torque works to close the grippers further. A small amount of pretension with a spring or rubber band should be applied to the jamming mechanism. This pretension keeps the jamming mechanism in contact with the cap. This design would have a clear advantage over the designs above. The amount of torque applied to the cap is not limited by a user's grip strength. The user just needs to apply a force to the opener's handle and that force would be translated into a gripping action by the jamming mechanism. A key test of this device would be to open a bottle without closing my hand.</p> With a jamming mechanism bottle opener, the limiting factor may become the user's grip on the bottle itself. I'd like to build a jamming mechanism for holding the bottle secure to a kitchen countertop or table. Caps have a smaller range of sizes compared with bottles. I suspect the challenge will be to make a mechanism that can hold bottles of all different sizes securely.</p> Additional Articles:</h2> Gallup N, Bow JK, Pearce JM. Economic Potential for Distributed Manufacturing of Adaptive Aids for Arthritis Patients in the U.S.</a> Geriatrics (Basel). 2018;3(4):89. Published 2018 Dec 6. doi:10.3390/geriatrics3040089</li> </ul> Adventures in Non-photorealistic Rendering 2020-01-15T00:00:00+00:00 </p> </p> Here's a quick experiment with a non-photorealistic rendering (NPR) of a heart using Blender. I created a custom shader to render 3D objects in the style of intaglio printing. You've likely seen intaglio before without knowing the name. It's the style of printmaking most associated with the portraits on currency.</p> </p> Inspiration for this project came from the Blender NPR YouTube channel and Blender community. This is very much a work in progress. I am working on gaining greater control over the line direction.</p> Diagnostic Tests, Visualized 2020-01-15T00:00:00+00:00 Diagnostic tests are invaluable tools in our medical arsenal, but they are not without pitfalls. These tests can be used to identify diseases in their early states or eliminate competing diagnoses to find the correct treatment course. However, no test is perfect. When a healthy person receives a positive result, this can result in anxiety, additional (often invasive) tests, and occasionally misdiagnosis/over-treatment.</p> Let's look at an example:</h2> The numbers for this example come from an open-access article.1</a></sup></p> 50% of people have the condition that we're screening for ( * "Prevalence" or $\Pi$)</li> 90% of people with the disease * will be correctly identified as * having the condition ( * "Sensitivity" or $S$)</li> 30% of people without the disease will be correctly identified as being healthy ("Specificity" or $Sp$)</li> </ul> If we give the screening test to 10,000 people, we would expect the following results:</p> </th> Have Condition</th> Healthy</th> Row Totals</th></tr></thead> Positive Test</strong></td> 4500 True Positive</td> 3500 False Negative</td> 8000</td></tr> Negative Test</strong></td> 500 False Negative</td> 1500 True Negative</td> 2000</td></tr> Column Totals</strong></td> 5000</td> 5000</td> 10000</td></tr> </tbody></table> For a given diagnostic test, there are many related parameters that we might care about. Some of these terms include prevalence, sensitivity, false negative rate, specificity, false positive rate, accuracy, likelihood ratio, and much more. I'm a visual person and wanted a way to represent these parameters and their relationships visually.</p> The visualization below demonstrates the performance of diagnostic tests with relevant parameters marked along the outside of the square.</p> The square represents all people that receive our diagnostic test</li> The left column represents the people with the condition.</li> The right column represents healthy people.</li> The green regions represent correct</em> test results.</li> </ul> </p> I'm working on an interactive version of this chart, so stay tuned. My hope is that being able to play with each parameter will result in a deeper intuition for how these parameters relate.</p> Referenced Articles:</h2> ^{1</sup> Maxim LD, Niebo R, Utell MJ. Screening tests: a review with examples. Inhal Toxicol. 2014;26(13):811-828. doi:10.3109/08958378.2014.955932</a></p> </div> Hodgkin-Huxley Model Using Julia 2020-01-15T00:00:00+00:00 For a biomedical engineer, studying the Hodgkin-Huxley model is a rite of passage—and rightly so. Alan Hodgkin</a> and Andrew Huxley</a> set out to reverse engineer the spiking behavior of our nerves, and their work won them the Nobel Prize. This apex of scientific achievement was only possible through the combination of exquisite biological technique, genius mathematical insight, and no small degree of patience.</p> I've modeled their equation in several programming languages: first in Matlab while in college, second using Python (when I couldn't afford a personal license of Matlab), and now here using Julia. It has become my "Hello World!" example for scientific computing. The winner?—Julia by a landslide!</p> Introduction to the Hodgkin-Huxley Model</h2> This introduction isn't meant to be exhaustive. If you would like to learn more about the electrical behavior of a cell membrane, I highly recommend Bioelectromagnetism - Principles and Applications of Bioelectric and Biomagnetic Fields by Jaakko Malmivuo & Robert Plonsey</a>.</p> The electrical behavior of a cell is not dictated by the flow of electrons. Instead, electricity flows through the body by the movement of ions. Thus the electrical behavior of a cell is dictated by ion concentrations and ion conductance. Ions can pass through the cell membrane by way of ion channels. Hodgkin and Huxley predicted that the conductance of these ion channels was dictated by activating "particles." Each of the particles could be active or inactive and the proportion of particles in the active state corresponds with the fraction of ion channels that are open.