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The emergence of chimera states in a network of nephrons
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The emergence of chimera states in a network of nephrons Jalal Khouhak , Zahra Faghani , Jakob L. Laugesen , Sajad Jafari PII: DOI: Reference:
S0577-9073(19)30971-2 https://doi.org/10.1016/j.cjph.2019.10.024 CJPH 993
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Chinese Journal of Physics
Received date: Revised date: Accepted date:
4 May 2019 12 October 2019 30 October 2019
Please cite this article as: Jalal Khouhak , Zahra Faghani , Jakob L. Laugesen , Sajad Jafari , The emergence of chimera states in a network of nephrons, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.10.024
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Highlights
The kidney plays an important role in our body. The kidneys consist of a large number of nephrons. The interaction between these nephrons induces different patterns in the network. A network of coupled nephrons is constructed and investigated. Emergence of chimera state is observed in this network.
The emergence of chimera states in a network of nephrons Jalal Khouhak a, Zahra Faghani a, Jakob L. Laugesen b, Sajad Jafari a,1 a b
Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran Department of Food Science, University of Copenhagen, Denmark
Abstract The kidney plays an essential role in our body, mainly by controlling secretion and reabsorption of water and salts. The kidneys consist of a large number of nephrons which are the functional units of the kidney. The interactions between these nephrons induce different behaviors which can be considered by a dynamical model. In this paper, a network of coupled nephron models and its dynamics is investigated. Numerical simulations of the network reveal various types of dynamical patterns depending on the coupling function and strength. One of the observed phenomenon is the emergence of chimera state. A chimera state is defined by the coexistence of coherent and incoherent groups in a network of identical oscillators. The occurrence of the chimera state can be related to the situation of disturbed synchronous oscillation of the TGFmediated proximal pressure.
Keywords: chimera state; multi-stability; kidney; nephron model.
1
Corresponding author. E-mail address: [email protected]
1. Introduction Networks of coupled oscillators are known as a class of interactive systems that can be found in a wide range of science branches such as physics, engineering, and especially, natural and biological systems [1-3]. The collective behavior of a set of coupled oscillators is mainly dependent on the coupling strength and mutual interactions between the oscillators. The type of coupling can also affect the collective behavior depending on whether it is global or local [4, 5]. The study of the collective behaviors in the biological networks can help in understanding different mechanisms in the living systems [6-8]. In 2002, two Japanese researchers, Kuramoto and Battogtokh, found a new dynamical behavior in a network of oscillators [9]. While evaluating the dynamical behaviors of a ring of non-locally coupled identical oscillators, they found a particular pattern with coexistence of coherent and incoherent oscillators. In 2004, Abrams and Strogatz called this phenomenon to chimera state [10]. Since the discovery of chimera, it has attracted many scientists and researchers in various fields [11, 12]. As well, many spatiotemporal patterns have been observed in heterogeneous systems ever since [13, 14] and various types of chimera state have been introduced. Kemeth et al. [15] classified the chimera state into three types, named stationary, breathing, and turbulent chimera. Zakharova et al. [16, 17] found a pattern composed of spatially coherent and incoherent oscillation death state and named it chimera death. Kapitaniak et al. [18] introduced a new pattern in which some oscillators escaped from the coherent cluster and called it imperfect chimera that has been observed in different systems [19, 20]. Wei et al. [21] investigated the network of AC induced Hindmarsh-Rose neurons and observed occurrence of non-stationary chimeras. Some researchers have concentrated on investigating the effect of different parameters’ values on the emergence of chimera state [22, 23]. For example, Omelchenko et al. [24] surveyed the effect of variations in the strength of coupling parameters, on the non-locally coupled Fitzhugh-Nagumo oscillators. The mammalian kidney has a crucial role in managing blood pressure variations. In other words, it has the duty of creating a proper environment for all cells in the body by controlling the excretion of salts and water. [25]. The kidney function depends on a variety of complex mechanisms, which are managed by nephron; the individual functional unit of the kidney. It controls all the related mechanisms. The nephron compensates for the changes in the arterial blood pressure with two different mechanisms [26, 27]. Firstly, the most important mechanism is called the tubuloglomerular feedback mechanism (TGF). It can generate self-sustained oscillations that are associated with a time delay of the fluid passing through the loop of Henle -a curved shape portion of the nephron that leads from the proximal tubule to the distal tubule. Another mechanism is called myogenic mechanism that is related to the smooth muscle cells in the arteriolar wall and responds to changes in transmural pressures. The model used in this investigation is mainly focused on the TGF mechanism. In TGF, the nephron controls the incoming blood flow by increasing and decreasing the radius of afferent arterioles [27-29]. Nonlinear characteristics of each single nephron, the TGF feedback mechanism, and more notable, the interaction between coupled nephrons can lead to various complex dynamical behaviors. Thus, it is likely to observe collective behaviors such as chimera state, in the network
of interacting nephrons. In this paper, at first a single nephron model and its bifurcation diagram are described to show different dynamical behaviors. Then, the mathematical equations of a network of coupled nephrons are given, and its collective behaviors are investigated.
2. The Nephron Model The model used in this study is a set of six coupled first-order differential equations which are as follows [30]:
𝑑𝑃𝑡 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 )
𝑑𝑅𝑎 = 𝑉𝑅𝑎 𝑑𝑡
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎 = −𝑑𝑉𝑅𝑎 + 𝑑𝑡 𝜔 𝑑𝜒1 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑑𝑡 𝑇
(1)
𝑑𝜒2 3 = (𝜒1 − 𝜒2 ) 𝑑𝑡 𝑇 𝑑𝜒3 3 = (𝜒2 − 𝜒3 ) 𝑑𝑡 𝑇
where 𝐹𝑓𝑖𝑙𝑡 implies the single nephron glomerular filtration rate, and 𝐶𝑡𝑢𝑏 represents the elastic compliance of the tubule. 𝐹𝐻𝑒𝑛 is the Henle flow, 𝑃𝑡 is the proximal tubular pressure, 𝑃𝑑 is the distal tubular pressures, and 𝑅𝐻𝑒𝑛 is the flow resistance in the loop of Henle. In this model, the reabsorption of water and salts, which takes place as the flow passes through the loop of Henle, is ignored to simplify the problem. 𝐹𝑟𝑒𝑎𝑏 and 𝑅𝐻𝑒𝑛 represent the reabsorption in the proximal tubule and the flow resistance respectively, and both are assumed to be constant. 𝐻𝑎 is the afferent hematocrit, 𝐶𝑎 and 𝐶𝑒 are the concentration of protein in the afferent and efferent plasma, respectively. 𝑃𝑎 and 𝑃𝑔 are the arterial and glomerular blood pressures respectively, and 𝑅𝑎 is the hemodynamic resistance of the afferent arteriole. 𝑃𝑣 shows the venous pressure and 𝑅𝑒 shows the efferent arteriolar resistance, with both being considered as constants. 𝜓 refers to the activation of the afferent arteriole, and T is the total delay time. 𝜒1 , 𝜒2 , and 𝜒3 are intermediate variables in the delay chain. Finally, 𝑃𝑎𝑣 is the average pressure in the active part of the arteriole [30].
