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18.3: Synchronizability - Mathematics


An interesting application of the spectral gap/algebraic connectivity is to determine the synchronizability of linearly coupled dynamical nodes, which can be formulated as follows:

[frac{dx_{i}}{dt} =R(x_{i}) +alpha{sum_{jepsilon{N_{i}}}(H(x_j) -H(x_i)}) label{(18.6)}]

Here (x_i) is the state of node (i), (R) is the local reaction term that produces the inherent dynamica lbehavior of individual nodes, and (N_i) is the neighborhood of node (i). We assume that (R) is identical for all nodes, and it produces a particular trajectory (x_s(t)) if there is no interaction with other nodes. Namely, (x_s(t)) is given as the solution of the differential equation (dx/dt = R(x)). (H) is called the output function that homogeneously applies to all nodes. The output function is used to generalize interaction and diffusion among nodes; instead of assuming that the node states themselves are directly visible to others, we assume that a certain aspect of node states (represented by (H(x))) is visible and diffusing to other nodes.

Eq. ef{(18.6)} can be further simplified by using the Laplacian matrix, as follows:

[frac{dx_i}{dt} =R(x_i) -alpha{L} egin{pmatrix} H(x_1) H(x_2) vdots H(x_n)end{pmatrix} label{(18.7)}]

Now we want to study whether this network of coupled dynamical nodes can synchronize or not. Synchronization is possible if and only if the trajectory (x_i(t) = x_s(t)) for all (i) is stable. This is a new concept, i.e., to study the stability of a dynamic trajectory, not of a static equilibrium state. But we can still adopt the same basic procedure of linear stability analysis: represent the system’s state as the sum of the target state and a small perturbation, and then check if the perturbation grows or shrinks over time. Here we represent the state of each node as follows:

[x_i(t) =x_s(t) +Delta{x_i(t)} label{(18.8)}]

By plugging this new expression into Eq. ef{(18.7)}, we obtain

[frac{d(x_s+Delta{x_i})}{dt} =R(x_s+Delta{x_i})- alpha{L} egin{pmatrix} H(x_s+Delta{x_1}) H(x_s+Delta{x_2}) vdots H(x_s +Delta{x_n})end{pmatrix} label{(18.9)}]

Since (∆x_i) are very small, we can linearly approximate (R) and (H) as follows:

[frac{dx_s}{dt} +frac{dDelta{x_i}}{dt} =R(x_s) +R'(x_s)Delta{x_i}-alpha{L} egin{pmatrix} H(x_s)+H'(x_s)Delta{x_1} H(x_s)+H'(x_s)Delta{x_2} vdots H(x_s)+ H'(x_s)Delta{x_n}end{pmatrix} label{(18.10)} ]

The first terms on both sides cancel out each other because xs is the solution of (dx/dt = R(x)) by definition. But what about those annoying (H(x_s))’s included in the vector in the last term? Is there any way to eliminate them? Well, the answer is that we don’t have to do anything, because the Laplacian matrix will eat them all. Remember that a Laplacian matrix always satisfies (Lh = 0). In this case, those (H(x_s))’s constitute a homogeneous vector (H(x_s)h) altogether. Therefore,( L(H(x_s)h) = H(x_s)Lh) vanishes immediately, and we obtain

[frac{dDelta{x}}{dt} =R'(x_s)Delta{x_i}-alpha{H'}(x_s)L egin{pmatrix} Delta{x_1} Delta{x_2} vdots Delta{x_{n}} end{pmatrix}, label{(18.11)}]

or, by collecting all the (∆x_i)’s into a new perturbation vector (∆x),

[frac{dDelta{x}}{dt} =(R'(x_s)I -alpha{H'}(x_s)L)Delta{x}, label{(18.12)}]

as the final result of linearization. Note that (x_s) still changes over time, so in order for this trajectory to be stable, all the eigenvalues of this rather complicated coefficient matrix ((R'(x_s)I −αH'(x_s)L)) should always indicate stability at any point in time.

We can go even further. It is known that the eigenvalues of a matrix (aX+bI) are (aλ_i+b), where (λ_i) are the eigenvalues of (X). So, the eigenvalues of ((R'(x_s)I −αH'(x_s)L) are

[-alpha{lambda_{i}}H'(x_s) +R'(x_s), label{(18.13)}]

where (λ_i) are (L)’sneigenvalues. The eigenvalue that corresponds to the smallest eigenvalue of (L), 0, is just (R'(x_s)), which is determined solely by the inherent dynamics of (R(x)) (and thus the nature of (x_s(t))), so we can’t do anything about that. But all the other (n − 1) eigenvalues must be negative all the time, in order for the target trajectory (x_s(t)) to be stable. So, if we represent the second smallest eigenvalue (the spectral gap for connected networks) and the largest eigenvalue of (L) by (λ_{2}) and (λ_n), respectively, then the stability criteria can be written as

[alpha{lambda_{i}}H'(x_{s}(t)) >R' (x_{s}(t)) qquad{ ext{for all t, and}} label{(18.14)}]

[alpha{lambda_n}H'(x_s(t)) >R'(x_s(t)) qquad{ ext{ for all t, }} label{(18.15)} ]

because all other intermediate eigenvalues are “sandwiched” by (λ_2) and (λ_n). These inequalities provide us with a nice intuitive interpretation of the stability condition: the influence of diffusion of node outputs (left hand side) should be stronger than the internal dynamical drive (right hand side) all the time.

Note that, although (α) and (λ_i) are both non-negative, (H'(xs(t))) could be either positive or negative, so which inequality is more important depends on the nature of the output function (H) and the trajectory (x_s(t)) (which is determined by the reaction term (R)). If (H'(x_s(t))) always stays non-negative, then the first inequality is sufficient (since the second inequality naturally follows as (λ_2 ≤ λ_n)), and thus the spectral gap is the only relevant information to determine the synchronizability of the network. But if not, we need to consider both the spectral gap and the largest eigenvalue of the Laplacian matrix.

