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\newcommand{\vsp}{\vspace{0.3cm}} \newcommand{\ds}{\displaystyle} \newcommand{\mod}{{\mbox{ mod }}} \newcommand{\m}{{\mathcal{M}}} \newcommand{\p}{{\mathcal{P}}} \newcommand{\q}{{\mathcal{Q}}} \newcommand{\1}{{\bf{1}}} \newcommand{\0}{{\bf{0}}} \begin{document}
\title{Operator Control of Inter-Neural Computing Machines}
\author{Mau-Hsiang~Shih %~\IEEEmembership{Member,~IEEE,} and~Feng-Sheng~Tsai %,~\IEEEmembership{Fellow,~OSA,}% <-this % stops a space
\thanks{%Manuscript received October 5, 2011; revised July 29, 2012; accepted August 3, 2012.
%Date of publication August 27, 2012; date of current version October 15, 2012.
This work was supported by the National Science Council of Taiwan.} %
\thanks{The authors are with the Department of Mathematics, National Taiwan Normal University,
88 Sec. 4, Ting Chou Road, Taipei 11677, Taiwan (e-mail: [email protected];
[email protected]).}% <-this % stops a space
%\thanks{Digital Object Identifier 10.1109/TNNLS.2012.2212455}% }
% The paper headers
\markboth{IEEE Transactions on Neural Networks and Learning Systems}%
%\markboth{}%
{SHIH AND TSAI: OPERATOR CONTROL OF INTER-NEURAL COMPUTING MACHINES}
%\IEEEpubid{0000--0000/00\$00.00~\copyright~2007 IEEE} \maketitle
A dynamics representation of neural population responses asserts that
motor cortex is a flexible pattern generator sending rhythmic, oscillatory signals to generate multiphasic patterns of movement. This raises a question concerning the design and the control of new computing machines
that mimic the oscillatory patterns and the multiphasic patterns seen in neural systems.
To address this issue, we design an inter-neural computing machine (INCM)
made up of plastic random inter-neural connections.
We develop a mechanical way to measure collective ensemble firing of neurons in INCM.
Two sorts of plasticity operators are derived from the measure of synchronous neural activity
and the measure of self-sustaining neural activity, respectively. Such plasticity operators conduct activity-dependent operation to modify network structure of INCM.
The activity-dependent operation meets the neurobiological perspective of Hebbian synaptic plasticity
and displays the tendency toward circulation breaking aiming to control neural population dynamics.
We call such operation ``operator control" of INCM and develop a population analysis of operator control
for measuring how well single neurons of INCM can produce rhythmic, oscillatory activity but,
at the level of neural ensembles, generate multiphasic patterns of population responses.
\end{abstract}
\begin{IEEEkeywords}
Adaptive plan, decirculation process, machine learning, multiphasic patterns, nonlinear dynamics, oscillatory patterns, plasticity
operators, population dynamics. \end{IEEEkeywords}
\IEEEPARstart{I}{n} the decade of the 2000s, the study of brain-machine interfaces (BMIs)
{has built up} a picture of how neuronal populations give rise to sensation, behavior or
other complex brain processes \cite{Nicolelis2001}--\cite{Quiroga}, %\cite{Nicolelis2001,Nicolelis2003,Nicolelis2009,Quiroga},
going beyond the analysis of single-neuron recording. It supports the concept of distributed neural coding,
which was first proposed by T. Young \cite{Young} and further popularized by D. O. Hebb \cite{Hebb}--\cite{Sejnowski}, %\cite{Hebb,Nicolelis,Sejnowski},
revealing that distributed neural ensembles define the physiological unit of
the mammalian CNS (central nervous system) \cite{Nicolelis2009,Averbeck}.
Some measures to clarify information extracted from distributed neural ensembles
are derived from the combination of information theory and decoding algorithms,
including, for example, the firing rates of neurons over some time windows \cite{Rieke},
the precise timing of neural spikes \cite{Optican,Steveninck}, and the correlations of neural ensemble firing \cite{Averbeck}.
{Through the use} of those measures, multiple motor parameters (position, velocity, force, direction, and so on) are recognized and then used to accurately predict various arm movements \cite{Nicolelis2001,Quiroga}.
Recently, a different representation of neural population responses has been proposed from the perspective of nonlinear dynamics \cite{Churchland}.
It has shown that motor cortex is a flexible pattern generator sending rhythmic, oscillatory signals to generate motion.
Hence, by adding two or more other rhythms of the motor neurons firing at a given moment,
the brain can create multiphasic patterns of movement.
In other words, neural population responses are to reflect underlying dynamics of rhythmic, oscillatory patterns
and display multiphasic moving incidentally.
Here we wish to construct an inter-neural computing machine (INCM)
that mimics the ``oscillatory patterns" and the ``multiphasic patterns" seen in neural systems.
INCM mainly consists of an evolutionary neural network (with random inter-neural connections)
and an adaptive plan which determines a dynamic clamp of network modification (plasticity operators) to
generate the decirculation process in neural network dynamics \cite{Shih2012}.
The plasticity operators suggest an activity-dependent mechanism to control populations of neurons,
relating to a comprehensive view about the neurobiological perspective of Hebbian synaptic plasticity \cite{Hebb}. In the literature, several forms of computing machines were developed for adaptive pattern recognition
\cite{Carpenter}--\cite{Widrow90}, %\cite{Carpenter,Minsky,Widrow90},
including Madaline \cite{Widrow,Winter}, backpropagation \cite{Rumelhart},
ART architectures \cite{Grossberg1976a,Grossberg1976b}, CAM networks \cite{Hopfield1982,Hopfield95},
and support vector machines \cite{Cortes,Scholkopf}.
The learning processes embedded in such computing machines are mainly based on the definition of an error-correction process, a steepest-descent process,
a competitive process, or an energy-minimization process,
iteratively changing inner network structure of computing machines and generating outputs that elicit the optimal distribution of
category consistent with the structure of input patterns \cite{Carpenter}.
Such learning processes focus on the conception of ``optimal control" of the assignment of a label to a given input pattern and have many practical applications in classification
regression \cite{Dufrenois,Slavakis}, and signal processing \cite{Nicolelis2001,Nicolelis2003,Widrow}.
But it remains stymied when trying to establish machines to perform dynamical features presenting in population responses of biological neurons.
To address this issue, we develop a population analysis of operator control of INCM
for measuring how well single neurons of INCM can produce rhythmic, oscillatory activity but,
at the level of neural ensembles, generate multiphasic patterns of population responses.
With such operator control, INCM is likely to be a platform trying to connect
multiphasic patterns of population responses
(for example, the amounts of approximatively simultaneous firing of neurons \cite{Quiroga})
with amalgams of rhythmic, oscillatory activity of single neurons. In Section \ref{IM} we describe the architecture of INCM.
In Section \ref{secalg} we define the conception of operator control of INCM.
We find feasible plasticity operators that are essentially linked to the neurobiological perspective of Hebbian synaptic plasticity.
Furthermore, an algorithm describing the execution process of operator control is established
in the adaptive plan of INCM.
Section \ref{sec2} is to represent circulation breaking under operator control of INCM.
Section \ref{tests} is to provide experimental tests for demonstrating
how well single neurons of INCM produce rhythmic, oscillatory activity but,
at the level of neural ensembles, generate multiphasic patterns of population responses.
A rigorous proof of circulation breaking under operator control of INCM is put in the Appendix.
The inter-neural computing machine (INCM) has four separate components,
denoted $\textrm{INCM}=(\mathcal{N},\mathcal{L},\Pi,\tau)$ (see Fig. \ref{fig1}), with the following functions.
Component $\mathcal{N}$ is an evolutionary neural network (with random inter-neural connections),
which receives input patterns and processes them into an output sending to components $\Pi$ and $\tau$.
Component $\mathcal{L}$ is an adaptive plan that can be executed during the adaptation of input patterns.
The adaptive plan serves as a program, a kind of algorithm that takes plasticity operators to modify network structure of
$\mathcal{N}$
and generates a memoried set of neuronal active states sending to $\Pi$.
Component $\Pi$ is an aggregation, which is hooked up to both $\mathcal{N}$ and $\mathcal{L}$.
When $\Pi$ is given the newly generated output of $\mathcal{N}$ and the newly generated memoried set of $\mathcal{L}$,
it merges and passes them to $\mathcal{L}$ for action.
Component $\tau$ is a time controller defining a slide window necessary for descrambling the neural activity counts
conducted by the combined system, $\mathcal{N}$ plus $\mathcal{L}$ plus $\Pi$.
INCM is remarkable in that
(i) input patterns are continuously exciting and altering INCM in a sequential fashion;
(ii) network structure of $\mathcal{N}$ is alternately changing under operator control;
(iii) the neural activity counts (the equivalent of the neuronal vote) are dynamically changing during the adaptation of input patterns. \subsection{Components $\mathcal{N}$, $\mathcal{L}$, and $\Pi$}
The original construct of the evolutionary neural network $\mathcal{N}$ was described in \cite{Shih}.
With the mathematical construct, we have shown that neurons can synchronize their activity underlying
an evolutionary network structure induced by Hebbian synaptic plasticity \cite{Hebb}.
Here, in contrast with the finding of the network structure for neural synchronization,
we represent $\mathcal{N}$ as an evolutionary neural network with random connection structure.
We will show that an adaptive plan exists for modifying any random connection structure of $\mathcal{N}$ such that
INCM can capture the dynamical feature presenting in neural population responses.
For network description,
name the {neurons} $1,2,\ldots,n$ in the evolutionary neural network $\mathcal{N}$.
The dynamical system of the $n$ coupled neurons is modeled by the nonlinear equation \cite{Shih2012,Shih}:
\begin{equation}\label{dynamics}
x(t+1)=H(x(t),\mathcal{P}(t)), \quad t=0,1,\ldots, \end{equation}
where $x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\in\{0,1\}^{n}$ is a vector of {neuronal active states}
denoting the population responses of neurons at time $t$,
$\mathcal{P}(t)=(A(t),s(t))$ is a vector of {control parameters} with $A(t)=(a_{ij}(t))\in M_{n}(\Real)$ denoting
the {evolutionary coupling matrix} of the network at time $t$ and $s(t)\subset\{1,2,\ldots,n\}$ denoting the neurons that adjust their activity at time $t$,
and $H(\cdot,\mathcal{P}(t))$ is a function whose $i$th component is defined by $$ [H(x,\mathcal{P}(t))]_{i}=\bbbone\left(\sum_{j=1}^{n}a_{ij} (t)x_{j}-b_{i}\right)\quad\mbox{if}~i\in s(t), $$ otherwise $[H(x,\mathcal{P}(t))]_{i}=x_{i}$,
where $b_{i}\in\Real$ is the threshold of neuron $i$ and
the function $\bbbone$ is the Heaviside function: $\bbbone(u)=1$ for $u\geq 0$, otherwise 0,
On each subsequent time $t=0,1,\ldots,$ the network $\mathcal{N} $ generates a vector of neuronal active states $x(t+1)$
according to $(\ref{dynamics})$.
