Hi. In this module, we're going to talk a little bit more about dynamic causal modeling, or DCM. We'll try to understand a little bit more about the mathematics behind it. So DCM attempts to model latent neuronal activation using hemodynamic time series. This is based on a neuronal model of interacting regions, supplemented with a forward model that tells how the neural activity is transformed into the observed hemodynamic response. So effective connectivity is parameterized in terms of the coupling among these latent neuronal activity in different regions, and we can estimate these parameters by perturbing the system and measuring the response. So here's a little cartoon illustration, we have two regions here, one and two. And there's a relationship between these two regions, so Z1 represents the neuronal activation in region one, and Z2 the neuronal activation in region two. And we see that activation in one gives rise to activation in two and vice versa. So this is a very simple model of what's going on in these two different brain regions. Now, however, the problem with this is that we can't measure neuronal activation using fMRIs, so we have to do something else. So what we do is we perturb the system here. So let's say that we have some sort of task and this gives rise to the perturbation as follows. So, u1 perturbs Z1, so it might increase the neuronal activation. And u2 changes the relationship between regions one and two. Now once we do that, this is going to give rise to changes in the hemodynamic response, which we can measure. And so the idea here is that changes in the neuronal activation gives rise to the changes in the hemodynamic response, which we can measure using fMRI. The goal here now is to take the values of Y1 and Y2, which we can measure, and figure out what was going on on the neuronal level. So in order to do this, we have to make some definitions. So let's define the neuronal states as z. So z is equal to z1, z2, up to zN, where we have N different regions. In our little cartoon, we have two regions, so z would just be z1 and z2. Here, the effective connectivity model is described by the following model. This a bi-linear equation that tells us how the neuronal activity at time t changes. And here z of t is the neuronal activity at time t, which is latent. And u t of y is the jth of J inputs at time t. These are known to us, since we know what we do to the system. The matrix A represents the first order connectivity among regions in the absence of input. And this specifies how regions are connected and whether these connections are uni- or bidirectional. The matrix C represents the intrinsic influence of inputs on the neuronal activation. This specifies how these inputs are connected to the different regions. Finally the matrices B of j represent the change in coupling induced by the jth input. And this specifies how connections are changed by the inputs. So we can see how these equations work by looking at our simple example again. So if we do this, we see that this bi-linear equation can written in the following form. So here we see that changes in the neuronal activation in the two regions, one and two, depend on the neuronal activations in the regions at the current time point, Z1 and Z2, and also on the inputs u1 and u2. Now, neuronal activation causes changes in the blood volume and deoxyhemoglobin that cause changes in the observed BOLD response. And so the hemodynamics in the DCM model is described using what's called an extended Balloon model, which involves a set of hemodynamic state variables, state equations and hemodynamic parameters, theta of h. So, this is what the Extended Balloon Model looks like, and it's not for the faint of heart. So it's a series of differential equations showing how changes in activity-dependent signal, flow induction, changes in blood volume, and deoxyhemoglobin lead to the changes in the hemodynamic response. So we're not going to go into this in detail, but just kind of wanted to give you a flavor that equations like this exist. Now, summarizing all of this information, we have different state equations that we're going to be using in DCM. So, we have the neuronal state, which is described by these neuronal activations, Zt, and these have parameters, we're going to call them theta of C. And then we have the hemodynamic states, which depends on s of t, v of t and q of t. And so the observed data is a function of q of t, of v of t. And we won't talk about this so much in detail, but I just want you to know that such a function exists, and this has parameters theta of h. So we can combine the neuronal and hemodynamic states, and get the following state-space model. And this a sort of standard state space model that people know how to solve. And basically in DCM analysis is performed using Bayesian methods. So normal priors are placed on the unknown thetas here and the posterior density is used to make inferences about the connections. Finally, model comparisons, the comparison between different suggested models, can be performed to determine whether the data favors one of the models over another. And so this is done, using Bayesian model comparisons. So, there's something called the model evidence, which is the probability of the data, given a specific model, and it can be computed in the following way. Now, in order to compare models one computes something called the Bayes factor. So the Bayes factor, we'll call this B i of j, is basically the ratio of the model evidence from model i divided by the model evidence for model j. So if Bij is large, and by large I mean bigger than one, than i is more likely than j. And if Bij is less than one, then j is more likely than i. Now, in order to compute this, various approximations exist, including negative free energy, AIC, or BIC. And this is all implemented in the SBM software that's very popular in the neuro-imaging community. Here's an example from the UCL group who's behind SBM, and here they're showing three different models here that they want to compare, and they use Bayes factors to compare the three different candidate DCMs and find the one that is the mostly likely. So, that's the end of this module. Here, I've just tried to give you a brief feel for what dynamic causal modeling is. Again, this is implemented in the SBN software. And it tends to be quite mathematical, but I just wanted to give you a basic feel for what's going on there so you could read papers about it and have a general idea understanding. Okay. In the next module we'll talk about Granger causality. I'll see you then. Bye. [SOUND]