Here is a simple example. Assume that a bunch of nodes are oscillating in an exponentially accelerating pace:
[frac{d heta_i}{dt} =eta{ heta_i} +alpha sum_{j epsilon{N_i} ( heta_{j} - heta_{i})} label{(18.16)}]

Here, (θ_i) is the phase of node (i), and (β) is the rate of exponential acceleration that homogeneously applies to all nodes. We also assume that the actual values of θi diffuse to and from neighbor nodes through edges. Therefore, (R(θ) = βθ) and (H(θ) = θ) in this model.

We can analyze the synchronizability of this model as follows. Since (H'(θ) = 1 > 0), we immediately know that the inequality ef{(18.14)} is the only requirement in this case. Also, (R'(θ) = β), so the condition for synchronization is given by

[alpha{lambda_2} > eta, ext{or} lambda_2 > frac{eta}{alpha}. label{(18.17)}]

Very easy. Let’s check this analytical result with numerical simulations on the Karate Club graph. We know that its spectral gap is 0.4685, so if (β/α) is below (or above) this value, the synchronization should (or should not) occur. Here is the code for such simulations:

Here I added a second plot that shows the phase distribution in a ((x,y) = (cos{θ},sin{θ})) space, just to aid visual understanding.

In the code above, the parameter values are set toalpha =2 andbeta =1, so (λ_2 0.4685 < β/α = 0.5) . Therefore, it is predicted that the nodes won’t get synchronized. And, indeed, the simulation result confirms this prediction (Fig. 18.3.1(a)), where the nodes initially came close to each other in their phases but, as their oscillation speed became faster and faster, they eventually got dispersed again and the network failed to achieve synchronization. However, ifbetais slightly lowered to 0.9 so that (λ_2 = 0.4685 > β/α = 0.45), the synchronized state becomes stable, which is confirmed again in the numerical simulation (Fig. 18.3.1(b)). It is interesting that such a slight change in the parameter value can cause a major difference in the global dynamics of the network. Also, it is rather surprising that the linear stability analysis can predict this shift so precisely. Mathematical analysis rocks!

Exercise (PageIndex{1})

Randomize the topology of the Karate Club graph and measure its spectral gap. Analytically determine the synchronizability of the accelerating oscillators model discussed above with (α = 2), (β = 1) on the randomized network. Then confirm your prediction by numerical simulations. You can also try several other network topologies.

Exercise (PageIndex{2})

he following is a model of coupled harmonic oscillators where complex node states are used to represent harmonic oscillation in a single-variable differential equation:

[ frac{dx_i}{dt} =iomega{x_i} +alpha sum{jepsilon{N_i}(x^{gamma}_{j} -x^{y}_{i})} label{(18.18)}]

Here i is used to denote the imaginary unit to avoid confusion with node index i. Since the states are complex, you will need to use (Re(·)) on both sides of the inequalities ef{(18.14)} and ef{(18.15)} to determine the stability.

Analyze the synchnonizability of this model on the Karate Club graph, and obtain the condition for synchronization regarding the output exponent (γ). Then confirm your prediction by numerical simulations.

You may have noticed the synchronizability analysis discussed above is somewhat similar to the stability analysis of the continuous field models discussed in Chapter 14. Indeed, they are essentially the same analytical technique (although we didn’t cover stability analysis of dynamic trajectories back then). The only difference is whether the space is represented as a continuous field or as a discrete network. For the former, the diffusion

is represented by the Laplacian operator (∇^2), while for the latter, it is represented by the Laplacian matrix (L) (note again that their signs are opposite for historical misfortune!). Network models allows us to study more complicated, nontrivial spatial structures, but there aren’t any fundamentally different aspects between these two modeling frameworks. This is why the same analytical approach works for both.

Note that the synchronizability analysis we covered in this section is still quite limited in its applicability to more complex dynamical network models. It relies on the assumption that dynamical nodes are homogeneous and that they are linearly coupled, so the analysis can’t generalize to the behaviors of heterogeneous dynamical networks with nonlinear couplings, such as the Kuramoto model discussed in Section 16.2 in which nodes oscillate in different frequencies and their couplings are nonlinear. Analysis of such networks will need more advanced nonlinear analytical techniques, which is beyond the scope of this textbook.


Enhancing robustness and synchronizability of networks homogenizing their degree distribution

A new family of networks, called entangled, has recently been proposed in the literature. These networks have optimal properties in terms of synchronization, robustness against errors and attacks, and efficient communication. They are built with an algorithm which uses modified simulated annealing to enhance a well-known measure of networks’ ability to reach synchronization among nodes. In this work, we suggest that a class of networks similar to entangled networks can be produced by changing some of the connections in a given network, or by just adding a few connections. We call this class of networks weak-entangled. Although entangled networks can be considered as a subset of weak-entangled networks, we show that both classes share similar properties, especially with respect to synchronization and robustness, and that they have similar structural properties.

Highlights

► We present rewiring procedure that produces networks with weak-entangled structure. ► We examine the structural properties, synchronization and robustness of the networks. ► We show that they share similar properties as entangled networks. ► Our procedure improves the synchronizability and robustness of a given network. ► The procedure exploits only the homogenization of the nodes’ degree.


18.3.2. Rules of Calculus¶

We now turn to the task of understanding how to compute the derivative of an explicit function. A full formal treatment of calculus would derive everything from first principles. We will not indulge in this temptation here, but rather provide an understanding of the common rules encountered.

18.3.2.1. Common Derivatives¶

As was seen in Section 2.4 , when computing derivatives one can oftentimes use a series of rules to reduce the computation to a few core functions. We repeat them here for ease of reference.

Derivative of constants. (fracc = 0) .

Derivative of linear functions. (frac(ax) = a) .

Power rule. (fracx^n = nx^) .

Derivative of exponentials. (frace^x = e^x) .

Derivative of the logarithm. (fraclog(x) = frac<1>) .

18.3.2.2. Derivative Rules¶

If every derivative needed to be separately computed and stored in a table, differential calculus would be near impossible. It is a gift of mathematics that we can generalize the above derivatives and compute more complex derivatives like finding the derivative of (f(x) = logleft(1+(x-1)^<10> ight)) . As was mentioned in Section 2.4 , the key to doing so is to codify what happens when we take functions and combine them in various ways, most importantly: sums, products, and compositions.

