Evelyn Lamb: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians to tell us about their favorite theorems. I’m your host Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. This is your other host.
Kevin Knudson: I’m Kevin Knudson, a professor of mathematics at the University of Florida. How’s it going?
EL: Great. I’m excited about a new project I’m working on that is appropriate to plug at the beginning of this, so I will. So I’ve been working on another podcast that will be coming out in the fall, may already be out by the time this episode is out. It’s with the folks at Lathisms, that’s L-A-T-H-I-S-M-S, which is a project to increase visibility and recognition of Hispanic and Latinx mathematicians. And our guest today is going to be a guest on that podcast too, so I’m very excited to introduce our guest, who is Erika Camacho. Hi, Erika. Can you tell us a little bit about yourself?
Erika Camacho. Sure. So I’m an associate professor at Arizona State University. My concentration is, well I’m a professor of applied mathematics, and my concentration is mathematical physiology, mainly focusing in the retina and modeling the retina and the deterioration of photoreceptors. And I’m in the west campus of Arizona State University, which is mainly focusing, it’s both a research and student focused institution, so it’s kind of like a hybrid between what you would call more of a research place and also a liberal arts education.
KK: Very nice. Which city is that in?
EC: We’re in Glendale, the west valley of Arizona, Phoenix greater area.
EL: I was in Arizona not too long ago, and the time zone is always interesting there because it’s exactly south of Utah, but I was there after Utah and most of the country went to daylight saving time, and most of Arizona doesn’t observe that, so it was kind of fun. I also went through part of the Navajo Nation there that does observe daylight saving time, so I changed time zones multiple times just driving straight north, which was kind of a fun thing.
EC: It is very confusing. Let’s say you have an event that you’re going to, and you’re driving to one where it’s say in some of the Navajo Nation, and you don’t realize that you might miss some of your event because of the time change. You’re just driving and you’re crossing the border where it changes to a different time zone. It takes a while to get adjusted to. I missed a flight one time for the same reason. I was not aware that Arizona didn’t observe daylight saving. Now I’m aware.
KK: I actually have a theory that someone could run a presidential campaign, and their sole platform is that they would get rid of daylight saving time, and they would win in a landslide.
EL: I mean, people have won on less.
EL: So Erika, we invited you here not to chat about time zones or presidents but to chat about theorems. So what is your favorite theorem?
EC: Before I say my favorite theorem, like I said, I am an applied mathematician. So I focus on modeling. And in modeling, there’s a lot of complexities, a lot of different layers and levels where you’re trying to model things. So many of the systems you’re trying to develop as you create this model tend to be nonlinear models. Many times I’m looking at how different processes change over time. So many of the processes I work with are continuous. So I work with differential equations, and they tend to be nonlinear. Sometimes that’s where the complexity comes in, trying to analyze nonlinear systems, and the most accurate way, the way that we’re going to get the most insight into some of the behavior we’re looking for in terms of physiological systems that relate to the retina and retinal degeneration, one of the things that we’re really looking at is what happens in the long run? How is it that photoreceptors degenerate over time, and can we do something to stop the progression of blindness or the progression of certain diseases that would cause the photoreceptors to degenerate? So we’re really asking what are the long-term solutions of the system, and how did they evolve over time? So we’re looking for steady states. We’re looking for what is their stability and what are the changes in the processes or the mechanisms that govern those systems, which usually are defined by the parameters that end up actually leading to a change in the stability of the equilibria? And they could take the system to another equilibrium that is stable now. So in physiological terms, to another pathological state, or another state that we could hopefully do a few strategies to prevent blindness. So that’s the setting of where I come from, and when you asked me this question, what is my favorite theorem, it was hard because as applied mathematicians we utilize different theory. And all the theory is useful, and depending on what the question is, then the mathematics that are utilized are very different.
So I thought, “What is the theorem that is utilized the most in the case where we’re looking at nonlinear systems and we’re trying to analyze them? And one of the most powerful theorems out there, which is one that has almost become addicting, that you use it all the time, is the Hartman-Grobman theorem. I say addicting because it’s a very powerful theorem. It allows us to take a nonlinear system and in certain cases be able to analyze it and be able to get an accurate depiction of what’s happening around the equilibrium point, what is the qualitative behavior of the system, what are the solutions of the system, and what is their stability. Because you’re looking at, in most cases, a continuous system, you can map it and be able to kind of piece it together.
