Asexual Awareness Week

Happy Ace Awareness Week! It’s four weeks into our ten-week quarter, and between my Machine learning class, working on the last edits for my paper, my job, and my board position, I have been making time to attend a weekly virtual space hosted by the LGBTQ center at my university: the Ace/Aro space. Today, in honor of Ace week, we had an Ace Awareness Social, which was incredibly empowering and affirming. I’m still processing my gratitude for this space, and I thought I would take some time to reflect on it a little.

This is a part of my identity that I haven’t discussed with most of my friends in-depth, but since it is global Ace Week, and this quarter is the first time in my life that I have actively sought out Ace-centered social spaces in my university, I thought I would dive in and try to unpack some of my experience. Keep in mind that this post is only one person’s experience, and I still have a lot to learn about myself and others who identify on the spectrum with different experiences. If you want to learn more, there are a myriad of more comprehensive resources. With that, I will get started on discussing the basic theory and my experiences.

From what I understand, the basic pillar behind Asexual Spectrum theory is the split attraction model. The main idea behind this is that romantic and sexual attraction are separate from one another. The Ace community exists on a spectrum; there are some asexual people who are romantic, who date and have romantic feelings and deep emotional connections and commitment with little or no sexual desire. There are also people who are aromantic, who have platonic relationships, no relationships, or purely sexual relationships. And there are people who lie somewhere in the middle, people who experience sexual and/or romantic attraction very rarely, and people who are demisexual or demiromatic, forming sexual or romantic attraction, respectively, only after forming a close emotional bond.

Typically, in mainstream culture, we tend to view romance and sex as closely intertwined, or at the very least, romance always seems to have some sort of sexual undertones. For me, these things have always been separate, and low sexual attraction has always been a point of isolation for me even in the queer community. With the hypersexual and sex-positive, colorful environment, even though I love it and cherish and respect it, I often feel there is something missing for me. It has been extremely affirming for me to find spaces this quarter to discuss these things and hear like-minded experiences.

Growing up, as many young women are, I was taught that if a boy or a man asked for my number or asked me on a date, to say “I’m flattered, but no, thank you.” I memorized these lines and recited them, without giving thought to what I wanted. I felt confused when girls my age started showing interest in boys, and during sleepovers, I would dread the inevitable question: “who do you like?” When boys showed interest in me, it flew completely over my head, and I only became aware of it when my friends pointed it out to me. I tell people that my first love was Chemistry, because once I discovered my passion for science, I didn’t have eyes for anything else. In my senior year of high school, the running joke amongst me and my friends was that I was asexual, because I was the only one in the group who had absolutely no interest in dating.

In college, things became a little more complicated. In my first year, for the first time, lived in a dorm with fifteen other girls. I was in a triple with two other girls, and the three of us were close with one of our floormates and adopted her as our “honorary roommate”. The four of us did almost everything together, from late-night study parties and microwave cooking to exploring gelato shops, restaurants, and beaches in San Diego. After almost a year of being friends, I started to develop a stronger emotional attachment to one of my roommates. At the time, I wasn’t aware that it was romantic, because I didn’t feel any sexual attraction towards her. She was a close friend, and we had edited each other’s essays for our college humanities courses and confided in each other about our vulnerabilities. She was the only person I had trusted with my high school trauma, and she seemed to be the only one who could understand the depth of my emotional experience. It was confusing because it didn’t feel to me like more than a friendship, except for the fact that things were suddenly awkward. I started to become shy around her. When we were hanging out in a group, I became self-conscious. I noticed her from the corner of my eye, and I talked to everyone else in the group but her. “Why are you being weird around me randomly?” she’d asked playfully. At some point, we talked about it. I didn’t understand my feelings at the time, but I explained that I felt more emotionally attached to her than my other friends. She told me that she would always be my friend, but that she preferred to set boundaries in friendships. It was confusing and painful at the time, but eventually, our relationship repaired itself and I grew from it.

I was still not willing to admit that my feelings were romantic, but a few months later, I questioned my sexuality. I downloaded dating apps and tried talking to women. I matched with a lesbian woman who was around my age, who was beautiful by all conventional standards, and who had similar interests and hobbies, including pets. On paper, it seemed perfect. We made plans to meet up with our dogs at a nearby dog park. I enjoyed talking to her and I enjoyed her company, but I didn’t feel any spark. We became friends and kept in touch, but it never became anything more than that. I concluded that it was just a phase, and I didn’t actually like women that way.

A couple of years later, I tried dating men, and was met with the same disappointing results. I liked them as people and understood they were attractive by society’s standards, but I didn’t feel any sparks. I remember telling my friend that I was confused because the guy I was seeing kept texting me between dates. I was going to see him in less than a week. Why was he so clingy? My friend burst out laughing and told me that was normal for guys you’re dating.

