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.

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.

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.

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.

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.

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.