 Research
 Open Access
Brain connectivity extended and expanded
 Obrad Kasum^{1, 2},
 Edin Dolicanin^{1, 2},
 Aleksandar Perovic^{1, 2} and
 Aleksandar Jovanovic^{1, 2}Email author
https://doi.org/10.1140/epjnbp/s403660150019z
© Kasum et al.; licensee Springer. 2015
 Received: 25 February 2015
 Accepted: 19 March 2015
 Published: 10 April 2015
Abstract
Background
The article is focused on the brain connectivity extensions and expansions, with the introductory elements in this section.
Method
In Causality measures and brain connectivity models, the necessary, basic properties demanded in the problem are summerized, which is followed by short introduction to Granger causality, Geweke developments, PDC, DTF measures, and short reflections on computation and comparison of measures.
Results
Analyzing model semantic stability, certain criteria are mandatory, formulated in preservation/coherence properties. In the sequel, a shorter addition to earlier critical presentation of brain connectivity measures, together with their computation and comparison is given, with special attention to Partial Directed Coherence, PDC and Directed Transfer Function, DTF, complementing earlier exposed errors in the treatment of these highly renowned authors and promoters of these broadly applied connectivity measures. Somewhat more general complementary methods are introduced in brain connectivity modeling in order to reach faithful and more realistic models of brain connectivity; this approach is applicable to the extraction of common information in multiple signals, when those are masked by, or embedded in noise and are elusive for the connectivity measures in current use; the methods applied are: Partial Linear Dependence and the method of recognition of (small) features in images contaminated with noise. Results are well illustrated with earlier published experiments of renowned authors, together with experimental material illustrating method extension and expansion in time.
Conclusion
Critical findings, mainly addressing the connectivity model stability, together with the positive effects of method extension with weak connectivity are summarized.
Keywords
 Granger causality
 Brain connectivity measure
 PDC
 DTF
 Measure comparison
 Weak brain connectivity
 Partial (Linear) Dependence
 Small object recognition
Background
Granger’s method (some extended application in [19]), [1012], has been in the focus of extensive research in neuroscience, expanded in various developments. We list some of the standardized connectivity measure terminology [1320], mentioning Granger – Geweke counterpart measure couples, contrary to Bacala  Sameshima [18] concept of causality measure counterpart. These renowned leading authors refer to “proper frequency domain counterparts to Granger causality”. Our correction is based on Geweke fundamental relation between temporal and frequency domain causality measures. We shortly focus our attention to the earlier analyzed measure comparisons, adding important argumentation.
We introduce methods of Partial (Linear) Dependence – PLD and (image) small object recognition in order to deal with the weak brain connectivity connectivity elusive or undetectable by the connectivity measures or heavily masked by noise, hopefully extending standard methods. These methods are applicable to both, frequency distributions, spectral like objects, and to frequencytime distributions, e.g. spectrograms, which expand the time point to dynamicsintime view. This article is partly extension of our work [4], from which we reproduce fragments necessary for developments and discussion here.
In some circumstances brain processes might exhibit behavior similar to stochastic systems or fluid dynamics. No matter how much such analogies and similarities might be fruitful, we better keep some reserve for subjecting the brain to either statisticians or plumbers. We should not forget that brain is a highly complex information processing system, with reach information flow between large number of coprocessing points, which is our basic initial hypothesis, better: axiom1. Then, obviously, the dynamics of connectivity patterns has essential role, which includes connectivity patterns and their time switching as well.
With broader application domains which include neurology research and practice, expanding the sophistication of involved models which already operate with connectivity arguments in the most sensitive segments, strongly influencing expert’s decision making, the demand for careful critical reinvestigation of theory and application has become continuously necessary. Establishing of neurological disorders, psychological evaluations, highly confident polygraphy are all of crucial significance for the subjects involved. Finally, we witnessed on a recent conference, an expert’s elaboration of evaluations concerning level of patient brain damage after a stroke, consciously lost according to the contemporary criteria, with bad prognosis and consequential termination reasoning and planning. We know that we do not know the circumstances so well in order to produce categorical conclusions in such matters.