The aggregation $\Pi$ merges the vector $x(t+1)$ with a {memoried set} of vectors of
neuronal active states $\mathcal{M}(t)$ generated by the component $\mathcal{L}$.
This forms a merging sequence of states $\Pi(\mathcal{M} (t),x(t+1))=\{\mathcal{M}(t),x(t+1)\}$.
This merging sequence is then received by the component $\mathcal{L}$, which denotes an {adaptive plan}
regarded as a function of the form \begin{equation}\label{algorithm}
\mathcal{L}(\mathcal{P}(t),\Pi(\mathcal{M}
(t),x(t+1)))=(\mathcal{P}(t+1),\mathcal{M}(t+1)), \end{equation}
$t=0,1,\ldots,$
where the vector of control parameters changes from $\mathcal{P} (t)$ to $\mathcal{P}(t+1)$
and the memoried set changes from $\mathcal{M}(t)$ to $\mathcal{M}(t+1)$
according to the activity of neurons in the merging sequence of states $\Pi(\mathcal{M}(t),x(t+1))$.
The adaptive plan generates the vector of control parameters $\mathcal{P}(t+1)$, which is then received by the network $\mathcal{N}$,
as well as generates the memoried set $\mathcal{M}(t+1)$, which is then received by the aggregation $\Pi$.
The computing of $\mathcal{N}$, $\mathcal{L}$, and $\Pi$ reveals the {feedback regulation} in INCM:
the vector of neuronal active states (generated by $\mathcal{N}$) as well as
the vector of control parameters (generated by $\mathcal{L}$) keep changing in time,
looping back on one another and giving rise to patterns of complexity.
\begin{figure}[t!] \center
\includegraphics[height=1.5in,width=3.3in]{archi.eps} \caption{Architecture of
$\textrm{INCM}=(\mathcal{N},\mathcal{L},\Pi,\tau)$,
where the components $\mathcal{N}$, $\mathcal{L}$, and $\Pi$ are to formulate
the collective dynamics of the evolutionary neural network,
and the component $\tau$ is to define a slide window necessary for descrambling the neural activity counts
conducted by the population responses of neurons.
INCM receives a sequence of input patterns $y^{k},k=0,1,\ldots,$ at time $T_{k},k=0,1,\ldots,$
and descrambles the neural activity counts $f_{\tau}(t+1)$ for each $t=0,1,\ldots.$
}\label{fig1} \end{figure}
\subsection{Component $\tau$}\label{comtau}
We require the time controller $\tau$ to define a {slide window} necessary for descrambling the neural activity counts
conducted by the population responses of neurons (see Fig. \ref{fig222} for an illustration).
Let $\mathcal{S}\subset\{1,2,\ldots,n\}$ denote a subset of neurons.
Due to the slide window, the output of INCM at time $t+1$ is defined by
\begin{equation}\label{eq01}
f_{\tau}(t+1)=\sum_{t-\tau+1< T\leq t+1}\sum_{i\in \mathcal{S}}x_{i}(T),\quad t=0,1,\ldots,
\end{equation}
which performs the equivalent of the neuronal vote among neurons in $\mathcal{S}$ \cite{Nicolelis2001}
or, in mathematical terms, the amount of approximatively
simultaneous firing of neurons in $\mathcal{S}$ \cite{Quiroga}. \begin{figure}[!t]
\center
\includegraphics[height=1.5in,width=3.3in]{Popcode.eps}
\caption{Slide window for descrambling the neural activity counts. Black squares in each row of this schematic illustration represent the activity of a single neuron over time.
There are $17$ coupled neurons whose activity is simultaneously recorded (i.e. the number of elements in the set $\mathcal{S}$ is $17$).
With the time controller $\tau=15$, the neural activity counts $f_{\tau}$ indicated by the slide windows at time $t+1$ (the left dashed rectangle),
at time $t+23$ (the middle dashed rectangle), and at time $t+45$ (the right dashed rectangle) are 110, 113, and 104, respectively. }\label{fig222}
\end{figure}
Input patterns are continuously exciting and altering INCM in a sequential fashion.
When INCM receives an input pattern $y^{k}$ at time $T_{k}$, INCM is designed to:
\begin{itemize}
\item[1)] \textit{Afference}: At time $t=T_{k}$,
\begin{itemize}
\item[a)] reset $x(t)$ as the input pattern $y^{k}$;
\item[b)] reset $\mathcal{M}(t)$ as the singleton $y^{k}$. \end{itemize}
\item[2)] \textit{Computing}:
For each $t=T_{k},T_{k}+1,\ldots,T_{k+1}-1$, \begin{itemize}
\item[a)] generate $x(t+1)$ by $(\ref{dynamics})$;
\item[b)] aggregate the merging sequence $\Pi(\mathcal{M} (t),x(t+1))$;
\item[c)] generate $\mathcal{P}(t+1)$ and $\mathcal{M}(t+1)$ by $(\ref{algorithm})$.
\end{itemize}
\item[3)] \textit{Efference}:
For each $t=T_{k},T_{k}+1,\ldots,T_{k+1}-1$, \begin{itemize}
\item[a)] reset the slide window at time $t+1$;
\item[b)] descramble the output $f_{\tau}(t+1)$ by $(\ref{eq01})$. \end{itemize}
\end{itemize}
In terms of a programming language, the operation of INCM can be formulated as follows:
\textrm{
\begin{itemize}
\item[] for $t$ = $0,1,\ldots,$
\item[] \quad\quad if ($t$ == $T_{\mathtt{k}}$ for $k$ = $0,1,\ldots$)
\item[] \quad\quad\quad $x(T_{\mathtt{k}})$ := $y^{\mathtt{k}} $;
\item[] \quad\quad\quad $\mathcal{M}(T_{\mathtt{k}})$ := $y^{\mathtt{k}}$;
\item[] \quad\quad end
\item[] \quad\quad $x(t+1)$ := $H(x(t),\mathcal{P}(t))$;
\item[] \quad\quad ($\mathcal{P}(t+1),\mathcal{M}(t+1))$ := $\mathcal{L}(\mathcal{P}(t),\Pi(\mathcal{M}(t),x(t+1)))$; \item[] \quad\quad $f_{\tau}(t+1)$ :=
$\sum_{t-\tau+1<T\leq t+1}\sum_{i\in \mathcal{S}}x_{i}(T)$; \item[] end
\end{itemize} }
As a result, INCM is to make a random net of neurons deal directly and adaptively
with the afference of an arbitrary sequence of input patterns, and then to descramble the neural activity counts denoting the amounts of approximatively simultaneous firing of neurons.
\section{Adaptive plan}\label{secalg}
To find an adaptive plan for the capture of the dynamical feature presenting in neural population responses,
we consider the process of {circulation breaking} in neural network dynamics:
the occurrence of a loop of neuronal active states leads to a change in control parameters,
which feeds back to reinforce neurons to tend to break the circulation of neuronal active states in this loop.
The decirculation process was first proposed in \cite{Shih2012}, in which a criterion that describes and quantifies perturbations of network structure and neural updating
was given.
Here we go further and consider that \begin{itemize}
\item[1)] when neurons $i,j$ tend to fire synchronously, or
\item[2)] when neuron $i$ repeatedly or persistently takes part in firing neuron $j$,
\end{itemize}
what are the activity-dependent changes that can take place in the connections between neurons $i,j$
coming up with a way to process circulation breaking in neural network dynamics?
We address such activity-dependent changes in neural connections because it is essentially linked to the neurobiological perspective of Hebbian synaptic plasticity \cite{Hebb}.
Hebbian synaptic plasticity has been considered to be the basis of memory and learning \cite{Nicolelis,Sejnowski,Strogatz}.
In the following, we introduce two measures to quantify synchronous activity
of neurons and self-sustaining activity of neurons, respectively. The two measures are to derive two sorts of plasticity operators, respectively,
useful in determining the above activity-dependent changes in neural connections.
\subsection{Measures of neural activity}\label{SecMea}
Denote by $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ a {loop} of states in $\{0,1\}^{n}$,
meaning that $p>1$, $x^{0},x^{1},\ldots,x^{p}$ $\in\{0,1\}^{n} $, $x^{0}=x^{p}$, and $x^{i}\neq x^{j}$ for some $i,j\in\
{1,2,\ldots,p\}$.
For each $m=0,1,\ldots,p$, we say that the state $x^{m}$ is in the position $m$ of the loop $\Omega$.
For each 01-string $x=x_{1}x_{2}\cdots x_{n}$, let \begin{eqnarray*}
&\1(x)=\{i;~x_{i}=1,~1\leq i\leq n\},&\\ &\0(x)=\{i;~x_{i}=0,~1\leq i\leq n\}.& \end{eqnarray*}
For two sets $U$ and $V$, we write $U\setminus V$ for the set-theoretical difference $\{x\in U;~x\not\in V\}$,
$U \triangle V$ for the {symmetric difference} $(U\setminus V)\cup(V\setminus U)$,
and $\sharp U$ for the number of elements of $U$. For every $i=1,2,\ldots,n,$
let \begin{eqnarray}\label{eqq1} &\hspace{-0.5cm}M_{i}(\Omega)=\{m;~i\in \1(x^{m-1})\triangle \ 1(x^{m}), m=1,2,\ldots,p\},&\nonumber\\ &\hspace{-0.5cm}M_{i}(\Omega)^{+}=\{m;~i\in \ 1(x^{m})\setminus \1(x^{m-1}), m=1,2,\ldots,p\},&\nonumber\\ &\hspace{-0.5cm}M_{i}(\Omega)^{-}=\{m;~i\in \1(x^{m-1})\setminus \1(x^{m}), m=1,2,\ldots,p\}.& \end{eqnarray}
Here $M_{i}(\Omega)$ denotes the collection of the positions $m$ of the loop $\Omega$ in which
neuron $i$ changes its state from $x^{m-1}_{i}=0$ to
$x^{m}_{i}=1$ or from $x^{m-1}_{i}=1$ to $x^{m}_{i}=0$, whereas $M_{i}(\Omega)^{+}$ (resp., $M_{i}(\Omega)^{-}$) denotes the collection of the positions $m$ of the loop $\Omega$ in which
neuron $i$ changes its state from $x^{m-1}_{i}=0$ to $x^{m}_{i}=1$
(resp., changes its state from $x^{m-1}_{i}=1$ to $x^{m}_{i}=0$).
