 OK. OK, so as I mentioned in the snapshot series, what we've done for many years is study chemical systems with the aim of getting some insights into biological systems. And the idea is that certain chemical systems like the Blusoff-Chabotinsky reaction show dynamics, exhibit dynamics, that is remarkably similar to dynamic you see in living systems. The difference is that the chemical system, this BZ reaction, or BZ, is very well-defined. The state variables are simple concentrations. The governing equations are well-established kinetics and transport laws, and that's the difference. So we can really delve in deeply, whereas in biological systems, oftentimes you don't even know the state variables, let alone the governing equations are sometimes even more difficult. So that's the theme. I'd like to start off thanking people because they're the most important. Mark Tinsley has been with me at WVU for many years, and he does most of the work. Annette Taylor is now at Sheffield, and she's involved in a lot of this. Jia Yonghuang is a former grad student, so is Fenghuang, and so is Simba Unkomo, and Yan Tots, and Harold Engel, are colleagues at TU Berlin. And of course, thank the funders without whom none of this would be possible. OK, so to get started, first I'd like to tell you a little bit some basics, and we'll start off by talking about reaction diffusion fronts. In all self-organizing processes, you need to have some sort of feedback, either positive feedback or negative feedback. In this case, I'm describing positive feedback, and this is as simple as it gets. A is a reactant, reacts with B to B to give 3B. So here's the rate law. The rate law for B depends upon B squared. So the more B is generated, the faster it's generated. And so that's like an explosion, except it's an isothermal explosion. And here's your reaction diffusion equation. Here's just fixed second law. And here you can make this one variable, because A and B have to add up to the same. And so when we look at this, the first thing that happens when we spike this with a little bit of B is B starts to spread out by diffusion and starts to grow. It's consuming A, and now it very rapidly goes up to its maximum. And now it's converting this A to B as it propagates with constant velocity and constant waveform. So that's how a reaction diffusion wave, the simplest of all, works. You can make this reaction more complex by just adding the decay of the activator, or autocatalyst, and making it an open system. So we always supply A and B. You can do this, and the engineers do this all the time with continuously stirred tank reactors. But open systems are very important, because we are all open systems. We take in energy, and we expel waste. And so in this case, what's going to happen, we spike this with A here, I'm sorry, B here. And but now this system oscillates in time. And what you're going to see is that the rise is similar, but we have a decay in the back. And what this allows, this is called a propagating pulse, this allows the successive waves. And so that's very important in all, especially living systems, nerve impulses. OK, so here's a famous picture by a famous person, Art Winfrey. Here is, these are thin films of Belusov-Chabotinsky's solution that I'll tell you more about as we go on. And the reaction is set off by imperfections, little inhomogeneities in the solution, or scratches on the glass, or whatever. You can see that they put out waves at different frequencies. If you take one of these waves and you break it, what happens is you get these counter-rotating spirals. And now you see that once things settle down, it's more or less a constant wavelength, no matter. So these don't depend upon any heterogeneity. Of course, spirals are very fundamental and everywhere. And so it's important to learn about them. How does a spiral form? Well, if you have an experiment, we do this in the hands-on sessions. We just kind of tilt it very carefully, and you break a wave. In a simulation, you can break a wave just by removing the top half of the simulation. And so what happens is that the red is the autocatalist, and here it diffuses a head and initiates autocatalysis. So it's self-propagating. And so in the planar region, it's all constant velocity. But at the tip, it can not only diffuse a head, it can diffuse out to the side, so there's less autocatalist, less activator. And so it isn't as fast. And so it falls back. And it falls back. It falls back. And at the tip, there's a zero velocity. And that's how spirals work. The blue is the recovery variable. So you can see the activator here, the inhibitor here. This is a cut through right here. So on this side, you can see that the autocatalist shoots up. In the BZ reaction, it increases by five orders of magnitude and concentration in a millisecond. And then it shoots back down, and the recovery variable resets the clock, basically. So that's the way spirals work. Most spirals and patterns are studied in planar configurations. And so one can also look in a non-planar configuration. Here is a bead similar to the ones that you just saw in the hands-on experiment, about 1.4 millimeters in diameter, about three times as large. And this has a spiral wave on it, starting at the north pole. And you can see through this bead. And so it comes around, crosses the equator, goes behind, crosses the equator again, and goes down to the south pole. And you can see in these snapshots that these are taken at every, what, 12.8 seconds. And bam, bam, bam, bam, bam. This is one period right here. And bam, bam, bam, bam, bam. This is another period. So it's kind of like a spherical barber pole or something like