Makni Salima, Beckmann Christian, Smith Steve, Woolrich Mark
Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.
Neuroimage. 2008 Oct 1;42(4):1381-96. doi: 10.1016/j.neuroimage.2008.05.052. Epub 2008 Jun 6.
In Penny et al. [Penny, W., Ghahramani, Z., Friston, K.J. 2005. Bilinear dynamical systems. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360(1457) 983-993], a particular case of the Linear Dynamical Systems (LDSs) was used to model the dynamic behavior of the BOLD response in functional MRI. This state-space model, called bilinear dynamical system (BDS), is used to deconvolve the fMRI time series in order to estimate the neuronal response induced by the different stimuli of the experimental paradigm. The BDS model parameters are estimated using an expectation-maximization (EM) algorithm proposed by Ghahramani and Hinton [Ghahramani, Z., Hinton, G.E. 1996. Parameter Estimation for Linear Dynamical Systems. Technical Report, Department of Computer Science, University of Toronto]. In this paper we introduce modifications to the BDS model in order to explicitly model the spatial variations of the haemodynamic response function (HRF) in the brain using a non-parametric approach. While in Penny et al. [Penny, W., Ghahramani, Z., Friston, K.J. 2005. Bilinear dynamical systems. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360(1457) 983-993] the relationship between neuronal activation and fMRI signals is formulated as a first-order convolution with a kernel expansion using basis functions (typically two or three), in this paper, we argue in favor of a spatially adaptive GLM in which a local non-parametric estimation of the HRF is performed. Furthermore, in order to overcome the overfitting problem typically associated with simple EM estimates, we propose a full Variational Bayes (VB) solution to infer the BDS model parameters. We demonstrate the usefulness of our model which is able to estimate both the neuronal activity and the haemodynamic response function in every voxel of the brain. We first examine the behavior of this approach when applied to simulated data with different temporal and noise features. As an example we will show how this method can be used to improve interpretability of estimates from an independent component analysis (ICA) analysis of fMRI data. We finally demonstrate its use on real fMRI data in one slice of the brain.
在彭尼等人的研究中[彭尼，W.，加哈拉马尼，Z.，弗里斯顿，K.J. 2005年。双线性动力系统。《英国皇家学会会报B：生物科学》360(1457) 983 - 993]，线性动力系统（LDSs）的一个特定案例被用于对功能磁共振成像（fMRI）中血氧水平依赖（BOLD）反应的动态行为进行建模。这个状态空间模型，称为双线性动力系统（BDS），用于对fMRI时间序列进行去卷积，以估计实验范式中不同刺激所诱发的神经元反应。BDS模型参数使用加哈拉马尼和辛顿提出的期望最大化（EM）算法进行估计[加哈拉马尼，Z.，辛顿，G.E. 1996年。线性动力系统的参数估计。技术报告，多伦多大学计算机科学系]。在本文中，我们对BDS模型进行了修改，以便使用非参数方法明确地对大脑中血流动力学反应函数（HRF）的空间变化进行建模。在彭尼等人的研究中[彭尼，W.，加哈拉马尼，Z.，弗里斯顿，K.J. 2005年。双线性动力系统。《英国皇家学会会报B：生物科学》360(1457) 983 - 993]，神经元激活与fMRI信号之间的关系被表述为使用基函数（通常为两个或三个）进行核展开的一阶卷积，而在本文中，我们主张采用空间自适应广义线性模型（GLM），其中对HRF进行局部非参数估计。此外，为了克服通常与简单EM估计相关的过拟合问题，我们提出了一种全变分贝叶斯（VB）解决方案来推断BDS模型参数。我们证明了我们的模型能够估计大脑每个体素中的神经元活动和血流动力学反应函数的有用性。我们首先研究这种方法应用于具有不同时间和噪声特征的模拟数据时的行为。作为一个例子，我们将展示如何使用这种方法来提高fMRI数据独立成分分析（ICA）分析估计的可解释性。我们最后展示了它在大脑一个切片的真实fMRI数据上的应用。
