# Robust-Output-Controlled Synchronization Strategy for Arrays of Pancreaticβ-Cells

Authors: David I. Rosas Almeida; Laura O. Orea Leon
Journal: Complexity
 (2018)

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## Abstract

This paper presents a synchronization strategy based on second-order sliding mode control, to obtain robust controlled synchronization in an array of uncertain pancreaticβ-cells. This strategy considers a synchronization scheme with a reference cell, which incorporates the desired dynamics, and an array of cells, which does not demonstrate adequate synchronization. The array may be formed by active and inactive cells having different strengths in gap junctions. For an array with three cells, we design the coupling signal considering that only the output of an active cell of the array is available. The coupling signal is the signum of the difference between the output of the reference cell and the output of an active cell in the array; this ensures exact synchronization in finite time between both cells. Then, this coupling signal is applied to the other cells in the array, and we establish the conditions required to be satisfied to obtain approximate synchronization between the reference cell and all other cells in the array. The performance of this technique is demonstrated by the results of numerical simulations performed for several cases of connections for an array with three cells and the reference cell. Finally, we show through a numerical simulation that this technique can be applied to arrays with many β-cells.


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## Body

## 1. Introduction

Diabetes mellitus is a syndrome of disordered metabolism caused usually by a combination of hereditary and environmental causes. Diabetes results in abnormally high blood sugar levels, a condition that is known as hyperglycemia. This is caused by an autoimmune attack onβ-cells secreted by the pancreas, classified as type I diabetes, or by the inadequate function of β-cells in counteracting the fluctuations of high and low blood glucose within the body, classified as type II diabetes [1–3].Biological systems such as the pancreas, which is a multicellular system, have a complex communication mechanism between cells to ensure proper functioning. In the case ofβ-cells, the coupling of electrical and metabolic signals between cells is required for adequate insulin secretion and effective glycemic control. Problems in connections have been implicated in both type I and type II diabetes [4–6].A normal dynamical behavior in plasma insulin levels is pulsatile for humans and other animals; however, when diabetes is present, this behavior is damaged. Further, experimental data show that the application of insulin in pulses is more efficient than constant insulin application for the control of plasma insulin levels in humans with diabetes; this suggests that type II diabetes may be caused, at least in part, by either a loss or an irregularity of plasma insulin pulsatile behavior. Because the insulin in plasma is oscillatory andβ-cells produce insulin oscillations of a similar frequency, it is desirable that an important amount of β-cells in pancreas must be synchronized; otherwise, no oscillations in the blood insulin level would occur [7, 8].Moreover, other experimental results show that pulsatile, also called bursting, electrical activity is present only whenβ-cells are analyzed in an array, also called islet, where the interconnections between cells are considered; when β-cells are analyzed in isolation, most of them are in an inactive state. On the other hand, if too many cells are inactive in an array, they might stop exhibiting pulsatile activity [9, 10].Thus, the synchronization phenomenon in the islets ofβ-cells has been extensively studied in recent decades. Some of these studies present dynamic models of the individual β-cells, as well as connection models that define the islets [8, 11]. One of the most recent works on this subject is [12], where a dynamic β-cell model based on fractional calculus is presented.Many studies have focused on the synchronization phenomenon based on dynamic models ofβ-cells and their connections. They present formal and numerical analyses of the conditions related to the presence of synchronization between the cells that form the islet, the type of synchronization with respect to parameters of the cells, and conditions for the presence of chaos [7, 9, 10, 13–15]. Studies on the dynamics of β-cells based on experimental measurements have also been performed, such as in [16–18].We present a synchronization strategy, based on second-order sliding mode control, in order to obtain robust controlled synchronization in an array of uncertain pancreaticβ-cells. This strategy considers a synchronization scheme with a reference cell, which ensures the desired dynamics, and an array of cells, which does not demonstrate adequate synchronization. The array may be formed by active and inactive cells with different strengths in gap junctions.For an array with three cells, we design the coupling signal considering that only the output of an active cell of the array is available. The coupling signal is the signum of the difference between the output of the reference cell and the output of an active cell in the array, which guarantees exact synchronization in finite time between both cells. Then, we apply this coupling signal to other cells in the array and establish sufficient conditions to obtain approximate synchronization between all cells in the array with the reference cell. We show the performance of this technique through numerical simulations for several cases of connections for the array with three cells and the reference cell. Finally, through numerical simulation, we show that this technique can be applied to arrays with manyβ-cells.The rest of this paper is organized as follows. Section2 establishes definitions regarding the synchronization of dynamic systems, the model of a β-cell, and a model of an array of β-cells, which are the basis of this work. Section 3 defines the synchronization problem that is resolved in this work. In Section 4, the synchronization technique for the synchronization of three β-cells is presented, which is based on the availability of only the measurement of the output of a cell of the array of β-cells. Numerical simulations for several cases of connections for the array with three β-cells and the reference cell are presented in Section 5. In Section 6, we present the application of the synchronization technique to an array with ten β-cells. Finally, in Section 7, we present the conclusions.