that. So why would anybody be interested in spiral waves on spheres? You have to ask that, right? Well, there's a lot of reason to be interested. And one of the most important reasons is our heart. You, the heart operates by the sinus node, which is a collection of cells that fire periodically. And there's an electromechanical wave that spreads through the atria and then goes down these percunji fibers to the ventricles. And this complex root gives rise to productive pumping of blood. And so normal pumping is like this. It's every a little bit less than a second or so. Here's the simulation. But occasionally, people get this pump, pump, pump, pump, pump in their heart. This is called tachycardia. And it looks like this in the electrocardiogram. And the reason it is occurring is because a wave broke, just like the wave we were discussing. And it started forming a spiral. And the spiral has a period that is shorter than the pacemaker. And so if it's atrial, it's not lethal. Even ventricular, it's not lethal. But the spirals can break up again, and then you're in trouble. Because your heart is working hard, but it's not doing anything useful. It's cardiac fibrillation. And this is a real experiment here. I highly recommend this website. This is from Flavio Fenton and Elizabeth Cherry. And they have done a fabulous job of putting up a website on heart dynamics. OK, so let's move on. One can do other things. Oh, no, I wanted to make another point. So what insights does this experiment give us? Well, this is a normal spiral wave at the North Pole, but the geometry forces an abnormal spiral wave at the South Pole. It is a sink. And so the heart is not a sphere, but it's not planar either. And so spirals must also have a sink of some sort. It turns out this is quite complicated. It involves a spiral that collapses periodically. And there's been a lot of theoretical work on this now. OK, so you could also ask, what is it like? We know that biological systems are cellular. So let's look at excitable media that's cellular. And I learned at some, oh, it was dynamic states in Budapest many, many years ago. I learned that sulfon membranes, ferroin, sticks very well to them. So I came back to WVU and I told Oliver Steinbach, who was with me at the time, that we should use a plotter to draw things with ferroin. He had a much better idea. He says, let's use an inkjet printer. And so we printed an array of triangles and put it on a gel of reactant with catalyst-free reactant and waited to see what happens. You can see that this array of triangles, the triangles are red, are ferroin, and the wave is blue, gives rise to these hexagonal structures. So the local geometry gives rise to global anisotropy. Probably more important, and that's true in the heart, too, because heart cells are elongated and your waves are distorted. But more importantly, these spirals form spontaneously. And I forgot to say, but you can kind of infer it from my description of the spiral, they're not spontaneous in a homogeneous system. You have to set up special initial conditions to get a spiral. You don't here in heterogeneous media. OK, so now I'd like to tell you about how you can use another version of the BZ reaction that's light sensitive to also do some interesting experiments that have some offer insights into biological systems. This special catalyst is a ruthenium compound. And what it does when it is exposed to light, it generates bromide. In this solution, bromide is the inhibitor of autocatalysis. So it serves as the inhibitor. And there's another chemical called bromisacid that's the activator. So we thought it'd be fun to look and see what noise does to propagating waves. And this is actually smooth. My loop in the video puts a stop to it. But we're basically taking a random, on the interval, intensities, putting them in different size blocks and then having a wave propagate through this. Over here is dark. And we make a spiral wave here. And then it propagates in here. And this is the mean zero average of the noise. We've set this up to be sub-excitable. And I'm going to talk more about what that means. But it means that it doesn't support sustained wave propagation. And so the wave comes in here, and it experiences this noise. OK, so here is an example of what we see. The noise level is basically from zero to the maximum that we can achieve. And so with zero noise, this is what a sub-excitable medium does. I just got through telling you that free ends of waves, chemical or heart or any other, curl up to form spirals. But if you make the medium less and less excitable, less and less active, you get to the point that they don't curl up anymore and they just contract. This is like if you had a piece of paper, you lit the corner and the flame burns through the paper. But then you do the experiment again and you use a squirter to make it moist. And you reach a point pretty soon that the wave doesn't propagate through. But it fails. You get wave failure, propagation failure. OK, and so you can see as we add noise, it goes a little bit further. Add more noise, it is sustained. Add more noise, our max, and it's not so good because it's so much that it starts to cause trouble. And so we can just see this. Actually, I think the overlays are more informative than the movies, but here's what the movies look like. And so we figured that the most important thing was to measure the signal at this point where there's no noise and use that as a measure of what the noise is doing. Yes? It's a spiral wave sending in successive waves. Over here there's a spiral that's sending in successive waves. No, the spiral is over here in that black box that I showed you