Although I’d had admiration crushes on my female teachers in the past, a lot of it had more to do with wanting to be like them and seeking their approval. The summer before my last year of college was the first time I had a crush on another woman, which made me realize that I definitely liked women. Suddenly, I became a clingy texter, much like the guy I had been seeing. I wanted to talk to her all the time, but I convinced myself that it was just because I admired her and really wanted to be her friend. There was so much shame and unpacking my own trauma that came with that experience that I denied it for several months. But finally, during the spring break before my last quarter of college, I watched the movie “Love, Simon,” a coming of age story about a teenage boy and his process of coming out to his friends and family. I was inspired by this movie, and I joined a support group in the LGBT center in my university, which helped me process my attraction to women.

About a month before graduating, I finally came out to one of my friends. She’d always teased me about men, and it never even occurred to her that I might like women, which made the experience scarier. But she was incredibly supportive, and she told me that she wanted to take me to a gay bar, which remains one of the only girl bars in California. When we reached the bar, she looked around at all the women and asked me “So, any prospects?” Although I was extremely moved by her support, the idea that one could be attracted to someone they’d never even spoken to baffled me.

When I came to grad school, I became a lot more involved in the LGBTQ+ community on campus. I made a lot of new queer girl friends, and I felt a sense of belonging from being in spaces where attraction to women felt so normal. But even in those spaces, I found myself suppressing parts of my identity in order to fit in. I realized that most people form attraction upon first glance, that they can immediately tell from looking at someone whether or not they would date them. This is something I’ve never been able to relate to. Even during the times when I’d experienced attraction to women, I never really had the thought that I wanted to date them or become physically intimate – it was more that I had affection for them and wanted to get to know them better – but on a different level than with friends. Although a lot of my attraction was based on feeling emotionally connected, I recently learned that in asexuality theory, there are, more complex forms of attraction, such as aesthetic attraction. You can be drawn to someone because of how they look, without the underlying desire to be physically intimate.

I’ve realized that part of the reason dating apps never seem to work for me is because the intimacy feels forced. In order to have genuine intimacy and a connection that doesn’t feel like a chore, I need to get to know someone on an emotional level, in a setting that doesn’t feel like there is pressure to date or escalate things physically. However, to avoid complicating my friendships by developing feelings, I have started keeping more emotional boundaries with all of my friends. Sometimes, I wonder if I will always be lonely. But today, something a woman in the Ace Awareness Social said really resonated with me, about demiromanticism, and wondering if romantic feelings arose simply because society puts pressure on us to “find someone.”

I’ve thought about that often, whether I really want to “find love” or I just romanticize the idea of falling in love because I’ve never experienced that deep, powerful feeling that they show in the movies, where you love someone with all their heart and they love you back in exactly the same way and you lean on each other and support each other through all your endeavors. I’ve spent so much of my life taking care of myself and focusing on school that I often feel that having someone else there would make me feel stifled. But then looking around and seeing everyone else in love makes me feel pressure to find “the one.” I know that at some point, most of my friends are going to find “the one” for them, and since society tends to place romantic relationships high on a pedestal over friendships, I sometimes fear that I’m going to be left behind.

When I first started grad school, I considered attending the weekly Ace Space, but it was at 11 am on Friday, and I always had a class or some meeting or the other during that time. This is the first quarter where I don’t have a full course load, and the online setting makes my schedule even more flexible. I’m very grateful that I’ve been able to find the time to squeeze it in. Attending the weekly space and this Ace Awareness Social has been extremely powerful, because I’ve realized that there are other students in my university who share my experiences, whose lives aren’t centered around physical intimacy and they’re okay with it. I’ve started to grow fond of this space where students get into passionate discussions about tea and animals and aesthetics and ice-cream flavors, as well as deeper discussions about self-discovery and finding our place in the world. With the pandemic, the election, uncertainty about the future, and navigating this perpetual feeling of being an outsider in society, it’s easy to feel lost. But during this month, the Ace community has been an anchor that has been keeping my hope and motivation alive, and for that, I am extremely thankful.

NSF Fellowship Application Tips

Around this time last year, I applied to the National Science Foundation Graduate Research Fellowship Program (NSF GRFP). It has a pretty low acceptance rate and is highly dependent on factors outside of our control, such as the review panelists we are randomly assigned and their moods on the day they read our statements. I knew it was a crapshoot, and I was mostly applying because, as a second year, it was my last chance to do so. Getting accepted for the fellowship was a very pleasant surprise. It has made me feel a lot more confident in my abilities and career goals and has made me somewhat more motivated to work through these very difficult times.