</p> Transfer Rate Coefficients</h2> In their model, there were three different activating particles: n, m, and h. The particles jump from their inactive state to their active state at a rate of $\alpha$. Similarly they become inactive again at a rate of $\beta$. Hodgkin and Huxley determined how these transfer rates changed as a function of the membrane voltage (or more accurately the difference between the membrane voltage and the resting potential). The particle kinetics and governing differential equation are shown below:</p> Kinetics:</h3> $$ n \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-n) $$</p> $$ m \overset{\alpha_m}{\underset{\beta_m}{\leftrightharpoons}} (1-m) $$</p> $$ h \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-h) $$</p> Differential Equations:</h3> $$ \frac{d n}{d t}=\alpha_n(1-n)-\beta_n n $$</p> $$ \frac{d m}{d t}=\alpha_m(1-m)-\beta_m m $$</p> $$ \frac{d h}{d t}=\alpha_h(1-h)-\beta_h h $$</p> The Setup</h2> The Julia code below imports the necessary packages and defines functions for the transfer rate coefficients for n, m, and h. We also define the steady state values for each of these activation particles to help define our initial conditions. Finally we set a few constants that will be used throughout the script.</p> using</span> Plots, DifferentialEquations, LaTeXStrings </span> </span>## Constants </span>const</span> C_m = </span>1 </span># μF/cm², membrane capacitance </span>const</span> G_NaMax = </span>120 </span># mS/cm², max conductivity of Na channel </span>const</span> G_KMax = </span>36 </span># mS/cm², max conductivity of K channel </span>const</span> G_L = </span>0.3 </span># mS/cm², leak conductivity </span>const</span> V_r = -</span>65 </span># mV, resting potential </span>const</span> V_Na = </span>50 </span># mV, Nernst voltage for Na </span>const</span> V_K = -</span>77 </span># mV, Nernst voltage for K </span>const</span> V_L = -</span>54.387 </span># mV, Nernst voltage for leak </span> </span>## Function Definitions </span>α_n(dV) = (</span>0.1 </span>- </span>0.01</span>dV)/(exp(</span>1 </span>- </span>0.1</span>dV)- </span>1</span>) </span>β_n(dV) = </span>0.125</span>/(exp(</span>0.0125</span>dV)) </span>α_m(dV) = (</span>2.5 </span>- </span>0.1</span>dV)/(exp(</span>2.5 </span>- </span>0.1</span>dV)- </span>1</span>) </span>β_m(dV) = </span>4</span>/(exp(dV/</span>18</span>)) </span>α_h(dV) = </span>0.07</span>/(exp(</span>0.05</span>dV)) </span>β_h(dV) = </span>1</span>/(exp(</span>3 </span>- </span>0.1</span>dV)+ </span>1</span>) </span> </span># Steady states </span>n_∞(dV) = α_n(dV)/(α_n(dV)+β_n(dV)) </span>m_∞(dV) = α_m(dV)/(α_m(dV)+β_m(dV)) </span>h_∞(dV) = α_h(dV)/(α_h(dV)+β_h(dV)) </span></code></pre> Differential Equations</h2> </p> This diagram shows the circuit model for a small patch of cell membrane. Each ion can only pass through a designated channel. The conductance of each ion channel is determined by the parameters G which is affected by the activation particles described above. Each ion also has a voltage associated with it called the Nernst voltage. In addition to the ion channel conductances, there are also leak currents and capacitive currents. From this circuit model, the final differential equation can be derived.</p> Hodgkin and Huxley solved for the membrane current per unit area as function of the ion conductance and membrane capacitance. Recall that the current across a capacitor depends on the capacitance and change in voltage over time. For my simulation, instead of solving for the membrane current, I am going to solve for the change in voltage over time. The membrane current will be a stimulus which I can adjust to cause the simulated cell to fire. Essentially this is the current that I am injecting into my simulated cell to cause an action potential.</p> $$ I_m=C_m \frac{d V_m}{d t}+(V_m-V_{Na}) G_{Na}+(V_m-V_K) G_K+(V_m-V_L) G_L $$</p> $$ \frac{d V_m}{d t} = \frac{I_{inj}(t) + (V_{Na} - V_m)G_{Na} + (V_K - V_m)G_K + (V_L - V_m) G_L}{C_m} $$</p> Julia is a very concise and high level language. As a result, it can often look like pseudocode. The differential equations defined above for the transfer rate coefficients and the membrane voltage can be seen directly in the HH_model function.</p> # Injected Current Function </span>I_inj(t) = </span>10 </span> (</span>5 </span>< t < </span>30</span>) </span> </span>function </span>HH_model</span>(u,p,t) </span> n, m, h, Vm = u </span> </span> </span># Update transfer rate coefficients, n, m, and h </span> </span># V_diff: membrane voltage - rest voltage </span> V_diff = Vm - V_r </span> dn = α_n(V_diff)(</span>1</span>-n) - β_n(V_diff)n </span> dm = α_m(V_diff)(</span>1</span>-m) - β_m(V_diff)m </span> dh = α_h(V_diff)(</span>1</span>-h) - β_h(V_diff)h </span> </span> </span># Update cell membrane voltage, Vm </span> G_K = G_KMax * n^</span>4 </span># Sodium conductance </span> G_Na = G_NaMax * h * m^</span>3 </span># Potasium conductance </span> dVm = ( I_inj(t) + (V_Na - Vm)G_Na + </span> (V_K - Vm)G_K + (V_L - Vm)*G_L ) / C_m </span> </span> [dn; dm; dh; dVm] </span>end </span> </span>## Run Model: </span>u</span>0 </span>= [n_∞(</span>0</span>); m_∞(</span>0</span>) ; h_∞(</span>0</span>); -</span>65.1</span>] </span>tspan = (</span>0.0</span>,</span>50.0</span>) </span>prob = ODEProblem(HH_model, u</span>0</span>, tspan) </span>sol = solve(prob, saveat=</span>0.01</span>); </span></code></pre> # Plotting </span>p</span>1 </span>= plot( </span> sol.t, sol[</span>4</span>,:], </span> legend=</span>false</span>, </span> ylabel="</span>Voltage [mV]</span>" </span>) </span>p</span>2 </span>= plot( </span> sol.t, I_inj.(sol.t), </span> legend=</span>false</span>, lc=</span>:red</span>, </span> ylabel="</span>Current</span>" </span>) </span>p</span>3 </span>= plot( </span> sol.t, sol[</span>1</span>:</span>3</span>,:]', </span> label=["</span>n</span>" "</span>m</span>" "</span>h</span>"], </span> color=[</span>:blue :orange :green</span>], </span> legend=</span>:topright</span>, </span> xlabel="</span>Time [ms]</span>", </span> ylabel="</span>Fraction Active</span>" </span>) </span> </span>l = grid(</span>3</span>, </span>1</span>, heights=[</span>0.4</span>, </span>0.2</span> ,</span>0.4</span>]) </span>plot( </span> p</span>1</span>, p</span>2</span>, p</span>3</span>, </span> layout = l, </span> size=(</span>600</span>,</span>600</span>), </span> lw=</span>3 </span>) </span></code></pre> </p>}