## 2. Preliminary Definitions

### 2.1. Synchronization Definitions

Considerk interconnected dynamical systems that form an array
(1)Σi:ẋi=fixi,t+Fix1,…,xk,xr,t,yi=hixi,i=1,…,k,and the following reference system
(2)Σr:ẋr=frxr,t,yr=hrxr,where xir∈ℜn are the state vectors, ẋ≡dx/dt, fir:ℜn×ℜ→ℜn are, in general, nonlinear vector fields, Fi⋅ are vector fields describing interconnections, yir∈ℜ are the outputs of the systems, and hir:ℜn→ℜ are smooth functions. We establish the following definitions based on [19].Let us considerl functionals ϵi:Y1×Y2×⋯Yk×Yr→ℜ,i=1,…,l. Here, Yj are the sets of all output functions.Definition 1.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr are synchronized with respect to the functionals ϵ1,…,ϵl if
(3)ϵiy1t,…,ykt,yrt=0,i=1,…,l.Definition 2.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr, with initial conditions x10,…,xk0,xr0, are asymptotically synchronized with respect to the functionals ϵ1,…,ϵl if
(4)limt→∞ϵiy1t,…,ykt,yrt=0,i=1,…,l.Definition 3.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr, with initial conditions x10,⋯,xk0,xr0, are asymptotically approximately synchronized with respect to the functionals ϵ1,…,ϵl if there is a small positive constant ζ such that
(5)limt→∞ϵiy1t,…,ykt,yrt≤ζ,i=1,…,l.
Now, we considerk interconnected dynamical systems with the following control inputs
(6)Σi:ẋi=fixi,t+Fix1,…,xk,xr,ui,t,yi=hixi,i=1,…,k,where xr is given by (2) and ui∈ℜ are the control inputs.Definition 4.
The problem of controlled asymptotic synchronization with respect to the functionalsϵ1,…,ϵl is that the control signals ui must be determined as a feedback function of the states x1,…,xk,xr such that (4) or (5) holds for the closed loop system.


### 2.2. Model of aβ-Cell

The model of aβ-cell is given by [11]
(7)ẏ=fy−w−x,ẇ=ω∞y−w,ẋ=εhy−x,where y∈ℜ denotes the membrane potential, and we consider that this state variable is the output of the cell; w∈ℜ is a channel activation variable; x∈ℜ is related to the concentration of intracellular calcium and ADP; and ε≪1 is a small positive parameter. Further,
(8)fy=−f3y3+f2y2+f1y,ω∞y=ω3y3+ω2y2−ω1y−ω0,hy=by+y0,In the sequel, we will use the parameters proposed in [10, 11]: f3=1/12, f2=3/8, f1=37/64, ω3=11/12, ω2=3/8, ω1=227/64, ω0=3, ε=0.0025, and b=4. If y0=0.954, the cell shows activity, and thus, β-cell is active. However, if y0=1.375, the solution of the system (7) converges to equilibrium, and we say that β-cell is inactive [10]. Figure 1 illustrates these behaviors; in the right column, the black lines denote the states of an active cell, and in the left column, the blue lines represent the states of an inactive cell.Figure 1
Behaviors ofβ-cells. The right and left columns correspond to active and inactive β-cells, respectively.


