text:
\section{Introduction}

Synchronization in systems of coupled oscillators has been investigated in various fields such as nonlinear dynamics, network science, and statistical physics and has applications in physiscs, biology, and technology \cite{BLE88,PIK01}. The examples of nontrivial collective dynamics in such ensembles include synchronous regimes in arrays of Josephson junctions \cite{WIE95} and lasers \cite{WIE90}, coordinated firing of cardiac pacemaker cells \cite{PES75}, synchronous emission of light pulses by a population of fireflies \cite{BUC68} and emission of chirps by a population of crickets \cite{WAL69}, and synchronization in ensembles of electrochemical oscillators \cite{KIS02}.

An important example of this class is related to the collective
dynamics of neuronal populations. Indeed, synchronization of
individual neurons is believed to play the crucial role in the
emergence of pathological rhythmic brain activity in Parkinson's
disease, essential tremor, and epilepsies; a detailed discussion of
this topic and numerous citations can be found in
Refs.~\onlinecite{TAS99,GOL01,MIL03}. Obviously, the development of
techniques for suppression of the undesired neural synchrony
constitutes an important clinical problem. Technically, this problem
can be solved by means of implantation of microelectrodes into the
impaired part of the brain with subsequent electric stimulation
through these electrodes \cite{CHK78,BEH91}. A fundamental
understanding of synchronization and its control can help to improve
the results of such a treatment by reducing at the same time its side effects.

Control of synchronization has so far focused on networks of identical nodes
\cite{ZHO08,LU09, LU12, SEL12, GUZ13, LEH14, SCH16}.
However, in realistic
networks the nodes will always be characterized by some diversity
meaning that the parameters of the different nodes are not identical
but drawn from a distribution.  It is well known that such
heterogeneities in the nodes can hinder or prevent synchronization and
that the coupling strength is a crucial parameter in this context
\cite{STR00,SUN09a}. Moreover, in most cases perfect synchronization -- in the sense
that the state of all nodes is identical at all times  -- is unfeasible in
the presence of  heterogeneities.

In order to grasp the complicated interaction of neurons in large
neural networks, those are often lumped into groups of neural
populations, each of which can be represented as an effective excitable
element that is mutually coupled to other elements
\cite{ROS04,ROS04a,POP05}. In this sense the simplest model which may
reveal features of interacting neurons consists of two or a few  coupled neural oscillators. Each of these can be represented by a simplified FitzHugh-Nagumo (FHN) system \cite{FIT61a,NAG62}.

In the present paper we develop a
control algorithm ensuring collective synchrony in heterogeneous
FitzHugh-Nagumo networks. Considerable attention is paid to studying
irregular coupling topologies; the motivation of this comes from
recent results in the area of neuroscience. Diffusion tensor magnetic
resonance imaging (DT-MRI) studies have revealed an intricate
architecture in the neuron interconnectivity of the human and
mammalian brain, which has already been used in simulations
\cite{VUK14}. The analysis of DT-MRI images (resolution of the order
of $0.5$ mm) has shown that the connectivity of the neuron axons
network represents a hierarchical geometry with fractal dimensions
varying between $2.3$ and $2.8$, depending on the local properties, on
the subject, and on the noise reduction threshold
\cite{KAT09,EXP11,PRO12}. The fractal connectivity dictates a hierarchical ordering in the
distribution of neurons which is essential for the fast and optimal
handling of information in the brain \cite{OME15}. To take this into account, we consider as an
application of our method a hierarchical
model which well represents these fractal properties of
the brain.

The rest of the paper is organized as follows:
Section~\ref{Sec:model_eq} introduces the model, while
Sec.~\ref{Sec:FHN_analys} analyzes the behavior of the FHN network: The
possible bifurcation scenarios are investigated for a unidirectional
ring topology, and the influence of the coupling strength on the
synchronizability of the network is discussed. In Sec.~\ref{Sec:analys_hier}, hierarchical
topologies are studied in dependence on different system
parameters. Section~\ref{Sec:control} develops the control algorithm and provides the simulation results
of controlling synchrony in different network topologies. Finally, we conclude with Sec.~\ref{Sec:concl}.

\section{Model equation}\label{Sec:model_eq}
The local dynamics of each node in the network is modeled by the
FitzHugh-Nagumo (FHN) differential equations \cite{FIT61a,NAG62}. The
FHN model is paradigmatic for excitable dynamics of type II, i.e., close to a Hopf
bifurcation \cite{LIN04}, a bifurcation which is not only
characteristic for neurons but also occurs in the context of other
systems ranging from electronic circuits \cite{HEI10} to
cardiovascular tissues \cite{WIN91,NAS04} and the climate system \cite{GAN02}. The
$i$th node of the network is described as follows:
\begin{equation}\label{m}
\begin{aligned}
\epsilon \dot{u_i}&=u_i-\frac{u_i^3}{3}-v_i+C\sum\limits_{j=1}^{N}G_{ij}[u_j(t-\tau)-u_i(t)], \\
\dot{v_i}&=u_i+a_i,\quad i=1,\dots,N,
\end{aligned}
\end{equation}
where $u_i$ and $v_i$ denote the activator and inhibitor variable of
the nodes $i=1,\dots,N$, respectively. $\tau$ is the delay, i.e., the
time the signal needs to propagate between node  $i$ and $j$. $\epsilon$ is a time-scale parameter and
typically small (here we will use $\epsilon=0.01$), i.e., $u_i$
is a fast variable, while $v_i$ changes slowly. The coupling matrix $\mathbf{G}=\{G_{ij}\}$
defines which nodes are connected to each other. We construct the
matrix $\mathbf{G}$ by setting the entry $G_{ij}$ equal to $1$~(or $0$) if
the $j$th node couples (or does not couple) into the $i$th node. The overall coupling strength is given by $C$.

In the uncoupled system ($C=0$), $a_i$ is a threshold parameter: For
$|a_i|>1$ the $i$th node is excitable, while for $|a_i|<1$
it exhibits self-sustained periodic firing. This is due to a
supercritical Hopf bifurcation at $|a_i|=1$ with a locally stable equilibrium
point for $|a_i|>1$ and a stable limit cycle for $|a_i|<1$.
Here, we assume that the threshold parameters $a_i$ are chosen from the
interval $|a_i-\mu|<\sigma$ for some $\mu$.  For the analytical
considerations presented in Sec.~\ref{Sec:FHN_analys}, we do not make any further
assumption on the probability distribution of the $a_i$, while
for the simulation in Sec.~\ref{Sec:analys_hier} we use a Gaussian
distribution with mean  $\mu$ where we discard values of $a_i$ for which
$|a_i-\mu|<\sigma$  is not fulfilled.
For $\mu=1$,  oscillatory and excitable nodes coexist.

\section{Analysis of FitzHugh-Nagumo network}\label{Sec:FHN_analys}
This section studies the behavior of a FHN network of heterogeneous
nodes in the absence of the controller.  First, we perform a linear stability
analysis of the equilibrium point to get insight into the possible
bifurcations. Second, we discuss for which  coupling strengths the
network synchronizes. Afterwards, we study the
impact of the coupling strength on the type of synchronization, i.e.,
whether synchronization takes place in the equilibrium or in an oscillatory
state.

\subsection{Linear stability of the equilibrium point}\label{Sec:Lin_stab}

The linear stability analysis follows the approach suggested in
Refs.~\onlinecite{DAH08c,SCH08,PLO15}. Consider the unidirectional
ring of $N$ nodes described by the FHN equation with heterogeneous
threshold parameters, i.e., system \eqref{m} coupled by the following
adjacency matrix
\begin{equation}\label{eq1}
\mathbf{G} = \begin{pmatrix}
0 & 1 & 0 &\cdots & 0 \\
0 & 0 & 1 &\cdots & 0 \\
\vdots & \vdots & \vdots &\ddots & \vdots \\
0 & 0 & 0 &\cdots & 1 \\
1 & 0 & 0 &\cdots & 0
\end{pmatrix}.
\end{equation}

The unique equilibrium point of the system \eqref{m} with coupling matrix \eqref{eq1} is given by $\mathbf{x}^*\equiv(u_1^*,v_1^*,\dots,u_N^*,v_N^*)^{\rm\scriptscriptstyle T}$, where $u_j^*=-a_j$, $v_j^*=-a_j+a_j^3/3+C(a_j-a_{(j+1)\bmod N})$ for $j=1,\dots,N$. Linearizing Eq.~\eqref{m} with matrix \eqref{eq1} around the equilibrium point $\mathbf{x}^*$ by setting $\mathbf{x}(t)=\mathbf{x}^*+\delta\mathbf{x}(t)$, one obtains

\begin{multline}\label{eq2}
\delta \mathbf{\dot x}=\frac{1}{\epsilon}\begin{pmatrix}
\xi_1 & -1 & 0 &0&0& \cdots&0& 0 \\
\epsilon & 0 & 0 &0&0& \cdots &0& 0 \\
0 & 0 & \xi_2 & -1  &0&\cdots&0 &0 \\
0 & 0 & \epsilon & 0&0&\cdots&0 &0\\
0 & 0 & 0 & 0&\xi_3&\cdots&0 &0\\
\vdots &\vdots &\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0 & 0 &0 &0&0&\cdots&\xi_N&-1\\
0& 0 &0&0&0&\cdots&\epsilon&0
\end{pmatrix}\delta\mathbf{x}(t)\\
+\frac{1}{\epsilon}\begin{pmatrix}
0 & 0 & C & 0 & 0 & \cdots&0 &0 \\
0 & 0 & 0 & 0 & 0 & \cdots&0 & 0\\
0 & 0 & 0 & 0 & C  &\cdots & 0& 0\\
0 & 0 & 0 & 0 & 0  &\cdots & 0 & 0\\
0 & 0 & 0 & 0 & 0  &\cdots & 0 & 0\\
\vdots &\vdots &\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
C & 0 &0 & 0  & 0 & \cdots& 0 & 0\\
0 & 0 & 0 & 0  & 0& \cdots& 0 & 0
\end{pmatrix}\delta\mathbf{x}(t-\tau),
\end{multline}
where $\xi_j=1-a_j^2-C$ for $j=1,\dots,N$. The substitution
\begin{equation}\label{eq3}
\delta \mathbf{x}(t)=\rm e^{\lambda t}\mathbf{q},
\end{equation}
where $\mathbf{q}$ is an eigenvector of the Jacobian matrix, leads to the characteristic equation for the eigenvalue $\lambda$:
\begin{equation}\label{eq4}
\begin{vmatrix}
\xi_1-\epsilon\lambda & -1 & C\rm e^{-\lambda\tau} &0& \cdots& 0&0 \\
\epsilon & -\epsilon\lambda & 0 &0& \cdots & 0 &0\\
0 & 0 & \xi_2-\epsilon\lambda& -1  &\cdots &0 &0\\
0 & 0 & \epsilon & -\epsilon\lambda&\cdots &0&0\\
\vdots &\vdots &\vdots&\vdots&\ddots&\vdots&\vdots\\
C\rm e^{-\lambda\tau} & 0&0&0 &\cdots&\xi_N-\epsilon\lambda&-1\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}=0.
\end{equation}