First, we observe, that in the contemporary connectivity modeling some procedures, computations and estimates need increased care in order to lead towards correct conclusions. It is shown that the neighborhood of zero is of accented importance in such evaluations and that unification of values with the difference below zero thresholds is necessary as the first step in computation and comparison evaluation. Harmonization of thresholds corresponding to measures involved in comparisons is an open issue requiring mathematically reasonable solutions. Computational stability is a general demand everywhere. Varying fundamental parameters in small neighborhoods of elsewhere published and established values, we perform detailed analysis of semantic stability of the deduced published exemplary connectivity models. When we face computational instability, if we deal with models of the real world, it immediately generates semantic instability and often a singularity. In this context, if connectivity graphs essentially change when computational differences of arguments occur within computational zero, then this is immediately reflected semantically as proportionally unstable maps of brain connectivity structures. This is not acceptable in any interpretation of experimental data and questions applicability of the brain connectivity models. We extract some characteristic examples involving wrong logic and those based on reasoning with insufficient care and precision.
Analysis of the operators involved  used in the measure computation and comparison by renowned authors, proved that the spectral maximum selected as the representative invariant for both DTF and PDC measures before their comparison is not justified and might lead to the not well founded or invalid conclusions. Suggestions for the improvements of used simplification operators in computation of measures and for complementary comparisons are given. Second, there is an attitude with respect to connectivity present at large, where the thicker connectivity arrows are proportionally more important than the thinner ones, with discrimination made following corresponding signal intensityenergy level. If we remember the above stated axiom, we have to become more sensitive towards connectivity concept in general and rephrase importance criterion, or rather erase it completely. In the information processes, a weaker energy process can be much more important than those at higher energies. Also, short or ultrashort messages might precede hierarchically the longer transports. Consequently, when building a brain connectivity model, we have to take in consideration all discernible data related to connectivity.
Even more, we should accept an extra hypothesis, axiom2: there are processes related to connectivity, which are indiscernible or hardly discernible from noise, with unknown importance for the brain functioning models established by connectivity measures in the current use.
Connectivity as currently performed and understood is omitting the essential temporal dimension. The conclusive connectivity graphs, the aims in connectivity estimations, are to be replaced or rather expanded in time dimension, since brain dynamic changes can be essential at the millisecond scale. This is for us the axiom3, which has to be respected. Even for short events, these graphs change or massively change in time and it is necessary to integrate their time dynamics into the model that should make sense, strongly analogous to the relation between individual spectrum and timespectrum, spectrogram. The former makes sense only when there is no intrinsic dynamics, thus, in highly stationary processes only. This does not apply to e.g. music: one cannot be aware of a melody, nor detect it applying single spectrum.
Finally, let us note that connectivity graphs present in recent research reports are usually structures with highly limited number of nodes, essentially a lot under currently achieved resolution of signal acquisition sensors (EEG, MEG, mixes). With earlier listed simplifications we are offered highly reduced graphs as brain connectivity models, which leads to oversimplified understanding of brain functioning. Certainly, in very short time, what we depict with 6 node or restricted 20 node graph today, with stronger currents as essential, thus with up to 20 directed graph links, when expanded with real weaker connectivity within currently achieved resolution, with added connectivity time dynamics in the range of millisecond resolution, will be represented with hundreds of nodes and higher number of more realistic links, probably easily reaching 2^{16} or 2^{20} links. Such combinatorial explosion is realistic. The forthcoming large size models would have to be generated, inspected, analyzed, compared, classified and monitored automatically, offering synthesis in higher abstraction synthetic invariants to experts. Who is modelling Internet functional or dysfunctional connectivity with up to 20 links? Brain is much more complex than the Internet. We are approaching the end of the connectivity modelling golden age, end of simple functionality explanations, we better be prepared for the change and work on it.
Method
We present connectivity measure terminology (expanded in the Appendix), then our enhancements applicable in weak brain connectivity, where the standard connectivity measures remain undecidable, followed by examples with published measure comparisons with extended scrutiny and examples of applications of methods focused on weak brain connectivity, ending with conclusions integrated into a Proposition. Signals and software used in experiments shown here are available at our web: http://www.gisss.math.rs/.
Causality measures, brain connectivity models
Initial points

