Comparing hemodynamic models with DCM

Abstract

The classical model of blood oxygen level-dependent (BOLD) responses by Buxton et al. [Buxton, R.B., Wong, E.C., Frank, L.R., 1998. Dynamics of blood flow and oxygenation changes during brain activation: the Balloon model. Magn. Reson. Med. 39, 855-864] has been very important in providing a biophysically plausible framework for explaining different aspects of hemodynamic responses. It also plays an important role in the hemodynamic forward model for dynamic causal modeling (DCM) of fMRI data. A recent study by Obata et al. [Obata, T., Liu, T.T., Miller, K.L., Luh, W.M., Wong, E.C., Frank, L.R., Buxton, R.B., 2004. Discrepancies between BOLD and flow dynamics in primary and supplementary motor areas: application of the Balloon model to the interpretation of BOLD transients. NeuroImage 21, 144-153] linearized the BOLD signal equation and suggested a revised form for the model coefficients. In this paper, we show that the classical and revised models are special cases of a generalized model. The BOLD signal equation of this generalized model can be reduced to that of the classical Buxton model by simplifying the coefficients or can be linearized to give the Obata model. Given the importance of hemodynamic models for investigating BOLD responses and analyses of effective connectivity with DCM, the question arises which formulation is the best model for empirically measured BOLD responses. In this article, we address this question by embedding different variants of the BOLD signal equation in a well-established DCM of functional interactions among visual areas. This allows us to compare the ensuing models using Bayesian model selection. Our model comparison approach had a factorial structure, comparing eight different hemodynamic models based on (i) classical vs. revised forms for the coefficients, (ii) linear vs. non-linear output equations, and (iii) fixed vs. free parameters, epsilon, for region-specific ratios of intra- and extravascular signals. Using fMRI data from a group of twelve subjects, we demonstrate that the best model is a non-linear model with a revised form for the coefficients, in which epsilon is treated as a free parameter.

Fig. 1

Schematic summary of the hemodynamic forward model in DCM. Experimentally controlled input functions u evoke neural responses x, modeled by a bilinear differential state equation, which trigger a hemodynamic cascade, modeled by 4 state equations with 5 parameters. These hemodynamic parameters comprise the rate constant of the vasodilatory signal decay (κ), the rate constant for autoregulatory feedback by blood flow (γ), transit time (τ), Grubb's vessel stiffness exponent (α), and capillary resting net oxygen extraction (ρ). The so-called Balloon model consists of the two equations describing the dynamics of blood volume (ν) and deoxyhemoglobin content (q) (light grey boxes). Integrating the state equations for a given set of inputs and parameters produces predicted time-series for ν and q which enter a BOLD signal equation λ (dark grey box) to give a predicted BOLD response. Here, we show the equations of the RBM_N(ε) model from this paper. For parameter estimation, an observation model is used that treats the observed BOLD response as a function of inputs and parameters plus some observation error (see main text).

Fig. 2

This figure shows a log-normal probability density function (solid line) with a mean of 1 and a variance of 0.5. For comparison, a Gaussian probability density function with identical mean and variance is shown (dashed line). Note that in contrast to the Gaussian, the support of the log-normal density is restricted to positive numbers, as indicated by the dotted vertical line.

Fig. 3

(A) Summary of the DCM used in this evaluation study (see Stephan et al., 2007a for details). This is a four-area model, comprising reciprocally connected lingual gyrus (LG) and fusiform gyrus (FG) in both hemispheres. Non-foveal visual stimuli (words) were presented in either the right (RVF) or left (LVF) visual field with a randomized stimulus onset asynchrony between 1.5 and 2.5 s during 24 s blocks; these were modeled as individual events driving contralateral LG activity. During the instruction periods, bilateral visual field (BVF) input was provided for 6 s; this was modeled as a box-car input to LG, in both hemispheres. Connections were modulated by task and stimulus properties (grey dotted lines). Intra-hemispheric LG→FG connections were allowed to vary during a letter decision (LD) task, regardless of visual field. In contrast, inter-hemispheric connections were modulated by task conditional on the visual field (LD|LVF and LD|RVF). (B) This figure provides an anecdotal example of how two different models (red solid line: RBM_N(ε); blue dashed line: RBM_L) fit measured BOLD data (black solid line). For this example, we chose the first 160 scans from the left fusiform gyrus in a single subject from this study. The x-axis denotes seconds, y-axis denotes percent signal change.

Fig. 4

Summary of the model comparison results. This graph compares the eight DCMs resulting from the factorial structure of our comparison (classical vs. revised coefficients, linear vs. non-linear BOLD equations, fixed ε vs. free ε) against a reference model (the original revised model by Obata et al., 2004, with fixed values for E₀ and ε.). The figure shows the log of the group Bayes factor (i.e., the log model evidence summed across the twelve subjects, minus the summed log-evidence for the reference model). Except for the reference model, all models treated E₀ as a free parameter. See Table 1 for interpretation of the model names.

Fig. 5

This figure demonstrates the effect that changing the resting oxygen extraction fraction, E₀, and the ratio of intra- to extravascular BOLD signal at rest, ε, has on the shape and amplitude of the hemodynamic impulse response as generated by the RBM_N (solid lines) and CBM_N (dotted lines) models. Colors indicate different values of ε (see legend). Vertical lines indicate the position of the response maximum.

Fig. 6

Posterior correlation matrix for the RBM_N(ε), averaged across all twelve subjects. Three correlations are of particular interest (see black rectangles). ε shows negative correlations (up to − 0.61; r_ε,C ) with the input parameters, C. A small subset of the A parameters (fixed connection strengths) also correlate negatively with ε (up to − 0.53; r_ε,A ). Importantly, ε is not strongly correlated with the parameters of main interest within DCM, i.e., with the parameters of the context-dependent modulation of connections, B. These correlations were mostly close to zero and maximally − 0.29 (r_ε,B ). θ^h = hemodynamic parameters (except ε).