The rubber hand illusion is a perceptual illusion in which participants experience an inanimate rubber hand as their own when they observe this model hand being stroked in synchrony with strokes applied to the person’s real hand, which is hidden. Earlier studies have focused on the factors that determine the elicitation of this illusion, the relative contribution of vision, touch and other sensory modalities involved and the best ways to quantify this perceptual phenomenon. Questionnaires serve to assess the subjective feeling of ownership, whereas proprioceptive drift is a measure of the recalibration of hand position sense towards the rubber hand when the illusion is induced. Proprioceptive drift has been widely used and thought of as an objective measure of the illusion, although the relationship between this measure and the subjective illusion is not fully understood. Here, we examined how long the illusion is maintained after the synchronous visuotactile stimulation stops with the specific aim of clarifying the temporal relationship in the reduction of both subjective ownership and proprioceptive drift. Our results show that both the feeling of ownership and proprioceptive drift are sustained for tens of seconds after visuotactile stroking has ceased. Furthermore, our results indicate that the reduction of proprioceptive drift and the feeling of ownership follow similar time courses in their reduction, suggesting that the two phenomena are temporally correlated. Collectively, these findings help us better understand the relationships of multisensory stimulation, subjective ownership, and proprioceptive drift in the rubber hand illusion.The online version contains supplementary material available at 10.1007/s00221-021-06211-8. The rubber hand illusion is a perceptual illusion in which participants experience an inanimate rubber hand as their own  and hadA right-handed male cosmetic prosthetic hand  was placed in front of the participants. A plastic divider was placed lateral to the rubber hand, behind which the participant’s real hand was placed in a relaxed position. The rubber hand, in view, and the real hand, hidden behind the divider more laterally, were placed 15 cm apart . The rubber hand was placed on the other side of the divider in the participant’s field of view. The black cloth was then placed over the participant’s right arm and the space between the rubber hand and the participant’s body as described above.In the first part, before the induction of the illusion, the participant performed the baseline hand localization task for the proprioceptive drift measure. This preillusion hand localization was performed by asking the participant to close their eyes and slide their left index finger along the horizontal ruler on the table and point to where they felt their right index finger was located. The experimenter asked the participant to close their eyes and placed the participant’s left index finger on the left side of the metal ruler . The participant was instructed to do one swift motion towards their right index finger and then stop and utter the word “here” when they felt that they were holding their left index immediately above their right index finger. The experimenter then noted the value and returned the participant’s left hand to a relaxed position on the table and subsequently asked the participant to open their eyes again. The whole hand localization task procedure, including moving the participant’s left hand to the metal ruler hand back to its relaxed position on the table and the experimenter noting the measurement, took approximately 10–20 s. We used this intermanual pointing task to probe proprioceptive drift, because it has been widely used and reproduced before  in line with earlier studies , most participants had their eyes closed throughout the measurements at 0, 20, 40 and 60 s, after which there was a pause long enough for them to have time to open their eyes. In experiment 2, by contrast, the participants had their eyes closed starting at the end of the visuo-tactile stimulation and throughout the 300-s-long testing phase. They only opened their eyes for one second (under the careful surveillance and instructions of the experimenter) to view the rubber hand immediately before the subjective ratings were obtained or the hand localization task was performed to make it clear to the participants that they should rate the feeling of ownership over the rubber hand , for which we pooled the data for each outcome measure and condition separately. We fitted various curves to this pooled data set  by first calculating a residual norm cost function and then finding the minima of this function using the function “fminsearch” (Nelder-Mead method) in MATLAB (version 2018b). The starting points entered into the function “fminsearch” were randomized real numbers between 0 and 10, and the function went through ten thousand iterations in the process of finding the minima of the cost function. Once we identified the best fitting curve type, we fitted this curve type to each participant’s individual data and extracted the slopes and intersect as well as the goodness of fit coefficients (R2). This fitting procedure was repeated 100 times for each participant with randomized starting points to find the parameters yielding the best fit while still conforming to the curve type and form of the best fitting curve. We then compared the coefficients for the “intersect”, decay and constant as well as the goodness of fit between the two outcome measures and conditions using 2 × 2 ANOVAs (proprioceptive drift/subjective illusion ratings & synchronous/asynchronous). We also used the fitted curves to calculate the “half-life” of the outcome measures, namely, the time in seconds it takes for the z-scores at timepoint 0 to decrease by 50 percent and compared these between outcome measures and conditions in a 2 × 2 ANOVA. Furthermore, we compared the intersects, decays, constants and half-lives of the two outcome measures in the synchronous conditions with Bayesian hypothesis testing to examine whether there was any support for the null hypothesis. We also ran a correlation analysis for the intersects, decays, constants and half-lives between the two outcome measures to see if the individual coefficient for one outcome measure could predict the coefficient for the other outcome measure.The second step of the analysis was to analyze the decay of the subjective illusion ratings and proprioceptive drift and compare their time courses. To this end, we first z score normalized synchronous–asynchronous difference for the two outcome measures. The resulting coefficients for the fitted curves were then compared between the two outcome measures using a Student’s t test as well as a Bayesian statistical test.In an additional analysis, the same curve selection and fitting procedure was performed for the U tests for each individual timepoint for the synchronous–asynchronous difference for the proprioceptive drift and subjective illusion ratings separately. The Mann–Whitney U test was used due to violations of the assumption of normality in the data. All statistical comparisons were performed using JASP software .For independent comparisons between the two experiments, we ran 2 × 2x6 ANOVAs for the proprioceptive drift and subjective illusion ratings separately with the factors experiment, synchrony and timing, followed by Mann–Whitney t tests were performed between the synchronous and asynchronous measurements for each timepoint , 20 s  and 120 s . Comparisons at all other timepoints were nonsignificant (Supplementary Table 1).To compare the proprioceptive drift between the synchronous and asynchronous conditions, int Fig. , panel AT tests were performed between the synchronous and asynchronous measurements for each timepoint  was analyzed with a 2 × 2 ANOVA, as were the R2 values and the half-lives (se methods). To illustrate these results, we have included a figure with the pairwise comparisons of the parameters  of the fitted curves for the synchronous conditions  showed no significant main effect of outcome measure  or synchrony  and no significant interaction . The analysis of the intersect values  or synchrony  and no significant interaction . The same pattern of results was observed for the decay coefficient , synchrony  and no interaction , indicating that the decay of the fitted curves in both outcome measures had similar steepness and temporal dynamics. Finally, the constant values for the fitted exponential decay curves , no significant main effect of synchrony  and no interaction . In an additional analysis, we calculated the half times of the fitted curves, which takes into account both the intersect, decay coefficient and constant of the exponential decay curve. The analysis revealed no significant main effects  or interaction , further strengthening the interpretation that the curves display similar decays between outcome measures. Bayesian hypothesis testing between the proprioceptive drift and subjective illusion ratings for each coefficient and half-life in the synchronous conditions showed weak to moderate support for the null hypothesis with Bayes factors (BF10) in the range 0.233–0.382 (Supplementary Table 3). The