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Interacting Oscillators for Energy Homeostasis
Junghyo Jo
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DOI: 10.22661/AAPPSBL.2018.28.4.57

Interacting Oscillators for Energy Homeostasis



Living systems consume energy for their information processing. Stable maintenance of energy availability is, therefore, essential for operating life. Energy homeostasis is regulated by α, β, and δ cells in the islets of Langerhans in the pancreas. The islet cells produce hormones in a pulsatile manner to regulate blood glucose levels. Furthermore, they communicate with each other within islets, and between islets, with unique spatial organization. In this review, we summarize recent advances in the mathematical modeling of the hierarchical cellular network, and discuss a potential design principle of the network.


Glucose and fat are primary fuels for life, akin to gasoline and diesel for cars. Therefore, living systems need to tightly control energy availability. Humans maintain approximately 5 mM of glucose in the blood. Considering the molecular weight of 180 g/mol of glucose, the total amount of blood in the average human body - about 5 L - contains approximately 4.5 g of glucose. To maintain this delicate mass of energy, our body is equipped with a very fine device. The malfunction of this device leads to metabolic disease, i.e., diabetes, from which approximately 10% of the world's population has been suffering [1].

These devices, the islets of Langerhans, which control glucose homeostasis, are scattered in the pancreas. They occupy 1% of the volume of the pancreas, but the islets receive 10% of the pancreatic blood flow. Each islet, on average, consists of a few hundred endocrine cells [2]. Since the human pancreas has about a million islets, the total mass of islets in the pancreas is estimated to be only 1 g. It is remarkable that a device of 1 g is tightly controlling 5 g of glucose in the body. For a marked comparison, brand-new insulin pumps, which have primitive functions compared to their natural counterparts, weight about 60 g.

The micro-organ of each islet consists of α, β, and δ cells. Pancreatic α cells secrete glucagon to increase blood glucose levels under a fasting state by breaking glycogen in the liver into glucose. On the other hand, β cells secrete insulin to decrease blood glucose levels after meals by synthesizing glycogen from glucose and stimulating glucose absorption into peripheral cells, including fat cells. Glucagon and insulin are counter-regulatory hormones for regulating glucose homeostasis. The third cell population, δ cells, secrete somatostatin, but their role in glucose regulation is still a mystery.

Like many hormones in human physiology, islet cells secrete the hormones in a pulsatile manner with a characteristic period of 5-10 minutes. The question as to why many biological phenomena show this kind of oscillation remains unanswered [3]. Oscillatory stimuli may have the benefit of preventing receptor desensitization as compared to continuous stimuli. Another possible benefit is that oscillatory signals can guarantee predictability for systems to respond. In any case, considering the oscillatory secretion of hormones, the phase coordination between cells and between islets must have a large impact in integrated hormone concentrations in blood.

It is interesting that the islet cells interact with each other to coordinate the phase relations between hormones. Given the cellular interaction, the spatial organization of islet cells must have functional implications. It is remarkable that islets cells are not randomly aggregated, but they are specially organized. In mouse islets, β cells are located in the islet core, whereas non-β cells are located in the islet mantle. However, in human islets, α cells are more abundant, and non-β cells also exist in the islet core. Recently it has been suggested that the structural difference between species originates from the fractional difference of islet cells rather than the differential adhesion between cell types [4]. Human islets have a partial mixture structure, neither ordered nor completely disordered.

Regarding the cellular interaction, islet cells have a special symmetry (Fig. 1): α cells stimulate hormone secretions of β and δ cells; β cells suppress α cells, but stimulate δ cells, and; δ cells suppress α and β cells. In analogy to human society, α cells are "poor" because they positively affect neighboring cells, but they are negatively affected by them. On the other hand, δ cells are "bad" because they negatively affect neighboring cells, but they are positively affected by them. Finally, β cells are "strange" because they positively affect δ cells affecting them negatively, while they negatively affect α cells affecting them positively [5].

The islet of Langerhans is an interesting cellular network. The role of the third cell population, δ cells, and the special symmetry of cellular interactions between α, β, and δ cells, their spatial organization, and the scattered distribution of individual islets in the pancreas are all interesting topics where intriguing questions can be asked. In this review, we summarize the possible answers for those questions by using mathematical models of coupled oscillators.

Islet cells - biological oscillators
Islet cells secrete hormones in a pulsatile manner, and the secreted hormones affect the activity of neighboring cells. These two observations motivated us to model the cell network as a network of coupled oscillators. First, we focus on the phase dynamics of α, β, and δ cells [6]. Considering the signs of interactions between them (Table 1), the phase dynamics follows


where ωα, ωβ, and ωδ are intrinsic frequencies of oscillations of glucagon, insulin, and somatostatin, or the activities of α, β, and δ cells. The interaction strengths between cells are defined to depend on the present cell activities, rα, rβ and rδ.