For every $i,j=1,2,\ldots,n$, denote by \begin{eqnarray}\label{eq33} &&\hspace{-0.8cm}\Upsilon_{ij}(\Omega)=\sharp(M_{i} (\Omega)^{+}\cap M_{j}(\Omega)^{+})+\sharp(M_{i} (\Omega)^{-}\cap M_{j}(\Omega)^{-})\nonumber\\[-1.5ex]\nonumber\\[-1.5ex] &&\hspace{0cm}-\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{-})-\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{+}), \end{eqnarray}
which can be regarded as a measure of {synchronous activity} between neurons $i,j$, that is,
if neurons $i,j$ tend to change their firing or quiescent states synchronously (resp., asynchronously) in $\Omega$,
then $\Upsilon_{ij}(\Omega)>0$ (resp., $\Upsilon_{ij}(\Omega)<0$) (see Fig. \ref{fig2}(a), (b)).
For every $i,j=1,2,\ldots,n$, denote by \begin{equation}\label{eq34}
\Gamma_{ij}(\Omega)=\min\{\sharp M_{i}(\Omega),\sharp M_{j} (\Omega)\},
\end{equation}
which can be regarded as a measure of {self-sustaining activity} of neurons $i,j$,
that is, if neuron $i$ or $j$ tends to maintain more self-sustaining firing or self-sustaining quiescent states in $\Omega$ than neuron $i'$ or $j'$,
then $\Gamma_{ij}(\Omega)<\Gamma_{i'j'}(\Omega)$ (see Fig. \ref{fig2}(c), (d)).
We refer to the resulting matrices $\Upsilon(\Omega)=(\Upsilon_{ij} (\Omega))$ and $\Gamma(\Omega)=(\Gamma_{ij}(\Omega))$ as the measure of synchronous activity and the measure of self-sustaining activity derived from the loop $\Omega$ of states in $\ {0,1\}^{n}$, respectively.
\center
\includegraphics[height=1in,width=3.5in]{measure3.eps}\\
\caption{Schematic illustration of the measures of neural activity in the loop $\Omega=[0101,0010,$ $1101,0010,0101]$.
Nodes in the outer (resp., inner) circle of $\rm{(a)}$ or $\rm{(c)}$ denote the circulation of neuronal active states $0,0,1,0,0$ (which are ranged in clockwise order) of neuron $1$
(resp., the circulation of neuronal active states $1,0,1,0,1$ of neuron $2$)
(white: quiescent states; black: firing states),
whereas nodes in the outer (resp., inner) circle of $\rm{(b)}$ or $\rm{(d)}$ denote
the circulation of neuronal active states $0,1,0,1,0$ of neuron $3$ (resp., the circulation of neuronal active states $1,0,1,0,1$ of neuron $4$).
$\rm{(a)},\rm{(b)}$ Dashed lines indicate neurons changing from firing states to quiescent states,
whereas thick solid lines indicate neurons changing from quiescent states to firing states.
In $\rm{(a)}$, we obtain $\Upsilon_{12}(\Omega)=2$ because neurons $1$ and $2$ change their firing or
quiescent states synchronously in the lower right and lower left quadrants.
In $\rm{(b)}$, we obtain $\Upsilon_{34}(\Omega)=-4$ because neurons $3$ and $4$ change their firing or
quiescent states asynchronously in all the quadrants.
$\rm{(c)},\rm{(d)}$ Gray zones indicate neurons maintaining self-sustaining quiescent states ($\rm{(c)}$: $\rm{I,III,V}$; $\rm{(d)}$: $\rm{I,III,VI,VIII}$),
whereas white zones indicate neurons maintaining self-sustaining firing states ($\rm{(c)}$: $\rm{II,IV,VI}$; $\rm{(d)}$:
$\rm{II,IV,V,VII}$).
Because neuron $1$ changes its states only twice in the circulation and all other neurons change their states four times in the
circulation
($\Gamma_{12}(\Omega)=2<\Gamma_{34}(\Omega)=4$),
we obtain that neuron $1$ tends to maintain more self-sustaining activity than other neurons,
that is, the gray zone $\rm{V}$ of $\rm{(c)}$
{passes more quadrants than others} of $\rm{(c)}$ and $\rm{(d)} $.
}\label{fig2} \end{figure}
\subsection{Plasticity operators}
Let $\Theta$ be a collection of loops in $\{0,1\}^{n}$.
Denote by $\langle\cdot,\cdot\rangle$ the Hilbert-Schmidt inner product in
$M_{n}(\Real)$, i.e. if $A=(a_{ij})$ and $B=(b_{ij})\in M_{n}(\Real) $,
then $\langle A,B\rangle={\rm tr}(AB^{T})=\sum_{i,j}a_{ij}b_{ij} $.
Let $\varepsilon>0$ and let \begin{eqnarray}\label{eq02} &&\hspace{-0.7cm}\mathcal{D}(\Theta)=\ {\mathcal{D}_{\mathtt{SY}} +\mathcal{D}_{\mathtt{SK}};~\mathcal{D}_{\mathtt{SY}}=\mat hcal{D}_{\mathtt{SY}}^{T}\in M_{n}(\Real),\nonumber\\ &&\hspace{0cm}\mathcal{D}_{\mathtt{SK}}=-\mathcal{D}_{\mat htt{SK}}^{T}\in M_{n}(\Real),\mbox{ and }\\
&&\hspace{0cm}\langle
|\mathcal{D}_{\mathtt{SK}}|,\Gamma(\Omega)\rangle\leq\langle \ mathcal{D}_{\mathtt{SY}},\Upsilon(\Omega)\rangle-\varepsilon \mbox{ for each loop }\Omega\in\Theta\},\nonumber
\end{eqnarray}
where $|\mathcal{D}_{\mathtt{SK}}|=(| (\mathcal{D}_{\mathtt{SK}})_{ij}|)$.
The matrices $\mathcal{D}_{\mathtt{SY}}$ can be regarded as one sort of plasticity operators induced by synchronous activity of neurons,
whereas the matrices $\mathcal{D}_{\mathtt{SK}}$ can be regarded as another sort of plasticity operators induced by self-sustaining activity of neurons
The set $\mathcal{D}(\Theta)$ collects all the combining operation of $\mathcal{D}_{\mathtt{SY}}$ and
$\mathcal{D}_{\mathtt{SK}}$,
which will feed back to determine a dynamic clamp of network modification on $\mathcal{N}$.
\begin{example}\label{ex1} In $(\ref{eq02})$, we may select
$\mathcal{D}_{\mathtt{SY}}=\mathcal{R}+\mathcal{R}^{T}$, where $\mathcal{R}=(r_{ij})\in M_{n}(\Real)$ satisfies
\begin{equation}\label{eq13}
r_{ii}\geq \sum_{1\leq j\leq n,j\neq i}|r_{ij}|
+\varepsilon\quad\mbox{for each }i=1,2,\ldots,n. \end{equation}
Then, for each loop $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ in $\Theta$,
we conclude from $(\ref{eq13})$ that $\langle
\mathcal{D}_{\mathtt{SY}},\Upsilon(\Omega)\rangle\geq\sum_{1\l eq m\leq p}\left(\sum_{i\in \1(x^{m})\triangle \1(x^{m-1})}\right.$ $\left.\left(r_{ii}-\sum_{j\in \1(x^{m})\triangle \1(x^{m-1}),j\neq i}\left|r_{ij}\right|\right)\right)\geq\varepsilon.$
For such a choice of $\mathcal{D}_{\mathtt{SY}}$, let \begin{equation}\label{eq30} \gamma_{\Theta}=\min\{\left(\langle \mathcal{D}_{\mathtt{SY}},\Upsilon(\Omega)\rangle-\varepsilon\rig ht)/||\Gamma(\Omega)||; \Omega\in\Theta\}, \end{equation} where $||\Gamma(\Omega)||=\langle \Gamma(\Omega),\Gamma(\Omega)\rangle^{\frac{1}{2}}$. Then for any choice of $S\in M_{n}(\Real)$ with $S=-S^{T}$, we set $\mathcal{D}_{\mathtt{SK}}=\alpha S$, where $\alpha\in\Real$ is such that
\begin{equation}\label{eq05}
||\mathcal{D}_{\mathtt{SK}}||=\langle
\mathcal{D}_{\mathtt{SK}},\mathcal{D}_{\mathtt{SK}}\rangle^{ \frac{1}{2}}
\end{equation}
$(\ref{eq13})$ and $(\ref{eq05})$ show a rule for selecting
$\mathcal{D}_{\mathtt{SY}}$ and $\mathcal{D}_{\mathtt{SK}}$ such that $\mathcal{D}(\Theta)$ is nonempty.
\end{example}
\textit{Remark 1 (Linking to Hebbian synaptic plasticity):} In INCM assume that
the dynamic clamp of network modification on $\mathcal{N}$ is defined by
$A(t+1)-A(t)=\mathcal{D}_{\mathtt{SY}}
(t+1)+\mathcal{D}_{\mathtt{SK}}(t+1) \in\mathcal{D} (\Theta(t+1))$
for $t=0,1,\ldots,$ where $\Theta(t+1)$ is a collection of loops generated by the running of INCM at time $t+1$.
Furthermore, we associate to a loop $\Omega$ in $\Theta(t+1)$ a selection of $\mathcal{D}_{\mathtt{SY}}(t+1)$ and
$\mathcal{D}_{\mathtt{SK}}(t+1)$ such that \begin{eqnarray}\label{eq03} &\mathcal{D}_{\mathtt{SY}}(t+1)_{ij}> 0&\mbox{ if }\Upsilon_{ij}(\Omega)>0,\nonumber\\[-1.5ex]\\[-1.5ex] &\mathcal{D}_{\mathtt{SY}}(t+1)_{ij}< 0&\mbox{ if }\Upsilon_{ij}(\Omega)<0\nonumber \end{eqnarray}
for every $i,j=1,2,\ldots,n$ and \begin{equation}\label{eq04}
|\mathcal{D}_{\mathtt{SK}}(t+1)_{ij}|\cdot \Gamma_{ij}(\Omega) \propto c,\quad c\in\Real
\end{equation}
for every $i,j=1,2,\ldots,n$ with $\Gamma_{ij}(\Omega)\neq 0$. The selection above reveals a comprehensive view about the neurobiological perspective of Hebbian synaptic plasticity: \begin{itemize}
\item[1)] The selection $(\ref{eq03})$ shows that the connection between neuron $i$ and neuron $j$
is temporarily strengthened (i.e. $\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}>0$)
when neurons $i,j$ tend to change their firing or quiescent states synchronously in the loop $\Omega$
(i.e. $\Upsilon_{ij}(\Omega)>0$).
This reveals the synchronous firing aspect of Hebbian synaptic plasticity,
which is often summarized as ``when coupled neurons fire synchronously, the connection between them is strengthened" \cite{Hebb,Strogatz}.