I was the only person in my department who applied last year, and most of the resources I used for my application came from online, and from the advice and examples of successful statements by my seniors in Queer organizations on campus I have been participating in. I am also very thankful for my advisor, who gave me thorough suggestions on my proposal. I know at least one person in my department is applying this month, and I would like to pay forward the support I’ve received in the ways I can. I have started a part-time position at my university campus Graduate Writing Center, where I will be reading other students’ statements and providing them with feedback and support. I will not be publishing my statements online, but I will provide some general suggestions and strategies that I have learned here. If you would like to see my statements, you can feel free to contact me, either through the contact form or through other means if we already know each other! If this is helpful for you, my application was in Mathematical Sciences – Mathematical Biology.

Tip # 1: Read the program solicitation. Read all of it. Make sure you understand exactly what they are looking for. The two main review criteria for this fellowship application are Intellectual Merit and Broader Impacts. Make sure you devote enough in your statements to the Broader Impacts portion, as this is a common shortcoming of many applications.

Tip # 2: This is not the time for modesty! If you were meeting a new friend for coffee or going on a date, it might be a good idea not to rattle off a laundry list of all your accomplishments. But you are not trying to get the review panel to like you as a person. You are trying to convince them that you are worth throwing government money at. Make sure to list everything you’ve ever done, especially when it comes to publications and presentations. I will say right now that I did not have any publications when applying, but I am currently working on a first author publication (fingers crossed that it’ll be submission-ready this month!). So I listed this tentative paper with the year 2020, and wrote In Preparation. I would highly recommend this, especially if you currently do not have any publications, or if you are in the process of preparing a first-author publication – which often carries more weight. Also, make sure to list every poster and/or oral presentation you’ve ever done, even if it was just a department-wide poster session or presentation and you don’t think it was a big deal. This is not the time to leave anything out.

Tip # 3: Make sure you give your letter of recommendation writers enough time to write letters, and make sure they are people who know you and know your research well. As a general rule, I would suggest asking them at least a month or three weeks in advance, although earlier is probably better. I would also suggest giving them reminders as the deadline approaches, as you want to make sure everything is submitted on time. It is probably not the best idea to ask a random professor that you never spoke to but got an A in their class. You want to have someone who can vouch for your abilities in research. In addition, make sure to mention to your writers about the Intellectual Merit and Broader Impacts review criteria. The reviewers will be looking for both these things in your letters as well as your statements. If they aren’t familiar with your outreach work, provide them with a CV and/or description of your activities. One thing that I think really helped my application was getting a letter from a woman who was a postdoc in my undergrad lab and is now a tenure-track faculty member. She could speak to my research abilities, but also about the conversations we had as fellow women in a field where there aren’t many (theoretical physics). I also mentioned in my personal statement how seeing that representation in my undergrad lab encouraged me to apply to graduate school and pursue a physics-centered research group.

Tip # 4: Create a narrative about your scientific journey. If you are applying for this fellowship, it is likely that you have a range of professional experiences before this point, whether it is working in a lab, industry, healthcare, or peer-led projects. There is probably something that you’ve gained from each of these experiences that have led you to the project you are proposing today. Make sure that everything you are listing somehow ties into skills or perspectives you’ve gained that have made you more able to conduct the project you are proposing. Make sure you don’t list anything without somehow tying it in to how it has shaped you as the researcher you are today.

Tip # 5: For Broader Impacts, while it might be helpful to mention your own personal adversities and minority status, what will be even more useful is to list the ways that you plan to uplift other marginalized groups on a broad level. If you are not a member of a marginalized group, talk about initiatives you’ve taken to support those in marginalized groups throughout your career, and how to plan to continue doing so as you progress. If you are a member of a marginalized group, a good way to mention it is to bring it up in the context of outreach organizations you’ve participated in, and how you plan to use representation to encourage others in STEM, such as recruiting people to your program and increasing retention by making workspaces safer for marginalized people. If you identify as LGBTQ+, but you have never participated in and do not plan to participate in identity-based orgs, I would suggest not including it. However, if you were inspired by a talk by an LGBTQ+ identifying faculty member and it has shaped your confidence and pursuit of your career in some way, that could be a powerful thing to include.

Tip # 6: If there are any gaps in your records, such as lack of publications due to time limitations in your undergraduate research or lower grades because of some personal and/or financial adversity, I would include some kind of explanation in your personal statement. For example, I included the two projects I was involved in during undergrad, which have stalled in the research group in favor of other projects and my contributions were never published. However, it is best not to make the hardship the focus of your statement and delve too much into it.  Instead, you can use this as a testament to your resilience and persistence, something that is incredibly important, as research is hard and will involve a lot of failures that you will have to be prepared to overcome. Remember the purpose of the application, which is convincing a panel of strangers who have never seen you to throw money at your project. You want to make sure that everything you include in your personal statement has some purpose that is highlighting either your intellectual merit or your potential to benefit society as a whole. The overall feel of the statements should be positive.