correlation analyses between the intersect, slope, constants and half-life of the curves for the proprioceptive drift and subjective illusion ratings showed no significant correlation (Supplementary Table 4).The analysis of the curve fits  in the range 0.250–0.361 .In a supplementary analysis to compare the outcome measures, the synchronous–asynchronous differences were calculated and then z-score normalized. Curves were fitted to the data points as described in the methods section, and the coefficients of the resulting exponential curves on the form of The first step of the analysis indicates that the subjective feeling of ownership persists for up to 300 s after the end of the visuo-tactile stimulation, whereas the proprioceptive drift is no longer significantly different between the synchronous and asynchronous after 40 s. Thus, the illusion is not immediately eliminated when the visuotactile stimulation stops but rather shows substantial persistence. In the second step of the analysis, we show that the proprioceptive drift and the subjective rating display a similar time course with decreasing values over time with similar slopes and half times of their fitted exponential decay curves. Thus, we found no support for our initial hypothesis that proprioceptive drift and subjective ownership should display significantly different time courses of reduction; in contrast, both measures seemed to follow a rather similar general decay function, albeit with the synchronous–asynchronous difference persisting for longer in the subjective illusion ratings.t tests was performed between the synchronous and asynchronous measurements for each timepoint , 20 s  and 300 s . Comparisons at all other timepoints were nonsignificant (Supplementary Table 6).As in experiment 1, a set of int Fig. , panel AT tests between the synchronous and asynchronous measurements for each timepoint , 20 s , 40 s , and 60 s . Comparisons at the 120 s and 300 s timepoints were nonsignificant (Supplementary Table 6).int Fig. , panel BR2 values see supplementary Table 7) showed no significant main effect of outcome measure  or synchrony  and no significant interaction . The analysis of the intersect values , no significant main effect of synchrony  or any significant interaction . The same pattern of results was seen for the decay coefficient , synchrony  and no interaction , indicating that the decay of the fitted curves in both outcome measures had similar steepness and temporal dynamics. Finally, the constant values for the fitted exponential decay curves , no significant main effect of synchrony  and no interaction . In an additional analysis, we calculated the half-lives of the fitted curves, which takes into account both the intersect, decay coefficient and constant of the exponential decay curve. The analysis revealed no significant main effect of outcome measure  but no main effect of synchrony  and no interaction . Bayesian hypothesis testing between the proprioceptive drift and subjective illusion ratings for each coefficient and half-life in the synchronous conditions showed weak to moderate support for the null hypothesis with Bayes factors (BF10) in the range 0.232–0.819 (Supplementary Table 3). The correlation analyses between the intersect, slope, constants and half-life of the curves for the proprioceptive drift and subjective illusion ratings showed no significant correlation (Supplementary Table 4).To compare the decay curves of the Z-scored proprioceptive drift measure and the subjective illusion ratings, we analyzed the coefficients and the half-lives of the fitted exponential decay curves Fig. . The ana10) in the range 0.256–0.304 .In addition, we analyzed the synchronous–asynchronous differences for the two outcome measures. Curves were fitted to the z-score normalized data points as described in the methods section, and the coefficients of the resulting exponential curves on the form of The results from experiment 2 show a similar overall pattern as the findings from experiment 1. The rubber hand illusion displays substantial permanence after the brush stroking ceases, with both subjective ownership and proprioceptive drift thereafter displaying a relatively slow decay. The first step of the analysis indicates that the proprioceptive drift is maintained for 40 s after the end of the visuo-tactile stroking before tapering off. An unexpected difference in the proprioceptive drift at 300 s is probably related to a negative drift in the asynchronous condition at this timepoint. Since this finding was only observed at this timepoint and was not seen in experiment 1, and we have no explanation for it, we will not consider it further in this study. For the subjective ratings, we show that the significant difference in the feeling of ownership between the synchronous and asynchronous conditions persists for at least 60 s after the end of the visuo-tactile stimulation. However, we note that the absolute values of ownership ratings in the synchronous condition were lower in experiment 2 than in experiment 1. The ratings decreased to a mean value of 0 after 20 s, while in experiment 1, affirmative mean ratings (above 0) were observed even after 300 s , experiment (experiment 1/experiment 2) and timing  for the proprioceptive drift measures and subjective illusion measures separately. For proprioceptive drift, we observed a main effect of experiment  and a main effect of synchrony  but no main effect of timing  and no significant interactions. For the subjective illusion ratings, we similarly observed a main effect of experiment  and a main effect of synchrony  but no main effect of timing  and significant interactions. The lack of significant interactions between the factors experiment and timing for both the proprioceptive drift  and subjective illusion ratings  indicates that the evolution of the measures is similar across experiments, whereas the lack of significant interactions between the factors experiment and synchrony for both the proprioceptive drift  and subjective illusion ratings  indicates that the synchronous–asynchronous difference is consistent across experiments.To compare whether there were any differences between experiments 1 and 2 in terms of the effect of the amount of hand viewing time on the proprioceptive drift and subjective ratings, we first conducted a 2 × 2 × 6 ANOVA of the W = 277.00, p = 0.034, rank-biserial correlation 0.385; Fig. To further analyze the differences, we conducted an independent sample interaction analysis, where the synchronous–asynchronous difference for the proprioceptive drift and subjective illusion ratings was calculated for each participant and each timepoint  in both experiments. We then compared each timepoint for both the proprioceptive drift and subjective rating in experiment 1 with its corresponding timepoint in experiment 2. Our results show that there is no difference in either the proprioceptive drift or the subjective ratings for timepoints 0, 20, 40, 60 and 120 s Fig. . For theWe conducted two experiments to investigate the relationship between proprioceptive drift and the feeling of ownership in the rubber hand illusion. In particular, we investigated how these two measures of the rubber hand illusion decayed after the end of the visuo-tactile stimulation. There were two main findings. First, we found that the rubber hand illusion showed a substantial degree of persistence after the visuotactile stimulation stopped. In both experiments, we found that significant effects on subjective ownership ratings could be observed for up to 300 s and for proprioceptive drift up to 40 s after we stopped applying the brush strokes. Second, we found that the subjective feeling of ownership of the rubber hand, measured through a questionnaire statement, and the proprioceptive drift, registered with a hand localization task, had similar time courses for their reduction. Furthermore, regardless of whether the participants looked at the rubber hand for most of the time during the experiment (experiment 1) or kept their eyes closed as much as possible (experiment 2), we observed similar time courses in the reduction of subjective ownership and proprioceptive drift when comparing the synchronous condition to the asynchronous control. Collectively, these results suggest that once the perceptual systems have “decided” that the rubber hand is one’s own and that the visual impressions from the artificial hand and somatosensory sensations from the hidden real hand should be bound together, the brain tends to persist with this model in the absence of strong counterevidence against this interpretation. Only after tens of seconds is there evidence of a slow process of reverting this illusory perception back to veridical bodily awareness with the gradual loss of subjective ownership and decreasing proprioceptive drift.z score normalized values), slopes or half times. We do, however, observe a slower decay of the subjective illusion ratings in the second experiment, which seems to be driven by the asynchronous conditions and the inexplicable result at the 300 s timepoint. In a previous study , within peripersonal space Supplementary file2 (DOCX 8645 KB)Below is the link to the electronic supplementary material.