I1. connectivity estimation separated from other properties of interest, e.g. “connectivity strength”;

I2. beside directed connectivity, separate treatment of bare connectivity – with no direction indication, when direction is more difficult to determine, and as a correctness test in the graphs deduced;

I3. more precise calculations and aggregations;

I4. scrutiny of involved operators;

I5. appropriate changes in the calculation and comparison procedures resulting in the more precise modeling of the connectivity structures and properties;

I6. special attention to the thresholds involved and related numeric zero which is basic for all other conclusions;

I7. stability analysis; stability of computations and model in the neighborhood of zero; stability wrt. all involved parameters.

I8. model stability in time;

I9. differential connectivity: inspection of deduced connectivity models by comparison to the structures deduced by other faithful connectivity measures and methods;

I10. harmonization of basic parameters of involved measures;

I11. proper definition of the reduction level (rounding filtered or “negligible” contents);

I12. connectivity graphs time expansion;

I13. alternative or additional approaches in model integration;
Granger causality, Geweke developments, PDC, DTF
All details on the method are available in the cited and other literature. All definitions and elements are briefly given in Appendix. When we have three variables x(t), w(t) and y(t), if the value of x(t+1) can be determined better using past values of all the three, rather than using only x and w, then it is said that the variable y Granger causes x, or Gcauses x. Here w is a parametric variable or a set of variables.
The theorem is for two var case. For the general case, authors announced soon publishing. Otherwise, we note that all important conclusions in their earlier papers, especially [18] are reaffirmed again in [21].
Computation and comparison of measures
Results and discussion
Preservation/coherence properties

P1. substructure invariance, i.e. restriction of a measure to a substructure should not change its range; thus, measure values on the intersection of substructures remains coherent.

P2. Structural stability; measure computation and comparisons/similarity estimates should be invariant to some degree of fluctuations of the involved operators (here P, sim, N _{ k }). These conditions should secure measure stability in extended, repeated and similar experiments.

P3. measure comparison should be stable in all involved parameters.

P4. continuity in models  similarity measures: small must remain small and similar has to remain similar in measure comparisons. The small difference of argument implies bounded difference of the result; this applies to predicate connected, with small shift of argument.
Structural properties of measures are determined on small objects  in the zero neighborhoods, whence the measure zero ideal is of key importance, which is the reason to list separately the zeroaxioms, ZAx, for either a single considered measure or for a set of compared measures:

Z0. substructure partitioning invariance (measure restriction to a substructure remains coherent);

Z1. fluctuations of operators involved in measure computation and comparison need be tolerant (continuity);

Z2. in similar circumstances numeric zero (significance threshold) should be stable quantity (to allow comparability of results);

Z3. comparison of a set of measures needs prior unification (argumentation necessary) of their zero thresholds (for otherwise, what is zero for one is not zero for other measures; consequently, the measure values which are identical for one measure, are discerned as different by other measures; that must cause problems);

Z4. in similar circumstances grading should be stable quantity;

Z5. measure values which are different by ≤ numericzero should remain identical in any posterior computation/grading, if applied (this is in accordance with the prior congruence on the ideal of zero measure sets);

Z6. values in any posterior computation/grading (if applied) should differ by no less than numericzero and grades should be of unified diameter; in this way values in posterior grading range are harmonics of numericzero;

Z7. final grading as a (small) finite projection of normalized range [0, 1] needs some conceptual harmonization with the standard additivity of measures; this step should involve fuzzification;

Z8. grading should be acceptable by various aspects present in the interpretation of related experimental practice (that means that the picture obtained using a projection/grading of [0, 1] range should not semantically be distant from the original picture – based on the [0, 1] range with, e.g. a sort of continuous grading);