Sum rule. (fracleft(g(x) + h(x) ight) = frac(x) + frac(x)) .

Product rule. (fracleft(g(x)cdot h(x) ight) = g(x)frac(x) + frac(x)h(x)) .

Chain rule. (fracg(h(x)) = frac(h(x))cdot frac(x)) .

Let us see how we may use (18.3.6) to understand these rules. For the sum rule, consider following chain of reasoning:

By comparing this result with the fact that (f(x+epsilon) approx f(x) + epsilon frac(x)) , we see that (frac(x) = frac(x) + frac(x)) as desired. The intuition here is: when we change the input (x) , (g) and (h) jointly contribute to the change of the output by (frac(x)) and (frac(x)) .

The product is more subtle, and will require a new observation about how to work with these expressions. We will begin as before using (18.3.6):

This resembles the computation done above, and indeed we see our answer ( (frac(x) = g(x)frac(x) + frac(x)h(x)) ) sitting next to (epsilon) , but there is the issue of that term of size (epsilon^<2>) . We will refer to this as a higher-order term, since the power of (epsilon^2) is higher than the power of (epsilon^1) . We will see in a later section that we will sometimes want to keep track of these, however for now observe that if (epsilon = 0.0000001) , then (epsilon^<2>= 0.0000000000001) , which is vastly smaller. As we send (epsilon ightarrow 0) , we may safely ignore the higher order terms. As a general convention in this appendix, we will use “ (approx) ” to denote that the two terms are equal up to higher order terms. However, if we wish to be more formal we may examine the difference quotient

and see that as we send (epsilon ightarrow 0) , the right hand term goes to zero as well.

Finally, with the chain rule, we can again progress as before using (18.3.6) and see that

where in the second line we view the function (g) as having its input ( (h(x)) ) shifted by the tiny quantity (epsilon frac(x)) .

These rule provide us with a flexible set of tools to compute essentially any expression desired. For instance,

Where each line has used the following rules:

The chain rule and derivative of logarithm.

The derivative of constants, chain rule, and power rule.

The sum rule, derivative of linear functions, derivative of constants.

Two things should be clear after doing this example:

Any function we can write down using sums, products, constants, powers, exponentials, and logarithms can have its derivate computed mechanically by following these rules.

Having a human follow these rules can be tedious and error prone!

Thankfully, these two facts together hint towards a way forward: this is a perfect candidate for mechanization! Indeed backpropagation, which we will revisit later in this section, is exactly that.

18.3.2.3. Linear Approximation¶

When working with derivatives, it is often useful to geometrically interpret the approximation used above. In particular, note that the equation

approximates the value of (f) by a line which passes through the point ((x, f(x))) and has slope (frac(x)) . In this way we say that the derivative gives a linear approximation to the function (f) , as illustrated below:


Lesson 1

Let's see how tape diagrams and equations can show relationships between amounts.

1.1: Which Diagram is Which?

Here are two diagrams. One represents (2+5=7) . The other represents (5 oldcdot 2=10) . Which is which? Label the length of each diagram.

Expand Image

Draw a diagram that represents each equation.

1.2: Match Equations and Tape Diagrams

Here are two tape diagrams. Match each equation to one of the tape diagrams.

Expand Image

1.3: Draw Diagrams for Equations

For each equation, draw a diagram and find the value of the unknown that makes the equation true.

You are walking down a road, seeking treasure. The road branches off into three paths. A guard stands in each path. You know that only one of the guards is telling the truth, and the other two are lying. Here is what they say:

  • Guard 1: The treasure lies down this path.
  • Guard 2: No treasure lies down this path seek elsewhere.
  • Guard 3: The first guard is lying.

Which path leads to the treasure?

Summary

Tape diagrams can help us understand relationships between quantities and how operations describe those relationships.

Expand Image

Diagram A has 3 parts that add to 21. Each part is labeled with the same letter, so we know the three parts are equal. Here are some equations that all represent diagram A:

(displaystyle 3oldcdot =21)

Notice that the number 3 is not seen in the diagram the 3 comes from counting 3 boxes representing 3 equal parts in 21.

We can use the diagram or any of the equations to reason that the value of (x) is 7.

Diagram B has 2 parts that add to 21. Here are some equations that all represent diagram B:

We can use the diagram or any of the equations to reason that the value of (y) is 18.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.


The Savvy PM Blog

Yes. The mathematics are simple when it comes to Agile, Scrum, XP, Lean, Kanban, and the exam for certification as a PMI-ACP, PMI’s Agile Certified Practitioner.

For example, what’s the optimum Scrum team size? Jeff Sutherland and Ken Schwaber state, in The Scrum Guide (July 2011), that the team should be no less than 5 (including the Product Owner and ScrumMaster), and no more than 9 (which includes the Product Owner and ScrumMaster if they also are executing work in the Product Backlog). So, the optimum team size is typically stated as 7 ± 2. Simple math!

Calculating how many story points can be accomplished in an iteration based on team velocity? Again, simple math. No calculus involved! Not even the Mean Value Theorem!

Here is another example of simple math: the PMI-ACP application requires the candidate to submit a minimum of 21 contact hours in Agile training to qualify for the certification. Try out this simple math: 18 + 3 = 21 !

Velociteach’s 2-Day Pass the PMI-ACP class provides 18 contact hours for the participant. The additionally required 3 contact hours are provided by the PMI-ACP Practice Test A, available via InSite, Velociteach’s eLearning facility. An applicant can obtain all 21 of the required contact hours via these two sources!

18 + 3 = 21 ! Simple math! Learn how to pass the PMI-ACP on your first try by combining knowledge from Andy Crowe’s new book, The PMI-ACP Exam: How to Pass on Your First Try, along with instructor and eLearning guidance – all brought to you by Velociteach, a brand that is trusted by thousands who have passed their PMP Exams on the first try using the Velociteach method.