This paper studies a system of Ordinary Differential Equations modeling a chemical reaction network and derives from it a simulation tool mimicking Loss of Function and Gain of Function mutations found in cancer cells. More specifically, from a theoretical perspective, our approach focuses on the determination of moiety conservation laws for the system and their relation with the corresponding stoichiometric surfaces. Then we show that Loss of Function mutations can be implemented in the model via modification of the initial conditions in the system, while Gain of Function mutations can be implemented by eliminating specific reactions. Finally, the model is utilized to examine in detail the G1-S phase of a colorectal cancer cell. Signaling networks are Chemical Reaction Networks (CRNs) consisting of an interconnected set of pathways, modeling the flow of chemical reactions initiated by information sensed from the environment through families of receptor ligands  by following standard procedures Feinberg  thus allIn the present work we start from a CRN modeling physiological  states and we show how it can be formally modified to simulate individual pathological conditions associated to particular classes of mutations. We compute and analyze the asymptotic stable states of both the systems, mimicking either a healthy or a pathological condition, in order to quantify the changes in the species concentrations induced by mutations. Since most cancer diseases result from an accumulation of a set of mutations , which are essential in the determination and interpretation of results  and Gain of Function (GoF) mutations  cells process information from external growth factors and the related answer  it admits a set of generators p, say weakly elemented.We say that a CRN is elemental or basic species.Borrowing the terminology of Shinar et al. , the speconservative, i.e. that there exists Condition (ii) in Definition Definition In general the set of elemental species of a network is not unique as it depends on the choice of the basis of the conservation vectors and, fixed a basis, multiple minors equal to the identity matrix may exist.Given a weakly elemented CRN, up to a change in the order of the components of Thus Eq.  may be rHenceforth we assume the CRN to be weakly elemented, with elemental variables position .p conservation equations is solved straightforwardly with respect to the elemental variables, thus yielding a parametric description of the affine space defined in .Given the initial condition mentclass2pt{minimmentclass2pt{minimFeinberg . Accordistoichiometric compatibility class (SCC) of Given a value mentclass2pt{minimLet s in Eq. . If \docEquation  simply fEquation  follows In this work, we are interested in computing the asymptotically stable states of system , that shA CRN is said to satisfy the global stability condition if for every stoichiometric compatibility class there exists a unique globally asymptotically stable state This means in particular that every trajectory with initial point on the given SCC tends asymptotically to the steady state a et al. , Feinbera et al.  and Yu aa et al. .A mutation consists essentially in a permanent alteration in the nucleotide sequence of the genome of a cell. Mutations play a fundamental role in cancer evolution 22) do noondition  holds thmentclass2pt{minimAs proved in Theorem i-th GoF mutation, i-th LoF mutation by computing the asymptotically stable state The procedure described by Definition ).If Eq.  holds anObviously, changing the order of the mutations affects the trajectory covered by the described iterative procedure but will lead to the same final steady state.In this section we apply the previous procedures to the analysis of a CRN which has been devised to provide a simplified description of how signals carried by a ligand from outside the cell are processed in order to determine the behavior of a possibly cancerous colorectal cell. The starting point of our analysis is the kinetic model originally proposed by Tortolina et al. , to modeHenceforth, we shall refer to this signaling network for healthy and CRC cells as CRC–CRN. The basic idea underlying the CRC–CRN approach developed in Tortolina et al.  is that a et al.  validateThe most relevant features of the CRC–CRN are summarized below; the full list of chemical species together with their abbreviated names, the set of chemical reactions, the system of ODEs, the fixed values of the rate constants, and the initial concentrations of the elemental variables in an ideal physiological state can be found in the Supplementary Materials of Tortolina et al. . The CRCmentclass2pt{minimWe have obtained a basis of It is seen by inspection that the CRC–CRN is weakly elemented, with 9 chemical species not involved in the conservation laws. Up to the 8 constant species, the list of elemental species coincides with that of the basic species of Tortolina et al. , which wc is a given constant; here NLK denotes the concentration of the elemental species “Nemo like kinase”  pathway, which represents a major cell proliferation signal transduction pathway from the cell surface to the nucleus. The MAPK pathway, following activation by the epidermal growth factor (EGF) protein, regulates the growth and division (proliferation) of cells, the process by which cells mature to carry out specific functions (differentiation), cell movement (migration), and the self-destruction of cells (apoptosis). In a normal cell, the BRAF protein is switched on and off in response to signals that control cell growth and development. In consequence of a mutation, a BRAF protein may be continuously active, transmitting messages to the nucleus even in the absence of these chemical signals. Thus, the overactive BRAF protein may contribute to the growth of cancers by allowing abnormal cells to grow and divide without the mediation of external signals (Armaghany et al. We investigated the impact on the CRC–CNR of a mutation resulting in the GoF of the elemental species BRAF.H is defined to include all the reactions involved in the deactivation of BRAFTo this end, following Definition As described in Tortolina et al. , Supplemmentclasspt{minimaentclass1pt{minimaSimilarly to what we have done in the previous section, we computed the asymptotically stable states of the trajectories obtained solving the system of ODEs defined by the stoichiometric matrix t in which the solution of the system is computed.Figure As a final remark, we observe that, as shown in Fig. entclass1pt{minimaIn this paper we have considered a system of ODEs modeling a CRN, with emphasis on the mutual relationships between moiety CLs and stoichiometric surfaces. CLs have led to the characterization of weakly elemented CRN and the definition of ideal states, which are crucial in the treatment of mutations. Confining attention to systems satisfying the global stability condition, we have considered two classes of mutations, LoF and GoF, often found in cancer cells. Mutations have been modeled as projection operators and basic properties of mutated networks have been analyzed, aiming at the determination of mutated equilibrium states as asymptotic steady limits of trajectories. The new model for mutations allows for a simple treatment of their combinations Furthermore, if the original and the projected CRNs satisfy the global stability condition, the resulting equilibrium is independent of the order in which mutations are combined.As an application of the previous results we have investigated a system of ODEs describing the G1-S phase of a CRC cell. Due to the huge number of variables involved, the analysis has been based on numerical simulations. We have found 81 independent, linear, moiety CLs; they have been applied to support the conjecture that CRC–CRN satisfies the global stability condition. It has also been found that the elemental chemical species coincide with the basic species defined by Tortolina et al.  on biochWe are aware that this computational approach to CRN is devised to capture few, although decisive, aspects of the G1-S transition point of a cell, as well as the modifications of the network induced by cancer mutations. We think that the model can provide a guide to future experimental research based on the interpretation of results of simulations, particularly in the case of the development and optimization of targeted therapies against already mutated cells. For example, if the simulation predicts a significant increase of the mutated concentration of a specific metabolite with respect to its physiological value, then that metabolite can be regarded as a potential drug target. More in general, an analysis of the mutated profile of the simulated CRN in the G1-S phase should provide hints about convenient choices of targets for available drugs, together with a framework for the simulation of their consequences (Facchetti et al. P, may be modeled by adding to the network a new pseudo-reaction of the form Possible development of our model may involve different directions. For example, we could extend it to examine alterations of mRNA induced by changes from physiological to mutated equilibrium (Castagnino et al. Feinberg , and thumentclass2pt{minimFeinberg .As a final comment, we remark that the rate constants in the CRC–CRN considered in this paper are assumed as known and estimated from the scientific literature (Tortolina et al.


We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid. This coexistence of large numbers of species has long been considered a paradox, a classical example being ‘the paradox of the plankton’3 in which many species of plankton coexist in the same ecological niche. This has motivated many community ecologists to try to solve the problem of species coexistence, often beginning with the premise that all individuals of all species live in a well-mixed, homogeneous, non-spatial habitat5. However, spatial heterogeneity  is one of the most obvious features of the natural world. In addition, heterogeneities in the spatial landscape have been shown to have profound effects on the dynamics of populations and the structure of communities7. Thus, the study of spatial effects on community structure has more recently become a central theme in ecology8.One of the central questions in community ecology is why there are so many different species of plants and animals coexisting in nature9 to form the complex networks that characterize ecological communities. These complex networks can take many structural forms ranging from linear hierarchies to intransitive tournaments, which describe interactions where there is no single best competitor, but rather the network involves at least one 3-cycle (or intransitive triad)10. This structural form determines whether an ecological community is able to thrive and persist or will go extinct as it evolves over time11. For example, a linear hierarchy associated with resources critical for a species’ survival undermine coexistence because at equilibrium the dominant competitor will exclude all others12. On the other hand, theoretical and experimental studies have shown that cyclic competition combined