Tip # 7: For your research statement, I would recommend organizing it in pieces. What I did was start off with a biological introduction, lead into a broad question, and three sub-projects that fall under the umbrella of addressing my broad question. I then created separate paragraphs for each of these three sub-projects. It can be helpful to use bold or italic font to highlight these themes, and the specific steps you plan to take to address these things. You want to show that you have thought about methods, and especially if you are already a grad student, show the panel that the institution you are in has the resources to help you carry out your project. The more clarity and organization you have in laying out your plan, the better. It could be helpful to provide a figure or an equation (if you are in a more mathematical field such as mine). Make sure to address broader impacts of the research, as well as potential broader impacts that come with communicating the research and recruiting and mentoring undergraduate students participating in your research.

Tip # 8: If you don’t get the fellowship, DON’T BE DISCOURAGED. It does not mean anything about you as a scientist. There are so many faculty members I admire and respect who have been rejected by this fellowship, but they still went on to be amazing scientists. There are peers of mine who deserve it just as much as I did, if not more. It is a very random process! I also know someone whose labmate applied one year and got rejected, and applied the next year with nearly the same application and got accepted. That just goes to show that getting accepted and rejected has so much to do with factors that are out of your control. It is always a good idea to try, because you never know (for the same reasons), but just know that even if you don’t get it, you are incredibly awesome and you can still do amazing science!

I hope this was helpful, and feel free to contact me for any feedback! Also, know that these tips are just one person’s opinion, and there are many more resources for advice and support! I will include some that I have personally used:

NSF GRFP Website

Tips Websites:

https://www.alexhunterlang.com/nsf-fellowship

http://www.malloryladd.com/nsf-grfp-advice.html

https://www.profellow.com/tips/8-tips-for-crafting-a-winning-nsf-grfp-application/

http://www.christineliuart.com/writing/2018/8/31/advice-for-applying-to-the-nsf-grfp

http://www.clairemckaybowen.com/fellowships.html

YouTube Videos:

Qualifying Exams

Ever since I found out what qualifying exams were, I was absolutely terrified. I remember being an undergrad listening to the grad students from my research group and my TA sections talking about “that test you have to take after the first couple years where you can be tested on literally anything in your field and if you fail, you get kicked out of grad school lol” and, as someone with low to medium key test anxiety, it sounded like my personal kind of hell. Even after going through the grad school application process, my entire future rested on a few hours and a few pieces of paper?

Our written quals are subject-based. We have five core courses: Deterministic Models in Biology, Modeling in Biology: Structure, Function, and Evolution, Stochastic Modeling in Biology, Biomedical Data Analysis, and Computational Algorithms. The qualifying exams for those subjects are offered at the end of August each year. Each subject exam can be assigned a PhD pass, a Masters pass (slightly lower level), or no pass. In order to pass the overall comprehensive exams and remain in the program, a student must get at least three PhD level passes and one Masters level pass. Each student gets two tries to get the required number of passes.

It might seem like these qualifying exams are just like final exams, since, after all, they are single exams self-contained in just five 10 week courses, right? Wrong! What I quickly learned when I entered grad school was how much all of these courses built on years and years of knowledge from high school and undergrad mathematics, how much of this knowledge was assumed background knowledge that was required in order to even begin to comprehend any of the lectures. I realized how kind my undergrad professors and TAs had been in taking the time to rehash material from basic algebra 2, trigonometry, and differential equations in office hours in order to help us understand more difficult material. I missed the warm embrace of assumed ignorance, as my graduate school professors were surprised, disappointed, and in some cases even mortally offended if students showed the slightest sign of rustiness in material we should have learned in our undergrad probability theory courses, in our numerical linear algebra courses, and in our complex analysis courses. It was intimidating, to say the very least, and I certainly did not pick up all of the material from the lectures the first time around. Aside from reviewing all my undergrad course notes and textbooks and completing all the core course problem sets on time, there was so much to do and so much to learn during the quarter with research, preparing for group meetings, and neuroscience electives. Throughout the year, the prospect of qualifying exams seemed to be looming over my head. To the put it in the most graceful and delicate way possible, I was terrified because I didn’t know s***.

During the first part of the summer, before my San Diego Pride trip, I spent my days in the lab, partly working on research and partly reviewing and rewriting all my notes from the lectures and the textbooks. After the trip, after recovering for a few days, I collected myself (physically and emotionally) and collected all the books and notes from undergrad that I thought would be useful in order to decode the notes that I had spent the first half of the summer writing. First, I went through all the problem sets that I had already done during the quarter. Looking at the solutions I had written up (most of which I had forgotten by this point), I tried to recall the theorems from undergrad courses I had used, and the corresponding textbooks that would have more detailed information I could review. After finding these textbooks all around the various bookshelves in the house, I went through the sections I thought would be useful. Below is a stack of the textbooks that I used during this process.