Z9. Connectivity graphs time expansion; it is necessary to introduce time dynamics in these observations.
Mathematical principles must be respected for consistency preservation; computations in repeated and similar experiments have to be comparable and stable. Measure computation and comparisons are complex, consisting of steps, some of which usually do not commute, demanding care and justification.
Enhancements
We will present two methods applicable to the weak brain connectivity, the case when a set of signals share a common information, which is either hardly detectable or even undetectable by direct observation or the connectivity measures in current use. The first is based on rather pragmatic property, partial linear dependence, PLD: for the set of functions (signals or vectors) S = {f _{ i } : i ∈ I} with the common domain, the set of restrictions to a subdomain is linearly dependent (wrt. some usual scalar product), while the complementary restrictions are linearly independent. PLD could be used to extract the common information easier. Then we might be looking for the maximal PLD sets, corresponding to different common information. If we generalize this slightly we come to PD in case when we have dependence which is nonlinear. The second approach exploits the methods originally introduced to eliminate clutter in radar images and to enhance small objects in images and is applicable to both signals and images. Since spectrograms are a sort of images, we can apply those methods to the spectrograms, spectrogram composites, or somewhat more general objects of the similar kind. Both methods are applicable to the noise contaminated signals.
PLD  PD
If applied to a given frequency, a reduced size set of frequencies, or a known frequency band, this method can supply good answers with not really complex calculations. Similar can be the case for the frequency distributions (spectra, composites, connectivity measures, spectrograms)  parts containing frequencies with poor signal to noise ratio, especially when multiple spectra or spectrograms are available.
In practice, all above integrals become finite sums, we might wish to use different indices in different occasions, which is why we distinguish them all. Obviously, we can have situations with the global index negligible while the local index is perceptible. Obviously, we can easily extend the above definitions to include some modulation fluctuations, or we could rephrase the concepts in order to fit better specific needs. In our practice that means that we can search for sets, the subsets of E, signals  electrode network measurements, which contain the same frequency component and select the optimal subsets in P(E). Clearly, the larger set of signals shares common information, the easier should be its extraction. However, with E at 300 now days, growing larger in time, the search for suitable, or larger G’s through P(E), having already 2^{298} elements, is quite a task, without a specializing guidance it will miss all optimums. With some previous knowledge on involved functionality, or starting with rather small sets – as seeds and expanding them we might be in a position to learn how to enlarge initial seeds. After preprocessing of connectivity for selectivity for certain application, we can use (linear) dependent channels to enhance the periodic component present in all members of a set G, which might be near or even below noise threshold in all inspected signals, as illustrated with the experiments below. The advantage is in the following property: the local processes contain independent components behaving quite randomly, the noise behaves randomly and random components will be zero flushed by the above criteria, in either composite spectra or composite spectrograms. Even without knowing at which frequencies interesting periodic patterns might be expected, the above method provides a high resolution spectral and spectrogram scanning. If there are artifacts which are characteristic for certain frequency bands, in case when the searched information is out of these bands with sufficient frequency separation, we might be able to localize and extract even the features embedded in the noise. Thus, e.g. in the composite spectra and spectrograms, the first index ED(g, λ) might easily converge to the numeric zero, while the second index ED(g, Λ), for certain spectral neighborhoods Λ of λ can locally amplify the hidden information, exposing it to perception. The same is even more obvious for spectrograms and products (ΠED, ΠEDS) indices, where we might exploit further properties of spectrogram (composite’s) features. When G is modulated by certain MS’s, we can still separate the carriers if known or well estimated, even within the same procedure, as above. The above procedure could be extended involving specific sorts of comb like filters, unions of the narrow band filters, which could enhance weak spectral components.
Application of image processing methods
A variety of problems in image processing is related to the contour – object detection, extraction and recognition. This we encountered in cytology preparations, variety of optical images, radar images, spectrogram features [4], mixing and concatenating processing methods.
Small object recognition
This simple discrimination reduces random noise significantly and exhibits signals together with residual noise. Performing procedure (19) thru (26), we generate filtered image with extracted signals. The method is adaptable, using two parameter optimization (minimax): minimal integral surface of detected objects, then maximization of the number of small objects.
Small object recognition using Kalman filter banks
Initially: \( {P}_0\left(x,y\right)={\widehat{X}}_0\left(x,y\right)=0,Q=1,R=100 \), where Q is the covariance of the noise in the target signal, R is the covariance of noise of the measurement. We put (depending on problem dynamics): the output filtered image in k ^{th} iteration is the matrix \( {\widehat{X}}_k \), the last of which is input in the procedure described by equations (19) to (26), finally generating the image with extracted objects. This method shows that the signal level could remain unknown if we approximately know statistical parameters of noise and statistics of measured signal to some extent. In our basic case we know that signal mean is somewhere between 0 and 255 and that it is contaminated with noise with unknown sigma.
Examples
Connectivity measure evaluations
In this section we focus to the final result of the connectivity measure computations – the brain connectivity directed graphs, as the main model representing brain connectivity patterns. Due to various technological and methodological limitations, contemporary mapping of brain activity using electroencephalography and magneto encephalography operates with a few hundreds of brain signals, thus, close to mega links. No doubt, this resolution will be continuously increasing, down to a few millimeters per electrode and better, all in 3D, increasing proportionally the cardinality of connectivity graphs, as discussed earlier. In a graph we define orbits of individual nodes: the kth orbit of a node a will consist of nodes whose distance via a directed path from node a is k (separate for both in/out paths). It is assumed that the connectivity graphs exhibit direct connections of processes which are directed. This was ambition of all scientists who proposed the connectivity measures in brain analysis; this is expectation of all scientists interpreting their experiments with computation of the connectivity measures. We just add that there might be scenarios where bare connectivity is decidable, while directed is hard to resolve.
Left side table corresponds to the first time slice of the experiment – related to Figure 1 ; each matrix coordinate has on top the PDC spectral maximum from the shadow spectrum at corresponding coordinates in Figure 1 left, below the spectral maximum for DTF  similarly obtained from Figure 1 right; the connectivity links are sorted column vise, i.e. in the first column are A10 links towards the areas defined as row names
A10  0.18  0.53  0.03  0.06  0.09  A10  0.77  0.32  0.49  0.33  0.31  
0.25  0.58  0.06  0.10  0.09  0.82  0.76  0.32  0.22  0.19  
A3  0.09  0.27  0.06  0.04  0.07  A3  0.09  0.49  0.32  0.06  0.14  
0.13  0.28  0.14  0.12  0.17  0.28  0.79  0.22  0.07  0.09  
A17  0.41  0.41  0.11  0.12  0.10  A17  0.44  0.36  0.11  0.07  0.11  
0.43  0.47  0.17  0.23  0.23  0.37  0.68  0.34  0.18  0.13  
CA1  0.66  0.39  0.23  0.32  0.11  CA1  0.14  0.29  0.53  0.44  0.25  
0.49  0.53  0.57  0.32  0.38  0.20  0.65  0.66  0.41  0.28  
CA3  0.27  0.46  0.33  0.39  0.67  CA3  0.24  0.26  0.54  0.28  0.52  
0.61  0.50  0.51  0.29  0.48  0.27  0.65  0.72  0.22  0.28  
DG  0.44  0.44  0.47  0.47  0.41  DG  0.10  0.09  0.19  0.53  0.30  
0.58  0.59  0.35  0.35  0.31  0.15  0.36  0.39  0.57  0.49  
A10  A3  A17  CA1  CA3  DG  A10  A3  A17  CA1  CA3  DG 