There are also some other numerical equations that are necessary to be solved and calculated before the differential equations. They are as follows [30]: 𝐹𝐻𝑒𝑛 =
𝑃𝑡 −𝑃𝑑 𝑅𝐻𝑒𝑛
𝐹𝑓𝑖𝑙𝑡 = (1 − 𝐻𝑎 ) (1 − 𝑃𝑔 = 𝑃𝑣 + 𝑅𝑒 (
𝐶𝑎 𝑃𝑎 − 𝑃𝑔 ) 𝐶𝑒 𝑅𝑎
𝑃𝑎 − 𝑃𝑔 − 𝐹𝑓𝑖𝑙𝑡 ) 𝑅𝑎
𝑃𝑜𝑠𝑚 = 𝑎𝐶 + 𝑏𝐶 2 1
𝐶𝑒 = 2𝑏 (√𝑎2 − 4𝑏(𝑃𝑡 − 𝑃𝑔 ) − 𝑎) 𝜓 = 𝜓𝑚𝑎𝑥 −
(2)
𝜓𝑚𝑎𝑥 − 𝜓𝑚𝑖𝑛 3𝜒 1 + exp[𝛼 (𝑇𝐹 3 − 𝑆)] 𝐻𝑒𝑛0
𝑅𝑎 = 𝑅𝑎0 [𝛽 + (1 − 𝛽)𝑟 −4 ] 𝑃𝑎𝑣 =
1 𝑅𝑎0 (𝑃𝑎 − (𝑃𝑎 − 𝑃𝑔 )𝛽 + 𝑃𝑔 ) 2 𝑅𝑎
4.7 𝑃𝑒𝑞 = 2.4 × 𝑒 10(𝑟−1.4) + 1.6(𝑟 − 1) + 𝜓 ( + 7.2(𝑟 + 0.9)) 1 + 𝑒 13(0.4−𝑟) Variables and parameters in Eq.2 are only shortly described here. For more detailed information, see Barfred et al. [25] and references therein. 𝐹𝐻𝑒𝑛 is determined by the difference between distal tubular pressure 𝑃𝑑 and proximal tubular pressure divided by 𝑅𝐻𝑒𝑛 , the flow resistance in the lop of Henle. 𝐹𝑓𝑖𝑙𝑡 is the glomerular filtration rate, 𝐻𝑎 is the afferent hematocrit, 𝐶𝑎 is the concentration of protein in the afferent, and 𝐶𝑒 is the concentration of protein in the efferent plasma. 𝑃𝑎 And 𝑃𝑔 are the arterial and glomerular blood pressures respectively, and 𝑅𝑎 , which has a significant role in the dynamical model, is the hemodynamic resistance of the afferent arteriole. The glomerular pressure 𝑃𝑔 is calculated by distributing the arterial to the venous pressure drop between the afferent and the efferent arteriolar resistance, where the venous pressure 𝑃𝑣 and the efferent arteriolar resistance 𝑅𝑒 are considered as constants parameters. Incorporating the relationship between the osmotic pressure 𝑃𝑜𝑠𝑚 and the protein concentration 𝐶 results in calculating the 𝐶𝑒 in the fifth auxiliary function. The sixth function demonstrates the description of glomerular feedback, which is an empirically based sigmoidal relation between 𝜓 and 𝜒3 where 𝜓𝑚𝑎𝑥 and 𝜓𝑚𝑖𝑛 are the maximum and
minimum values of the activation. 𝛼 stands for the slope of the feedback curve, and S is the displacement of the curve along the flow axis. The seventh auxiliary function represents the dynamical behavior of arteriolar resistance where 𝑅𝑎0 is a standard value of the arteriolar resistance, and r is the radius of the active part of the vessel normalized to its resting value [31]. In the eighth and ninth functions, 𝑃𝑎𝑣 is the average pressure in the active part of the arteriole, and 𝑃𝑒𝑞 is the equilibrium value of 𝑃𝑎𝑣 . The value of parameters and constants are represented in Table 1.
Table 1. List of model parameters 𝑃𝑎 = 13.3𝐾𝑃𝑎
𝐶𝑎 = 54𝑔/𝑙
𝑃𝑣 = 1.3𝐾𝑃𝑎
𝑙 a = 21.7𝑃𝑎( ) 𝑔
𝑃𝑑 = 0.6𝐾𝑃𝑎
𝑙 2 b = 0.39𝑃𝑎 ( ) 𝑔
𝑅𝑎 = 2.4𝐾𝑃𝑎
ω = 20𝐾𝑃𝑎𝑆 2
𝑅𝑒 = 1.9𝐾𝑃𝑎
d = 0.04/𝑠
𝑅𝐻𝑒𝑛 = 5.3𝐾𝑃𝑎
β = 0.67
𝐶𝑡𝑢𝑏 = 3.0𝑛𝑙/𝐾𝑃𝑎
𝜓𝑚𝑎𝑥 = 0.44
𝐻𝑎 = 0.5
𝜓𝑚𝑖𝑛 = 0.2
𝐹𝑟𝑒𝑎𝑏 = 0.3𝑛𝑙/𝑠
𝜓𝑒𝑞 = 0.38
𝐹𝐻𝑒𝑛0 = 0.2𝑛𝑙/
The bifurcation diagram of intratubular pressure, for 𝛼 varying from 5 to 16, is shown in Fig. 1. From this figure, it is clear that for the values of 5 < 𝛼 < 11.5, the dynamical behavior of the system is periodic, and for 𝛼 > 11.5, the dynamic is chaotic, and the bifurcation diagram goes towards period-doubling root to chaos [30].
0.9
t
P (mmHg)
0.8 0.7 0.6 0.5 0.4 4
6
8
10
12
14
16
Fig.1. Bifurcation diagram of intratubular pressure for a single nephron.
