Marko Gosak

Pacemaker-driven stochastic resonance on diffusive and complex networks of bistable oscillators

We study the phenomenon of stochastic resonance on diffusive, small-world and scale-free networks consisting of bistable overdamped oscillators. Important thereby is the fact that the external subthreshold periodic forcing is introduced only to a single oscillator of the network. Hence, the forcing acts as a pacemaker trying to impose its rhythm on the whole network through the unit to which it is introduced. Without the addition of additive spatiotemporal noise, however, the whole network, including the unit that is directly exposed to the pacemaker, remains trapped forever in one of the two stable steady states of the local dynamics. We show that the correlation between the frequency of subthreshold pacemaker activity and the response of the network is resonantly dependent on the intensity of additive noise. The reported pacemaker-driven stochastic resonance depends most significantly on the coupling strength and the underlying network structure. Namely, the outreach of the pacemaker obeys the classic diffusion law in the case of nearest-neighbor interactions, thus being proportional to the square root of the coupling strength, whereas it becomes superdiffusive by an appropriate small-world or scale-free topology of the interaction network. In particular, the scale-free topology is identified as being optimal for the dissemination of localized rhythmic activity across the whole network. Also, we show that the ratio between the clustering coefficient and the characteristic path length is the crucial quantity defining the ability of a small-world network to facilitate the outreach of the pacemaker-emitted subthreshold rhythm. We additionally confirm these findings by using the FitzHugh?Nagumo excitable system as an alternative to the bistable overdamped oscillator.

Functional connectivity in islets of Langerhans from mouse pancreas tissue slices

We propose a network representation of electrically coupled beta cells in islets of Langerhans. Beta cells are functionally connected on the basis of correlations between calcium dynamics of individual cells, obtained by means of confocal laser-scanning calcium imaging in islets from acute mouse pancreas tissue slices. Obtained functional networks are analyzed in the light of known structural and physiological properties of islets. Focusing on the temporal evolution of the network under stimulation with glucose, we show that the dynamics are more correlated under stimulation than under non-stimulated conditions and that the highest overall correlation, largely independent of Euclidean distances between cells, is observed in the activation and deactivation phases when cells are driven by the external stimulus. Moreover, we find that the range of interactions in networks during activity shows a clear dependence on the Euclidean distance, lending support to previous observations that beta cells are synchronized via calcium waves spreading throughout islets. Most interestingly, the functional connectivity patterns between beta cells exhibit small-world properties, suggesting that beta cells do not form a homogeneous geometric network but are connected in a functionally more efficient way. Presented results provide support for the existing knowledge of beta cell physiology from a network perspective and shed important new light on the functional organization of beta cell syncitia whose structural topology is probably not as trivial as believed so far.

Prevalence of stochasticity in experimentally observed responses of pancreatic acinar cells to acetylcholine

Calcium ions play an important role in both intra- and intercellular signaling. In pancreatic acinar cells intracellular Ca(2+) regulates exocytotic secretion and fluid secretion. In this paper we study the typical experimental traces of Ca(2+) responses in pancreatic acinar cells obtained in response to the physiological agonist acetylcholine. To determine whether they are stochastic or deterministic in nature, we analyze the traces with methods of nonlinear time series analysis. In particular, by performing surrogate data tests and employing a determinism test for short time series, we show that the responses of pancreatic acinar cells to acetylcholine are stochastic with only faintly expressed deterministic features. Presented results thus corroborate the notion that mathematical models should take stochasticity explicitly into account when describing intra- and intercellular processes, and that indeed further efforts should be directed toward this subject.

Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus.

The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure.

Network science of biological systems at different scales: A review

Network science is today established as a backbone for description of structure and function of various physical, chemical, biological, technological, and social systems. Here we review recent advances in the study of complex biological systems that were inspired and enabled by methods of network science. First, we present research highlights ranging from determination of the molecular interaction network within a cell to studies of architectural and functional properties of brain networks and biological transportation networks. Second, we focus on synergies between network science and data analysis, which enable us to determine functional connectivity patterns in multicellular systems. Until now, this intermediate scale of biological organization received the least attention from the network perspective. As an example, we review the methodology for the extraction of functional beta cell networks in pancreatic islets of Langerhans by means of advanced imaging techniques. Third, we concentrate on the emerging field of multilayer networks and review the first endeavors and novel perspectives offered by this framework in exploring biological complexity. We conclude by outlining challenges and directions for future research that encompass utilization of the multilayer network formalism in exploring intercellular communication patterns in tissues, and we advocate for network science being one of the key pillars for assessing physiological function of complex biological systems-from organelles to organs-in health and disease.