(*) 1. set common zero = 0.2;

2. apply N _{1} operator: provides PDC power maximum, for all frequencies, for a given pair of input nodes;

3. apply N _{2} operator: provides DTF power maximum for a given pair of input nodes, for all frequencies;

4. apply P operator (the same operator P) for both PDC and for DTF were applied as projections (the five value grading, corresponding to connectivity degree, after calculation of spectral maxima);

5. Difference of the graded maxima is exhibited as visualized difference –a pair of connectivity graphs depicting all pairs of signals in the respective time slices, 2 s each.

2. apply N _{1} operator: provides PDC power maximum, for all frequencies, for a given pair of inputs;

3. apply N _{2} operator: provides DTF power maximum, for all frequencies, for a given pair of inputs;

4. perform zero ideal congruence for PDC; i.e. identify the corresponding values from previous step whose difference ≤ zero;

5. perform zero ideal congruence for DTF;

6. perform zeroideal congruence for PDC and DTF corresponding values;

7. generate the graph of connectivity difference;

8. apply P operator (the same projector operator P) for both PDC and for DTF, on their respectivegraphs (optional).

zerothreshold: common as in (*), varying over values which were present in the above mentioned similar, related experiments.

zero unification  performed prior to grading, consequently, avoiding that the small (difference) becomes bigger or big, just because ranges of measures are replaced (simplified) by coarser than original smooth [0,1]range;
The difference of PDC power maximum and DTF power maximum coordinate vise
A10  −0.07  −0.05  −0.03  −0.04  0.0  A10  −0.05  −0.44  0.17  0.11  0.12  
A3  −0.04  −0.01  −0.08  −0.08  −0.10  A3  −0.19  −0.30  0.10  −0.01  0.05  
A17  −0.02  −0.06  −0.06  −0.11  −0.13  A17  0.07  −0.32  −0.23  −0.01  −0.02  
CA1  0.15  −0.14  −0.34  0.0  −0.27  CA1  −0.06  −0.36  −0.13  0.03  −0.03  
CA3  −0.34  −0.04  −0.18  0.10  0.19  CA3  −0.03  −0.39  −0.18  0.06  0.24  
DG  −0.14  −0.15  0.45  0.12  0.10  DG  −0.05  −0.27  −0.20  −0.04  −0.19  
A10  A3  A17  CA1  CA3  DG  A10  A3  A17  CA1  CA3  DG 
After the above basic convergence of the two measure comparison, we should not omit the following divergence, basically maintaining the same sort of procedure, just introducing the slight variations in the same kind of argumentation.