Exercises 18.3

Ex 18.3.1 Find an $f$ so that $ abla f=langle 2x+y^2,2y+x^2 angle$, or explain why there is no such $f$. (answer)

Ex 18.3.2 Find an $f$ so that $ abla f=langle x^3,-y^4 angle$, or explain why there is no such $f$. (answer)

Ex 18.3.3 Find an $f$ so that $ abla f=langle xe^y,ye^x angle$, or explain why there is no such $f$. (answer)

Ex 18.3.4 Find an $f$ so that $ abla f=langle ycos x,ysin x angle$, or explain why there is no such $f$. (answer)

Ex 18.3.5 Find an $f$ so that $ abla f=langle ycos x,sin x angle$, or explain why there is no such $f$. (answer)

Ex 18.3.6 Find an $f$ so that $ abla f=langle x^2y^3,xy^4 angle$, or explain why there is no such $f$. (answer)

Ex 18.3.7 Find an $f$ so that $ abla f=langle yz,xz,xy angle$, or explain why there is no such $f$. (answer)

Ex 18.3.8 Evaluate $dsint_C (10x^4 - 2xy^3),dx - 3x^2y^2,dy$ where $C$ is the part of the curve $x^5-5x^2y^2-7x^2=0$ from $(3,-2)$ to $(3,2)$. (answer)

Ex 18.3.9 Let $<f F>= langle yz,xz,xy angle$. Find the work done by this force field on an object that moves from $(1,0,2)$ to $(1,2,3)$. (answer)

Ex 18.3.10 Let $<f F>= langle e^y,xe^y+sin z,ycos z angle$. Find the work done by this force field on an object that moves from $(0,0,0)$ to $(1,-1,3)$. (answer)

Ex 18.3.11 Let $<f F>= left langle <-xover (x^2+y^2+z^2)^<3/2>>,<-yover (x^2+y^2+z^2)^<3/2>>,<-zover (x^2+y^2+z^2)^<3/2>> ight angle.$ Find the work done by this force field on an object that moves from $(1,1,1)$ to $(4,5,6)$. (answer)


18.3: Synchronizability - Mathematics

Ye shall not enter into the kingdom of heaven. —The force of the words as spoken to the Twelve can hardly be exaggerated. They were disputing about precedence in the kingdom, and in that very dispute they were showing that they were not truly in it. It was essentially spiritual, and its first condition was abnegation of self. Even the chief of the Apostles was self-excluded when he gloried in his primacy. The words at least help us to understand the more mysterious language of John 3:3 John 3:5, as to the “new birth” of water and the Spirit, which one, at least, of the disputants must, in all likelihood, have heard.

Matthew 18:3-4 . And said, Verily I say unto you — What I say is an undoubted and most important truth, a truth which you ought not only firmly to believe but seriously to lay to heart: except ye be converted — Turned from these worldly and carnal views and desires and become like little children — “Free from pride, covetousness, and ambition, and resemble them in humility, sincerity, docility, and disengagement of affection from the things of the present life, which excite the ambition of grown men,” ye shall be so far from becoming the greatest in my kingdom, that ye shall not so much as enter into it. Observe well, reader, the first step toward entering into the kingdom of grace is to become as little children: lowly in heart, knowing ourselves utterly ignorant and helpless, and hanging wholly on our Father who is in heaven, for a supply of all our wants. We may further assert, (though it is doubtful whether this text implies so much,) except we be turned from darkness to light, and from the power of Satan to God: except we be entirely, inwardly changed, and renewed in the image of God, we cannot enter into the kingdom of glory. Thus must every man be converted in this life, or he can never enter into life eternal. Whosoever therefore shall humble himself — He that has the greatest measure of humility, joined with the sister graces of resignation, patience, meekness, gentleness, and long-suffering, shall be the greatest in Christ’s kingdom: whosoever rests satisfied with the place, station, and office which God assigns him, whatever it may be, and meekly receives all the divine instructions, and complies with them, though contrary to his own inclinations, and prefers others in honour to himself, — such a person is really great in the kingdom of heaven, or of God.

The verb means to change or turn from one habit of life or set of opinions to another, James 5:19 Luke 22:32. See also Matthew 7:6 Matthew 16:23 Luke 7:9, etc., where the same word is used in the original. It sometimes refers to that great change called the new birth or regeneration Psalm 51:13 Isaiah 60:5 Acts 3:19, but not always. It is a general word, meaning any change. The word "regeneration" denotes a particular change the beginning to live a spiritual life. The phrase, "Except ye be converted," does not imply, of necessity, that they were not Christians before, or had not been born again. It means that their opinions and feelings about the kingdom of the Messiah must be changed. They had supposed that he was to be a temporal prince. They expected he would reign as other kings did. They supposed he would have his great officers of state, as other monarchs had, and they were ambitiously inquiring who should hold the highest offices. Jesus told them that they were wrong in their views and expectations. No such things would take place. From these notions they must be turned, changed or converted, or they could have no part in his kingdom. These ideas did not fit at all the nature of his kingdom.

And become as little children - Children are, to a great extent, destitute of ambition, pride, and haughtiness They are characteristically humble and teachable. By requiring his disciples to be like them, he did not intend to express any opinion about the native moral character of children, but simply that in these respects they must become like them. They must lay aside their ambitious views and their pride, and be willing to occupy their proper station - a very lowly one. Mark says Mark 9:35 that Jesus, before he placed the little child in the midst of them, told them that "if any man desire to be first, the same shall be last of all and servant of all." That is, he shall be the most distinguished Christian who is the most humble, and who is willing to be esteemed least and last of all. To esteem ourselves as God esteems us is humility, and it cannot be degrading to think of ourselves as we are but pride, or an attempt to be thought of more importance than we are, is foolish, wicked, and degrading.

Mt 18:1-9. Strife among the Twelve Who Should Be Greatest in the Kingdom of Heaven, with Relative Teaching. ( = Mr 9:33-50 Lu 9:46-50).