IMG_2345

For me personally, I found that the most gaps in my knowledge were in probability and linear algebra, as my Stochastic Modeling and Computational Algorithms classes (both taught by the same professor) took a lot of the theorems and proofs I learned in those courses for granted.

Something I really came to appreciate through studying for these exams was the sheer intellectual brilliance of my professor who taught my Stochastic Modeling and Computations Algorithms courses. He had written textbooks for these courses, and I am ashamed to admit that during the classes, I had skipped over many of the proofs and examples in the book. A fourth-year student in my department and in my lab, one of the few people who entered my program with more of a biology background than a math background, shared some advice on passing this professor’s exams, for someone with less confidence in their mathematical abilities. “Read all the examples and proofs in the textbook. Make sure you can understand how he got to the conclusions. His books are very dense and compact and he skips a lot of steps. Make sure you know how to fill in the gaps.” This seemed like a daunting task, but this older student (bless his soul) also provided me with a 75 page stack of his notes on the textbook examples from the time when he was studying for quals three years ago, where he filled in the gaps, and I could use them as a reference in case I got stuck. For example, my professor used things like binomial theorem and Taylor expansion approximations to condense a lot of the equations, things that weren’t immediately obvious upon first glance. It was a daunting task, and I didn’t get through the textbooks cover to cover. But I got through a significant portion of the chapters that were more emphasized in the courses, and in the end, I felt like a stronger applied mathematician. My eye had gotten better at recognizing when to use these little tricks to simplify expressions and approximate.

One of the courses, the Structure, Function, and Evolution course, was taught by my own PI, which meant it would be important for me to pass this particular subject because I do want to remain in his lab. The interesting thing about his course is that it was not mathematically the most challenging, although there were some complicated PDEs there when we started talking about diffusion and population genetics. During the classes, when we had problem sets due, his office hours would be completely full with students from the course probing him what exactly he was trying to ask with the questions and trying to decode his convoluted wording. According to the older students in our program, the main difficulty about his exam was interpreting the questions. After looking at some of the past exams, I noticed some general themes, as he tended to ask questions that bridged concepts we learned earlier in the class, relating to network theory and geometry, and later concepts in population genetics.

Biomedical Data Analysis was one that I felt least prepared for, as during the class, we had focused a lot on using R to extract statistical parameters from datasets and fit models to the data, but not much on deriving statistical results. Particularly for me, since I did not have much of a statistics background in undergrad, I felt even more overwhelmed by the unfamiliar vocabulary that my professor assumed we had learned in kindergarten. I studied for this one by making a lot of use of statistics videos on YouTube, which proved to be more useful in aiding my conceptual understanding than our textbook. In addition, our professor was kind enough to host a review session at the beginning of August, which clarified some of the confusion I had.

The older students in our department had told us that very few students in our department had obtained PhD level passes on one of the exams, the Deterministic Models in Biology Course. It was taught by a notoriously tough professor with a background in Physics and a joint appointment in the Mathematics department, and many of the homework problems he had assigned didn’t even have analytical solutions. Since my main goal was to stay in the program and I only had a couple months to study for these exams, I followed the advice of the older students and spent the majority of my time on the other subjects, leaving only a couple of days to study for that one.

Before August, I had spent most of my time studying at home, since I did not want to lug all my undergrad textbooks around on campus. However, I believe that in preparation for exams as intense as these ones, it can be very helpful to study with others to get some feedback and test understanding. During August, I spent a lot of my time studying with my classmate Janet* (name has been changed for privacy). She is the only other PhD student in my year and already has a medical degree and had studied in China, where early math education was much more advanced, so I definitely worried that our study groups would end up being her incredible brain carrying a lot of my dead weight. I had declared my major in math late, and didn’t know what proof by induction was until the latter half of my second to last year of college. Meanwhile, she already knew how to apply proof by induction while crawling out of the womb (okay, this *might* be a *slight* exaggeration, but truly not that far off). However, I think we got a really good, productive, mutually beneficial flow going when we were studying together in August.

At first, I spent the whole day studying in the office, but after a while, I realized that being around the older students stressed me out more than it helped. One of my pet peeves was when they would try to quiz me on random facts from some of the courses, shouting at me things like “Hey, quick, under what conditions can you add the powers when multiplying matrix exponential? When the power matrices commute, duh! Those are easy points you’re missing!” Of course, I didn’t know how to conjure these facts on the spot, but I felt that I did know more than it seemed from my blank looks, because after thinking about it for a moment, I could even conjure a proof for that fact. Although these students were well-intentioned, I knew what worked best for me, and it was not being holed up in the office all day, subject to this stressful banter that left me feeling discouraged about my prospects for the exams.

