 So Mike showed us an astonishing diversity of eyes and discovered a lot of exquisitely well-adapted novel and optical mechanisms. And of course, as we've heard from Daan, all of these have evolved in order that animals can have the vision they need. Now, evolution is driven both by the benefits of a particular adaptation, but it's also influenced by costs. So, tiny, lots of animals have very small eyes, because Daan Eric says every animal has the eye it's needs, and they're very small. They have grotty little eyes because they don't need very good vision. And so the costs of a good way of the benefits. And so I think a lot of people would agree that if we're to better understand these adaptations, we should be looking at eyes through a cost benefit lens. And Mike was very good at making new instruments that enabled us to see new stuff we've never seen before. And so I'm offering as my tribute tonight a new instrument, which is a cost-benefit lens with slight resolution. I think Mike would like the idea that this is totally new stuff that hasn't been published. So we start with what makes a good eye. The New Testament says that there are three factors. The first is the point of thread blanching, which is what you would call is the blur. You want that to be as narrow as possible. So you get a nice sharp image. The second is that you want your photoreceptors spaced close together at small angles so that you can resolve the difference between nearby objects. And the third is that you want to catch lots of photons to make the photon noise. The variance in the photon catch goes up as the number of photons. And it's photon noise which makes things look horrible and speckly at very low light levels when you don't have very many photons. So you want to get a good signal to noise ratio by increasing your photon catch. And we know partly through, we go and look in the Old Testament revival, you'll find all the equations you need and to work out that these three quality factors depend on the physical dimensions of the eye. So if you want the photoreceptors to subtend smaller angles, then you want to make either the radius of a compound eye or the focal distance of a lens eye make that longer so that now the angle is reduced. If you want to make the lower circle narrower, you want to make the lens larger in diameter because that's the equation for the ultimate limit to the resolving power of a lens which is departure. And if you want a better signal to noise ratio, you want to make a longer photoreceptor which catches more of the photons that come in because that will increase as the square roots of the number of photons. So everybody knows that there is an expense, a cost to having vessel creation. It is that you need a much bigger eye and that's very good for a lot of purposes. And in fact, people have used eye radius, eye focal length or axial length length from top to bottom or the area of the corner of the eye as measures of cost because they reflect bigger equals better. But these measures have rather poor resolution because many eyes are, for example, are not spherical. This compound eye, although it is part of a sphere, it's in fact a shell only an outer shell of a sphere and this inner part of the eye isn't occupied at all. And many so-called and many simple eyes are actually tubular, for example, or distorted and we're the multiple man. And the second and maybe even more serious problem is that when an animal invests in an eye, it invests in both photoreceptors and in some optical mechanism for focusing the light into the photoreceptors. And in these simple measures, the costs of the optics and photoreceptors for all that together. And so we're not really seeing very much about costs and benefits. And so I'm going to introduce a new measure of cost, which is quite simple to derive. In principle, there are some equations, some geometry called specific volume, which is the volume of devoted to an angle as per angle of visual space, not the total volume of the eye, but the volume per angle of space that you're looking at. And I think that improves and we'll see that it improves the cost-benefit lenses resolving power. So the best way to introduce this is just to show how it's done. So we'll take the acquisition compound eye of an insect, which has an array of lenses, and under at least focus the light onto photoreceptors here, which then absorb them. And you can see that this eye is locally spherical. And the volume of the optics is simply the volume of this outer shell. And the volume of the photoreceptor array is simply the volume of this next shell in here. And there are equations for geometry, which give you those and the volume of the entire eye. Now the interesting thing about these equations is that they contain the factors R and D, which is and delta pi. R is D and delta pi. So they contain the radius of the eye, D and delta pi, because that's R, and the focal distance focal length, and the lens, and the length of the photoreceptors. And those are all the factors which determined our three quality factors. Longer lens, bigger D, and longer F gives you better spatial resolution, and the longer L gives you a better signal to noise ratio. So these measures and costs are directly related to the ability of the eye to form and capture an image. So they can, they link the volumes directly linked to the R3 quality factors, the blur, the signal to noise ratio, and the angular density of photoreceptors. And these equations given by physics and geometry, and by physiology, and they're very, very simple. This equation here, that the signal to noise maximum is a quarter times the square root of some constant times the length, needs a little bit of explanation, because it may be unfamiliar to you, to many people in the room. But this is a discovery made by photoreceptor physiologists, that the number of microvilli that a photoreceptor makes in an insect eye determines its signal to noise ratio. And the principle is quite, that sounds really scary, but the principle is quite simple. So the