Ch. 5 Bayesian Treatment of Neuroimaging Data Will Penny and Karl Friston Ch. 5 Bayesian Treatment of Neuroimaging Data Will Penny and Karl Friston 18 Dec summarized by Soo-Jin Kim Contents 1. Summary 2. The General Linear Model (GLM) 3. Parameter Estimation 1.ML for GLM 2.Bayes Rule for GLM 4. Posterior Probability Mapping (PPM) 5. Dynamic Causal Modeling (DCM) 2008, SNU Biointelligence Lab,2 Summary A general issue in the analysis of brain imaging data the relationship between the neurobiological hypothesis one posits and the statistical models adopted to test that hypothesis fMRI: functional maps showing which regions are specialized for specific functions General linear model Analysis of functional specialization Posterior probability maps (PPM) Analysis of functional integration Dynamic causal modeling (DCM) 2008, SNU Biointelligence Lab,3 Design matrix Design matrix for study of attention to visual motion four columns: Photic stimulation, Motion, Attention and a constant rows : time points in the imaging time series (360 images) The relative contribution of each of these variables can be assessed using standard least squares or Bayesian estimation. Classical inferences : T or F statistics Bayesian inferences: posterior or conditional probability 2008, SNU Biointelligence Lab,4 Designed effect Time point General Linear Model The general linear model is an equation that expresses an observed response variable y in terms of a linear combination of explanatory variables X 2008, SNU Biointelligence Lab,5 where y is a T 1 vector comprising responses at, e.g. T time points, X is a T K design matrix, is a vector of regression coefficients, and e is a T 1 error vector GLMs are fitted to each voxel using ML or OLS estimators. Temporal Basis Functions (1/2) Functional MRI using blood oxygen level dependent (BOLD) contrast provides an index of neuronal activity indirectly via changes in blood oxygenation levels Temporal basis functions are important because they enable a graceful transition between conventional multilinear regression models with one stimulus function per condition and FIR models with a parameter for each time point following the onset of a condition. 2008, SNU Biointelligence Lab,6 Temporal Basis Functions (2/2) The convolution model for fMRI responses takes a stimulus function encoding the supposed neuronal responses and convolves it with an HRF to give a regressor that enters into the design matrix 2008, SNU Biointelligence Lab,7 Parameter Estimation: ML Maximum Likelihood N ( x; m, ) specifies a multivariate Gaussian distribution over x with mean m and covariance . The likelihood specified by the GLM described in the previous section is given by p(y|) = N(y;X ,C e ), where C e is an error covariance matrix. The maximum-likelihood (ML) estimator is given by [25]: ML =(X T C e -1 X) -1 X T C e -1 y For other modalities, C e = 2 I often suffices. The ML estimate then reduce to the ordinary least squares (OLS) estimator OLS =(X T X) -1 X T y 2008, SNU Biointelligence Lab,8 Parameter Estimation: Bayesian Bayes rule for GLMs The Bayesian framework allows us to incorporate our prior beliefs about parameter values. If our beliefs can be specified using the Gaussian distribution P( )=N(; p, C p ), where p is the prior mean and C p is the prior covariance, then the posterior distribution is [18] p( |Y ) = N(; ,C), Where C -1 =X T C e -1 X+C p -1 (posterior precision) =C(X T C e -1 y+C p -1 p (posterior mean) In the abscence of prior information, i.e. C p -1 = 0, the above estimate reduces to the ML estimate. If the priors are correct, then Bayesian estimation is more accurate than ML estimation. 2008, SNU Biointelligence Lab,9 Bayesian estimation: Nonlinear case 2008, SNU Biointelligence Lab,10 Posterior Probability Mapping (PPM) Posterior probability map ( p( > |Y ) > ): images of the probability that an activation exceeds some specified threshold, given the data Posterior distribution ( p( |Y ) ) : probability of getting an effect, given the data 2008, SNU Biointelligence Lab,11 Activation threshold Probability Here, we used an effect-size threshold of 0.7% of whole-brain mean, and a probability threshold of 0.95. SPM vs. PPM Maximum intensity projections (MIPs) of SPM (left) and PPM (right) for the fMRI study of attention to visual 2008, SNU Biointelligence Lab,12 SPM vs. PPM The difference bwt SPM and PPM The SPM identifies a smaller number of voxels than the PPM The SPM appears to have missed a critical and bilaterally represented part of the V5 complex The SPM is more conservative because the correction for multiple comparisons 2008, SNU Biointelligence Lab,13 Dynamic causal modeling (DCM) Dynamic causal modeling (DCM) is used to make inferences about functional integration and has been formulated for the analysis of fMRI time series in Friston et al. Current DCMs for fMRI comprise a bilinear model for the neurodynamics and an extended balloon model for the hemodynamics The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model () z y 2008, SNU Biointelligence Lab,14 DCM: bilinear state equation The neurodynamics are described by the multivariate differential equation This is known as a bilinear model because the dependent variable, , is linearly dependent on the product of u j and z. intrinsic connectivity m external inputs (known) system state direct inputs modulation of connectivity state changes 2008, SNU Biointelligence Lab,15 DCM: balloon model In brief, for the i th region, neuronal activity z i causes an increase in the vasodilator signal s i that is subject to autoregulatory feedback. Inflow f i responds in proportion to this signal with concomitant changes in blood volume v i and deoxyhemoglobin content q i. This process converts neuronal activity in the i th region z i to the hemodynamic response. The parameters = {A,B,C, h} are updated until convergence. 2008, SNU Biointelligence Lab,16 DCM for the fMRI study of attention to visual motion The interesting aspects of this connectivity involve the role of motion and attention in exerting bilinear effects. The effect of motion in the visual field was modeled as a bilinear modulation of the V1 to V5 connectivity The effect of attention was allowed to modulate the backward connections from IFG and SPC. 2008, SNU Biointelligence Lab,17