This roadblock turned out to be a blessing in disguise, because I soon fell into the easy routine of going to the campus at around 7 am, spending the day reviewing past exams in the Biomedical Library until 3 in the afternoon. From around 3-5 pm, I would go up to the office to discuss these exams with Janet. I found that though she helped a lot with the more probability theory related problems, I was also able to help her a lot with my PI’s convoluted wording in his past exams due to the language barrier. Plus, I felt that after working for my PI for almost a year, I got a sense of how his brain worked and the kinds of questions he was asking. I was glad that I could contribute to these study sessions as well as gain from them. I think this process of studying improved my work ethic and anxiety management, forced me to review individual undergrad courses and bring them together in ways that I didn’t know existed, and improved my confidence in problem-solving. Something I think about a lot is how in undergrad, I took a variety of applied math courses, but learned about mathematics mostly from a theoretical perspective without truly understanding how to apply what I learned to research. I think that the process of studying for these five courses in-depth helped me understand not only what mathematical tools are available, but how to use them in real biomedical problems and why they’re important.

Finally, at the end of August, the exams began. We had three, spaced out days of exams. The first two days, we had two exams each. The first day was Stochastic Modeling in the morning and Computational Algorithms in the afternoon. The second day was Structure, Function, and Evolution in the morning and Biomedical Data Analysis in the afternoon. The last day was just Deterministic Models in the afternoon. For each session, we got a 30 minute reading period, where we could read the exam questions and ask the professors for any clarification about the wording of the questions. Then, the three of us were split off into three separate rooms on the floor. I was assigned the classroom where most of our courses had occurred, which was encouraging because I had read research claiming that recall of material during exams can be enhanced if the exam takes place in the same room where learning occurred (to be fair, though, most of my learning had occurred during the summer at home, in the Biomedical Library, and in the office rather than the classroom). We were allowed to eat and drink during the exams, and the older students were very nice and brought us chocolates and water the day of our first exam.

I will admit that after every single exam, I felt terrible and slightly violated, although none more than the last exam, for which I didn’t even finish half of the questions. The good thing is that for a lot of the exams, it was not necessary to answer all of the questions to completion get a PhD pass; it was more important to show how we are thinking – something I had been trained to do since my elementary school math (“show your work!” is permanently etched in my brain).

After the exams, I took a yoga class with one of my college friends, ran a lot, swam a lot, bought all my textbooks, binders, notebook paper, and replenished pencils for my fall classes, worked on my poster for a quantitative and computational biology retreat where I’m presenting at the end of September, and went to a Diversity in STEM Conference in Irvine where I got to catch up with a friend who is a PhD student there. It was busy, but I needed to keep busy so I wouldn’t keep thinking about my anxiety about the results.

A week later, much earlier than I was expecting, I got the results: I got a PhD level pass in all the exams except Deterministic Models in Biology – I got no pass in that subject. I learned that Janet also got no pass in that exam, and since she’s one of the smartest people I know, in a twisted way, it made me feel a little validated that it’s not like only dumb people get “no pass” or something! (I’m saying this slightly in jest, as I do recognize it as a toxic thought, but it will take some more time to train myself to not have these thoughts.) I will be taking a 2 quarter sequence in Mathematical Physics in the Physics department this coming year, so hopefully, I will fill some of the gaps in my knowledge on that side of Biomathematics. Overall, I’m pretty happy with my results in all the other classes, thrilled that I get to stay in the program and continue working on the project I’ve been working on, looking forward to my last year of courses – all very interesting elective courses I chose because of their relevance to my research – and very much looking forward to meeting all the new grad students in my department (there are five, and mostly other women, by the way, which makes me happy).

This coming Monday is our Department Orientation for the new students, and at noon, there is a potluck where everyone from the department meets the new students. I remember last year, when I was a first-year coming into the department, the Vice Chair announced that both the two second years had passed their qualifying exams. It might seem silly, but during my pre-exam anxiety and habitual catastrophic thinking, I remember thinking about how that if I didn’t pass my quals, it would be announced to all the new entering students, and then I would have to go through this same process again with the first-years next summer. I’m very relieved this will not be the case.

Overall, although I know I will probably forget most of what I learned during this summer, it was helpful for me to have a broad idea about the vast breadth of tools in applied mathematics – knowledge that I will be building on this year in my applied math and physics elective courses. I might not remember the details of how to solve every type of problem by hand, but generally knowing what kind of tools are available, I believe, will make me more informed and better able to come up with ideas to tackle new problems in my research in the years to come. The details are things that I can learn on the fly, as needed.