EL: So it’s been a long time since I took any differential equations. I’m a little embarrassed, or did any differential equations.
KK: Me too.
EL: So can you tell us a little more about the setting of this theorem?
EC: So the Hartman theorem, like I said, is a theorem that allows us to study dynamical systems in continuous time. It’s very powerful because it gives us an accurate portrayal of the flow, solutions of the nonlinear system in a neighborhood around a fixed point, the equilibrium, the steady state. So I’m going to be using fixed point, equilibrium, and steady state interchangeably. In some cases, and in the cases where it does help, is in the cases where the equilibrium that we’re looking at, the eigenvalues of the linearized system, or the nonlinear system that we’re looking at, actually has nonzero real part. In other words, we’re looking at hyperbolic equilibrium points. That’s when we could actually apply this system.
EC: This theorem, otherwise we cannot. That is, for certain cases? The standard technique is you look at your nonlinear system, you linearize it through a process, and you’re able to then shift your equilibrium point to the origin, and now you’re considering the linearized system, and that system, the Jacobian you obtain through the linearization has eigenvalues that have nonzero real part. Then you’re able to apply the Hartman theorem, which tells you that there is this homomorphism from the nonlinear system, the flow and the solutions of the nonlinear system, locally, to the actual linear system. And now everything that you get that you would normally be able to see analyzed in a linear system locally, you’re able to do it for the nonlinear system. So that’s where the powerful thing comes in. Like I said, the gist of it is that the solutions of the nonlinear system can actually be approximated by a linear system, but only in a neighborhood of the equilibrium point. And this is only in the case where we have hyperbolic equilibrium fixed points. But that is very powerful because that allows us to really get a handle on what’s going on locally in a neighborhood of the steady state. For us, we’re looking at, say, how certain diseases progress in the long run, where are we heading? Where is the patient heading, in terms of blindness? And it really allows us to be able to move in that direction in terms of understanding what is going on. And like I said, it’s powerful not just because it’s telling us about the stability, but it’s actually telling us the qualitative structure of the solution and the behavior, right, of your solutions, locally are the same in the linear case and the nonlinear case because of this topological equivalence.
KK: That’s pretty remarkable. But I guess the neighborhood might be pretty small, right?
EC: Right. The neighborhood is small.
EC: In nonlinear systems, you have plenty of different equilibrium points around those neighborhoods, right, but again remember that your solutions in the phase space are changing continuously, so you are able to kind of piece together what is going on, more or less, but for sure you know what’s going on in the long term behavior, you know what’s going on around that neighborhood, and for given initial conditions, which is really key in math applications because sometimes we’re asking what happens for different initial conditions. What are the steady states? What do the solutions look like in the long run? What do things look like, what is going on for different initial conditions?
KK: So if you’re modeling the retina, how many equations are we talking? How big are these systems?
EC: Well, that’s the thing. In the very most simplified case, where you’re able to divide the photoreceptors into the rods and cones, then you have two populations.
EC: And in one of the cases we’re looking at the flow of nutrients, so we are also considering the retinal pigment epithelium cells, which is another population, so you have three equations in that case. So that’s a more simplistic situation, but it’s a situation where we have been able to really get a sense of what’s going on in terms of degeneration in these two classes of photoreceptors that undergo a mutation. So one of the diseases I work on is retinitis pigmentosa, and the reason why that is a very complicated case that we haven’t been able to really get a handle on and be able to come up with better therapies and better ways of stopping degeneration of the photoreceptors—in fact there is no cure for stopping photoreceptors from degenerating—is because the mutation happens in the rods. The rods are the ones that are ill. Yet the cones die, which are perfectly healthy. And trying to understand how is it that the rods actually are communicating with the cones that ends up also killing them is an important part, and with a very simplistic model for an undiseased case, we were able to actually, before biologically this link was discovered, that in fact the photoreceptors produced this protein that is called the rod-derived cone viability factor, that helps the cone survive, and we were able to show that mathematically just by analyzing the equilibria and being able to look at different things in the long run, and the invariant spaces, and being able to show what we know just by basic biology of what happened to the rods and the cones and then realized that the communication had to be a one-way interaction from the rods to the cones. So that’s one of the models that we have. And then once we had that handled, we were able to introduce the disease and look at a four-dimensional system.