3. Biological Coupling and simulation results To consider the interaction and physiological coupling between nephrons, the effect of each nephron on the others was taken into account by coupling the activation parameters of smooth muscle cells in the arterial wall for all the nephrons. In other words, the activation parameter 𝜓 of each nephron was assumed to affect the others. It is because of vascularly propagated signals that the propagation of signal for each nephron affects the others. So, for each nephron, a coupling term was added to its activation parameter, which is the effect of its neighboring nephrons. This term is calculated as follows [32]: 𝑁
∆𝜓𝑗 = ∑ 𝜃 𝜓𝑖 exp(−𝜆(𝐿𝑗𝑖 + 𝐿𝑔 ))
(3)
𝑖=1,𝑖≠𝑗
∆𝜓𝑗 is the coupling term added to the TGF activation parameter of each nephron. 𝜓𝑖 is the TGF activation for 𝑖 𝑡ℎ nephron which is obtained from the mathematical model of each single nephron as stated before. 𝐿𝑗𝑖 is the distance from the origin of afferent arteriole for the 𝑗 𝑡ℎ nephron to its origin for the 𝑖 𝑡ℎ nephron, and 𝐿𝑔 is the length of the afferent arteriole which was considered constant and identical for all nephrons in the network. 𝜆 is a parameter that increases or decreases the strength of exponential damping of coupling behavior related to the interaction of
neighboring nephrons. 𝜃 is a constant parameter used to consider the effect of various coupling strength based on physiological considerations [32]. To construct the network, 𝑁 oscillators of Eq. (1) were coupled on a one-dimensional ring. The coupling was non-local, and each oscillator was coupled with its 2𝑃 nearest neighbors. The equations for the network in the effect of considering a new definition for the auxiliary equation of activation parameter are given as:
𝑑𝑃𝑡 𝑖 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 )
𝑑𝑅𝑎 𝑖 = 𝑉𝑅𝑎 𝑖 𝑑𝑡
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎𝑖 = −𝑑𝑉𝑅𝑎 𝑖 + 𝑑𝑡 𝜔 𝑑𝜒1 𝑖 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑖 𝑑𝑡 𝑇
(4)
𝑑𝜒2𝑖 3 = (𝜒1𝑖 − 𝜒2 𝑖 ) 𝑑𝑡 𝑇 𝑑𝜒3 𝑖 3 = (𝜒2𝑖 − 𝜒3𝑖 ) 𝑑𝑡 𝑇 𝜓 = 𝜓𝑚𝑎𝑥 −
𝜓𝑚𝑎𝑥 −𝜓𝑚𝑖𝑛
3𝜒3 𝑖 −𝑆)] 𝑇𝐹𝐻𝑒𝑛0
1+exp[𝛼(
+ ∆𝝍𝒋 ,
Here, 𝑖 = 1,2, … , 𝑁 is the index of 𝑡ℎ𝑒 𝑖 𝑡ℎ oscillator, parameter 𝑑 is the strength of coupling, and 𝑃 describes the number of coupling neighbors for each oscillator. In all simulations, the value of 𝑃 = 20, 𝑁 = 100, and 𝜃 = 0.5 are constant. Results are presented in figures 2 and 3. As mentioned in section 2, by changing the bifurcation parameter 𝛼 of a single nephron model, different dynamical behaviors can be obtained, such as chaotic and periodic behaviors. The intention was to use different values of 𝛼 and observe network patterns by varying coupling strength. The initial conditions were chosen randomly. First, 𝛼 was set at a value of 8(𝛼 = 8 ) at which a single nephron shows period-2 behavior. Figure 2(a) shows the spatiotemporal pattern obtained for 𝜆 = 6 and 𝐿𝑔 = 0.1. According to this figure, there are multiple coherent clusters and some incoherent oscillators, which seem to be non-static in time. Therefore, this state can be called a multi-headed chimera state. The multiheaded chimera state consists of some subdivisions of coherent and incoherent clusters [33]. The time snapshot corresponding to this state is shown in Fig. 2(b).
Then, 𝛼 was set at 10 (𝛼 = 10) at which a single nephron exhibits period-2 behavior. When 𝜆 = 7 and 𝐿𝑔 = 0.05, an interesting state (as illustrated in Fig. 2(c)) was observed. As can be seen, the nephrons are divided into two synchronized clusters. It can be seen that there are some solitaries in between clusters. So, this state can be named as imperfect cluster synchronization. Figure 2(d) shows the time snapshot of this state at 𝑡 = 2950.
Fig. 2. a) Spatiotemporal pattern of the network for α = 8, θ = 0.5, λ = 6 and Lg = 0.1 which shows multi chimera state. b) the time snapshot at t = 2950 referring to (a). c) Spatiotemporal pattern of the network for α = 10, θ = 0.5, λ = 7 and Lg = 0.05 which shows imperfect cluster synchronization. d) the time snapshot at t = 2950 referring to (c)
In the next step, the 𝛼 values at which a nephron exhibits chaotic behavior were examined. First, 𝛼 was fixed at 15 and then at 16. The spatiotemporal pattern and time snapshot of the network in the case of 𝛼 = 15, 𝜆 = 5, and 𝐿𝑔 = 0.09 are depicted in Fig. 3(a) and (b). According to Fig. 3(b), the nephrons are divided into six synchronized clusters. It can be seen that there are some solitaries among the clusters. So, this state is also named as imperfect cluster synchronization. Figures 3(c) and (d) display the spatiotemporal pattern and time snapshot of the network when 𝛼 = 16, 𝜆 = 10, and 𝐿𝑔 = 0.02. These figures each represent six synchronized clusters, four of
which are phase-synchronized. There are also some solitaries in between the clusters. So this state is an imperfect cluster synchronization as well.
Fig. 3. a) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟓, 𝜽 = 𝟎. 𝟓, 𝝀 = 𝟓 𝒂𝒏𝒅 𝑳𝒈 = 𝟎. 𝟎𝟗 which shows imperfect cluster synchronization. b) the time snapshot at 𝒕 = 𝟐𝟗𝟓𝟎 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟔, 𝜽 = 𝟎. 𝟓, 𝝀 = 𝟏𝟎 𝒂𝒏𝒅 𝑳𝒈 = 𝟎. 𝟎𝟐 which shows imperfect cluster synchronization. d) the time snapshot at 𝒕 = 𝟐𝟗𝟓𝟎 referring to (c).
