April 2019 Grad School Life Updates

I originally planned to update this blog every week or so during school, but as soon as the quarter started, things got super busy and it was easy to put this off. Hopefully, I will be better about it this quarter!

To give some background, ever since I started thinking about applying to grad programs, I knew that I wanted to come to my school and program, Biomathematics. I did a lot of research on different aspects of the programs, and even more after I was invited to the interview weekends. I chose this place based on a lot of factors, including academic fit, future goals, advisors, general feel of the program, location, and LGBTQ+ friendliness of the campus.

The program has been wonderful so far and has even surpassed my expectations. It is a pretty tiny program, only 15 grad students total, so the classes are very small and everyone in the program knows each other. Every Thursday, the grad students, some of the students who work for our professors but are from neighboring departments such as Math and Biostatistics, and postdocs all go to a nearby bar, Barney’s, for “pub night”, where they basically drink beer, spill (metaphorical) tea, and relieve stress. In my experience with the students, they have all been incredibly helpful, friendly, and inclusive. I have been careful about sharing personal information with them and thus have only come out to one person in my program so far. I hope that I can make closer friendships with the other students over time.

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Grad school classes have been an adjustment in a lot of ways. On one hand, there is a lot more material covered per class – there have been times when the entirety of a math class I had taken in undergrad was covered in just two lectures. Moreover, it is impossible to get all the required background simply from attending class, and it’s necessary to do a lot of extra reading. One thing that has been surprising for me is that in my program’s core courses, as well as the neuroscience course I took, it didn’t seem as difficult as it was in undergrad to get good grades (despite the material being a lot more daunting). I think this is probably because in undergrad, there were more tricks on exams that were designed to weed people out, and now, the focus is on learning, asking questions that may or may not have answers, and being self-motivated to seek extra references for more information, but we aren’t being directly or comparatively evaluated for those things.

Another difference is that there is a lot more emphasis on reading papers and critical thinking, such as proposing potential experiments or critically examining the presentation of data and results in published papers. Some of my core biomathematics courses had homework problems that had no analytic solutions, or that there were multiple possible approaches, and the professors just wanted to see us come up with ideas, defend our assumptions, and solve as far as analytically (or numerically) possible. This is obviously quite different from undergraduate mathematics or chemistry classes, where there are standard solutions to most classical problems either in the back of the book or somewhere on the internet! But I suppose it is moving more reflective of problems in research that have not been previously solved.

I have particularly enjoyed the aspect of courses that involve choosing papers to review for final presentations, and it has allowed me to explore applications of mathematics and computation to neuroscience and has made me more excited about research. When I was in undergrad, although I studied in a theoretical physics group that looked at neuron dynamics, I wasn’t sure if I was doing it only because that was the main opportunity that came my way, but not out of real passion. I think I was too stressed about the prospect of grad school at the time to really develop my passion in research. However, I have always found myself drawn to related topics for class projects and during our department seminars. Biomathematics is a broad field, and I was originally considering exploring the statistical genetics route that is popular in my department, but after starting here, I think that my interests truly lie in neuroscience and mathematical physics, and I am now much more certain in choosing my research focuses and courses.

My department has many course requirements (4 core biomath courses, 2 biomath electives, 6 applied math courses, and 6 biology courses), and as a result, unlike some of the more experimentally focused departments like biology and engineering, they encourage us to focus on coursework and passing the qualifying exams during the first year. We don’t have official research rotations, and we don’t have to decide on an advisor until the end of the second year. However, all of my classmates have started working with potential advisors.

Although I unofficially attended research meetings in fall quarter, this winter quarter was my first official quarter of directed research. At the same time, one of my core courses was taught by my potential advisor (or PI, although my friends who are not in science keep thinking I mean “private investigator” when I use that term). He was an amazing lecturer; he wasn’t the kind of professor who continuously spews information while we furiously try to scribble everything down, but he led us to certain ideas by asking questions. One thing I really like about working with him, both through the course and during the research meetings and updates, is that although his work is clearly mathematically oriented (his background is in particle physics – interestingly, just like my PI in undergrad), unlike a lot of mathematicians and physicists, he has a very conceptual and biologically relevant approach. Some people in our program prefer more mathematical rigor, but for me, it seemed to be a perfect blend.