\item[2)] The selection $(\ref{eq04})$ shows that
if neuron $i$ or $j$ tends to maintain more self-sustaining firing or self-sustaining quiescent states than
neuron $i'$ or $j'$ in the loop $\Omega$ (i.e. $\Gamma_{ij} (\Omega)<\Gamma_{i'j'}(\Omega)$),
then the change of connections between neurons $i,j$ takes place more easily
than that between neurons $i',j'$
(i.e. $|\mathcal{D}_{\mathtt{SK}}(t+1)_{ij}| >|\mathcal{D}_{\mathtt{SK}}(t+1)_{i'j'}|$).
This reveals the self-sustaining firing aspect of Hebbian synaptic plasticity,
which is stated as ``when an axon of cell $A$ is near enough to excite a cell $B$
and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells" \cite{Hebb}.
\end{itemize}
\subsection{Algorithm describing the execution process of operator control}
\begin{table*}[t!]
\caption{Suppose that $\Pi(\mathcal{M}(t),x(t+1))=\{x(t-4),x(t-3),\ldots,x(t+1)\}$.
$\rm{(a)}$ Since $x(t+1)\neq x(\kappa)$ for each $\kappa=t-4,t-3,\ldots,t$, we set $\eta(t+1)=-\zeta_{\max}$.
$\rm{(b)}$ Since $x(t+1)=x(t)$, we set $\eta(t+1)=t$. But $ [x(\eta(t+1)),x(t+1)]=[100,100]$ is not a loop.
$\rm{(c)}$ Since $x(t+1)=$ $x(t-3)$, we set $\eta(t+1)=t-3$, which points to the starting point of the loop
$[x(t-3),x(t-2),\ldots,x(t+1)]=[001,010,011,100,001]$.}\label{tab1} \begin{center} \footnotesize
\begin{tabular}{c||cccccc||c} \hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
$\Pi(\mathcal{M}(t),x(t+1))$&4)$ & 3)$ & 2)$ & $x(t-1)$ & $x(t)$ & $x(t+$x(t-1)$ & $\eta(t+$x(t-1)$\\\hline
(a)&000 & 001 & 010 & 011 & 100 & 101 & $-\zeta_{\max}$\\\hline (b)&000 & 001 & 010 & 011 & 100 & 100 & $t$\\\hline
(c)&000 & 001 & 010 & 011 & 100 & 001 & $t-3$\\\hline \end{tabular}
\end{center} \end{table*}
The algorithm that takes plasticity operators to modify network structure of $\mathcal{N}$
can be expressed by the function \begin{equation}\label{LDSS}
\mathcal{L}(\mathcal{P}(t),\Pi(\mathcal{M}
(t),x(t+1)))=(\mathcal{P}(t+1),\mathcal{M}(t+1)), \end{equation}
$t=0,1,\ldots,$ defined by three main components: \begin{itemize}
\item[1)] \textit{Initialization}: At time $t=0$, set
\begin{itemize}
\item[a)] a real number $\varepsilon>0$;
\item[b)] positive integers $\rho$, $\zeta_{\min}$, $\zeta_{\max}$ with $\zeta_{\min}<\zeta_{\max}$;
\item[c)] a collection of loops $\Theta(0):=\emptyset$. \end{itemize}
\item[2)] \textit{Loop collection}:
Establish a recursive procedure on each time $t=0,1,\ldots.$ \begin{itemize}
\item[a)]
For the given $\Pi(\mathcal{M}(t),x(t+1))$ in $(\ref{LDSS})$, which is rewritten by $x(t'),x(t'+1),\ldots,x(t+1)$, $t'\leq t$, set
\begin{eqnarray*}
&&\hspace{-1cm}\eta(t+1):=\max\{\kappa;~x(t+1)= x(\kappa),\\ &&\hspace{2cm}\kappa=t',t'+1,\ldots,t\}
\end{eqnarray*}
if there exists $\kappa\in\{t',t'+1,\ldots,t\}$ such that $x(t+1)=x(\kappa)$; otherwise $-\zeta_{\max}$.
Here $\eta(t+1)$ is a time index pointing to the starting point of the loop $[x(\eta(t+1)),x(\eta(t+1)+1),\ldots,x(t+1)]$
when $\eta(t+1)\neq -\zeta_{\max}$ and $\eta(t+1)\neq t$ (see Table \ref{tab1}). \item[b)] Set \begin{eqnarray*} &&\hspace{-0.7cm}\mathcal{M}(t+1)\\ &&\hspace{-0.5cm}:=\left\{ \begin{array}{l} x(t+1)\mbox{ if }\zeta_{\min}<t+1-\eta(t+1)\leq\zeta_{\max},\\ \Pi(\mathcal{M}(t),x(t+1))\mbox{ otherwise}.\\ \end{array} \right. \end{eqnarray*} \item[c)]
If $\zeta_{\min}<t+1-\eta(t+1)\leq\zeta_{\max}$, then set the loop \begin{equation}\label{eq23}
\Omega(t+1):=[x(\eta(t+1)),x(\eta(t+1)+1),\ldots,x(t+1)]; \end{equation}
Let $t+1-\eta(t+1)$ denote the length of the loop $\Omega(t+1)$ when $\Omega(t+1)\neq\emptyset$.
\item[d)]
Set the collection of loops \begin{equation}\label{eq24} \Theta(t+1):=\{\Omega(\kappa)\neq\emptyset;~\kappa= t-\rho+1,\ldots,t+1\}. \end{equation} \end{itemize} \item[3)] \textit{Adaptation}: For each $t=0,1,\ldots,$ \begin{itemize}
\item[a)]
if $\Omega(t+1)\neq\emptyset$, then set $$
\mathcal{D}(t+1):=
\mathcal{D}_{\mathtt{SY}}(t+1)+\mathcal{D}_{\mathtt{SK}} (t+1)\in\mathcal{D}(\Theta(t+1));
$$
otherwise $\mathcal{D}(t+1)$ is selected to be the null matrix in $M_{n}(\Real)$;
\item[b)] set $A(t+1):=A(t)+\mathcal{D}(t+1)$, $s(t+1)\subset\ {1,2,\ldots,$ $n\}$, and
the vector of control parameters $$
\mathcal{P}(t+1):=(A(t+1),s(t+1)). $$
\end{itemize} \end{itemize}
As a result, the function $\mathcal{L}$ is a kind of algorithm that defines a dynamic clamp of network modification (the adjustment of control parameters with plasticity operators)
\section{Circulation breaking}\label{sec2}
In Section \ref{secalg} we describe the adaptive plan which defines the conception of operator control of INCM.
In this section, we wish to show that operator control of INCM essentially involves the decirculation process in neural network dynamics.
For this, denote by $\Omega=[x^{0},x^{1},\ldots,x^{p}]$ a loop of states in $\{0,1\}^{n}$.
For every $i,j=$ $1,2,\ldots,n$, we assign an integer, denoted $c_{ij}(\Omega)$,
according to the rule: $$
c_{ij}(\Omega)=x^{0}_{j}(x^{0}_{i}-x^{1}_{i})+x^{1}_{j} (x^{1}_{i}-x^{2}_{i})
+\cdots+x^{p-1}_{j}(x^{p-1}_{i}-x^{p}_{i}). $$
We refer to the resulting matrix $C(\Omega)=(c_{ij}(\Omega))$ as the {decirculating map} of $\Omega$.
For example, let
$\Omega=[1111100000,0011$ $111000, 0000111110, 0111110000, 0001111100, 1111100000].$ Then \begin{eqnarray*} &&\hspace{-0.7cm}C(\Omega)=\\ &&\hspace{-0.7cm}\left( \begin{array}{rrrrrrrrrr}
1 & 1 & 1 & 0 & ~~0 & -1 & -1 & -1 &0 & ~~0 \\ 1 & 2 & 2 & 1 & 0 & -1 & -2 & -2 &-1 & 0 \\
0 & 1 & 2 & 1 & 0 & 0 & -1 & -2 &-1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 &-1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 \\ -1 & -1 & -1 & 0 & 0 & 1 & 1 & 1 &0 & 0 \\ -1 & -2 & -2 & -1 & 0 & 1 & 2 & 2 &1 & 0 \\ 0 & -1 & -2 & -1 & 0 & 0 & 1 & 2 &1 & 0 \\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 1 &1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 \\ \end{array}
\right).
\end{eqnarray*}
With the decirculating map $C(\Omega)$,
we are bound to consider the linear functional $A\To\langle A, C(\Omega)\rangle$
on the Hilbert space $M_{n}(\Real)$ of all real $n\times n$ matrices endowed with
the Hilbert-Schmidt inner product $\langle\cdot,\cdot\rangle$. \begin{theorem}[Tendency toward Circulation
Breaking]\label{th111}
Consider $\textrm{INCM}=(\mathcal{N},\mathcal{L},\Pi,\tau)$ with the adaptive plan $\mathcal{L}$ described in Section {\rm\ref{secalg}}.
Let $t_{1},t_{2},$ $\ldots,t_{\mu}\in\Nature$ be increasing time steps
and let $\Omega$ be a loop of states in $\{0,1\}^{n}$. Suppose that
\begin{eqnarray}\label{eq16}
&&\hspace{-0.4cm}\Omega=\Omega(t_{1}+1)=\Omega(t_{2}+1)=\cdots=\Om ega(t_{\mu}+1)\nonumber\\
&&\in\bigcap_{t_{1}\leq t\leq t_{\mu}}\Theta(t+1), \end{eqnarray}
where $\Omega(t_{l}+1)=[x(\eta(t_{l}+1)),x(\eta(t_{l} +1)+1),\ldots,x(t_{l}+1)]$ for $l=1,2,\ldots,\mu$.
Then
\begin{eqnarray}\label{eq18}
&&\hspace{-1.9cm}\langle A(t_{1}+1), C(\Omega)\rangle<\langle A(t_{2}+1), C(\Omega)\rangle\nonumber\\ &&\hspace{0.9cm}<\cdots<\langle A(t_{\mu}+1), C(\Omega)\rangle, \end{eqnarray} where \begin{eqnarray*} A(t_{l+1}+1)=A(t_{l}+1)+\sum_{t_{l}< t\leq t_{l+1}}\mathcal{D}(t+1) \end{eqnarray*}
for each $l=1,2,\ldots,\mu-1$.
Furthermore, if $A(t_{\mu}+1)$ displays the threshold \begin{equation}\label{eq19}
\langle A(t_{\mu}+1), C(\Omega)\rangle\geq0, \end{equation}
then $\Omega(t+1)\neq \Omega$ for $t=t_{\mu}+1,t_{\mu} +2,\ldots,t_{\mu}+\rho$.
\end{theorem} \begin{figure}[t!] \center
\includegraphics[height=1.2in,width=3in]{circu55.eps}
\caption{Schematic illustration of the process of circulation breaking in neural network dynamics.
The nodes denote the vectors $x(t)$ of neuronal active states flowing in the direction indicated by the dashed arrows $1$, $2$, and $3$.