4. Mathematical coupling and simulation results Also from a merely mathematical point of view, a new coupling is considered between nephrons in the network, for which there is no biological evidence yet, or maybe has not been identified until now. However, this kind of coupling was taken into account already; hence, the main equation for the new network becomes as follows:
𝑑𝑃𝑡𝑖 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 ) +
𝑑 2𝑃
∑𝑖+𝑃 𝑗=𝑖−𝑃(𝑉𝑅𝑎𝑗 − 𝑉𝑅𝑎𝑖 )
(5)
𝑑𝑅𝑎 𝑖 = 𝑉𝑅𝑎 𝑖 𝑑𝑡 𝑖+𝑃
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎𝑖 𝑑 = −𝑑𝑉𝑅𝑎 𝑖 + + ∑ (𝑉𝑅𝑎 − 𝑉𝑅𝑎𝑖 ) 𝑗 𝑑𝑡 𝜔 2𝑃 𝑑𝜒1𝑖 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑖 𝑑𝑡 𝑇
𝑗=𝑖−𝑃
𝑑𝜒2𝑖 3 = (𝜒1 𝑖 − 𝜒2 𝑖 ) 𝑑𝑡 𝑇 𝑑𝜒3 𝑖 3 = (𝜒2𝑖 − 𝜒3 𝑖 ) 𝑑𝑡 𝑇
Where 𝑑 is coupling strength, as mentioned in section 2, by changing the bifurcation parameter 𝛼 of the single nephron model, different dynamical behaviors can be obtained, such as chaotic and periodic behaviors. We use different values for 𝛼 and observe network patterns by varying the coupling strength. The initial conditions are chosen randomly. First, 𝛼 is set at 𝛼 = 9 at which a single nephron shows period-2 behavior. Figure 4(a) shows the spatiotemporal pattern obtained for 𝑑 = 0.3. According to this figure, there are multiple coherent clusters and some incoherent oscillators in between. Therefore, this state can be called a multiheaded chimera state. The time snapshot corresponding to this state is shown in Fig. 4(b). Next, 𝛼 was set at 𝛼 = 11, at which a single nephron exhibits period-4 behavior. When 𝑑 reaches 0.6, a particular state (as sketched in Fig. 4(c)) is observed. As can be seen, the nephrons are divided into six synchronized clusters which can be divided into two in-phase synchronous groups. The two groups are also anti-phase synchronous with each other. It can be seen that there are some solitaries in between the clusters; therefore, this state is imperfect cluster synchronization. Figure 4(d) shows the time snapshot of this state at 𝑡 = 2950. The same type of spatiotemporal patterns is also observed for 𝛼 = 6 𝑎𝑛𝑑 𝛼 = 13 which are not demonstrated here.
Fig. 4. a) Spatiotemporal pattern of the network for 𝜶 = 𝟗 and 𝒅 = 𝟎. 𝟑 which shows multi chimera state. b) the time snapshot at t=2910 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟏 and 𝒅 = 𝟎. 𝟔 which shows imperfect cluster synchronization. d) the time snapshot at t=2950 referring to (c).
In the next step, the 𝛼 values at which a nephron exhibits chaotic behavior were examined. First, 𝛼 was fixed at 15 and then at 16. The spatiotemporal pattern and time snapshot of the network in the case of 𝛼 = 15 are depicted in Fig. 5(a) and (b). According to Fig. 5(b), there are two synchronous clusters, and each oscillator is located in one of them. So, in this case, the pattern shows cluster synchronization. The same results are obtained for 𝛼 = 8 and 𝛼 = 14. Figures 5(c) and (d) represent the spatiotemporal pattern and time snapshot of the network when 𝛼 = 16. These figures illustrate cluster synchronization, but as shown in the spatiotemporal pattern, the cluster boundaries are not static in time.
Fig. 5. a) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟓 and 𝒅 = 𝟎. 𝟑, which shows cluster synchronization. b) the time snapshot at t=2930 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟔 and 𝒅 = 𝟎. 𝟒 which shows cluster synchronization. d) the time snapshot at t=2970 referring to (c).
Discussion In previous studies on the collective behaviors of a group of interacting and coupled nephrons, many behaviors have been identified and investigated. These behaviors include synchronization behavior, quasiperiodicity and chaos in a nephron tree [32], synchronization of tubular pressure oscillation in interacting nephrons [34], synchronization effects which occur among neighboring
nephrons [35], multistable dynamics of coupled nephrons [36], and some other nearly related and similar researches such as [37]. However, it is anticipated that in a complex system such as a kidney, with its complex dynamical behavior, more dynamical states can emerge. In previous studies, it has been approved that the nephrons with the synchronized pressure are coupled with each other [38]. As a result of this interaction, the oscillation of the TGF-mediated proximal pressure is synchronized. It has been proposed that the “cross-talk” between the TGF signals from the different nephrons, that is transmitted along the afferent arterioles, and probably also along a part of the interlobular artery results in the interaction [38]. Further studies show that in-phase synchronization is observed in the data analysis of the rats with normal blood pressure. Also, the chaotic phase synchronization between adjacent nephrons is found in the rats with elevated blood pressure [39]. Regarding the experiments, we could conclude that interacted nephrons oscillators oscillate synchronously in typical situations. Thus the chimera states in the network of coupled nephrons may illustrate the abnormal situation or a malfunction in kidney task.
Conclusion In this paper, a ring network of non-locally coupled nephrons was studied. The bifurcation diagram of the incorporated model shows that a single nephron can exhibit different periodic and chaotic behaviors due to changes in bifurcation parameters. Many bifurcation parameters can be considered and affect the dynamical state of each nephron. These parameters may all have pieces of physiological evidence such as variations in the delay of the Henle loop, or variations in TGF gain such as T and α. Here, the spatiotemporal patterns of the network were investigated in both modes of periodic and chaotic nephrons dynamics, by changing the coupling strength in two different coupling functions. The results showed various patterns by changing the parameters. It was obtained that when the bifurcation parameter is set at the periodic region, the network represents multi chimera state or imperfect cluster synchronization, and when each nephron is set at the chaotic region, by setting the model parameter, the cluster synchronization appears in the network. Thus, the emergence of chimera states is observed in the model in a state which is assumed as the normotensive mode. Declaration of interests
The authors declare that they have no competing interests.