### 2.3. Synchronization Scheme

The synchronization scheme is shown in the connection graph in Figure2, where the green pentagon represents an active cell, red pentagons denote inactive cells, and the yellow circle indicates the reference cell. As shown in this graph, we can establish the presence or absence of couplings between the cells in the array and the reference cell. In addition, the coupling strength in each coupling can be determined.Figure 2
Example of a connection graph; the green pentagon, red pentagons, and yellow circle, respectively, indicate an activeβ-cell, the inactive β-cells, and the reference β-cell.We then consider an array ofk pancreatic β-cells, called nodes, where each cell is modeled by (7) by adding a coupling input ui to the first equation, as follows:
(9)Σi:ẏi=fyi−wi−xi+ui,ẇi=ω∞yi−wi,ẋi=εbyi+y0i−xi,where
(10)ui=∑j=1,j≠ikgjiyj−yi+ρivi,for i=1,…,k; the array could consist of k1≤k active cells and k−k1 inactive cells.The term∑j=1,j≠ikgjiyj−yi represents the natural coupling, gap junctions, between the i‐th β-cell and the other cells in the array, and gji≥0 is the coupling strength; if gji=0, there is no coupling between the Σj and Σi cells. In this work, we assume that in general, the values of the constants gji may be different for each coupling.The termρivi represents the artificial control coupling that is designed to generate synchronization between all cells in the array with a reference β-cell. In addition, we consider a reference β-cell given by
(11)Σr:ẏr=fyr−wr−xr,ẇr=ω∞yr−wr,ẋr=εbyr+y0r−xr.Withvi=0, the cells in the array may exhibit different behaviors depending on the value of gji and the number of active and inactive cells [15]. Figure 3 shows the behavior in an array with ten β-cells, of which five are active and five are inactive, gji=gij>0, and vi=0 for all i,j; in this case, there is no synchronization.Figure 3
Behavior of an islet with tenβ-cells of which five are active and five are inactive.


## 2.1. Synchronization Definitions

Considerk interconnected dynamical systems that form an array
(1)Σi:ẋi=fixi,t+Fix1,…,xk,xr,t,yi=hixi,i=1,…,k,and the following reference system
(2)Σr:ẋr=frxr,t,yr=hrxr,where xir∈ℜn are the state vectors, ẋ≡dx/dt, fir:ℜn×ℜ→ℜn are, in general, nonlinear vector fields, Fi⋅ are vector fields describing interconnections, yir∈ℜ are the outputs of the systems, and hir:ℜn→ℜ are smooth functions. We establish the following definitions based on [19].Let us considerl functionals ϵi:Y1×Y2×⋯Yk×Yr→ℜ,i=1,…,l. Here, Yj are the sets of all output functions.Definition 1.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr are synchronized with respect to the functionals ϵ1,…,ϵl if
(3)ϵiy1t,…,ykt,yrt=0,i=1,…,l.Definition 2.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr, with initial conditions x10,…,xk0,xr0, are asymptotically synchronized with respect to the functionals ϵ1,…,ϵl if
(4)limt→∞ϵiy1t,…,ykt,yrt=0,i=1,…,l.Definition 3.
We say that outputsy1t,…,ykt,yrt of the systems Σ1,…,Σk,Σr, with initial conditions x10,⋯,xk0,xr0, are asymptotically approximately synchronized with respect to the functionals ϵ1,…,ϵl if there is a small positive constant ζ such that
(5)limt→∞ϵiy1t,…,ykt,yrt≤ζ,i=1,…,l.
Now, we considerk interconnected dynamical systems with the following control inputs
(6)Σi:ẋi=fixi,t+Fix1,…,xk,xr,ui,t,yi=hixi,i=1,…,k,where xr is given by (2) and ui∈ℜ are the control inputs.Definition 4.
The problem of controlled asymptotic synchronization with respect to the functionalsϵ1,…,ϵl is that the control signals ui must be determined as a feedback function of the states x1,…,xk,xr such that (4) or (5) holds for the closed loop system.