photoreceptor absorbs light using a column of photopigment with photosensitive membrane. And this column is made out of tiny little fingers pushed out and packed together called microvilli. And Roger Harding colleagues showed that each microvillus processes the signal from a single photon when it's absorbed. But it takes time to process that signal. So a single minus microvillus can only process as a maximum about 100 photons per second. So a very bright light. So as you increase the light intensity, so this is the rate at which photoreceptor, this is measurements that I made with John Anderson, where we were able to estimate the rate at which a photoreceptor was absorbing photons. And then looking at the noise, work out the rate at which it must be transducing these photons, converting them into little signals. And you can see at low light levels, the rate increases with the rate at which pigment is absorbing in these microvilli. But then you get to higher light levels because it takes time to process a signal, some of the microvilli are not able to process the photons that they've just absorbed because they're already processing much. And so the rate at which you're transducing photons falls below the rate at which you're absorbing photons. And it turns out that this means that the signal to noise ratio stopped climbing. And in fact, it goes through a maximum, which is this value of the quarter of the square root of the number of microvilli. And that comes from both the theory of the binomial statistics of this absorption process, but also has been shown in experiments and by modeling. So this is a hard limit. And it's a real limit. And this makes it very convenient because this is a very simple equation here. So for our first model of this cost-benefit lens, we can use this very simple equation if we consider just live photoreceptors, which are what these are, which are operating under very bright light conditions when this happens. And you'll see that in this experiment, we looked at two photoreceptors, one of which is a long photoreceptor, 250 microns long, and another one is shorter. And you can see that the longer one does indeed transduce more photons in bright light. So it all properly works. So the other question, the obvious next question is, well, how well does do our two measures of cost, the volume per solid angle of the optics and the volume per solid angle of the photoreceptor array, how well do they represent the real costs of the eye? Well, the first cost is space. And of course, this isn't exact. The second is mass. It's heavy. So it's reasonable to assume that the density is about one because everything in biology is made of water to a first approximation. So that's a reasonable approximation. Materials, it's a bit dodgy to assume they're all the same. But the cost is probably relatively small compared to the other ones. So for a first task through the problem, our lens won't have perfect resolution, but it might still show us some. The real problem is with energy. So there are two energy costs. One is the energy cost of carrying the eye as you're moving around in flight. And that's proportional to mass, especially when you're flying. That's okay. But these photoreceptors consume more energy per unit volume of photoreceptor than most tissues in the body because they have to use a lot of energy to convert all those photons into electrical signals. So this is not going to work. We're going to have to find some way of accounting for the extra energy consumption of photoreceptors. And the way in which we do this is we adopt a method which was used by biologists who are interested in weapons. So these are the weapons of stag beetles. So the male has this giant jaws here, which it uses to fight other males and to display to females. So how much does that cost? And what they simply did was say, okay, when the stag beetle flies, it's going to cost energy to lift this thing off the ground and carry it around. And so you take the specific metabolic rate. That's the energy per gram of the flying insect. That's the cost of flying, energy cost of flying a gram measured in the laboratory. And you simply multiply that by the weight of this weapon. And that gives you the energy used to fly that weapon around. And so we apply this to the eye. In the eye, people have measured the specific metabolic rate is oxygen consumption per gram. And these people here did fantastic experiments where they measured this for the eyes of a blowfly. So we can work out from their values what the oxygen consumption per microvillages. And then if we divide that by the specific metabolic rate, like in flying beetles, we get an equivalent mass. And then we convert that because the density is one to a volume. And so we've converted energy to volume through this trick. And we calculate that from very little data on this, but our best estimate is that the energy ranges from between 0.05 and 0.5 cubic microns per microvillage, which is quite significant because the volume of the microvillage, the true volume of the microvillage, is only 0.2 microns per unit. So the total specific cost of our eye is going to be the specific cost of the optics volume, the specific cost, the photorescent of volume, and this energy surcharge added on here, where this is a factor the energy per microvillage is an equivalent volume. Okay, so now we have all of the equations that make our lens work. Let's apply it and see if we can see anything new. And we looked at published data on the compound eyes of dialing flies because this is the best dataset available. And we made some simplifying assumptions and they all perform roughly the same function. So that's kind of what we're going to do. And so what we're going to do is we're going to model eyes and we're going to on the computer and we're going to make eyes of different configurations, but at the same cost. So sorry, they're the same configuration that all opposition eyes, but they have different dimensions for their components, that they all have all of these eyes that we construct