with dispersal of species through space may facilitate the maintenance of diversity in a community due to the inability of one species to exclude all the others13. Such cyclic competition, analogous to the classical rock–paper–scissors (RPS) game, have been found to exist in a variety of real ecosystems including plant systems15, marine benthic systems16 and microbial populations18. Cyclic competition also plays a role in the mating strategy of side-blotched lizards19, the overgrowth of marine sessile organisms20, competition between mutant strains of yeast21 and in explaining the oscillating frequency of lemming populations22.Spatial structure and dynamics determine the way the interactions among species living within a given habitat are interconnected23. Early theoretical studies on cyclic competition among three species are based on the Lotka–Volterra model of ordinary differential equations (ODEs), which ignores the effects of the spatial domain and predicts a solution of unstable periodic dynamics that leads to the extinction of two of the species after a short transient25. However, numerous theoretical models have shown that the three competing species can coexist indefinitely if ecological processes such as dispersal, migration and cyclic competitive interactions occur over small spatial scales26. Space therefore plays a central role in mediating ecological dynamics and the use of spatial models in ecology has grown enormously over the last two decades.Non-transitive competition introduces interesting and complicated dynamics in time and space28. Models that treat space explicitly can be classified according to whether population sizes, space, and time are treated as discrete or continuous entities. Among these, continuous space reaction–diffusion partial differential equations7, metapopulation patch models29 and individual-based cellular automata31 are the principal approaches that have been used in studies of spatially structured populations. Our focus in this paper is on the metapopulation framework.Various models have been used to explore the effects of spatial structure on population dynamics. However, the most fundamental distinction between the different models is the way in which the spatial dimension is represented. Some models treat space explicitly, giving some sort of description at each spatial location at any given time, or implicitly, incorporating parameters that vary with spatial scale or following only the percent cover of different species across the landscape32 in 1969 to describe the population dynamics of insect pests in farmlands, has been broadly used in conjunction with networks to assess the viability of populations that persist in fragmented landscapes linked by dispersal and in modelling large-scale spatial transmission of emerging diseases.Metapopulation theory constitutes a useful framework for explicitly incorporating spatial effects in models for population dynamics in ecology. Whereas classical population models treat local populations as closed systems (no immigration or emigration), this assumption is probably not valid for many species. Movement of individuals among populations is common and can have profound effects on the dynamics of local populations. Metapopulation models describe an open system in which extinction and persistence of local populations depend on the movement of individuals among a set of habitat patches. This concept, introduced by Levins33. Many species often inhabit discrete areas of the landscape  where demographic processes occur within patches, and dispersal occurs between them. The movement of organisms between patches leads to local differences in colonization and extinction rates, which can influence the spatial distribution of a species over time34.Thus, a metapopulation consists of local populations, living in spatially discrete habitats (patches)—where population dynamics takes place and homogeneous mixing of individuals is assumed—connected to each other through dispersal or migration35, the authors assume that migrations between patches are random with the migration rate constant being equal to the reciprocal of the number of dispersal links from a given patch to the other patches. They therefore define a dispersal graph to be homogeneous if all nodes (patches) have the same number of links, otherwise the graph is considered heterogeneous.In typical metapopulation modelling, the combination of habitat patches and their connectivity  can be seen as a network (or directed graph), where patches represent the nodes of the network and dispersal routes represent the edges. Each edge of the graph is assigned a nonnegative constant known as the rate constant of migration. In36. For example, numerical simulations by Nagatani et al.35 have shown that in a metapopulation model for the rock–paper–scissors (RPS) game, the dynamics are significantly different when the dispersal graph is homogeneous compared to when the graph is heterogeneous. Specifically, their simulations show that the coexistence equilibrium within each patch is asymptotically stable in the case of a heterogeneous graph, while the same equilibrium remains neutrally stable in the case of a homogeneous graph (as in the single patch case). That is, heterogeneity leads to the indefinite coexistence of all three species with abundances equal to the coexistence equilibrium values, whereas homogeneity leads to the perpetual coexistence of all three species with periodically oscillating abundances . However, as far as we know, these numerical results have been provided without mathematical proof.For many ecologists, the central question is how the structure or connectivity pattern of the underlying graph influences the dynamics of the metapopulation. Many model-based and empirical studies have shown that the persistence of metapopulations is influenced by connections between habitat patches35 are not only valid for RPS competition systems, but also extend to a broader class of competition networks. We first broaden the class of competition networks under consideration by giving a more general definition of homogeneity/heterogeneity that is based on the adjacency matrix of the dispersal graph without imposing any restrictions on the values of the migration rate constants. Then by combining concepts and results from game theory, chemical reaction network theory (CRNT) and dynamical systems, we provide a mathematical proof for the numerical observations in35, generalizing the scope of application to cover this broader class of competition networks within the patches. Specifically, we first consider a mean-field ODE model that describes the intra-patch dynamics in the absence of migration. The equilibrium point of this mean-field model is closely related to the Nash equilibrium (NE) of a symmetric zero-sum game and thus by analyzing the NE of the game, the behaviour of the coexistence equilibrium within each patch can be deduced. Then we borrow from CRNT the concept of detailed-balancedness38, a key feature for the kind of metapopulation models used in35, but not made explicit in it. We define a metapopulation model to be detailed-balanced if there exist positive equilibrium species proportions (or relative abundances) for which the overall migration rate of each species between any two patches is zero. Then, by assuming that our metapopulation model is detailed-balanced, we provide a mathematical proof for the numerical observations of35, which shows that these numerical observations are not only valid for a three-species cyclic competition system, but also apply to any n-species tournament for which a coexistence equilibrium exists.Here, we show, mathematically, that the numerical observations ofNotation: Below is a list of symbols and notations used in this paper. In this section, we highlight some terminologies, methods, and results from literature that will be used in this paper.i dominates species j at a rate j to node i in the graph i outcompetes species j at a rate j outcompetes species i. If We shall assume, like in many theoretical approaches, that, within each patch, each species can compete against every other species and that this competition is asymmetric such that for any pair of species, one is dominant over the other. These interactions can be represented by the paradigmatic reactions:mentclass2pt{minimi and In a single patch system, we assume that the population dynamics are governed by a mean-field model that tracks the deterministic time evolution of the proportions of competing species assuming that: (1) there are a large number of competing individuals within the patch and (2) the contact rates between members of the competing species follow a mass-action rule. Thus, the intra-patch dynamics are described by a system of ordinary differential equations (ODE system):40. Here, the n species represent a set of pure strategies available to each player and i at time t. A strategy for a player is a vector n pure strategies at time t. A strategy is pure if it is a unit vector, otherwise it is mixed. A strategy The properties  of the coexistence equilibrium of the mean-field ODE system  can be dfolk theorem of evolutionary game theory41, which states that; mentclass2pt{minimmentclass2pt{minimIt has already been shown that every symmetric matrix game admits at least one NE41, so that System .Since mentclass2pt{minim42. Kaplansky provided necessary and sufficient conditions for the game to be completely mixed43. That is, a two-player, symmetric, zero-sum game is completely mixed if its payoff matrix a, b, c are different from zero and have the same sign. The unique optimal strategy is then given by 44.In this work, we are interested in System  for whic45.Finding the stable coexistence equilibrium of  is thus 46, we show that if System  in incidence matrix48) is defined as follows:detailed balanced if it admits a thermodynamic equilibrium. In other words, a detailed-balanced single species reversible reaction network is one for which there exists an equilibrium We associate a finite asymmetric directed graph reaction . The incribed by , then5\dA reversible single species mass action chemical reaction network is detailed balanced if and only if37, Remark 3.1), from condition  constant of proportionality being the same across species. This constant of proportionality will be referred to as the rate constant associated with the migration. It is assumed that if there is migration between two given patches, then it is bidirectional, i.e., the rate constant of migration from j to k is strictly positive if and only the same holds for the migration from k to j. Just like in the case of a reversible single-species chemical reaction network, inter-patch migrations may be described by a weighted symmetric directed graph j to patch k. Finally, it is also assumed that the graph connected, i.e., there is a path between every two distinct vertices of the graph.Mathematical models based on traditional metapopulation theory usually assume that the metapopulation is made up of many neighboring spatially homogeneous habitat patches connected via dispersal. Consider an interconnected network