My advisor has done a lot of previous work on cardiovascular networks and the scaling of radius and length of individual vessels across levels of the network. I came to visit him before applying to the program, and when I told him that I was interested in neuroscience, he said that he could imagine the possibility of applying the same methods of analysis to study neuronal networks. Since I came to the group, I have been working on formulating this problem, solving for scaling ratios using Lagrange multipliers (more details about this method in my First Quarter Research Progress post), and analyzing data, both from images and quantitative data from 3D reconstructions of neurons. I have reformulated this problem so that instead of minimizing the power loss due to dissipation, I am minimizing conduction time. For neurons, one of the major evolutionary driving factors is the speed in conducting signals. For example, if you touch something hot like a stove, it would be helpful to have this sensory information relayed as soon as possible so you can pull your hand away before burning it! I have also been reading some papers from the fifties about conduction velocity in neurons and the effects of myelination (fatty layers that provide insulation for nerve fibers) on this speed, and have recently incorporated the degree of myelination as a parameter. I am also looking to modify the space filling constraint to fit neuronal systems, but I am not quite sure how to do this yet. Taking neuroscience courses concurrently with this project is helpful because sometimes I will get random ideas from class that I might be able to translate to math in a way that I can incorporate it into my model. Sometimes, I watch videos of talks by researchers in biology about dendritic morphology and structural neuroscience and feel somewhat overwhelmed, because I am obviously making a lot of simplifying assumptions and not taking into consideration factors such as genetic influences.

Overall, although research is messy and involves a lot of seeking information from various fields, as well as catching up on basic electrodynamics, fluid mechanics, and neuroscience that I never learned in a class, I am enjoying it a lot. This is my first time having my own project, as in undergrad I was for the most part working as a minion, completing menial coding tasks for grad students’ projects. My office mate in my undergrad research group, now a fourth-year grad student in the same group, came to visit me over spring break and told me I seemed a lot more confident than I was last year. Which is strange to me because I feel more overwhelmed and confused the more I learn! I suppose the “confidence” might come from accepting that I don’t know everything, or even a lot, and I’m more comfortable with being uncomfortable, if that makes any sense at all.

As I anticipated, making friends has been quite difficult for me in grad school. It was especially difficult in fall quarter, when I avoided going to LGBTQ+ specific events out of fear of the unknown, mostly, and just went to the weekly department pub nights every now and then, and spent the rest of my time shut up in my own room. My department mates are wonderful and lovely, but aside from the fact that I am not hugely into drinking, the conversations were centered around heteronormative romantic experiences, and I found myself feeling isolated a lot of the time – especially since I’m not out to most of them. When I talked to my mom about it over winter break, she suggested that I add queer org meetings to my schedule rigidly, with the same priority as classes, just so that I could feel more of a sense of community. I decided that this was a good idea, as mental health is an important thing to commit to.

In winter quarter, I regularly attended two queer orgs. One of these is called QSTEM, or Queers in STEM. It was founded by a second year PhD student in Geochemistry who identifies as a gay man. This org is mostly other graduate students, and the vast majority of them are men, which is not entirely unexpected. I have enjoyed participating in social events such as board game nights and ice cream socials. They also have a lot of outreach opportunities, which I hope I have time to get more involved in as my courses finish up and some time is freed up.

The second org I attended was called Queer Girl, and is only open to women and non-binary people. I was the only one there who wasn’t an undergrad, but was a nice social space to discuss things like queer representation in media (or the lack thereof, especially when it comes to women) – it gave me the opportunity to talk about Shay Mitchell in Pretty Little Liars and a random Korean webtoon I found called “Fluttering Feelings.” There’s definitely a lot I could learn from these women, as they would talk about their sexuality openly, which is something I’ve never been comfortable doing. Being around other women like me helped normalize my experiences a little. One of the coordinators of the group was a fellow Asian woman from San Diego (when I went to undergrad), and it was nice to meet someone I could vent to about missing San Diego and people always assuming we’re straight (being Asian/South Asian and having long hair is a surefire way to convince everyone you’re straight).

One of the social events in this club was a trip to Cuties Coffee, a queer owned and themed coffee shop in East Los Angeles that is designed to be a daytime, sober space for queer socialization and an alternative to the gay bars in West Hollywood. I loved visiting this place so much that I have now made it part of my weekend routine – I go there from around noon to four almost every Saturday to either study for classes or work on coding for research. I have included a picture from that day, and used the rainbow pride flag emojis to cover faces for the privacy of the other org members.

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I can’t stress how important it has been for me to have a queer sober space to go to, as I would say I’m pretty far on the introverted side of the spectrum and I never quite feel comfortable meeting new people in bars or nightclubs. (I still mostly keep to myself, drink my coffee/tea, and study during my trips to Cuties, but I hope I will cross the barrier of talking to strangers soon!). At the beginning of winter quarter, I went to West Hollywood a few times to check out the gay bars and nightclubs. Although I love walking on the main strip in West Hollywood, and enjoyed the experience to some extent, it’s not ideal for me because 1) the bars and clubs are largely catered towards gay men – Wednesdays are the only nights specifically for women, and there are no specific clubs for women, and 2) for some reason, being in these spaces where I’m (theoretically) approaching random strangers who are making snap judgments and impressions about me solely based on my physical appearance spiked some of my body insecurities, and to be honest, that’s not a headspace I want (or need) to be in. Right now, the focus for me is on meeting new queer friends and building community, and I’m grateful for these multiple sober spaces I have had access to this quarter.

