Nonlinear temporal dynamics of the cerebral blood flow response

Abstract

The linearity of the cerebral perfusion response relative to stimulus duration is an important consideration in the characterization of the relationship between regional cerebral blood flow (CBF), cerebral metabolism, and the blood oxygenation level dependent (BOLD) signal. It is also a critical component in the design and analysis of functional neuroimaging studies. To study the linearity of the CBF response to different duration stimuli, the perfusion response in primary motor and visual cortices was measured during stimulation using an arterial spin labeling technique with magnetic resonance imaging (MRI) that allows simultaneous measurement of CBF and BOLD changes. In each study, the perfusion response was measured for stimuli lasting 2, 6, and 18 sec. The CBF response was found in general to be nonlinearly related to stimulus duration, although the strength of nonlinearity varied between the motor and visual cortices. In contrast, the BOLD response was found to be strongly nonlinear in both regions studied, in agreement with previous findings. The observed nonlinearities are consistent with a model with a nonlinear step from stimulus to neural activity, a linear step from neural activity to CBF change, and a nonlinear step from CBF change to BOLD signal change.

Figure 1

Diagram of the transformation from stimulus presentation to BOLD response. At each step in the model, nonlinearities may be introduced.

Figure 2

Timing of imaging relative to stimulus presentation. Images are acquired every 2 sec (TR = 2 sec), alternating between control and tag conditions. The presented stimulus is repeated with a period of 2n + 1 seconds (n = 3 above) so that over four consecutive stimulus cycles, a control and a tag image are acquired at each second of the stimulus cycle. The gray arrows illustrate tag and control images that are gathered with 1‐sec resolution over a 2‐sec window of the stimulus cycle.

Figure 3

Average flow and BOLD signal measurements. Each experiment consisted of 16 repetitions of three durations of motor or visual stimulus (2, 6, and 18 sec) followed by a resting period with no stimulus (19 sec). The measurements shown are the intersubject average response to a single cycle of stimulus. Responses are expressed as percent signal change from baseline.

Figure 4

Linear, time‐invariance analysis of motor and visual flow data shown in Figure 3. Plot titles indicate the duration of the short‐ and long‐duration responses used in each analysis. Linearity was tested by appropriately replicating, shifting, and summing the measured response to a short‐duration stimulus. If flow is a linear transformation of the stimulus, this superposition (solid gray lines) should predict the measured response to the long‐duration stimulus (dotted black lines). A mismatch of the predicted and measured long‐duration responses indicates a nonlinear response. The flow response in the motor cortex appears to be nearly linear (with a slight overprediction of the 6‐sec response), whereas the flow response in the visual cortex behaves nonlinearly.

Figure 5

Linear, time‐invariance analysis of motor and visual BOLD data shown in Figure 3. Plot titles indicate the duration of the short‐ and long‐duration responses used in each analysis. Linearity was tested by appropriately replicating, shifting, and summing the measured response to a short‐duration stimulus as in Figure 4. The BOLD response in both the visual and motor cortices is nonlinearly related to stimulus duration.

Figure 6

Fractional moment differences (mean ±1 SD) calculated according to Eq. 1 for the zeroth, first, and second moments (top to bottom) from individual subject data. Intersubject averages of the linearity comparisons are shown in Figures 4 and 5. A response is nonlinear if M_n is significantly different from zero for any n. The motor CBF response is not significantly nonlinear, but the visual CBF and BOLD responses are in general nonlinear.

Figure 7

Model for the steps from block stimulus presentation to BOLD response (based on the pathway introduced in Fig. 1). The neural response is modeled as a nonlinear function of the presented stimulus in which an initial strong response decays to a lower steady‐state value (Eq. 2). This neural response is convolved with a flow impulse response model (the gamma‐variate function of Eq. 3) to give the CBF response. This model for the CBF response is a nonlinear function of the stimulus, but a linear transformation of the neural response. Finally, the step from CBF to BOLD is described by a scaling curve that saturates with increasing flow.

Figure 8

Model functions fit to the motor and visual flow data shown in Figure 3. These functions describe the steps from stimulus to neural response (left subplot) and neural response to flow response (right subplot). The convolution of these two forms (described by Eqs. 2 and 3) were simultaneously fit to the measured flow response to 2, 6, and 18‐sec stimuli in the motor area (τ_n = 0.25, t_d = 0.65, a = 0.25, τ_h = 1.25, c = 65, m = 3) and the visual area (τ_n = 0.5, t_d = 1.5, a = 3.0, τ_h = 1.5, c = 54, m = 3). Of the parameters that effect the functional form of the model (a, τ_n, m, and τ_h), only a differs significantly across the two fits. The (nearly linear) motor flow data is fit by a neural response with only a small overshoot (a = 0.25), whereas the (highly nonlinear) visual flow data is fit by a neural response with a large overshoot (a = 3.0). Also note that the flow impulse responses fit to the motor and visual data are quite similar, with FWHM of 5.2 sec (motor) and 6.2 sec (visual).

Figure 9

Flow model fits (solid gray lines) and measured flow responses (dotted black lines). The flow models are generated from the functions shown in Figure 8 by varying the input stimulus duration.

Figure 10

Illustration of the interaction between neural adaptation nonlinearities and the saturation nonlinearity of the BOLD effect at high levels of flow. The flow signal is assumed to be scaled according to the saturation curve (solid gray line) to produce the BOLD signal. For small adaptation nonlinearities (a nearly linear flow response such as our motor measurements, a = 0.25), the flow change measured for short‐duration stimuli is small (here, 22%) compared to the flow response to long‐duration stimuli (65%). In this case, the effect of BOLD saturation is strong, and a linear prediction of the long‐duration BOLD response based on the short‐duration BOLD response (upper dashed black line) will yield a strong overprediction. On the other hand, a strong adaptation nonlinearity (i.e., a strongly nonlinear flow response such as in our visual data, a = 3.0) will cause even short‐duration flow responses to nearly reach the plateau value. The saturation nonlinearity has little effect in this case (lower dashed line).

Figure 11

Simulations of the linearity analyses shown in Figures 4 and 5 using the flow models shown in Figure 8 and the BOLD saturation curve shown in Figure 10. The models were used to generate flow and BOLD responses to 2‐ and 18‐sec stimuli. The curves shown are the “actual” 18‐sec response (in black) and the 18‐sec response that is predicted from the 2‐sec response (in gray). The top figures (labeled a = 0.25) are generated from the flow model fit to the motor flow data. Here, a weak nonlinearity in the flow response is transformed into a strong BOLD nonlinearity by the BOLD saturation effect. The bottom figures (labeled a = 3.0) are generated from the visual flow model. The strong adaptation nonlinearity in the flow response leads to a weak nonlinearity in the step from CBF change to BOLD change. Note the similarity of these curves to the corresponding figures marked “2/18 s” in Figures 4 and 5.