of m-dimensional diagonal matrix whose i in patch j across the entire metapopulation, then the net migration rate i from other patches to patch j is given byi in patch j in the absence of migration. Since the dominance relationships among the species ,,detailed of type  for whicn species and m patches can be described by mn differential equationshomogeneous, otherwise we call it heterogeneous. Whether a balanced metapopulation model is homogeneous or not can be checked from the adjacency matrix We have seen earlier in “, System  can be rrom Eqs. –13), it, it\docu35, the authors assume that migrations from one patch to other patches are random with a probability of migration (or migration constant) equal to the reciprocal of the number of dispersal links from a patch to other patches. They thus define a dispersal graph to be homogeneous if all nodes have the same degree (number of links), otherwise the graph is heterogeneous. With this definition, homogeneity, in general, is equivalent to the existence of cycles in the dispersal graph, whereas heterogeneity is equivalent to their absence. However, with our new definition, it is clear that this is not necessary. An example of such a case is shown in Fig. InIn this section, we present a theorem that gives an expression for a coexistence equilibrium of a balanced metapopulation model. Before we state our main theorem in this section, we need the following lemma.Letdenote the incidence matrix of a finite connected directed graphand letdenote a diagonal matrix with positive diagonal entries. For anyit holds thatMoreoverif and only ifwhereAssume that the 48 that Since the graph We now state the main theorem of this section.A balanced metapopulation model described by System , .We now prove the local stability of the unique coexistence equilibrium corresponding to the balanced metapopulation model . For theConsider the balanced metapopulation model  is positive on V, we haveLet From Eq. , it foll. System . Assume s System . Since \s System , we haveCase 1: The model is heterogeneous, i.e., the vectoris not parallel toIn this case, Eq.  will be . System  consistst System  is persiCase 2: The model is homogeneous, i.e.In this case, Eq.  takes th. System  consistsThe above results can be illustrated by simulating System  for the 35 are valid. This has been done, firstly, by giving a more general definition of homogeneity/heterogeneity of a metapopulation model that is based on the nature of the adjacency matrix of its inter-patch migrations rather than on the number of dispersal links as was previously done in35. Secondly, we have performed a stability analysis of the coexistence equilibrium of a metapopulation model whose inter-patch migration is assumed to be detailed balanced and whose intra-patch dynamics is governed by a mean-field ODE system with a coexistence equilibrium. Detailed balancing, a well-known concept in chemical reaction network theory (CRNT), is a key feature of many metapopulation models and previous studies had not explicitly explored this concept. The motivation behind using concepts from CRNT is based on the fact that much of the interesting dynamical behavior observed in biological systems can be understood by analyzing the underlying chemical components and CRNT provides a unified mathematical approach to the study of chemical processes. The assumption of detailed balancing thus allows us to view the inter-patch migrations as a detailed balanced single species mass action reaction network and make use of the already available results on the latter.In this paper, we have expanded the class of competition networks for which the numerical observations of Nagatani et al.46, coupled with LaSalle’s invariance principle, it is shown that: (1) if the model is heterogeneous, then the coexistence equilibrium is locally asymptotically stable; it is globally stable if the considered metapopulation is persistent; (2) if the model is homogeneous and persistent, then the dynamics of the model is analogous to that of a single well-mixed patch; in this case, the coexistence equilibrium is neutrally stable. These results provide a mathematical support for the numerical results of35 and demonstrate that the numerical observations extend beyond the three-species cyclic systems to a larger class of networks.The results show that the considered metapopulation model admits a unique coexistence equilibrium. By making use of the Lyapunov function constructed in35 and most metapopulation models, the above results have been achieved by examining the simplified case in which the patches are assumed to be homogeneous and contain a local population in which individuals are well mixed. In addition, it is assumed that all the patches contain the same species. Although this assumption is far from real metapopulations, it makes the mathematical analysis tractable and provides a starting point for future analysis of more realistic metapopulation models. However, the results reaffirm the intuition held by many community ecologists that spatial heterogeneities in the landscape can have profound effects on the dynamics of populations within an environment. In the metapopulation framework, these heterogeneities come in many forms ranging from differences in dispersal rates among the patches to patch size distributions, among others. Also the fact that fragmented habitats are more likely to present some level of spatial heterogeneity underlies the importance of the metapopulation framework in studies for nature conservation.It should, however, be noted that, as in


Dressings for restoring organ defects and/or hemostasis in the injury site are being actively applied in operational units. These dressings are used in various surgeries and are widely represented in the foreign and domestic markets of medical products. Many local implants have different levels of hemostatic activity, which requires standardization of the algorithm of choice and the methods of their study.in vitro have been considered and evaluation criteria of their physical, chemical and organoleptic properties in vitro have been proposed. This will allow a researcher to choose optimal variants of samples for further experiments on biological models more effectively as well as to save funds, time and reduce the number of experiments in vivo.Here the methods of studying the performance properties of hemostatic implants  Abdominal surgery today shows a tendency to organ-preserving operative treatment of parenchymal organs  injuries due to their high functional significance , 2. TradThere are several methods provided both by foreign and national manufacturers. Their effectiveness depends on the approach to the production of morphological basis of samples, the basis usually being a sponge structure of animal (collagen) or synthetic  origin . This spDespite the increase in the number of surgeries and a growing need of surgical units in hemostatic implants used for local hemostasis, the common algorithm for evaluating effectiveness of these materials has not been established yet.in vitro methods; step II is a method of an “acute” in vivo experiment which is necessary for identifying the duration of bleeding and the volume of blood loss; step III is a method of “chronic” in vivo experiment — investigation of intensity of reaction of animal tissues to the means of subcutaneous implantation and in an injury model of parenchymal organs, etc. These steps allow for a comprehensive study of hemostatic capacity of sponge hemostatic dressings and elimination of possible negative effects of their application.We proposed  an algorin vivo [in vitro.The majority of modern researchers compare hemostatic means based on the findings obtained in experiments in vivo . HoweverBy the term “performance characteristics of medical products” we understand a set of properties that reveal a possibility of applying this medical product and using it effectively . For theCurrently, only a part of the methods provided below is regulated by the corresponding documents (GOST), represented in the State Pharmacopeia, etc. These methods have not changed from the moment of their development, but they are still relevant especially when used from a comprehensive perspective.sorption parameters. These parameters characterize the underlying property of dressings — adhesion [1=–F2. Evaluation of the adhesion force is the most important for the development of the technology to change the product properties and enhance its effectiveness [Investigation of performance characteristics of hemostatic implants should be started with the evaluation of adhesion , 16. Acctiveness .The study of adhesion is a complicated multi-sided multi-step process when one takes into consideration not only the properties of contact surfaces , but also their contact environment, in particular, humidity , 19.Investigation of these characteristics can be done with calculation methods based on such key parameters as length, width, and weight of samples. It does not require any expensive and hard-to-access equipment.lМ (g/cm) is calculated according to the formula:During the analysis linear and surface density is identified –22. The lm is the weight of the point sample (g); L is the average length of the point sample (cm).where sM (g/cm) is calculated according to the following formula:The surface density b is the average width of the point sample (cm); lm is the weight of the point sample (g); L is the average length of the point sample (cm).where Some authors –25 suggeV) of through pores and one-side open pores is as follows:And the formula of volume  is which is equal to 0.8812 g/cm3) [where 2 g/cm3) .P) is calculated according to the formula:The total sponge porosity (V1 is the volume of air-dry sample (4 cm3); V2 is the volume after soaking in the solvent medium.where openР), i.e. the volume of pores contacting with the environment is calculated as follows:An important step of testing is identifying an ability of samples to absorb moisture , 28. Opem1 is the weight of the initial sample with the size of 2×2×1 cm; m2 — is the weight of the sample after it has been soaked in water for 4 h.where actW) is calculated according to the formula:To identify the moisture content of samples a group of researchers –31 propoZ1 is the weight before drying; Z2 is the weight after the second drying. The arithmetic mean is taken for the result.where To identify a sorbing capacity of implants the following methods are used , 33. TheactК), sorbed by each sample is identified as follows:There are several variants of doing this investigation –36. For n is the weight of water sorbed by the sample (g); m is the weight of the sample (g). The arithmetic mean of three samples is the sponge sorbing capacity.where Alam et al.  suggest In study  after thh) is measured in 10 min from the moment of contact of the colored liquid and the zero mark. For h the highest point of the contour of the soaked dressing is taken. Capillarity  is calculated according to the formula К=h/10.The sorbing capacity of dressings can be studied with the methods of evaluation of capillarity — by increasing or decreasing the level of