Fig. 1: Interaction signs between islet cells. Red arrows represent positive interactions, whereas blue bar-headed arrows represent negative interactions.

For example, the interaction from β to α cells is proportional to rβ / ra ; as the affector cell is more active and the receiver cell is less active, the interaction becomes stronger. The phase dynamics is then determined by the intrinsic factor and the interaction factor of which balance is tuned by the parameter K.

The relative coordination between the three cell types can be analyzed by the dynamics of relative phases, x = θα-θβ and y = θα-θδ. Their dynamics can be deduced from Eq. (1) as follows


To obtain this result, we assumed that the intrinsic frequencies of α, β, and δ cells are identical (ωα=ωβ=ωδ=ω), since the hormonal oscillations have more or less similar periods, of 5 to 10 minutes. In addition, the activity of cells (or hormone secretion) is mainly governed by blood glucose concentration. Thus we examined three glucose conditions: (i) a low glucose condition where α cells are most active (rα=1, rβ=rδ=0.2); (ii) a normal glucose condition where the three cell types are equally active (rα=rβ=rδ=1); and (iii) a high glucose condition where β cells are most active (rα=0.2, rβ=1, rδ=0.5). In the low glucose condition, the phases of three cell types are synchronized in phase (Fig. 2a). On the other hand, in the high glucose condition, α cells are out of phase with β cells, and out of phase with δ cells (Fig. 2b). In contrast to the high and low glucose conditions that have single phase coordination, at the normal glucose condition, the relative phases, x = θαβ and y = θαδ, have oscillatory solutions, alternating in-phase and out-of-phase coordination with clockwise and counterclockwise directions (Fig. 2c).

Fig. 2: Phase coordination between islet hormones. (a) Phase plane and vector flows of phase differences (x = θαβ, y = θαδ) and (b) time trajectories of hormone levels Hσ = rσ(1+cos θσ) for σ ∈ {α, β, δ} cell types at a low glucose condition (rα =1, rβ = rδ = 0.2). (c, d) The same plots at a high glucose condition (rβ = 1, rα = 0.2, rδ = 0.5). (e) Phase plane at a normal glucose condition (rα = rβ = rδ =1). (f) Clockwise and (g) counterclockwise cyclic trajectories of hormone levels at the normal glucose condition.

Intra-islet networks and cellular synchronization
It is a great simplification that islet cells are approximated as oscillators. This idea has allowed us to explore the cellular network further. Equation (1) considers a coarse-grained network with single α, β, and δ cells. Real islets have a few hundred cells with different fractions and structures of α, β, and δ cells, depending on the species of animal. Mouse islets have about 80-90% β cells, 5-10% α cells, and 1-5% δ cells, and β cells are mostly located in the islet core, while non-β cells are located on the islet mantle. In contrast, human islets have 60% β cells, 30% α cells, and 10% δ cells, and they are mixed together in the human islets [7]. Since the three-dimensional coordinates of islet cells have been measured, the governing rule for islet organization has been theoretically explored [4]. The conclusion is that the differences between species in islet structure originates from the fractional difference of islet cells, rather than the cell-to-cell adhesion difference. The islets have a conserved rule that homotypic cells attract more strongly than heterotypic attraction. Human islets are not random aggregates of islet cells, but they have a structure that is a partial mixture, neither completely ordered nor disordered.

Given the coordinates of islet cells, a realistic network of islet cells can be explored. The phase dynamics of islet cells in the network is governed by


which is a generalized equation of Eq. (1). Here we used a generalized notation σk ∈ {α, β, δ} for the cell type of the k-th cell. The sign of interaction from cell type σj to cell type σk is denoted by the matrix Aσkσj (Table 1). Note that we include the autocrine interaction between the same cell type with Aαα=Aββ=Aδδ=1. To consider a simpler situation, we first explored the network of α and β cell only, ignoring the third minor cell type, δ cells.

Since the islet is composed of many cells, the synchronization of hormone secretion from the many cells has a great impact in glucose regulation. Therefore, we examined the synchronization between cells depending on islet structures. In particular, the core-mantle structure of mouse islets was compared to a completely sorted structure and a random mixture structure (Fig. 3).

Table I. The interaction signs between islet cells. Positive (+), negative (-), and unknown (.) interactions.