We can express the $(2N-1)$th row of the obtained matrix as a sum of two
rows rewriting Eq.~\eqref{eq4}  as
\begin{multline}\label{eq5}
\begin{vmatrix}
\xi_1-\epsilon\lambda & -1 & C\rm e^{-\lambda\tau} &0& \cdots& 0 & 0\\
\epsilon & -\epsilon\lambda & 0 &0& \cdots & 0& 0 \\
0 & 0 & \xi_2-\epsilon\lambda& -1  &\cdots &0& 0 \\
0 & 0 & \epsilon & -\epsilon\lambda&\cdots &0& 0\\
\vdots &\vdots &\vdots&\vdots&\ddots&\vdots&\vdots\\
0 & 0&0&0&\cdots &\xi_{N}-\epsilon\lambda&-1\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}\\
+\begin{vmatrix}
\xi_1-\epsilon\lambda & -1 & C\rm e^{-\lambda\tau} &0& \cdots& 0 & 0\\
\epsilon & -\epsilon\lambda & 0 &0& \cdots & 0& 0 \\
0 & 0 & \xi_2-\epsilon\lambda& -1  &\cdots &0 & 0\\
0 & 0 & \epsilon & -\epsilon\lambda&\cdots &0& 0\\
\vdots &\vdots &\vdots&\vdots&\ddots&\vdots&\vdots\\
C\rm e^{-\lambda\tau} & 0&0&0&\cdots &0&0\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}=0.
\end{multline}

Denote by $I_1$ the first matrix in the Eq.~\eqref{eq5} and by $I_2$
the second one, respectively. Firstly, we consider the determinant of
matrix $I_1$. We express the first row of $I_1$ as the sum of the rows
$(\xi_1-\epsilon\lambda,  -1,  0, 0, \ldots, 0, 0)$
and $(0, 0,  C\rm e^{-\lambda\tau},0, \ldots, 0, 0)$. In this way,
$I_1$ is written as the sum of two determinants. The first of these
two determinants is Laplace expanded
starting with its first row. As a result, we can rewrite $I_1$ as
\begin{multline}\label{eq6}
\det I_1 =\epsilon(1-\xi_1\lambda+\epsilon\lambda^2)\\
\times\begin{vmatrix}
\xi_2-\epsilon\lambda & -1 & C\rm e^{-\lambda\tau} &0& \cdots& 0 & 0\\
\epsilon & -\epsilon\lambda & 0 &0& \cdots & 0& 0 \\
0 & 0 & \xi_3-\epsilon\lambda& -1 &\cdots &0& 0 \\
0 & 0 & \epsilon & -\epsilon\lambda&\cdots &0& 0\\
\vdots &\vdots &\vdots&\vdots&\ddots&\vdots&\vdots\\
0 & 0&0&0&\cdots &\xi_{N}-\epsilon\lambda&-1\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}\\
+\begin{vmatrix}
0 & 0 & C\rm e^{-\lambda\tau} &0& \cdots& 0 & 0\\
\epsilon & -\epsilon\lambda & 0 &0& \cdots & 0 & 0\\
0 & 0 & \xi_2-\epsilon\lambda& -1  &\cdots &0 & 0\\
0 & 0 & \epsilon & -\epsilon\lambda&\cdots &0& 0\\
\vdots &\vdots &\vdots&\vdots&\ddots&\vdots&\vdots\\
0 & 0&0&0&\cdots &\xi_{N}-\epsilon\lambda&-1\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}.
\end{multline}
Evidently, the determinant of the second matrix in the sum \eqref{eq6} equals zero. By repeating the same procedure, we derive
\begin{equation}\label{eq7}
\det I_1 = \epsilon^N\prod\limits_{j=1}^{N}(1-\xi_j\lambda+\epsilon\lambda^2).
\end{equation}

Now let us  consider the determinant of matrix $I_2$.  Again, we use a
Laplace expansion; this time starting with the $(2N-1)$th row we get
\begin{multline}\label{eq8}
\det I_2 = C\rm e^{-\lambda\tau}\\
\times\begin{vmatrix}
 -1 & C\rm e^{-\lambda\tau} &0&0& \cdots& 0& 0 \\
 -\epsilon\lambda & 0 &0&0& \cdots & 0& 0 \\
 0 & \xi_2-\epsilon\lambda& -1 &C\rm e^{-\lambda\tau} &\cdots &0 & 0\\
 0 & \epsilon & -\epsilon\lambda&0&\cdots &0& 0\\
\vdots &\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&0&\cdots&0&0\\
0&0&0&0&\cdots&\epsilon&-\epsilon\lambda
\end{vmatrix}.
\end{multline}
This matrix has dimension $(2N-1)\times(2N-1)$ and can be expanded by the last column:
\begin{multline}\label{eq9}
\det I_2 = -\epsilon\lambda C\rm e^{-\lambda\tau}\\
\times\begin{vmatrix}
 -1 & C\rm e^{-\lambda\tau} &0&0& \cdots& 0& 0  \\
 -\epsilon\lambda & 0 &0&0&\cdots & 0& 0  \\
 0 & \xi_2-\epsilon\lambda& -1 &C\rm e^{-\lambda\tau} &\cdots &0& 0  \\
 0 & \epsilon & -\epsilon\lambda&0&\cdots &0& 0 \\
\vdots &\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&0&\cdots& -1 & C\rm e^{-\lambda\tau} \\
0&0&0&0&\cdots&-\epsilon\lambda&0\\
\end{vmatrix}.
\end{multline}

We continue expanding by the last column and finally obtain
\begin{equation}\label{eq10}
\det I_2 =-\epsilon^NC^N\lambda^N\rm e^{-\lambda\tau N}.
\end{equation}
Using Eqs.~\eqref{eq7} and \eqref{eq10} we obtain the characteristic equation for system \eqref{eq2}:
\begin{equation}\label{eq11}
\prod\limits_{j=1}^{N}(1-\xi_j\lambda+\epsilon\lambda^2)-(C\lambda\rm e^{-\lambda\tau})^N=0.
\end{equation}

We can neglect $\epsilon$ since $\epsilon\ll1$ holds, i.e., $\epsilon\approx0$ in the following.  Substituting $\lambda=i \omega$, which holds at the stability boundary given by a Hopf bifurcation, into the Eq.~\eqref{eq11} yields
\begin{equation}\label{eq12}
\prod\limits_{j=1}^{N}(1-\xi_j i\omega)=(C i\omega\rm e^{-i\omega\tau })^N.
\end{equation}

We take the squared absolute value of both sides of Eq.~\eqref{eq12}
leading to
\begin{equation}\label{eq13}
\prod\limits_{j=1}^{N}(1+\xi_j^2\omega^2)=(C \omega)^{2N}.
\end{equation}
Equation~\eqref{eq13} can be expressed as
\begin{equation}\label{eq14}
\left(\prod\limits_{j=1}^{N}\xi_j^2-C^{2N}\right)\omega^{2N}+\mathbf{P}(\omega^2)=0,
\end{equation}
where $\mathbf{P}(\omega^2)$ is the polynomial of $(N-1)$th degree and with positive coefficients. For $\prod_{j=1}^{N}\xi_j^2>C^{2N}$ Eq.~\eqref{eq14} and hence  Eq.~\eqref{eq12} has no solution for real valued $\omega$ and, thus, no Hopf bifurcation will take place. Taking the square root of this inequality and resubstituting $\xi_j=1-a_j^2-C$ yields
\begin{equation}\label{eq15}
\left|\prod\limits_{j=1}^{N}(1-C-a_j^2)\right|>|C|^{N}.
\end{equation}
This inequality defines the values of threshold parameters $a_j$ where
a Hopf bifurcation is impossible, i.e., the equilibrium point is
stable. This result is a generalization of the inequality obtained in
Ref.~\onlinecite{PLO15} for an unidirectional ring topology. For
sufficiently large values of the coupling strength and in the presence
of a large number of excitable nodes, i.e., $a_i>1$, the inequality
\eqref{eq15} is fulfilled, hence the whole network is in an excitable
state. Analytical treatment of this problem for other network
topologies is nontrivial. However, we expect qualitatively similar
results for other network topologies, for example with symmetric
coupling matrices, as will be discussed in
Sec.~\ref{Sec:coupl_strength_analys} and checked by the simulation in
Sec.~\ref{Sec:analys_hier}.

\subsection{Synchronization analysis}\label{Sec:synch_analys}

In this section, we study the conditions under which network~\eqref{m}
synchronizes. We will show that the coupling strength is the crucial
parameter determining the synchronizability of the network. Here, we will focus on the case of $\tau=0$, i.e., a
coupling without delay. This allows for an analytic
treatment of the problem. The simulations in Sec.~\ref{Sec:analys_hier} will show
that the results obtained for $\tau=0$ hold in very good approximation
for $\tau>0$ though the delay might influence the transient behavior.

Consider the FHN network \eqref{m} with heterogeneous threshold
parameters but without delay, i.e., $\tau=0$,
\begin{equation}\label{f1}
\begin{aligned}
\epsilon \dot u_i&=u_i-\frac{u_i^3}{3}-v_i+C\sum\limits_{j=1}^{N}G_{ij}[u_j-u_i],\\
\dot v_i&=u_i+a_i.
\end{aligned}
\end{equation}
We assume that the connectivity graph $\Gamma$ of network \eqref{f1} is connected
and undirected, i.e., its adjacency matrix $\mathbf{G}$ is symmetric
and has no zero rows. By averaging over all nodes we obtain an
averaged trajectory described by
\begin{equation}\label{f2}
\begin{aligned}
\epsilon \dot {\bar u}&=\bar u-\psi(u_1,\dots,u_N)-\bar v,\\
\dot {\bar v}&=\bar u+\bar a,
\end{aligned}
\end{equation}
where $\bar u = \frac{1}{N}\sum_{j=1}^{N}u_j$, $\bar v =
 \frac{1}{N}\sum_{j=1}^{N}v_j$, $\bar a =  \frac{1}{N}\sum_{j=1}^{N}a_j$, and
$\psi(u_1,\dots,u_N) =  \frac{1}{3N}\sum_{j=1}^{N}u_j^3$. It will be shown that
this averaged trajectory  approximates well the synchronized behavior.

In the following, we want to investigate how synchrony spreads in a
network. For this purpose we consider a network where all but one node
(in the following: node $i$)
are in synchrony, and evaluate the conditions under which this node
will synchronize with the other nodes. This approximates the
situation where a population of neurons is affected by pathological
synchronization during epileptic seizure and the question arises under
which condition the synchrony spreads.

To this end, let us introduce a \textit{leader} system:
\begin{equation}\label{f3}
\begin{aligned}
\epsilon \dot u_L&= u_L-\frac{u_L^3}{3}-v_L,\\
\dot v_L&=u_L+\bar a,
\end{aligned}
\end{equation}
which describes a  FHN system without coupling and a threshold
equal to the mean  $\bar a=  \frac{1}{N}\sum_{j=1}^{N}a_j$. The dynamics of the \textit{leader} system
approximates well the dynamics of the synchronized network \eqref{f1}
which can easily be seen by comparing Eq.~\eqref{f3} with  Eq.~\eqref{f2}.

Let us now consider the $i$th node of the network \eqref{f1}, i.e.,
the node which is not yet synchronized with the rest of the  network. Assume that
node $i$ has $n_i$ connections with its neighbors.

Note that in this case the coupling to node $i$ can be approximated by
$\sum_{j=1}^NG_{ij}u_j\sim n_iu_L$. Keeping
this in mind, we subtract the first row of Eq.~\eqref{f3} from the
first one of  Eq.~\eqref{f1}, and the second one from the second one,
respectively.  In addition we make the following substitution
\begin{equation}\label{f4}
\begin{aligned}
\delta_{u}(t)&=u_i(t)-u_L(t)+a_i-\bar a, \\ \delta_{v}(t)&=v_i(t)-v_L(t)+d,
\end{aligned}
\end{equation}
where $d=(1-Cn_i)(a_i-\bar a)$ is a constant, and finally obtain
\begin{equation}\label{f5}
\begin{aligned}
\epsilon \dot\delta_{u}&=(1-Cn_i)\delta_{u}-\delta_{v}-(\delta_{u}-a_i+\bar a)\phi, \\
\dot\delta_{v}&=\delta_{u},
\end{aligned}
\end{equation}
where $\phi(t)=\frac{1}{3}(u_i(t)^2+u_i(t)u_L(t)+u_L(t)^2)$, $\phi(t)\ge0~\forall t$ is a nonnegative function.