liquid in capillaries . During the investigation one measures the rate of solution rise in a glass tube densely filled with samples in a specific way. For this purpose samples with the same weight are shaped into bands. A glass tube is densely filled with these bands from the zero mark. Then this tube is placed into a vessel filled with colored water in such a way that the liquid is at the marked level. The liquid height Q=: –1): K(t)=InmQ/(mQ–Q), where Q is the amount of liquid sorbed by 1 g of substance for the time t, mQ is the maximum amount of the sorbed liquid (highest swelling);constant of swelling rate (min3): ρapp=m/V.apparent density for porous sponges  are obtained. Information about the degree of electrical conductivity is obtained with current spectroscopy. The character of relief is identified by the mean deviation of the sample surface from the isoline in 100 randomly selected points.in vitro with the above-mentioned methods stated [The investigation of polymer membranous implants s stated  that themechanical methods to characterize such property as elasticity [Properties of hemostatic dressings are studied with asticity , 47. It asticity ). These asticity –50.The tested sample must be rectangular in its cross-section, have no superficial film and visible defects. Two lines marking the working area are applied on the samples with a marker with parallel plates. The internal distance between the lines must be 25–50 mm with error margin of ±1% , 52.Furthermore, it is recommended to model samples with a die cutter used for cutting details and rubber blanks.After the sample has been prepared according to the regulatory acts and conditions of the experiment it is fixed into the clams of the machine . The claTests on preliminary indention are carried out in the following way. A sample is placed on the bearing surface of the machine in such a way that its center is under the indenter center. The samples that have slots on one side must be located in such a way that this side is near the bearing surface –60. LoadThe remained deformation of compression is defined by the difference between the initial and end thickness of the tested object. The end thickness is the value obtained from the sample compression for the given period of time (72 h), at the specific temperature mode (20±2°С), humidity (65±5%) and the time of reconstitution of the initial thickness. When the load is removed, changes in the sample thickness are identified –64.Before the elasticity test the central part of the sample is marked to limit the length of the test surface which must be at least 50 mm for rectangular samples in accordance with GOST 29104.1-91 , 65–67.The nominal strength and percentage elongation at break are set with the method of stretching with constant load. The thickness of the object is measured at five evenly located or at two random points in sample area. The thickness of test samples should not differ by more than ±2% –70.Е=σ/ε, where σ is tension; ε is elongation at break.Then parameters of elasticity and plasticity are calculated. Elastic modulus (Young’s modulus) characterizes an ability of material to resist deformity , 72. It By plasticity we understand an ability of material to change shape and sizes without being destroyed through exposure to external force. Plasticity is measured by such parameters as percentage elongation (contraction) δ and lateral contraction (expansion) ψ during a static test for stretching (contraction) –75:l0 and kl are the lengths of the sample before and after break, F0 and kF are the square area of cross-section of the sample before and after destruction.where The conclusion about material plasticity is made according to the correlation of the obtained findings with the following criteria: flexible — δ>5%, fragile — δ<5%. However, ideally, these criteria in physics and mechanics of materials are applicable only to solid substances and bodies  –78.qualitative (identification of presence of substances with the given level of purification stated by manufacturer) and quantitative . It is significant to study traces of substances  used in the technological process — reagents that must not be present in the finished product. Improper storage conditions or quality of reagents can considerably affect the end properties of implants, their effectiveness potentiate tissue reaction of the macroorganism and therefore reduce biological inactivity of the material. The present of “extra”  substances can lead to the development of paradox allergic reactions, or formation of teratogen and oncogene properties of hemostatic dressings [Chemical methods of evaluating properties of sponge hemostatic dressings can be divided into ressings , 80.One of the simplest and the most available and representative from the practical point of view methods is the method of identification of water extract reaction –83. The Organoleptic methods deserve special attention. They are universal and available. Physical and chemical properties of sponge hemostatic dressings are identified instrumentally or by calculation, while organoleptic (manipulation) properties should be identified by invited experts (expert method), i.e. practicing surgeons –88. Evalin vitro methods. However, this evaluation will be incomplete due to considerable inaccuracies compared to in vivo experiments as a macroorganism is a complex open system with multiple variables that cannot be taken into consideration today even by high-capacity computing equipment.Development and investigation of sponge hemostatic dressings is performed in different directions. Various approaches to modification of dressings are being developed to potentiate their effectiveness –100. It,in vitro will help a researcher to understand the sample choice at the initial stages of work. Algorithmization will give general understanding of mechanisms of action of hemostatic dressings which can be further used in comparison of their effectiveness and finally will alleviate the choice of prospective tested samples from the wide range of hemostatic products.The algorithm suggested by us  and the


The aim of the study was to analyze factors predicting (a) the transition to parenthood for female same‐sex couples in Sweden and (b) which partner is the birth mother for the first and (any) second child.Longitudinal studies in which couples become parents are rare for same‐sex couples in any context, even though these families are increasing. Childbearing in lesbian couples is an interesting case for testing theories linked to family utility maximization and household bargaining, as these couples can often choose who will carry a child.Discrete‐time event history and linear probability models are estimated on Swedish population register data (1995–2016) to analyze couples' transitions to first and second birth and the choice of birth mother.The higher the household income and partners' educational levels, the more likely couples are to become parents. However, within‐couple income gaps are small, and income and education are unrelated to the choice of first‐birth mother. Couples are more likely to have a second child and to switch birth mothers if both are highly educated or the first social mother is highly educated.Factors predicting which couples become parents are similar in same‐sex and different‐sex couples. In same‐sex couples, short‐term within‐couple specialization is of little relevance for who becomes the birth mother. Analyses of the transition to a second birth suggest that long‐term planning matters for who becomes the first‐ and second‐birth mother. In recent decades, families have emerged who previously either faced difficulties forming and/or were not legally recognized. Among them are female same‐sex parent families. Same‐sex parenthood was legally recognized in Sweden in 2003, but even before then, registered partner couples and parents could be identified in population registers. Far from all parents register as partners—and not all couples want to become parents—but research indicates that registered partnership (1995–2008) and marriage (from 2009) has been increasingly common among couples who become parents  become parents, the decision regarding To date, we know of no research on the association between socioeconomic factors and the transition to joint parenthood in lesbian couples. In addition, earlier research on within‐couple specialization is predominantly cross‐sectional, retrospective, and based on either surveys or in‐depth interviews with a small group of individuals. Sociodemographic characteristics may be misrepresented in these data due to systematic interview participation and nonresponse  the probability that female, married, or registered partner same‐sex couples become parents by giving birth to a child and (b) who will be the birth mother in these couples, that is, the person who carries the first and (any) second child.In the following, we first discuss theories and earlier research on which couples become parents (between‐couple factors) and second, the choice of birth mother in these couples (within‐couple factors). Thereafter, the theoretical background is summarized, and hypotheses are formulated before we turn to a description of the data and the empirical models. The results are presented, and conclusions are discussed in the two final sections.Focusing on nonfinancial reasons for having children, same‐sex couples and different‐sex couples tend to state similar reasons for wanting to become parents, such as expecting increased happiness, well‐being, and life fulfillment , resulting in more modest legal differences between marriage and cohabitation in Sweden than in many other contexts.For lesbian couples, the costs of assisted procreation or adoption add to the regular expenditures of having children. Couples may choose to privately arrange home insemination, but if they do, the social mother must adopt the child to be legally recognized as a parent in Sweden, and in order to adopt, she has to be married to the birth mother. Neither adoption  nor marriage is needed if the couple conceives at a certified clinic, yet the mothers need to cohabit. The major legal differences between marriage and cohabitation have to do with inheritance rights and the way property is divided in case of a divorce . Some choose to use clinics abroad even for their first child. Couples may go to Denmark to conceive as a way to reduce the wait, obtain more detailed information about the donor, or be able to choose an anonymous donor (the latter is not an option in Sweden). Couples who use clinics abroad pay for the full treatment, making this solution more expensive than using a Swedish clinic. Worth keeping in mind is that becoming parents and having a child is a well‐thought‐out, planned choice for lesbian couples. No couples become pregnant by accident. Although a comparison to different‐sex couples is not the focus of this study, this most likely contributes to a stronger link between income and parenthood in same‐sex compared with different‐sex couples. Earlier research shows that the birth mother in a same‐sex couple on average has a higher income than