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A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

Collective beta cell activity in islets of Langerhans is critical for the supply of insulin within an organism. Even though individual beta cells are intrinsically heterogeneous, the presence of intercellular coupling mechanisms ensures coordinated activity and a well-regulated exocytosis of insulin. In order to get a detailed insight into the functional organization of the syncytium, we applied advanced analytical tools from the realm of complex network theory to uncover the functional connectivity pattern among cells composing the intact islet. The procedure is based on the determination of correlations between long temporal traces obtained from confocal functional multicellular calcium imaging of beta cells stimulated in a stepwise manner with a range of physiological glucose concentrations. Our results revealed that the extracted connectivity networks are sparse for low glucose concentrations, whereas for higher stimulatory levels they become more densely connected. Most importantly, for all ranges of glucose concentration beta cells within the islets form locally clustered functional sub-compartments, thereby indicating that their collective activity profiles exhibit a modular nature. Moreover, we show that the observed non-linear functional relationship between different network metrics and glucose concentration represents a well-balanced setup that parallels physiological insulin release.

The problem of making a network of dynamical systems synchronize onto a common evolution is the subject of much ongoing research in several scientific disciplines. It is nowadays a well-known fact that the synchronization processes are gradually in influenced by the interaction topology between the dynamically interacting units. A complex coupling configuration can significantly affect the synchronization abilities of a networked system. However, the question arises what is the optimal network topology that provides enhancement of the synchronization features under given circumstances. In order to address this issue we make use of a network model in which we can smoothly tune the topology from a highly heterogeneous and efficient scale-free network to a homogeneous and less efficient network. The network is then populated with Poincaré oscillators, a paradigmatic model for limit-cycle oscillations. This oscillator model exhibits a parameter that enables changes of the limit cycle attraction and is thus immediately related to flexibility/rigidity properties of the oscillator. Our results reveal that for weak attractions of the limit cycle, intermediate homogeneous topology ensures maximal synchronization, whereas highly heterogeneous scale-free topology ensures maximal synchronization for strong attractions of the limit cycle. We argue that the flexibility/rigidity of individual nodes of the networks defines the topology, where maximal global coherence is achieved.

Doktorska disertacija zajema raziskave na področju kolektivne dinamike mrežno sklopljenih oscilatorjev. Razdeljena je na dva dela. V prvem delu analiziramo, kako dinamične lastnosti oscilatorjev in struktura mreže sovisno vplivata na kolektivno dinamiko. Pokažemo, da je kolektivna dinamika fleksibilnih oscilatorjev najbolje koordinirana, ko so oscilatorji povezani v primeru široko skalno mrežo. Oscilatorji z močno disipativno dinamiko, ki implicira rigidnost, pa dosežejo najvišjo raven sinhronizacije v skalno neodvisnih mrežah. Pojav analitično razložimo in rezultate ponazorimo z različnimi matematičnimi modeli, ki vključujejo tako zvezne kakor tudi diskretne oscilatorje, ter izkazujejo različne stopnje dinamične kompleksnosti. Pri analizi kolektivne dinamike upoštevamo tudi hitrost širjenja signalov med vozlišči v mreži. Ugotovimo, da obstaja tako optimalna mrežna topologija kakor tudi optimalna hitrost širjenja signalov med vozlišči mreže, pri kateri je raven kolektivne sinhronizacije najvišja. Ugotovitve in metodologijo iz naših teoretičnih študij v drugem delu disertacije apliciramo na sistem povezanih celic beta v Langerhansovih otočkih trebušne slinavke miši, ki predstavlja z vidika fiziologije metabolnih procesov izredno pomemben predmet preučevanja. Mrežno povezane celice beta, katerih poglavitna naloga je izločanje inzulina in s tem uravnavanje koncentracije glukoze v krvi, analiziramo ob podpori eksperimentalnih podatkov, izmerjenih pri različnih koncentracijah glukoze. Naši rezultati kažejo, da se celice beta povezujejo v lokalne funkcionalne skupnosti. Njihova segregiranost pa se v splošnem manjša z naraščajočo koncentracijo glukoze. S postopnim povečevanjem koncentracije glukoze postanejo v otočku tudi vse bolj izražene lastnosti široko skalnih mrež malega sveta. S tem rezultati doktorske disertacije prispevajo k razlagi fiziološkega pomena učinkovitost mrežne povezanosti celic beta in nakazujejo možnosti patoloških sprememb, ki so posledica sprememb v medcelični komunikaciji. This doctor thesis is both theoretical and applicative. In the theoretical part of the thesis, we examine how the interplay of dynamical features of oscillators and structural properties of complex networks affect the collective behavior of the system. We show, that weakly dissipative and flexible oscillators synchronize best in a broad scale network topology, whereas on the other hand strongly dissipative and rigid oscillators exhibit maximal synchronization in a scale-free network topology. We provide an analytical explanation for this phenomenon and validate it by implementing various continuous as well as discrete mathematical models that exhibit different levels of dynamical complexity. In the continuation, we additionally investigate how speed of signal transmission in the network affects the collective dynamic of the system. Our results show that besides an optimal network topology, also an optimal information transmission speed exists, at which the system reaches the highest degree of global synchronization. In the second part we apply the findings and the methodology from our theoretical studies to the examination of the collective pancreatic beta cell activity in the islets of Langerhans, which represents the main mechanism for the regulation of blood glucose homeostasis by the secretion of the hormone insulin. We show that the beta cells dynamics is not synchronized on the global scale of the whole islets. Instead, the cells form local clusters of synchronized activity which tend to get less segregated under higher stimulatory glucose concentrations. Furthermore, higher glucose concentrations also lead to the presence of broad scale small world connectivity patterns in the functional beta cell network. The main findings thereby shed light on the physiology and collective behavior of the islets of Langerhans and point out the possibilities of pathological changes associated with changes in the intercellular communication pathways.

