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\section{Specific Comments from Rick:}

Page 1: ``Wavelet analysis can be viewed as an alternative to classical {\it %
Windowed Fourier Analysis}.'' This is certainly true, but I do not see that
it has anything to do with the present paper. Windowed Fourier Analysis is
not (to the best of my knowledge) used for channel detection.

Amir Reply: When  the concept of the wavelet transform is presented
usually people ask how it relates to the classical Fourier Transform.
This statement is only to say how it relates to Fourier.

Page 1: ``This framework provides us with novel robust algorithms that will
apply state-of-the-art signal and stochastic processing techniques to some
of the cutting edge problems of molecular biology ..." This certainly seems
to overstate what is presented in this paper. By implication, it suggests
that what follows is likely to be better in some sense than previously used
techniques.

Amir reply: Before the overview I added the following:
"In this paper, on the application of multiresolution
analysis to channel detection, we, so far, do not out perform
the existing methods. The purpose here
is to point at a new direction in channel
detection that has a big potential and future.

Thus it is premature, at this point, to claim substantial
victory over other methods such as the threshold method.

So far we can state that the threshold method, for example, and the
preliminary wavelet analysis are comparable in performance.
On the otherhand, the threshold method, for example, seems to have reached
its upper bound while multiresolution has a big future ahead."

Page 2: ``Our results demonstrate the feasibility and utility of multiscale
analysis.'' In my opinion, the results do NOT demonstrate the utility of
multiscale analysis in the sense that nothing presented shows any obvious
advantage of this approach. The utility of this approach is assumed, but not
clearly demonstrated.

I am not saying that multiscale analysis is not useful; I am saying its
utility has not been demonstrated in this manuscript.

Amir reply: In this paper we demonstrate, clear cut, that we get the
same results as the threshold. Its utility was demonstrated very clearly
in this paper.

Page 2: ``One advantage of wavelet analysis is that it is surprisingly
undemanding of computer resources and so can be implemented in real time.''
This has certainly not been demonstrated in the paper. Clearly from the
description of the algorithm, this approach must require much more computer
time than is required by simple threshold detection. If we wish to say
otherwise, we must support our opinion with results.


Amir Reply: They are very easy to implement and they running real-time
on PC/486 33 MHz. Most of the work is simple convolution.
With smart data structures, as explained later in the paper, the
process is trivial. All my programs are running on the PC.

Page 3: ``We believe that the wavelet transform theory has a good chance to
succeed in many applications where the classical Fourier analysis has failed
or is not adequate." I do not believe that this has anything to do with this
paper.

Amir Reply: This belongs to the general statements on wavelets and 
has nothing to do with channel detection.

Page 5: ``We examine the performance of our algorithm on models with white
noise and with filtered $f^2$ noise." This is the only time $f^2$ noise is
mentioned in this paper, none of the results appear to have come from
channel currents embedded in $f^2$ noise.

Amir reply: This is correct and we don't claim otherwise.

Page 5: ``It should be noted that the filtering rounds the corners of the
signal, spreading them on about 1.7 sample points." Delete.

Amir reply: was deleted.

Pages 5-7: ``Training for Level Estimates -- Preprocessing Step''. I wonder
what is gained from this section. There are certainly a variety of ways of
accomplishing the level estimation, and it is not demonstrated that this is
the best or quickest method. It would not work in the presence of any
significant drift and so might have to be supplemented in many real
application.

Amir reply: Like everything in life for each thing 
"...There are certainly a variety of ways ofaccomplishing the 
level estimation, ...". We don't claim that there are no other ways.
There are many others. It is up to you to come with them and 
to write papers about that. This is the method that at least three
respected people thought to utilize here. You can come with other
methods. It is fine with us. Probably, in later stages this is 
going to be eliminated alltogether. But for now this is our 
solution.

Page 8, Figure 1: As I have already pointed out, operationally, all that is
being done here is really a high pass filter (differentiation) followed by a
series of low pass filters, with the output of each low pass filter being
what is used at each level. This should be stated.

Amir reply: You can point out many things you want again and again but this
is not what is being done here: In each scale (level) there is a low-pass
and high-pass. Then the low pass is again convulted with low and high ...

So I am not inserting your statement.