For the exposition, see on [1323]Mr 9:33-50.

except ye be converted or turned from that gross notion of a temporal kingdom, and of enjoying great grandeur, and outward felicity in this world and from all your vain views of honour, wealth, and riches,

and become as little children: the Arabic renders it, "as this child" that is, unless ye learn to entertain an humble, and modest opinion of yourselves, are not envious at one another, and drop all contentions about primacy and pre-eminence, and all your ambitious views of one being greater than another, in a vainly expected temporal kingdom things which are not to be found in little children, though not free from sin in other respects,

ye shall not enter into the kingdom of heaven: ye shall be so far from being one greater than another in it, that you shall not enter into it at all meaning his visible, spiritual kingdom, which should take place, and appear after his resurrection, upon his ascension to heaven, and pouring forth of the Spirit: and it is to be observed, that the apostles carried these carnal views, contentions, and sentiments, till that time, and then were turned from them, and dropped them for, upon the extraordinary effusion of the Holy Spirit, they were cleared of these worldly principles, and understood the spiritual nature of Christ's kingdom which they then entered into, and took their place in, and filled it up with great success, without envying one another having received the same commission from their Lord, and Master: so that these words are a sort of prophecy of what should be, as well as designed as a rebuke to them for their present ambition and contentions.

(b) An idiom taken from the Hebrews which is equivalent to repent.

Matthew 18:3. Εἴ τις ἀπέχεται τῶν προαιρετικῶν παθῶν , γίνεται ὡς τὰ παιδία , κτώμενος διʼ ἀσκήσεως , ἅπερ ἔχουσι τὰ παιδία ἐξ ἀφελείας , Euthymius Zigabenus.

To turn round ( στραφῆτε , representing the μετάνοια under the idea of turning round upon a road), and to acquire a moral disposition similar to the nature of little children —such is the condition, without complying with which you will assuredly not ( οὐ μή ) enter , far less be able to obtain a high position in, the Messianic kingdom about to be established. The same truth is presented under a kindred figure and in a wider sense in John 3:3 John 3:5 ff. the divine agent in this moral change, in which child-like qualities assume the character of manly virtues , is the Holy Spirit comp. Luke 11:13 Luke 9:55.

Matthew 18:3. ἐὰν μὴ στραφῆτε : unless ye turn round so as to go in an opposite direction. “Conversion” needed and demanded, even in the case of these men who have left all to follow Jesus! How many who pass for converted, regenerate persons have need to be converted over again, more radically! Chrys. remarks: “We are not able to reach even the faults of the Twelve we ask not who is the greatest in the Kingdom of Heaven, but who is the greater in the Kingdom of Earth: the richer the more powerful” (Hom. lviii.). The remark is not true to the spirit of Christ. In His eyes vanity and ambition in the sphere of religion were graver offences than the sins of the worldly. His tone at this time is markedly severe, as much so as when He denounced the vices of the Pharisees. It was indeed Pharisaism in the bud He had to deal with. Resch suggests that στραφῆτε here simply represents the idea of becoming again children, corresponding to the Hebrew idiom which uses שׁוּב = πάλιν ( Aussercanonische Paralleltexte zu Mt. and Mk. , p. 213).— ὡς τὰ παιδία , like the children, in unpretentiousness. A king’s child has no more thought of greatness than a beggar’s.— οὐ μὴ εἰσέλθητε , ye shall not enter the kingdom, not to speak of being great there. Just what He said to the Pharisees ( vide on chap. Matthew 5:17-20).

3 . be converted ] Literally, be turned . The Greek word is used in a literal sense, except here and Acts 7:39 Acts 7:42.

shall not enter ] much less be great therein.

Matthew 18:3. Καὶ εἶπεν , and said ) By asking who is the greatest? each of the disciples might offend himself, his fellow-disciples, and the child in question. The Saviour’s words (Matthew 18:3-20) meet all these offences, and declare His own and His Father’s anxiety for the salvation of souls. We perceive hence the connection between the different portions of His speech.— ὡς τὰ παιδία , as little children ) They must possess a wonderful degree of humility, simplicity, and faith to be proposed as an example to adults. Scripture exhibits everywhere favour towards little children.— οὐ μὴ εἰσέλθητε , ye shall not enter ) So far from being the greatest, ye shall not even enter therein. He does not say, “ye shall not remain,” but, “ye shall not enter,” so as to repress their arrogance the more.

The word converted has acquired a conventional religious sense which is fundamentally truthful, but the essential quality of which will be more apparent if we render literally, as Rev., except ye turn. The picture is that of turning round in a road and facing the other way.

But the double negative is very forcible, and is given in Rev. in nowise. So far from being greatest in the kingdom of heaven, ye shall not so much as enter.


Tree-Seed Algorithm (TSA) has good performance in solving various optimization problems. However, it is inevitable to suffer from slow exploitation when solving complex problems. This paper makes an intensive analysis of TSA. In order to keep the balance between exploration and exploitation, we propose an adaptive automatic adjustment mechanism. The number of seeds can be defined in the initialization process of the optimization algorithm. In order to further improve the convergence rate of TSA, we also modify the change model of seed numbers in the initialization process with randomly changing from more to less. With the improvement of two mechanisms, the main weakness of TSA has been overcome effectively. Based on the above two improvements, we propose a new algorithm-Sine Tree-Seed Algorithm (STSA). STSA achieves good results in solving high-dimensional complex optimization problems. The results obtained from 24 benchmark functions confirm the excellent performance of the proposed method.

The authors are grateful to the financial support by the National Natural Science Foundation of China (no. 61572225), Natural Science Foundation of the Science and Technology Department of Jilin Province, China (no. 20180101044JC), the Social Science Foundation of Jilin Province, China (nos. 2019B68, 2017BS28), the Foundation of the Education Department of Jilin Province, China (nos. JJKH20180465 kJ) and the Foundation of Jilin University of Finance and Economics (no. 2018Z05).


Etomidate: A Complementary Diagnostic Tool for Pre-Surgical Evaluation in Temporal Lobe Epilepsy

How to Cite: Vega-Zelaya L, Sanz-Garcia A, Ortega G J, de Sola R G, Pastor J. Etomidate: A Complementary Diagnostic Tool for Pre-Surgical Evaluation in Temporal Lobe Epilepsy, Arch Neurosci. 2016 3(4):e34915. doi: 10.5812/archneurosci.34915.