Fig. 3: Cellular organization and synchronization. Spatial organization of α (red) and β (green) cells: (a) sorted, (b) core-mantle, and (c) mixture structures. Depending on the relative cellular activities rβ/rα, α and β cells have different degrees of synchronization (0: complete desynchronization, 1: perfect synchronization) given (d) sorting, (e) core-mantle, and (f) mixture structures.

The degree of synchronization can be measured by the standard order parameter for oscillators of σ cell types,


Note that given local interaction, homotypic cells are sometimes aligned with a phase difference of π. To consider this, the factor 2 is introduced in Eq. (4). Later we will relax this assumption when the angular symmetry is broken in the phase dynamics. Given the completely sorted structure, islet cells show mostly synchronized states independent of the glucose condition (Fig. 3d). However, the core-mantle structure shows phase synchronization and desynchronization of islet cells depending on the glucose condition (Fig. 3e). At high/low glucose conditions, islet cells show collective behavior of synchronization and produce concentrated hormone secretion within local time windows, while at normal glucose conditions, islet cells show desynchronization and hide the action of hormones by maximally suppressing the local synchronization. The flexibility of cellular synchronization is diminished in the mixture structure that generates more desynchronized states (Fig. 3f).

In addition to the structural effect, signs of cellular interactions can also affect cellular synchronization. In the partial mixture structure, the asymmetric interaction between α and β cells allows for controllable synchronization, whereas mutual attraction and mutual repulsion lead to more synchronized behavior. Finally, in the presence of δ cells, the additional complexity in the cellular interactions further suppresses the degree of cellular synchronization [5].

Inter-islet networks and glucose homeostasis
Islets are scattered throughout the pancreas, and they are physically separated to each other. Considering that the human pancreas has approximately one million islets, the (de)synchronization of their hormone secretion must have a great impact in human physiology. It has been a long-standing puzzle to determine how the many islets show coordinated hormone secretion. Two major hypotheses are (i) direct control by the central nervous system [8] and (ii) entrainment by the rhythmic change of glucose concentration [9]. These hypotheses are not mutually exclusive. If the hormone secretions from the numerous islets in the pancreas are not synchronized, their integrated hormone secretion would have shown flat profiles. Another example of in vitro experimental evidence is that the activities of isolated islets can be entrained to secrete hormones synchronously by oscillatory glucose stimuli [10, 11]. To understand the interplay between glucose regulation by islets and islet response to glucose stimuli, a closed loop model of glucose and islet hormones has been proposed (Fig. 4).

Fig. 4: Pancreatic islets and glucose regulation. In the pancreas, α and β cells in physically-separated islets secrete glucagon and insulin, respectively, responding to a blood glucose level G. The integrated glucagon Hα and insulin Hβ, secreted from individual islets, play the counter-regulatory role for increasing and decreasing glucose levels. The glucose level is also perturbed by external glucose input I(t).

First, the amplitude and phase dynamics of for α, β, and δ cells within the n-th islet is governed by glucose levels:


where fσ(G) and gσ(G) are amplitude and phase modulation factors for σ ∈ {α, β, δ} cell type depending on glucose concentration G. The amplitude equation leads to a stationary amplitude level that is a sigmoidal function consistent with experimental observations:


Basically, more glucagon is secreted at low glucose, whereas insulin and somatostatin are secreted more at high glucose. Here, somatostatin has a lower threshold than insulin. The phase modulation factors control the duration of active and silent phases of the hormone oscillations, depending on cell type and glucose:


which make α cells have a longer active phase at low glucose, and β and δ cells to have a longer active phase at high glucose. The phase modulation breaks the rotational symmetry, and defines maximal and minimal hormone secretion at θσ =0 and π, respectively.

Equation (5) does not include the cellular interaction within an islet. Now the interaction can be considered as


where again the interaction matrix Aσσ, specifies the interaction sign from σ' cell type to σ cell type. The glucose regulation by glucagon and insulin can be described by


where λ is the scale parameter determining how much and how effectively hormones regulate glucose levels. Glucagon and insulin are the counter-regulatory hormones that increase and decrease blood glucose levels. The removal of glucose by insulin is proportional to the present glucose level, while the production of glucose is independent on the present glucose level. The total hormone secretion from N islets are defined as


where a unity is introduced to prevent the negativity of hormone levels. Finally, Eq. (9) considers external glucose input such as meals, I(t). Equations (8) and (9) then complete the description of the glucose regulation circuit. When external glucose stimulates islets, insulin level Hβ increases and glucagon level Hα decreases to decrease glucose levels, and work oppositely when glucose levels decrease during exercise or sleep. Given this setup, important questions are (i) how stable is the glucose regulation; (ii) how much hormones are consumed for the regulation; and (iii) how are these properties dependent on the design of the local interaction between α, β, and δ cells.