Progressive glucose stimulation of islet beta cells reveals a transition from segregated to integrated modular functional connectivity patterns

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.

Cellular diversity promotes intercellular Ca2+ wave propagation.

Calcium ions are an important second messenger in living cells. Calcium signals in form of waves serve as a means of intercellular communication and thus represent a vibrant subject for experimental and theoretical investigations. Here we study the role of cellular variability on the occurrence of Ca2+ wave propagation in a net of diffusively coupled cells. Dynamics of individual cells is simulated by a mathematical model for Ca2+ oscillations. Structural diversity of cells is introduced via variations of the bifurcation parameters, which signify cell sensitivity for external stimulation. Remarkably, for sufficient values of variability Ca2+ waves emerge, which are mostly ordered for intermediate variability strength. We analyze the spatial profile via the autocorrelation function, which confirms a resonance-like response due to the cellular variability. Thus, the reported phenomenon is a novel observation of diversity-induced spatial coherence resonance in a tissue-like media.

Importance of cell variability for calcium signaling in rat airway myocytes

Calcium signaling controls several essential physiological functions in different cell types. Hence, it is not surprising that different aspects of Ca(2+) dynamics are in the focus of in-depth and extensive investigations. Efforts concentrate on the development of proper theoretical models that would provide a unified description of Ca(2+) signaling. Remarkably, experimentally recorded Ca(2+) signals exhibit a rather large diversity, which can be observed irrespective of the cell type, measuring techniques, or the nature of the signal. Our goal in the present study therefore is to present a theoretical explanation for the variability observed in experiments, whereby we focus on caffeine-induced Ca(2+) responses in isolated airway myocytes. By employing a stochastic model, we first test whether the observed variability can be attributed to intrinsic fluctuations that are a common feature of biochemical reactions that govern Ca(2+) signalization. We find that stochastic effects, within ranges that correspond to actual conditions in the cell, are far too modest to explain the large diversity observed in experimental data. Foremost, we reveal that only cell variability in theoretical modeling can appropriately describe the observed diversity in single-cell responses.

Topologically determined optimal stochastic resonance responses of spatially embedded networks

We have analyzed the stochastic resonance phenomenon on spatial networks of bistable and excitable oscillators, which are connected according to their location and the amplitude of external forcing. By smoothly altering the network topology from a scale-free (SF) network with dominating long-range connections to a network where principally only adjacent oscillators are connected, we reveal that besides an optimal noise intensity, there is also a most favorable interaction topology at which the best correlation between the response of the network and the imposed weak external forcing is achieved. For various distributions of the amplitudes of external forcing, the optimal topology is always found in the intermediate regime between the highly heterogeneous SF network and the strong geometric regime. Our findings thus indicate that a suitable number of hubs and with that an optimal ratio between short- and long-range connections is necessary in order to obtain the best global response of a spatial network. Furthermore, we link the existence of the optimal interaction topology to a critical point indicating the transition from a long-range interactions-dominated network to a more lattice-like network structure.

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells.

Self-sustained oscillatory dynamics is a motion along a stable limit cycle in the phase space, and it arises in a wide variety of mechanical, electrical, and biological systems. Typically, oscillations are due to a balance between energy dissipation and generation. Their stability depends on the properties of the attractor, in particular, its dissipative characteristics, which in turn determine the flexibility of a given dynamical system. In a network of oscillators, the coupling additionally contributes to the dissipation, and hence affects the robustness of the oscillatory solution. Here, we therefore investigate how a heterogeneous network structure affects the dissipation rate of individual oscillators. First, we show that in a network of diffusively coupled oscillators, the dissipation is a linearly decreasing function of the node degree, and we demonstrate this numerically by calculating the average divergence of coupled Hopf oscillators. Subsequently, we use recordings of intracellular calcium dynamics in pancreatic beta cells in mouse acute tissue slices and the corresponding functional connectivity networks for an experimental verification of the presented theory. We use methods of nonlinear time series analysis to reconstruct the phase space and calculate the sum of Lyapunov exponents. Our analysis reveals a clear tendency of cells with a higher degree, that is, more interconnected cells, having more negative values of divergence, thus confirming our theoretical predictions. We discuss these findings in the context of energetic aspects of signaling in beta cells and potential risks for pathological changes in the tissue.