## 2.2. Model of aβ-Cell

The model of aβ-cell is given by [11]
(7)ẏ=fy−w−x,ẇ=ω∞y−w,ẋ=εhy−x,where y∈ℜ denotes the membrane potential, and we consider that this state variable is the output of the cell; w∈ℜ is a channel activation variable; x∈ℜ is related to the concentration of intracellular calcium and ADP; and ε≪1 is a small positive parameter. Further,
(8)fy=−f3y3+f2y2+f1y,ω∞y=ω3y3+ω2y2−ω1y−ω0,hy=by+y0,In the sequel, we will use the parameters proposed in [10, 11]: f3=1/12, f2=3/8, f1=37/64, ω3=11/12, ω2=3/8, ω1=227/64, ω0=3, ε=0.0025, and b=4. If y0=0.954, the cell shows activity, and thus, β-cell is active. However, if y0=1.375, the solution of the system (7) converges to equilibrium, and we say that β-cell is inactive [10]. Figure 1 illustrates these behaviors; in the right column, the black lines denote the states of an active cell, and in the left column, the blue lines represent the states of an inactive cell.Figure 1
Behaviors ofβ-cells. The right and left columns correspond to active and inactive β-cells, respectively.


## 2.3. Synchronization Scheme

The synchronization scheme is shown in the connection graph in Figure2, where the green pentagon represents an active cell, red pentagons denote inactive cells, and the yellow circle indicates the reference cell. As shown in this graph, we can establish the presence or absence of couplings between the cells in the array and the reference cell. In addition, the coupling strength in each coupling can be determined.Figure 2
Example of a connection graph; the green pentagon, red pentagons, and yellow circle, respectively, indicate an activeβ-cell, the inactive β-cells, and the reference β-cell.We then consider an array ofk pancreatic β-cells, called nodes, where each cell is modeled by (7) by adding a coupling input ui to the first equation, as follows:
(9)Σi:ẏi=fyi−wi−xi+ui,ẇi=ω∞yi−wi,ẋi=εbyi+y0i−xi,where
(10)ui=∑j=1,j≠ikgjiyj−yi+ρivi,for i=1,…,k; the array could consist of k1≤k active cells and k−k1 inactive cells.The term∑j=1,j≠ikgjiyj−yi represents the natural coupling, gap junctions, between the i‐th β-cell and the other cells in the array, and gji≥0 is the coupling strength; if gji=0, there is no coupling between the Σj and Σi cells. In this work, we assume that in general, the values of the constants gji may be different for each coupling.The termρivi represents the artificial control coupling that is designed to generate synchronization between all cells in the array with a reference β-cell. In addition, we consider a reference β-cell given by
(11)Σr:ẏr=fyr−wr−xr,ẇr=ω∞yr−wr,ẋr=εbyr+y0r−xr.Withvi=0, the cells in the array may exhibit different behaviors depending on the value of gji and the number of active and inactive cells [15]. Figure 3 shows the behavior in an array with ten β-cells, of which five are active and five are inactive, gji=gij>0, and vi=0 for all i,j; in this case, there is no synchronization.Figure 3
Behavior of an islet with tenβ-cells of which five are active and five are inactive.


## 3. Synchronization Objective

Consider an array ofk β-cells defined by (11), in which k1 cells are active and k2 cells are inactive; k=k1+k2. Further, consider that the cells in this array do not present adequate synchronization.Based on the previously mentioned definitions, and given a connection graph, the issue at hand is to design coupling signalsvi such that asymptotic synchronization, specifically,
(12)limt→∞yrt−yit=0,∀i∈1,…,k,limt→∞yit−yjt=0,∀i,j∈1,…,k,i≠j,or approximate synchronization, specifically,
(13)yrt−yit≤ζ,∀i∈1,…,k,(14)yit−yjt≤ζ,∀i,j∈1,…,k,i≠j,t≥T0,can be obtained.