Now we’re looking internally at the metabolic process inside the cones because there’s a metabolic process. So the rods produce this protein. How is that protein taken by the rods, and what does it do once it’s inside the rods? For that we really need to look inside the metabolic process and the kinetics of the cones and also the rods. There, if you’re just considering the cones, you’re looking at 11-12 differential equations.
EC: With many parameters. So at that point we’re going to a much higher dimension. And that’s where we currently are. But that has given us a lot of insight, not just in how the rods help the cones but how is it that other processes are being influenced, getting affected? And again, where the Hartman-Grobman theorem applies is to autonomous systems, where time is not explicit in the equations.
KK: This is fascinating.
EL: Math gets this kind of rap for being really hard, but then you think, like, math is so much simpler than this biological system. Your rods being sick make your cones die!
EC: But I think the mathematics is essential. There’s a big cost in taking certain experiments to the lab, just to be able to understand what is going on. There’s a cost, there’s a time dependence, and math bypasses that. So once you have a mathematical model that is able to predict things. That’s why you start with things that are already known. Many times the first set of models that I create are models that show what we already know. They’re not giving any new insight. It’s just to show that the foundation is ready and we can build on it, now we can introduce some new things and be able to ask questions about things we don’t know. Because once we are able to do that, really it’s able to guide us to places, or at least indicate what kinds of lab studies and experiments should be run and what kinds of things should be focused on. And that’s one of the things we do. For example, one of the collaborators I work with is the Vision Institute of Paris, so the institute and the director there, and the director of genetics as well. And we have this collaboration where I think working together has really helped guide their experiments and their understanding of where they should be looking, just as they helped me really understand what are the types of systems we need to consider and what are the things that we can neglect, that we don’t have to really focus on? And I think that’s the thing, mathematics is really powerful to have in biological system, I think.
EC: And my favorite theorem can be used to gain insight into photoreceptor degeneration in very complicated systems. Another thing that’s interesting about the Hartman-Grobman theorem is that one of the things that is really powerful is that you don’t have to find a solution, a solution to the nonlinear system, to get an understanding of what’s going on and get an insight into the qualitative behavior of those solutions. And I think that is really powerful. Do you have any questions?
EL: So, I mean, a lot. But something I always think is interesting about applied mathematicians is that often they end up working in really different application areas. So did you start out looking at retinas and that kind of biological system, or did you start out somewhere else in applied math and gradually move your way over there?
EC: So when I started in applied math, what I really liked was dynamical systems. Yes, the first project I worked on as a graduate student was actually looking at the cornea and how different light intensities affect the developing cornea. And for that I really had to learn about the physiology of the eye and the physiology of the retina. But then I did that for graduate school, and initially once I went out of graduate school, in my postdoc I was working on how different fanatic groups get formed.
EL: Oh wow, really different application.
EC: Which was in Los Alamos. I was looking at what are the sources of power that allow groups that can become terrorists, for example, to really become strong. What are the competing forces? So it was more a sociological application, but again using dynamical systems to try to understand it. Later on I moved on to a more general area of math biology, looking at other different systems and diseases, but then I went back through an undergraduate project in an REU. Usually the way I work with undergraduates is I make them be the ones that ask the question, that select the application. And I tell them, you have to go learn all about it because you have to come and teach me. And then from there I’m going to help you formulate the questions that can be put into a mathematical equation and that can be modeled somehow. And they were very interested, they wanted to do something with a PDE. And they thought, well, something with the retina. And they thought that retinitis pigmentosa would be perfect for modeling with a PDE and being able to analyze it that way. And then as I learned more about the disease, the interesting thing is that when the cones begin to die, like the rods are the ones that are sick, but when the cones begin to die, there is no spatial dependence anymore. They don’t die in a way that you can see this spatial dependence. It’s really more random, and it’s more dependent on the fact that there is this lack of protein that is not being synthesized by the rods anymore. Many times what happens is there is this first wave of death of photoreceptors where the rods die, and when most of then die, when 90 percent of them are gone, then the cones begin to die. And then there are all these other things about, yes, you can think about wakes and the velocity of them, but there is not this spatial dependence. Initially it is, but that’s when only the rods are dying. But when we are really interested in asking the questions about why the cones die, there’s not that case anymore.
