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between integrated C4VR output and each imaging data were plotted on the photo
image of the ventral medullary surface (Fig. 12.2d), and there were few pixels whose
values were over 0.4. In contrast, when signals were moving time averaged (bin
width
3 pixels, respiratory related activities
became visible (Fig. 12.2f, g). Many pixels whose correlation coefficient values
were more than 0.5 were seen widely (Fig. 12.2h).
=
7) and spatially averaged by 3
×
12.2.2 Modeling
Let us consider a model which estimates the respiratory motor output from respira-
tory related optical signals. It is natural to assume that the respiratory motor output
is the sum of an estimation function of optical signals derived from pixels involved
in respiratory neuronal activities:
N
i = 1 g i ( x i ( t )) .
y (
t
)=
(12.1)
Here, y ,
,
x j , and N represent the estimated C4VR motor output, the estimation
function, optical time series data in pixel i , and the number of respiratory related
pixels, respectively. However, the number of respiratory related pixels is several
hundreds, and it is not practical to deal with (12.1) because too many numbers of
parameters must be determined. Instead, we consider a model where we estimate
the respiratory motor output from optical signals derived from a specific set of res-
piratory related pixels. The number of pixels taken into account N 0 is very small as
compared to N , and in the extreme case, N 0 can be 1.
g i
N 0
j = 1 f j ( x j ( t )) .
y (
t
)=
(12.2)
In this case, it is essential to develop methods to determine the estimation func-
tion and to select the specific set of pixels. First, let us consider the case where N 0
is 1, i.e., a single-input single-output (SISO) model. The model must satisfy the
following conditions:
(1) The respiratory motor output is not activated unless the optical signal within
the pre-BoC exceeds a certain threshold.
(2) The pre-BotC region is activated earlier and deactivated later than the respi-
ratory motor output.
To satisfy these conditions, we consider a nonlinear dynamic model consisting of
a sigmoid function and a delayed first-order transfer function (STF model; sigmoid
and transfer function model),
Ke Ls
1
1
y (
s
)=
Ts ×
e ( x ( s ) a ) ,
(12.3)
+
1
+
 
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