The three cell types have a total of six directional interactions between them. Therefore, if one considers three simple possibilities of interaction types (none, positive, negative), then there is a total of 729 (=36) possible interaction networks that can be defined, including two completely anti-symmetric interaction networks (Fig. 5). The stability of glucose regulation and hormone consumption can be probed by measuring the relative glucose fluctuation |δG|/G and total hormone level Hα+Hβ. Among 729 possible networks, the native network is ranked in the top ten in terms of stable glucose regulation and efficient hormone consumption [12].

Fig. 5: Two anti-symmetrically interacting networks. Native islet network (left).

Fig. 6: Glucose regulation and inter-islet synchronization. Glucose regulation, hormone levels, and inter-islet synchronization under a normal glucose condition (I=0) until 100 min and a high glucose condition (I=14 mM/min) after 100 min (blue regions). G0=7 mM. ρinter = (ραinter+ ρβinterδinter)/3. Phase planes at the last row represent the phase differences of individual islets before (left panel, black) and after (right panel, blue) the glucose infusion. Two different islet networks are considered: no interactions between islet cells (left column) and native interactions (right column). N = 1000 islets, 2π/ω= 5±1 min, K = 0.4 min-1, μ = 0.1 min-1, λ = 0.001 min-1 are used. Initial values of the variables are rnσ(0) ∈ [0.25, 0.75], θnσ(0) ∈ [0.2π], and G(0)=7 mM.

The phase modulation factor gσ(G) in Eq. (5) makes every islet respond to the global signal of glucose oscillation G, and the common response makes them entrain and synchronize. The synchronization between islets can be measured by another order parameter,


for σ ∈ {α, β, δ} cell types. In the absence of the cellular interaction (Aσσ'=0), every islet becomes easily entrained by the global stimuli of the oscillatory glucose, and they become synchronized (Fig. 6). However, in the presence of the native interaction, islets resist synchronization. This is possible because the three attractors of phase coordination (Fig. 2) makes each islet sit on different attractors, which helps them resist glucose entrainment. Another cyclic anti-symmetric network (Fig. 5) shows the same conclusion. On the other hand, in a high glucose condition, the three attractors lose their stability, and islets become more synchronized. Therefore, the natural design of islet cell interactions allows the controllability of (de)synchronization. At normal glucose conditions, the system suppresses inter-islet synchronization to prevent unnecessary hormone action, while at high/low glucose conditions, the system uses synchronization to concentrate the total hormone secretion from numerous islets.


This review summarizes recent developments of islet modeling. The hormonal oscillation and the paracrine interaction between islet cells can be appropriately described by the model of coupled oscillators. Based on the oscillator model, we developed a series of models to understand the intra-islet network, the inter-islet network, and glucose regulation. We then found that the partial mixture organization of islet cells, the physically separate islet distribution in the pancreas, and the anti-symmetric interaction between islet cells seem to be "designed" to impose controllable coordination of endocrine hormones for stable glucose regulation.

The original Kuramoto model considers positive interactions between oscillators to describe various synchronization phenomena [13]. Two populations of oscillators have been considered recently [14]: 1) attractive oscillators, called "conformists," are affected positively by neighbors, and; 2) repulsive oscillators, called "contrarians," are affected negatively by neighbors. The system of pancreatic islets may be a natural extension of the Kuramoto model with three populations of oscillators, α, β, and δ cells.

In biological phenomena, synchronization is not always desirable. For example, epilepsy is caused by the excessive synchronous neuronal activity. Therefore, controllable synchronization is necessary, and recently this subject has been intensively studied [15]. Here, the pancreatic islets serve as an important biological example for demonstrating controllability. In addition to the system of pancreatic islets for glucose regulation, oscillation, synchronization, and adaptation to external stimuli are also essential features to grasp in order to understand how suprachiasmatic nuclei generate circadian rhythms [16]. Therefore, the model of coupled oscillators may play an even more important role in our understanding biological rhythms.

Acknowledgements: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning, the Ministry of Education (2016R1D1A1B03932264) and the Max Planck Society, Gyeongsangbuk-Do and Pohang City.


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Junghyo Jo is an assistant professor at Korea Institute for Advanced Study. After earning his PhD from Seoul National University in 2007, he was a postdoc at the National Institutes of Health in the US, and worked at the Asia Pacific Center for Theoretical Physics as a Junior Research Group Leader. His research field is computational biology.