at the same cost. So believe it or not, these two eye regions have the same cost. That's because when you go out and everything gets longer than you would in the center of the sphere. And this one has invested heavily in lenses and less heavily in photoreceptors and vice versa. And the result is going to be that here you're going to have very sharp image, but it's going to be very noisy. But here the image is more blurred, but it's much more reliable. Now it's quite difficult to look at these two things and say, well, which one of those do I really prefer, which is best? So we want a measure of performance and you could use lots of measures of performance because we're talking about these universal factors that influence everything in the quality of the image of the eye. You could use some of Dan's new measures or you could use small target detection or something. But we chose to use information capacity in bits of information per solid angle per second because this is a general measure of all of the information that's available for you to see. And in fact, it's the one that Dan used when he was free of paper and I would read it if we had time in my preference. So okay, so we make all of these eyes all have the same cost and then we work out how much information for each eye is going to be able to capture what its capacity is, and then we plot them out and compare them. And what you get is this rather grand thing which in other areas of physiology is called a performance surface, that's just surface across morpho space where morpho space is just all of the possible configurations where I didn't make the same cost. And you can see that there is an optimum, a best configuration which maximizes performance, but the optimum is rather broad. These red areas performance is 95% of maximum. Now this is bad news for people who like to write papers saying that everything in the world is optimized, but good news but insects because it means you can specialize your eye for better signal to noise ratio by sacrificing optics or by crisper images by sacrificing signal to noise ratio without losing more than 5% of your general performance vision. And one of the things that people have always observed is that in these compound eyes in particular everything is exquisitely fine tuned. There are loads and loads of yet minor adjustments and this makes sense because in fact if you make a minor adjustment for one function you're not going to suffer greatly for another function until this makes minor adjustments much more profitable. Well does it tell us anything useful about the structure of eyes? Well we can apply our method to work out how much volume is given to the photoreceptors in a compact fly eye as a function, as a percentage of the total volume of the eye. And you can see that in these flies between 50 and 80% of the volume is photoreceptors. And the theory says that you want a zero energy cost, you want to give a very large volume of the eyes to photoreceptors because they're cheaper. If you increase the energy cost the volume you devote reduces but it's still high and the data falls somewhere between these two theoretical predictions. So we conclude that in fact lots of volume is being given to photoreceptors because this is an efficient way so this bit makes an efficient eye. And of course the photoreceptors, the volume of photoreceptors is long because the photorus is high because the volume is high because the photoreceptors are very long. They get to be something like flies 300 microns long. A single cone in your retina, the equivalent structure is only something like 20 microns long. So these are really long photoreceptors and in fact, as Don Eric has shown, these acquisition insect eyes have the world's longest photoreceptors even though they operate in purely bright light. And this helps to explain why. This is an efficient way of using more resources. And so what you see is that as you invest more in the eye you have larger lenses, narrower angles between omitidia and that gives you better spatial acuity and the length of the photoreceptors increases. So in fact the investment in photoreceptors is matched to the investment in optics. So acuity and length increase in step. And this is seen almost everywhere you look in acquisition of our eyes. So if these acquisition, if this is a particular character of acquisition eyes that they have really long photoreceptors compared to simple eyes, then if we build an equivalent model of a simple eye, which we do just in this very here, you should see that the, you should find that the simple eye photoreceptors, which are the solid nine, the theoretical predictions, they should be much shorter. And indeed they are zero energy costs. The difference is from there to there. And for high energy costs, the difference is from there to there. But you see another thing. You see in the theory that when you impose an energy cost, a small energy cost, it has a tiny influence. I've got these two the wrong way around. That should be point for, oh no, I've got the right way out. It has very little influence on the optimum length. But when you apply the same energy cost to the simple eye photoreceptors and imply an even smaller energy cost, you find that there's a very big chain. So the simple eyes are much more sensitive to photoreceptor energy costs. And that's simply, and that's basically because as you see from this comparison here, in the simple eye, the photoreceptors are more closely packed. So in the compound eye, the volume costs of the photoreceptors are much higher relative to the energy energy costs. And so this has makes a profound influence on the design of the eye. And you can see this when you take the photoreceptor energy costs in our models as a function of eye size, where this is photoreceptor energy as a percentage of the total cost of the model eye. And in a compound eye, it starts off quite high. And then as you make the eye bigger, it becomes smaller and smaller. And that's because these photoreceptors are getting spaced further and further