KK: It’s sort of a uniformly distributed death pattern, as it were? What I love about this is, you know, here’s a problem that basically a second-year calculus student can understand in some sense. You have two populations. We teach them this all the time. You have two populations, and they’re interacting in some way. What’s the long-term behavior? But there are still so many sophisticated questions you can ask and complicated systems there. Yeah, I can see why your undergrads were interested in this, because they understood it immediately, at least that it could be applied. And then they brought this to you, and now you’re hopefully going to cure RP, right?
EC: Right, well another thing is that you can understand, and you can use math that is not very high-level to start to get your hands dirty. And for example, now that we’re looking at this multi-level layer where you’re looking at the molecular level and also at the cellular level, then you’re really asking about multi-scale questions and how can we better analyze the system when we have multiple scales, right? And then there are sometimes questions about delay. So the more focused and the more detailed the model becomes, the more difficult the mathematics becomes.
EC: And then there are also questions, for example, without the mathematics, there’s a lot of interesting mathematics going on, I’m sorry, I mean without the biology, that you could analyze with mathematics. We did a project like that with a collaborator where the parameter space was not really relevant biologically, but the mathematics was very interesting. We had all this different behavior. We had not just equilibrium points, but we had periodic solutions, torus, we had all of this, and what is going on? And a lot of this happened in a very small region, and it just became more of a mathematical kind of analysis rather than just a biological one.
EL: Yeah, very cool. So another part of this program is that we like to ask our guest to pair their theorem with something, you know, food, beverage, music art, anything like that. So have you chosen a pairing for the Hartman-Grobman theorem?
EC: I thought about it a lot, because like I said it’s such a powerful theorem, and I go back to the idea of it’s addicting. I think anyone who’s worked in dynamical systems in the nonlinear case in a continuous timeframe definitely utilizes this theorem. It comes to a point where we are doing it automatically. So I thought, what is something that I consider very addicting, yet it looks very simple, right? It’s elegant but simple. But once you have it, it’s addicting. And I could not think of anything else but the Tennessee whiskey cake. Have you ever had it?
EL: No, but it sounds dangerous.
EC: It is delicious. It’s funny, I don’t like whiskey, and I had it when I went to San Antonio to give a talk one time. I was like, well, okay, everyone wanted it, so I decided to go with it. I usually pick chocolate because that is my favorite.
EL: Yeah, that’s my go-to.
EC: I love chocolate. So I said, well, let me try it. It was the most delicious thing. Now I want to be able to bake it, make it. I had a piece, and I want more.
KK: So describe this cake a little bit. Obviously I get that it has bourbon in it.
EC: The way it’s served is it’s served warm, and it has vanilla ice cream. It has nuts, and it has this kind of butterscotch or sometimes chocolate sauce over it. And it’s very moist. It has those different layers. I also think, right, in terms of complexity, it has these different layers. In order to get a sense of the power of it, you have to kind of go through all the layers and have all of them in the same bite. And I feel like that with the Hartman theorem, right, that the power of it is really to apply it to something that has nonlinearity, that is really complex, and something that you know you might not be able to get a handle on the solutions analytically, but you still want to be able to say what is going on, what is the behavior, where are we heading? To somehow be able to infer what the solutions are through a different means, to be able to go around, and it gives you that kind of ability.
KK: And this is where whiskey helps.
EC: Well the whiskey’s the addicting part, right?
EL: So have you made this cake at all, or do you usually order it when you’re out?
EC: Usually I order it when I’m out. But I want to make it. So my mom’s birthday is coming up on August 3rd, and I’m going to try to make it. I was telling my husband, “We’re going to have to make it throughout the next few days because I’m pretty sure we’re going to go through a few trials.”
EC: I can never get it right.
EL: Even the mistakes will be rewarding, just like math.
KK: And again the whiskey helps.
EC: But that’s an interesting question. I thought, what can I pair it with? And the only thing I could think of was something that’s addicting or something that has multiple layers but that all of them have to be taken at once, that you’re allowed to look at all of them at once.
EL: This fits perfectly.
KK: Sounds great. Well I’ve learned a lot today. I’ve never thought about modeling the eye through populations of rods and cones, but now that you say it, I guess sure, of course. And now I have to look up Tennessee whiskey cake.
EL: Yeah, it’s really good. You should try it.
KK: I’m going to go do that.
EL: It’s almost lunch here, so you’re definitely making me hungry.
KK: Well thanks a lot for joining us.
EL: Thanks a lot for being here.
EC: Well thank you so much for having me here. I really enjoyed it.