earlier. And it's moving like this. Yes, and this is a channel that it can only take a slice of it because it can't propagate outside of this channel. It can be, it's generally, it should be perpendicular, but things aren't always perfect. OK, and so what happens is if you look at this, if you have a cell size that's 4 pixels by 4 pixels, you increase the noise, you increase the noise, and you reach a maximum of about 60%. And then it starts to fall off because the noise is so much that it's breaking the waves. And some propagate well, some do not. And you can also say this is reflected in the error bars here. So this is a lot like stochastic resonance, but it's not exactly like stochastic resonance. So we call it stochastic resonance-like. And if you make the box size smaller, you have to go to a higher noise level to reach the maximum, and then it's cut off. So how is this relevant to biological systems? Well, Frank Moss and Peter Young teamed up with Anne Cornell Bell and her student and looked at cultured glial cells. Now you can infuse canate into glial cells. And if you put 10 millimolar, you don't see anything. If you put 100 millimolar, you just see noise. But somewhere in between, 50 millimolar, you see these objects appear that are circular waves, propagate out, and fail. And so you do see that this and canate makes the glial cells. It's a very noisy because it's all ion channel transport. And so they felt that this was important. They felt that our experiments reinforced their interpretation. It's all that it just depends on your interpretation. Another example of waves in biological systems, I wanted to just show you this because it's so beautiful. Eric Newman and Kathleen Soss did these experiments. You can initiate waves mechanically, chemically, or electrically. This is mechanical. This is just a pipette. And this is the calcium waves in retinal glial cells. And so you can see it propagates out beautifully. Here in the bottom panels, this shows a time lapse of propagating out. Then they tried to initiate it at 180 seconds later, and it was refractory. It wouldn't initiate because it was in the waved tail. And then at 270 seconds later, it propagated out beautifully again. Now, their cover picture, you can see this is the five panels that we just kind of looked at, made to look nicer. But you can see that it's a sub-excitable medium. First off, it fails. And then there's propagation failure. And you can also see that they didn't say anything about noise. And I don't want to say too much other than it looks kind of noisy to me. OK, so one of the things we wanted to do was try to figure out exactly what sub-excitable means. And we saw this picture before. And you kind of see that sub-excitable means that there is no sustained propagation. There is propagation, but not sustained, if you have a wave with free ends. And so we thought, well, let's try to understand this a little better. And so we have a similar setup to the noise experiment. But now we're just going to, we have this channel that we can have a spiral wave. We can have a big spiral wave over here. And this is the channel it comes into. And then this is a region that we can adjust the excitability by adjusting the light. And I realize now that I didn't tell you a critical thing. And that is, well, I did. I said that this catalyst, the photosensitive catalyst, it reacts with bromelonic acid to give bromide, which is an inhibitor. So the higher the light intensity, the lower the activity or the excitability, the lower the light intensity, the higher the activity. And so we make this very high so that the wave cannot propagate in here, but we can adjust this. OK, so that's our experiment. What do we find? Well, what we find is that for a particular background light intensity, if we use a simple, linear, I guess it's actually negative feedback because of this inverse higher the intensity, lower the excitability, it's a very simple linear feedback. And so what happens is if this wave starts to grow, we see that it's growing. And so A increases, that means that the light intensity increases so it shrinks back. If it shrinks too much, you can see that A decreases, so the light intensity decreases, and the wave grows back. And so the wave size is fundamentally important in this system. And so A is just, you just count the pixels as it goes through a window to get the area. Small A is the gain, and B is the offset. And the offset, in our case, is the background light, the excitability, to determine the excitability. And so what does this give us? It gives us a locus of stabilized waves that looks like this. This is increasing light intensity, so it's decreasing excitability. OK? Now if you are on this locus somewhere, if you're exactly on it, you stay on it. If you have a feedback algorithm, you also stay on it. But if you move over to here, it's a sub-excitable medium, and it is not sustained, and it collapses. On this side, if we should move over here, it forms a spiral wave, and it happily rotates. What happens is you get closer and closer to this locus, this wave, wavelength, it's longer and longer, and it basically becomes a planar wave. And in the asymptote, it's unbounded and is an infinite planar wave. This dotted line is the 1D collapse of the wave, and that's the absolute threshold for wave propagation. And so the sub-excitable medium is this region right here. And that's actually a pretty fundamental thing we learned. Now that we have these waves, we can make them do what we want them to do. This is not a pickle. This is a wave segment, and it's in an excitability gradient. And the gradient is perpendicular to the normal norm of the