Another extracurricular activity I participated in this winter was a club that does educational outreach in the form of presenting posters about various neuroscience to elementary through high school students to get them excited about learning about the brain. I was part of this Committee called Project Glia, which is responsible for designing and creating posters. I really wanted a way to keep in touch with my art – it can be extremely cathartic and rewarding, and I also want to catch up on the neuroscience background I never had in undergrad for my research, so this was the perfect opportunity for me. I designed this poster for “Music and the Brain”, and I was working with two undergrads who did a lot of the neat typography and shading. The director of Project Glia is a senior undergrad who happens to be taking one of my current graduate neurosciences classes with me, The Biology of Learning and Memory.

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One thing I found strange in participating in these activities is that sometimes the undergrads I interact with seem to look up to me in a way, or think that I know things because I am a grad student. One of the students was talking to me the other day about imposter syndrome and comparing yourself to other people, and I ended up saying something along the lines of “Oh, I totally understand that feeling because I used to do that too. But honestly, you can drive yourself crazy comparing yourself to other people – I know because I have done it too, but I realized it was no longer serving me, and I realized I don’t have to be this ‘star student’ to still enjoy what I’m doing.” “That’s SO true,” she had responded sincerely, and meanwhile I was internally panicking during this entire interaction. It was different than listening to a friend, someone who considers me a colleague, and I was suddenly aware of the power dynamic and how much responsibility I had. I think because I’m currently a woman in a grad school program in a related field, some of these women who have goals of grad or med school see me as a sort of safe person to vent to who knows what it’s like to go through this kind of application process and how demoralizing it can be. I was quite nervous about saying the right thing, and having the right mix of relatability and encouragement – all without sounding too preachy or pretentious. When I talked about this later at pub night later with a sixth-year in my program, someone who has significant teaching experience, he reiterated that I have the power to reduce these young women’s imposter syndrome in STEM simply by listening to them and encouraging them. Which is exciting, but also intimidating, because just a year ago, I was that undergrad.

Anyways, that is the (long-winded) gist of the updates of my grad school life over the past quarter. I have some ideas for future, more focused posts, but hope to update more often with these topics as they come up! Until then, I have an exam coming up in my cell neurobiology course, a data analysis assignment, and a research presentation coming up. Wish me luck!

First Quarter Research Progress and Ideas

To be honest, I spent most of my first quarter of graduate school on classes, seminars, and getting adjusted to the new environment. However, I did start attending research meetings in a group I am interested in, and I have some ideas for a potential project. I am very excited about beginning this project, and I hope that this coming quarter, I will be able to make more progress. Luckily, there is a postdoc in the group who is also excited about it, and he has been very thorough in providing me with papers to read and feedback on my work. I will begin to describe my progress briefly.

The group I have been working in studies a wide range of systems such as predator-prey dynamics, multi-drug interaction, the relationship between sleep and metabolic rate, and cardiovascular networks. Since there are so many diverse projects happening in our group, our group meetings are split by topic. The sub-group I joined focuses on networks. So far, they have been mostly focusing on cardiovascular networks. They develop models that describe these networks, such as the scaling laws that describe changes in the radius and length of vessels across levels of the network. Then, they test these models against data extracted from 3D images.

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Since my primary interest in biology is in neuroscience, I approached the group to find out if there were any projects in neuroscience. The PI told me that although there are currently no projects in neuroscience in this group, there are mathematical similarities between neuronal networks and cardiovascular networks, and he saw a future in extending the image analysis of cardiovascular networks to neurons.

We can think of a network of neurons like the cardiovascular system, a resource distribution network that is subject to biological and physical constraints.  Deriving a power law relationship between radius and length of successive levels of a vascular network relies on minimizing the power lost due to dissipation while maintaining the assumptions that the network is of a fixed size, a fixed volume, and space filling. This calculation is carried out using the method of Lagrange multipliers, and assuming that the flow rate is constant. The power loss due to dissipation in the cardiovascular network is P = \dot{Q_0}^2 Z_{net}, where \dot{Q_0} is the volume flow rate of blood and Z_{net} is the resistance to blood flow in the network. For a neuronal network, we will use an analogous equation, P = I_0^2 R_{net}, where I_0 is the current, and R_{net} is the resistance to current flow in the network. We will carry out the Lagrange multiplier calculations in a similar fashion to the calculations for cardiovascular networks.

For cardiovascular networks, we use the Poiseuille formula for resistance, which is the hydrodynamic resistance to blood flow in the network. According to this formula, the impedance at a level k in the network is given by Z_k = \frac{8 \mu l_k}{\pi r_k^4}. We can reduce \frac{8 \mu}{\pi} to a single constant C, so this is equivalent to Cl_k r_k^{-4}. Thus, the resistance is proportional to the product of some powers of the length and the radius. If we want to consider a general formula for the resistance, we can consider a formula with powers p and q of of length and radius respectively. That is, our resistance formula at level k is R_k = \Tilde{C} l_k^p r_k^q.