Diagnostic Test Visualization: Part 2

2021-01-04T00:00:00+00:00

View the full Julia source for this post</a>

I wanted to revisit a visualization that I made in a previous post</a> which shows various parameters of diagnostics tests. I glossed over an issue in that previous post that I hope to address here. For a typical screening test, the visualization would be nearly unreadable because important regions on the graph would be too small to see. The box in the visualization is divided vertically by the prevalence of the condition. People with the condition are on the left and people without the condition are on the right. If the prevalence of the condition is low (say 1%), then the people with the condition (both true positives and false negatives) take up just a tiny sliver of the graph. The same problem occurs with very high prevalence for people without the condition. Similarly if the sensitivity or specificity is close to either 0 or 1, one of the boxes vanishes into a sliver.

To a New Scale</h2>

Instead of fundamentally altering the diagram, I figured I could rescale the axes. Logarithmic scaling is for quitters</a> and doesn't meet my needs, so I need a new scaling function. Probability is bounded between 0 and 1. I want to expand the regions on the ends of this range nearest to 0 and 1. On the low end, I need to distinguish two anomalous events with probabilities like 1% and 0.1%. Similarly, I would like the scale to distinguish two events which are nearly certain with probabilities like 99% and 99.9%. The scaling function should have an inverted "S" shape that gets steeper near 0 and 1. The steep regions of the graph have the effect of mapping a narrow range of inputs (depicted on the x-axis) into a larger region of output (depicted on the y-axis).

using</span> Plots
</span>
</span>logit(x) = log(x/(</span>1</span>-x))
</span>
</span>begin
</span>	xx = </span>0</span>:</span>0.001</span>:</span>1.0
</span>	plot( logit, xx, label="</span>Logit function</span>", lw=</span>2</span>)
</span>	plot!( xlim=(-</span>0.1</span>,</span>1.1</span>), ylim=(-</span>4.1</span>,</span>4.1</span>),
</span>		legend=</span>:topleft</span>, framestyle=</span>:zerolines</span>,
</span>		size=(</span>400</span>,</span>400</span>)
</span>	)
</span>end
</span>savefig(joinpath(@</span>__DIR__</span>, "</span>output</span>", "</span>diagnostic_tests_02_fig2.svg</span>"))
</span></code></pre>
</p>
At first it seems like logit function</a> (the inverse of the logistic function) might be a good fit. This maps values between 0 and 1 to $-∞$ and $+∞$. The one annoyance is that it takes a bounded range and maps it to an unbounded range. I wanted all the relevant features of my visualization to stay within a box so having a rescaling capable of shooting points off to infinity was not ideal.</p>
The function below satisfies all my requirements. The amount of scaling at the ends of the range is determined by a parameter $k$. One really convenient feature is that to invert the scaling, you only need to apply the same function with the reciprocal parameter.</p>
$$ f(x; k) = \frac{1}{1 + \left(\frac{1}{x} - 1 \right)^k} $$</p>
$$ f(\ f(x; k);\ \frac{1}{k})= x $$</p>
I have to credit an answer from user Ron on math.StackExhange.com</a> with this equation. The question posted in that thread was relating to a sigmoidal function for an AI application, but Ron's equation is precisely what I was looking for.</p>
I've found that a scaling factor of $\frac{1}{5}$ is a really good balance. With this scaling factor probabilities of 0.1%, 1%, 10%, and their inverses get remapped to close to 10% divisions. This means I can create nearly regularly spaced ticks marks that fall on these preferred numbers.</p>
using</span> Plots
</span>
</span>scale_fwd(x, k=</span>5</span>) = </span>1 </span>/ (</span>1 </span>+ ((</span>1</span>/x) - </span>1</span>)^(</span>1</span>/k))
</span>scale_rev(x, k=</span>5</span>) = </span>1 </span>/ (</span>1 </span>+ ((</span>1</span>/x) - </span>1</span>)^(k))
</span>
</span>begin
</span>	xx = </span>0</span>:</span>0.001</span>:</span>1.0
</span>	plot( scale_fwd, xx, label="</span>k=5</span>", color=</span>:dodgerblue1</span>)
</span>	plot!(scale_rev, xx, label="</span>k=1/5</span>", ls=</span>:dash</span>, color=</span>:dodgerblue1</span>)
</span>	plot!(xx, xx, lw=</span>2</span>, ls=</span>:dot</span>, color=</span>:gray</span>, label="</span>x=y</span>")
</span>	plot!(aspect_ratio=</span>:equal</span>, 
</span>		xlim=(-</span>0.1</span>,</span>1.1</span>), 
</span>		ylim=(-</span>0.1</span>,</span>1.1</span>),
</span>		legend=</span>:topleft</span>,
</span>		framestyle=</span>:zerolines</span>,
</span>		size=(</span>600</span>,</span>600</span>),
</span>		lw=</span>2</span>,
</span>	)
</span>    savefig(joinpath(@</span>__DIR__</span>, "</span>output</span>", "</span>diagnostic_tests_02_fig3.svg</span>"))
</span>end
</span></code></pre>
</p>
New Version of the Visualization</h2>
I recreated my visualization from before but with axes rescaled. Take note of the light dashed lines and axis labels. The prevalence is only 2.2% but this doesn't collapse the left side of the plot into a sliver. It goes without saying that this graph isn't meant to be used as a quantitative comparison.</p>
Logarithmic scales are ubiquitous for a reason; they have a very nice property that turns curves depicting exponential decays or growth into straight lines. I'm not sure if there is a deeper mathematical elegance to the scale I've chosen or if there is another scaling function that's better suited. My primary goal is to have a readable chart that helps me better understand diagnostic tests.</p>
</p>
I could have stopped with the image above, but I wasn't quite satisfied. I wanted my visualization to be interactive. I fired up an interactive Pluto.jl</a> notebook and created a visualization which reacts to input from a few sliders. The graphic was made with Luxor.jl</a>. Luxor.jl</a> isn't meant to be interactive, but Pluto.jl</a> notebooks are reactive. When I update the value of a slider, the notebook automatically updates all the affected code including the code which generates the visualization. I may explore using Javis.jl</a> in the future for smoother animations. I'm pretty pleased with the outcome and there's still a lot more that could be done.</p>
</p>
Wrapping Up</h2>
This project is mostly a vehicle for me to better remember the details of diagnostic testing. I've been pretty unsatisfied with the descriptions and visualizations on this topic. Creating my own visualization gives me a motivation to dive into terminology of diagnostic tests and learn new skills (such as creating interactive visualizations).</p>
There's still more work that could be done. For this first plot, I've chosen to divide the plot first by people with and without the condition (i.e. splitting vertically by the prevalence). The vertical segments are subsequently split based on the results of their test. I chose this order because the most commonly discussed parameters of a diagnostic test—sensitivity, specificity, and prevalence—can be read directly from the graph. You could just as easily flip the order of these cuts. That is, first cut the square horizontally by people's test results (negative and positive) and then cut the horizontal segments by people with and without the condition. On this alternative plot, parameters including positive predictive value (PPV), false discovery rate (FDR), false omission rate (FOR), and negative predictive value (NPV) can be read off from the graph directly.</p>

