function with a knot at 0.4, see figure 14. The error process is again generated by a
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Figure 11 Samples path of 500 observations withµA andµB as trend function and different noise parametersdand σ2.
Gaussian FARIMA(0,d,0) process with d ∈ {−0.3,−0.1,0,0.1,0.3}. The variance of the noise process is always one. To study the effect of the normalisation, we start with a variant of the usual Bayesian information criterion
IC1(p) = log ˆσ2
+p· 1
2 ·log(n) n .
For each d, we iterate through the sample sizes n = 50,100,150, . . . ,350,400. For each sample size n and d, we simulate 300 sample paths and we fit a linear spline withl knots (l = 1,2,3) to each sample path. On the basis of IC1 we then choose the optimal model and count the frequency of each model and sample size. When fitting the spline models, we always restrict the minimal distance of the knots by h−1(n) = 1
10√
log(n) Finally, we plot the frequency of each model against the sample size (figure 15). To study the effect of the different normalisations we repeat this analysis with three variants of the Bayes
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Figure 12 Estimated variances of least squares estimators (◦) in log-log scale for µA (η = 0.2, a4 = 1,300, d ∈ {−0.3,−0.1,0.1,0.3} and σ2 ∈ {16,64}. Solid line indicates the theoretical variance as predicted by 5.2.10.
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Figure 13 Estimated variances of least squares estimators (◦) in log-log scale for µB (η = 0.5, a4 = 480,d∈ {−0.3,−0.1,0.1,0.3} andσ2∈ {16,64}. Solid line indicates the theoretical variance as predicted by 5.2.10.
information criterion:
IC2(p) = log ˆσ2
+p·1
2 · log(n) n1−2d IC3(p) = log ˆσ2
+p·2· log(n) n IC4(p) = log ˆσ2
+p·2· log(n) n1−2d
As can been in figure 15, the empirical results for IC1 and IC2 are in line with theorem 5.3.11. In the case of long memory, only the information criterion with the normalisation n1−2d lead to a consistent estimation whereas in case of antipersistence the normalisation n1 leads to a consistent estimation. Moreover, multiplying the penalisation term by a constant has a huge impact on the finite sample properties, although it does not affect the asymptotic properties. Numerical results are given in tables 25 and 26 in the appendix.
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Figure 14 Sample path of 100 observation with linear trend function as used in simulation study.
6.2.1.
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Figure 15 Results for information criteriaIC1,IC2,IC3 andIC4
Application
We now return to the starting point of this work, namely the analysis of glomerular activity patterns. We study the glomerular activity patters obtained in an experiment which was conducted to examine the effect of octopamine on odour responses in projection neurons of the antennal lobe. Octopamine, a biogenic amine, is “the structural and functional analog of the vertebrate neurotransmitter noradrenaline”, see [RMS+, page 1], and plays a crucial role in many physiological processes. In particular, it is of great importance in odour learning in bees, see [RMS+, page 1-2].
7 Experimental procedure
A detailed description of the experiment is given in [RMS+]. Figure 16 provides an schematic illustration. In total, 38 bees were involved in the experiment, with each bee receiving the usual staining protocol described earlier. To analyse the effect of octopamine, the sample was divided into three different treatment groups: group 1 (untreated, n=13), group 2 (treated, n=13) and group 3 (treated control, n=12). The first group received no additional treatment apart from the staining protocol. In the second group the octopamine receptor dsAmOA1 was downregulated by means of RNA interference in order to study the role of this specific receptor. The bees in group 3 received the same treatment as in group 2, except that injected dsRNA (dsFRED) had no specific effect on the cell biology.1 Rather, this group acts as a control for the general effects of an RNA interface treatment on the response behaviour. The actual imaging experiment consists of four stimulation series with each series in turn consisting of six odour pulses (table 2). To assess the effect
1This dsRNA interferes with the expression of a developmental protein from Drosophila (FRED), see [RMS+].
138
of octopamine on the response behaviour, the level of octopamine within the antennal changes with each series.
The odour pattern remains the same for each series. At the end of each stimulation block, 125 seconds of background noise is recorded (omitted in table 2, but indicated by the broken time lines in figure 16).
