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where a , K , L , and T represent threshold, gain, dead time (delay) and time constant,
respectively. The parameter values were determined so that the variance of estima-
tion error was minimized. Given that the sampling interval is
Δ
t , the dead time L
is an integer multiple of
Δ
t , i.e., where l is an integer. For (12.3) is rewritten in a
discrete form:
K
y (
e Δ t / T y
e Δ t / T
n
)=
(
n
1
)+(
1
)
e ( x ( n l 1 ) a ) .
(12.4)
1
+
2 is expressed as
For a given l , the variance of estimation error
σ (
e
)
N
k = 1 ( y ( n ) y ( n ))
1
N
2
l
2
σ
(
e
)=
.
(12.5)
l
are estimated by one of the nonlinear
optimization method, the sequential quadratic programing method [14, 13]. Finding
l that minimizes
Then, a l C T l and K l that minimize
σ
(
e
)
l gives the optimal set of parameter values for a , K , L , and T as
σ
a l , K l , T l
l .
Next, we consider a multi-input single-output (MISO) model where the output
is estimated by the weighted sum of STF model estimates applied to optical time
series data at each pixel, expecting the improvement of the estimation.
,
Δ
t
I
i = 1 w i
K i e L i s
1
1
y
=
e ( x i a i ) ×
T i s x i (
s
) .
(12.6)
+
+
1
Here we estimate w i , the weight coefficient, using the same nonlinear optimiza-
tion method so that the estimation error variance is minimized.
12.2.3 Classification of Optical Signals Based on Activation
Timing
We classified optical signals into five categories based on the timing of the onset
of activation, the timing when the activation reached its peak, the timing when the
activation subsided to the resting state, and the magnitude of variation (Fig. 12.3a).
The timings were evaluated relative to the respiratory motor activity. Figure 12.3b
exemplifies the five activation patterns with relation to the respiratory motor activity.
Note that the respiratory motor activity and optical time series data were artificially
composed in these examples. Type-1 pixels correspond to pixels within the pFRG,
whereas Type-2 pixels correspond to those within the pre-B otC, which more directly
contribute to the respiratory motor output. Type-3 and Type-4 are also respiratory
related pixels, but are assumed to poorly contribute to the respiratory motor activity.
Type-5 is pixels that show by chance behaviors similar to the respiratory motor
activity.
 
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