There are two loops $\Omega$ and $\Omega'$ of neuronal active states alternately occurring
at time steps $t_{1},t_{1}',t_{2},t_{2}',\ldots,t_{\mu},t_{\mu}'$, that is,
$\Omega(t_{l}+1)=[x(\eta(t_{l}+1)),x(\eta(t_{l} +1)+1),\ldots,x(t_{l}+1)]=\Omega$
and
$\Omega(t_{l}'+1)=[x(\eta(t_{l}'+1)),x(\eta(t_{l}'+1)+1),\ldots,x(t_ {l}'+1)]=\Omega'$ for each $l=1,2,\ldots,\mu$.
The network structure of $\mathcal{N}$ will be modified by the adaptive plan $\mathcal{L}$
in the sequential order $A(t_{1}+1),A(t_{1}'+1),\ldots,A(t_{\mu} +1),A(t_{\mu}'+1)$.
By Theorem $\ref{th111}$, this implies two increasing sequences $\langle A(t_{l}+1), C(\Omega)\rangle<\langle A(t_{l+1}+1), C(\Omega)\rangle$
and $\langle A(t_{l}'+1), C(\Omega')\rangle<\langle A(t_{l+1}'+1), C(\Omega')\rangle$ for each $l=1,2,\ldots,\mu-1$.
Suppose that the increasing sequence $\langle A(t_{l}+1), C(\Omega)\rangle, l=1,2,\ldots,\mu,$
displays the threshold $\langle A(t_{\mu}+1), C(\Omega)\rangle\geq0$.
Then Theorem $\ref{th111}$ implies that
the circulation of neuronal active states in the loop $\Omega$ will be broken temporarily, that is,
$\Omega(t+1)\neq \Omega$ for $t=t_{\mu}+1,t_{\mu} +2,\ldots,t_{\mu}+\rho$.
}\label{figcircu} \end{figure}
The proof of Theorem \ref{th111} can be found in the Appendix. Theorem \ref{th111} shows that, when the loop $\Omega$ of neuronal active states occurs at time $t_{1}+1,t_{2}+1,$ $\ldots,t_{\mu}+1$,
the dynamic clamp of network modification defined by the adaptive plan $\mathcal{L}$
will conduct an increasing sequence induced by the decirculating map $C(\Omega)$ in $(\ref{eq18})$,
which may display the threshold $(\ref{eq19})$ with $\Omega(t+1)\neq \Omega$ for $t=t_{\mu}+1,t_{\mu} +2,\ldots,t_{\mu}+\rho$, showing that neurons in INCM
can alter their connections with the tendency toward circulation breaking (see Fig. \ref{figcircu}).
\section{Experimental tests for operator control}\label{tests} An experimental result is reported to validate that
single neurons of INCM can produce rhythmic, oscillatory activity but,
at the level of neural ensembles, generate multiphasic patterns of population responses.
For this, we consider the parametric setting of INCM defined by Example \ref{ex1} and Table \ref{tab2}.
With the parametric setting, plasticity operators will be selected in the execution process of operator control
and, based on Theorem \ref{th111},
INCM will tend to break the circulation of neuronal active states in the loops of length greater than or equal to 12
(this is because $\zeta_{\min}=12$, $\zeta_{\max}=100$, and $T_{k}=100k$),
generating a testing sample of operator control of INCM until the running time $t=100000$.
\begin{table*}[t!]
\caption{Parametric setting of INCM.}\label{tab2} \begin{center} \footnotesize
\begin{tabular}{cc} \hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... &\vspace{-0.25cm}\\ Parameters&Setting\\\hline\hline &\vspace{-0.25cm}\\ $n$&$17$\\\hline &\vspace{-0.25cm}\\ $t$&$0,1,\ldots,100000$\\\hline &\vspace{-0.25cm}\\ $k$&$0,1,\ldots,1000$\\\hline &\vspace{-0.25cm}\\ $T_{k}$&$100k$\\\hline &\vspace{-0.25cm}\\ $\tau$&$15$\\\hline &\vspace{-0.25cm}\\ $\varepsilon$&$0.0001$\\\hline &\vspace{-0.25cm}\\ $\rho$&$5000$\\\hline &\vspace{-0.25cm}\\ $\zeta_{\min}$&$12$\\\hline &\vspace{-0.25cm}\\ $\zeta_{\max}$&$100$\\\hline &\vspace{-0.25cm}\\ $b=(b_{1},b_{2},\ldots,b_{n})$&$b_{i}\in(-4,4),$ $i=1,2,\ldots,n$\\\hline &\vspace{-0.25cm}\\
$x(0)$&randomly chosen from $\{0,1\}^{n}$\\\hline &\vspace{-0.25cm}\\
$A(0)=(a_{ij}(0))$&$a_{ij}(0)\in(-15,15)$ if $i\neq j$; otherwise $a_{ii}(0)=-22$\\\hline &\vspace{-0.25cm}\\ $x(t+1)$&equation $(\ref{dynamics})$\\\hline &\vspace{-0.25cm}\\ $\Omega(t+1)$&equation $(\ref{eq23})$\\\hline &\vspace{-0.25cm}\\
$\mathcal{R}(t+1)$&(i) $r_{ij}(t+1)\in(-0.0006,0.0006)$ if $i\neq j$; otherwise $r_{ii}(t+1)=\sum_{1\leq j\leq n,j\neq i}|r_{ij}(t+1)| +\varepsilon$;\\
$=(r_{ij}(t+1))$&(ii) $r_{ij}(t+1) > 0$ if $\Upsilon_{ij} (\Omega(t+1))>0$; $r_{ij}(t+1) < 0$ if $\Upsilon_{ij} (\Omega(t+1))<0$\\\hline &\vspace{-0.25cm}\\ $\mathcal{D}_{\mathtt{SY}}(t+1)$&$\mathcal{R} (t+1)+\mathcal{R}(t+1)^{T}$\\\hline &\vspace{-0.25cm}\\ $\Theta(t+1)$&equation $(\ref{eq24})$\\\hline &\vspace{-0.25cm}\\ $\gamma_{\Theta(t+1)}$&equation $(\ref{eq30})$\\\hline &\vspace{-0.25cm}\\
$S(t+1)=(s_{ij}(t+1))$&(i) $S(t+1)=-S(t+1)^{T}$; (ii) $|s_{ij} (t+1)|\cdot \Gamma_{ij}(\Omega(t+1)) =1$ if $s_{ij}(t+1)\neq 0$ and $\Gamma_{ij}(\Omega(t+1))\neq 0$\\\hline
&\vspace{-0.25cm}\\ $\mathcal{D}_{\mathtt{SK}}(t+1)$&$(\gamma_{\Theta(t+1)}/|| S(t+1)||)\cdot S(t+1)$\\\hline &\vspace{-0.25cm}\\ $A(t+1)$&$A(t)+\mathcal{D}_{\mathtt{SY}} (t+1)+\mathcal{D}_{\mathtt{SK}}(t+1)$\\\hline &\vspace{-0.25cm}\\
$s(t+1)$&reset $s_{i}(t+1)=1$ with probability $0.9$, $i
=1,2,\ldots,n$; $s(t+1)=(s_{1}(t+1),s_{2}(t+1),\ldots,s_{n}(t+1)) $\\\hline
\end{tabular} \end{center} \end{table*}
To show that single neurons of INCM can produce rhythmic, oscillatory activity,
we consider the base-10 representation $y(t)$ of any $01$-string $x(t)$ distributed averagely in the interval $[0,100]$,
that is, $$
y(t)=\left(\sum_{1\leq i\leq n}2^{i-1}x_{i} (t)\right)\cdot\left(\frac{100}{2^{n}}\right). $$
The upper-half parts in Fig. \ref{figactivity}(a), (b), and (c)
depict the distribution of $y(t)$ from time $t=0$ to $t=1000$, from time $t=49000$ to $t=50000$, and from time $t=99000$ to
$t=100000$, respectively.
Comparing the distribution of $y(t)$ in Fig. \ref{figactivity}(a) with that in Fig. \ref{figactivity}(c),
we obtain that $y(t)$ tends to gather around the points $53$, $60$, $68$, $72$, and $80$ as time $t$ tends to $100000$.
The gathering of $y(t)$ signifies the rhythm of neuronal active states,
meaning that single neurons oscillate in isolation but cooperate as a whole to generate rhythm.
\begin{figure}[t!] \center
\includegraphics[height=1.85in,width=3.3in]{01.eps}\\ \includegraphics[height=1.85in,width=3.3in]{50.eps}\\ \includegraphics[height=1.85in,width=3.3in]{100.eps}\\ \caption{Distribution of neuronal active states and the corresponding neural activity counts.
(a) Time increases from $t=0$ to $t=1000$. (b) Time increases from $t=49000$ to $t=50000$. (c) Time increases from $t=99000$ to $t=100000$.}\label{figactivity}
\end{figure}
To show that INCM can generate multiphasic patterns of population responses,
we consider the neural activity counts defined by the slide window of the component $\tau$ (see Section \ref{comtau}),
that is, $$
f_{\tau}(t+1)=\sum_{t-\tau+1< T\leq t+1}\sum_{i\in \mathcal{S}}x_{i}(T),\quad t=0,1,\ldots,
$$
where $\tau=15$ and $\mathcal{S}=\{1,2,\ldots,17\}$. The lower-half parts in Fig. \ref{figactivity}(a), (b), and (c)
depict the distribution of the neural activity counts from time $t=0$ to $t=1000$, from time $t=49000$ to $t=50000$, and from time $t=99000$ to $t=100000$, respectively.
Comparing all the distribution of the neural activity counts in Fig. \ref{figactivity},
we obtain no remarkable difference except the multiphasic patterns and
{the increase of the averages}
(In Fig. \ref{figactivity}(a), (b), and (c), the averages of the neural activity counts are $105$, $110$, and $125$, respectively).
Multiphasic patterns of the neural activity counts exhibit a great variety of neuronal votes (approximatively simultaneous firing) among neurons,
comprised of distinct functional and transient neural ensembles for generating behavioral outputs \cite{Nicolelis2009,Quiroga}.
Whereas, the increase of the averages of the neural activity counts exhibits more and more participation of firing neurons in the
generation of behavioral outputs. \begin{figure}[t!] \center \includegraphics[height=1.4in,width=1.68in]{101.eps} \includegraphics[height=1.4in,width=1.68in]{102.eps}\\ \includegraphics[height=1.4in,width=1.68in]{103.eps} \includegraphics[height=1.4in,width=1.68in]{104.eps}\\ \includegraphics[height=1.4in,width=1.68in]{105.eps} \includegraphics[height=1.4in,width=1.68in]{106.eps} \caption{Simulation results of operator control of INCM. For the testing sample described in Fig. $\ref{figactivity}$,
we obtain $\rm{(a)}$ the increasing polynomial curve fitting to the averages of the neural activity counts;
$\rm{(b)}$ the high variances of the neural activity counts; and $\rm{(c)}$ the increasing polynomial curve fitting to the clusters of neuronal active states.