Experimental setup and extraction of the glomerular time series follow the general description provided above. As the identification and extraction algorithm is applied to the whole recording of each bee at once, the set of identified glomeruli remains the same throughout the different series for a specific bee. However, the number of identified glomeruli varies largely among the bees, see table 1. A detailed list of the identified glomeruli in each individual is given in the appendix, see table 27, 28 and 29. To make the sample more homogeneous, we thus keep only those glomeruli which were identified in more than 75 percent of the individuals and discard the others. This gives us a subset of seven glomeruli, namely glomerulus 17, 33, 36, 42, 47, 48 and 602. As can be seen in table 1, these glomeruli are identified in most bees with a reasonable rate. Again, the complete data can be found in the appendix, see table 30, 31 and 32.
Distribution of the number of identified glomeruli Number of identified
glomeruli 4 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25
Number of bees 1 1 1 1 4 2 1 4 3 3 6 2 2 2 2 1 1 1
Distribution of the number of identified glomeruli (reduced set: 17, 33, 36, 42, 47, 48, 60)
Number of identified glomeruli 2 4 5 6 7
Number of bees 1 2 1 11 23
Table 1 Distribution of the number of identified glomeruli among different bees
8 Descriptive analysis
For each identified glomerulus in each individual an optical recording of every odour pulse is available. Each recording consists of a time series of 110 observations (sampling rate: 5 Hz), with the stimulus taking place after 30 observations (6 seconds). In most cases, the response to an odourant stimulus is a fast increase followed by a gradual decline.
In some cases, however, there is no increase or even a decrease after the stimulus visible.
Typically, the visible response patterns last for 15-20 observations. Hence, in order to simplify the modeling, our analysis focuses on the first 45 observations. This observation
2For the nomenclature of the glomeruli, see [GMM99].
Group 3 (dsFRED, n=12) Group 1 (untreated, n=13) Group 2 (dsAmOA1, n=13)
38 Bees
1st series of odour pulses (0 mM octopamine)
2nd series of odour pulses (1 mM octopamine)
4th series of odour pulses (0 mM octopamine)
Legend
3rd series of odour pulses (10 mM octopamine)
1-hexanol (10 - 2 M) 1-heptanone (10 - 2 M)
1-nonanol (10 - 2 M) 1-nonanol (10 - 3 M) 1-nonanol (10 - 4 M) 1-nonanol (10 - 2 M)
Time (with breaks)
Figure 16 Schematic illustration of the experiment. Grey background of the second and third series of odour pulses indicate a higher concentration of octopamine. The broken time lines between the series indicate breaks of 125 seconds were only background noise was measured. Coloured bars indicate the odour pulses.
odour concentration of odour concentration of octopamine First stimulation set
1-nonanol 10−2 M 0 mM
1-hexanol 10−2 M 0 mM
1-heptanone 10−2 M 0 mM
1-nonanol 10−2 M 0 mM
1-nonanol 10−3 M 0 mM
1-nonanol 10−4 M 0 mM
Second stimulation set
1-nonanol 10−2 M 1 mM
1-hexanol 10−2 M 1 mM
1-heptanone 10−2 M 1 mM
1-nonanol 10−2 M 1 mM
1-nonanol 10−3 M 1 mM
1-nonanol 10−4 M 1 mM
Third stimulation set
1-nonanol 10−2 M 10 mM
1-hexanol 10−2 M 10 mM
1-heptanone 10−2 M 10 mM
1-nonanol 10−2 M 10 mM
1-nonanol 10−3 M 10 mM
1-nonanol 10−4 M 10 mM
Fourth stimulation set
1-nonanol 10−2 M 0 mM
1-hexanol 10−2 M 0 mM
1-heptanone 10−2 M 0 mM
1-nonanol 10−2 M 0 mM
1-nonanol 10−3 M 0 mM
1-nonanol 10−4 M 0 mM
Table 2 Protocol of odour impulses
period is also chosen by [RMS+]. To estimate the trend function, we fit a linear least squares spline with two knots to each trace. The position of the first knot is fixed at t=30 (the time of the stimulation), the position of the second knot is estimated by means of least squares estimation, for more computational details see section 9.3. Figure 8 shows a typical sample for various bees, glomeruli, odours and different levels of octopamine together with the estimated trend functions. It is hypothesised that octopamine enhances
“positive answers”, i.e. if a signal of glomerulus shows an increase in response to the stimulus then this increase should be stronger under a higher concentration of octopamine.
In the next section we will precisely define what we mean by “positive answer”.