So we'll start with talking about the two process model which I'm sure you have heard about. Now just to remind you, the two process model basically represents the interaction of the main forces that control or influence sleep wake cycling. Namely, the 24-hour circadian rhythm and the homeostatic sleep process. So the model consists of process S, which reflects sleep propensity. Namely, the fact that after we've been up for a while, we get sleepy. And physiologically, it's then to reflect this power of slow wave activity in the EEG that's measured during non-REM sleep. So in the model, this process S increases during waking with an exponential profile as shown over here, and then decreases during sleep exponentially. So the second component of the model is process C, the circadian rhythm. And that's represented in the model. Here, by this sinusoidal change in the propensity for falling asleep or waking up. And in the model, they interact, as shown in this panel, by the thresholds for when waking and sleeping happen are modulated by the circadian rhythm. And so, we're to think about how this model works. Namely, at the beginning of waking, so here in this slide, process S starts to increase during waking. And then at some point, when it hits this upper threshold of the circadian rhythm, sleep should occur and then process S starts to decrease. And when it reaches the lower threshold set by process C, by the circadian process, then waking should occur. Now this model has been very influential in that, it's actually been able to capture a lot of experimental and behavioral results. Namely, as this panel shows, been able to capture the circadian variation in sleep duration depending on the time of sleep onset. So for example, if perhaps, you stay awake longer than when you were supposed to, when the model would have said you should go to sleep and you fall asleep then, you end up having a shorter sleep duration because of the circadian rhythm. So this model has been very successful in capturing this behavior as well as others' behaviors. But let's take a minute and just see actually, what is this model? How is this model actually represented? So process S is given by some exponential equations here, where process S increases during wake according to the first line in this equation and decreases during sleep. according to the second line. And process C is actually just given by a sinusoidal function that is, with some amplitude for the upper and lower thresholds and some skewness given by this phase function. And in setting up, developing the two process model, the parameters, the different parameters, the time scale of the decay of process S, or the increase in process S and the parameters of the process C were determined by fitting the model to a slow wave activity data. So as I said, this model has been very influential as namely and some of the contributions are that it's triggered numerous experimental studies. Namely, behavioral studies to probe these interactions of sleep homeostasis and circadian rhythm. And it's triggered experimental studies into physiological correlates for process S and you've learned about that perhaps adenosine might be a physiological correlate for the sleep homeostatic drive. It's also provided a framework to interpret different types of experimental results such as, the effects of sleep deprivation, isolation from time cues, and shift-work. However, there are some limitations. Namely that this model is really only phenomenological. So for example, what is process S? At this point, it's just the sort of the intuitive qualitative phenomenon of the fact that we get sleepy and after we sleep, we're able to stay awake longer. Also, how are the thresholds for process C that instigate the initiation or the determination of the wave states. How are they actually enacted? There's no physiology presented in the model. So in particular, this interaction has no physiological basis for interaction is given. And additionally, the model doesn't include other sleep states such as the REM sleep state and any known details of the circadian pacemaker. So there are some serious limitations to this model, but in its defense, this is what we could describe as a conceptual model. It really is more of an elaborated kind of story model. A model that we could talk about in terms of words that can qualitatively capture experimental results. But it really is too simple to allow an analysis of underlying physiological mechanisms. However, again, I just want to say that this two process model has been greatly influential and has contributed to our understanding of sleep and circadian processes.