We define the objective function as follows:

    \[P = I_0^2 R_{net} + \lambda V + \lambda_M M + \sum_{k=0}^{N} \lambda_k n^k l_k^3 \]

This objective function arises from the fact we want to minimize power loss, the first term, while imposing the three constraints that correspond to the last three terms: size, volume, and space filling. Each constraint corresponds to a Lagrange multiplier. The last constraint comes from the fact that a resource distribution network must feed every cell in the body. This, each branch at the end of the network feeds a group of cells called the service volume, v_N, where N is the terminal level, and the number of vessels at that level is N_N, so the total volume of living tissue is V_{tot} = N_N v_N. If we assume that this argument holds over all network levels, we have N_N v_N = N_{N-1} v_{N-1} = ... = N_0 v_0. We assume that the service volumes vary in proportion to l_k^3, so the total volume is proportional to N_kl_k^3. Our objective function has N terms related to space filling, since the space filling constraint must be satisfied at each level k. We assume that the branching ratio is constant, so the number of vessels at level k is n^k. We can define the volume as \sum_{k=0}^N N_k \pi r_k^2 l_k.

Note that we are defining the constraints the same we we did for vascular networks, but it is unclear whether these assumptions are accurate for neuronal networks. However, for the sake at arriving at a preliminary theoretical result for the scaling of neuronal networks, we will keep constraints.
The total resistance at each level is the resistance for a single vessel divided by the total number of vessels, that is, R_{k, tot} = \frac{\Tilde{C} l_k^p r_k^q}{n^k}. The net resistance of the network is the sum of the resistances at each level, so R_{net} = \sum_{k = 0}^N \frac{\Tilde{C} l_k^p r_k^q}{n^k}. If we define new Lagrange multipliers, \lambda' = \pi \lambda, we can rewrite the objective function as follows:

    \[P = I_0^2 \sum_{k = 0}^N \frac{\Tilde{C} l_k^p r_k^q}{n^k} + \lambda' \sum_{k=0}^N n^k r_k^2 l_k + \lambda'_M M + \sum_{k=0}^{N} \lambda'_k n^k l_k^3 \]

To normalize further, we can divide by the constant I_0^2\Tilde{C}, since the current is constant, and absorbing this constant into new definitions of the Lagrange multipliers, we get:

    \[P = \sum_{k = 0}^N \frac{l_k^p r_k^q}{n^k} + \Tilde{\lambda} \sum_{k=0}^N n^k r_k^2 l_k + \Tilde{\lambda}_M M + \sum_{k=0}^{N} \Tilde{\lambda}_k n^k l_k^3 \]

To find the radius scaling ratio, we will minimize P with respect to r^k, at an arbitrary level k, and set the result to 0. Thus, we can find a formula for a Lagrange multiplier and derive the scaling law.

So we have:

    \[\frac{dP}{dr_k} = \frac{l_k^p qr_k^{q-1}}{n^k} + 2 \Tilde{\lambda} n^k r_k l_k = 0 \]

Solving for the Lagrange multiplier, we have:

    \[\Tilde{\lambda} = -\frac{qr_k^{q-1}l_k^p}{2n^{2k} r_k l_k} = \frac{\frac{-q}{2}}{n^{2k}l_k^{1-p}r_k^{2-q}}\]

Since this is a constant, the denominator must be constant across levels. So

    \[\frac{n^{2(k+1)}l_{k+1}^{1-p}r_{k+1}^{2-q}}{n^{2k}l_{k}^{1-p}r_{k}^{2-q}} = 1\]

It is useful to consider the case where the resistance is related to the length linearly, that is, for p =1. Thus, we obtain the scaling ratio:

    \[\frac{n^{2(k+1)}r_{k+1}^{2-q}} {n^{2k}r_{k}^{2-q}} = 1 \rightarrow \frac{r_{k+1}}{r_k} = n^{\frac{-2}{2-q}}\]

To find the length scaling ratio, we will minimize P with respect to l^k, at an arbitrary level k, and set the result to 0. Thus, we can find a formula for a Lagrange multiplier, using the formula above, and derive the scaling law.