the mother in a different‐sex couple .In the empirical analyses, the transition to parenthood is seen as a two‐step process where the decision to become parents is made before the couple decides who will be the birth mother. Hence, which married (or registered partner) couples become parents is empirically analyzed first. Based on the literature review, the expectation is that the partner with the lower income and/or education is most likely to be the birth mother (see Table the more resourceful partner will be the (first) birth mother ) provided by Statistics Sweden are analyzed. Data cover the entire population and is available annually until 2016. A total of 4,251 female couples who married in the period 1995 to 2016 were identified. A subsample of these (see below for exclusions) were followed annually from the year of marriage until they had a first child and a second child, if any. The first analysis, the parenthood analysis (P1), follows couples from registered partnership or marriage and onwards to analyze which couples become parents. The second analysis, the first birth mother analysis (M1), analyzes which partner in these couples is the birth mother of the couple's first child. Age is used to discriminate between the two partners, and the younger partner is chosen as the index person. In the third analysis, the second child and parenthood analysis (P2), we follow the parental couples from first birth onward and analyze which ones have a second child. Finally, the fourth analysis, the second birth mother analysis (M2), estimates the likelihood that both children in two‐child families have the same birth mother.Swedish population register data  analysis. The M2 analysis is run on a sample of 510 two‐child families.SEs are clustered at the couple level.A discrete‐time event history model is used for the analysis of which couples become parents where the event is the birth of the couple's first (P1) child and second (P2) child. Episodes are censored in 2016 (the last year of observation) or at the time of divorce, emigration, or death of any of the partners. As the dataset consists of couple‐years, Linear probability models (LPMs) are estimated to analyze which partner is the birth mother in couples who become parents (M1) and have a second child (M2). The analyses are cross‐sectional and estimated in the year of birth of the couple's first and second child, respectively. The M1 outcome is a binary variable indicating that the younger partner (the index person) is the birth mother of the child, and the M2 analysis indicates whether the birth mother of the first child is also the birth mother of the second child.Marriage cohort is included in the P1 and P2 analyses. In P1, it is divided into the categories 1995–2002, 2003–2004, 2005–2008, and 2009–2016. The cut‐off points correspond to the first year that same‐sex partners could adopt (2003), get assisted procreation at a Swedish clinic (2005), and marry (2009). In P2, the two first categories are collapsed because of the small number of couples who had a second child these years (20 in 1995–2002 and 37 in 2003–2004). The partners may have had children before they entered the current marriage. Hence, dichotomous variables indicate whether any of the partners have a previous biological child. These variables  are not included in the P2 and M2 analysis, as very few of the parents who have a second child have children from before .Duration of marriage is included in the P1 and M1 analyses and acts as the time variable in the discrete‐time event history analysis of first parenthood. It is divided into the categories 0–1, 2–3, and >3 years of marriage. The P2 and M2 analyses instead include years since first child birth. It is divided into the categories 0–1, 2, 3, 4, and >4 years in P2, and time varying. In M2, years since first child birth is measured in the year of birth of the second child, and because of small cell sizes, the variable is divided into the categories 0–2 years, 3, and >3 years. Age of the younger partner  and of the first birth mother  in the year of marriage is categorized into 18–30, 31–35, and >35 years of age. The analyses also include the age difference; in the P1 and M1 analyses, whether the older partner is at least 4 years older than the younger partner, and in the P2 and M2 analyses, whether the first birth mother is at least 4 years younger or 4 years older than the first social mother.household income (log)). The logarithmic transformation reduces the likelihood that influential outliers bias the results. Household income is time‐varying and lagged by 1 year in the P1 analysis and measured the year before birth in the P2 analysis. The M1 analysis includes the younger partner's income and her relative income (her income as a share of the household income) the year before birth. The measure is relevant for estimating short‐term family utility and partners' specialization in paid and unpaid work. For those who have their child early in the year, the income the year before may be influenced by pregnancy‐related absences from work. Therefore, additional analyses (available on request) were estimated with income measured 2 years before birth, with similar results as those presented later. The M2 analysis includes the income and relative income of the first birth mother the year before first birth in order not to let the use of parental leave influence the two measures.The P1 and P2 analyses include the logarithm of the sum of the partners' annual gross work incomes .Figure Most couples had their first child early in the marriage. The average time span between marriage and childbirth was 1.5 years Table . Figure n < 20).The partners in the couple were equally likely to be the birth mother of the first child if the child was born in the first year of marriage Figure . From thFigure Table In the parental couples where only one partner was highly educated, the first birth mother was somewhat more likely than the social mother to be the one with higher education.Childbearing in female‐same sex couples was more common among couples and partners who had not previously had a child, and the first birth mother was slightly less likely than the social mother to have had a previous child. Among the parental couples, 38% had a second child during the studied period. The same partner gave birth to both children in 55% of the couples who had a second child Table . RegardiIn the following, the results of the event‐history analyses and LPMs are presented. Cell sizes for categorical variables used in these analyses can be found in The results of the discrete time, event history model of parenthood (P1), and the LPM of first birth mother (M1) are displayed in Table As expected, couples were less likely to have a child the older the younger partner was. The age difference was, for most, not decisive for their likelihood of becoming parents, although there is a negative association (significant at the 10 % level). Childbearing was positively associated with household resources; if one or both partners were highly educated, they were more likely to have a child than if neither was highly educated, and the probability that they became parents increased with household income.The choice of birth mother in the couples who become parents is analyzed in an LPM (M1). The probability that the younger partner carried the child increased with her age (implying that the likelihood that the older partner gave birth decreased with her age). If there was a large age difference in the couple (at least 4 years), the younger partner was more likely than the older partner to be the first birth mother.The choice of the younger partner as the birth mother was positively associated with the older partner having a child previously, but unrelated to the younger partner's previous biological parenthood, all else being equal.Partly because the M1 analysis focuses on couples who, on average, were more highly educated than those who did not transition to parenthood (cf. P1), education was unrelated to the choice of birth mother, as was the younger partner's income and relative income the year before childbirth. Hence, the likelihood that a couple had a child was related to income and educational resources, whereas the choice of birth mother for this child was not.Table The main independent variables in Table The final analysis in Table Returning to the theoretical expectations and hypotheses, the first hypothesis received support, as couples with higher educational and financial resources were more likely to transition to parenthood. The second and third hypotheses, linked to within‐couple short‐term  and long‐term financial reasoning (relative resources/bargaining), did not receive any clear support. Within‐couple resource differences were small in the couples who transitioned to parenthood, which may explain why factors other than income and education matter for who becomes the birth mother. Nevertheless, among those with two children, taking turns giving birth was more common for couples in their upper thirties and in couples where the social mother or both parents were more educated. This may provide tentative evidence for the long‐term family utility perspective  second child. Earlier research .Among the couples who (decide to) become parents, the choice of birth mother may theoretically be linked to short‐term or long‐term family utility. The short‐term perspective predicts a within‐couple division of reproductive behavior and paid work that is in line with Becker's theory on specialization. In other words, the lowest earning partner should be the birth mother for the first and any second child (Hypothesis 2). The long‐term family utility perspective predicts that the higher earning, or more educated partner, is the one who is chosen as the first birth mother as she has the better position in the labor market and thereby can “afford” to be pregnant and take leave (Hypothesis 3). If the parents go on to have a second child, the less resourceful partner will have had time to accumulate more work experience, and the long‐term family utility perspective is strengthened if she carries the second child.In the empirical models, higher educated couples are more likely to transition to a first birth (Model P1) but a partner's income and higher level of education is unrelated to her likelihood of bearing the couple's first child (M1). However, if a couple has a second child, the parents are more likely to switch birth mothers if the first child's social mother or both mothers are highly educated. Hence, the long‐term family utility perspective may influence the decision of who will be the birth mother in families where career‐related costs of childbirth are higher. Within‐couple income differences are small in lesbian couples. Still, opportunity costs of childbirth and leave‐taking differ by educational level, and birth mothers often take the first and the longest parental leave . This research has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme awarded to Marie Evertsson (Grant Agreement No. 771770).AppendixS1. Supporting Information.Click here for additional data file.