We now introduce the following Lyapunov function
\begin{equation}\label{f6}
V(t,\mathbf{\Delta}(t))=\frac{\epsilon\delta_{u}^2(t)}{2}+\frac{\delta_{v}^2(t)}{2},
\end{equation}
where $\mathbf{\Delta}=(\delta_{u},\delta_{v})$, and find its
derivative according to Eq.~\eqref{f5}
\begin{multline}\label{f7}
\dot
V(t,\mathbf{\Delta}(t))=\delta_{u}[(1-Cn_i)\delta_{u}-\delta_{v}-(\delta_{u}-a_i+\bar
a)\phi] \\
+\delta_{v}\delta_{u}=(1-Cn_i)\delta_{u}^2-(\delta_{u}^2-(a_i-\bar a)\delta_{u})\phi.
\end{multline}

The first term in Eq.~\eqref{f7} is negative for $C>1/n_i$. With
$\phi(t)\ge0~\forall t$ we note that the second term in Eq.~\eqref{f7} is
nonpositive for $|\delta_{u}|>2\sigma$, since $|a_i-\mu|<\sigma$ and therefore $|a_i-\bar a|<2\sigma$.

In conclusion, the Lyapunov function derivative is negative and the Lyapunov function \eqref{f6} decreases.
Thus, we obtain the inequality $C>1/n_i$ as a  sufficient condition that the $i$th neuron
synchronizes with the other nodes with a  level of precision
equal to  $2\sigma$, i.e., $|u_i(t)-u_L(t)+a_i-\bar a |<2\sigma$ holds for $t>t_c$, where $t_c$ is the time
transients need to decay.

In the case that more than one node is not synchronized with the rest
of the network, an analytic
treatment is difficult in the presence of heterogeneous threshold
parameters.  However, the previous results in this Section suggest that in this case
the network synchronizes if $C>1/n_{\text{min}}$ where
$n_{\text{min}}$ is the minimal degree of the network, i.e., $n_{\text{min}}\le n_j$,
$j=1, \ldots, N$ and $n_j$ denotes the degree of the $j$th
node. In the synchronized state, the activator and
inhibitor variables fulfill $u_j-u_L \approx a_j-\bar
a$ and $v_j-v_L \approx (1-Cn_j)(a_j-\bar a)$, respectively.

If the inequality $C>1/n_{\text{min}}$ does not hold, the network might
desynchronize in which case control is needed to enforce a synchronized state.  A
control scheme will be discussed in Sec.~\ref{Sec:control}.

\subsection{Influence of  the coupling strength
  on the type of synchronization}\label{Sec:coupl_strength_analys}

In the last subsection, we have discussed the conditions under which the
network synchronizes  but have not specified in which state -- oscillatory
or excitatory -- the synchronization takes place.
Here, we investigate the influence of the coupling strength on the
type of synchronized dynamics.

 To this end  let us consider the $i$th neuron in network \eqref{m} and
 rewrite it as
\begin{equation}\label{f11}
\begin{aligned}
\epsilon \dot u_i&=u_i-\frac{u_i^3}{3}-v_i,\\
\dot v_i&=u_i+a_i-C\sum\limits_{j=1}^{N}G_{ij}[\dot u_j(t-\tau)-\dot u_i(t)],
\end{aligned}
\end{equation}
where we use the following substitution  $v_i \rightarrow v_i-C\sum_{j=1}^{N}G_{ij}[u_j(t-\tau)- u_i(t)]$.

The behavior of the i$th$ node without coupling depends on the
threshold parameter $a_i$. From Eq.~\eqref{f11} it follows that the
coupling effectively acts as a shift of the threshold. If the $j$th
node is excitable, then the derivative of its activator equals
zero. Therefore, only nodes in the oscillatory state influence the
effective threshold $a_i-C\sum\limits_{j=1}^{N}G_{ij}[\dot u_j(t-\tau)-\dot u_i(t)]$.

Suppose that $a_i>1$, i.e., the i$th$ node is excitable. If the
coupling strength $C$ is too small, then the i$th$ node remains in
the excitable regime. Increasing the coupling strength forces the  $i$th node to exhibit self-sustained
periodic firing. If we now further increase the coupling strength,
the absolute value of the term $|a_i-C\sum_{j=1}^{N}G_{ij}[\dot u_j(t-\tau)-\dot u_i(t)]|$ becomes larger
than $1$ for large time intervals meaning that the $i$th node stops to oscillate.
Thus, if the coupling is too high, the whole system is
excitable. To summarize the results of this section, if we want to synchronize the FHN network in the
oscillatory regime, we should choose the coupling sufficiently large
for synchronization, however, not so huge that we leave the
oscillatory regime and synchronize the network in the equilibrium point.

\section{Analysis of hierarchical network topology}\label{Sec:analys_hier}

In this section, we consider a network with specific hierarchical
architecture and study numerically how the network synchronization depends on different
system parameters as the coupling strength $C$, the delay $\tau$,
the variance of threshold parameters $\sigma^2$, and the topology. For the
simulations we use the normally distributed threshold parameters $a_i$
with mean $\mu=1$ and variance $\sigma^2>0$ meaning that
oscillatory and excitable nodes coexist. We restrict
ourselves to  threshold parameters $a_i$ from the
interval $|a_i-\mu|<\sigma$. The study of
different hierarchical architectures in the neuron connectivity is
motivated by MRI results of the brain structure which show that the
neuron axons networks spans the brain area fractally and not
homogeneously \cite{KAT09,EXP11,KAT12, OME15}. In the rest of this
paper the word ``fractal'' will be employed to denote
hierarchical structures of finite order $n$, since the human brain
has finite size and does not cover all orders. This is in contrast to
the exact definition of a fractal set where  $n\to\infty$.

Simple hierarchical structures can be constructed using the classic
Cantor fractal construction process on a ring network \cite{MAN83, FED88}. Using the
iterative bottom-up procedure to construct the Cantor set, we create a
symbolic sequence consisting of $0$ and
$1$
hierarchically nested . Starting with a base pattern $B$ containing $b$ symbols ($0$ or
$1$) we iterate it $n$ times and obtain a system of size
$N=b^n$. During each iteration step the symbols $1$ and $0$ are
replaced by the base pattern $B$ and a series of $b$ zeros,
respectively. Thus, we get the string $G_1$ of size $N=b^n$. The
resulting string $G_1$ defines the connections of all neurons to
the first node.  We now add a $0$ before the first symbol of $G_1$, i.e., the
first node has no connection with itself. In this extended string, a
$1$ at position $i$, $i=2,\ldots,  b^n+1$, represents a link between  the  $i$th  node
and the first one, while a $0$ means that these two nodes do not
couple. To construct the coupling matrix $\mathbf{G}$, we regard
$G_1$ as the first row in the matrix $\mathbf{G}$. Each of the
following rows are obtained by shifting the preceding
row cyclically by one element resulting in a $(b^n+1)\times (b^n+1)$ circulant matrix, i.e.,
$G_{ij}=G_{1,(j-i+N)\bmod N}$, $i,j=1,\ldots,  b^n+1$.
The matrix constructed in this way contains a hierarchical
distribution of gaps with a variety of sizes. The following equation
describes the first three rows of a coupling matrix with hierarchical topology with the base $B=[1~0~1]$, $b=3$, $n=2$:
\begin{equation}\label{f12}
\mathbf{G} = \begin{pmatrix}
0 & 1 & 0 &1 &0 & 0 &0&1&0&1 \\
1&0 & 1 & 0 &1 &0 & 0 &0&1&0\\
0&1&0 & 1 & 0 &1 &0 & 0 &0&1\\
\cdots &\cdots &\cdots & \cdots &\cdots&\cdots&\cdots&\cdots&\cdots&\cdots
\end{pmatrix}.
\end{equation}

The number of times the symbol $1$ appears in the base, denoted by
$c_1$, defines formally the fractal dimension $d_f$ of the structure,
as $d_f=\ln c_1/\ln b$. This measure $d_f$ describes perfectly the
fractal structure when the number of iterations $n\to\infty$. Note that for symmetric base patterns the adjacency matrices of networks
constructed according to this procedure are always symmetric and, thus, the results
obtained in Sec.~\ref{Sec:FHN_analys} hold for $\tau=0$ and in good
approximation for $\tau>0$.

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig1.jpg}}
\end{minipage}
\caption{Dependence of the activator synchronization index $\Delta_u$
  on the coupling strength $C$ for the FitzHugh-Nagumo network given
  by Eq.~\eqref{m} with Cantor-type hierarchical topology. The black dashed line indicates the synchrony condition $\Delta_u<2 \sigma$. Parameters:
  $B=[1~0~1]$, $b=3$, $n=4$, $N=82$, $\tau=1.5$, $\epsilon=0.01$. The
  threshold parameters $a_i$ are normally distributed with mean $\mu=1$ and standard deviation $\sigma=0.1$, truncated at $a_i=\mu\pm\sigma$. Initial conditions: $u_i(t)=0$, $v_i(t)=0$, $i=1,\dots,N$,  for $t\in[-\tau,0]$.}
\label{figC}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig6.jpg}}
\end{minipage}
\caption{Dependence of the activator oscillation amplitude $A_u$ of the first node on the coupling strength $C$ for the FitzHugh-Nagumo network given by Eq.~\eqref{m} with Cantor-type hierarchical topology. Parameters and initial conditions as in Fig.~\ref{figC}.}
\label{figAmp}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig2.jpg}}
\end{minipage}
\caption{Dependence of the activator synchronization index $\Delta_u$ on the standard deviation $\sigma$ of threshold parameters $a_i$ for the FitzHugh-Nagumo network given by Eq.~\eqref{m} with Cantor-type hierarchical topology. The black dashed line indicates the synchrony condition $\Delta_u<2 \sigma$. Parameters: $C=0.065$. Other parameters and initial conditions as in Fig.~\ref{figC}.}
\label{figSig}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig3.jpg}}
\end{minipage}
\caption{Dependence of the activator synchronization index $\Delta_u$ on the coupling strength $C$ and the standard deviation $\sigma$ of threshold parameters $a_i$ for the FitzHugh-Nagumo network given by Eq.~\eqref{m} with Cantor-type hierarchical topology. The white dashed curve  indicates the synchrony boundary $\Delta_u=2 \sigma$. To the right of this curve synchronization holds. Parameters and initial conditions as in Fig.~\ref{figC}.}
\label{figCSig}
\end{figure}

The simulations were carried out in the {\it Matlab} environment using the function $\it dde23$.

In the current study we use as base pattern $B=[1~0~1]$, i.e.,
$b=3$.  We perform $n=4$ iterations yielding a network of size $N=b^4+1=82$ nodes. For this base the value of $c_1$
(number of times the symbol $1$ is encountered within the base $B$)
equals $2$, i.e., the connectivity matrix has a fractal dimension of
$d_f=\ln2/\ln3=0.6309$. After $n=4$ iterations, each node is connected
to  $n_i=c_1^n=2^4=16$ other nodes, while no connection to the
remaining $b^n+1-n_i=66$ elements exists. From the results obtained in
Sec.~\ref{Sec:synch_analys}, we expect the network to synchronize at $C>1/n_i=0.0625$.