So we have:

    \[\frac{dP}{dl_k} = \frac{pl_k^{p-1}r_k^{q}}{n^k} + \Tilde{\lambda} n^k r_k^2 + 3\Tilde{\lambda_k} n^k l_k^2 = 0 \]

Solving for the Lagrange multiplier, we have:

    \[\Tilde{\lambda_k} = \frac{-\frac{pl_k^{p-1}r_k^{q}}{n^k} - \Tilde{\lambda} n^k r_k^2}{3n^k l_k^2}\]

Substituting \Tilde{\lambda}, as calculated before:

    \[\Tilde{\lambda_k} = \frac{-\frac{pl_k^{p-1}r_k^{q}}{n^k} + \frac{q r_k^2}{2n^{k}l_k^{1-p}r_k^{2-q}} }{3n^k l_k^2} = \frac{(\frac{q}{2} - p)pr_k^q l_k^{p-1}}{3n^{2k} l_k^2} = \frac{q-2p}{6} \frac{1}{n^{2k}l_k^{3-p}r_k^{-q}}\]

Since this is a constant, the denominator must be constant across levels. So

    \[\frac{n^{2(k+1)}l_{k+1}^{3-p}r_{k+1}^{-q}}{n^{2k}l_{k}^{3-p}r_{k}^{-q}} = 1\]

In the case where p=1, we have

    \[ \frac{n^{2(k+1)}l_{k+1}^{2}r_{k+1}^{-q}}{n^{2k}l_{k}^{2}r_{k}^{-q}} = 1\rightarrow (\frac{l_{k+1}}{l_k})^2 = n^{-2} (\frac{r_{k+1}}{r_k})^q\]

Substituting the scaling law for radius, we have:

    \[ (\frac{l_{k+1}}{l_k})^2 = n^{-2} (n^{\frac{-2}{2-q}})^q \rightarrow \frac{l_{k+1}}{l_k} = n^{-1 - \frac{q}{2-q}} \rightarrow \frac{l_{k+1}}{l_k} = n^{\frac{-2}{2-q}} \]

We can test these calculations for our vascular networks calculation, where q = -4. Our scaling laws for radius and length are \frac{r_{k+1}}{r_k} = \frac{l_{k+1}}{l_k} = n^{-1/3}, as expected.

We will now attempt to repeat these calculations using a resistance formula specific to neuronal networks.

We think of the resistance to blood flow as the resistance due to the viscosity of the fluid. For neuronal networks, we can think of axons and dendrites as wires through which current is flowing. The resistance as the resistance to current flow through the “wire” due to intrinsic properties of the wire. The resistance is given by R_k = \frac{\rho l_k }{A}, where A is the cross-sectional area of the wire, and l_k is the length of the segment at that level. \rho is the intrinsic resistivity of the axon or dendrite, and we are assuming that \rho is constant, meaning that the material is uniform. If we assume that the axons or dendrites are cylindrical, we can define the cross-sectional area as \pi r_k^2 for level k, so the resistance for level k is given by R_k = \frac{\rho l_k }{\pi r_k^2}.

Assuming that the branching ratio is constant, the number of branches at each level is n^k, and the total resistance at each level is R_{k,tot} = \frac{\rho l_k }{\pi r_k^2 n^k}. The net resistance is the sum across all levels, that is R_{net} = \sum_{k=0}^N\frac{\rho l_k }{\pi r_k^2 n^k}.

Our objective function for this case can be derived in a similar manner as in the general case, setting \Tilde{C} = \frac{\rho}{\pi}, setting p = 1, and q = -2, based on the constants and powers for our specific resistance equation. Thus, we have the objective function

    \[P = \sum_{k = 0}^N \frac{l_k}{r_k^2 n^k} + \Tilde{\lambda} \sum_{k=0}^N n_k r_k^2 l_k + \Tilde{\lambda}_M M + \sum_{k=0}^{N} \Tilde{\lambda}_k n^k l_k^3 \]

To find the radius scaling ratio, we will minimize P with respect to r^k, at an arbitrary level k, and set the result to 0. Thus, we can find a formula for a Lagrange multiplier and derive the scaling law.

So we have:

    \[\frac{dP}{dr_k} = \frac{-2l_k}{n^k r_k^3} + 2 \Tilde{\lambda} n^k r_k l_k = 0 \]

Solving for the Lagrange multiplier, we have:

    \[\Tilde{\lambda} = \frac{1}{n^{2k}r_k^{4}}\]

Since this is a constant, the denominator must be constant across levels. So

    \[\frac{n^{2(k+1)}r_{k+1}^{4}}{n^{2k}r_{k}^{4}} = 1\]

Thus, we can solve for the scaling ratio:

    \[ \frac{r_{k+1}}{r_k} = (n^{-2})^{1/4} = n^{-1/2}\]

To find the length scaling ratio, we will minimize P with respect to l^k, at an arbitrary level k, and set the result to 0. Thus, we can find a formula for a Lagrange multiplier, using the formula above, and derive the scaling law.