Abstract

Context: Temporal lobe epilepsy is the most frequent drug-resistant epilepsy. It has a high success rate in surgical treatment, provided that the epileptic zone (EZ) was accurately localized through a pre-surgical evaluation and removed. Pharmacological activation inducing interictal activity is tested as a complementary method in the pre-surgical diagnosis, although with nonspecific results and limited safety, due to poorly tolerated side effects.

Evidence Acquisition: Etomidate is a well-tolerated, fast onset and rapid decline drug with a few side effects. Studies conducted to evaluate the safety and usefulness of etomidate to identify the EZ showed that etomidate activates irritative zone only in the areas where spikes previously appeared in basal conditions, besides having a high coefficient of lateralization for ictal onset zone (IOZ). Regarding the analysis of the topography of the voltage sources, it is shown that interictal, ictal and etomidate-induced activities greatly overlap, indicating that the biophysical mechanisms are similar, and the cortical areas where all types of activities appear are likely the same or closely related. In addition, from the point of view of complex networks, etomidate produces very similar changes in the limbic network to those occurring during temporal seizures i.e. an impaired connectivity in the ipsilateral side to the IOZ.

Results: All findings suggest that etomidate, in a specific manner, activates the neural and biophysical mechanisms of spontaneous epilepsy.

Conclusions: This technique can be used as a diagnostic tool during the pre-surgical evaluation of patients with TLE to define the region resected during epilepsy surgery with confidence.

1. Context

Temporal lobe epilepsy (TLE) is the most frequent drug-resistant epilepsy. It represents approximately two thirds of the intractable seizure population requiring surgical management (1). Surgical treatment for these patients is typically a safe, effective and well-established option, with a success rate of 70% to 90% (2).

The best surgical outcomes are obtained when the epileptic zone (EZ) (3) is accurately localized during the pre-surgical evaluation. The EZ usually includes the ictal-onset zone (IOZ), the cortical region where seizures start and, in a variable and not well-defined degree, the interictal or irritative zone (IZ), the area showing interictal epileptiform discharges (IEDs). The ancillary tests used during the pre-surgical evaluation include (2, 4) video-electroencephalography (v-EEG), magnetic resonance imaging (MRI), single photon emission-computed tomography (SPECT) and positron emission tomography (PET). When results of the tests are not functionally and anatomically consistent, invasive recordings, such as foramen ovale (FO), subdural or depth electrodes are required (5). Pharmacological activation inducing or increasing interictal activity is also used as a complementary method, together with v-EEG to improve accuracy in the pre-surgical diagnosis. The tested drugs include: methohexital (6), clonidine (7), pentylenetetrazol (8), thiopental (9) and opiates (10). Nevertheless, the results were not specific and adverse effects were poorly tolerated, decreased safety and precluded the use of many drugs. Etomidate is a non-barbiturate imidazole derivative hypnotic agent with a rapid onset, a short duration of action, and minor side effects associated with intravenous perfusion. It acts as a selective modulator of the gaminobutyric acid receptor A (GABAA). It is shown that etomidate can be safely used to activate epileptogenic activity (11-16)

In fact, Pastor et al. (2010) showed that etomidate induced interictal spiking activity ipsilateral to the ictal onset zone (IOZ) and can correctly lateralize 95% of the patients with TLE. Furthermore, as this drug facilitates the reliable identification of the IOZ, it could be used to diagnose patients who do not experience seizures during v-EEG recordings or influence decisions regarding the placement of intracranial electrodes. In a recent work, authors showed that etomidate perfusion induce changes in the underlying epileptic network related to the ones found during spontaneous seizures (17).

The current paper reviewed the evidence found to date, about the effects of etomidate in the interictal and ictal activities and epileptic network in patients with temporal lobe epilepsy, in an effort to support its potential value as an additional diagnostic tool for the preoperative assessment in patients with drug-resistant epilepsy.

2. Evidence Acquisition

2.1. Bioelectrical Activity Induced by Etomidate

The changes induced after etomidate administration onto scalp EEG was described in an accurate way (14, 18, 19). In stage 1, small increases in amplitude and frequency were observed in the scalp EEG, followed by a generalized and high-amplitude delta activity (stage 2). Besides, studies using scalp EEG and foramen ovale (FO) electrode recordings described the high-voltage spikes and sharp-waves superimposed on this pattern (14). This interictal activity never appeared in different areas from the ones included in the IZ, i.e. they appeared only in the areas where interictal epileptiform discharges appeared in basal conditions. In the study, they compared the lateralization induced by etomidate with the IOZ identified spontaneous seizures recorded by VEEG + FO. Lateralization induced by etomidate perfusion was assessed through lateralization coefficient (LC), comparing the frequency of IED (spikes/min) in the left and right areas, considering the mesial and lateral areas.

Etomidate correctly identified the temporal lobe in 21 of 22 patients (Figure 1A). In addition, there was a strong correlation between the lateral and mesial values of the LC.

A, scalp and EFO record (left, top and right, bottom) during a spontaneous crisis originated in right mesial temporal region B, record in the same patient during etomidate administration (arrow). In the right mesial temporal region a marked increase in irritative activity is observed C, box detail records in scalp. Note the increase in scalp generalized delta activity in response to etomidate (right).

Likewise, the activity was grouped from the temporal areas where seizures occurred, and, the activity from areas where seizures did not occur. In order to access the kinetics of activity after etomidate infusion they used the equivalent to the first derivative for the discrete time-series of the frequency of spikes.

It was found that activity increased in both the mesial and lateral epileptic areas, although the increase was higher in the mesial region. Conversely, practically no increase was observed in the non-epileptic areas following etomidate administration (Figure 1B and 1C). Furthermore, the epileptic region displayed a higher frequency and also faster kinetics.