Page 8: ``We perform several cycles of Gaussian wavelet smoothing by
applying the low-pass part of the wavelet transform." This is a needlessly
confusing way of stating that the data was simply passed through simple
Gaussian low-pass FIR filters. In fact, the three filters used could have
been replaced by a single filter (3 Gaussian filters with -3 dB bandwidths
of 1, 2, and 4 used in series can be replaced with a single filter with a -3
dB bandwidth of 0.873).


Amir reply: This is the way we are doing that. As you mentioned before, 
there are many ways to do things and we choose this way. 

Page 8: ``In the future we plan to eliminate the smoothing procedure in the
beginning in order to detect short events.'' This should either actually be
tried or not it should not be mentioned at all. Since without smoothing the
SNR will initially be much worse, eliminating smoothing actually may not
work.

Amir reply: The smoothing is not necessary. The multiresolution 
anaylsis has all the information burried inside and if we know 
(I guess we know now - but we don't want to insert it in the current
version) how to read it correctly the smoothing can be eliminated.



Page 9: ``The well known Canny edge detector ... " This is not well known to
Physiologists.

Amir Reply: You are right. I changed the wording there.

Page 9: What is a Lipschitz exponent? This is not well known to
Physiologists.

Amir Reply: I added some explanations as footnote.


Page 11: ``we detect the modulus maxima by finding the points where the
discrete wavelet transform is larger than its two closest neighbor values
and strictly larger than at least one of them''. This sounds to me like it
will detect {\sf every }noise peak (including very small ones), as well as
signal peaks. This would seem time consuming and not very efficient. Perhaps
I do not completely understand; operationally, what exactly is being done?

Amir reply: You are completely right. It detects 
`` every noise peak (including very small ones)''. But this is the part of 
the main process. He detects small ones (I did mention all of them 
in the  figures such as figure 7 to shorten the presentation.
Almost every other point is a source for a chain but most of them not
strong to survive our measures. 
 But these
small ones  disappear while we are going deeper in the scales (going 
to coarse level)  because they too small (weak) to survive the
decomposition. Therefore, the algorithm is doing denoising. I mentioned
couple of times and figure 2 on page 13 demonstrates this procedure because
in level 7 there are less oscillations than those in level one. There are
less peaks.


I added this type of remarks in line 790.

Page 11: I do not understand what criterion is used for ``chaining'' points
across several scales. $E.g.$, it is stated that ``locations 6000, 6001,
6000, 5999, 5990, 5977 are chained according to the algorithm already
described''. I understand that each location is from a different scale, but
the locations differ by up to 24 points (6001 vs. 5977). {\sf How is it
determined that these all represent the same feature?}

Amir reply: The criterion for chaining is exaplained very clearly. Because
we don't do decimation and we use symmetric filters we know each
point in each scale how it propgates from one scale to another.
It explained in length in the paper.

Page 12, Table 2: I still do not understand why the $a_i{}^{ref}$ values
increase as the level number increases. What sort of normalization is being
used? Obviously, since each new level can be obtained simply by low pass
filtering the previous level, the amplitude of peaks would normally decrease
with additional filtering.


Amir reply: We look at $a_i{}^{ref}$ that are located in the high-pass and
not in the low-pass part of the signal. By the theory, since we take clean
signal the amplitude (in the high-pass grows). In the presence of noise
this is a different story. We can see, for instance in figure 7, that
when noise is present column 4 "A=" the numbers grows till certain level
and then decrease.


Page 12, equation 11: why $is<r_i{}^{ref}\times 0.2$ selected, $i.e.$, why
0.2?

Amir reply: In order to detect an edge we want to see how much the 
computed values along the decomposition are differ from the clean 
``ideal'' signal. We alowed divergence from the ``ideal'' one by 
a value of 0.2.

Page 14: Why only start considering the value of $A_{max}$ at level 4?

Amir reply: In the first levels we don't know for sure if this a noise or
an edge. The more we go in the decomposition we get rid of the noise 
as demonstrated clearly in  Figure 2. Only the strong ones survive to
the last level with some strength that we see in column 4 A= figure 7.
These are the candidates to be classified as edges.
Therefore, we start from level 4, since from there we start to see
a clearer view of the edges. 

When we will drop the smoothing, we will go 10 and more levels in the
decomposition and we will probably start to apply the tests from
level 6 or even 7.

Page 15: ``Threshold verification" What is the point of this section? I
think it is unnecessary.

Amir reply: was deleted.