So we have:

    \[\frac{dP}{dl_k} = \frac{1}{n^k r_k^2} + \Tilde{\lambda} n^k r_k^2 + 3\Tilde{\lambda_k} n^k l_k^2 = 0 \]

Solving for the Lagrange multiplier, we have:

    \[\Tilde{\lambda_k} = \frac{-\frac{1}{n^k r_k^2} - \Tilde{\lambda} n^k r_k^2}{3n^k l_k^2}\]

Substituting \Tilde{\lambda}, as calculated before:

    \[\Tilde{\lambda_k} = \frac{-\frac{1}{n^k r_k^2} - \frac{1}{n^{k}r_k^{2}} }{3n^k l_k^2} = - \frac{2}{3n^{2k}l_k^2 r_k^2}\]

Since this is a constant, the denominator must be constant across levels. So

    \[\frac{n^{2(k+1)}l_{k+1}^{2}r_{k+1}^{2}}{n^{2k}l_{k}^{2}r_{k}^{2}} = 1\]

Thus, substituting in the scaling ratio for radius, we can solve for the scaling ratio for length:

    \[(\frac{l_{k+1}}{l_k})^2 = n^{-2} (\frac{r_{k+1}}{r_k})^{-2} = n^{-2} (n^{-1/2})^{-2} = n^{-1} \rightarrow \frac{l_{k+1}}{l_k} = n^{-1/2}\]

Note that these scaling laws are consistent for the theoretical predictions from our general formulas, for q = -2.

Some of the assumptions we have made for the purpose of these calculations are as follows:

  • The current flow is constant across all levels of the network
  • The axons and dendrites are cylindrical
  • The material of the axons and dendrites is uniform and can be linked to a constant of specific resistivity
  • The network has a fixed size
  • The network is contained within a fixed volume
  • The network is space filling
  • The branching ratio is constant

Particularly in the case of the volume and space-filling constraints, and the constant branching ratio, it is unclear if a neuronal network has the same properties that we assume hold for vascular networks. In addition, it is unclear whether it is reasonable to assume that the current flow is constant. Thus, it might be worth reexamining these constraints and assumptions to add more biologically realistic and relevant ones.

Moreover, instead of focusing on this optimization problem of minimizing power loss, it might be more fruitful to examine a different optimization problem, such as minimizing the time for a signal to travel from one end to another end of the network.

These scaling laws give us some preliminary ideas to work with. We can try using image analysis techniques to measure length and radii of segments of axons and dendrites across levels in images and see whether information extracted from the data supports our theoretical conclusions.

 

References

Savage, Van M., Deeds, Eric J., Fontana, Walter. (2008). Sizing up Allometric Scaling Theory. PLOS Computational Biology.

Johnston, Daniel, Wu, Samuel Miao-Sin . (2001). Foundations of Cellular Neurophysiology. MIT Press.

Undergraduate Research Experience

My most important experience in undergrad was working in a group in theoretical physics studying neurons, both on the level of individual neurons and beginning to build simple models for neuronal networks. My group studied a range of nonlinear dynamical systems, and my research focused on dynamics at the molecular level.

When I first began working in the group, my primary prior experience had been undergraduate coursework in chemistry. I had taken only lower-level undergrad courses in math, physics, and to a lesser extent, biology, and my only programming experience was one week of an online course in Python. It definitely didn’t feel like enough at first, and it was an extremely steep learning curve. After my two years of working there, I picked up a lot of skills in programming, learned some basic neuroscience and physics concepts, was able to put the material from my coursework in mathematics, numerical analysis, and programming into practice, and most importantly, learned how to teach myself new material on the fly.

The data I had access to for my research was current and voltage data from current clamp experiments. This means that during the experiment, a current was injected into a cell, and the resulting potential was measured at discrete time intervals of 0.02 milliseconds. Although we only had data from one of the variables, since the dynamical equation of voltage depends on the dynamics of the gating variables and a set of parameters such as the maximal conductances of the ion channels, we can extract this information from the voltage time series. We do this by minimizing a cost function, which has terms for both measurement error and model error. We fix the measurement error and begin with an initial model error, obtaining an initial guess for the minimum, and then me slowly enforce the model constraints until we arrive at a global minimum. We use this state to estimate the most likely values of parameters and time series for the variables.

The first project I worked on was estimating parameter values for induced human neurons. Our experimental collaborators in neuroscience were able to create these cells by converting human skin stem cells to cells with neuronal properties. They were able to obtain current and voltage data through current-clamp experiments. The goal of the project was to estimate parameters for both healthy cells and cells from Alzheimer’s patients. In comparing the results, if we are able to find separation in the parameter space, we might even use this to classify unknown cells based on their current and voltage activity. Moreover, we can learn more about the dynamics and modify our model for induced human neurons as needed.

To test the validity of our estimates, we use the parameter estimates at the end of our time window and use the model to integrate forward the voltage equation, obtaining a time series prediction for voltage. If these predictions match the data closely, we can place more confidence in our estimates.

Using the simple NaKL model, where we were only considering sodium and potassium currents, we got the following results for predictions:

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As we can see, although the model predicts the spiking regions well, the subthreshold regions are less accurate. As a result, I tried adding a hyperpolarization-activated inward current to the voltage equation, which added two more variables to the system. The results of the predictions using the estimated parameters were as follows:

predict71

Another project I started working on was modeling the network of neurons in HVC, the premotor nucleus of a songbird called a zebra finch. Songbirds are good models for human language learning because male songbirds spend their youth listening to a tutor, producing syllables and listening to themselves, and eventually establishing a pattern of song syllables unique to themselves.