## 4. Design of Coupling Signals for an Array with Threeβ-Cells and a Reference β-Cell

In this section, we propose a strategy to synchronize the array shown in Figure2, where the models of Σ1, Σ2, and Σ3 cells are given by
(15)ẏ1=fy1−w1−x1+g21y2−y1+g31y3−y1+ρ1v1,ẇ1=ω∞y1−w1,ẋ1=εby1+y01−x1,ẏ2=fy2−w2−x2+g12y1−y2+g32y3−y2+ρ2v2,ẇ2=ω∞y2−w2,ẋ2=εby2+y02−x2,ẏ3=fy3−w3−x3+g13y1−y3+g23y2−y3+ρ3v3,ẇ3=ω∞y3−w3,ẋ3=εby3+y03−x3,and the model of the reference cell Σr is given by (11).Consider that the measurement of the output of only one cell is available, for exampley1 from Σ1 cell, and that both Σr and Σ1 are actives; y0r=y01. To design the coupling signal v1, to synchronize Σ1 cell with Σr cell, we define the error variables e1r=y1−yr, ε1r=w1−wr, and ϵ1r=x1−xr, whose dynamics are given by
(16)ė1r=f1e1r−ε1r−ϵ1r−g21e21−g31e13−f3y13−yr3+f2y12−yr2+ρ1v1,ε̇1r=−ω1e1r−ε1r+ω3y13−yr3+ω2y12−yr2,ϵ̇1r=εbe1r−ϵ1r−εby01−y0r,considering that
(17)f1e1r−ε1r−ϵ1r−g21e21−g31e13−f3y13−yr3+f2y12−yr2<δ,we propose the coupling signal v1 as
(18)v1=−csigne1r.Then, the dynamics ofe1r are given by
(19)ė1r=f1e1r−ε1r−ϵ1r−g21e21−g31e13−f3y13−yr3+f2y12−yr2−ρ1csigne1r,if ρ1c>δ, the variable e1r converges to zero in finite time owing to the existence of a sliding mode in the discontinuity surface e1r=0. To prove this, we use the Lyapunov function as
(20)Ve1r=12e1r2,and its derivative is
(21)V̇e1r=e1rf1e1r−ε1r−ϵ1r−g21e1r−g31e13−f3y13−yr3+f2y12−yr2−ρ1ce1r,≤δ−ρ1ce1r.This proves thate1r=0 is an attractive surface. However, we also have
(22)lime1r→0+ė1r<0,lime1r→0−ė1r>0,such that convergence to e1r=0 occurs in finite time. Then, the equivalent control ueq is given by ė1r=0 when e1r=0:
(23)−ε1r−ϵ1r−g21e21−g31e13−ueq=0ueq=−ε1r−ϵ1r−g21e21−g31e13.The zero dynamics of system (16) is given by
(24)ε̇1r=−ε1r,ϵ̇1r=−ϵ1r−εby01−y0r.Becausey01=y0r, the zero dynamics converge asymptotically to zero, and we can conclude that the reference cell Σr and the cell Σ1 are asymptotic synchronized and, as a consequence, in steady state,
(25)ueq=−g21e21−g31e13.Now, since we have access to the output of only one cell, specifically,Σ1 in the considered case, we propose that the coupling signals v2 and v3 are equal to v1; then, we analyze the behavior of the error variables e2r=y2−yr, ε2r=w2−wr, ϵ2r=x2−xr, e3r=y3−yr, ε3r=w3−wr, ϵ3r=x3−xr, e23=y2−y3, ε23=w2−w3, and ϵ23=x2−x3. Assuming that y1=yr in finite time, the dynamics of the error variables are given by
(26)ė2r=−g12−f1e2r−ε2r−ϵ2r−g32e23+ρ2v1+f3yr3−y23+f2y22−yr2,ε̇2r=−ω1e2r−ε2r+ω3y23−yr3+ω2y22−yr2,ϵ̇2r=εbe2r−εϵ2r+εby02−y0r,e3r=g23e23−g13−f1e3r−ε3r−ϵ3r+ρ3v1+f3yr3−y33+f2y32−yr2,ε̇3r=−ω1e3r−ε3r+ω3y33−yr3+ω2y32−yr2,ϵ̇3r=εbe3r−εϵ3r+εby03−y0r,ė23=−g32+g23−f1e23−ε23−ϵ23−g12e2r+g13e3r+ρ2−ρ3v1+f3y33−y23+f2y22−y32,ε̇23=−ω1e23−ε23+ω3y23−3y33+ω2y22−y32,ϵ̇23=εbe23−εϵ23+εby02−y03.The coupling signalv1 is given by (18); however, we can substitute it by the equivalent control given in (25). Then, we obtain the matrix representation of system (26) as follows:
(27)ϕ̇=Aϕ+γ⋅,where vector ϕ is given by
(28)ϕ=e2rε2rϵ2re3rε3rϵ3re23ε23ϵ23T,The matrixA is defined as
(29)A=a11−1−1a1400−g3200−ω1−10000000εb0−ε000000a4100a44−1−1g2300000−ω1−10000000εb0−ε000a7100a7400a77−1−1000000−ω1−10000000εb0−ε,where
(30)a11=−g12+ρ2g21−f1,a14=ρ2g31,a41=−ρ3g21,a44=−g13−f1−ρ3g31,a71=−g12+ρ2−ρ3g21,a74=g13+ρ2−ρ3g31,a77=−g32+g23−f1.Finally, vectorγ⋅ is given by
(31)γ⋅=f3yr3−y23+f2y22−yr2ω3y23−yr3+ω2y22−yr2εby02−y0rf3yr3−y33+f2y32−yr2ω3y33−yr3+ω2y32−yr2εby03−y0rf3y33−y23+f2y22−y32ω3y23−y33+ω2y22−y32εby02−y03.The disturbance termγ⋅, in general, satisfies
(32)γ⋅≤σ0+σ1ϕ,where σ0 and σ1 are constants. Then, if matrix A is a Hurwitz matrix, the synchronization objectives (13) and (14) are satisfied; therefore, the array is approximate synchronized with the reference β-cell.