Page 15: ``This procedure $[i.e.$, threshold detection] can work
independently well in certain cases for single ionic channels. But it is
limited to single channel, while the proposed algorithm is more general in
the sense that it can be extended to two or more ionic channels." Threshold
detection can readily be extended to more than one channel.

Amir reply: was deleted.


Page 20, ``Experimental results": This section is very weak. Equations 14
and 15 are too obvious to need to be included. No significant results are
actually presented.

Amir reply: I am leaving these equations in place. Good that they are 
obvious. I thought that the other are also obvious but you said that
this community is not fimilar with it.

I am adding some more explanantions to the ``Experimental results''.

Page 21, Table 4: It seems that the only difference in the rows is the open
channel amplitude (2.0 going down to 1.10). The number of levels of
smoothing is always 3, and the {\sc rms} noise before smoothing is always
0.87; however, the {\sc rms} noise {\it after} smoothing is different in
each case. How is this possible? In the first row the {\sc rms} noise
actually {\it increases} (from 0.87 before smoothing to 0.90 after
smoothing); again how is this possible? Looking at the last two rows, the
value of A is 1.10 in each case and the {\sc rms} noise is 0.87 before
smoothing in each case, but the {\sc SNR} before smoothing is reported in
one case to be 1.26 and in the other case 1.14. Clearly in both cases it is
1.26. Perhaps A in the last column was intended to be 1.0. Even so, there
seems to be something wrong with this table. In addition, none of the
``results'' listed tabulate anything about the performance of anything but
the training algorithm, and the very questionable results of a smoothing
operation. In the results illustrated in Figures 3-6, smoothing increases
the {\sc SNR} from 1.45 to roughly 12; why does smoothing have so little
effect in Table 4?

Amir reply: A should be 1.0 and not 1.10 and this is fixed.

Page 23: ``Better smoothing and detection should be possible, for example,
using a Harr (rectangular function) basis." Without some justification for
this statement, it should be eliminated.

Amir reply: this is suggestion for the future.

Page 23: ``... oversample in general, hoping that the steep nonlinear
dependence of wavelet resolution on number of samples ...'' This must be
justified or eliminated. There is no obvious reason this should help in this
situation.

Amir reply: this is suggestion for the future. 

Page 23, paragraph starting with ``Another possibility is to eliminate the
smoothing procedure ...'' Again, justify this or eliminate the entire
paragraph. Mentioning ``more sophisticated denoising algorithms'' without
any hint of what they are adds nothing to the paper.

Amir reply: this is suggestion for the future.  I added more information
about that.

Page 23: ``In a later stage, we plan to study the decay of the noise across
the scales of a multi-resolution analysis.'' This can be done trivially by
simple analytical techniques.

Amir reply: I don't know how to do it. If you know, it will be nice
addition to the paper if you can add
" This can be done trivially by
simple analytical techniques." 


Figures 7-9: At no place in these figures is the utility of multiscale
resolution clearly demonstrated. Every accepted event would have been
accepted at level 2; there may be some improvement in terms of locating the
edge correctly, but this is not clear.{\bf \ This does not mean that the
multiscale aspect of the algorithm does not help, but simply that it has not
been demonstrated. }Some demonstration{\sf \ that multiscaling helps} would
seem necessary, since requiring that features `survive' through many levels
can, in fact, cause errors. It can cause brief events to be missed.

Amir reply: After reading that I wonder if you understand the algorithm.
This is the core of the algorithm.

I continue to wonder whether the ``remarkable orthogonality properties'' of
wavelets have anything to do with what is actually done here in this paper.
I don't see why they would.

Amir reply: We don't need the orthogonalty in this paper.

In order to make the paper represent what has been done (without depending
on things we have not done), some comments concerning the performance of the
algorithm relative to more traditional approaches would seem appropriate.
The paper should also use some plain English to describe operationally what
actually has been done and what is actually being presented.

I think several things are important:

1. Much of the algorithm can be replaced with a high pass filter followed by
a series of low pass filters. In fact, this is what it is.


Amir reply: Sorry!  wrong ....

2. High pass filtering (differentiation) reduces signal to noise ratio for
channel data provided that the noise is white or has a PSD that increases
with increasing frequency. This is a penalty associated with this type of
edge detection. I can provide details of this later if you wish.

Amir reply: Please do.

3. Something is required to demonstrate that the multiscale aspect of the
algorithm significantly improves results.


Amir reply: We comment on that many times all the way already.