IMG_9192.PNG

Within HVC, there are three types of neurons. The HVC_{Ra} neurons lead to the premotor pathway for the song, the HVC_{X} neurons are essential for learning and memory, and the HVC_I neurons have inhibitory connections with the other two types of neurons.

We built a simple model of the connections with the following assumptions, determined from the results of in vivo experiments:

  1. HVC_I neurons have only inhibitory connections with the others
  2. HVC_{RA} and HVC_X neurons have only excitatory connections with HVC_I neurons
  3. HVC_{RA} have a sequence of excitatory connections with each other that store the bird’s own song
  4. There are no direct connections between HVC_{RA} and HVC_X neurons
  5. There can be multiple inhibitory connections on a single HVC_X neuron
  6. The auditory input, which is converted to a current, directly influences all of these neurons to some extent

Below is an illustration of the simplest form of our model, with only three neurons of each type:

hvcnetworkdiagram

When I was working in the group, we did not yet have experimental data. However, we attempted to create simulated data with pre-determined parameters and use our methods to estimate them. We planned to use the results of these twin experiments to design experiments for our collaborators.

We used song recordings from the lab and extracted pressure wave data from the mp3 files, and then used a transfer function to convert this to a current. Then, we used this current and parameters values we determined, integrating the model’s dynamical equations and obtaining time series data for voltage and the gating variables. In this model, there are nine neurons, and each of these has its own voltage equation and corresponding gating variable equations.

I was only able to complete the twin experiments for this simple model before coming to grad school, but during my time in the group, I developed a script in C that would automatically write the model equations and organize the relevant information into the files we need for data assimilation.

IMG_9193

My code makes use of the connection matrix, where the column on the left refers to the presynaptic neuron and the column on the right refers to the postsynaptic neuron, and the synaptic connections strengths are either 0, signaling no connection, or 1, signaling a connection. The code asks the user to manually list the connections using coordinates.

The code can easily be modified for more complex models, such as varying the size of the connection matrix, or varying the strengths of the synaptic connections. When I first wrote the files for data assimilation for this model with a network that has three neurons of each type, it took a couple weeks to complete manually, with some trial and error. My hope that this code will make it more efficient to run twin experiments for larger and more complex models.

I am happy with the research experiences I have had in undergrad, and I feel that it has prepared me to approach independent research here in graduate school. However, our models are very simple and not very biologically realistic. Since my program has a greater emphasis on not only physics, but biological training, I will be able to understand the properties and behavior of neurons at a deeper level, and develop models that are not simply mathematically elegant, but capture the essence of the biology as accurately as possible.

 

References

Armstrong, E., Abarbanel, H. D. (2016). Model of the songbird nucleus HVC as a network of central pattern generators. Journal of neurophysiology, 116(5), 2405-2419.

Daou, A., Ross, M., Johnson, F., Hyson, R., Bertram, R. (2013). Electrophysiological characterization and computational models of HVC neurons in the zebra finch. Journal of neurophysiology, 110, 1227-1245.

Long, M. A., Jin, D. Z., Fee, M. S. (2010). Support for a synaptic chain model of neuronal sequence generation. Nature, 468(7322), 394.

Mooney, R., Prather, J. F. (2005). The HVC microcircuit: the synaptic basis for interactions between song motor and vocal plasticity pathways. Journal of Neuroscience, 25(8), 1952-1964.

Williams, H. (2004). Birdsong and singing behavior. Annals of the New York Academy of Sciences, 1016(1), 1-30.

High School Science and Research Introduction

When I was younger, the word “scientist” immediately brought to my mind the image of a frenzied, frizzy-haired man in a lab coat. Although I definitely experienced the innate curiosity about the natural world that many children have, I could never picture myself, a goofy teenage girl who loved wearing dresses and writing stories and drawing animals, pursuing science professionally.

This changed in high school, in my tenth-grade chemistry class, when I fell in love with chemistry. Chemistry consumed my life, eating away at my sleep, invading my dreams, and even infiltrating my creative endeavors. I spent my free time drawing brightly-colored tests tubes, atomic models, and molecules, looking up experiments on youtube, and using what I was learning in the classroom to conduct mini experiments in the kitchen. Although I am now pursuing a more theoretical branch of science, I tell people that chemistry was my first love, because the way my thoughts were utterly consumed with this newfound interest was very similar to the way movies portray the infatuation period of a new crush. It was in that class that, for the first time, I began to consider the possibility of a scientific career.

Below, I have shown a painting I made, in the style of Rajasthani Miniature paintings. I tried to show the underlying molecules in Starbucks coffee and nature, including caffeine, oxygen, nitrogen, and water.

