So the next conceptual model we'll talk about is the reciprocal interaction model for REM sleep cycling. Now, this model was motivated by the identification of the REM active neurons in the LDT and the PPT that expresses acetylcholine. And their interaction with the wake-active or REM-off neurons, in the locus coeruleus and the dorsal raphe. And what the synaptic rejections between these populations indicates that norepinephrine and serotonin from the LC and the DR actually inhibit the REM-on neurons. Where the acetylcholine from the REM-on neurons ends up exciting these locus coeruleus and the dorsal raphe. And so it sets up what the deeming this inhibition one way and excitation the other way sets up this reciprocal interaction between these populations. And this interaction can actually generate cycling behavior between these populations. Where in a cyclic rhythm one set of populations will be active and the other silent. And then the other populations will be active and the original ones will be silent. And so I put this, talking about the reciprocal interaction model in terms of the conceptual model. But they have actually presented a mechanistic model to actually illustrate the cycling dynamics that can be generated by this reciprocal interaction. And McCarley and Hobson, what they turn to for this mechanistic model is a set of classic equations in mathematical biology known as the Lotka-Volterra equations. Which were originally designed to describe the interaction of a predator species and a prey species. So, here are some of these will be the most complicated equations we'll look at. I think some of the last equations that we'll look at, but not to worry, we'll talk about them slowly and in detail. So in this model, we've got two quantities, X being the activity or the firing rates in the REM-on population, and Y being the activity or firing rates, in the REM-off populations. But in the original predator-prey model, X would indicate the size of the population of the prey population, say the rabbits. And Y would indicate the size of the predator population, say the wolves. And how the model illustrates their interactions is in terms of these differential equations. So the equations contain derivatives, but everybody's had calculus back in high school. Think back, what a derivative says is what is the rate of change of the quantity X and the rate of change of the quantity Y. And what these equations give us is that what do these rate of changes depend on. And what it depends on is what's on the right-hand side of the two equations. So what we can see is that by looking at equations we can determine some of the assumptions of the model. Now, in terms of the REM-on activity in the model, say X or the size of the prey population, say the rabbits/. What we see is that the rate of change of X depends on X itself with this parameter a. And this is a positive term, meaning that, [COUGH] the population will grow basically, due to birth rates. But the population will decay for this term with the minus sign due to its interaction with Y or with, in this case, the predator species. So that the predation by Y reduces the prey's population growth in proportion to the sizes of the population, so in proportion to X and Y. Now, the other assumption is that if we look at the activity of Y, which would be our REM-off activity in the model. Or the size of the predator species, is that the predator population grows with its interaction with the prey species, because the wolves would eat the rabbits and their population would grow. But then there is its own death rate having to do with just the natural death in the population. And so a solution to these two equations shows that we can get this cycling behavior. So that's what's shown in this plot here, where the black curve is the X variable or the REM-on activity, and the red curve is the Y variable or the REM-off activity. And we can generate this sort of cycling behavior where one of the variables is high, while the other variable is low. So this Lotka-Volterra, sort of formalism, reflects this kind of reciprocal interaction in that the Y, the REM-off activity, has an inhibitory effect on X. And the X activity here in this equation, has an excitatory effect on Y to generate these cyclings. However, one limitation is that in this standard Lotka-Volterra case, is that we could, depending on what values there are for X and Y, our cycling solutions can change. And so what McCarley and Hobson proposed is modifications to this Lotka-Volterra framework in order to achieve just one oscillating solution. One stable oscillating solution, which is called a limit cycle. And so here are their modified equations. But what we can see is that they've added a few more constraints to the model. Some functions of X in some of the terms, here, or functions of Y and they've added also a circadian rhythm modulation term in this term here. But however, the original structure of the model is unchanged where Y has an inhibitory effect on X and X has an excitatory effect on Y. So what they did in this study is they could actually fit all those parameters and constraints on the model to actual firing rates measured in the REM-on populations and in the REM-off populations. So these two plots from their paper, on the Y axis, they show the firing rate or the discharge activity in spikes per second. Where the smooth curves are the model solutions and then the more histogram-like curves are the data that's been measured. And so they could fit the parameters so that the model replicated, the changes in firing rates of the REM-on populations and the REM-off populations over a REM cycle. And then could generate, basically, a pattern of REM sleep cycling over an eight hour period that replicated the timing of human REM sleep. So in this plot, the solid line indicates the X activity, where the dashed line indicates the Y activity and the portions of time that are marked out in black show when REM sleep would occur. So this model is able to replicate REM sleep cycling on time scales similar to human sleep. And they were also able to replicate the change in REM sleep bout duration over the night. Where the first REM sleep duration is usually shorter and over the next consecutive REM sleep episodes the duration increases. And they could match the durations to some data that's been measured. Another thing they were able to do with this model, is to investigate when REM sleep occurred at different circadian phases. And show the different patterns of how the REM bout durations changed, cycle by cycle, depending on when sleep onset occurs, what circadian phase of sleep onset occur. And so could use the model to replicate data for when sleep onset occurred near the maximum of core body temperature, which is a reflection of the circadian cycle. And could also capture when sleep onset occurred near the temperature minimum and what the durations of the REM bouts were in that case. So this model has been very influential in the field. It's been able to illustrate that the monoaminergic inhibition from the locus coeruleus and the the dorsal raphe and the cholinergic excitation from the LDT and the PPT. Could generate cycling behavior in REM sleep cycling behavior similar to what's been observed experimentally. It's provided as a mechanistic model that can replicate these dynamics of the REM cycling over time. And it's properties, namely the properties of the model, the limit cycle solutions, can explain some of these differences in the first REM episode bout durations under different conditions of different circadian phases. This reciprocal interaction model has basically remained an accepted framework for understanding REM sleep cycling. And has motivated many experiments into the influence of neurotransmitters, in particular, acetylcholine, on REM sleep. However, there are some limitations. Namely, that the structure of the firing rate interactions may not be very physiological. So, for example, if we think about neuronal populations, that are interacting via synaptic interactions. Would the firing rates in the populations actually change in proportion to the product of both the presynaptic and the postsynaptic firing rates? Which is basically the underlying assumption of how the reciprocal interaction model is actually implemented. So that while the idea, the framework of the reciprocal interaction, excitatory inhibitory reciprocal interaction, may be accurate. How it's actually implemented in this particular model doesn't really reflect the physiology. Also, the parameter values aren't necessarily based on any experimental measurements. They were fit to fit the data, but they couldn't be experimentally measured. And the additional constraints that they added in the limit cycle model are surely reasonable. But it's hard to sort of tie them into a particular experimental or physiological mechanisms that have been determined. However, as rebuttal and in the defense of this model, I'll just provide this quote that they give. A positive feature of modeling is to encourage the gathering of more data in areas, which a model highlights as important, but whose empirical basis is unclear. So, again, it's this idea of what a model can contribute is that it can point to areas where more experimental work needs to be done to understand the underlying mechanisms. So, next, we'll talk about the last conceptual model is a competing idea about mechanisms for REM sleep.