 synaptic plasticity. Now because this is such a diverse audience, I'll just briefly review. There's many types of synaptic plasticity, but I'm really going to focus on long-term synaptic plasticity, which is really a long-lasting and activity dependent change in synaptic strength. And when we talk about long-lasting, we mean an hour or more, an hour, eight hours, as long as we can measure it very often. And there's generally two types, potentiation, which is an increase, and depression, which then is a decrease in synaptic strength. And we're interested in this because it shares some characteristics with behavioral learning, such as persistence, just like we learn and remember things for a long time, and also activity dependence is similar to experience dependent learning. And we also see spatial specificity in various molecules, and that also varies with calcium. It's restricted to the stimulated spine, and this is just showing that with these different RAS members, you get different degrees of specificity. So the CDC 42 basically only extends for a few microns, whereas row A and H-RAS extend for a further distance. So we need to really capture these different time scales and different spatial scales in order to understand synaptic plasticity. And so here is my version of the diagram, and this is really only a subset of the diagram that Mary showed, showing some of these G-protein-coupled receptors on the spine or on the dendrite, and the molecules that they activate, some of them are still bound to the membrane, such as the dental aid cyclase, though some of them can diffuse, such as cyclic AMP. So we have very diverse spatial scales. We have glutamate receptors located on spine heads, and spines are so small that when we talk about concentration, we're talking about tens of molecules. And so these molecular interactions occur stochastically. But there are some of our G-protein-coupled receptors might be on the dendrites, not on the spines, so we can't really model just a single spine. We need to model the dendrites, diffusion between spines and dendrites. Some molecules do diffuse, some don't diffuse. And so one of the things we did in my lab is try to create an algorithm, a computer algorithm, so we could efficiently simulate this diverse time scale and spatial scale. And what we created was basically a spatial extension to Gillespie's tau-leap algorithm. So the tau-leap algorithm allows multiple reaction events and multiple diffusion events to occur during each time step. And so you can think of it as we have a particular space, and we are going to divide this into a mesh, because this is a mesoscopic algorithm. And then we're going to calculate what is the probability that, say, a molecule in this compartment leaves or a molecule in this compartment reacts with another molecule. And we do that using these standard equations for reaction propensity and diffusion propensity. And then we'll use a binomial random number to calculate how many of these molecules actually leave the compartment. And because binomial random numbers are expensive to generate, we store that in a lookup table, and then we can use uniform random numbers. So at each time step, we only need to choose a single random number to say how many molecules are going to react or diffuse. And this is just one of the validations. Now this is an approximate algorithm, so it's not exact. It's not going to exactly reproduce the Gillespie algorithm, but if you don't take too large a time step, you can see that here we're going to reproduce the deterministic solution, which is in black or gray, and we'll also reproduce smalden. And so these fluctuations with molecule A, of which there are relatively large numbers, and especially what we're calling molecule B, of which there are very small numbers, 0 to 10, shows that we do agree well with these other solution systems. So now we're going to use this software to address the question of what controls spatial specificity and what controls sensitivity to temporal pattern. And our goal is to actually do these simulations for minutes to hours in dendritic structures with spines that are 100 or even 500 microns long. Well, what we're doing right now is only 10 to 15 minutes and maybe only 20 to 30 microns, because we still need to make the algorithm yet more efficient or alternatively perhaps parallelize it. But this is one of the structures that I'll be using for these simulations. And what we want to answer first, we'll start with temporal specificity, which is how do these temporal patterns select for either depression or potentiation of synaptic strength. And now I'm going to switch, and I'm talking about the striatum, not the hippocampus. And unlike the hippocampus, in the striatum, both a high-frequency stimulation and a low-frequency stimulation produces long-term depression. And in fact, it's very difficult to produce potentiation in the striatum in the presence of a normal amount of magnesium. But what is observed in the striatum as well as other brain regions is a theta frequency of activity in vivo, and that actually gets stronger during learning. So one possibility was that a theta burst type of stimulation would produce potentiation, because that's actually what's happening in the brain. And so this is just showing an example of, say, a high-frequency stimulation and a theta burst, where we give four pulses, and then we wait a little bit, and then give four more pulses. So first we did the experiment to see whether we can select for LTP in the striatum with temporal pattern. And so this shows we're recording in the dorsal medial striatum over here, and we used 50 hertz within the burst, and then we tried different theta bursts. And we tried five hertz. That's what they use in the hippocampus. It doesn't really work very well. Well, if you look at those in vivo measurements, you see that it's closer to the theta is 7 to 11 hertz. So we tried 10 and a half hertz, and that gave us a very nice potentiation, which lasted for at least two hours. And then eight hertz also gave us a nice potentiation, though by two hours it was not really much more than the five hertz. And then we wanted to see if we could get this in the dorsal lateral striatum, which is where they're more commonly producing synaptic depression. And so this is just showing, again, that 50 hertz within the burst in dorsal medial and 11 and a half hertz theta, and we get this nice potentiation. And if we actually increase within the burst to 100 hertz, again, that's what they do in the hippocampus, we do get LTP, but it's not as strong. We do not get it dorsal lateral, but if we use our optimal paradigm laterally, we get a smaller amount of potentiation. So now we can use this stimulation paradigm to say which molecules exhibit this temporal specificity. And for this particular case, we