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The emergence of chimera states in a network of nephrons Jalal Khouhak , Zahra Faghani , Jakob L. Laugesen , Sajad Jafari PII: DOI: Reference:
S0577-9073(19)30971-2 https://doi.org/10.1016/j.cjph.2019.10.024 CJPH 993
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
4 May 2019 12 October 2019 30 October 2019
Please cite this article as: Jalal Khouhak , Zahra Faghani , Jakob L. Laugesen , Sajad Jafari , The emergence of chimera states in a network of nephrons, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.10.024
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Highlights
The kidney plays an important role in our body. The kidneys consist of a large number of nephrons. The interaction between these nephrons induces different patterns in the network. A network of coupled nephrons is constructed and investigated. Emergence of chimera state is observed in this network.
The emergence of chimera states in a network of nephrons Jalal Khouhak a, Zahra Faghani a, Jakob L. Laugesen b, Sajad Jafari a,1 a b
Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran Department of Food Science, University of Copenhagen, Denmark
Abstract The kidney plays an essential role in our body, mainly by controlling secretion and reabsorption of water and salts. The kidneys consist of a large number of nephrons which are the functional units of the kidney. The interactions between these nephrons induce different behaviors which can be considered by a dynamical model. In this paper, a network of coupled nephron models and its dynamics is investigated. Numerical simulations of the network reveal various types of dynamical patterns depending on the coupling function and strength. One of the observed phenomenon is the emergence of chimera state. A chimera state is defined by the coexistence of coherent and incoherent groups in a network of identical oscillators. The occurrence of the chimera state can be related to the situation of disturbed synchronous oscillation of the TGFmediated proximal pressure.
Keywords: chimera state; multi-stability; kidney; nephron model.
1
Corresponding author. E-mail address: [email protected]
1. Introduction Networks of coupled oscillators are known as a class of interactive systems that can be found in a wide range of science branches such as physics, engineering, and especially, natural and biological systems [1-3]. The collective behavior of a set of coupled oscillators is mainly dependent on the coupling strength and mutual interactions between the oscillators. The type of coupling can also affect the collective behavior depending on whether it is global or local [4, 5]. The study of the collective behaviors in the biological networks can help in understanding different mechanisms in the living systems [6-8]. In 2002, two Japanese researchers, Kuramoto and Battogtokh, found a new dynamical behavior in a network of oscillators [9]. While evaluating the dynamical behaviors of a ring of non-locally coupled identical oscillators, they found a particular pattern with coexistence of coherent and incoherent oscillators. In 2004, Abrams and Strogatz called this phenomenon to chimera state [10]. Since the discovery of chimera, it has attracted many scientists and researchers in various fields [11, 12]. As well, many spatiotemporal patterns have been observed in heterogeneous systems ever since [13, 14] and various types of chimera state have been introduced. Kemeth et al. [15] classified the chimera state into three types, named stationary, breathing, and turbulent chimera. Zakharova et al. [16, 17] found a pattern composed of spatially coherent and incoherent oscillation death state and named it chimera death. Kapitaniak et al. [18] introduced a new pattern in which some oscillators escaped from the coherent cluster and called it imperfect chimera that has been observed in different systems [19, 20]. Wei et al. [21] investigated the network of AC induced Hindmarsh-Rose neurons and observed occurrence of non-stationary chimeras. Some researchers have concentrated on investigating the effect of different parameters’ values on the emergence of chimera state [22, 23]. For example, Omelchenko et al. [24] surveyed the effect of variations in the strength of coupling parameters, on the non-locally coupled Fitzhugh-Nagumo oscillators. The mammalian kidney has a crucial role in managing blood pressure variations. In other words, it has the duty of creating a proper environment for all cells in the body by controlling the excretion of salts and water. [25]. The kidney function depends on a variety of complex mechanisms, which are managed by nephron; the individual functional unit of the kidney. It controls all the related mechanisms. The nephron compensates for the changes in the arterial blood pressure with two different mechanisms [26, 27]. Firstly, the most important mechanism is called the tubuloglomerular feedback mechanism (TGF). It can generate self-sustained oscillations that are associated with a time delay of the fluid passing through the loop of Henle -a curved shape portion of the nephron that leads from the proximal tubule to the distal tubule. Another mechanism is called myogenic mechanism that is related to the smooth muscle cells in the arteriolar wall and responds to changes in transmural pressures. The model used in this investigation is mainly focused on the TGF mechanism. In TGF, the nephron controls the incoming blood flow by increasing and decreasing the radius of afferent arterioles [27-29]. Nonlinear characteristics of each single nephron, the TGF feedback mechanism, and more notable, the interaction between coupled nephrons can lead to various complex dynamical behaviors. Thus, it is likely to observe collective behaviors such as chimera state, in the network
of interacting nephrons. In this paper, at first a single nephron model and its bifurcation diagram are described to show different dynamical behaviors. Then, the mathematical equations of a network of coupled nephrons are given, and its collective behaviors are investigated.
2. The Nephron Model The model used in this study is a set of six coupled first-order differential equations which are as follows [30]:
𝑑𝑃𝑡 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 )
𝑑𝑅𝑎 = 𝑉𝑅𝑎 𝑑𝑡
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎 = −𝑑𝑉𝑅𝑎 + 𝑑𝑡 𝜔 𝑑𝜒1 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑑𝑡 𝑇
(1)
𝑑𝜒2 3 = (𝜒1 − 𝜒2 ) 𝑑𝑡 𝑇 𝑑𝜒3 3 = (𝜒2 − 𝜒3 ) 𝑑𝑡 𝑇
where 𝐹𝑓𝑖𝑙𝑡 implies the single nephron glomerular filtration rate, and 𝐶𝑡𝑢𝑏 represents the elastic compliance of the tubule. 𝐹𝐻𝑒𝑛 is the Henle flow, 𝑃𝑡 is the proximal tubular pressure, 𝑃𝑑 is the distal tubular pressures, and 𝑅𝐻𝑒𝑛 is the flow resistance in the loop of Henle. In this model, the reabsorption of water and salts, which takes place as the flow passes through the loop of Henle, is ignored to simplify the problem. 𝐹𝑟𝑒𝑎𝑏 and 𝑅𝐻𝑒𝑛 represent the reabsorption in the proximal tubule and the flow resistance respectively, and both are assumed to be constant. 𝐻𝑎 is the afferent hematocrit, 𝐶𝑎 and 𝐶𝑒 are the concentration of protein in the afferent and efferent plasma, respectively. 𝑃𝑎 and 𝑃𝑔 are the arterial and glomerular blood pressures respectively, and 𝑅𝑎 is the hemodynamic resistance of the afferent arteriole. 𝑃𝑣 shows the venous pressure and 𝑅𝑒 shows the efferent arteriolar resistance, with both being considered as constants. 𝜓 refers to the activation of the afferent arteriole, and T is the total delay time. 𝜒1 , 𝜒2 , and 𝜒3 are intermediate variables in the delay chain. Finally, 𝑃𝑎𝑣 is the average pressure in the active part of the arteriole [30].