We now consider a network with a delay $\tau=1.5$ and a standard
deviation of $\sigma=0.1$ for the threshold parameters $a_i$. We change
the coupling strength $C$ from $0$ to $0.1$ and check whether the network
synchronizes. To characterize the degree of synchrony, we introduce
the activator synchronization index
$\Delta_u=\max_{i=1}^{N}|u_i(t^*)-\bar u(t^*)+a_i-\bar a|$, where we define synchrony by the condition $\Delta_u<2\sigma$. Here we
use $t^*=30\ge t_c$, where $t_c$ is the characteristic transient time, to analyze the network after the decay of all
transients. The results of the simulation are presented in
Fig.~\ref{figC}, where the synchronization index vs. the coupling
strength is shown. One can see that there is a critical coupling strength $C^*\approx0.045$,
such that for $C>C^*$ synchronization holds with some level of
precision. To distinguish whether the network is in an oscillatory or excitable
state we  introduce the activator oscillation amplitude of the first
node $A_u$ as the difference between the maximum and minimum values of
the activator after the transient. Figure~\ref{figAmp} shows the
dependence of the activator oscillation amplitude of the first node
$A_u$ on the coupling strength $C$ showing that for $C>0.06$ the
$1$st node is in the excitable state. Thus, by combining the results of
the simulations shown in Figs.~\ref{figC} and~\ref{figAmp} we conclude
that for $0.045<C<0.06$ we obtain synchronization in an oscillatory state.
The results of the simulations confirm the idea discussed in Sec.~\ref{Sec:coupl_strength_analys}.

Now we fix the delay $\tau=1.5$ and the coupling strength $C=0.065$ and
change the standard deviation $\sigma$ from $0$ to $0.5$. The results
of the simulation are shown in Fig.~\ref{figSig}. As anticipated, the synchronization error increases
as $\sigma$ is enlarged. For all $\sigma$ the activator synchronization
index is below $2\sigma$ marked by a black dashed line. This is the level of precision of
synchronization predicted in Sec.~\ref{Sec:synch_analys}. Note that for
$\sigma<0.1$ the activator synchronization index $\Delta_u$ is close
to zero. To raise the level of precision for $\sigma>0.1$, we can increase the coupling strength $C$.

Figure~\ref{figCSig} shows the value of the activator synchronization
index $\Delta_u$ in the $C$-$\sigma$ plane. The black area corresponds
to parameter values where the network synchronizes. The value
$C>1/n_\text{min}=0.0625$ approximates well the threshold between
synchronization and desynchronization as predicted in
Sec.~\ref{Sec:synch_analys}. For increasing $\sigma$, $\Delta_u$
increases showing that synchronization with less precision takes place.

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig4.jpg}}
\end{minipage}
\caption{Dynamics of 82 FitzHugh-Nagumo
  systems given by Eq.~\eqref{m} with Cantor-type hierarchical topology. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters:  $C=0.065$, $\tau=6$.  Other parameters and initial conditions as in Fig.~\ref{figC}.}
\label{figTau}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig5.jpg}}
\end{minipage}
\caption{Dynamics of 1297 FitzHugh-Nagumo
  systems given by Eq.~\eqref{m} with Cantor-type hierarchical topology. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters:
  $B=[1~0~1~0~0~0]$, $b=6$, $n=4$, $N=1297$, $C=0.04$. Other parameters and
  initial conditions as in Fig.~\ref{figC}.}
\label{figTop}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{fig7.jpg}}
\end{minipage}
\caption{Dependence of the activator synchronization index $\Delta_u$
  on the number of nodes $N$ for the FitzHugh-Nagumo network given by
  Eq.~\eqref{m} with bidirectional ring topology and local coupling, Eq.~\eqref{adj}. Parameters: $\tau=1.5$, $\epsilon=0.01$, $C=1$. The
  threshold parameters $a_i$ are normally distributed with mean $\mu=1$ and standard deviation $\sigma=0.1$. Initial conditions: $u_i(t)=0$, $v_i(t)=0$, $i=1,\dots,N$,  for $t\in[-\tau,0]$.}
\label{figN}
\end{figure}

We now study the influence of the delay $\tau$ on the dynamics of
the network. To this end, we fix the coupling strength at $C=0.065$
and  the standard deviation at $\sigma=0.1$. For these parameters we
always obtain synchronization, however, the delay affects the transient time.  For example, for $\tau=6$ the transient
time takes approximately $110$ units of time (see
Fig.~\ref{figTau}). Figure~\ref{figTau}a shows that during the
transient time, the network already synchronizes but with a lower
frequency than the final one.
Note that many authors have emphasized the
importance of the transient phenomena that arise in neural networks
with delay \cite{GRO99,MIL10,MIL12,PAK98,QUA11}. However, for the
chosen parameters we have not observed these phenomena

Next we consider how synchronization depends on the base pattern $B$. A systematic study of the influence of different base patterns has been performed for networks of identical Van der Pol oscillators in the context of chimera states \cite{ULO16}. In agreement
with the synchronization condition  $C>1/n_\text{min}$ (cf. Sec.~\ref{Sec:synch_analys}), the numerical simulations show that synchronization
depends only on the number of elements equal to $1$ in each row
of the connectivity matrix $\mathbf{G}$. We carry out simulations with
different bases of $b=6$ elements with $n=4$ iteration and $c_1=2$
nonzero entries, and obtain synchronization in all cases for
$C=0.065$. As an example we show in Fig.~\ref{figTop} the results of simulations
of the  FHN network with $C=0.04$ and the base pattern $B=[1~0~1~0~0~0]$. Note that
for $C=0.065$ and base pattern $B=[1~0~1~0~0~0]$ there is also synchronization
(not shown here), however, the network is in the excitable regime, because
the coupling strength is too high for the  network to be in the oscillatory regime as discussed in Sec.~\ref{Sec:coupl_strength_analys}.

Finally, we investigate how the size of the network influences its
synchronizability. Since in the case of hierarchical
topologies the number of nodes $N$ is given by $b^n+1$ where $b$ is the length
of the base and $n$ the number of iterations, the hierarchical network
does not allow for increasing $N$ in equally-sized small
steps. Therefore, we choose here a bidirectionally locally coupled ring  given
by the following adjacency matrix
\begin{equation}\label{adj}
\mathbf{G} = \begin{pmatrix}
0 & 1 & 0 &\cdots & 1 \\
1 & 0 & 1 &\cdots & 0 \\
\vdots & \vdots & \vdots &\ddots & \vdots \\
0 & 0 & 0 &\cdots & 1 \\
1 & 0 & 0 &\cdots & 0
\end{pmatrix}.
\end{equation}
Thus, each node has only two (bidirectional) connections for any size of the system.
We fix the coupling strength at $C=1$ and  the standard deviation at
$\sigma=0.1$. For $C=1>1/n_\text{min}=0.5$ we expect synchronization (cf. Sec.~\ref{Sec:synch_analys}).
The results of the simulation are presented in
Fig.~\ref{figN}, where the synchronization index vs. the number of
nodes is shown. One can see that $\Delta_u<2\sigma$, $\forall N$. This means that the network
is synchronized with precision $2\sigma$ for all $N$. We conclude that
synchronization does not depend on the network size but only on the
coupling strength and the number of connections to each node of the
network. Note, however, that the quality of synchronization, i.e., the
value of $\Delta_u$ slightly depends on the network size.

\section{Control of synchronization in FitzHugh-Nagumo network}\label{Sec:control}
In this section we study how to control synchronization
in the case where the synchronization condition $C>1/n_\text{min}$  is not
met. For this purpose a  mean field control, i.e., adding the term
$I=\frac{1}{N}\sum_{i=1}^{N}u_i$ to each node, has been considered, for
example in Refs.~\onlinecite{ROS04,ROS04a}.
However, here we suggest a different approach. We will use
the control $I(t)$ in  the form
\begin{equation}\label{f8}
I(t)=\gamma u(t),
\end{equation}
where $\gamma$ is a control gain and $u$ is an activator value of the \textit{master} system, which is described by the following equations
\begin{equation}\label{f9}
\begin{aligned}
\epsilon \dot u&=u-\frac{u^3}{3}-v,\\
\dot v&=u+a \\
\end{aligned}
\end{equation}
where $v$ is an inhibitor value and $a$ is a desired threshold.
Compared to a mean field control this approach has the advantage that
it is not necessary to measure and average some output variable.
Furthermore, even if all oscillators are in the excitable regime the
control ensures synchronization in an  oscillatory state if we choose
$|a|<1$ in Eq.~\eqref{f9}.

In the following, we will use $\gamma=0.3$ and $a=0.9$, i.e.,the
\textit{master} system is in the oscillatory regime.
We will demonstrate the control in the form Eq.~\eqref{f8} on different network topologies.

\subsection{Bidirectional ring topology}\label{Sec:ring_net_control}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic1a.jpg}}
\end{minipage}
\caption{Dynamics of 500 FitzHugh-Nagumo
  systems given by Eq.~\eqref{m} with a locally coupled ring topology Eq.~\eqref{adj} without control, i.e., $I(t)=0$. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters: $N=500$,
  $C=0.1$, $\tau=1.5$, $\epsilon=0.01$. The threshold parameters $a_i$
  are normally distributed with mean $\mu=1$ and standard deviation $\sigma=0.1$, truncated at $a_i=\mu\pm\sigma$. Initial conditions: $u_i(t)=0$, $v_i(t)=0$, $i=1,\dots,N$,  for $t\in[-\tau,0]$.}
\label{fig2}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic1b.jpg}}
\end{minipage}
\caption{Control of synchronization in 500 FitzHugh-Nagumo
  systems with a locally coupled ring topology Eq.~\eqref{adj} by the external stimulus
  $I(t)$ according to Eqs.~\eqref{f8} and \eqref{f9}. (a) and (b): time series of the activator and the inhibitor of all nodes, respectively. Parameters: $\gamma=0.3$, $a=0.9$. Other parameters and initial conditions as in Fig. \ref{fig2}.}
\label{fig3}
\end{figure}

We start our consideration with a one-dimensional ring of $N$
delay-coupled FHN oscillators, where each element is coupled to its
neighbors on both sides, i.e., the system \eqref{m} with the connectivity matrix
$\mathbf{G}$ given by Eq.~~\eqref{adj}.

For the simulation we consider the case of $N=500$ nodes. We use
a coupling strength $C$ equal to $0.1$. Without control the network
does not synchronize as can be seen in  Fig.~\ref{fig2}, where the
simulation of a network without control, i.e., $I(t)=0$, is shown.
Clearly, there is no synchronization between the activator and inhibitor values.

To synchronize this network we use the control in the form of
Eqs.~\eqref{f8} and \eqref{f9}. The result is shown in
Figure~\ref{fig3}:  The
 control goal is achieved and synchronization holds for the activators (Fig.~\ref{fig3}a) and the inhibitors (see Fig.~\ref{fig3}b).

\subsection{Hierarchical network topology}\label{Sec:hier_net_control}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic2a.jpg}}
\end{minipage}
\caption{Dynamics of 730 FitzHugh-Nagumo
  systems with Cantor-type hierarchical topology without control, i.e., $I(t)=0$. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters: $n=6$, $N=730$, $C=0.001$. Other parameters and initial conditions as in Fig.~\ref{figC}.}
\label{fig4}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic2b.jpg}}
\end{minipage}
\caption{Control of synchronization in 730 FitzHugh-Nagumo
  systems with Cantor-type hierarchical topology by external stimulus $I(t)$
  according to Eqs.~\eqref{f8} and \eqref{f9}. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters: $\gamma=0.3$, $a=0.9$. Other parameters and initial conditions as in Fig. \ref{fig4}.}
\label{fig5}
\end{figure}

Next we investigate a hierarchical topology with base pattern $B=[1~0~1]$ and
iteration number $n=6$, i.e., the coupling matrix $\mathbf{G}$ has the
dimension $N\times N$ with $N=3^6+1=730$ (for the construction of a
hierarchical network
see Sec.~\ref{Sec:analys_hier}). We use a very weak coupling strength
$C$ equal to $0.001$. Figure~\ref{fig4} shows the results of the
simulation of network behavior without control, i.e., $I(t)=0$. Clearly,
 no synchronization between the nodes takes place.