For the simulation taken over 1000 testing samples of operator control of INCM,
we obtain that $\rm{(d)}$
{the average polynomial curve} fitting to the averages of the neural activity counts is increasing;
$\rm{(e)}$ the average polynomial curve fitting to the variances of the neural activity counts is between 26.8 and 29.5;
and $\rm{(f)}$ the average polynomial curve fitting to the clusters of neuronal active states is increasing.}\label{figtrain}
\end{figure}
As a result, the upper-half and the lower-half parts in Fig. \ref{figactivity}(a), (b), and (c) together suggest that
decirculation is a simple dynamical process that enforces more and more single neurons in INCM
to cooperate as a whole to generate rhythm as well as to generate multiphasic patterns of the neural activity counts.
To demonstrate the above assertion, we consider the following quantities:
for $k=1,2,\ldots,1000$ we define the average of the neural activity counts from time $100(k-1)$ to $100k$ by
$$
Ave(100k)=\left(\sum_{100(k-1)\leq t< 100k}f_{\tau} (t+1)\right)/100,
$$
the variance of the neural activity counts from time $100(k-1)$ to $100k$ by \begin{eqnarray*} &&\hspace{-0.8cm}Var(100k)\\ &&\hspace{-0.5cm}=\left(\sum_{100(k-1)\leq t< 100k}\left(f_{\tau} (t+1)-Ave(100k)\right)^{2}\right)/100, \end{eqnarray*}
and the cluster of neuronal active states from time $100(k-1)$ to $100k$ by
\begin{eqnarray*} &&\hspace{-0.8cm}Clu(100k)\\ &&\hspace{-0.5cm}=\left(\sharp\{t;~ \theta(t+1)>0.08,~100(k-1)\leq t< 100k\}\right)/100, \end{eqnarray*} where $\theta(t+1)=\left(\sharp\{l;~|y(l+1)-y(t+1)|<3,~100(k-1)\right.$ $\left.\leq l< 100k\}\right)/100$
is a quantity measuring the cluster degree of the point $y(t+1)$. Fig. \ref{figtrain} shows the simulation results of operator control of INCM.
It reveals more and more participation of firing neurons, the high variances of the neural activity counts, and the gathering of neuronal active states
in INCM, validating how well INCM can mimic neural population dynamics under operator control.
\section{Conclusion}
In the search for principles of pattern generation {of INCM},
the operator control approach is presented that encompasses the construct of plasticity operators,
the process of circulation breaking, and the experiment of INCM mimicking neural population dynamics.
The novelty of building INCM rests on its practical ability to
take plasticity operators to modify random inter-neural connections and to control neural population dynamics.
Such level of operator control is a way beyond anything
that can currently be achieved in actual neural systems---controlling one or two neurons can be
done with a dynamic clamp, for example, but not with a whole population.
This suggests an initial but critical step toward the construct and the control of neural network computing machines
regarding the use of oscillatory and multiphasic patterns in neural population dynamics.
\appendix[Proof of Theorem \ref{th111}]
To prove Theorem $\ref{th111}$, we need to show that \begin{itemize}
\item[1)] the Hilbert-Schmidt inner product $\langle A, C(\Omega)\rangle$
is invariant under certain perturbations of the matrix $A\in M_{n} (\Real)$;
\item[2)] a surgical operation on $\Omega$ exists for counteracting the complexity for computing
$\langle A(t_{l+1}+1)-A(t_{l}+1), C(\Omega)\rangle$. \end{itemize}
To prove 1), we consider the {symmetric perturbative matrix}
$\widetilde{D}^{ij}_{\delta}=({\widetilde{d}^{ij}}_{rl})\in M_{n} (\Real)$ defined by
$i,j\in\{1,2,\ldots,n\}$, $\delta\in\Real$, and \begin{equation}\label{eq08}
{\widetilde{d}^{ij}}_{rl}=\delta~\mbox{ if } (r,l)=(i,j) \mbox{ or } (r,l)=(j,i);
\mbox{ otherwise } {\widetilde{d}^{ij}}_{rl}=0, \end{equation}
where $r,l\in\{1,2,\ldots,n\}.$
\begin{lemma}[Invariance of Perturbations]\label{lem3}
Let $\Omega=[x^{0},x^{1},$ $\ldots,x^{p}]$ be a loop of states in
$\{0,1\}^{n}$. Let $A\in M_{n}(\Real)$ and let $i,j\in\{1,2,\ldots,n\}$.
If %$(i,j)$ does not lie in the set \begin{equation}\label{teq08}
(i,j)\not\in \bigcup_{1\leq m\leq p}\left((\1(x^{m-1})\triangle \ 1(x^{m}))\times (\1(x^{m-1})\triangle \1(x^{m}))\right), \end{equation}
then for any $\delta\in\Real$ $$
\langle A+\widetilde{D}^{ij}_{\delta},C(\Omega)\rangle =\langle A,C(\Omega)\rangle.
\end{lemma} \begin{IEEEproof}
Let $\delta\in\Real$ and fix $i,j\in\{1,2,\ldots,n\}$ satisfying $ (\ref{teq08})$.
Suppose that $M_{i}(\Omega)=\emptyset$ or $M_{j} (\Omega)=\emptyset$.
Then %Since $\Omega$ is a loop of states in $\{0,1\}^{n}$, it is readily seen that
$c_{ij}(\Omega)=-c_{ji}(\Omega)=0$ and
$\langle A+\widetilde{D}^{ij}_{\delta},C(\Omega)\rangle=\langle A,C(\Omega)\rangle.$
On the other hand, suppose that $M_{i}(\Omega)\neq\emptyset$ and $M_{j}(\Omega)\neq\emptyset$.
Write
$M_{j}(\Omega)=\{j_{1},j_{2},\ldots,j_{\sharp M_{j}(\Omega)}\},$ where $j_{1}<j_{2}<\cdots<j_{\sharp M_{j}(\Omega)}$ and
$\sharp M_{j}(\Omega)$ is even.
Suppose that all the elements in $M_{j}(\Omega)$ are such that \begin{eqnarray}\label{eqq2} &x^{j_{1}}_{j}=x^{j_{3}}_{j}=\cdots=x^{j_{(\sharp M_{j} (\Omega)-1)}}_{j}=0,& \nonumber\\[-1.5ex]\\[-1.5ex] &x^{j_{2}}_{j}=x^{j_{4}}_{j}=\cdots=x^{j_{\sharp M_{j} (\Omega)}}_{j}=1.&\nonumber \end{eqnarray}
Consider the shift function $\sigma$ on $\{1,2,\ldots, p\}$ given by \begin{equation}\label{eq55}
\sigma(k)\equiv k+1\mod p \end{equation}
for each $k=1,2,\ldots,p.$
For each $k,k'\in\{1,2,\ldots,p\}$, define the set \begin{equation}\label{eq56}
[k,k']=\{k,\sigma(k),\ldots,\sigma^{\lambda(k,k')}(k)\}, \end{equation}
where $\lambda(k,k')=k'-k$ if $k'\geq k$; otherwise $\lambda(k,k')=p+k'-k$.
\begin{eqnarray}\label{eq57}
&(k,k']=[k,k']\setminus\{k\},~[k,k')=[k,k']\backslash\ {k'\},&\nonumber\\
&(k,k')=[k,k']\backslash\{k, k'\},& \end{eqnarray}
and $j_{(\sharp M_{j}(\Omega)+1)}=j_{1}$. Then the computation of $c_{ij}(\Omega)$ can be expressed by
\begin{eqnarray*}
&&\hspace{-1cm}c_{ij}(\Omega)
=\sum_{1\leq k\leq \sharp M_{j}(\Omega)}
\left(\sum_{m\in(j_{k},j_{k+1})}\left(x_{j}^{m}x_{i}^{m}-x_{j}^{m-1}x_{i}^{m}\right)\right.\nonumber\\
&&\hspace{0.5cm}+\left.\left(x_{j}^{j_{k}}x_{i}^{j_{k}}-x_{j}^{j_{k}-1}x_{i}^{j_{k}}\right)\right).
\end{eqnarray*}
Furthermore, by $(\ref{eqq1})$ and $(\ref{eqq2})$, we obtain that
for each $k=1,2,\ldots,$ $\sharp M_{j}(\Omega),$ if $ (j_{k},j_{k+1})\neq\emptyset$ then \begin{eqnarray*} \sum_{m\in(j_{k},j_{k+1})} \left(x_{j}^{m}x_{i}^{m}-x_{j}^{m-1}x_{i}^{m}\right)=0. \end{eqnarray*} Therefore, by $(\ref{eqq2})$, we see that \begin{eqnarray}\label{eqij} &&\hspace{-1.3cm}c_{ij}(\Omega)=\sum_{k=2,4,\ldots,\sharp M_{j}(\Omega)}x_{i}^{j_{k}} -\sum_{k=1,3,\ldots,\sharp M_{j}(\Omega)-1}x_{i}^{j_{k}}\nonumber\\ &&=\sum_{k=2,4,\ldots,\sharp M_{j}(\Omega)} \left(x_{i}^{j_{k}}-x_{i}^{j_{k+1}}\right). \end{eqnarray}
To compute $c_{ji}(\Omega)$, first let \begin{equation}\label{eq26}
\Lambda=\{k;~[j_{k},j_{k+1})\cap M_{i}(\Omega)\neq\emptyset\}. \end{equation}
for each $k\not\in\Lambda,$ if $[j_{k},j_{k+1})\neq\emptyset,$ then $$ \sum_{m\in[j_{k},j_{k+1})} \left(x_{i}^{m}x_{j}^{m}-x_{i}^{m-1}x_{j}^{m}\right)=0. $$
Therefore, the computation of $c_{ji}(\Omega)$ can be expressed by \begin{eqnarray}\label{eqq9} &&\hspace{-1.3cm}c_{ji} (\Omega)=\sum_{k\in\Lambda}\sum_{m\in[j_{k},j_{k+1})} \left(x_{i}^{m}x_{j}^{m}-x_{i}^{m-1}x_{j}^{m}\right)\nonumber\\ &&+\sum_{k\not\in\Lambda}\sum_{m\in[j_{k},j_{k+1})}\left(x_{i} ^{m}x_{j}^{m}-x_{i}^{m-1}x_{j}^{m}\right)\\ &&=\sum_{k\in\Lambda}\sum_{m\in[j_{k},j_{k+1})} \left(x_{i}^{m}x_{j}^{m}-x_{i}^{m-1}x_{j}^{m}\right).\nonumber \end{eqnarray}
Let $k\in\Lambda$ and let \begin{equation}\label{eq25}
[j_{k},j_{k+1})\cap M_{i}(\Omega)=\ {i_{k_{1}},i_{k_{2}},\ldots,i_{k_{r(k)}}\} \end{equation}
such that for each $l=1,2,\ldots,r(k)$, $\
{i_{k_{l}},i_{k_{l+1}},\ldots,i_{k_{r(k)}}\}\subset[i_{k_{l}},j_{k+ 1}).$
Since $x_{i}^{m-1}=x_{i}^{m}$
for each $m\in[j_{k},j_{k+1})\setminus M_{i}(\Omega)$, we have \begin{eqnarray}\label{eqq10} &&\hspace{- 1cm}\sum_{m\in[j_{k},j_{k+1})}\left(x_{i}^{m}x_{j}^{m}-x_{i}^{m-1}x_{j}^{m}\right)\nonumber\\ &&\hspace{0.5cm}=\sum_{1\leq l\leq r(k)} x_{j}^{i_{k_{l}}}\left(x_{i}^{i_{k_{l}}}-x_{i}^{i_{k_{l}}-1}\right).