apart. In the simple eye, but high energy costs, even though the photoreceptors are very, very short, the energy cost is now the dominant cost in the eye. It's over 60%. And that's because the photoreceptors are really tight in the simple eyes. So what have we seen? So I hope we've seen that this very pompous microcosm is deriving performance services over more space is useful. We actually do see stuff with this lens that we haven't seen before. And we see that photoreceptor costs shape the entire eye because they are competing for the resources that are invested in the eye. And this, so what this means is that if you spend more on photoreceptors, you have to spend less on optics. So the optics are all smaller. So you can't really look at the two in isolation. And we also see that the impact of the energy costs depends on eye time. So you can't come along and say, well, people have found this thing about energy costs in an in-second line. And it's going to have the same effect on the lizard's eye because they're in geometries. And finally, we see that when we look at our eyes, we find that the investments in optics and photoreceptors are very carefully matched to maximize efficiency. And this is a new principle of the design of eyes. Now, looking ahead, I think our cost-benefit lens can evolve to improve its resolution. And we know how this is done. You have to adapt it to get rid of all sorts of defects and artifacts and actually. And so to begin with, it can be adapted to view eyes of any shape or size. We have formulated this approach and shown that it's useful using these compound eyes where the geometry is very simple. But the principle is that the shape and size of a component in the eye and its constituents determine the eye's performance. So in principle, you can apply this approach to any eye component of arbitrary shape or size. And we now have fantastic methods for imaging the structure of eyes within the eyes and wonderful modeling techniques which can tell you how weird blocks behave optically. The second thing is we applied it to very bright light levels to simplify our proof of principle. But in fact, my co-author on the study, Bran and Andeth, and I, Heras, and I, he's the Andeth Heras. So anyway, Bran, Heras, and I developed a very simple simulation in which you can work out how many photons are given a ray of microvilli is absorbing at any light level. So we're not just stuck to looking at bright light, you can look at dim light too. And I can give you a practical example of how you might want to apply this. So this is Darn, Heras, and Helmut's famous paper from 1996 where they tested the Darwin's theory of the evolution of the lens eye that you start off with a plate a little. And then you bend it slightly and that gives it some vestige of tiny angular sensitivity and you bend it more and more. So you end up with a little pinhole and that works like a pinhole camera. And then you introduce a lens here and that starts to focus the light and that makes it much, much better. And Mike Land famously, when talking about this, famously said that this is a critical stage and any lump of stuff put in there would improve the performance of the eye, even the lump of snot. So we could apply this method to Mike's lump of snot. And I think it would work very well. So I have lots of thank yous to say. So first thank you to Bran, who in fact did all of the modeling and formulated the project with me. And also to lots of people I've worked for, some of whom are in this room on the visual ecology of photoreceptors and photoreceptor costs as an important factor in our design. And I also have to think thank all of you in the room who are visual neuroscientists who share the same interests because we are, it's such great fun to work with you all. And the reason is because we're proud in our work but we're not proud in people, proud as people. We like to share things and we like to work with each other. And we just want to discover stuff because it's important and it's fun. It should all be fun. And I think we have to give a big thank you to Mike for that. Years and years ago we were discussing, I was discussing with a group of colleagues rivalry in scientific fields and how dreadful it could be. And this person told me that this particular branch of chemistry, I won't reveal it to you, was notorious for everybody being really cutthroat and nasty to each other and trying to steal things and publish them first and all this sort of stuff. And he said it was because the first two people who discovered the technique that got this field going were bitter rivals and really jealous of each other. And that tradition has just carried on. And I think Mike has played a seminal role in creating a tradition in our field which has made it more productive and emphasises something that science should be fun. So thank you, Mike. So there are any questions? Any questions? Yeah, hi, good to see you. This is a political construction, of course, I suppose, or building complex structures and so on. And the other is the primary structure. Yeah, the second part of it implies wide enough management in terms of provision of energy and collection of waste and so on. So it's, my guess is that we were talking about the command at the second. We were talking about the retinas energy bill. So the current feeling in the field is that you make this eye and you build this brilliant optics and then you build this brilliant retina which is adapted and you put them together and you don't actually care about how much you pay for each of them. And the energy bill is just something annoying that you have to pay on top of everything else and it's going to be big but that's, hey, that's vision. The answer to your second, so you make a very good point that photoreceptors are not only more active from second to second as they are transducing, they cost much more to maintain because stuff is continually turning over, particularly the visual pigment and stuff wears out. And of course they're more difficult, they're more expensive to