Tolerance Analysis with Julia

2020-05-20T00:00:00+00:00

Intro</h2>
This project started when I was reviewing the Excel engineering calculators found on MITCalc.com</a>. MITCalc has many useful calculators for sale for designing gears, springs, beams, and much more. They also have many generic engineering tools including a spreadsheet for performing tolerance analysis. Being both frugal and curious, I set out to see if I could replicate this function with Julia. I've posted the results here for other frugal, curious engineers.
Disclaimer: I've done this as a proof of concept to fulfill my own curiosity—use your own judgment to determine if this tool will work for you and your application.

Background on Tolerances</h2>
If you are making only a few parts and assembling the parts yourself, make the parts to fit. If you watch a master woodworker, they'll often use layout calipers or use one piece to mark another (i.e. relative dimensioning). If there's another party involved in the manufacturing or you are making many parts, you need to consider the tolerances of your dimension. Dimensions and tolerances are a way to communicate intent. There are several different ways a dimension may be presented on the drawing of a part:

Limit of size — This type of tolerance lists two dimensions: the smallest and the largest acceptable dimension. The dimensions are either written from smallest to largest left-to-right, or the larger dimension is on top of the smaller dimension.</li>
Equal bilateral form — This is the type of tolerance most people are familiar with. The dimension has a nominal value followed by the acceptable variation. The dimension may be larger or smaller by equal amounts.</li>
Unequal bilateral form — Like the equal bilateral form, the dimension has a nominal value followed by the acceptable variation, but in this case the variation is not symmetrical. The dimension may be larger or smaller on either side by a different amount.</li>
Unilateral form — In this case the dimension is only allowed to be either larger or smaller than the nominal dimension.</li>
Unless Otherwise Specified (UOS) — A part drawing will often have a table of tolerances in the title block. The tolerance of each dimension can be specified by the number of significant digits given.</li> </ol>
Decimal Places</th> Format</th> Standard Tolerance</th></tr></thead>