There are different theories about why etomidate activates the irritative area one could be explained by the fact that etomidate almost exclusively acts on the β2 and β3 subunits of the GABAA receptor at clinical concentrations (20). Studies on rats show that β2 subunits seem to be preferentially located on GABAergic interneurons and excessive activation of these receptors could cause disinhibition of cortical activity and seizures. Studies on cultured astrocytes show that etomidate inhibits glutamate uptake, increasing the extracellular glutamate concentration to a level that can escape the synaptic cleft and activate extra-synaptic receptors. As a consequence, irritative activity increases (21). Besides, examination of slices obtained from patients with epilepsy revealed a decrease in the reversal potential of Cl - anions (22). This change could induce a depolarization instead of hyperpolarization after GABA release, driving the irritative activity.

2.2. Voltage Sources in Mesial Temporal Lobe

An early study aimed to examine different electrophysiological properties of voltage sources in the records obtained through FO electrodes (23) in mesial TLE. The primary purpose of the study was to ensure that pharmacologically induced irritative activity was generated by the same cortical regions observed under basal conditions. It was found that etomidate could induce irritative activity in the mesial temporal region in an extremely specific manner, i.e. triggering the same cortical structures responsible for spontaneous IED.

A more recent study by the authors also proposed to analyze the electrophysiological properties of interictal activity induced by etomidate (24), and consider whether it matches or not with interictal activity at baseline conditions, it also considered its relationship with ictal activity. A classical electrostatic theory was applied to derive mathematical expressions of IED (i.e. spikes and sharp waves) recorded using FO (25, 26). An infinite and homogenous volume conductor and an isotropic medium were assumed.

It was found that the scattering and iequiv were similar for interictal and pharmacologically induced activities (Figure 2A), suggesting that the single source model could reliably explain interictal and recorded etomidate-induced spiking using FO electrodes in TLE.

Furthermore, the spatial distribution of the voltage sources responsible for interictal baseline activity was similar to the ones obtained for etomidate-induced activity. In addition, there was a close correlation between areas where irritative activity occurred under basal conditions and the ones induced through etomidate. The analysis of normalized sources showed that 76.9% of the patients show localised distribution of voltage sources at either the interictal baseline or in the presence of etomidate. A more scattered distribution along the axis z, with IP20 - 75 > 20 mm (Rank: 21, 5 - 28,3 mm), was observed in 22.1% of the patients. A linear relationship was also found among the three functional states (Figure 2B).

In order to analyze the topographical relationship between pharmacologically induced activity and that of the IOZ, the degree of superposition was established among sources obtained from interictal, ictal and pharmacologically induced activities (24). Thus, the zr-plane or configuration space, was divided into small non-overlapping patches of Δx × Δy, where Δx = 1 mm, Δy = 1 mm and area = 1 mm 2 . This tessellation covers the entire theoretical surface. The active area was defined (where current-source appears, irrespective of the functional state) in two steps: (i) patches where spiking activity 1 spike/mm 2 and (ii) spikes less than a minimum distance (dmin) from other spikes. The dmin = 1.44 mm was selected (Equation 1). In this way, spurious spikes were excluded from the active area and the total area where the current sources appear for each type of activity and the superposition between them was obtained.

There was a great degree of superposition of the current-source topography for interictal, ictal and pharmacologically induced activities. For each pair of activities, the area and the spikes of superposition were interictal-ictal: 10.0 ± 1.5 mm 2 (31%) and 74.7 ± 12.6 spikes/ mm 2 (12%) interictal-etomidate induced: 7.3 ± 1.4 mm 2 (24%) and 74.7 spikes/ mm 2 (12%) and ictal-etomidate induced: 4.7 ± 1.0 mm 2 (24%) and 30.0 spikes/mm 2 (45%), respectively.

2.3. Seizure Networks Dynamics and Etomidate

A recently published work addressed the very relevant issue whether temporal lobe seizures produce network changes comparable to the ones elicited by etomidate administration (17). Scalp and FOE recordings of nine patients with temporal lobe epilepsy were analyzed employing analysis of complex networks. To evaluate the global aspects of the cortical dynamics, several variables in network theory were calculated (see Vega-Zelaya et al., 2015b for more details): the average path length (APL), density of links (DOL), average clustering coefficient (ACC), modularity (Mod) and spectral entropy (SE).

The APL is calculated as the average of all the shortest paths in terms of the number of steps along the network nodes between every pair of vertices in the network. In this sense, low values for the APL imply efficient and fast communication across the network functional topology. The DoL is the ratio of the actual number of edges in the network to the number of all possible edges between the network nodes. A network with many links or a high density of links implies highly synchronous behavior at both short and long distances. A high DoL decreases the APL in the network because a greater number of edges allows for a greater number of alternative paths between the two nodes. The ACC characterizes local connectedness in a network by measuring how well neighbours are connected in a given node. Modularity (Mod) measures how well a given partition or division in a community in a complex network corresponds to a natural or expected sub-division. The averages for each community were calculated, and the overall spectral entropy was defined as the average over all communities.

Two different groups were evaluated. The first group corresponded to the whole network (scalp + FOE), and the second one corresponded to the mesial sub-networks (FOE).

Based on the previous published results regarding ictal network dynamics (27, 28), it was expected to find that the ACC and DoL would increase with respect to the pre-ictal values both during the seizure and etomidate administration. Since the increase in the DoL would increase the available paths, it was assume that they lower the APL. Also it was expected that the increase in DoL would make intra-regional links more available and lower the modularity. Likewise, since anesthetics are known to lower SE (29), it was proposed that SE would decrease in both cases.

Considering the whole network, it was observed that etomidate fits better to the previous statement, except for modularity (Figure 3A). In four of nine patients, the APL did not decrease during etomidate administration. As expected, SE decreased after etomidate administration in all patients and also decreased during the seizure (Figure 3B), implying that the frequency content during the seizure reduced to a few relevant ones. During the seizures, the network measures demonstrated disparate results. DoL and Mod best followed the previous statement (in seven out of nine seizures) and the worst measures were APL and ACC.