\end{eqnarray}
Furthermore, by $(\ref{eqq1})$ and $(\ref{eq25})$, we have \begin{eqnarray}\label{eq28}
&&\hspace{-1cm}x_{i}^{i_{k_{l}}-1}=x_{i}^{i_{k_{l+1}}}~\mbox{ for each }l=1,2,\ldots,r(k)-1,\nonumber\\[-1.5ex]\\[-1.5ex]
&&\hspace{-1cm}x_{i}^{i_{k_{l}}}=x_{i}^{i_{k_{l+1}}-1}~\mbox{ for each }l=1,2,\ldots,r(k)-1.\nonumber
\end{eqnarray}
Thus, for $r(k)$ is even, we have \begin{equation}\label{eq91}
\sum_{1\leq l\leq r(k)}x_{j}^{i_{k_{l}}}\left(x_{i}^{i_{k_{l}}}-x_{i}^{i_{k_{l}}-1}\right)=0.
\end{equation}
For $r(k)$ is odd, we have \begin{equation}\label{eq92}
\sum_{1\leq l\leq r(k)}x_{j}^{i_{k_{l}}}\left(x_{i}^{i_{k_{l}}}-x_{i}^{i_{k_{l}}-1}\right)
=x_{j}^{i_{k_{r(k)}}}\left(x_{i}^{i_{k_{r(k)}}}-x_{i}^{i_{k_{r(k)}}-1}\right).
\end{equation}
Let $\bar{\Lambda}=\{k;~k\in\Lambda, k \mbox{ is even},\mbox{ and } r(k)\mbox{ is odd}\}.$
Combining $(\ref{eqq1})$, $(\ref{eqq2})$, $(\ref{eqq9})$, $ (\ref{eqq10})$, $(\ref{eq91})$, and $(\ref{eq92})$ gives \begin{eqnarray}\label{eqji} &&\hspace{-1cm}c_{ji}(\Omega)=0+\sum_{k\in\Lambda, r(k)\mbox{ is odd}}x_{j}^{i_{k_{r(k)}}}\left(x_{i}^{i_{k_{r(k)}}}-x_{i}^{i_{k_{r(k)}}-1}\right)\nonumber\\ &&=0+\sum_{k\in\bar{\Lambda}}\left(x_{i}^{i_{k_{r(k)}}}-x_{i}^{i_{k_{r(k)}}-1}\right). \end{eqnarray}
Now, fix $k\in\{2,4,\ldots,\sharp M_{j}(\Omega)\}$, and consider the term
\begin{eqnarray*} \lambda_{ij}(k)= \left\{
\begin{array}{ll} x_{i}^{j_{k}}-x_{i}^{j_{k+1}}&\mbox{ if } ~k\not\in\bar{\Lambda}\\ x_{i}^{j_{k}}-x_{i}^{j_{k+1}}+x_{i}^{i_{k_{r(k)}}}-x_{i}^{i_{k_{r(k)}}-1}&\mbox{ if }~k\in\bar{\Lambda}\\ \end{array} \right. \end{eqnarray*}
deriving from the partial sum of $(\ref{eqij})$ and $(\ref{eqji})$. Then, by $(\ref{eqq1})$, $(\ref{eq26})$, and $(\ref{eq28})$, we have $\lambda_{ij}(k)=0$.
This shows that $$ c_{ij}(\Omega)+c_{ji}(\Omega) =\sum_{k=2,4,\ldots,\sharp M_{j}(\Omega)}\lambda_{ij}(k)=0 $$ and, hence, $\langle A+\widetilde{D}^{ij}_{\delta},C(\Omega)\rangle =\langle A,C(\Omega)\rangle,$
proving the assertion. \end{IEEEproof}
Fig. $\ref{figSug}$ depicts the surgical operation on $\Omega$, which conducts Lemma \ref{lem4}.
By using Lemma \ref{lem3} and Lemma \ref{lem4},
we can counteract the complexity of computation described in 2). \begin{lemma}[Surgical Operation on Loops]\label{lem4}
Let $\Omega=[x^{0},$ $x^{1},\ldots,x^{p}]$ be a loop of states in $\{0,1\}^{n}$
and let $i,j\in\{1,2,$ $\ldots,n\}$ with $i\neq j$. Suppose that
$$
(i,j)\in \bigcup_{1\leq m\leq p}\left((\1(x^{m-1})\triangle \ 1(x^{m}))\times (\1(x^{m-1})\triangle \1(x^{m}))\right) $$
and
M_{i}(\Omega)\cap M_{j}(\Omega)=\ {m_{1},m_{2},\ldots,m_{\varsigma}\} \end{equation}
with $1\leq m_{1}<m_{2}<\cdots<m_{\varsigma}\leq p$. Construct
$\widetilde{\Omega}^{ij}=[y^{0},y^{1},\ldots,y^{p+\varsigma}] $
as a loop of states in $\{0,1\}^{n}$ according to the surgical operation on $\Omega$:
\begin{eqnarray}\label{eeq31}
&y^{m}=x^{m} &\mbox{ for each }m=0,1,\ldots,m_{1}-1,\nonumber\\
&y^{m+k}=x^{m} &\mbox{ for each }m=m_{k},m_{k} +1,\ldots,m_{k+1}-1\nonumber\\
&&\mbox{ and } k=1,2,\ldots,\varsigma-1,\nonumber\\ &y^{m+\varsigma}=x^{m} &\mbox{ for each } m=m_{\varsigma},m_{\varsigma}+1\ldots,p, \end{eqnarray} and \begin{eqnarray}\label{eeq35} &&\hspace{-1cm}y^{m_{k}+k-1}_{l}\nonumber\\ &&\hspace{-0.6cm}=\left\{ \begin{array}{rl} 1&\mbox{if }l\in\left(\1(x^{m_{k}-1})\cap\ {i\}\right)\cup\left(\1(x^{m_{k}})\cap\{j\}\right)\\ 0&\mbox{otherwise} \end{array}\right. \end{eqnarray}
for each $l=1,2,\ldots,n$ and $k=1,2,\ldots,\varsigma$. Then
\begin{equation}\label{eq31}
(i,j)\not\in \bigcup_{1\leq m\leq p+\varsigma}\left((\1(y^{m-1})\triangle \1(y^{m}))\times (\1(y^{m-p+\varsigma}\left((\1(y^{m-1})\triangle \
1(y^{m}))\right) \end{equation}
and for any $\delta\in\Real$ and $A\in M_{n}(\Real)$ \begin{equation}\label{eq35}
\langle A+\widetilde{D}^{ij}_{\delta},C(\widetilde{\Omega}^{ij})\rangle =\langle A,C(\widetilde{\Omega}^{ij})\rangle. \end{equation} \end{lemma} \begin{figure}[t!] \center \includegraphics[height=1.3in,width=3.5in]{sugry.eps} \caption{Surgical operation on $\Omega=[x^{0},x^{1},\ldots,x^{p}]$.
$\rm{(a)}$ For clarity, $\Omega=[01,10,11,00,01]$, $i=1$, and $j=2$ are shown here.
Nodes in the outer (resp., inner) circle denote the circulation of
neuronal active states $0, 1, 1, 0, 0$ (which are ranged in clockwise order) of neuron $i$
(resp., the circulation of neuronal active states $1, 0, 1, 0, 1$ of neuron $j$) (white: quiescent states; black: firing states).
Gray areas indicate all the positions $m$ (here $m=1$ or $3$) such that
$(i,j)\in\left(1})\triangle \1(x^{m}))\times (\1(x^{m-1})\triangle \1(x^{m}))\right)$.
$\rm{(b)},\rm{(c)}$ Surgical operation on $\Omega$ comprises three main steps.
First, keep all the nodes in the outer and inner circles (expressed by $(\ref{eeq31})$).
Second, cut all the curves in the gray areas (indicated by ``$\times$").
Third, insert new nodes to connect the cut curves (indicated by arrows and expressed by $(\ref{eeq35})$).
Then we obtain the loop
$\widetilde{\Omega}^{ij}=[01,00,10,11,10,00,01]$ satisfying $ (\ref{eq31})$,
showing that no gray areas exist in $\rm{(c)}$. }\label{figSug}
\end{figure} \begin{IEEEproof}
The assertion $(\ref{eq35})$ follows from Lemma \ref{lem3} and $ (\ref{eq31})$,
so we need to prove $(\ref{eq31})$. Let $i,j\in\{1,2,\ldots,n\}$ with $i\neq j$. For each $m\in\{1,2,\ldots,p+\varsigma\}$.
Suppose that $m=m_{k}+k-1$, where $k=1,2,\ldots,\varsigma$. Then, by $(\ref{eeq31})$ and
$(\ref{eeq35})$, we have \begin{eqnarray}\label{eeq56} &&\hspace{-0.6cm}\1(y^{m-1})\triangle \ 1(y^{m})=(((\1(x^{m_{k}})\cap\{j\})\setminus \1(x^{m_{k}-1}))\nonumber\\ &&\cup(\1(x^{m_{k}-1})\setminus \ {i\}))\cap(((\1(x^{m_{k}})\cap\{j\})\setminus \1(x^{m_{k}-1}))\nonumber\\ &&\cup(\1(x^{m_{k}-1})\setminus(\1(x^{m_{k}})\cap\{j\}))). \end{eqnarray}
Suppose that $m=m_{k}+k$, where $k=1,2,\ldots,\varsigma$. Then, by $(\ref{eeq31})$ and $(\ref{eeq35})$, we have
\begin{eqnarray}\label{eeq57} &&\hspace{-0.6cm}\1(y^{m-1})\triangle \ 1(y^{m})=((\1(x^{m_{k}})\setminus(\1(x^{m_{k}-1})\cap\ {i\}))\nonumber\\ &&\cup((\1(x^{m_{k}-1})\cap\{i\})\setminus \ 1(x^{m_{k}})))\cap((\1(x^{m_{k}})\setminus\{j\})\nonumber\\ &&\cup((\1(x^{m_{k}-1})\cap\{i\})\setminus\1(x^{m_{k}}))). \end{eqnarray}
Since $i\neq j$, it follows from $(\ref{eeq56})$ and $(\ref{eeq57})$ that
$(\1(y^{m-1})\triangle$ $\1(y^{m}))\cap\{i\}=\emptyset$, establishing
$$
(i,j)\not\in 1})\triangle \1(y^{m}))\times (\1(y^{m-1})\triangle \1(y^{m})).
$$
On the other hand, suppose that $m=1,2,\ldots,p+\varsigma$ with $m\neq
m_{k}+k-1$ and $m\neq m_{k}+k$ for each $k=1,2,\ldots,\varsigma$.