There are also some other numerical equations that are necessary to be solved and calculated before the differential equations. They are as follows [30]: 𝐹𝐻𝑒𝑛 =
𝑃𝑡 −𝑃𝑑 𝑅𝐻𝑒𝑛
𝐹𝑓𝑖𝑙𝑡 = (1 − 𝐻𝑎 ) (1 − 𝑃𝑔 = 𝑃𝑣 + 𝑅𝑒 (
𝐶𝑎 𝑃𝑎 − 𝑃𝑔 ) 𝐶𝑒 𝑅𝑎
𝑃𝑎 − 𝑃𝑔 − 𝐹𝑓𝑖𝑙𝑡 ) 𝑅𝑎
𝑃𝑜𝑠𝑚 = 𝑎𝐶 + 𝑏𝐶 2 1
𝐶𝑒 = 2𝑏 (√𝑎2 − 4𝑏(𝑃𝑡 − 𝑃𝑔 ) − 𝑎) 𝜓 = 𝜓𝑚𝑎𝑥 −
(2)
𝜓𝑚𝑎𝑥 − 𝜓𝑚𝑖𝑛 3𝜒 1 + exp[𝛼 (𝑇𝐹 3 − 𝑆)] 𝐻𝑒𝑛0
𝑅𝑎 = 𝑅𝑎0 [𝛽 + (1 − 𝛽)𝑟 −4 ] 𝑃𝑎𝑣 =
1 𝑅𝑎0 (𝑃𝑎 − (𝑃𝑎 − 𝑃𝑔 )𝛽 + 𝑃𝑔 ) 2 𝑅𝑎
4.7 𝑃𝑒𝑞 = 2.4 × 𝑒 10(𝑟−1.4) + 1.6(𝑟 − 1) + 𝜓 ( + 7.2(𝑟 + 0.9)) 1 + 𝑒 13(0.4−𝑟) Variables and parameters in Eq.2 are only shortly described here. For more detailed information, see Barfred et al. [25] and references therein. 𝐹𝐻𝑒𝑛 is determined by the difference between distal tubular pressure 𝑃𝑑 and proximal tubular pressure divided by 𝑅𝐻𝑒𝑛 , the flow resistance in the lop of Henle. 𝐹𝑓𝑖𝑙𝑡 is the glomerular filtration rate, 𝐻𝑎 is the afferent hematocrit, 𝐶𝑎 is the concentration of protein in the afferent, and 𝐶𝑒 is the concentration of protein in the efferent plasma. 𝑃𝑎 And 𝑃𝑔 are the arterial and glomerular blood pressures respectively, and 𝑅𝑎 , which has a significant role in the dynamical model, is the hemodynamic resistance of the afferent arteriole. The glomerular pressure 𝑃𝑔 is calculated by distributing the arterial to the venous pressure drop between the afferent and the efferent arteriolar resistance, where the venous pressure 𝑃𝑣 and the efferent arteriolar resistance 𝑅𝑒 are considered as constants parameters. Incorporating the relationship between the osmotic pressure 𝑃𝑜𝑠𝑚 and the protein concentration 𝐶 results in calculating the 𝐶𝑒 in the fifth auxiliary function. The sixth function demonstrates the description of glomerular feedback, which is an empirically based sigmoidal relation between 𝜓 and 𝜒3 where 𝜓𝑚𝑎𝑥 and 𝜓𝑚𝑖𝑛 are the maximum and
minimum values of the activation. 𝛼 stands for the slope of the feedback curve, and S is the displacement of the curve along the flow axis. The seventh auxiliary function represents the dynamical behavior of arteriolar resistance where 𝑅𝑎0 is a standard value of the arteriolar resistance, and r is the radius of the active part of the vessel normalized to its resting value [31]. In the eighth and ninth functions, 𝑃𝑎𝑣 is the average pressure in the active part of the arteriole, and 𝑃𝑒𝑞 is the equilibrium value of 𝑃𝑎𝑣 . The value of parameters and constants are represented in Table 1.
Table 1. List of model parameters 𝑃𝑎 = 13.3𝐾𝑃𝑎
𝐶𝑎 = 54𝑔/𝑙
𝑃𝑣 = 1.3𝐾𝑃𝑎
𝑙 a = 21.7𝑃𝑎( ) 𝑔
𝑃𝑑 = 0.6𝐾𝑃𝑎
𝑙 2 b = 0.39𝑃𝑎 ( ) 𝑔
𝑅𝑎 = 2.4𝐾𝑃𝑎
ω = 20𝐾𝑃𝑎𝑆 2
𝑅𝑒 = 1.9𝐾𝑃𝑎
d = 0.04/𝑠
𝑅𝐻𝑒𝑛 = 5.3𝐾𝑃𝑎
β = 0.67
𝐶𝑡𝑢𝑏 = 3.0𝑛𝑙/𝐾𝑃𝑎
𝜓𝑚𝑎𝑥 = 0.44
𝐻𝑎 = 0.5
𝜓𝑚𝑖𝑛 = 0.2
𝐹𝑟𝑒𝑎𝑏 = 0.3𝑛𝑙/𝑠
𝜓𝑒𝑞 = 0.38
𝐹𝐻𝑒𝑛0 = 0.2𝑛𝑙/
The bifurcation diagram of intratubular pressure, for 𝛼 varying from 5 to 16, is shown in Fig. 1. From this figure, it is clear that for the values of 5 < 𝛼 < 11.5, the dynamical behavior of the system is periodic, and for 𝛼 > 11.5, the dynamic is chaotic, and the bifurcation diagram goes towards period-doubling root to chaos [30].
0.9
t
P (mmHg)
0.8 0.7 0.6 0.5 0.4 4
6
8
10
12
14
16
Fig.1. Bifurcation diagram of intratubular pressure for a single nephron.