V diplomski nalogi primerjamo prenos signalov v bioloških in električnih sistemih. Osredotočili smo se na znotrajcelično signalizacijo, kjer obstajajo signalne poti, sestavljene iz proteinskih kaskad. Z matematičnim modelom smo opravili različne numerične simulacije in analizirali vlogo proteinskih kaskad pri prenosu signala v znotrajceličnem prostoru. Te vloge smo primerjali z različnimi električnimi sistemi, katerih delovanje je dobro znano in ga lahko opišemo matematično. Namen takšnih primerjav je bolje razumeti vloge signalnih poti v znotrajceličnem prostoru. Primerjave temeljijo na numeričnih izračunih in primerjavah diferencialnih enačb, ki tvorijo matematični modeli za opis delovanja določenega sistema. V diplomskem delu je pokazano, da lahko proteinski kaskadi dodelimo lastnosti stikala, ki ojačuje signal in deluje kot nizkofrekvenčno prepustno sito. Pokazali smo tudi, da lahko z uporabo pozitivne povratne zanke aktivnemu stikalu povečamo ojačanje, zmanjšamo mejno frekvenco in povečamo njegovo nestabilnost. Kadar v sistem vključimo negativno povratno zanko, povečamo stabilnost sistema, povečamo mejno frekvenco in zmanjšamo ojačanje. Izdelali smo tudi biološka logična vrata, ki temeljijo na logičnih operacijah v digitalni elektroniki. Z uporabo teh logičnih vrat smo izdelali še spominsko vezje, ki lahko pomni 1 bit podatkov. In this thesis signal transduction pathways in biological and electrical systems are analyzed and compared. The main goal is to analyze intercellular signalization, in particular the pathways consisting of protein cascades. The protein cascades are studied by using mathematical models of signal transduction systems. The biological systems are compared with analogous electrical systems, which are well known and can easily be mathematically described. The main idea of comparing these systems is to gain better knowledge of the functioning of signaling pathways in the intercellular space. The comparison is based on numerical simulations and direct looking for analogies of the differential equations consisting the mathematical models of particular systems. We show that the protein cascades can function as switches that amplify signals and also serve as a low-frequency filter. Furthermore, we show that by introducing a positive feedback loop into an active switch, we can increase its gain, reduce the cutoff frequency and increase the instability of the system. On the other hand, if a negative feedback is applied, the system becomes more stable, the cutoff frequency rises and its gain drops. We also constructed biological logical gates, which are based on logical procedures used in digital electronics. On the basis of these gates, we designed a memory module, which can store 1 bit of information.

Type 2 diabetes mellitus (T2DM) has been associated with insulin resistance and the failure of β-cells to produce and secrete enough insulin as the disease progresses. However, clinical treatments based solely on insulin secretion and action have had limited success. The focus is therefore shifting towards α-cells, in particular to the dysregulated secretion of glucagon. Our qualitative electron-microscopy-based observations gave an indication that mitochondria in α-cells are altered in Western-diet-induced T2DM. In particular, α-cells extracted from mouse pancreatic tissue showed a lower density of mitochondria, a less expressed matrix and a lower number of cristae. These deformities in mitochondrial ultrastructure imply a decreased efficiency in mitochondrial ATP production, which prompted us to theoretically explore and clarify one of the most challenging problems associated with T2DM, namely the lack of glucagon secretion in hypoglycaemia and its oversecretion at high blood glucose concentrations. To this purpose, we constructed a novel computational model that links α-cell metabolism with their electrical activity and glucagon secretion. Our results show that defective mitochondrial metabolism in α-cells can account for dysregulated glucagon secretion in T2DM, thus improving our understanding of T2DM pathophysiology and indicating possibilities for new clinical treatments.

Type 2 diabetes mellitus (T2DM) has been associated with insulin resistance and the failure of β-cells to produce and secrete enough insulin as the disease progresses. However, clinical treatments based solely on insulin secretion and action have had limited success. The focus is therefore shifting towards α-cells, in particular to the dysregulated secretion of glucagon. Our qualitative electron-microscopy-based observations gave an indication that mitochondria in α-cells are altered in Western-diet-induced T2DM. In particular, α-cells extracted from mouse pancreatic tissue showed a lower density of mitochondria, a less expressed matrix and a lower number of cristae. These deformities in mitochondrial ultrastructure imply a decreased efficiency in mitochondrial ATP production, which prompted us to theoretically explore and clarify one of the most challenging problems associated with T2DM, namely the lack of glucagon secretion in hypoglycaemia and its oversecretion at high blood glucose concentrations. To this purpose, we constructed a novel computational model that links α-cell metabolism with their electrical activity and glucagon secretion. Our results show that defective mitochondrial metabolism in α-cells can account for dysregulated glucagon secretion in T2DM, thus improving our understanding of T2DM pathophysiology and indicating possibilities for new clinical treatments.