1</td> `0.x</code></td>` `± 0.2"</code></td></tr>`
2</td> `0.0x</code></td>` `± 0.01"</code></td></tr>`
3</td> `0.00x</code></td>` `± 0.005"</code></td></tr>`
4</td> `0.000x</code></td>` ± 0.0005"</code></td></tr> </tbody></table> In general, use equal bilateral tolerance, but there are exceptions to that rule. Remember, dimensions and tolerances communicate intent. Consider the case where you want an eighth-inch hole, but the application permits the hole to be larger more readily than it permits the hole to be smaller. You may want to specify the dimension as something like $\varnothing\ 0.125 + 0.005/-0.001"$. If instead you were to specify the dimension as $\varnothing\ 0.127 \pm .003"$, this might signal to the manufacturer that they would need a special $0.127"$ tool. In general, holes produced by standard drills are over-sized, so a wise machinist might know to use a quarter-inch drill when given a dimension of $\varnothing\ 0.127 \pm .003"$. However, $\varnothing\ 0.125 +0.005/-0.001"$ more clearly communicates your intent: a $0.125"$ hole. Tolerance Analysis</h2> Tolerance analysis is the process of calculating how error might accumulate in dimensions with tolerances. There are two main ways to calculate tolerance: worst-case analysis and statistical analysis. Worst-case analysis is just what it sounds like: you consider each dimension at its largest and smallest acceptable values and you calculate the upper and lower bounds for the combined dimensions. Since most manufacturing processes will produce real dimensions following a normal distribution, it's extremely unlikely that anything close to the worst-case scenario will come true. The power of the worst-case analysis is in its simplicity. It boils down to just adding up the dimensions twice: once for the lower limit dimensions and again for the upper limit dimensions. Statistical tolerance analysis is much more powerful. Instead of assuming each part is at the limits of its acceptable variation, statistical tolerance analysis assumes each dimension is sampled from a statistical distribution. For the sake of simplicity, we'll assume that each dimension is sampled from a normal distribution centered on the nominal dimension and that the tolerance represents three times the standard deviation ($3\sigma$ on either side for a total of $6\sigma$). When we assume the dimensions are normal distributions, we can use the root sum squared (RSS) method to combine their tolerances as standard deviations. Note: Not all manufacturing processes will produce dimensions with a normal distribution. The methods to calculate the tolerance stack-up get more complicated; special software has been developed to deal with these cases. I plan to look into various probabilistic programming packages in Julia (Soss.jl</a>, Turing.jl</a>, and Gen.jl</a>) to see if they can be used for this type of tolerance analysis. Consider the simple case of stacking two parts together. The left, purple part has a dimension of $40 \pm 0.5 mm$.</li> The right, blue part has a dimension of $25 \pm 0.1 mm$.</li> </ul> Worst-Case Tolerance Analysis</h3> Using worst-case analysis, we can calculate the upper and lower bound of the stack. The smallest acceptable dimensions of each part are 39.5 and 24.9 for a total of 64.4. The largest acceptable dimensions of each part are 40.5 and 25.1 for a total of 65.6. The total stack will fall between 64.4 and 65.6—put another way, $64 \pm 0.6 mm$. The worst-case analysis produces a more conservative (larger) tolerance compared with the RSS method. Statistical Tolerance Analysis — RSS Method</h3> Using the RSS method, we sum the standard deviations using the following formula: $$ \sigma_{total}=\sqrt{\sum_{i=1}^{n} \sigma_{i}^{2}}\ = \sqrt{0.5^2 + 0.1^2}\ =\ 0.5099.. $$ The nominal dimensions are summed normally to produce a tolerance stack up of $65 \pm 0.51 mm$. These processes are not terribly complicated for simple geometries, but the complexity of these calculations explodes rapidly with more dimensions. Measurements.jl</h2> using Measurements a = 4.5 ± 0.1; b = 3.8 ± 0.4; @show 2a + b </code></pre> 2a + b = 12.8 ± 0.45 </code></pre> Measurements.jl</a> is a package in Julia that takes care of uncertainty propagation for you. Values with uncertainty can be specified with ±</code>, as we would hope. In this case, the ±</code> symbol is a custom infix operator that creates a new type called a Measurement. Through the unreasonable effectiveness of multiple dispatch, Measurements.jl</a> extends the basic operations of Julia, so all calculations done on a Measurement will look the same as if they were just a float. More details about Measurements.jl</a> can be found in the package documentation. Here is a basic example of how Measurements.jl</a> works: Tolerance Analysis with Measurements.jl</h2> using Measurements dim1 = 40 ± 0.5; dim2 = 25 ± 0.1; @show dim1 + dim2 </code></pre> dim1 + dim2 = 65.0 ± 0.51 </code></pre> Consider our simple example above. The following code is able to replicate the statistical tolerance analysis method without the hassle. There's no special methods to call. The complexities of propagating uncertainty though addition are handled by extending the definition of addition to quantities with uncertainty. Okay, that's too easy. Let's try a problem with more teeth. This example was taken from an example in the documentation for the MITCalc.com</a> tolerance analysis calculator. Imagine a part with two holes. The positions of the two holes are determined by chains of dimensions. We would like to know the center-to-center distance and tolerance between the two holes. Our simple calculation methods above quickly become unwieldy. The MITCalc example suggests first deriving the equations below: $$ X=B-D+(C-E) \cdot \cos (A)-G \cdot \sin (A) $$ $$ Y=(C-E) \cdot \sin (A)+G \cdot \cos (A)-F $$ $$ Z=\sqrt{X^{2}+Y^{2}} $$ Ignore those equations. There's an easier way. Performing statistical tolerance analysis on this part becomes trivial with Measurements.jl</a>. Simply define each of the dimensions as a vector. For example, dimension $F$ is really a vector with a Measurement for one of its values: $$ F = \begin{bmatrix}0 \44.95 \pm 0.05\end{bmatrix} $$ using Pkg; Pkg.add("Measurements") using Measurements, LinearAlgebra # Define the degree symbol to convert from degree to radians ° = pi / 180; # Create a quick function to rotate vectors vec_rotate(θ, vector) = [cos(θ) -sin(θ); sin(θ) cos(θ)] * vector A = (30 ± 0.2)°; B = [120, 0]; C = vec_rotate(A, [120, 0]); D = [25, 0]; E = vec_rotate(A, [-25, 0]); F = [0, 44.95 ± 0.05]; G = vec_rotate(A, [0, 44.95 ± 0.05]); path1 = B + C + G + E; path2 = F + D; Z = path1 - path2; @show norm(Z) # The length of Z </code></pre> Once all the vectors are defined, determining Z is easy. Z is just a vector derived from all the other dimension vectors. The length of Z can be found by taking its norm. The result is $160.26 \pm 0.24$. norm(Z) = 160.26 ± 0.24 </code></pre> Wrapping Up</h2> Smaller Details:</h3> I've only demonstrated doing the RSS statistical tolerance analysis method. I may write a quick function to do worst-case analysis using Measurements.jl</a>—stay tuned.</li> I've converted a unilateral dimension ($45 +0/-0.1$) in the example above into equal bilateral dimensions ($44.95 \pm 0.05$).</li> I've bodged a little convenience function to rotate vectors. I would like to make a more generic function, but it wasn't worth the hassle for this proof of concept. Let me know if you know of a better solution.</li> I've also defined the degree symbol as π/180. I made it a habit to write that at the beginning of all Julia scripts, because it's ridiculously convenient to just write 30°</code> instead of 30π/180</code>.