## 5. Numerical Simulations

This section reports on the numerical simulation results of the synchronization of the array with threeβ-cells and a reference cell, shown in Figure 2, considering the different coupling situations between cells. During the first 1000 s of all simulations described in this section, v1=0, and no natural synchronization is observed for all cases. After the first 1000 s, the coupling signal (18) is applied, with ρ1=1 and c=2.5.First, we analyze the synchronization between the reference cellΣr and the cell Σ1. Figures 4 and 5 show that the state variables of the Σ1 cell converge to the state of the Σr cell; the convergence of y1 to yr is in finite time, and the convergence of w1 and x1 to wr and xr is in asymptotic form. The coupling signal v1 is shown in Figure 5.Figure 4
Behavior of state variables of theΣr and Σ1 cells before and after the application of coupling signal v1.Figure 5
Trends of synchronization errors betweenΣr and Σ1 cells before and after the application of coupling signal v1.Now, we first consider a configuration of the array wherein all connections are present, and the parametersgi,j are g21=0.8, g31=5, g12=3, g32=2, g13=10, g23=2, ρ1=1, ρ2=1, and ρ3=1. For this case, matrix A has the eigenvalues λ1,2=−5.0430±4.6293i, λ3=−1.9982, λ4,5=−0.9097±0.1036i, λ6=−0.1156, λ7=−0.0467, and λ8,9=−0.0035±0.0012i. Consequently, we obtain approximate synchronization, as is seen in Figure 6.Figure 6
Outputsyi and errors eir, i=1,2,3, for g21=0.8, g31=5, g12=3, g32=2, g13=10, g23=2, ρ1=1, ρ2=1, and ρ3=1.Now, consider that the connections between the reference cellΣr and the Σ2 and Σ3 cells are lost; ρ2=ρ3=0, as is seen in Figure 7. For this case, matrix A has the eigenvalues λ1,2=−7.1660±3.3107i, λ3=−1.9905, λ4,5=−0.8937±0.0572i, λ6=−0.1035, λ7=−0.0522 and λ8,9=−0.0037±0.0007i. Subsequently, this array exhibits approximate synchronization as well, as is evident in Figure 8.Figure 7
Connection graph for case in whichρ2=ρ3=0.Figure 8
Outputsyi and yr and errors eir, i=1,2,3, for g21=0.8, g31=5, g12=3, g32=2, g13=10, g23=2, ρ1=1, and ρ2=ρ3=0.Finally, we consider an array configuration wherein some connections between cells are lost; in particular,g21=g32=0 and also ρ2=0. Such an array is shown in Figure 9. For this case, matrix A has the eigenvalues λ1,2=−3.0792±2.5873i, λ3=−2.8697, λ4,5=−0.8410±0.1989i, λ6=−0.5458, λ7=−0.0089, and λ8,9=−0.0042±0.0023i, which produce approximate synchronization, as is evident in Figure 10.Figure 9
Connection graph for the case in whichρ2=0 and g21=g32=0.Figure 10
Outputsyi and yr and errors eir, i=1,2,3, for g21=0, g31=5, g12=3, g32=0, g13=10, g23=2, ρ1=1, ρ2=0, and ρ3=1.