To synchronize this network we use again the control $I(t)$ given by
Eqs.~\eqref{f8} and \eqref{f9}.  Figure~\ref{fig5} presents the results of a
 simulation of the FHN network \eqref{m} with hierarchical topology
 and the external stimulus $I(t)$. The
 control goal is achieved and synchronization holds (see the time
 series of the activators and inhibitors in Fig.~\ref{fig5}(a), and (b), respectively).

\subsection{Random network topology}\label{Sec:rand_net_control}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic3a.jpg}}
\end{minipage}
\caption{Dynamics of 500 FitzHugh-Nagumo
  systems with a random topology without control, i.e., $I(t)=0$. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters: $N=500$. The nonzero entries are drawn from a Gaussian distribution with mean
$\mu_C=1$ and variance $\sigma_C^2=1$, truncated at $a_i=\mu\pm\sigma$. Parameters: $C=0.0001$. Other parameters and initial conditions as in Fig. \ref{fig2}.}
\label{fig6}
\end{figure}

\begin{figure}
\begin{minipage}[h]{1\linewidth}
\center{\includegraphics[width=1\linewidth]{pic3b.jpg}}
\end{minipage}
\caption{Control of synchronization in 500 FitzHugh-Nagumo
  systems with a random topology by external stimulus $I(t)$ according
  to Eqs.~\eqref{f8} and \eqref{f9}. (a) and (b): time series of the activator
  and the inhibitor of all nodes, respectively. Parameters: $\gamma=0.3$, $a=0.9$. Other parameters and initial conditions as in Fig. \ref{fig6}}
\label{fig7}
\end{figure}

Last we consider a network of $N=500$ nodes with random topology. We
use a sparse matrix with link density equal to $0.3$. This means there are
approximately $0.3N^2$ normally distributed nonzero entries.  These
nonzero entries are drawn from a Gaussian distribution with mean
$\mu_C=1$ and variance $\sigma_C^2=1$, truncated at $a_i=\mu\pm\sigma$. This process results in a
weighted random network. Note that
in this case the average node degree is much higher compared to the
previously studied topologies. Thus, only for very weak
coupling strengths we anticipate  non-synchronous dynamics (see
Sec.~\ref{Sec:synch_analys}), in which case the need of control
arises.

In fact, for $C=0.0001$ no synchronization takes place as can be seen
in Fig.~\eqref{fig6}. Applying the control in the form of
Eqs.~\eqref{f8} and \eqref{f9} synchronizes the network as  shown for
 the activators and inhibitors in Figs.~\ref{fig7}(a) and (b), respectively.

\section{Conclusion}\label{Sec:concl}
We have considered synchronization and its control in networks of heterogeneous FitzHugh-Nagumo systems, a
neural model which is considered to be generic for excitable
systems close to a Hopf bifurcation. To this end, we have investigated
networks with heterogeneous threshold parameters.
It is well known that networks with
heterogeneous nodes are much less likely to synchronize than networks of
identical nodes. Furthermore, synchrony will take place in a state
where the trajectories of the different nodes are not identical but
small deviations can be observed. To counteract this effect of
heterogeneity we have proposed an algorithm for controlling synchrony.

First,  we have generalized the analytic conditions for the
occurrence of the Hopf bifurcations obtained in
Refs.~\onlinecite{SCH08,PLO15} for unidirectional locally coupled ring networks. In this way, we have been able to determine the threshold values for which
the system is in the excitable regime.
Afterwards, we have considered the case that all but one node are in
synchrony and derived a critical coupling strength depending on the minimum node degree of the network.
Above the critical coupling strength the node will synchronize with
the rest of the network. However, if the coupling strength becomes too
large, synchronization still holds but amplitude death sets in. Thus,
synchronization in an oscillatory state is only possible for
intermediate coupling strengths. We have studied synchronization in
networks with Cantor-type hierarchical topologies. The study of different
hierarchical architectures in the neuron connectivity is motivated by
MRI results of the brain structure which shows that the neuron axon network spans the brain area fractally and not homogeneously. The results of these numerical investigations match with our analytical synchronization condition.

Based on the result that the network does not synchronize for too small
coupling strengths, we have suggested to add the same
external stimulus to all nodes to ensure synchronization in networks
where it is absent without control. We have considered the activator value of a master neuron as an external stimulus.
We have then applied the control algorithm to different types of networks
(bidirectional ring networks, hierarchical networks, random
networks), and the simulation results have shown synchronization. Since we use the activator value of the neuron in the oscillatory
state, we suppose that it is possible to consider other periodic
functions with appropriate frequency as a control to ensure the
synchronization, and we have confirmed this in simulations (not
shown in the paper). Furthermore, we suggest that our control algorithm
and obtianed synchronization conditions can also help with study of desynchronization of the network in cases where this is desirable, for
example in the case of abnormal synchrony in the brain as in epilepsy
or Parkinson disease.
 Given the paradigmatic nature of the FitzHugh-Nagumo system, we expect our investigations to be applicable in a wide range of excitable systems, for instance, to the other models for neural spiking \cite{WIL99,IZH03,NAU08}.

\begin{acknowledgements}

This work is supported by the German-Russian Interdisciplinary
Science Center (G-RISC) funded by the German Federal Foreign Office
via the German Academic Exchange Service (DAAD). J.L. and E.S.
acknowledge support by Deutsche Forschungsgemeinschaft (DFG) in the
framework of SFB 910. S.P. and A.F. acknowledge support by SPbSU (Grant No. 6.38.230.2015) and by the Goverment of Russian Federation (Grant  No. 074-U01). Section~\ref{Sec:FHN_analys} was performed in IPME RAS, supported solely by RSF (Grant No. 14-29-00142).
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


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Ultimately, you want to look at hiring tree trimming companies to get the expertise that they have to offer. At Metro Tree, we have a certified arborist who can help you with tree selection, identify tree problems, and much more. We will take care of the trees on your property so you don't have to be bothered with it. We bring professional landscapers and state of the art equipment over so all you have to do is call and schedule the service.


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Organisations & People
Search State Records Office of Western Australia
SeriesS4864 - FARM RECORDS - CLAREMONT MENTAL HOSPITAL
File01 - Royal Agricultural Society of WA - Royal Show prize certificates (15)
File02 - Department of Agriculture Stock Branch - 26 horse delivery consignment notes.
File03 - Australian Illawarra Shorthorn Society - two calf entry certificates
File04 - Invoices & other documents re hiring of horses from the Department of Agriculture to Claremont Asylum.
File05 - Miscellaneous documents re cows at the Claremont Hospital Dairy Farm.
Series S4864 - FARM RECORDS - CLAREMONT MENTAL HOSPITAL
FARM RECORDS - CLAREMONT MENTAL HOSPITAL
1903-08-01-1972-12-31 (Creation)
Various - see descriptive note
AU WA A543 - CLAREMONT MENTAL HOSPITAL (1903-08-01 - 1972)
In 1901, a Parliamentary Select Committee was appointed to inquire into the mental health system in Western Australia. The Committee found that Fremantle Asylum was inadequate in many respects and recommended it be closed. As a result, a new mental health facility in Claremont - Claremont Hospital for the Insane - was established. In 1903, Government Reserve H8636 (394 acres) was set aside for this purpose.
By August 1903, temporary buildings had been established on the site and some initial patients transferred. Female patients remained at the Fremantle Asylum until 1908, after which they were transferred to Claremont.
The Claremont facility was Western Australia's principal mental hospital until its closure in 1972.
It housed the majority of the State's mental patients, both male and female adults (in separate wings) and children. Admission was by certification under the Lunacy Act 1903 or the Inebriates Act 1912, although 'voluntary boarders' were also admitted for limited periods of time. In the absence of community facilities, most patients were long-term.
Patients were admitted for a wide range of physical as well as mental disorders, including developmental disability, old age, alcoholism and serious infectious diseases that caused delirium. War veterans with psychiatric injuries were also housed in Claremont until separate facilities were built, including Stromness (1918-1926) and Lemnos Hospitals (1926-1995).
The hospital operated a small farm within its premises and registers of herds and farm produce are included in the records held by the State Records Office.
In 1933 the Hospital's name was changed to Claremont Mental Hospital, and then in 1967 to Claremont Hospital. In 1972, the Hospital was closed and divided into two separate campuses: Swanbourne Hospital for psychogeriatric patients and adults with developmental disabilities, and Graylands Hospital for acute psychiatric patients.
AU WA A1223 - WHITBY FALLS HOSPITAL (1897-01-01 - 1972-01-01)
A one thousand acre property for Whitby Falls Hospital was purchased from a William Paterson by the State government in 1897 for the sum of 7, 000 pounds. Fremantle Asylum by this time had become over-crowded and it was decided to move some psychiatric patients to Whitby Falls. The farmhouse on the property was renovated to accomodate 50 patients and the first 12 patients were placed there on 12 July 1897.
Despite a proclamation on 12 September 1900 declaring Whitby Falls an asylum, it was eventually deemed too remote a location and a new asylum was built at Claremont instead.
Whitby Falls Hospital became an auxiliary to Claremont Hospital and a cooperative dairy farm and cattle breeding program was implemented by the two hospitals.
In 1914, Whitby Falls was gazetted a home for male alcoholics under the Inebriates Act 1912. The Hospital reverted again to take psychiatric patients from Claremont in 1918.
Due to the small number of patients at Whitby Falls Hospital, the Under Secretary declared in 1932 that "as an institution Whitby hardly justified it's existence (financially)" and it was declared "first and foremost a farm". Whitby's agricultural program was used as occupational therapy and rehabilitation for patients placed there but it also became a profitable operation that included poultry, dairy, vegetable, orchard, piggery and beef production.
In 1958, the government built a new hospital on the grounds of Whitby Falls. The original farmhouse was left unused until demolition in 1971.
In 1972, Claremont Hospital was divided into Graylands Hospital and Swanbourne Hospital. Whitby Falls became an annexe to Graylands and was de-gazetted from being an approved hospital to a hostel. The Armadale Health Service supplied psychiatric and allied health services to the hostel from the 1980's.
(Administrative history reference: Mental Hospitals Department, "Whitby Falls Mental Hospital, Control and Development of", file 23/1964, State Records Office of WA, consignment 3614).
Records of the dairy farm operated at the Claremont Mental Hospital and Whitby Falls Hospital including information on herds, production, and sales.
Confirm Item Request: FARM RECORDS - CLAREMONT MENTAL HOSPITAL
AU WA A543 - CLAREMONT MENTAL HOSPITAL (Creator)
AU WA A1223 - WHITBY FALLS HOSPITAL (Creator)