</li> This method is also compatible with Unitful.jl</a>. This means that you could give units to each dimension vector and the calculation proceeds without any additional monkey business. I haven't shown this here, but it does work. Using Unitful.jl</a> may be particularly useful if you're compelled to use mixed imperial and metric dimensions.</li> </ul> Future Work</h3> There is still a lot that could be done. I've considered creating a package to perform tolerance analysis, but for now I'm happy with using Measurements.jl</a>. More refined statistical tolerance analysis processes could probably still use Julia, but I believe I'll need to use one of the probabilistic programming packages (Soss.jl</a>, Turing.jl</a>, and Gen.jl</a>) to perform uncertainty propagation with arbitrary distributions. Here are some features that could be part of a Julia package specifically for tolerances: Supporting all of the tolerance types above, including a way to parse a dimension written as a string.</li> Support for multiple different tolerance analysis methods, including worst-case, RSS, Monte Carlo simulations, and others.</li> Support for units (a must!).</li> Integration with Plots.jl</a> (this would be interesting).</li> The ability to specify standard tolerances (e.g., ANSI or ISO hole tolerance standards)</li> </ul> If you end up using this, let me know! I'd be really curious to hear what you think. Assistive Bottle Opener 2020-01-17T00:00:00+00:00 Biomedical engineering was a natural choice for me. It combines my knack for engineering with my desire to help people. Biomedical engineers are more or less guaranteed to work on projects intended to help people. The spark of altruism at the beginning of a project can slowly fade as a project gets longer. This project came out of a desire to work on a smaller project where I could see that spark through from beginning to end. Inspiration for this project comes from two places. First, I stumbled upon an article</a> discussing the cost savings from 3D printing assistive devices. Second, I stumbled across MakersMakingChange.com</a> where such devices are posted. Starting Point</h2> I can't claim credit for this design. I believe I downloaded this design from MyMiniFactory</a>, though there appears to be several versions of this opener floating around on the web. Armed with my Prusa i3 MK3</a>, I printed this design as a starting point for my prototyping. This version works well for bottles within a small range of cap diameters. There were many bottle caps in my kitchen with smaller caps than this opener could accommodate. I also wanted to improve the aesthetics of this device if I could. The hard surface transitions and utilitarian design doesn't make for an inviting object to hold. I liked the curved profile of the handles. The outside of the curve finds the center of my palm nicely making for a secure grip. Prototype 1 - Complete Failure</h2> My next thought was to add a conical taper to the cap-gripping portion. I increased the height of the cap-gripping section to have a gently tapering cone. As I was designing this, I suspected that the opener would simply slide upward as the grips were closed. This was indeed the case. If there was enough friction and the taper angle of the cone were shallow enough, I'm pretty sure this design would work. I wanted this design to be printable in a variety of different materials. Prototype 2 - Progress</h2> With this new prototype I added three different sections of different diameters. This opened most of the caps that I tested. I added a taper between each section of different diameters. This taper helps with 3D printing. The added benefit is that if the cap is slightly larger than one of the sections, a small amount of downward pressure will expand the opener to allow the cap to fit. There were a few issues that I wanted to improve: The handles could be a bit longer to get the right amount of leverage and to improve the grip comfort.</li> The force required to close the opener is higher than I would like. This is mostly because of how thick the plastic is around the cap-gripping portion. This section acts like a large live hinge.</li> The gaps between the three different diameter sections were too large. More diameters would work better.</li> </ul> Prototype 3 - Ready for Testing</h2> I added several more sections of different diameters to the cap-gripping section. The handles were lengthened and material was taken away from around the cap-gripper. This prototype works wonderfully. There are only a few bottles that it doesn't work on. If a bottle has a small cap and the neck of the bottle is much wider, the bottle opener can run into the bottle's neck before the cap gets to the right section of the cap-gripper. Next steps are to give it to a family member that has trouble opening bottles to get their feedback. Stay tuned. Other Ideas - Jamming</h2> I would like to try a design based on a jamming principle next. That is, when the bottle opener is torqued against the cap, the applied torque works to close the grippers further. A small amount of pretension with a spring or rubber band should be applied to the jamming mechanism. This pretension keeps the jamming mechanism in contact with the cap. This design would have a clear advantage over the designs above. The amount of torque applied to the cap is not limited by a user's grip strength. The user just needs to apply a force to the opener's handle and that force would be translated into a gripping action by the jamming mechanism. A key test of this device would be to open a bottle without closing my hand. With a jamming mechanism bottle opener, the limiting factor may become the user's grip on the bottle itself. I'd like to build a jamming mechanism for holding the bottle secure to a kitchen countertop or table. Caps have a smaller range of sizes compared with bottles. I suspect the challenge will be to make a mechanism that can hold bottles of all different sizes securely. Additional Articles:</h2> Gallup N, Bow JK, Pearce JM. Economic Potential for Distributed Manufacturing of Adaptive Aids for Arthritis Patients in the U.S.</a> Geriatrics (Basel). 2018;3(4):89. Published 2018 Dec 6. doi:10.3390/geriatrics3040089</li> </ul> Adventures in Non-photorealistic Rendering 2020-01-15T00:00:00+00:00 Here's a quick experiment with a non-photorealistic rendering (NPR) of a heart using Blender. I created a custom shader to render 3D objects in the style of intaglio printing. You've likely seen intaglio before without knowing the name. It's the style of printmaking most associated with the portraits on currency. Inspiration for this project came from the Blender NPR YouTube channel and Blender community. This is very much a work in progress. I am working on gaining greater control over the line direction. Diagnostic Tests, Visualized 2020-01-15T00:00:00+00:00 Diagnostic tests are invaluable tools in our medical arsenal, but they are not without pitfalls. These tests can be used to identify diseases in their early states or eliminate competing diagnoses to find the correct treatment course. However, no test is perfect. When a healthy person receives a positive result, this can result in anxiety, additional (often invasive) tests, and occasionally misdiagnosis/over-treatment. Let's look at an example:</h2> The numbers for this example come from an open-access article.1</a> 50% of people have the condition that we're screening for ( * "Prevalence" or $\Pi$)</li> 90% of people with the disease * will be correctly identified as * having the condition ( * "Sensitivity" or $S$)</li> 30% of people without the disease will be correctly identified as being healthy ("Specificity" or $Sp$)</li> </ul> If we give the screening test to 10,000 people, we would expect the following results: </th> Have Condition</th> Healthy</th> Row Totals</th></tr></thead> Positive Test</td> 4500 True Positive</td> 3500 False Negative</td> 8000</td></tr> Negative Test</td> 500 False Negative</td> 1500 True Negative</td> 2000</td></tr> Column Totals</td> 5000</td> 5000</td> 10000</td></tr> </tbody></table> For a given diagnostic test, there are many related parameters that we might care about. Some of these terms include prevalence, sensitivity, false negative rate, specificity, false positive rate, accuracy, likelihood ratio, and much more. I'm a visual person and wanted a way to represent these parameters and their relationships visually. The visualization below demonstrates the performance of diagnostic tests with relevant parameters marked along the outside of the square. The square represents all people that receive our diagnostic test</li> The left column represents the people with the condition.