Nonlinear event-related responses in fMRI
Summary (4 min read)
INTRODUCTION
- This paper is about evoked hemodynamic responses in functional magnetic resonance imaging (fMRI) and how the measured blood oxygen level dependent (BOLD) effects can be related to underlying neuronal activity.
- This model was subsequently elaborated in the context of the general linear model (2) (3) (4) .
- The importance of this work relates to understanding the nonlinear relationship between evoked responses and sensory or behavioral parameters (such as presentation rate) and the implied constraints imposed upon experimental design and analysis.
- From a data analysis perspective, the framework described in this paper can be seen as a generalization of linear approaches that characterize hemodynamic responses, evoked by single events, in terms of basis functions of peri-stimulus time (9).
- In the second, event-related experiment, words were presented in isolation.
Nonlinear System Identification
- Neuronal and neurophysiological dynamics are inherently nonlinear and lend themselves to modeling by nonlinear dynamic systems.
- The problem of characterizing the relationship between the stimulus function or neuronal activity u ( t ) and the hemodynamic response fit) reduces to estimating the kernel coefficients h".
- In their experience, the nature of fMRI data does not permit t.he use of such techniques, so the authors have adopted a standard least squares approach.
- A special case of this is the Poisson form adopted in Friston et al. (1) that corresponds to a single gamma density with equal mean and variance.
- To accommodate these slight shifts in time, the authors often supplement the basis functions with their temporal derivatives (Fig. 1 ).
Experimental Design and Data Acquisition
- The authors apply the theory presented above to fMRI time series obtained from a single normal male subject during passive listening to words presented alone or continuously at different rates.
- The data were acquired at 2 Tesla using a Magnetom VISION (Siemens, Erlangen) whole body MRI system, equipped with a head volume coil.
- After discarding initial scans (to allow for magnetic saturation effects), each time series comprised 1200 (first study) and 1000 (second study) volume images with 3-mm isotropic voxels.
- In the first, epoch-or rate-related experiment, the subject listened to monosyllabic or bisyllabic concrete nouns (i.e., dog, radio, mountain, gate) presented at five different rates (10, 15, 30, 60, and 90 worddmin) for epochs of 34 s (20 scans), intercalated with periods of rest.
- The five presentation rates were successively repeated according to a Latin Square design.
Data Preprocessing
- The data were analyzed with SPM96 (Wellcome Department of Cognitive Neurology, http://www.fil.ion. ucl.ac.uk/spm).
- The time series were realigned, corrected for movement-related effects, and spatially normalized into the standard space of Talairach and Tournoux (13) using the subject's coregistered structural Tl scan (14, 15).
- The data were spatially smoothed with a 5-mm isotropic Gaussian kernel and temporally smoothed with a ,8 -s Gaussian kernel.
- Because the authors also smoothed the design matrix, the temporal smoothing does not affect the kernel or response function estimates (2) .
Epoch-Related Responses
- The data were analyzed using a design matrix that included the explanatory variables (convolved time series) in Eq. [3].
- The remaining columns contain the constant (used to estimate go) and other effects designated as confounds (low-frequency artifacts, global effects, and so on).
- The left-hand columns contain the explanatory variables of interest x,(t) and x,(t).x,(t), where x,(t) is word presentation rate u(t) convolved with t h e basis functions b,(t) in Fig. 1 .
- The second-order kernel (lower panel) is presented in image format.
Event-Related Responses
- By specifying a stimulus function u(t) that models the occurrence of a single word, the authors can use Eq. [I] and the kernel estimates in Fig. 3 to simulate the hemodynamic response of this brain region to single word.
- This response is "simulated" using a model whose parameters were determined without ever presenting single words in isolation (i.e,, the Volterra series model based on the rate experiment).
- A validation of the model can be effected in terms of the empirically determined event-related response to actual single words using the second experiment.
- The basis functions were the gamma functions used above (solid lines in Fig. 1 ) and their derivatives (dotted lines in Fig. 1 ).
- The striking similarity between the empirically observed event-related response and that predicted on the basis of the Volterra kernels ho, hl, and hZ obtaining from the rate experiment (upper panel) can be considered a validation of the estimation procedure and the underlying model.
NONLINEAR ASPECTS OF EVOKED RESPONSES
- To assess the significance of the nonlinear response components (due to hZ), over and above the first-order components, the authors repeated the analysis of the rate experiment, treating the first-order effects (i.e., the contributions determined by h') as confounds.
- The resulting SPM(F} is shown in Fig. 6 and implicates both periauditory regions and the left posterior superior temporal region (Wernicke's area), suggesting that nonlinear effects are not only prevalent but very significant (P < 0.001 corrected) in this experimental design.
- The Form of the Kernel Estimates and Implications for the "Structure" of Nonlinear Effects.
- The overall form is very similar and this suggests a simple form for the underlying nonlinear model of evoked responses: where f(.) is a nonlinear scalar function.
where ~' ( T , , T , ) has been replaced by h ' ( ~~) . h ~( ~~) .
- This simpler model is equivalent to convolving the stimulus function with a first-order kernel (i.e., a linear "latent" hemodynamic response function) and then taking some nonlinear (e.g. , second-order polynomial) function of the result.
- The distinction between the general form implied by the Volterra series and this simpler form is depicted in Fig. 7 .
- It is pleasing to note that this simple form was adopted by Vazquez and No11 (10) in their nonlinear characterization of evoked visual responses.
- These authors assumed a Gaussian form for the kernel, but still Note that this SPM{F} has been thresholded at half the value employed for the equivalent SPM{.
- F} testing for the firstand second-order effects in Fig. 2 .
SPM{F} -Nonlinear effects
- The authors results speak to the appropriateness of the general form for their model.