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.

IMG_0054

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.

IMG_0079

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.

0-2

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.

IMG_9207

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.

Network Dynamics, Biophysics, and Mental Illness

This past fall was my first quarter of graduate school, and one of our core courses was Deterministic Models in Biology. For our final project, we chose a quantitative biology paper on a topic of our interest and presented on it to the class. The paper I chose was a review paper, Psychiatric Illnesses as Disorders of Network Dynamics by Daniel Durstewitz, Quentin J.M. Huys, and Georgia Koppe. My undergraduate research focused on the dynamics of neurons at the molecular level, and this paper helped me connect it to specific characteristics of mental illnesses.

This paper proposes that since observable cognitive and emotional states rely on the underlying dynamics of neuronal networks, we should use Dynamical Systems Theory (DST) to characterize, diagnose, and develop therapeutic strategies for mental illness.

The central idea of DST is that there is a set of differential equations that evolve in time. A set of dynamical equations could look as follows:

    \[\frac{dx_1}{dt} = \dot{x_1} = f_1(x_1, ... , x_M, t; \boldsymbol{\theta} )\]

    \[\frac{dx_2}{dt} = \dot{x_2} = f_1(x_1, ... , x_M, t; \boldsymbol{\theta})\]

    \[\vdots \]

    \[\frac{dx_M}{dt} = \dot{x_M} = f_M(x_1, ... , x_M, t; \boldsymbol{\theta})\]

The variables x_1, x_2, ... x_M represent the dynamical variables such as voltage or neural firing rate. These equations describe how each of these variables change over time. \boldsymbol{\theta} represents parameters, fixed values that are properties of the system that do not change over time.

We define a fixed point as the point at which the derivatives of all of the variables are equal to 0. Fixed points are stable if activity converges towards them, and unstable if activity diverges from them. Stable fixed points are called attractors. We can define the basin of attraction as the set of points from which activity converges towards the attractor.

The figure below shows an example of a phase plane, a representation of a space spanned by the two variables of a system. Note that it is possible to use dimensionality reduction methods to obtain visual representations for higher dimensional systems. The arrows show the activity of the system. The blue and orange curves represent nullclines, and along each of these curves, the derivative of one of the variables is 0. The green line represents the barrier between the two basins of attractions. It is possible to cross over this barrier as a result of either external influences or random fluctuations.

IMG_9208

I will discuss some basic neuroscience before going into the dynamics of mental illnesses. There are many ion currents that pass through a neuron membrane such as sodium, potassium, and calcium. The dynamics of these ions are driven by electrochemical gradients. Spiking activity occurs when there is a rapid influx of sodium ions, producing the spike followed by an efflux of potassium ions, returning the membrane potential to the threshold potential.

We can think of a neuron membrane as a capacitor, where positive and negative charges are accumulated on either side. The current is the rate of charge flowing per time, I = \frac{dq}{dt}, and the charge of a capacitor is defined as q = CV. The current through the membrane is this I_m = C_m \frac{dV_m}{dt}. We can think of this system as the circuit shown below:

IMG_9210Because of charge conservation, the sum of the currents across the capacitor and each of the resistors must be 0. In mathematical terms, this is C_m \frac{dV_m}{dt} = -\sum_i I_i.

If we approximate each of these currents as ohmic, they will satisfy Ohm’s law, V = IR, meaning that the current is proportional to the difference between the membrane voltage and the threshold voltage by a factor of 1/R, or in other words, the conductance.

If the conductance were constant over time, these would be linear. However, the conductance depends on the proportion of ion channels that are open and the proportion of channels that are closed, called the gating variables. For example, a sodium current can be described as

I_{Na} = g_{max}m^3h(V_m - E_{Na})

In this system, m and h are the gating variables, and they vary from 0 to 1, and g_{max} is the maximal conductance.

We can think of the dynamical equations for the gating variables as the result of a mass equation. Consider the reaction

Closed \rightleftharpoons Open

Suppose \alpha is the rate of opening of a channel, or the forward reaction above, and \beta is the rate of closing, the reverse reaction above, and both of these rates depend on the voltage. If m represents the proportion of channels that are open, the derivative over time is equal to the  forward rate times the concentration of reactants minus the reverse rate times the concentration of products. In other words:

\frac{dm}{dt} = \alpha(V_m)(1-m) - \beta (V_m)m

Another form of this dynamical equation commonly seen in the literature is:

\frac{dm}{dt} = \frac{m_{\infty}(V_m) - m}{\tau_{Na}(V_m)}

\tau_{Na} is the voltage-dependent time constant, and m_{\infty} is the steady-state proportion of open channels as a function of voltage.

