Application of artificial neural network to fMRI regression analysis

Abstract

We used an artificial neural network (ANN) to detect correlations between event sequences and fMRI (functional magnetic resonance imaging) signals. The layered feed-forward neural network, given a series of events as inputs and the fMRI signal as a supervised signal, performed a non-linear regression analysis. This type of ANN is capable of approximating any continuous function, and thus this analysis method can detect any fMRI signals that correlated with corresponding events. Because of the flexible nature of ANNs, fitting to autocorrelation noise is a problem in fMRI analyses. We avoided this problem by using cross-validation and an early stopping procedure. The results showed that the ANN could detect various responses with different time courses. The simulation analysis also indicated an additional advantage of ANN over non-parametric methods in detecting parametrically modulated responses, i.e., it can detect various types of parametric modulations without a priori assumptions. The ANN regression analysis is therefore beneficial for exploratory fMRI analyses in detecting continuous changes in responses modulated by changes in input values.

Introduction

In functional magnetic resonance imaging (fMRI) studies, searches for the occurrence of signal changes that correlated with certain events are conducted to detect brain activations. In the general linear model (GLM) approach (Friston et al., 1995), a regressor representing the canonical hemodynamic response function (HRF) is used to detect such correlations. However, if the shape of the hemodynamic response differs greatly from the pre-assumed shape (Aguirre et al., 1998, Miezin et al., 2000) or an unknown process mediates such correlations, we cannot detect those correlations. In some cases, a combination of regressors, the canonical HRF, its temporal derivative, and a dispersion derivative, is used to absorb this diversity of response shape (Friston et al., 1998a, Friston et al., 1998b). Although this method can absorb minor changes in the canonical HRF, it cannot absorb all diversity and the response variability is still a problem.

Various other approaches, which do not assume the shape of the HRF a priori, have been proposed including selective averaging (Dale and Buckner, 1997), smooth FIR filters (Goutte et al., 2000), and non-parametric Bayesian estimation of the HRF (Marrelec et al., 2003). The primary advantage of these methodologies, which are called non-parametric methods because no parametric models of the HRF are used, is that even when the response functions diverge from region to region or subject to subject, correlated responses can be detected.

In this study, we proposed another ‘semi-parametric’ method for fMRI analysis: the use of a very general class of functional forms to build more flexible models (Bishop, 1995). The proposed method uses a feed-forward layered artificial neural network (ANN) to describe a non-linear dynamic system of hemodynamic response. Some event over its recent history is used as an input and the BOLD signal from a particular voxel is used as a supervised signal. This type of artificial neural network is called a multi-layer perceptron (MLP). It is known that for an infinite number of hidden units, an MLP with one or more hidden layers whose output functions are sigmoid functions can approximate any continuous function to any degree of accuracy (Funahashi, 1989, Hornik et al., 1989). Thus this method can perform a non-linear regression analysis between BOLD signals and events without explicit modeling of the response function.

In the MLP, hidden units receive input vector x_in multiplied by an adjustable weight matrix W_hidden-in and bias value b_hidden. Hidden units have a general class of transfer function, sigmoid function tanh, and transfer the input value to the output layer.

$${\mathit{x}}_{\text{hidden}}=\tanh ({\mathbf{W}}_{\text{hidden−in}}{\mathit{x}}_{\text{in}}+{\mathit{b}}_{\text{hidden}})$$

At the output layer, the outputs of the hidden layer x_hidden are multiplied by an adjustable weight matrix W_out-hidden and a bias value b_out is added.

$$y={\mathbf{W}}_{\text{out−hidden}}{\mathit{x}}_{\text{hidden}}+{\mathit{b}}_{\text{out}}$$

y is a network output. Thus, an MLP approximates the regression function y = f(x_in), by a convolution of sigmoid functions with adjustable input weights. Hereafter, we call this regression the ANN regression.

The ANN regression produces results known to be equivalent to the Volterra kernel method (Volterra, 1959) without explicit definition of the kernel functions (Wray and Green, 1994). Friston et al., 1998b, Friston et al., 2000 have already applied the Volterra series approach to fMRI analysis. In their approach, the kernel functions are explicitly modeled (Friston et al., 2000). From the viewpoint of MLP, this approach can be seen as using fewer pre-assumed transfer functions at the hidden layer and only a restricted number of input-hidden connections. In contrast, the proposed ANN regression has no explicit form of response functions and uses fully adjustable connections between the input and hidden layers. Because an adequately trained network is equivalent to the Volterra series expansion with all the dimensions of that system (Wray and Green, 1994), the ANN regression provides more flexibility than using only a restricted set of Volterra kernels. Furthermore, such a network can describe any relations between input and output series.

The ANN regression analysis has another advantage over non-parametric methods. ANN regression can detect any type of response modulation by the input values. This type of analysis, which is known as parametric modulation analysis, deals not only with the existence of activation, but also with how that activation is modulated by the parameters of the event. To detect parametric modulation using non-parametric methods, we have to explicitly model the shape of the modulations; otherwise we can detect only linear modulations. The ANN regression, in contrast, does not require explicitly modeling of the modulation shape. Furthermore, any continuous modulation can be detected, owing to the ability of the ANN regression to adjust to any continuous functions.

This paper presents a method of ANN regression that can be applied to fMRI studies. In fMRI analyses, autocorrelation noise involved in the BOLD signal time course (Bullmore et al., 1996, Purdon and Weisskoff, 1998, Woolrich et al., 2001) may be a problem for the ANN regression, because it can fit any relations between events and BOLD signals. We considered this problem in analyzing null-task fMRI data and synthetic white noise data. In the following section, we describe the application of the ANN method to a practical case: a memory-guided saccade task. Because this task includes various responses with different shapes, it is a good example of how the ANN method can fit various shapes of responses. Finally, to explore the potential of the ANN method, a parametric modulation analysis was performed. Using a synthetic data set, we compared the ANN method and a non-parametric method and their ability to detect non-linear parametric modulations.