</li> The right column represents healthy people.</li> The green regions represent correct test results.</li> </ul> I'm working on an interactive version of this chart, so stay tuned. My hope is that being able to play with each parameter will result in a deeper intuition for how these parameters relate. Referenced Articles:</h2> ^{1 Maxim LD, Niebo R, Utell MJ. Screening tests: a review with examples. Inhal Toxicol. 2014;26(13):811-828. doi:10.3109/08958378.2014.955932</a> </div> Hodgkin-Huxley Model Using Julia 2020-01-15T00:00:00+00:00 For a biomedical engineer, studying the Hodgkin-Huxley model is a rite of passage—and rightly so. Alan Hodgkin</a> and Andrew Huxley</a> set out to reverse engineer the spiking behavior of our nerves, and their work won them the Nobel Prize. This apex of scientific achievement was only possible through the combination of exquisite biological technique, genius mathematical insight, and no small degree of patience. I've modeled their equation in several programming languages: first in Matlab while in college, second using Python (when I couldn't afford a personal license of Matlab), and now here using Julia. It has become my "Hello World!" example for scientific computing. The winner?—Julia by a landslide! Introduction to the Hodgkin-Huxley Model</h2> This introduction isn't meant to be exhaustive. If you would like to learn more about the electrical behavior of a cell membrane, I highly recommend Bioelectromagnetism - Principles and Applications of Bioelectric and Biomagnetic Fields by Jaakko Malmivuo & Robert Plonsey</a>. The electrical behavior of a cell is not dictated by the flow of electrons. Instead, electricity flows through the body by the movement of ions. Thus the electrical behavior of a cell is dictated by ion concentrations and ion conductance. Ions can pass through the cell membrane by way of ion channels. Hodgkin and Huxley predicted that the conductance of these ion channels was dictated by activating "particles." Each of the particles could be active or inactive and the proportion of particles in the active state corresponds with the fraction of ion channels that are open. Transfer Rate Coefficients</h2> In their model, there were three different activating particles: n, m, and h. The particles jump from their inactive state to their active state at a rate of $\alpha$. Similarly they become inactive again at a rate of $\beta$. Hodgkin and Huxley determined how these transfer rates changed as a function of the membrane voltage (or more accurately the difference between the membrane voltage and the resting potential). The particle kinetics and governing differential equation are shown below: Kinetics:</h3> $$ n \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-n) $$ $$ m \overset{\alpha_m}{\underset{\beta_m}{\leftrightharpoons}} (1-m) $$ $$ h \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-h) $$ Differential Equations:</h3> $$ \frac{d n}{d t}=\alpha_n(1-n)-\beta_n n $$ $$ \frac{d m}{d t}=\alpha_m(1-m)-\beta_m m $$ $$ \frac{d h}{d t}=\alpha_h(1-h)-\beta_h h $$ The Setup</h2> The Julia code below imports the necessary packages and defines functions for the transfer rate coefficients for n, m, and h. We also define the steady state values for each of these activation particles to help define our initial conditions. Finally we set a few constants that will be used throughout the script. using Plots, DifferentialEquations, LaTeXStrings ## Constants const C_m = 1 # μF/cm², membrane capacitance const G_NaMax = 120 # mS/cm², max conductivity of Na channel const G_KMax = 36 # mS/cm², max conductivity of K channel const G_L = 0.3 # mS/cm², leak conductivity const V_r = -65 # mV, resting potential const V_Na = 50 # mV, Nernst voltage for Na const V_K = -77 # mV, Nernst voltage for K const V_L = -54.387 # mV, Nernst voltage for leak ## Function Definitions α_n(dV) = (0.1 - 0.01dV)/(exp(1 - 0.1dV)- 1) β_n(dV) = 0.125/(exp(0.0125dV)) α_m(dV) = (2.5 - 0.1dV)/(exp(2.5 - 0.1dV)- 1) β_m(dV) = 4/(exp(dV/18)) α_h(dV) = 0.07/(exp(0.05dV)) β_h(dV) = 1/(exp(3 - 0.1dV)+ 1) # Steady states n_∞(dV) = α_n(dV)/(α_n(dV)+β_n(dV)) m_∞(dV) = α_m(dV)/(α_m(dV)+β_m(dV)) h_∞(dV) = α_h(dV)/(α_h(dV)+β_h(dV)) </code></pre> Differential Equations</h2> This diagram shows the circuit model for a small patch of cell membrane. Each ion can only pass through a designated channel. The conductance of each ion channel is determined by the parameters G which is affected by the activation particles described above. Each ion also has a voltage associated with it called the Nernst voltage. In addition to the ion channel conductances, there are also leak currents and capacitive currents. From this circuit model, the final differential equation can be derived. Hodgkin and Huxley solved for the membrane current per unit area as function of the ion conductance and membrane capacitance. Recall that the current across a capacitor depends on the capacitance and change in voltage over time. For my simulation, instead of solving for the membrane current, I am going to solve for the change in voltage over time. The membrane current will be a stimulus which I can adjust to cause the simulated cell to fire. Essentially this is the current that I am injecting into my simulated cell to cause an action potential. $$ I_m=C_m \frac{d V_m}{d t}+(V_m-V_{Na}) G_{Na}+(V_m-V_K) G_K+(V_m-V_L) G_L $$ $$ \frac{d V_m}{d t} = \frac{I_{inj}(t) + (V_{Na} - V_m)G_{Na} + (V_K - V_m)G_K + (V_L - V_m) G_L}{C_m} $$ Julia is a very concise and high level language. As a result, it can often look like pseudocode. The differential equations defined above for the transfer rate coefficients and the membrane voltage can be seen directly in the HH_model function. # Injected Current Function I_inj(t) = 10 (5 < t < 30) function HH_model(u,p,t) n, m, h, Vm = u # Update transfer rate coefficients, n, m, and h # V_diff: membrane voltage - rest voltage V_diff = Vm - V_r dn = α_n(V_diff)(1-n) - β_n(V_diff)n dm = α_m(V_diff)(1-m) - β_m(V_diff)m dh = α_h(V_diff)(1-h) - β_h(V_diff)h # Update cell membrane voltage, Vm G_K = G_KMax * n^4 # Sodium conductance G_Na = G_NaMax * h * m^3 # Potasium conductance dVm = ( I_inj(t) + (V_Na - Vm)G_Na + (V_K - Vm)G_K + (V_L - Vm)*G_L ) / C_m [dn; dm; dh; dVm] end ## Run Model: u0 = [n_∞(0); m_∞(0) ; h_∞(0); -65.1] tspan = (0.0,50.0) prob = ODEProblem(HH_model, u0, tspan) sol = solve(prob, saveat=0.01); </code></pre> # Plotting p1 = plot( sol.t, sol[4,:], legend=false, ylabel="Voltage [mV]" ) p2 = plot( sol.t, I_inj.(sol.t), legend=false, lc=:red, ylabel="Current" ) p3 = plot( sol.t, sol[1:3,:]', label=["n" "m" "h"], color=[:blue :orange :green], legend=:topright, xlabel="Time [ms]", ylabel="Fraction Active" ) l = grid(3, 1, heights=[0.4, 0.2 ,0.4]) plot( p1, p2, p3, layout = l, size=(600,600), lw=3 ) </code></pre> }