redpajama_set_name: RedPajamaCommonCrawl


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Article Highlights
Real Time With... IPC APEX EXPO: ICAPE Group Gears up For Show Time
01/13/2022 | Nolan Johnson, PCB007
CES 2022: Half Virtual, Still Valuable, and Here's Why
01/13/2022 | Dan Feinberg, I-Connect007
Happy Holden's Five Must-Reads of 2021
01/10/2022 | Michelle Te, I-Connect007
New Book From Isola Highlights Importance of Material Selection
In "The Printed Circuit Designer's Guide to... High Performance Materials," the latest release from I-007eBooks, readers will learn how to overcome challenges associated with choosing the right material for their specific application. Author Michael Gay of Isola provides a clearer picture of what to know when determining which material is the most desirable for which products. PCB materials and DFM expert Mark Thompson says, "I love this book, particularly the sections on the effects of the glass weave, the history of laminate, and the difference between Dk and effective Dk."
We're driving headlong into 2022, and there's a lot going on in the world of PCB design, fabrication, and assembly. It's trade show season now. IPC APEX EXPO starts in less than two weeks, and we're all getting ready for what is, for many, the first trade show in years. Do you know where you put your business cards after your last show in 2020? I keep half a box of biz cards in my glove compartment; I'm not going to make that mistake again.
ICAPE Group's Marc L'Hoste speaks with Editor Nolan Johnson in this pre-show interview. Marc is the vice president of Western Americas, and he outlines what visitors can expect to learn about in the ICAPE booth, including manufacturing facilities recently added to the portfolio, new technologies, and a new corporate branding. If you can't make it to IPC APEX EXPO, don't worry. We'll be bringing you interviews with the engineers, managers and technologists who are making a difference in our industry.
Living in a Material World: High-Speed Design Strategies
Any discussion about high-speed PCB design techniques would be incomplete without considering the properties and requirements of the materials. Your material selection drives much of your design strategy when you're operating at 28 gigabits per second or faster. We recently spoke with high-speed design expert Lee Ritchey of Speeding Edge, and electronic materials veteran Tarun Amla of Avishtech and Thintronics, about the relationship between advanced PCB materials and high-speed design techniques. They discuss the challenges facing designers and engineers working with materials at speeds that were considered unreachable not long ago, and what designers need to know about material selection as board speeds continue rising toward the stratosphere.
Another CES has wrapped up, and while not fully back to its glittery self, the show still managed to create quite a buzz in the electronics world. Attendance was about half of its typical 150,000 as hundreds of companies still touted their wares and I'm excited to share with you what I discovered. I reported on the show from the safety of my home office, which was a bit of a letdown because I really enjoy walking the aisles of the show, visiting the exhibits of both large and small companies.
Sunstone Growing With Supply Chain Strategies
Barry Matties and Nolan Johnson talk to Sunstone Circuits' Matt Stevenson, Kelly Atay, and Dawn DelCastillo about the current supply chain challenges and how they are adapting. They also share insights on market conditions, including trends in reshoring and marketing strategies related to reshoring.
Real Time With… IPC APEX EXPO 2022: Blackfox Mixes It Up with Training Opportunities
Andy Shaughnessy speaks with Jamie Noland, master IPC trainer and marketing manager for Blackfox Institute, about current and upcoming training opportunities and what you can expect from them at the show. Blackfox has expanded its offerings to reach a wider audience, including some very basic classes for beginners.
Over the past year, I-Connect007 has published more than 4,000 news items, nearly 300 articles, and 350 columns from the leading expert voices in the industry. We also have interviewed dozens of fabricators, assemblers, and designers on the most relevant and pressing topics in the ever-evolving printed circuit board industry. So, it only made sense to ask Happy Holden about his picks for the best reads at I-Connect007 in 2021.
CES 2022 Has Begun, But Not as Expected
This annual consumer electronics show in Las Vegas, which went virtual last year, still touts more than 2,200 exhibitors, but some of the biggest companies have modified their plans, and even the show itself is shorter. Many major exhibitors like Microsoft, Google, AMD, and Intel have either canceled or modified their in-person CES plans. Preshow events like CES Unveiled and the CES Media Days were still planned for Jan. 3 and 4, and Showstoppers will proceed. The change in the show's length follows two weeks of companies announcing their pivot to virtual-only plans for CES 2022.
A Tour of the New Isola Factory: Investing in North America
Barry Matties recently toured Isola's new 118,000-square-foot low-volume, high-mix manufacturing facility in Chandler, Arizona. The facility's construction began in 2020 and, like many building projects, was delayed because of the COVID-19 pandemic. Building a new factory during the pandemic was certainly a challenge but the team persevered and the new greenfield facility is now operating to meet the needs of the North American market. During the tour, Barry met with Travis Kelly, Ed Kelley, Sean Mirshafiei, Jenny Inocencio, and Walt Niziolek. They covered a range of topics, including lab operations, hiring, the supply chain, R&D, and the concept of the new facility. The factory was designed to extensively utilize automation to improve efficiencies and, as Travis Kelly stated, "to de-risk the labor shortage issue."
Catching Up with Allen Keeney of Johns Hopkins University
I have always been fascinated by research labs, especially those tied to major universities. These are the true leaders of innovation and invention and at the very top of the PCB industry. So, when I met Allen Keeney, chief engineer of the Advanced Electrical Fabrication Group at the Johns Hopkins Applied Physics Laboratory (APL) in Laurel, Maryland, I jumped at the chance to talk with him. You will enjoy this look at another facet of our PCB industry.
Rogers: Mergers, Automotive, and a Bright Future
Pete Starkey visits with Ingmar van der Linden, market segment manager at Rogers Corporation in the Netherlands, about the products his company was demonstrating during productronica. Pete asks Ingmar about the automotive industry, particularly Rogers' involvement in producing materials for sensors that create a safer driving experience. They also discuss the merger with DuPont, the trend toward material technologies and how to enable them.
Alun Morgan's productronica Slide Show
I-Connect007 has been bringing you interviews with a variety of technologists who attended last month's productronica show in Munich, Germany. As part of our continuing coverage of productronica 2021, we present this exclusive slide show of photos taken by our good friend Alun Morgan, technology ambassador for Ventec International Group and a fantastic photographer in his own right. Rumor has it that Lufthansa made him buy a separate seat for that giant camera of his.
Technical Track Opportunities at IPC APEX EXPO 2022
Matt Kelly, IPC chief technologist, discusses the growth of the technical conference tracks in this year's IPC APEX EXPO program, as well as market dynamics that are influencing the topics presented at this year's event. "We are thrilled with the number of abstract submissions we have received. As of today, we have over 150 submissions, which may be the highest number we've ever received. It shows the appetite and the interest people have to present and share their knowledge."
On the (Show) Road Again
Much has happened the past two years—politically, technically, and socially—that has greatly changed the path forward from what we had envisioned before the pandemic. Finally, here we are again, on the road between two full, in-person exhibition and technology events, AWE and CES. And what is the metaverse? Meta means transcendence and verse means universe, so we are approaching the capability of entering a virtual universe that is not reality but does exist outside of reality. What does that mean? It's not just the future of Facebook, although Facebook and other social media sites will play a major role. In my opinion, the metaverse is a combination of the real world that you and those you communicate with occupy, while the XR world is generated by your devices.
IPC APEX EXPO 2022: What to Expect Regarding COVID Restrictions
As IPC is adhering to California Department of Public Health Guidelines, proof of COVID vaccination or a negative test will be required to attend IPC APEX EXPO 2022. IPC has partnered with InHouse Physicians (iHP) for attendees to upload their proof of vaccine or negative COVID test results in advance of arrival at IPC APEX EXPO. Once information is uploaded via the iHP link, a trained medical professional will review the information within 24 hours and provide an email confirmation with a green check mark indicating they've been cleared for entry.
Industry Market Drivers, Inflation, and the Supply Chain
In this wide-ranging interview, Shawn DuBravac, chief economist for IPC, discusses a variety of market drivers and pressures that are affecting PCB manufacturing and assembly. He also shares his thoughts on the relationship between inflation, wages, and the current supply chain challenges—and what all this may mean to your bottom line in 2022.
Test and Inspection: Competitive Advantage and Cost of Doing Business
In this interview, Charlie Capers of Zentech breaks down the changing requirements he's seeing in test and inspection in the industry, and the growing importance of investing in testing for the future growth and success of your company. According to Capers, "When you collect statistics on defects, you will start to see a trend that says, for example, we've got a lot of shorts and opens, meaning that our solder paste process is not tight. If you're collecting statistics and really studying them, you will see trends in your processes."
Test and Inspection: Far Beyond Opens and Shorts
Gardien Vice President Todd Kolmodin talks about test and inspection market drivers from his perspective as a test service provider. Andy Shaughnessy and Happy Holden go down the "microvia rabbit hole" with Todd, as well as explore how OEM design requirements are driving test and inspection functionality and processes. When board layer counts and feature densities force longer test times, the tradeoffs to profitability for manufacturers become time and accuracy. Minimizing time while maximizing accuracy calls for new methods, which Kolmodin explains.
ICT Autumn Seminar Review: Live in 2021!
Faced with the choice between a real or virtual event, Bill Wilkie took a calculated risk. It has been many long months since members of the Institute of Circuit Technology had gathered together under one roof, but a fair-sized bunch of industry stalwarts braved the weather and the threat of COVID, descended on the Manor Hotel in Meriden, UK, on Nov. 30, and applauded Bill's decision to go live. They were not disappointed; the program was superb, the atmosphere upbeat, and the networking opportunity priceless.
John Mitchell: Focus on the Future
In a far-ranging interview, Dr. John Mitchell and Barry Matties discuss the upcoming IPC APEX EXPO, IPC programs, and the challenges (and opportunities) facing our industry. If you think Dr. Mitchell's assessment would be pessimistic, you'd be wrong. We're an industry that enjoys and needs to work together, and we've been held at bay for too long. "Throughout 2021, at every conference I've attended, people are really excited about seeing each other again," said Mitchell. "And since APEX EXPO is one of the premier events for the industry, that is the thing I'm happiest about—getting people back together again, getting them excited."
EMS Leadership Summit Expands Its Reach
Nolan Johnson talks with Tracy Riggan, senior director of solutions, and Mark Wolfe, executive EMS advisor, both at IPC. They discuss details for the EMS Leadership Summit planned for IPC APEX EXPO 2022, how the summit program has changed from years past, and who should attend. Hint: Directors and above, along with any "rising star" managers. "It's important in those discussions to have a wide breadth of ideas and learnings from different levels and roles of organizations," said Riggan. "We're expanding that conversation by bringing in younger leaders, maybe those in management positions staged to be the next presidents and vice presidents of their organizations."
Altix Discusses Trends in U.S., European and Asian Markets
In this interview with European Editor Pete Starkey, Altix President Jerome Van Straaten, and Sales and Marketing Director Frederic Baradel discuss their strategies for addressing customers in the U.S. Europe, and Asia, as well as the trends they're seeing in technology around the globe. Check back for more coverage of productronica, as well as news about the upcoming conferences and trade shows in 2022.
It hardly feels like the end of 2021, but here we are, looking at a new year and IPC APEX EXPO and AltiumLive coming up in less than two months. If you don't count days lost during the holidays, it's more like one month before showtime. Hey, at least we have some live events to attend now. This week, we have some association and trade show news, as well as articles on data management and the diminished role of U.S. chip makers in the global electronics supply chain. We're almost behind the eight-ball here, folks, and we can't build chip factories overnight. Let's get a move on!
Ron Lasky: A Perspective on Writing About Solder Defects
Nolan Johnson speaks with Dr. Ron Lasky About "The Printed Circuit Assembler's Guide to... Solder Defects." Ron is a full professor at Dartmouth College, but works part-time at Indium Corporation, helping their customers solve defects. This I-Connect007 eBook, he says, is a compilation of all he's learned over the years as well as the deep technical knowledge of the team at Indium.
Design to Production Flow: DFT and Test Coverage Using Industry 4.0 Principles to Produce Good Products
Achieving design for test (DFT) can be challenging for both design and test groups, as sometimes both expect that the other will be the one to manage DFT. The design and test groups might be in the same organization, or they could be an OEM vs. an EMS company. It works best if both the design and test groups are engaged in the process of DFT and trying to achieve the goal of the best test coverage and lowest rate of field returns.
EIPC Technical Snapshot Review: Semi-additive Processes
The development of ultra-high-density PCBs and packaging substrates using semi-additive and additive manufacturing processes was the theme of the 13th Technical Snapshot webinar presented by EIPC on November 24. It was introduced and moderated by technical director Tarja Rapala-Virtanen. Daniel Schulze, application engineering manager at Dyconex in Switzerland, gave the opening presentation, "Advanced high density rigid packaging substrates for RF and miniaturization." He explained that with their long-established capabilities in ultra-high-density PCBs in flex, rigid-flex and rigid multilayer technologies, it was logical for Dyconex to apply their expertise to the development of specialist IC substrates.
Alex Stepinski: A Philosophical View
"My philosophy is to rely more on sensors throughout the process to measure things non-destructively, then build a model for how you're going to perform, and just validate against the model," said Alex Stepinski. "It's the next step slowly happening worldwide. For instance, we've had 2D AOI for many years. Sometimes, this is complemented with electrical tests. Now, you start to see more 3D AOI happening. You see them putting more sensors on the AOI equipment for direct measurement. Then, you also have the traditional signal integrity testing."