wave velocity. And so what we do, as I said, light, it's photophobic, and so it goes to the dark side. And here is our experiment again. It's kind of a very similar experiment. And so you can do all kinds of things with this. One thing you can do is you can design a trajectory that you want the wave to follow, these hypotrochoid trajectories. Here's a circle. Here's a three-lob. Here's a four-lob hypotrochoid. And so these are simulations. They're not as interesting as the experiments, because in the experiments you can see that it starts to get off track. Someone opened the lab door and let some light in or something like that. And the algorithm adjusts the gradient proportional to the current position and the desired position, the target position. And so the further you are away from where you are supposed to be, the harder you're driven back. And so it follows it. And that's just one of the things you can do. Well, of course, we are really happy to get this in science and so forth. But then you go through, you're doing the galley proofs and do you want to send out a cover picture? And Eugene Mahiluk, who was my PhD student and stayed on for post-doc, he brought me pictures that he did the natural colorings. And ruthenium isn't nearly as pretty as the pheromone. It's orange and green, and it really looks bad in a figure. And so then he made red and blue, like pheromone would be. But it was not very truthful. But who cares about that? And finally he came and he says, I've made for you the most obnoxious color combination that I can think of. And so he gives it to me and I said, I like it. And so it's all marketing. And it means very little other than marketing. But we were very happy when it got on the cover. I asked this cover manager, why did they selected it? And they said, they've never seen anything like it. It may be because it was so garish. So you can look at one of these. Here is a four lobed hypo trochoid. This is simulation. It wouldn't be so perfect in the experiment. And then you can take a random walk. You just select the gradient from a uniform probability distribution on equal time intervals. And the purplish one has the waves. So you can see how they go. But there's also a completely different yellow and blue trajectory depending upon your initial conditions. And I think it would also depend as well on the noise seed. And so on this, you can see that the wave does kind of a Brownian walk. OK, here's the experiment. Here's our little channel that the waves come into. And it's not as magnificent as the simulations, but it's experiment. And so that's really nice. OK, so more recently, we wanted to take a look at kind of moving away from waves, look instead at discrete oscillators. And this whole avenue for us was inspired by this gorgeous paper by Sylvia de Monte and Paris and her student, and Preben Sordenson and Copenhagen and his student. And it all started long time ago, this is a long, long time ago, 1976 by the biologist Aldridge and Pi. And they found that if you have a stirred suspension of yeast cells, if the number density is high enough, they will oscillate in synchrony. So this yeast cell system became a paradigmatic system because it is like fundamental cell signaling. They have to be talking to each other to oscillate in synchrony. But their experiment in the paper was they decreased the number density by dilution of the medium. And at a point, it suddenly stopped in unison, stopped oscillating. There are two possibilities. One possibility is at that point, it's oscillating fine over here. 0 is the point that's critical number density. They just come defaced, become defaced, and the global signal is flat. That's one possibility. That's what everybody believed, actually. The other possibility is that at this critical number density, all the individual oscillators change states and relax to their steady states. And this was intriguing because it's very much like quorum sensing. Quorum sensing in bacteria, you have behavior, one behavior, and then you reach a critical density, say in vibrofissurei, and you suddenly they start chemiluminescing. Likewise, in seromonas argonosa, you form biofilm at a critical density. And this is kind of a very bad thing if it happens in your lungs. And so with the Sorensen study, the Copenhagen and Parris study, I should say, they tested this. They had a reactor. They're pumping in these yeast, so they're all the same age and so forth. And they studied this. And they had good evidence that it was the quorum sensing transition. And so we thought, well, we should probably look at this with the BZ discrete oscillators. And Annette Taylor and Mark Tinsley had done a beautiful study earlier than this 2007 paper one year. And they found synchronization in a solution, a suspension of these beads. And so you can see that basically Z is your catalyst, X is your activator, and Y is your inhibitor. The catalyst is immobilized, just like in the hands-on session. The really nice thing about this system is that in a 2 by 2 cuvette, you can have about 100,000 oscillators. And so you really get a lot of oscillators. This, what I described to you is global coupling. The X and the Y go into the solution and are felt by all the other oscillators. So here is our experiment. It's a very simple experiment. I need to hurry up. We look at it with a camera and with a fast shutter speed. We can freeze frame things. Here's an oscillation in black and white, unfortunately. And here are the oscillations in the images. And here's the electrochemistry. Also telling us about the oscillations. OK, so we can talk about exchange rate. And for us, it's just stirring rate. But you do