Abstract
This paper presents a synchronization strategy based on second-order sliding mode control, to obtain robust controlled synchronization in an array of uncertain pancreatic β-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.
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 on β-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 of β-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 and β-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 when β-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 of β-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 of β-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 pancreatic β-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 many β-cells.
The rest of this paper is organized as follows. Section 2 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
Consider interconnected dynamical systems that form an array and the following reference system where are the state vectors, , are, in general, nonlinear vector fields, are vector fields describing interconnections, are the outputs of the systems, and are smooth functions. We establish the following definitions based on [19].
Let us consider functionals . Here, are the sets of all output functions.
Definition 1. We say that outputs of the systems are synchronized with respect to the functionals if
Definition 2. We say that outputs of the systems , with initial conditions , are asymptotically synchronized with respect to the functionals if
Definition 3. We say that outputs of the systems , with initial conditions , are asymptotically approximately synchronized with respect to the functionals if there is a small positive constant such that Now, we consider interconnected dynamical systems with the following control inputs where is given by (2) and are the control inputs.
Definition 4. The problem of controlled asymptotic synchronization with respect to the functionals is that the control signals must be determined as a feedback function of the states such that (4) or (5) holds for the closed loop system.
2.2. Model of a β-Cell
The model of a β-cell is given by [11] where denotes the membrane potential, and we consider that this state variable is the output of the cell; is a channel activation variable; is related to the concentration of intracellular calcium and ADP; and is a small positive parameter. Further,
In the sequel, we will use the parameters proposed in [10, 11]: , , , , , , , , and . If , the cell shows activity, and thus, β-cell is active. However, if , 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.
2.3. Synchronization Scheme
The synchronization scheme is shown in the connection graph in Figure 2, 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.
We then consider an array of pancreatic β-cells, called nodes, where each cell is modeled by (7) by adding a coupling input to the first equation, as follows: where for ; the array could consist of active cells and inactive cells.
The term represents the natural coupling, gap junctions, between the β-cell and the other cells in the array, and is the coupling strength; if , there is no coupling between the and cells. In this work, we assume that in general, the values of the constants may be different for each coupling.
The term 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
With , the cells in the array may exhibit different behaviors depending on the value of 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, , and for all ; in this case, there is no synchronization.
3. Synchronization Objective
Consider an array of β-cells defined by (11), in which cells are active and cells are inactive; . 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 signals such that asymptotic synchronization, specifically, or approximate synchronization, specifically, can be obtained.
4. Design of Coupling Signals for an Array with Three β-Cells and a Reference β-Cell
In this section, we propose a strategy to synchronize the array shown in Figure 2, where the models of , , and cells are given by and the model of the reference cell is given by (11).
Consider that the measurement of the output of only one cell is available, for example from cell, and that both and are actives; . To design the coupling signal , to synchronize cell with cell, we define the error variables , , and , whose dynamics are given by considering that we propose the coupling signal as
Then, the dynamics of are given by if , the variable converges to zero in finite time owing to the existence of a sliding mode in the discontinuity surface . To prove this, we use the Lyapunov function as and its derivative is
This proves that is an attractive surface. However, we also have such that convergence to occurs in finite time. Then, the equivalent control is given by when :
The zero dynamics of system (16) is given by
Because , the zero dynamics converge asymptotically to zero, and we can conclude that the reference cell and the cell are asymptotic synchronized and, as a consequence, in steady state,
Now, since we have access to the output of only one cell, specifically, in the considered case, we propose that the coupling signals and are equal to ; then, we analyze the behavior of the error variables , , , , , , , , and . Assuming that in finite time, the dynamics of the error variables are given by
The coupling signal 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: where vector is given by
The matrix is defined as where
Finally, vector is given by
The disturbance term , in general, satisfies where and are constants. Then, if matrix 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 three β-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, and no natural synchronization is observed for all cases. After the first 1000 s, the coupling signal (18) is applied, with and .
First, we analyze the synchronization between the reference cell and the cell Figures 4 and 5 show that the state variables of the cell converge to the state of the cell; the convergence of to is in finite time, and the convergence of and to and is in asymptotic form. The coupling signal is shown in Figure 5.
Now, we first consider a configuration of the array wherein all connections are present, and the parameters are , , , , , , , , and . For this case, matrix has the eigenvalues , , , , , and . Consequently, we obtain approximate synchronization, as is seen in Figure 6.
Now, consider that the connections between the reference cell and the and cells are lost; , as is seen in Figure 7. For this case, matrix has the eigenvalues , , , , and . Subsequently, this array exhibits approximate synchronization as well, as is evident in Figure 8.
Finally, we consider an array configuration wherein some connections between cells are lost; in particular, and also . Such an array is shown in Figure 9. For this case, matrix has the eigenvalues , , , , , and , which produce approximate synchronization, as is evident in Figure 10.
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 ten β-cells, given by (9), including five active and five inactive cells, with the reference cell (11) using the coupling signal (18). For this simulation, , , , , and .
During the first 750 s in the simulation, the coupling signal 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 ; 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 and also approximately synchronize, but states form two groups as was predicted from the internal dynamics; however, it is noted that the errors between them stay bounded.
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 pancreatic β-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 active β-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 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 real β-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 matrix in the complex plane depends on the values of the parameters intrinsic to the array , 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 and of the artificial coupling signals introduced in the array to obtain the desired synchronization in the array.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.