- These sorts of insights may help to identify the biophysical level at which nonlinearities are expressed in fMRI.
- Simple nonlinear forms are also important from the point of view of system identification using optimization techniques, because there are fewer parameters to estimate (and their relationship to the system in question is often more apparent).
- It should be noted that the simplification implicit in Eq. [4] does not help in the context of framework adopted here.
Interactions Between Stimuli
- The authors examine how the nonlinear effects identified in the previous sections come to shape the responses to different stimuli.
- This captures the essence of nonlinear responses, in the sense that interesting nonlinearities (above and beyond a simple nonlinear mapping from neuronal input to hemodynamic response) involve interactions over time.
- Of course, these simulated responses are only predictions and suggest some interesting experiments for empirical verification.
- It can be seen that this response is attenuated markedly, with an augmented undershoot, in relation to the response obtained when the stimulus is presented in isolation (broken line).
- The responses here are simply the integral under the evoked response curve during word presentation.
Variation in Responses Over the Brain
- In general, the kernel estimates for other brain regions were very similar in form.
- These first three principal components are shown in the lower panel.
- The first corresponds to the canonical hemodynamic response with a very quick onset and early peak at 3 s.
- The negative scores are most pronounced in posterior temporal, parietal, and extrastriate regions (lower panel).
- This is a real phenomenon and can be demonstrated as such by looking at the event-related responses in the posterior superior temporal region using independent data from the second study.
DISCUSSION
- The response to the second word when preceded by the first ( line), obtained by subtracting the response to the first word from the response to both, and when presented alone .
- The line represents the simulated responses using the second-order hemodynamic response function in Fig. 3 , and the dots correspond to the observed responses at the voxel in question.
- Note the nonlinear and inverted U relationship between integrated response and word presentation frequency.
- These kernel coefficients can be thought of as high-order or nonlinear extensions of linear convolution or "smoothing" functions and, therefore, represent a nonlinear characterization of the hemodynamic response function.
- A number of techniques and observations have been presented in this paper, and the authors will now review and extend some of the more important issues.
What Are the Implications of this Work for Experimental Design and Analysis?
- The first thing to note is that there is a fundamental distinction between positing the same nonlinear hemodynamic response function that can account for varying responses to stimuli presented at different rates, and a series of rate-dependent, linear hemodynamic response functions.
- This simply involves separating the firstorder terms into a set of columns for each rate (Fig. 12 , upper right).
- In the linear analysis the authors relegate these interactions to formal differences among the rate-specific response functions (and would normally try to characterize these differences post hoc).
- In the linear analysis, the authors have discounted second-order effects in the hope that a suitably shaped first-order response function can model all the nonlinearities inherent in the real response.
- While this is justifiable for epochs of a fixed rate and length, it may not be for epochs that endure over different periods of time.
Which Then is the Most Appropriate Anaiysis to Use?
- This question is only posed in parametric experimental designs (20) when some experimental parameter is varied (for example, rate of stimulus presentation, response rate, duration of task, etc.).
- The numbers on the response functions denote the presentation rate in wordslmin.
- Is possible to implement the nonlinear analysis above with existing tools (e.g., SPM96), the construction of the design matrices is complicated, and a simple linear analysis may be quite sufficient for most purposes.
- The SPM{F} One aspect of the techniques presented in this paper is the use of the SPM(FJ to make inferences about the significance of the response, in terms of the response kernel coefficients.
- As the models of hemodynamic responses become more sophisticated, the number of parameter estimates involved changing continuously and does not conform to a series of discrete levels.
- In general, however, the results that obtain from the two approaches would be similar, and the question reduces to one of implementational expediency and simplicity in describing the results.
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Citations
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Cites background or methods from "Nonlinear event-related responses i..."
...We have demonstrated previously (Friston et al., 1997) that there are significant nonlinear components in hemodynamic responses; however, these are expressed when stimuli are presented close together in time (such that the response to one stimulus is modulated by the response to a preceding stimulus)....
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...We have recently described how to detect the hemodynamic responses evoked by single events in functional magnetic resonance imaging (fMRI) using linear (Josephs et al., 1997) and nonlinear (Friston et al., 1997) models....
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...We have demonstrated previously (Friston et al., 1997) that there are significant nonlinear components in hemodynamic responses; however, these are expressed when stimuli are presented close together in time (such that the response to one stimulus is modulated by the response to a preceding…...
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...This model was subsequently elaborated in the context of the general linear model (Friston et al., 1995a, b; Worsley and Friston, 1995) and has been employed recently in the analysis of event-related responses (Josephs et al., 1997; Friston et al., 1997)....
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...The techniques we have used (Josephs et al., 1997; Friston et al., 1997) to identify these changes model responses in terms of basis functions of peristimulus time, using the general linear model for statistical inference (in this case a multiple regression analysis)....
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Cites methods from "Nonlinear event-related responses i..."
...…with respect to w. Intuitively, the partial derivative captures how the RPE would change if it were computed according to a different value of w (Friston et al., 1998); in this case, it is just the difference between the RPEs computed with respect to model-based and model-free action values....
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...The second timeseries involved subtracting these TD prediction errors from the RPEs that would arise if the predictions had been model-based rather than model free (Daw, in press; Friston et al., 1998; Wittmann et al., 2008)....
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...Friston and colleagues used an extension of onvolution methods with nonlinear basis functions to ccount for nonlinearities in auditory response to words poken at different rates (Friston et al., 1998)....
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...…f w w r s v b t b m v a p f s f E d t n l r a t w s a u 417DECONVOLUTION OF BOLD fMRI IMPULSE RESPONSE een found to be approximately linear (Boynton et al., 996), and this has been the basis for most of the vent-related fMRI analysis to date (Buckner et al., 996; Cohen, 1997; Friston et al., 1998)....
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