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In my senior year, I took a course my high school offered called Studies in Scientific Research, in which students worked on year-long independent research projects of their own choosing. Because of my fascination with caffeine (likely due to my own frequent consumption), I tried testing the effects of caffeine in various forms on fruit fly populations. Fruit flies are useful because their nervous systems are similar to those of humans, and because they move through generations quickly, enabling us to observe potential genetic mutations caused by caffeine. I found that populations in which I introduced a small amount of caffeine had smaller larvae, and in some cases, delayed generational cycles. With large amounts of caffeine, the fruit fly populations almost immediately disappeared.

While I was waiting for fruit fly populations to develop, I also experimented with the caffeine power I had and was able to make interesting looking crystals using different solvents.

Since I conducted this research independently and with little guidance from faculty, and being an average high school student, I was very inexperienced, there is a good chance that some of these rather qualitative results were subject to confirmation bias. However, it was a good introduction for me to the world of scientific research, where information is multifaceted and messy, unlike information presented in the classroom and in textbooks.

The image below is a self-portrait I painted in senior year, of course, featuring fragments of the caffeine molecule.

Screen Shot 2019-01-06 at 7.53.26 PM

Although I am no longer studying chemistry, I have realized that a lot of the critical thinking skills I developed in my AP Chemistry class have helped me in my process in undergraduate coursework and even research, and I will always hold fascination and interest for scientific studies on the molecular level. My hope, also, is that through this blog, I will be able to keep in touch with my artistic side, using art as a tool in my personal process as a developing scientist.

Introduction

Hello to everyone reading this!

I am a first-year grad student in California, and I decided to create this blog to document some of my experiences on this path working towards becoming a scientist. I would like to use this page as a record of some of my ideas in research, as well as some personal reflections about research, classes, teaching experiences, social experiences, and pursuing hobbies.

My department, Biomathematics, is a small basic science research department within the school of medicine. It focuses on theoretical, computational, and statistical modeling in biology and biomedicine. My research interests are in neuroscience, and the project I have been starting to work on this year focuses on applying tools from physics and applied mathematics to model neuronal networks.

I started college as a chemistry major, but after a while, I realized that while I loved the theoretical side of chemistry, experiments were very much not my strong suit. I changed my major to mathematics/applied science, which allowed me to take theoretical chemistry classes along with a set of courses in applied mathematics. I have always found myself interested in neuroscience, and in my last two years of undergrad, I worked in a research group that studied neurons and neuronal networks from the perspective of theoretical biophysics. There, I picked up a lot of skills in programming and applied mathematics. More than anything, I learned how to find the background information I need for a given task, which has been tremendously helpful transitioning into graduate school. 

Aside from curiosity, my main motivation to study neuroscience comes from a desire to improve our understanding of the brain and mental health from a quantitative perspective. Mental health diagnoses are often based on self-reported qualitative data such as questionnaires, which are imprecise and very susceptible to bias. I believe that a greater understanding of the brain and cognitive processes from a theoretical perspective could not only better inform diagnosis and therapeutic intervention, but could reduce the stigma surrounding mental illness. Mental illnesses such as depression, anxiety, attention-deficit hyperactive disorder (ADHD), and obsessive-compulsive disorder (OCD) are often not taken as seriously as physical illness. As a result, many people suffer in silence and do not seek the treatment they need. A greater understanding of the mechanistic aspects of cognitive disorders is, I believe, a step towards recognizing the biological basis of mental illnesses and validating the health concerns of those affected. It motivates me to think about this as a long term goal, and that my studies in science are not only for my own benefit, but towards the benefit of society as a whole.

During my free time, I like finding content on the internet, in the form of blogs, art, and youtube videos. Since school is obviously a large part of my life, I like content about college and graduate school. I have found some content about life as a STEM student in graduate school, and I have felt a sense of inspiration and motivation from watching others working towards their research goals while simultaneously pursuing their hobbies. However, since my field, the interface between biology and mathematics, is relatively new, I rarely find content from students who are studying similar things that I can relate to. So I decided that if it doesn’t already exist, why not create it?

I believe it will be helpful for me to have a record of my progress in learning the material I need to know for my research, and writing things out in a pedagogical way would probably aid my own understanding of the things I’m learning. I also think that it will help me hold myself accountable, not only for my research progress, but also towards personal goals and hobbies, such as drawing and painting, swimming, dance, making new friends, and putting myself out there in the queer community.

It is likely that this blog will mostly be for my own record, and maybe some of my friends who might be interested in what I am doing. However, part of the reason I found it difficult to identify with other people in STEM is that I am often surrounded by peers who are very different from me. I have often benefitted from meeting other women, people of South Asian origin, and queer people in my field, and I know from my own experience how important representation is. I would love to know if anyone relates to any part of my experience, so please do not hesitate to contact me.

Here’s to a fruitful new year, and I am excited to begin this new journey.