A, scattering along the antero-posterior theoretical axis B, linear regression of the location of current sources for pairs of functional states, shown in the same colors. The linear regression of interictal/ictal is y = 1.09x - 0.50, r = 0.930, for interictal/etomidate y = 1.01x + 0.01, r = 0.922 and for ictal/interictal y = 0.856x + 0.952, r = 0.867.

Therefore, it was found that the changes showed similar behavior in both situations: an increase in the density of links (DoL) and a decrease in the spectral entropy (SE). The other measures did not show higher coincidences.

Regarding the mesial sub-networks, etomidate and seizures produce similar changes (Figure 6). Specifically, during etomidate, APL increases and DoL decreases on the ipsilateral side of the seizures than on the contralateral side, mimicking the effect of seizures on these network measures. Previous findings already showed impaired connectivity in the ipsilateral side in patients with TLE (30-32). Thus, decreased connectivity was shown in the mesial structures, ipsilateral to the SOZ, compared with the contralateral side during resting-state fMRI studies (30, 31) and during the inter-ictal stage through electrophysiological methods (32). All these results were consistent with recent reports suggesting that a loss of connectivity within specific network structures are involved in seizure generation (33, 34).

A, Inter-ictal epileptiform discharge B, ictal onset C, pharmacologically induced activity.

A, ictal stage, the x-axis marks the time related to seizure onset (thick, vertical, solid line) B, etomidate, the x-axis marks the time related to the end of etomidate perfusion end (thick, vertical, solid line).

A, ictal stage, The x-axis marks the time related to seizure onset (thick, vertical, solid line). B, etomidate, the x-axis marks the time related to the end of etomidate perfusion (thick, vertical, solid line). The patient outcome was Engel I.

3. Results

Etomidate might be considered as a useful tool to identify the EZ and IOZ in patients with TLE. It is a safe, efficient and well tolerated drug. The described side-effects are not severe and can be monitored adequately by scalp EEG. The most frequent ones include, myoclonus affected small distal muscles, pain, paradoxical and euphoric reactions (14). Hemodynamic effects are observed (35) with higher doses and include tachycardia and increases in mean arterial pressure (19).

This method can activate the EZ, inducing irritative activity only in the areas where spontaneous activity was recorded (14). Additional evidence shows that interictal, ictal and etomidate-induced activities greatly overlap indicating that the physiophatological mechanisms are similar, and that presumably the cortical areas corresponding to each of the activities are likely the same or closely related (24). These findings are of great significance, since etomidate might facilitate the identification of different patients regarding the activity topography, and it can be used for new surgical approaches, e.g., high-definition radiosurgery in patients with well-localized sources. An important issue to emphasize is that etomidate induced changes are produced with a low probability to eliciting seizures (14). This is an important advantage because it was observed that etomidate reproduce the same biophysical characteristics at the same location as the seizures do, without triggering an ictal episode.

Moreover, it is proved that etomidate uses the same neural network that the seizures do. Therefore, the findings show that the changes etomidate produces in the limbic network are very similar to the ones occurring during temporal seizures. The most remarkable finding in relation to the epileptogenic network is that mesial synchronization ipsilateral to the IOZ is reduced compared with those of the contralateral side under basal conditions in both situations implying that etomidate and seizures produce similar changes in the mesial sub-networks. From the viewpoint of a functional brain network in epilepsy, etomidate could be used in research to increase knowledge about network dynamics during seizure activity.

4. Conclusions

This method offers several benefits as a diagnostic tool during the pre-surgical evaluation of patients with TLE. It may help to confirm results derived from other methods. It can facilitate diagnosis in patients without seizures during v-EEG recording. Besides, it has the potential to influence decisions regarding the need to use intracranial electrodes, and also the best placement of those electrodes. Considering all the mentioned points this technique can be used to better define the region resected during epilepsy surgery.

Acknowledgements

The current work was financed by the Ministerio de Sanidad FIS PI12/02839 and was partially supported by FEDER (Fonds Européen de Developpement Economique et Regional) and PIP 11420100100261 CONICET (Consejo Nacional de Investigaciones Científicas y Tecnicas).

Footnotes

  • Authors’ Contribution: Study concept and design, Lorena Vega-Zelaya and Jesus Pastor analysis and interpretation of data, Lorena Vega-Zelaya, Ancor Sanz-Garcia, Guillermo J Ortega and Jesus Pastor drafting of the manuscript, Lorena Vega-Zelaya and Jesus Pastor critical revision of the manuscript for important intellectual content, Jesus Pastor Statistical analysis, Lorena Vega-Zelaya, Guillermo J Ortega and Jesus Pastor administrative, technical, and material support, Ancor Sanz-Garcia, Guillermo J Ortega and Rafael G de Sola.
  • Funding/Support: This work was financed by the Ministerio de Sanidad FIS PI12/02839 and was partially supported by FEDER (Fonds Europeen de Developpement Economique et Regional) and PIP 11420100100261 CONICET (Consejo Nacional de Investigaciones Científicas y Tecnicas).

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Early use of thrust manipulation versus non-thrust manipulation: a randomized clinical trial

The purpose of this study was to investigate the comparative effectiveness of early use of thrust (TM) and non-thrust manipulation (NTM) in sample of patients with mechanical low back pain (LBP). The randomized controlled trial included patients with mechanically reproducible LBP, ≥ age 18-years who were randomized into two treatment groups. The main outcome measures were the Oswestry Disability Index (ODI) and a Numeric Pain Rating Scale (NPRS), with secondary measures of Rate of Recovery, total visits and days in care, and the work subscale of the Fears Avoidance Beliefs Questionnaire work subscale (FABQ-w). A two-way mixed model MANCOVA was used to compare ODI and pain, at baseline, after visit 2, and at discharge and total visits, days in care, and rate of recovery (while controlling for patient expectations and clinical equipoise). A total of 149 subjects completed the trial and received care over an average of 35 days. There were no significant differences between TM and NTM at the second visit follow-up or at discharge with any of the outcomes categories. Personal equipoise was significantly associated with ODI and pain. The findings suggest that there is no difference between early use of TM or NTM, and secondarily, that personal equipoise affects study outcome. Within-groups changes were significant for both groups.