3. Biological Coupling and simulation results To consider the interaction and physiological coupling between nephrons, the effect of each nephron on the others was taken into account by coupling the activation parameters of smooth muscle cells in the arterial wall for all the nephrons. In other words, the activation parameter 𝜓 of each nephron was assumed to affect the others. It is because of vascularly propagated signals that the propagation of signal for each nephron affects the others. So, for each nephron, a coupling term was added to its activation parameter, which is the effect of its neighboring nephrons. This term is calculated as follows [32]: 𝑁
∆𝜓𝑗 = ∑ 𝜃 𝜓𝑖 exp(−𝜆(𝐿𝑗𝑖 + 𝐿𝑔 ))
(3)
𝑖=1,𝑖≠𝑗
∆𝜓𝑗 is the coupling term added to the TGF activation parameter of each nephron. 𝜓𝑖 is the TGF activation for 𝑖 𝑡ℎ nephron which is obtained from the mathematical model of each single nephron as stated before. 𝐿𝑗𝑖 is the distance from the origin of afferent arteriole for the 𝑗 𝑡ℎ nephron to its origin for the 𝑖 𝑡ℎ nephron, and 𝐿𝑔 is the length of the afferent arteriole which was considered constant and identical for all nephrons in the network. 𝜆 is a parameter that increases or decreases the strength of exponential damping of coupling behavior related to the interaction of
neighboring nephrons. 𝜃 is a constant parameter used to consider the effect of various coupling strength based on physiological considerations [32]. To construct the network, 𝑁 oscillators of Eq. (1) were coupled on a one-dimensional ring. The coupling was non-local, and each oscillator was coupled with its 2𝑃 nearest neighbors. The equations for the network in the effect of considering a new definition for the auxiliary equation of activation parameter are given as:
𝑑𝑃𝑡 𝑖 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 )
𝑑𝑅𝑎 𝑖 = 𝑉𝑅𝑎 𝑖 𝑑𝑡
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎𝑖 = −𝑑𝑉𝑅𝑎 𝑖 + 𝑑𝑡 𝜔 𝑑𝜒1 𝑖 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑖 𝑑𝑡 𝑇
(4)
𝑑𝜒2𝑖 3 = (𝜒1𝑖 − 𝜒2 𝑖 ) 𝑑𝑡 𝑇 𝑑𝜒3 𝑖 3 = (𝜒2𝑖 − 𝜒3𝑖 ) 𝑑𝑡 𝑇 𝜓 = 𝜓𝑚𝑎𝑥 −
𝜓𝑚𝑎𝑥 −𝜓𝑚𝑖𝑛
3𝜒3 𝑖 −𝑆)] 𝑇𝐹𝐻𝑒𝑛0
1+exp[𝛼(
+ ∆𝝍𝒋 ,
Here, 𝑖 = 1,2, … , 𝑁 is the index of 𝑡ℎ𝑒 𝑖 𝑡ℎ oscillator, parameter 𝑑 is the strength of coupling, and 𝑃 describes the number of coupling neighbors for each oscillator. In all simulations, the value of 𝑃 = 20, 𝑁 = 100, and 𝜃 = 0.5 are constant. Results are presented in figures 2 and 3. As mentioned in section 2, by changing the bifurcation parameter 𝛼 of a single nephron model, different dynamical behaviors can be obtained, such as chaotic and periodic behaviors. The intention was to use different values of 𝛼 and observe network patterns by varying coupling strength. The initial conditions were chosen randomly. First, 𝛼 was set at a value of 8(𝛼 = 8 ) at which a single nephron shows period-2 behavior. Figure 2(a) shows the spatiotemporal pattern obtained for 𝜆 = 6 and 𝐿𝑔 = 0.1. According to this figure, there are multiple coherent clusters and some incoherent oscillators, which seem to be non-static in time. Therefore, this state can be called a multi-headed chimera state. The multiheaded chimera state consists of some subdivisions of coherent and incoherent clusters [33]. The time snapshot corresponding to this state is shown in Fig. 2(b).
Then, 𝛼 was set at 10 (𝛼 = 10) at which a single nephron exhibits period-2 behavior. When 𝜆 = 7 and 𝐿𝑔 = 0.05, an interesting state (as illustrated in Fig. 2(c)) was observed. As can be seen, the nephrons are divided into two synchronized clusters. It can be seen that there are some solitaries in between clusters. So, this state can be named as imperfect cluster synchronization. Figure 2(d) shows the time snapshot of this state at 𝑡 = 2950.
Fig. 2. a) Spatiotemporal pattern of the network for α = 8, θ = 0.5, λ = 6 and Lg = 0.1 which shows multi chimera state. b) the time snapshot at t = 2950 referring to (a). c) Spatiotemporal pattern of the network for α = 10, θ = 0.5, λ = 7 and Lg = 0.05 which shows imperfect cluster synchronization. d) the time snapshot at t = 2950 referring to (c)
In the next step, the 𝛼 values at which a nephron exhibits chaotic behavior were examined. First, 𝛼 was fixed at 15 and then at 16. The spatiotemporal pattern and time snapshot of the network in the case of 𝛼 = 15, 𝜆 = 5, and 𝐿𝑔 = 0.09 are depicted in Fig. 3(a) and (b). According to Fig. 3(b), the nephrons are divided into six synchronized clusters. It can be seen that there are some solitaries among the clusters. So, this state is also named as imperfect cluster synchronization. Figures 3(c) and (d) display the spatiotemporal pattern and time snapshot of the network when 𝛼 = 16, 𝜆 = 10, and 𝐿𝑔 = 0.02. These figures each represent six synchronized clusters, four of
which are phase-synchronized. There are also some solitaries in between the clusters. So this state is an imperfect cluster synchronization as well.
Fig. 3. a) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟓, 𝜽 = 𝟎. 𝟓, 𝝀 = 𝟓 𝒂𝒏𝒅 𝑳𝒈 = 𝟎. 𝟎𝟗 which shows imperfect cluster synchronization. b) the time snapshot at 𝒕 = 𝟐𝟗𝟓𝟎 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟔, 𝜽 = 𝟎. 𝟓, 𝝀 = 𝟏𝟎 𝒂𝒏𝒅 𝑳𝒈 = 𝟎. 𝟎𝟐 which shows imperfect cluster synchronization. d) the time snapshot at 𝒕 = 𝟐𝟗𝟓𝟎 referring to (c).