## 6. Numerical Analysis for an Array with Ten Cells

From the results presented in previous sections, we can formulate the hypothesis that the coupling signal (18) may produce approximate synchronization in arrays with more β-cells.In this section, we present the results of the numerical simulation in which the proposed synchronization strategy is applied to an array with tenβ-cells, given by (9), including five active and five inactive cells, with the reference cell (11) using the coupling signal (18). For this simulation, gij=1, ρi=1, i,j=1,…,10, i≠j, and c=2.During the first 750 s in the simulation, the coupling signalv1 is equal to zero. After this time, the coupling signal is applied. Figures 11 and 12 show that the β-cells in the array are not synchronized when v1=0; however, when the coupling signal is applied, all cells exhibit approximate synchronization within a short period. Further, in Figure 11, we can see that states wr and wi also approximately synchronize, but states xi form two groups as was predicted from the internal dynamics; however, it is noted that the errors between them stay bounded.Figure 11
States of the referenceβ-cell Σr and the β-cells in the islet Σi, before and after application of the coupling signal v1.Figure 12
Errors between the outputs ofβ-cell Σr and the β-cells in the islet Σi, before and after application of the coupling signal v1.


## 7. Conclusions

The controlled synchronization strategy proposed in this paper is a robust solution to, in theory, the problem of the synchronization of an array of pancreaticβ-cells. The strategy uses a discontinuous function that leads to a sliding mode phenomenon, and the control equivalent method is used to prove the occurrence of synchronization in finite time, when only the measured output of an active cell is known, and approximate synchronization occurs among the rest of the β-cells in the array.The principal advantage of this technique is that it only needs the output of one activeβ-cell and establishes the necessary conditions to ensure approximate synchronization between the array of β-cells and the reference cell considering several conditions of couplings gij and the number of active and inactive cells in the array.A possible disadvantage is that the coupling signal has high-frequency components that may cause damage in a realβ-cell islet. Therefore, low-pass filters can be incorporated to obtain the equivalent control, which can then be applied to the cells in the array.On the other hand, if this proposal goes from theory to practice, we must consider that the location of the eigenvalues of matrixA in the complex plane depends on the values of the parameters intrinsic to the array gji, which may be unknown. Then, a possible strategy to obtain approximate synchronization is to apply the coupling signal to as many cells as possible, active and inactive, and tune the parameters c and ρi of the artificial coupling signals introduced in the array to obtain the desired synchronization in the array.



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*Source: 5174981-2018-11-05.xml*