The dynamical equation for voltage for the simple NaKL model is as follows:

    \[\frac{dV}{dt} = g_{Na}m^3h(E_{Na}-V) + g_K n^4 (E_K -V) + g_L (E_L - V) + I_{inj}C^{-1}\]

Neuronal networks are the result of multiple neurons connected to one another through synapses. Pre-synaptic neurons deliverer chemicals, called neurotransmitters, to post-synaptic neurons. Some neurotransmitters are excitatory, such as NMDA (N-Methyl-D-aspartic acid), meaning they increase the likelihood of spiking activity, and others are inhibitory, such as GABA (gamma-aminobutyric acid), meaning that they decrease the likelihood of spiking activity. To describe the networks of neuronal networks, each individual neuron has a voltage equation as illustrated above, with additional terms relating to its synaptic currents. These currents depend on the synaptic conductance, the difference between the membrane voltage and the threshold voltage, the strengths of the synaptic connections, and the fraction of open channels for each receptor. The dynamical equation for the fraction of open channels usually depends on properties of the presynaptic neuron.

So far, the variables we have been considering have been the voltage and the gating variables. In order to discuss the dynamics of mental illness, we must think about another important variable: firing rate. This simply describes the rate of voltage spikes over time. Below is an example of a phase plane, where the vertical axis is the average firing rate of inhibitory neurons, and the horizontal axis is the average firing rate of excitatory neurons.

IMG_9209

In this system, the fixed points can be thought of as memories or goal-states, and we can use this system to consider the effects of the underlying dynamics on working memory or decision making. Increasing the depth of the basin of attraction can have the effect of increasing the stability of the state, while flattening the basin of attraction reduces the stability of the state.

This paper highlights the key role of dopamine in altering these attractor dynamics. Stimulating the D1 dopamine receptors has the effect of increasing firing activity of both excitatory (NMDA) and inhibitory (GABA) neurons. This alters the parameters of the system, in particular, the strengths of synaptic connections, over time. As a result, the basins of attraction are deepened, and the state is more stable and robust to external perturbations or noise fluctuations.

Stimulation of the D2 dopamine receptors has the opposite effect, flattening the basins of attractions. These flat attractor landscapes could lead to disorganized or spontaneous thoughts that can be experienced as hallucinations that are characteristic of schizophrenia. This can also explain the high distractibility in attention-deficit hyperactive disorder (ADHD). On the other hand, Obsessive Compulsive Disorder (OCD), a disorder characterized by rumination, invasive and recurrent obsessions and compulsions, can be linked to deep basins of attractions that are robust to potential distractors. Major Depressive Disorder characterized by a coexistence of rumination and a negative mood with lack of concentration and distractibility, and one can think of it as an imbalance between multiple attractor states.

The main point this review paper aims to illustrate is that in order to characterize and develop treatments for mental illnesses, one must consider the underlying network dynamics. The suggested role of dopamine in altering the depth of basins of attractions suggest that we might try to target the dynamics of schizophrenia patients, for example, through dopaminergic drugs.

I found the process of reading this review paper and the sources it cited extremely helpful for me in improving my understanding of neurons, neuronal networks, biophysics, and nonlinear dynamics, and linking my previous understanding of neurons to cognitive processes, something that I had not fully understood before. Because the review paper goes over the general information, I read many of the papers it cited to find the basis behind some of its claims. However, I still do not clearly understand the mechanism behind the changes in the attractor dynamics. I would like to learn more about how the parameters are changed, and how these changes, in turn, alter the attractor landscapes.

At this point, I believe that the connection between these dynamics and mental illnesses as presented in this paper seems rather speculative. However, I think that as more data is collected and analyzed, and further models are developed to understand the dynamics of neuronal networks, we can glean more insight towards understanding and developing treatments for mental illnesses.

References:

Durstewitz, D., Huys, Quentin J. M., Koppe, Georgia. (2018). Psychiatric Illnesses as Disorders of Network Dynamics. doi: https://arxiv.org/pdf/1809.06303.pdf

Durstewitz, D. (2009). Implications of synaptic biophysics for recurrent network dynamics and active memory. Neural Networks, 22(8), 1189-1200.

Durstewitz, D., Seamans, J. K. (2008). The dual-state theory of prefrontal cortex dopamine function with relevance to catechol-o-methyltransferase genotypes and schizophrenia. Biological Psychiatry, 64(9), 739-749.

Durstewitz, D. (2006). A few important points about dopamine’s role in neural network dynamics. Pharmacopsychiatry, 39(S 1), 72-75.

Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience: MIT Press.

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

Rolls, E. T., Loh, M., Deco, G. (2008). An attractor hypothesis of obsessive-compulsive disorder. European Journal of Neuroscience, 28(4), 782-793. doi: 10.1111/j.1460-9568.2008.06379.x

Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering: CRC Press.

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.