build because they're much more complicated cells. Now all of those costs are accommodated, can be accommodated within that energy factor. It's just a question, this is one of the improvements to the lens that we would like to see. We need a much better understanding of actually the costs that are involved. Of course the thing is, you know, once you built a rubbish lens, then there's now selective pressure to actually improve the quality of that lens because the rubbish lens tells you, hey, this eye has become more useful and it could get even more useful still. So I hope people will start looking in fact more critically and in more depth of a lot of these costs and some of our preconceptions could well be overturned. Okay. I wanted to give you an example of something in the world which is such a bad, you know, and you're greatly missing big leadership in this kind of thing. Oh, I haven't really, yeah, that's a really good question. It would have been that you're still evolving, what does that mean, your maths? Well, there's plenty of things, the fits are not perfect, it's a trend, because there's plenty of things we haven't taken account of. The maths is right, it's the numbers that you put into the maths, which could be wrong. And there might be a better way of actually dealing with the costs in the first place, but until you take this method and break it, then it's a start. As I say, it's a lens, it's not terribly good, but it's better than the previous lenses and it, given time and effort, it will evolve. Oh, so. Okay, I have one question. I think about the signal to noise ratio in photography, right, with the extra net exposure on that, and is it possible that some of the animals live in a slow world? This is another thing that actually we worked on, and that is that fast vision is more expensive than slow vision, because the photoreceptors have to use more energy to change their signals very quickly to keep up with what's going on. And we actually know a lot about the biophysics of that. And so you can equate the time result of the photoreceptors, which is a factor in this information capacity with energy consumption. We didn't build that into the model because it just makes the microscope that much more of a lens, that much more complicated. And we really wanted to show that this was a reasonable way to start moving, but yes, that would be an obvious next step. And I think the method would deal with that, the equations can deal with it, and we know the mistakes. The problem is when you're building in more space, every time you add a new dimension, everything goes up as the power of whatever it is you're looking at. And so you end up with some very, very elaborate models. And probably what we need is Mike to come along and boil it all down to some very simple equations that you can write on the back of an envelope. And once upon a time, I did this, but it didn't work. Yeah, I want to say one more thing about Mike making science fun. He always said, you don't start your research without hypothesis. People haven't seen what's out there, but he went out on the ship, he went out to have a good time, see what was in the water, and then you started getting new ideas. That's what's fun science is when people are something new and not discovering or even knowing something, really it's true. So, thanks. So I've got a question. I think it's naive, not bondage. I couldn't really think much of it. It's pretty good that as well as you. So that jumping spider with different sized eyes, it's using different cost-benefit systems. I'm just wondering what those why did what kind of function with these different sized eyes? So does that relate to the jumping spider's eyes and different sizes? We actually have a pretty good idea about either the sizes and why they have eye movements. So the frontal eyes, or as they're medial eyes, whatever you are, get confused. They have great big lens, fantastic resolving power. And they have very long focal lengths. So the interceptor angles are very small. So they have very fantastic social discrimination. Their discrimination is about one-fifth as good as ours, which is amazing. But is that about right? I went to a lecture on this. But it's also a very small part of the vision here. And it's for the thing is that eye is as long as the spider's body, almost. So it has this tiny little patch of focus, which are attached to the lens by this long sock. And as Mike showed, muscles move this little end of the sock around. So it looks in different directions. And just as the icing on the cake, before the light reaches the photoreceptors, it's refracted by a depression in the photoreceptor layer, which is equivalent to the foveal pit that you find in birds and in birds of prey, and also in comedians. And McIntyre and Williams show that this acts, this makes, converts the eye. It acts as a diverging thing. So it converts the eye into a Galilean telescope. A pair of opera glasses increases the magnification. The thing that all these three animals share in common is that their eyes touch some parts of the body. So you can't make them any longer in length. So they develop this telephoto pit. And then the other eyes get progressively smaller and smaller as they go more progressively. And that's the equivalent of your visual impurity falling off as you move towards the periphery, because you need those other eyes to centralize what they should be looking at. So they do optic flow and all of these sorts of things. And in fact, the casual observation suggests that in the longer focal length, well, in fact, there's a very nice paper just come out recently on the development of these eyes and spiders by Elka Bushbar. And as the eyes grow, they get longer focal lengths. And as the focal lengths get longer, the photoresence gets longer, which is what our model predicts. And also within a single instar, the eyes with the bigger lenses and longer focal lengths at longer distances. So that kind of fits on what we've got. I didn't pay him to ask that. Thanks, everybody for listening.