If you're in the United States like I am, and if you celebrated with a traditional Thanksgiving dinner, like my family did, then you just might have fallen asleep sitting up in a chair (or in my case, stretched out on the couch) and missed some of the highlights in the news this week. Never fear, I woke up early to bring the top five news items you should know for this week. We have industry bookings and sales news from IPC, some news from the flex sector, an explanation of induction lamination from Happy Holden, and two different takes on "sustainability."
Happy Thanksgiving From the I-Connect007 Team
In the United States, the Thanksgiving holiday is a celebration of gratitude and appreciation for the harvest and for the people close to us. Thankfulness, however, transcends just one day of official observance. The fourth Thursday of November is when the U.S. officially celebrates Thanksgiving Day and as we take time to observe this holiday, the I-Connect007 team wishes to offer our thanks to you, our global readers and contributors: designers, fabricators, engineers, assemblers, quality and process control gurus, chemists, physicists, supervisors, managers, entrepreneurs, business owners, standards writers, industry experts, and more. You breathe life into the vital, thriving, world-changing electronics industry. You are the real story.
RBP Discusses New Management and Plans for the Future
Ernie Litynski, president of RBP Chemical Technology and Dan Carey, incoming executive vice president, brief Nolan Johnson on the recent organizational changes that brought Carey on board. Both Litynski and Carey outline how they see these changes benefiting their customers and partners, including in the defense and aerospace sectors. As they explain, it's their expertise at helping solve problems that customers value most.
This month—and this year, really—has just flown by. As someone mentioned to me recently, "There are basically only three full work weeks before New Year's Day." Yikes. Thanksgiving is next week, which means most of your customers and suppliers will be hard to reach. Unless there's a problem, that is! In that case, they'll find you, even if you're carving a turkey with the kids and grandkids. So, with a short week on the horizon, most of us are in overdrive now, trying to get ahead of the game.
New Methods for Quantifying PCB Design Weaknesses and Manufacturing Challenges
Gerry Partida explains, "The industry is at a new point in evolving how we look at building boards. Our industry has historically built boards and then tried to find a test for them. Then, when they found a test for it, they figured out that it needed to be analyzed before they built the board. We did this with electrical test. We built boards and down the road, as people started asking, "Why am I buying bad boards? We should electrically test them," electrical test was introduced, reluctantly, into the test part of manufacturing printed circuit boards by suppliers or fabricators. Then they embraced it. But when we started testing boards, we did comparison tests."
DuPont Discusses Acquisition of Rogers Corporation
Nolan Johnson gets the latest news from Avi Avula, vice president and general manager of Interconnect Solutions at DuPont, regarding the recent announcement of DuPont's intent to acquire Rogers Corporation. Avula outlines the motivation for the acquisition, and how the two companies complement each other. Avula also gives some detail on how this might affect customers and business operations, under the proviso that it is early in the process and that many details have not yet been worked out.
Fascinating Opportunities in Flying Probe Testing
Pete Starkey visits with Peter Brandt, director of sales, Europe and Japan for atg Luther & Maelzer, who reflects upon joining the Mycronic group and goes on to discuss market and technology trends in testing, and what to expect from atg at this year's productronica. "There are big challenges regarding accuracy and measurement methods for our flying probe. But this is a high-volume business. This means we only participate in smaller production quantities, because you can say 99.9% of this market for electrical testing will be covered by dedicated testers, using high-density, and high-cost fixtures," Brandt says.
I-Connect007 Editor's Choice: Five Must-Reads for the Week
Welcome to Friday in mid-November. If you're celebrating Thanksgiving, have you made up your menu and bought the turkey? It will be here before you know it. Around here, we're talking financial news and I've selected my top five financial news items of the week. There was plenty to choose from and my picks this week highlight a few of the most-read financial news pieces from our industry, as well as a couple high-interest technology related pieces. We also released our latest webinar, an 11-part series from the expert at GEN3. Definitely check that one out!
Aismalibar on Thermally-Conductive Solutions
Pete Starkey chats with Eduardo Benmayor about the products Aismalibar will be bringing to productronica 2021, and how the pandemic gave the company time to really work on internal processes.
Honoring Those Who Served
On this Veteran's Day holiday, the I-Connect007 staff takes a moment to honor those men and women over the years, who served to protect and defend their country. Just on the I-Connect007 staff, we have loved ones who served, ranging from World War I to present-day active-duty military. The holiday isn't intended to be a political statement, but rather a reflection on service, duty, and personal sacrifice. In some cases, the ultimate sacrifice. Our staff has submitted images of veterans who are near and dear to their hearts. Check them out!
Do You Really Need to Buy New Equipment?
When we began looking into CapEx strategies for PCB manufacturing, we ran into a few company owners who were proud of making their old fabrication equipment last for decades. We recently spoke about CapEx planning with Alex Stepinski, former VP of GreenSource Fabrication, who built a zero-wastewater PCB fabrication facility from the ground up. As Alex explains, your old equipment might actually last for decades, and if you need new equipment, you might find what you're looking for in another industry.
Taiyo Discusses New Inkjet Technology and Solder Mask for Flex
At PCB West, Nolan Johnson spoke with Brian Wojtkiewicz of Taiyo about some of the company's new inkjet technology, which will be manufactured in the U.S. Brian also discusses a new solder mask that has been developed especially for the flexible circuit segment of the industry. "They basically want to keep all the same specifications with regular solder mask, so we're trying to keep all of that: the printability of it and how it looks. We're getting a lot more interest on the colors. They just don't want green anymore. They want it to be standard blue, reds, and whites and stuff like that."
Real Time With... productronica: SÜSS MicroTec on Supply Chain Options
Dr. Luca Gautero, product manager at SÜSS MicroTec, discusses marketing strategies for inkjet printing of solder mask, and explains how collaborative development along the supply chain has introduced new options for the designer and demonstrated the real benefits of this technology to the OEM.
It's merger mania! Yes, there's been a lot of M&A activity lately in our industry over the past few months. This week was no different when it comes to merger activities. In this week's selections, we have news about DuPont's acquisition of Rogers, Nano Dimension's merger with Essemtec, and NCAB's acquisition of Elmatica. We also have news from SMTA International, and an interview with Joe Clark of DownStream Technologies about their new flex and rigid-flex technology.
NCAB Discusses Recent Merger with Elmatica
I-Connect007's Nolan Johnson catches up with Anders Forsén, chief financial officer at NCAB Group, to get an update on the recently announced acquisition of Elmatica by NCAB. In this audio interview from the NCAB headquarters in Sweden, Anders details how he sees Elmatica and NCAB working together to deliver a stronger solution for customers.
DFM 101: PCB Via Structures
One of the biggest challenges facing PCB designers is not understanding the cost drivers in the PCB manufacturing process. This article is the latest in a series that will discuss these cost drivers (from the PCB manufacturer's perspective) and the design decisions that will impact product reliability.
Polar Instruments: Pandemic and Parts Shortage Lead to More R&D Time
During PCB West, I met with Geoffrey Hazelett, VP of sales for Polar Instruments. He explained how the pandemic actually helped Polar's R&D in the long run, and why the current parts shortages may offer a similar benefit for OEMs, who can now spend more time in what he calls "forced R&D."
I-Connect007 Editor's Choice: Five Must Reads for the Week
This week's top picks, as influenced by readership interest, don't seem to hold to a particular theme. Previously, I've referred to this as a potpourri, and now that I've done that once, I can't really do so again, can I? How about a mixed bag? No, I don't think so. While the news is certainly a diverse assortment, this just feels too obvious.
EIPC Technical Snapshot Review: Packaging Trends and a PTFE Masterclass
In circumstances that forced the postponement of its live conferences, seminars and workshops, EIPC has continued to provide its platform for the exchange and dissemination of market knowledge and technical information to the European interconnection and packaging industry. October 22 brought the 12th in the sequence, introduced and moderated by Martyn Gaudion, EIPC board member and CEO of Polar Instruments.
Bring Our Industry Back? Shift Your Attitude
Dan Beaulieu discusses some comments he made in a recent column and why this is an opportunity for our industry in North America to retool, refocus, and make one fundamental shift in our plan to bring our industry back.
Real Time With... SMTAI: Anaya Vardya of American Standard Circuits
Anaya Vardya, president and CEO at American Standard Circuits, speaks with Nolan Johnson ahead of the SMTA International exposition. Vardya shares why American Standard Circuits will be exhibiting at the show, and outlines the topics ASC will be presenting on the show floor.
There's a lot going on in the industry this week, and my top five picks have a little bit of everything.
Soldering to Aluminum With Mina
At DesignCon, I caught up with Divyakant Kadiwala of Averatek Corporation. He spent a few minutes explaining a special process developed by Averatek that allows technologists to solder to aluminum.
Barry Matties interviewed Thomas Walsh and Travis Houchin of Integrated Process Systems regarding industry trends they're seeing in capital expenditure and the changes they have noticed in their customers' requirements.
Well, William Shatner finally did the field research for his role as Captain James T. Kirk this week, going sub-orbital on the Blue Origin launch platform. No, I'm not making fun of Shatner's flight, quite the contrary. He's the first actor in the entire Star Trek franchise to go into space while still living. And at 90 years old, he's now the oldest person to make the trip as well!
Making Process Decisions in a Greenfield
While SEL is new to manufacturing PCBs, we have more than 35 years of experience in manufacturing. In addition to assembling our own PCBs and building completed units, we also manufacture many of the critical components used in our products and solutions, such as transformers and plastics.
This week we have quite a potpourri for you. Our newest columnist, Kim O'Neil of Prototron Circuits, offers details on setting up your own quality management system (QMS). Emmy-winning tech writer David Pogue will be a keynote at IPC APEX EXPO, which I think is a great move on the part of IPC. We have an article from Brent Fischthal of Koh Young America on the value of investing in the latest AOI equipment, which can save you time and money by eliminating false calls.
Tribal Knowledge and Design Data Formats
I recently began to investigate "tribal knowledge" in PCB and PCBA operations. My goal was to determine whether anything of value would be lost if it were not recorded or preserved for future industry technologists. In the face of modern material science and industry standards, the descriptive title "tribal knowledge" suggests that one must determine what is fact, what is fiction, and what may just be plain embellishment, coincidence, or even magic. In many cases, tribal knowledge seems to be composed of memories of problems solved due to trial and error, coupled with observations of resulting causes and effects.
Once again, seemingly like clockwork, the controversy erupts. It's such a divisive issue, and it seems no one can be undecided in this debate; each of us is expected to take a stand. All of society—including the media—asks, "Which side are you on?"
24 Essential Skills for Engineers: The Story Behind the Book
In this interview with I-Connect007's own Happy Holden about his newest book, 24 Essential Skills for Engineers, which he wrote over the span of his career, he highlights some particular moments from his time working at HP and as CTO of Foxconn which inspired many of the book's chapters. Happy explains why he covered engineering skills as well as "soft skills" such as problem-solving and communication—skills which are keys to succeeding as an engineer.
From DesignCon: Tektronix's Pavel Zivny on TDECQ
I caught up with Pavel Zivny, a system engineer at Tektronix in Portland, Oregon. Pavel was happy to share some of the technical aspects of the paper he was presenting at DesignCon.
Every fall, the action in the world of PCB design and manufacturing begins to pick up again. No more out-of-office email replies; summer vacation is over, and it's time to get back at it. This week's Top Five is a veritable pastiche of columns, articles, and news from the electronics industry.
Catching up With Schaltbau's Jim Tullo
As the electric vehicle market expands, it is useful to hear from someone with a goal of creating a significant position in that market. Schaltbau Group is one of those companies. In fact, one might say because of its string position in the rail marketplace, the company has been dealing in EVs for years. I sat down recently with Jim Tullo, president of Schaltbau's North American operations, to talk about their current and future business. I strongly advise you to dig into this one.
From DesignCon: Victory Giant Pushing the Technology Envelope
Victory Giant's Executive Vice President and CTO George Dudnikov let me in on the growth happening within his company, which includes a much larger Industry 4.0 facility and the capability to do pulse plating up to a 25:1 aspect ratio.
From DesignCon: Ohmega's New Home with Quantic
At this year's DesignCon, Nolan Johnson stops to visit with Bruce Mahler, who details the exciting changes happening with Ohmega and Ticer after being acquired by Quantic, and what that means for the market opportunity of OhmegaPly and TCR.
In some weeks, readership numbers are pretty even, which means I make my picks based on "excellence" or "achievement" of some sort. Other weeks, you readers "go viral" on a few key news items, making the selection process numerically driven. This was one of those weeks. Here, then, are the five most-read news items. Of course, leading the way are our esteemed technical editors, Pete Starkey and Dan Feinberg. These gentlemen are examples of my opening statement, as both enjoy huge popularity with readers precisely because they are so good at reporting on the industry.
ICT Autumn 2021 Webinar Review: High-voltage Testing and Advanced Antenna Materials
The Institute of Circuit Technology continues to deliver first-class on-line events. This year's ICT Autumn Webinar on September 7 presented papers by leading experts on high-voltage testing and advanced antenna materials. It was introduced and moderated by ICT Chair Emma Hudson.
Ucamco Improves Communication With Gerber Job Editor Notes in Mandarin
Ucamco recently announced that its Gerber Job Editor can now accept notes in Mandarin. I asked Karel Tavernier, managing director of Ucamco, to discuss the move to add Mandarin notes to the Gerber Job Editor, and what this means for CAM department users who may speak little or no English.