4. Mathematical coupling and simulation results Also from a merely mathematical point of view, a new coupling is considered between nephrons in the network, for which there is no biological evidence yet, or maybe has not been identified until now. However, this kind of coupling was taken into account already; hence, the main equation for the new network becomes as follows:
𝑑𝑃𝑡𝑖 𝑑𝑡
=
1 𝐶𝑡𝑢𝑏
(𝐹𝑓𝑖𝑙𝑡 − 𝐹𝑟𝑒𝑎𝑏 − 𝐹𝐻𝑒𝑛 ) +
𝑑 2𝑃
∑𝑖+𝑃 𝑗=𝑖−𝑃(𝑉𝑅𝑎𝑗 − 𝑉𝑅𝑎𝑖 )
(5)
𝑑𝑅𝑎 𝑖 = 𝑉𝑅𝑎 𝑖 𝑑𝑡 𝑖+𝑃
𝑃𝑎𝑣 − 𝑃𝑒𝑞 𝑑𝑉𝑅𝑎𝑖 𝑑 = −𝑑𝑉𝑅𝑎 𝑖 + + ∑ (𝑉𝑅𝑎 − 𝑉𝑅𝑎𝑖 ) 𝑗 𝑑𝑡 𝜔 2𝑃 𝑑𝜒1𝑖 3 = 𝐹𝐻𝑒𝑛 − 𝜒1 𝑖 𝑑𝑡 𝑇
𝑗=𝑖−𝑃
𝑑𝜒2𝑖 3 = (𝜒1 𝑖 − 𝜒2 𝑖 ) 𝑑𝑡 𝑇 𝑑𝜒3 𝑖 3 = (𝜒2𝑖 − 𝜒3 𝑖 ) 𝑑𝑡 𝑇
Where 𝑑 is coupling strength, as mentioned in section 2, by changing the bifurcation parameter 𝛼 of the single nephron model, different dynamical behaviors can be obtained, such as chaotic and periodic behaviors. We use different values for 𝛼 and observe network patterns by varying the coupling strength. The initial conditions are chosen randomly. First, 𝛼 is set at 𝛼 = 9 at which a single nephron shows period-2 behavior. Figure 4(a) shows the spatiotemporal pattern obtained for 𝑑 = 0.3. According to this figure, there are multiple coherent clusters and some incoherent oscillators in between. Therefore, this state can be called a multiheaded chimera state. The time snapshot corresponding to this state is shown in Fig. 4(b). Next, 𝛼 was set at 𝛼 = 11, at which a single nephron exhibits period-4 behavior. When 𝑑 reaches 0.6, a particular state (as sketched in Fig. 4(c)) is observed. As can be seen, the nephrons are divided into six synchronized clusters which can be divided into two in-phase synchronous groups. The two groups are also anti-phase synchronous with each other. It can be seen that there are some solitaries in between the clusters; therefore, this state is imperfect cluster synchronization. Figure 4(d) shows the time snapshot of this state at 𝑡 = 2950. The same type of spatiotemporal patterns is also observed for 𝛼 = 6 𝑎𝑛𝑑 𝛼 = 13 which are not demonstrated here.
Fig. 4. a) Spatiotemporal pattern of the network for 𝜶 = 𝟗 and 𝒅 = 𝟎. 𝟑 which shows multi chimera state. b) the time snapshot at t=2910 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟏 and 𝒅 = 𝟎. 𝟔 which shows imperfect cluster synchronization. d) the time snapshot at t=2950 referring to (c).
In the next step, the 𝛼 values at which a nephron exhibits chaotic behavior were examined. First, 𝛼 was fixed at 15 and then at 16. The spatiotemporal pattern and time snapshot of the network in the case of 𝛼 = 15 are depicted in Fig. 5(a) and (b). According to Fig. 5(b), there are two synchronous clusters, and each oscillator is located in one of them. So, in this case, the pattern shows cluster synchronization. The same results are obtained for 𝛼 = 8 and 𝛼 = 14. Figures 5(c) and (d) represent the spatiotemporal pattern and time snapshot of the network when 𝛼 = 16. These figures illustrate cluster synchronization, but as shown in the spatiotemporal pattern, the cluster boundaries are not static in time.
Fig. 5. a) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟓 and 𝒅 = 𝟎. 𝟑, which shows cluster synchronization. b) the time snapshot at t=2930 referring to (a). c) Spatiotemporal pattern of the network for 𝜶 = 𝟏𝟔 and 𝒅 = 𝟎. 𝟒 which shows cluster synchronization. d) the time snapshot at t=2970 referring to (c).
Discussion In previous studies on the collective behaviors of a group of interacting and coupled nephrons, many behaviors have been identified and investigated. These behaviors include synchronization behavior, quasiperiodicity and chaos in a nephron tree [32], synchronization of tubular pressure oscillation in interacting nephrons [34], synchronization effects which occur among neighboring
nephrons [35], multistable dynamics of coupled nephrons [36], and some other nearly related and similar researches such as [37]. However, it is anticipated that in a complex system such as a kidney, with its complex dynamical behavior, more dynamical states can emerge. In previous studies, it has been approved that the nephrons with the synchronized pressure are coupled with each other [38]. As a result of this interaction, the oscillation of the TGF-mediated proximal pressure is synchronized. It has been proposed that the “cross-talk” between the TGF signals from the different nephrons, that is transmitted along the afferent arterioles, and probably also along a part of the interlobular artery results in the interaction [38]. Further studies show that in-phase synchronization is observed in the data analysis of the rats with normal blood pressure. Also, the chaotic phase synchronization between adjacent nephrons is found in the rats with elevated blood pressure [39]. Regarding the experiments, we could conclude that interacted nephrons oscillators oscillate synchronously in typical situations. Thus the chimera states in the network of coupled nephrons may illustrate the abnormal situation or a malfunction in kidney task.
Conclusion In this paper, a ring network of non-locally coupled nephrons was studied. The bifurcation diagram of the incorporated model shows that a single nephron can exhibit different periodic and chaotic behaviors due to changes in bifurcation parameters. Many bifurcation parameters can be considered and affect the dynamical state of each nephron. These parameters may all have pieces of physiological evidence such as variations in the delay of the Henle loop, or variations in TGF gain such as T and α. Here, the spatiotemporal patterns of the network were investigated in both modes of periodic and chaotic nephrons dynamics, by changing the coupling strength in two different coupling functions. The results showed various patterns by changing the parameters. It was obtained that when the bifurcation parameter is set at the periodic region, the network represents multi chimera state or imperfect cluster synchronization, and when each nephron is set at the chaotic region, by setting the model parameter, the cluster synchronization appears in the network. Thus, the emergence of chimera states is observed in the model in a state which is assumed as the normotensive mode. Declaration of interests
The authors declare that they have no competing interests.
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