(***) 1. Set zero separately for each of {PDC, DTF};

2. apply N _{1} operator: provides PDC power maximum, for all frequencies, for a given pair of inputs;

3. apply N _{2} operator: provides DTF power maximum, for all frequencies, for a given pair of inputs;

4. perform zero ideal congruence for PDC;

5. perform zero ideal congruence for DTF;

6. generate the graph of connectivity difference.
Clearly, either the argumentation shown here was unknowable to the authors of [18] or they purposely selected the huge bias in the compared measures thresholds, in order to optimize the targeting conclusions.
Experiments with PLD and small object recognition
Here we briefly present potential of the methods, starting with PLD application examples, following with application of small object recognition methods. Some of the work was developed in [5].
π({sig1, sig2}, 1 kHz) > 0, π({sig3, sig4}, 3 kHz) > 0, while π({sig1/2, sig3}, λ) = 0 and
π({sig1/2,sig4}, λ) = 0,
Conclusions
Consistently and consequently with the initial points and preservation properties and the presented examples and argumentation, the following more or less compact conclusion on connectivity measure computation, comparison, comprising methods for the weak connectivity, incorporating remarks from [4] in order to group together the complete set is formulated as
Proposition 1
 1.
Separation of different properties. We proposed here to detach and separately investigate connectivity from connectivity degree. We propose further to distinguish between directed connectivity and nondirected connectivity. There are different situations in which it could be possible to establish the last without solving directed connectivity, especially in the case of weak connectivity.
 2.
Differentiated properties could be investigated with different methods.
 3.
Partial linear dependence and method of small object recognition. They can determine graph substructures with the shared information, which is their contribution in case when the connectivity measures do not resolve the issue for whatever reason. They can be used to generate time expansion of connectivity structures, exposing model’s time dynamics.
 4.
The PLD indices might be of special interest if there is any prior knowledge on where the masked information, as a frequency pulse or trigonometric polynomial components might be.
 5.
Covering noise. Both enhancing methods might resolve the common information in case when it is indiscernibly embedded in noise or e.g. spectral neighborhood.
 6.
Time delay. Both methods might uncover substructures with common information with tracking in time, resolving possibly present time delays.
 7.
Comparison/computational sequence. The corrected comparisons of DTF and PDC, for connectivity only, as performed above with published data, ultimately show very reduced differences of two measures for zerothreshold received from the analyzed and similar published experiments (most importantly above the zerothreshold, exhibiting connected structures versus those which are not), thus confirming that if analyzed with computation and comparison procedure corrections proposed here, the connectivity structures are much less different than it was demonstrated in [17,18], as presented in the graphs to the left, Figures 3 and 4, versus Figures 5 and 6 (left graphs) with original common zero threshold. However, strictly performed original comparison method opens room for large connectivity difference, and large connectivity model oscillations (Figures 3, 4, 5, 6, 7 and 8). This is even more accented with the asymptotic study by authors, where 0.01 is very reasonable PDC threshold. This zero threshold eliminates the difference of measures in the offered examples with the author’s comparison methodology, against original aims and involves other undesirable properties.
 8.
The instability of PDC – DTF comparisons is most significantly due to the possibly different values of statistical significance of PDC, ultimately corresponding to different sampling.
 9.Aggregation prior to comparison; functionally related frequencies. The above analysis was performed, maintaining strictly the reasons and methodology performed by authors of the original analysis [17,18]. Here we have to stress that performed as it is and with our interventions in the original evaluations as well, PDC and DTF measure comparison was not performed directly on the results of these measures computations, thus, comparing directly DTF_{ ij }(λ) and PDC_{ ij }(λ), (for relevant λ ' s)  the results of measurements at each couple of nodes for each frequency in the frequency domain, but, instead, as in the cited articles with the original comparisons, the measurement of differences of these two connectivity measures was performed on their synthetic representations  their prior “normalizations”  aggregations, obtained as the(where rng(Sp) is the effective spectral  frequency range) and, in the original, on their further coarser projections.$$ \max \left\{\mu \left(i,j,\lambda \right)\ \Big\ \lambda \in rng(Sp)\right\} $$
 10.
Essential departure from connectivity comparison analysis. In this way, in comparison of these measures the authors had substantial departure from original connectivity measurement computations for PDC and DTF which cannot be accepted without detailed further argumentation.