4. Something is also required to demonstrate that this approach has any
advantages at all over more traditional methods. Nothing presented (except
implication) suggests that it does:

Improved signal to noise handling ability is NOT demonstrated. In fact,
noise handling ability of this algorithm should be somewhat poorer than
thresholding.

The amount of computer time required for the algorithm is NOT less than
traditional thresholding. It seems clear that this algorithm will be slower
than thresholding.

Amir reply: You are wrong. As it is now, it running realtime on PC/486.

The algorithm does NOT proceed without prior knowledge of channel current
amplitude. In this sense it is identical to thresholding.

The training algorithm is NOT better than a number of alternative approaches.

Amir reply: so what ...

Because of the need for training the overall algorithm is NOT insensitive to
drift. This throws out one of the main advantages of differentiating the
data.

If some possible advantages of this approach can be defined, then it can be
considered worthwhile. If such advantages can not be defined and discussed,
then there is no real contribution here. An exploration of what wavelets
have to offer to channel detection is worthwhile. However, this paper in its
present form appears to indicate that the answer is ``Nothing''.

If there are advantages then they should be stated (they are not in this
paper). If not, then the negative conclusion is the result of the
exploration to date, and, if anything at all is published, it should clearly
state this conclusion. The assertion that wavelets have great promise in
this application needs to be backed up; at present this is an article of
faith rather than an established fact.

\section{General Comments from Rick on Manuscript}

Overall, I am not very happy with the manuscript. The paper generally
successfully describes the algorithm used (although sometimes in a rather
confusing way). However, nothing is presented that suggests that there is
any advantage to using this algorithm. Of course, there is the usual amount
of hyperbole, such as ``This framework provides us with novel robust
algorithms that will apply state-of-the-art signal and stochastic processing
techniques to some cutting edge problems ... '', but nothing presented backs
up such assertions. Indeed, nothing whatsoever in the way of any
`benchmarks' concerning the performance of the algorithm is presented. Yet
it is obvious at a glance from Figures 5 or 6 that the results presented
could have been easily matched with {\bf simple} thresholding.

Amir reply: I don't intend to invest more time in this paper. This is 
the best I can do and now we have to go to the second phase to improve the 
algorithm: throw away smoothing, training, and use more internal
multiscale analysis to  get short events and more than single channel.

We can n't in this stage please everyone. If you are not happy you can
take out your name from the paper. Otherwise, in this point I don't have time
and resources to improve it. 

\section{Rick's Comments on Amir's letter of October 30:}

I certainly agree that it is not necessary for the method described to
presently be able to out-perform more traditional approaches in any
particular area of performance $(e.g.$, noise handling ability) in order to
be worthwhile---provided that {\it some} advantage of the new approach is
demonstrated. If no advantage is presented, then I am much less convinced
that there is any contribution from the method. Thus the description of some
sort of advantage of the approach seems important to the paper. It is
certainly not sufficient (in my opinion) to assert without evidence that
future improvements are somehow inevitable.

If the assertion ``We can even find examples that defeat the threshold
method" is true, then I would strongly urge that one or more such examples
be presented. So far, I have not seen such an example.

The actual algorithm used is not altogether new to channel detection.
Differentiation followed by low pass filtering has certainly been used
previously, and is hardly a novel idea. Mutiresolution analysis (to the best
of my knowledge) has {\em not} been used previously for channel detection;
however, its utility must actually be demonstrated (not just assumed).

It is certainly a mistake to assert that years of effort have been invested
in the threshold method of channel detection. It is certainly true that this
method has been used for years, but it is a very simple and straight-forward
method that no one has spent years developing. The actual time spent
developing the technique for channel current detection is probably very
limited. It is also definitely true that ``the threshold method seems to
have reached its upper bound''. But it did so a long time ago. It is a
simple method, whose performance can be predicted quite precisely. This has
been true for many years. However, it has not been demonstrated in the case
of channel current detection that ``multiresolution has a big future
ahead'', or that ``its potential is huge''. This may be true---or it may
not. What I think is needed is some demonstration of this future or
potential, not just faith that it exists.

What I strongly feel is needed is an explanation of the possible advantages
of any new technique. Such a discussion should be concise and concrete and
not rely on possible future improvements unless the likelihood of their
success can be clearly demonstrated.

\end{document}