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BREAKING: San Antonio Spurs Sign New Player
By Missy Evans Last updated Jul 24, 2022
On Saturday, Shams Charania of The Athletic reported that the San Antonio Spurs had agreed to a deal with Jordan Hall.
Charania: "Undrafted Saint Joseph's G/F Jordan Hall has agreed on a two-way NBA contract with the San Antonio Spurs, sources tell @TheAthletic @Stadium."
Hall played his college basketball for Saint Josephs, and he averaged a very impressive 14.1 points, 6.1 rebounds and 5.8 assists per game in 30 games last season.
The Spurs are a young team, who is in a total rebuilding mode, so this could be a good signing for them considering he is only 20-years-old.
At 6'8″, he is extremely versatile and can play multiple positions on the floor.
Last season, the Spurs had a surprisingly decent season where they actually made the play-in tournament.
While they lost to the New Orleans Pelicans in the play-in tournament, they at least gave themselves a chance to make the playoffs, which was a shock on its own.
They were led by talented guard Dejounte Murray, who made his first career All-Star after being drafted in 2016.
However, they traded away Murray to the Atlanta Hawks during the offseason.
The Spurs will likely be one of the worst teams in the entire league next season, but they notoriously have one of the best organizations in all of the NBA (and sports).
Essentially, they have proven that they can build a team the right way, and there is no question that they will find themselves back in the playoffs mix at some point soon.
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Amazing shaped sewing box by unknown Dutch manufacturer, 1950. Amazing made piece, fully metal besides the handles they are made of plastic. Heavy quality and in fantastic condition. Whoever designed this piece had a very good look at the designs of Willy van der Meeren as the legs and a-symmetrical shapes are very much the same. The legs are made of industrial blue lacquered folded metal, the casing is lacquered off white and the folding doors are in blue, red and yellow. The floor protectors are made of white rubber and also the sides of the handles are made in white rubber, the door knobs are made of plastic. I have rarely seen this beautiful sewing or storage cabinet in my years of collecting but this is really something special.


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Armenian dramatic baritone Gevorg Hakobyan studied at the Komitas State Conservatory in Yerevan with Maestro Sergey Danielyan. In 2008, Gevorg was awarded both the Gold Medal and First Prize at the prestigious First International Pavel Lisitsian Baritone Competition in Moscow, and in 2011 was recognised as a Merited Artist of the Republic of Armenia.
Gevorg starts the 2019/20 season as Scarpia at Teatro Regio Torino in Tosca. A sought-after guest of the Bolshoi Theatres, Gevorg then returns to Bolshoi by invitation of Tugan Sokhiev as Tomsky in Pique Dame, Gryaznoi in The Tsar's Bride, Ibn-Hakia in Iolanta and Mazeppa (title role). Gevorg then returns to Oper Frankfurt as Don Carlo di Vargas (La forza del destino).
Recent highlights include Ibn-Hakia (Iolanta) at Palau de les Arts Valencia, Alfio (Cavalleria Rusticana) at Teatro Carlo Felice Genoa and The Bolshoi Theatre as Gryaznoi in Rimsky-Korsakov's The Tsar's Bride and Tomsky in Tchaikvosky's Pique Dame and the Mariinsky Theatre as Scarpia (Tosca).
Highlights of previous seasons include Tomsky (Pique Dame) and Scarpia (Tosca) at Stuttgart Opera, debut at Teatro dell'Opera di Roma as Scarpia (returning for Cavalleria rusticana and Pagliacci), The Tsar's Bride and Pique Dame at the Bolshoi Theatre, Simon Boccanegra (title role) at the Mariinsky Theatre under the baton of Valery Gergiev, Baron Scarpia (Tosca) and Gérard (Andrea Chénier) at Theater Kiel, Nabucco (title role) at Terme di Caracalla Rome and Scarpia at Teatro Communale di Bologna under Valerio Galli.
Gevorg has an ever-growing international reputation, performing regularly at multiple opera houses and theatres including the Mikhailovsky Theatre, Bolshoi Theatre, Teatro Lirico di Cagliari, Teatro Massimo di Palermo, Armenian National Academic Theatre of Opera and Ballet, Kiel Opera House, Hong Kong Cultural Centre and the Stars of the White Nights International Festival. In addition, Gevorg performs frequently at the Teatro Lirico in Cagliari, where he has already appeared in The Legend of the Invisible City of Kitezh and the Maiden Fevroniya (Fiodor), Eugene Onegin (title role), Cavalleria Rusticana (Alfio), Pagliacci (Tonio), I Shardana (Norace), and Cherevichki (Bes) and at Palau de les Arts Valencia in Aida (Amonasro), Cavalleria Rusticana (Alfio), Madama Butterfly (Sharpless) under Lorin Maazel and in Simon Boccanegra (Paolo Albiani) with Placido Domingo in the title role.


Fox News Channel host Megyn Kelly insisted Monday night that won't back down to Donald Trump and will "continue doing my job," after he complained about her being "unfair" to him during last week's GOP presidential debate.
Some critics construed the remarks to mean Trump was blaming Kelly's pointed questions on her having her period, but Trump denied that.
Kelly also acknowledged Trump's front-runner status and said he was worthy of his position at the top of the Republican field.
"Mr. Trump is an interesting man who has captured the attention of the electorate," she said.
Meanwhile, the presidential hopeful is scheduled to go on "Fox & Friends" Tuesday morning for his first interview since attacking Kelly and the network.
Later on Tuesday evening, Trump is expected to appear on Sean Hannity's Fox News Channel program.
The "Fox & Friends" announcement came after Trump said he made peace with the network's chairman.
"Roger Ailes just called. He is a great guy & assures me that 'Trump' will be treated fairly on @FoxNews. His word is always good!" Trump tweeted.
Earlier in the day, Trump was still blasting Kelly on MSNBC's "Morning Joe" and NBC's "Today" saying, "She should really be apologizing to me, you want to know the truth," he said.
That demand, of course, went nowhere.
Trump's next campaign stop is in Michigan, where a standing-room-only crowd topping 2,000 is expected to hear him at an expo center in Birch Run.


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Popular 'Diet' Ingredient Linked to Leukemia & Lymphoma, Study Shows
Written by bestfolkmedicine Published on October 29, 2018
30 years after its ban has been removed for pharmaceutical interests, one of the most popular artificial sweeteners known as Aspartame continues being one of the most debatable topics ever.
Even though its dangers are well-documented, a lot of people remain unaware of its history, its effects, and its ingredients, according to The Truth Soldier. Many only know that it is an FDA-approved alternative to sugar which has to be safe since otherwise, it would not be on the market.
But, things are not as simple as that. To find out more about how aspartame came to be an FDA-approved sweetener and why you should eliminate it from your diet for good, check out this article.
5 Major Issues Concerning Aspartame
According to Healthy Holistic Living, here are the 5 most important reasons why you need to stay away from aspartame as much as possible:
When this chemical enters the body, it gets transformed into formaldehyde, a chemical associated with cancer- aspartame is consisted of 3 compounds and it is a synergistically toxic chemical, that is, the sum of its 3 parts is more toxic than each of them separately. Methanol, one of the compounds, unlike the one from veggies, alcoholic drinks, and fruits, is not accompanied by ethanol which prevents poisoning and thus, it may lead to leukemia, DNA damage, and lymphoma.
Despite evidenced toxicity, it got approved by an FDA commissioner close to Reagan- early safety studies linked aspartame with seizures and death and it was therefore banned. But, the company which produced aspartame at that time, Searle, waited for a new FDA commissioner from the government of Reagan to be appointed and to remove the ban so that they can continue with the mass production of this chemical.
It may lead to brain damage- this chemical has the power to cross the blood-brain barrier and when its levels in the body are too high, the brain cells are attacked with surplus calcium, resulting in neural cell damage and cell death, which consequently leads to permanent brain damage.
It is produced from feces of GMO E. Coli- unfortunately; feces of GMO E.coli bacteria are used for the production of this harmful artificial sweetener. Modified with specific genes, E. Coli is made so that it can produce high amounts of an enzyme by-product which produces phenylalanine necessary for the creation of aspartame.
It can contribute to metabolic syndrome and obesity- when consumed in excessive amounts; aspartame can lead to weight gain and diabetes without any relation to caloric intake. Studies have found it to be even worse than sucrose in terms of encouraging weight gain whereas another research suggests that it causes hormonal changes which result in elevated appetite and sugar cravings.
THE TRUTH SOLDIER
artificial sweetener aspartame brain damage calories dangerous substance diabetes DNA damage E.coli formaldehyde GMOs metabolic syndrome obesity research sucrose sugar weight gain
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