simulations, and you really worry about the details of the exchange rate. And as you step up the density, number density, in this little staircase, you see that you have this noisy signal and small, noisy. Now it looks like real oscillations, low amplitude. And then finally, it gets to large amplitude. If you look at the millivolts, the electrochemistry, it's not zero, but then you have kind of a gradual rise. The images say that about 20% here, we're measuring here the fraction of the beads that are excited. And so about 20% are not excited. Do I have this backwards? Yeah, 20% are different. I think that they're not excited, and the period is not very informative. So this is really the signature of a classic Kuramoto transition. It's oscillating. They're all oscillating out of phase here, out of phase. But then they phase synchronize, and you have this Kuramoto-like rise. So if you change the stirring rate to now, oh, I'm sorry. I have to go this way. Oh no, I'm OK. OK, so now, OK, yes. OK, I'm sorry. I got mixed up where I was. OK, now we're going to look at higher stirring rate, higher exchange rate. And you can see now that there are just no, everything is not oscillating over here. Over here, this is really thick as pea soup almost. It's oscillating in synchrony. And so if you look at this, these are now simulations, you can analyze this by looking at the exchange rate and the number density. And these two experiments took place in two different regimes. At low exchange rate, low stirring, and if you increase the number density, you would have this gradual synchronization. But it would still be oscillating when it was unsynchronized, like a chromo to a transition. But at high exchange rate, it's not oscillating at all. It's flat. And then you have this jump to oscillations. This is kind of like a first order, second order phase transition. But what it tells us about the, you know, I think what we learned that was really important here is that even though Sorensen and De Monte found this avenue in the yeast experiments, there's no reason that they shouldn't find this avenue as well for synchronization if the parameters are accessible. And this is true for all of these quorum sensing type transitions. One should also see the gradual quorum auto type transition as well if the parameters are accessible. So I think that that really did add some insights into the quorum sensing, quorum auto transition phenomenon. I want to finish up by talking about Chimera states and Chimera states were first discovered by quorum auto in the talk talk in 2002. And Strogatz in 2004 named it and also did a lot of bifurcation analysis on it. And Chimera is the Greek mythology. It's pictured as incongruous animals glued together, a lion with a goat coming out the back and a serpent tail and so forth. These should not exist together. And so in the oscillator systems, and most of these studies have been done with quorum auto phase oscillators, you can have a system, say with 100 oscillators. Some of the oscillators are synchronized, perfectly synchronized, and the rest of the oscillators are completely unsynchronized, and they coexist. Even though the oscillators are identical and they're coupled to each other identically, that's kind of an amazing phenomenon. And so it's a new state in synchronization that people became interested in. Chima and quorum auto in the same year, 2004, as Strogatz coined Chimera, they discovered spiral Chimeras, which are really wonderfully exotic. So I'm just going to, we don't need to spend time on this because I don't want to run out of time. It's kind of the same setup, except now instead of a reaction diffusion system, we focus the light on each individual oscillator, and this is actually important. In all these experiments, we have a different BZ recipe so that light is not inhibitory, light is excitatory. So we shine light, and this catalyst causes bromous acid to be generated, and you get phase advances. In fact, these are phase resetting oscillators. And that's the only difference. And so our coupling here, this is important, our coupling, this is the light you would shine on oscillator J. This is the background light. This is the transmitted light intensity of oscillator J. This is the transmitted light intensity of all the other oscillators from J minus n to J plus n. And you just take the difference, multiply it by a constant. That constant looks like this. Here's the constant. It's Kermoto invented this coupling, and it's basically exponential fall off. And so if this is oscillator J in this row of cherries, the coupling strength is indicated by the thickness of their stems. So the coupling falls off as you go away from J. And every oscillator has exactly the same coupling. Now, this kappa is the decay constant of the coupling. And with high kappa, you have very short range coupling. With low kappa, you have very long range coupling. And we found somewhere in the middle, like 0.4, gives us a nice result. This is a local order parameter. Basically, it looks at the phases theta around oscillator J within a coupling radius of m to the left and m to the right. OK, so here is experiments that are one-dimensional. It's a one-dimensional chimera where you have 40 oscillators, I believe, that are in a circle. So you have periodic boundary conditions. And you can see this goes five times faster, so you can see this a periodicity. This is synchronized. In chemistry, the synchronized always is faster because firing together makes things reset faster. It's just the opposite in phase oscillators. But this is the a periodic region. And that's essential for a chimera. So I don't want to spend too much time