 11.
Comparison of connectivity in unrelated frequencies. If the parts of frequency domain are related to different processes which are (completely) unrelated, for example, if one spectral band is responsible for movement detection in BCI, while the other is manifested in the deep sleep, then, depending on the application, either can be taken as representative, but most often we will not take such individual maxima of both as representing quantity in the functional connectivity analysis; however, on the restrictions such procedure can be completely reasonable. If we look closely at the corresponding coordinates in the distribution matrices in Figure 1, for the compared measures we will find examples of frequency maxima distant in the frequency domain or even in the opposite sides of frequency distributions (e.g. (5,1) – first column fifth row; then (4,2), (5,2) or (6,5)).
 12.
Essential abandoning of the frequency domain in PDC, DTF measure computation and comparison. Obviously, we are approaching the question: when we have advantage of frequency measures over the temporal domain measures. In the comparison of DTF and PDC via their aggregations as explained above, much simpler insight is obtained; supplementary argumentation is necessary: for the aggregation choice, stability estimation complemented with comparison of DTF and PDC counterparts, for which we would propose their Ginverses DTF and PDC, as we introduced them.
 13.
Zero thresholds and connectedness. Maintaining original or corrected computational sequence in measure comparison, note that DTF computation and PDC computation are performed independently. Consequently, each of these two measures computations should apply the corresponding appropriate zero threshold, thus determining zeroDTF and zeroPDC independently for each of the computations, with independent connectedness conclusion for each measure for each pair of nodes. Then, the connectivity graphs would be statistically correct. However, if the two zeroes differ substantially, that might cause paradoxal results in comparisons. Possible zero threshold harmonization  unification would be highly desirable, as in e.g. [18] and other cited articles, but it must be properly derived.
Our fundamental concepts, models and comprehension cannot depend on the sample rate.
 14.
Aggregations over frequency domain. Note that frequency distributions exhibited in Figure 1 for PDC and DTF are somewhere identical, somewhere similar/proportional and somewhere hardly related at all  as the consequence of different nature of these two measures (which is established by other numerous elaborations). The same is true for the spectral parts above zero threshold.
Consequently, if the comparison of the two measures connectivity graphs was performed at individual frequencies or narrow frequency bands, the resulting graphs of differences in connectivity would be more fateful; they would be similar to those presented at certain frequencies, but would differ much more on the whole frequency domains. Obviously, connectivity at certain frequency or provably related frequencies is sufficient connectivity criterion; such criterion is valid to establish that compared measures behave consistently or diverge.
 15.
Brain dynamics and connectivity measures. Spectral time distributions – Spectrogram like instead of spectral distributions are necessary to depict brain dynamics. In the cited articles, dynamic spectral behavior is nowhere mentioned in measure comparison considerations, but it is modestly present in some examples of brain connectivity modeling – illustrating PDC applicability to the analysis focused on specific event  details in [17,18]. Trend change: in [27] authors recently started using matrices of spectrogram distributions instead of matrix distributions as in Figure 1. This is gaining popularity.
 16.
Spectral stability analysis. Comparison of PDC and DTF as in here analyzed articles, shows no concerns related to frequency distribution stability /spectral dynamics and comparison results. It is clear that comparisons based on individual frequency distributions are essentially insufficient, except in proved stationary spectra, and that local time history of frequency distributions – spectrograms, need to be used instead. Brain is not a static machine with a single step instruction execution.
 17.
Characterization theorem for (dis)connectedness [21]. Here we have simply sensitive play of quantifiers. By contraposition of the statement of the characterization theorem, involving information PDC and DTF as cited above, we obtain equivalence of the following conditions
o) the nodes j, i are connected;
a’) ∃ λ(λ ∈ [−π, π] ∧ iPDC_{ ij }(λ) ≠ 0);
b’) ∃ λ(λ ∈ [−π, π] ∧ iDTF_{ ij }(λ) ≠ 0);
c’) ∃ λ(λ ∈ [−π, π] ∧ f _{ j → i }(λ) ≠ 0);
and similarly with other conditions in the list.
Observe conditions a’) and b’).
 18.
Note that λ is independently existentially quantified above. That would suggest that iPDC and iDTF simultaneously confirm the existence of connectivity from j to i. However they might do it in totally unrelated frequencies, which could make that equivalence meaningless, similarly as discussed in 9. This is all wrong.
