Impact of BVF on single-region model estimation
We now compare and evaluate the respective impact of the realistic and assumed BVFs
on hemodynamic model estimation within a single-region. Firstly, we chose the maximally
activated voxel in the left primary motor cortex (LPM) on the basis of the analyzed
fMRI data from SPM5 as the ROI (Fig. 2) and then defined the cluster based on faces and edges excluding corners in order
for this voxel to have six neighbors. We extracted the ultimate time series to be
analyzed by averaging over the time series of seven voxels. This procedure allowed
the model parameters and state-space functions for each of the two subjects to be
estimated. Furthermore, for the sake of simplicity, we assumed that the neural parameter
had the same value throughout all trails: , where denotes the number of trials (i.e., here). A control random search algorithm was applied in the parameter estimation
procedure 25].
Fig. 5. Estimated BOLD signal (a) and reconstructed physiological states (b) for the maximally activated voxel of subject 2. For comparison, model estimation
was also performed with the typically assumed value of . The real value of this voxel was 0.0308. It is also evident from this subject that differences
in the estimated physiological states are relevant to deviations from the actual BVF
value
Fig. 6. Results of a DCM analysis applied to the finger tapping experiment. The value indicates
the connection strength ( in Eq. 3) in DCM. The coupling parameters calculated with the real are shown alongside the corresponding connections. The values in brackets indicate
the deviations from parameters estimated when using the assumed . in the LPM, in the RPM, and the assumed in both areas. and represent external inputs to the system, and are the hemodynamic observations, and arrows indicate connections
Figures 4 and 5 show the BOLD signal and underlying physiological variables of the two subjects for
the real derived from CBV imaging in the maximally activated voxel. The estimated BOLD signal
and state variables for an assumed value of are also drawn in Figs. 3 and 4 (as dashed lines). The comparison indicates that the assumed and true could produce similar BOLD estimates in terms of magnitude and shape, with only a
slight distinction in the plateau period. This result is consistent with those of
previous studies involving the balloon model. However, we also found a large difference
between the assumed and actual values in terms of the reconstructed physiological states. We can conclude that the
intensity of changes in the underlying state variables with the assumed were double those with the true ; that is, underestimating produced an overestimation of the physiological state variables. Moreover, Figs.
4 and 5 indicate that a larger difference between actual and hypothetical , resulted in a greater difference between estimated physiological state. This means
that attention should be paid to ensuring that a realistic is used in model estimation. The presence of a larger amount of blood in an activated
voxel magnifies the effects induced by neuronal activity, lead to an excessive signal
for that voxel and unrealistic activity predictions. Similar BOLD changes in a voxel
associated with larger veins will change f, v, and q less than for a voxel with a smaller blood fraction. Most activation detection techniques
are only capable of indicating the neural activity from changes in BOLD signal or
activity maps, and they do not direct infer whether the underlying physiological variation
is closely related to and actually reflects neural activity. Under this circumstance, the use of an arbitrary
value of will influence the spatial specificity of fMRI signals in statistical testing. However,
we can assume that functional activated regions induced by an experimental event rather
than large regional amounts of blood and the employment of an unrealistic are suitable when fMRI signal estimation and activation detection are exclusively
needed.
Table 1 indicates that the uncertainty of induces changes in other parameters, with exerting a complicated, nonlinear, and inconsistent influence on the entire hemodynamic
process. Table 1 also indicates that has a greater influence on the estimated neuronal efficacy parameter than on the other parameters ( is 0.3910 with the true , and 0.9089 with the hypothetical ). A previous study found that the uncertainty of model output was more sensitive
to variation of than those of other parameters, except 12]. The defined represents the efficacy with which neural activity causes an increased BOLD signal.
As a consequence, if we could use the true , the estimated could offer a better and more intuitive reflection of the activation level, enhancing
the functional specificity of fMRI.
Table 1. Model parameters estimated using the true value () and a typical assumed value () for the maximally activated voxels of two subjects
Impact of BVF on dynamic causal models
As for balloon model research, dynamic causal modeling (DCM) has been introduced to
explore effective connectivity based on hemodynamic observations 8], 9]. DCM extends the balloon model from a single region to multiple regions by utilizing
a multiple-input, multiple-output system. Single-region model estimation supposes
that the extrinsic experimental input consistently accesses all brain regions and
that a certain brain area only receives input in this way ( in Eq. 1), whereas DCM assumes that responses ( in Eq. 3) are elicited by two distinct inputs sources: the extrinsic influence of the sensory
input ( in Eq. 3) and the intrinsic influence of the interaction regions ( in Eq. 3). In other words, DCM uses estimated neural activities (internal and external) to
evaluate the causal correlation among brain areas. While the uncertain has an important influence on parameter in the hemodynamic model, it is interesting to know how the influences DCM. In this study we therefore also investigated the effect of on DCM.
We constructed the simplest two-region hierarchical system in order to demonstrate
the significant effect of BVF on the DCM system. From the two brain areas that interact
with and influence each other, we could measure the observed BOLD signals that each
of the two regions produced, the relationship can be expressed as follows:
(3)
where and are the neuronal dynamics in two regions, and represent external inputs to the system, and represent the internal connectivity within a region without input, and encode the fixed inter-region connectivity without input, and and embody the extrinsic influences of input on neuronal activity. One can augmented
the state vector consisting of the model parameters at two regions by concatenating
them into a single higher dimensional state space and the measurement vector was also
expanded to include two observations in two areas 8]. In the experiment, we adopted a 0–1 square-wave function as two inputs, and the
system output was two time series from two regions, and . While attempting to determine the dimension of the parameters, a more efficient
filtering strategy was used to deal with the model estimation problem 26], 27]. The estimation scheme employed for DCM is formally identical to that reported previously
5], 15]. The results of this analysis are presented in Fig. 5, in which the effective connections are presented as directed black arrows along
with coupling parameters calculated with the real and assumed . In order to construct the model system, we chose two regions in the left primary
(LPM) and the right primary motor cortex (RPM) containing the two maxima of the activation
map. The output region-specific time series comprised all adjacent (based on faces
and edges but not corners) voxels of each maximum (a total of seven voxels), the location
is shown in Fig. 2. The conflicts between the motor preparation were interpreted as inhibitory connections
between the LPM and RPM 28], 29]. The fixed connectivity from the RPM to the LPM is actually slightly weaker than
that from the LPM to the RPM. This indicates that backward influences (RPM to LPM)
are stronger than forward connections (LPM to RPM). Furthermore, the fixed connectivity
in the RPM is stronger than that in the LPM, indicating that the right path-way is
used more frequently than the pathway on the left side. From Fig.6 we conclude that the two different have different impacts, with the largest deviation being about for the strength of the visual input to the LPM or RPM.