Then, by $(\ref{eeq31})$, we can find $m'\in\{1,2,\ldots,p\}\setminus\
{m_{1},m_{2},\ldots,m_{\varsigma}\}$ such that \begin{equation}\label{eeq44}
\1(y^{m-1})\triangle \1(y^{m})=\1(x^{m'-1})\triangle \1(x^{m'}). \end{equation}
Since $m'\not\in \{m_{1},m_{2},\ldots,m_{\varsigma}\}$, it follows from $(\ref{qq10})$ and $(\ref{eeq44})$ that
\begin{eqnarray*}
&&\hspace{-0.8cm}(i,j)\not\in (\1(x^{m'-1})\triangle \ 1(x^{m'}))\times (\1(x^{m'-1})\triangle \1(x^{m'}))\\ &&=1})\triangle \1(y^{m}))\times (\1(y^{m-1})\triangle \1(y^{m})),
\end{eqnarray*} proving the assertion. \end{IEEEproof}
We now proceed to the proof of Theorem \ref{th111}. \textit{Proof of Theorem {\ref{th111}}:}
Let $l\in\{1,2,\ldots,\mu-1\}$. Denote by $$ \chi=\{t;~\Omega(t+1)\neq\emptyset,~t_{l}< t\leq t_{l+1}+\rho\}\neq\emptyset. $$
Since $\mathcal{D}(t+1)$ is the null matrix for each $t$ with $t_{l}< t\leq t_{l+1}$ but $t\not\in\chi$,
we have \begin{equation}\label{teq15} A(t_{l+1}+1)-A(t_{l}+1)=\sum_{t_{l}< t\leq t_{l+1},t\in\chi}\mathcal{D}(t+1). \end{equation} Fix $t\in\chi$. Denote by $C_{\mathtt{SY}}(\Omega)=\frac{1}{2}\left(C(\Omega) +C(\Omega)^{T}\right)$ and
$C_{\mathtt{SK}}(\Omega)=\frac{1}{2}\left(C(\Omega)-C(\Omega)^{T}\right)$ the symmetric part and
the skew-symmetric part of $C(\Omega)$, respectively. Then \begin{eqnarray}\label{meq45} &&\hspace{-1.24cm}\langle \mathcal{D}(t+1), C(\Omega)\rangle\nonumber\\ &&\hspace{-0.7cm}=\langle \mathcal{D}_{\mathtt{SY}}(t+1), C_{\mathtt{SY}}(\Omega)\rangle + \langle \mathcal{D}_{\mathtt{SK}}(t+1), C_{\mathtt{SK}} (\Omega)\rangle. \end{eqnarray} \textit{Claim 1:} \textit{ \begin{eqnarray*} &&\hspace{-0.8cm}\langle \mathcal{D}_{\mathtt{SY}}(t+1), C_{\mathtt{SY}}(\Omega)\rangle=\sum_{1\leq i\leq n}\mathcal{D}_{\mathtt{SY}}(t+1)_{ii}\cdot \frac{1} {2}\cdot\sharp M_{i}(\Omega)\nonumber\\
&&+\sum_{1\leq i,j\leq n,i\neq j}\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}\cdot\frac{1}{2}\left(\sharp(M_{i}(\Omega)^{+}\cap M_{j}(\Omega)^{+})\right.\nonumber\\
&&+\left.\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{-})\right)\nonumber\\
&&-\sum_{1\leq i,j\leq n,i\neq j}\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}\cdot\frac{1}{2}\left(\sharp(M_{i}(\Omega)^{+}\cap M_{j}(\Omega)^{-})\right.\nonumber\\
&&+\left.\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{+})\right).\nonumber
\end{eqnarray*}}
~~To deduce this, for each $i,j\in\{1,2,\ldots,n\}$, consider the symmetric perturbative matrix
$\widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}}(t+1)_{ij}}$ defined by $(\ref{eq08})$.
Then
\langle \mathcal{D}_{\mathtt{SY}}(t+1), C_{\mathtt{SY}} (\Omega)\rangle
=\sum_{1\leq i\leq j\leq n}\langle
\widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}}(t+1)_{ij}}, C(\Omega)\rangle.
\end{eqnarray}
For each $i,j\in\{1,2,\ldots,n\}$ with $i\neq j$,
if $M_{i}(\Omega)\cap M_{j}(\Omega)=\emptyset$, then $$
(i,j)\not\in \bigcup_{1\leq m\leq p}\left((\1(x^{m-1})\triangle \ 1(x^{m}))\times (\1(x^{m-1})\triangle \1(x^{m}))\right). $$
Thus, by Lemma \ref{lem3}, we have
$\langle \widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}},C(\Omega)\rangle=0$ and, hence, \begin{eqnarray}\label{teq05} &&\hspace{-0.7cm}\langle \widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}},C(\Omega)\rangle =\mathcal{D}_{\mathtt{SY}}(t+1)_{ij}\cdot\left(\sharp(M_{i} (\Omega)^{+}\cap M_{j}(\Omega)^{+})\right.\nonumber\\ &&\hspace{-0.5cm}+\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{-})-\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{-})\nonumber\\ &&\hspace{-0.5cm}\left.-\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{+})\right).~\hspace{-1.3cm} \end{eqnarray}
Alternatively, if $M_{i}(\Omega)\cap M_{j}(\Omega)\neq\emptyset$, then
$$
(i,j)\in \bigcup_{1\leq m\leq p}\left((\1(x^{m-1})\triangle \ 1(x^{m}))\times (\1(x^{m-1})\triangle \1(x^{m}))\right). $$
Consider the surgical operation on $\Omega$ defined by Lemma $\ref{lem4}$.
$\widetilde{\Omega}^{ij}=[y^{0},y^{1},\ldots,y^{p+\varsigma}] .$
Furthermore, by Lemma \ref{lem4}, we have \begin{equation}\label{eq12} \langle \widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}},C(\Omega)\rangle=\langle \widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}},C(\Omega)-C(\widetilde{\Omega}^{ij})\rangle. \end{equation}
According to $(\ref{eeq31})$ and $(\ref{eeq35})$, we deduce \begin{eqnarray}\label{eeq33} &&\hspace{-1cm}c_{ij}(\Omega)-c_{ij} (\widetilde{\Omega}^{ij})=\sum_{1\leq k\leq\varsigma}x^{m_{k}-1}_{j}\left(x^{m_{k}-1}_{i}-x^{m_{k}}_{i}\right)\nonumber\\ &&-\sum_{1\leq k\leq\varsigma}y^{m_{k}+k-1}_{j}\left(y^{m_{k}+k-1}_{i}-y^{m_{k} +k}_{i}\right)\nonumber\\ &&-\sum_{1\leq k\leq\varsigma}y^{m_{k}+k-2}_{j}\left(y^{m_{k}+k-2}_{i}-y^{m_{k}+k-1}_{i}\right) \end{eqnarray} and \begin{eqnarray}\label{eeq34} &&\hspace{-1cm}c_{ji}(\Omega)-c_{ji} (\widetilde{\Omega}^{ij})=\sum_{1\leq k\leq\varsigma}x^{m_{k}-1}_{i}\left(x^{m_{k}-1}_{j}-x^{m_{k}}_{j}\right)\nonumber\\ &&-\sum_{1\leq k\leq\varsigma}y^{m_{k}+k-1}_{i}\left(y^{m_{k}+k-1}_{j}-y^{m_{k} +k}_{j}\right)\nonumber\\ &&-\sum_{1\leq k\leq\varsigma}y^{m_{k}+k-2}_{i}\left(y^{m_{k}+k-2}_{j}-y^{m_{k}+k-1}_{j}\right). \end{eqnarray} For $k=1,2,\ldots,\varsigma$,
we compute each partial sum on the right hand sides of $ (\ref{eeq33})$ and $(\ref{eeq34})$
in cases $m_{k}\in M_{i}(\Omega)^{+}\cap M_{j}(\Omega)^{+}$, $m_{k}\in M_{i}(\Omega)^{-}\cap M_{j}(\Omega)^{-}$,
$m_{k}\in M_{i}(\Omega)^{+}\cap M_{j}(\Omega)^{-}$, and $m_{k}\in M_{i}(\Omega)^{-}\cap M_{j}(\Omega)^{+}$. Then, by $(\ref{eq12})$, we obtain
\begin{eqnarray}\label{teq06} &&\hspace{-0.7cm}\langle \widetilde{D}^{ij}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}},C(\Omega)\rangle=\mathcal{D}_{\mathtt{SY}} (t+1)_{ij}\cdot\left(\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{+})\right.\nonumber\\ &&\left.+\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{-})-\sharp(M_{i}(\Omega)^{+}\cap M_{j} (\Omega)^{-})\right.\nonumber\\ &&\left.-\sharp(M_{i}(\Omega)^{-}\cap M_{j} (\Omega)^{+})\right). \end{eqnarray}
Furthermore, since $\Omega$ is a loop of states in $\{0,1\}^{n}$, we have \begin{eqnarray}\label{teq04} \langle \widetilde{D}^{ii}_{\mathcal{D}_{\mathtt{SY}} (t+1)_{ii}},C(\Omega)\rangle =\mathcal{D}_{\mathtt{SY}}(t+1)_{ii}\cdot\frac{1}{2}\cdot\sharp M_{i}(\Omega) \end{eqnarray}
for each $i=1,2,\ldots,n$.
Combining $(\ref{teq03})$, $(\ref{teq05})$, $(\ref{teq06})$, and $ (\ref{teq04})$ proves Claim 1.
\textit{Claim 2:} \textit{
\begin{eqnarray*}
&&\hspace{-1cm}\langle \mathcal{D}_{\mathtt{SK}}(t+1), C_{\mathtt{SK}}(\Omega)\rangle\nonumber\\
&&\geq-\sum_{1\leq i,j\leq n}\frac{1}{4}\left(\min\{\sharp M_{i} (\Omega),\sharp M_{j}(\Omega)\}\right.\nonumber\\
&&\left.\cdot |\mathcal{D}(t+1)_{ij}-\mathcal{D} (t+1)_{ji}|\right)\nonumber\\