 19.
The equivalence of a’) and b’) clearly contradicts the nonequivalence of PDC and DTF, which is extensively verified in the cited very detailed analysis of Sameshima and collaborators, since these are the special cases of iPDC and iDTF. However, the statement of the theorem is true for the two var case only, when the orbits are reduced to 1^{st} orbits only. In this case cumulative influence reduces to the direct influence, with no transitive nodes. This generates a limit for the theorem generalizations.
 20.Zero threshold in iPDC and iDTF. Authors in [21] do not mention zero thresholds at all. As shown multiply above, in practice it has to be determined. Again, as in detail discussed above, note that the same problems are equally present here. E.g. computationally we could easily haveNobody will like that.$$ 0<i{\mathrm{DTF}}_{ij}\left(\lambda \right)=i{\mathrm{PDC}}_{ij}\left(\lambda \right)=0. $$
 21.
Recent DTF based connectivity graphs with simplified orbits. In the recent publications and conference reports of research teams using DTF as connectivity measure [2729], presenting even rather complex brain connectivity graphs involving rather numerous nodes, majority of graphs contain practically only 1^{st} orbits, which is the case when deficiencies of DTF are significantly masked since cascade connectivity is hidden, graphs are not faithful, departing seriously from reality.
 22.
Both DTF and PDC measures are not applicable in real time applications like Brain Computer Interfaces – BCI, where the will generated patterns in brain signals are recognized and classified by a number of direct methods. Some of methods related to weak connectivity are applicable in real time.
 23.
The DTF based connectivity diagrams where the zero is chosen arbitrarily high or much higher than the established zero threshold and where connectivity is restricted to a single narrow band, intentionally reduce the number of really connected connectivity links by large amount, offering highly distorted facts that are established by DTF. The similar holds for synthetic spectrogram connectivity matrices. If the methodology of [18] and [21] was used, one could not deduce less than 10 times more connected nodes in the “memory” task and the “cognitive” task using DTF, which is strongly inconsistent with the presented connectivity diagrams – factual proofs, which the DTF authors derived from the supplied matrices. In all experiments DTF converges towards Adams Axiom.
 24.
The DTF has been making a number of serious problems since its invention. The authors have been continuously making efforts to solve the problems inventing newer modifications of DTF, adding additional measures, or applying arbitrary restrictions to their connectivity measure in order to reach connectivity diagrams which should look more faithful. Hardly had they succeeded in these intentions.
Clearly, without careful mathematical consideration and argumentation connectivity graphs, in here cited and many others published articles are of shaken fatefulness and need supplementary corroboration. Connectivity measures are different enough that the question of their logical coherence is appropriate. This is elaborated through measure comparisons. Here the published comparison of DTF and PDC measures is discussed in some detail as an illustrative example, giving enough material for this issue to be more carefully investigated. As verified on a number of nontrivial synthetic systems, connectivity conclusions by DTF are not well founded, while PDC has good capacity in precise structural description, confirming PDC superiority to DTF measure. Quite often PDC ≤ DTF, but it does not hold generally, hence PDC is not a general refinement of DTF and these two measures are essentially different, especially if compared frequency – pointwise. When applied to real neurologic data with the original methodology, the methods seem to be highly semantically unstable generating large model structural oscillations with possible PDC threshold variation. Quite generally on published data, when thresholds are roughly harmonized PDCDTF connectivity differences vanish, opposite to the conclusions published in [18]. Comparison after frequency aggregations leads to wrong conclusions on functional connectivity, unless appropriate modulators are involved. The zero threshold harmonization when comparing measures is a difficult and challenging issue which ought’s to be solved properly, prior to measure computations and comparisons in general.
Two methods, the Partial Linear Dependence, PLD and the method of small object recognition are added for enhanced connectivity problem treatment, comprising time expansions, with examples of their contribution in cases when the shared information is masked or embedded in noise. The number of innovative alternative approaches is growing; aiming to overcome certain difficulties they are successfully applied in demanding applications e.g. [31,32].
Declarations
Acknowledgments
The work on this paper was supported by Serbian Ministry of Education projects III41013, ON174009 and TR36001.
Authors’ Affiliations
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