on the one-dimensional work, but I do want to tell you how this relates to biology. And the main way is, at least it's thought, I should say, that it's relevant to unihemispheric sleep, which is found in a lot of animals, dolphins, seals, whales, sea lions, and a lot of birds. And I've seen recently in lizards as well. And so especially animals of prey, here's this seal, and he's got one eye open looking for prey. And the way he does this, he's got to sleep sometime. And so he has one hemisphere asleep, and the other is awake. And there's very good evidence that this is true. In the one hemisphere, you have slow wave sleep, and in the other hemisphere, you have high frequency EEG. Dolphins or whales, they have to do the same thing because they have to know when to come up for air. Some of these birds fly thousands of miles, and they have to get some rest sometime. And so that's thought to be the relevance. So you can also look at these spiral wave chimeras. They can get very odd. You have looping behavior and so forth. And it's different on the two different cores. You can see the cores are different here. And so they're not like regular spirals. Why are they called chimeras? Because the core is made up of asynchronous oscillators, and the rest of the domain is made up of synchronized traveling waves. And so it's a really interesting type of chimera state. Well, we also found when we did this work that even weirder things can happen. Namely, one core can kind of just go crazy, start splitting up and multiplying and so forth. And it's just unlike any spiral type stuff you've ever seen. And so a couple of years ago, I was on sabbatical in Berlin working with Jan Tots and Harold Engel. And we decided, let's try to get to this in an experiment. And they have a wonderful machine lab at TU Berlin. And so Jan had the shop precision micro machine. This is a big array, 2,816. And you have to get these BZ oscillators, so the hole has to be the right size. And we first tried water solutions. You couldn't get them in because of the surface tension. So we wound up using methanol. We could get them in with methanol. And so you look at this. This is just nothing but a glare from light. It doesn't mean anything. But this just started. I don't know what it's. OK, I'll just show you this. This is not very interesting. They're unsynchronized at first. And then they form a spiral chimera. Well, this is actually the uncoupled. This is just asynchronous. But oh, no, I'm mistaken. This is actually the chimera. But to deal with this, we need to form a virtual array. And basically what that means is indexing. We find the oscillators that are well behaved. Some are doubly occupied. Some are dead. So we throw those away. And we look at all the good ones. And we put them, number them, put them in an array. And so we form a 40 by 40 array with 1,600 oscillators and a virtual array. And we initiate it by periodically forcing it in kind of a conical arrangement so that you have phase singularity in the middle. And then the chimera spiral starts. And you can kind of see in the middle. This is just raw data. Here's that glare over there. And you can kind of see that they're aperiodic. OK, so let me finish up quickly. So here's the same information in a kind of a, this is the raw data again. You saw the early forcing. This is just period. You can see that it's kind of aperiodic and kind of, more or less, periodic in the spiral. And here are the phases. And this is the order parameter. So this is a well-behaved chimera spiral. But oh, and I've actually forgot a very important detail. In the early work, Kermoto and Strogatz and many, many other theoretical papers, a phase frustration term was added to make all this happen so that the synchronization isn't so strong. But actually it was Abhijit Sen and Gotam Sethia who were instructors in this school, in its first school in 2008 in India. They started using a time delay instead. And it made a lot of sense because it takes time to deliver the signal. And as you change the time delay, things change rapidly. That is the key variable. Oops. And so if I can only get this movie to play, here we go. Okay, so now we've increased this delay. And now odd things happen. It starts to split the core. And so we found what we had experimentally, we found what we saw earlier. And you can see this in this plot. Here is the period of the spiral wave which increases with delay. And it increases because there are phase resetting oscillators. This is the average period of the core and then this is the size, the fraction of the a periodicity in the medium. And finally, here are some movies Mark Tinsley made of this is a splitting one. This video is long. So I'm gonna jump ahead. Let's jump to there. You can see that it's splitting off little spirals and it keeps splitting, keeps splitting. And now it forms about nine. But that of course is dependent on everything, initial conditions, et cetera. Now you look at these difference in time scales. This is 8700. This is 2000. There's another domain when you increase this delay a little bit more that really looks, it's just, what happens is it just goes too fast to split. And it just takes over the domain. And so this is also another very interesting thing. But another thing actually that requires further work, there's a lot of order in this disorder. And so that's a whole nother thing to do. So in the middle, that's the order parameter and blue is low order and yellow is high order. Okay, that's local order parameter. And I think with that, oh well, just to check that this isn't a special thing with the BZ reaction. It also happens in the 50 Nogumo. And with that, thanks.