just used a subset of reactions, which I'll call the GQ-coupled pathway. So in our model, we have calcium influx, as if through NMDA receptors. We have metabotropic glutamate receptors, type 1, group 1, and that's coupled to this GQ protein, which binds to phospholipase C. And the phospholipase C is actually activated by both G alpha Q and calcium, and that produces diacylglycerol and IP3. Now the diacylglycerol can either bind to dag lipase, which is a molecule that produces the endocannabinoid 2AG. And this is required for long-term depression. Many other experiments have shown that. But the diacylglycerol can also bind to this protein kinase C, which is transiently activated by calcium. And this has been shown at least to be important for a chemical type of LTP, though not other types. So we created this model and we wanted to validate it with a different type of data. So there's something called depolarization-induced suppression of inhibition, or DSI. And in these experiments, they have shown that if you depolarize, sorry, if you depolarize the neuron, say, for 0.1 second, you don't see really much to decrease in the inhibition. But if you depolarize the neuron for one second or five seconds, you will see a brief transient suppression of an inhibitory input. And we sort of see that in our model. We're actually looking at 2AG production as representing the depression. We see very little 2AG with a 0.1 second influx of calcium. And it's bigger for one second and quite large for five seconds. Well, this group, and this is a paper from the data, is from Uchigashima. And what they also showed is that if you add a molecule called THBG, which activates metabotropic glutamate receptors, you see an enhancement in this suppression. And so this shows that if you give a 0.1 second or a one second depolarization, you see the enhancement in the suppression. But with a five second stimulation, you don't. Because most likely you've saturated something with a five second depolarization by itself. So this shows what's happening in the model. And with a one second depolarization, this is just reproducing this trace up here. Now if we add DHBG, we see an enhancement of the 2AG produced. And now we want to compare how much is produced with the DHBG compared to normal. And so here's our ratio. And this has a very similar shape as the experimental data. And so we see an enhancement with 0.1 second and one second. And with five second, we see saturation. So now we can simulate the model using our LTP induction stimuli. And we chose to use a 20 hertz stimulation or low frequency because paper recently out of Chrysler's lab shows that this type of LTD, but not the 100 hertz requires 2AG. The 100 hertz does require endocannabinoids, but it's a different type. And we don't have that type in our model. So this is the 2AG production. And the black trace shows our theta burst stimulation. And the red trace shows 20 hertz. And it's fairly similar quantity here. And we're just using this in this smaller structure with a single spine. But now if we look at the protein kinase C, we see that theta burst produces a much larger increase, a much larger amount of PKC than does 20 hertz. And so we think that LTP is resulting because the PKC effect dominates, not because there's no 2AG production. So we think now we can say you have a molecular signature of our LTP. And this is just summarizing. There's very little difference in the 2AG with our stimulation paradigms, but a large difference in PKC. So our molecular signature is a ratio of PKC to 2AG, which is larger, for much larger for theta burst than for 20 hertz. But we also want to validate, is PKC really required for this type of LTP? So here's an experiment. This is our theta burst with no PKC inhibition. Now if we add a PKC inhibitor calorithrin, we see that we do not get LTP anymore. And this is just our non-stimulated control with calorithrin in the bath to show that this effect is not just some non-specific decrease in our ability to produce LTP. Well, what about spatial specificity? So now I want to use this larger morphology where we have multiple spines and we're only applying our glutamate and calcium to this one spine, as well as, well this is a mistake, not glutamate, but calcium is applied to other places on the dendrite. And so this plot is showing time on this x-axis, the spine number on the y-axis, and then the darkness represents the 2AG concentration in this case. And we see very little spatial specificity. For theta burst, you can make out here's when the theta burst is happening and you see a little bit of 2AG production, which extends throughout the morphology, though it's not as much over here. With 20 hertz, you can also make that out, though it's not quite as visible on this plot. Well, what about the protein kinasee? We actually see a dramatic spatial specificity. This is our theta burst and 20 hertz, and this is the stimulated spine and you see a very high elevation in PKC and much lower increase in these other spines. And with 20 hertz, we see much less of an elevation. So now we can summarize that using our ratio of PKC to 2AG. And so this is, we're looking at area under the curve to represent sort of total activity. And you can see with spine one, we have a very high ratio. And with these other spines for theta burst, we actually don't see much of a ratio. And it's very similar to our 20 hertz. And we see something similar. This is the PKC. If we're just looking at, instead of the ratio, the activity of the molecules individually. And so we suggest that means that when you're giving this stimulation, any stimulated spine will exhibit LTP, but the other spines might actually exhibit LTD. So this might be a form of homeosynaptic plasticity. But what we really need to do now is combine this small subset of pathways with our other pathways, the dopamine activated pathways, because there are interaction terms and that could change things a little bit, such as the PKA that gets activated by dopamine might actually be suppressing the GQ pathways and might be suppressing endocannabinoid production. And even though we need the GQ for PKC, this is where spatial location, again, becomes important. The location of these different molecules, whether it's been happening in the spine or in the dendrite, would influence which pathways get activated. And then I just want to conclude by giving credit to the people who did the work and thanking the funding agents. And this model was mostly done by Bohang Kim, who was a postdoc in the lab, and Lane Wallace, who was a visiting faculty in the lab at the same time. And Juan Yilco has been working on some of the software development. And Rodrigo also did the validation of the NeuroRD software. And Sarah Haas did those lovely experiments that we're now trying to publish. Thank you.