Animated Demo</h2>
</p>

Color Simplification</h2>

More Results</h2>
</div>

Similar Curves</h2>
I'd be remiss to end this article without mentioning some very similar curves.</p>

The lemniscate of Bernoulli</a></li>
The convict curve</a></li> </ul>
References:</h2>

Mathcurve.com LINTEARIA</a></li>
Mathcurve.com ELASTIC CURVE</a></li> </ul>

References:</h2>

Mathcurve.com LINTEARIA</a></li>
Mathcurve.com ELASTIC CURVE</a></li> </ul>

Wrapping Up</h2>

Assistive Bottle Opener

Additional Articles:</h2>

Gallup N, Bow JK, Pearce JM. Economic Potential for Distributed Manufacturing of Adaptive Aids for Arthritis Patients in the U.S.</a> Geriatrics (Basel). 2018;3(4):89. Published 2018 Dec 6. doi:10.3390/geriatrics3040089</li> </ul>

Adventures in Non-photorealistic Rendering

Diagnostic Tests, Visualized

Hodgkin-Huxley Model Using Julia

Kinetics:</h3>
$$ n \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-n) $$</p>
$$ m \overset{\alpha_m}{\underset{\beta_m}{\leftrightharpoons}} (1-m) $$</p>
$$ h \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-h) $$</p>

Differential Equations:</h3>
$$ \frac{d n}{d t}=\alpha_n(1-n)-\beta_n n $$</p>
$$ \frac{d m}{d t}=\alpha_m(1-m)-\beta_m m $$</p>
$$ \frac{d h}{d t}=\alpha_h(1-h)-\beta_h h $$</p>

Braden Eliason

Chopstick Fruit Bowl

Circle Mesh Extrude: Vibe-Coding a Real FeatureScript Feature with Claude

Vibe-Coding My First FeatureScript Feature: Zonohedron

A For Loop Hiding in Onshape's Linear Pattern

Kroki + Pluto = ❤

Graph Vizualization in Pluto Notebooks

Isometric Pixel Art in Blender

Monte Carlo Light Transport

Solving Beam Deflection

My Favorite Shape

Diagnostic Test Visualization: Part 2

Household Prints

Cellphone Camera Microscope Adapter

Mechanical Rope Tensioner

Dithering in Blender

Send Nodes</h2>
Here is the Blender image compositor node setup used to produce the results above.</p>
</p>
</p>

More Results</h2>
</div>
</p>

Pursuit Curves in Julia

Tolerance Analysis with Julia

Braden Eliason

Chopstick Fruit Bowl

Circle Mesh Extrude: Vibe-Coding a Real FeatureScript Feature with Claude

Vibe-Coding My First FeatureScript Feature: Zonohedron

A For Loop Hiding in Onshape's Linear Pattern

Kroki + Pluto = ❤

Graph Vizualization in Pluto Notebooks

Isometric Pixel Art in Blender

Color Simplification</h2>

More Results</h2> </div>

Monte Carlo Light Transport

Solving Beam Deflection

My Favorite Shape

Similar Curves</h2> I'd be remiss to end this article without mentioning some very similar curves.</p>

Diagnostic Test Visualization: Part 2

Household Prints

Cellphone Camera Microscope Adapter

Mechanical Rope Tensioner

Dithering in Blender

Send Nodes</h2> Here is the Blender image compositor node setup used to produce the results above.</p> </p> </p>

More Results</h2> </div> </p>

Pursuit Curves in Julia

Tolerance Analysis with Julia

Wrapping Up</h2>

Assistive Bottle Opener

Adventures in Non-photorealistic Rendering

Diagnostic Tests, Visualized

Hodgkin-Huxley Model Using Julia

Kinetics:</h3> $$ n \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-n) $$</p> $$ m \overset{\alpha_m}{\underset{\beta_m}{\leftrightharpoons}} (1-m) $$</p> $$ h \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-h) $$</p>

Differential Equations:</h3> $$ \frac{d n}{d t}=\alpha_n(1-n)-\beta_n n $$</p> $$ \frac{d m}{d t}=\alpha_m(1-m)-\beta_m m $$</p> $$ \frac{d h}{d t}=\alpha_h(1-h)-\beta_h h $$</p>

More Results</h2>
</div>

Similar Curves</h2>
I'd be remiss to end this article without mentioning some very similar curves.</p>

The lemniscate of Bernoulli</a></li>
The convict curve</a></li> </ul>
References:</h2>

Mathcurve.com LINTEARIA</a></li>
Mathcurve.com ELASTIC CURVE</a></li> </ul>

Send Nodes</h2>
Here is the Blender image compositor node setup used to produce the results above.</p>
</p>
</p>

More Results</h2>
</div>
</p>

Kinetics:</h3>
$$ n \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-n) $$</p>
$$ m \overset{\alpha_m}{\underset{\beta_m}{\leftrightharpoons}} (1-m) $$</p>
$$ h \overset{\alpha_n}{\underset{\beta_n}{\leftrightharpoons}} (1-h) $$</p>

Differential Equations:</h3>
$$ \frac{d n}{d t}=\alpha_n(1-n)-\beta_n n $$</p>
$$ \frac{d m}{d t}=\alpha_m(1-m)-\beta_m m $$</p>
$$ \frac{d h}{d t}=\alpha_h(1-h)-\beta_h h $$</p>