The ubiquity of mobile devices fosters the combined use of ecological momentary assessments (EMA) and mobile crowdsensing (MCS) in the field of healthcare. This combination not only allows researchers to collect ecologically valid data, but also to use smartphone sensors to capture the context in which these data are collected. The TrackYourTinnitus (TYT) platform uses EMA to track users’ individual subjective tinnitus perception and MCS to capture an objective environmental sound level while the EMA questionnaire is filled in. However, the sound level data cannot be used directly among the different smartphones used by TYT users, since uncalibrated raw values are stored. This work describes an approach towards making these values comparable. In the described setting, the evaluation of sensor measurements from different smartphone users becomes increasingly prevalent. Therefore, the shown approach can be also considered as a more general solution as it not only shows how it helped to interpret TYT sound level data, but may also stimulate other researchers, especially those who need to interpret sensor data in a similar setting. Altogether, the approach will show that measuring sound levels with mobile devices is possible in healthcare scenarios, but there are many challenges to ensuring that the measured values are interpretable. Smart mobile devices  are becoming increasingly ubiquitous. Their capabilities allow the combined use of ecological momentary assessments (EMA) and mobile crowdsensing (MCS) in the healthcare domain to not only collect qualitative longitudinal and ecologically valid data, but also to use sensors of smartphones as well as connected external sensors  to capture the context in which these data are collected . For exahttps://www.trackyourtinnitus.org/, accessed on 1 October 2021), a central backend for data storage, and a mobile application available for both Android and iOS. The mobile apps assess users’ individual tinnitus perceptions  by asking them to complete tinnitus EMA questionnaires at different times of the day [The TrackYourTinnitus (TYT) mobile platform uses EMA and MCS to track a user’s individual tinnitus. Tinnitus is the perception of an internal sound in the ears in the absence of a corresponding external sound. As symptoms are subjective and vary over time, TYT was created to monitor and evaluate the variability of these symptoms in the daily life of tinnitus affected patients or interested users . The pla the day . In addi the day . The det the day . The oveSPL, dB(A) or dB(C) values would circumvent this issue. To encounter that sound sensor values measured by a smartphone require further considerations in healthcare scenarios is also recognized by other works than TYT [Identification of an experimental setting that can be used to learn more about the interpretation possibilities of the collected TYT sound level values.In addition to the latter point, in the best case, the experiment should be appropriate to enable us to compare all sound level values across the different smartphone devices from different manufacturers and different mobile operating systems.Conduction of the experiment without the use of an expensive sound laboratory, with the goal to foster and facilitate the overall reproducibility.However, the data available in the TYT database  do not cthan TYT . From a than TYT , for exathan TYT  also hasthan TYT . HoweverThe TYT database was analyzed to identify the mobile device models that contributed the most environmental sound measurement data.The analysis of the database showed that more detailed device information is available for Android devices. For this reason, it was decided to use Android devices for the experiment.A sample of the identified device models was selected and acquired .A new mobile application was developed that mimics the behavior of the TYT app with respect to the sound measurement. More specifically, the app was implemented with the specific focus on the sound measurement but using the same software functions as TYT .The selected device models were equipped with this mobile application.For the evaluation of the smartphone devices equipped with the app, a sound signal was generated, for which the volume was adjusted to different sound levels using a professional calibrated sound level meter (SLM). Based on this setting, the values captured by the mobile app on the different mobile devices were recorded.Finally, the results were used to derive equations for the different device models that, in turn, can be used to transform the measurement data in the database into  comparable dB(C) values.Based on these requirements, different scenarios have been discussed. In the end, the following approach  was conceived to make the described values usable and comparable:How these steps were carried out in practice and what results were achieved are discussed in the following sections. In Measuring sound levels with smartphones has been a topic of research for some time. There are both scientific and commercial implementations of apps that perform sound measurements. In addition, studies evaluating the accuracy and precision of these apps can be found in the literature. Moreover, the ability of smartphones to perform sound level measurements in the environment as well as their calibration has been investigated and discussed in a thorough manner. Finally, there are works that deal with large data sets of sound levels measured with smartphones.SPL value by adding a constant calibration value. A built-in calibration tool can be used to determine this value using a constant pink noise [eq), C-weighted peak sound pressure levels (LCpk), as well as other values for different sampling periods [SPL and equivalent continuous sound level (Leq) values [NoiseMap  is an Annk noise . The iOSnk noise  was deve periods . Ambicit periods  is a mob periods . The acc periods . The Cit periods  mobile a) values . Further) values ,16,17) a) values ,19,20). ) values . There a) values ,27,28,29) values ,27,28,29) values ,24,28,29) values ,27,28,29) values , or repr) values ,28. Resu) values ,25. In g) values ,25. It h) values .Moreover, the ability of smartphones to perform environmental sound level measurements in general has been extensively discussed in the literature ,32,33,34den) estimation is proposed by [Furthermore, the calibration of smartphones for sound measurements and different approaches in this regard have been discussed in this context ,37,38,39Finally, works that deal with large data sets of sound levels measured with smartphones can be found in the literature. For example, interpolation ,41 and sHowever, to the best of our knowledge, the evaluation of an pre-existing large data set of uncalibrated environmental sound level amplitude values measured with smartphone sensors has not yet been considered in the literature. In this context, the chosen approach of making the data set of sound measurements comparable and interpretable by taking a sample of devices from this data set, calibrating them, and deriving corresponding equations is a novelty. Furthermore, none of the existing related works considers the assessment of environmental sounds measured with smartphone sensors, or smartphone sensor measurements in general, in the context of tinnitus.First, the materials and methods used to perform the experiments in the scope of the work at hand are described. In this context, the data set used for the initial analysis is outlined. Furthermore, the selection of hardware and software components used for the experiments is described. Finally, the experimental setup and procedure are delineated.Build.MODEL from the android.os.Build API , which can be used to uniquely identify the respective device model  were selected based on the following conditions:There are more than 500 entries containing sound measurements for the user.The reported tinnitus loudness (see ) is flucThe sound measurement is fluctuating and appears plausible .For the second analysis, the data set was analyzed on a Google Pixel 2 was used simply because it was available to the experimenters. This resulted in the five devices shown in Finally, the identified devices from both analyses were combined, resulting in eight devices, as highlighted in testo 815 by Testo SE & Co. KGaA is used. It allows measurements in the range of 32 to 130 dB and a frequency range of 31.5 to 8000 Hz. The SLM supports frequency weightings A and C. Its accuracy is ±0.5 dB under reference conditions at 94 dB and 1000 Hz in accordance with Class 2 of IEC 60,942 [PeakTech Prüf- und Messtechnik GmbH was used to calibrate the SLM. The accuracy of the calibrator is ±0.5 dB under reference conditions at 23 As a reference sound level meter (SLM) for the performed sound measurements the C 60,942 , with a GigaWorks T20 Series II by Creative connected to a notebook was used. The Online Tone Generator by Tomasz P. Szynalski [As a sound source for the experiments, the speaker of the zynalski  was usedMediaRecorder.getMaxAmplitude method to capture the “maximum absolute amplitude that was sampled since the last call to this method” [In order to mimic the behavior of the TYT app for the experiments, the corresponding code for the sound measurement was extracted and integrated into a new sound measurement mobile application. In addition, this allows to implement a more convenient way of extracting the results, as well as more insights into various parameters of the sound measurement. Equivalent to the TYT app, the sound measurement application utilizes the previously described  method”  every 50SPL, dB(A) and dB(C) at 1000 Hz are all equal and may therefore be used interchangeably for measurements at this frequency. For purposes of clarity, dB(C) is used for the remainder of this paper. To promote and facilitate the overall reproducibility, it was decided against a professional sound laboratory in favor of a simpler test environment for the experiments. Thus, for the measurement range, a lower limit of 50 dB(C) was chosen because the background noise in the test environment was measured at approximately 46 dB(C). 80 dB(C) was chosen as upper limit to avoid hearing damage for the experimenter . A distance of 30 cm between sound source and SLM/smartphone was chosen due to spatial restrictions to avoid reflections in the test room.Before conducting the actual experiments, various measurements were taken with different frequencies (125–2000 Hz), frequency weightings (A & C), distances to the sound source, and different smartphones to find the optimal settings for the experiments. The measurements indicate that—using the correct settings—the smartphones measure sound frequency-independently in the study’s frequency range of 125–2000 Hz, allowing a single frequency to be used for the experiments. The final settings are shown in The experimental setup is shown in Before conducting the experiments, the SLM is calibrated with the calibrator to account for the room conditions such as temperature and air pressure. Thereby, the calibrator is attached to the SLM and turned on, producing a sound at 94 dB and 1000 Hz. The SLM is then configured to measuring range 50–100 dB, time weighting “Fast” (the measured samples are averaged every 125 ms) and frequency weighting A. The SLM is then potentially fine-tuned until the display also shows 94 dB.The experimental procedure is structured as follows and was repeated for each of the five smartphones.The tone generator software is used to create a 1000 Hz sinus signal (pure tone) with the speaker.The volume is then adjusted until the SLM shows the desired sound pressure level.Subsequently, the measurement is started on the smartphone. As described in The steps 1.–3. are repeated for 5 dB increments between 50 and 80 dB(C) (an explanation for the measuring range can be found in the first paragraph of this subsection), resulting in seven values per smartphone.The final experiments resulted in a total of 35 values. The results are shown in The results of the experiments are then analyzed in terms of their interpretation. In this context, first, the experimental results are used for a logarithmic regression to derive respective equations for the different device models. Second, the legacy application code of the TYT app is analyzed for relevant implementation errors and poor design decisions that should be improved. Finally, the derived equations are used to transform the existing data in the TYT database into  comparable dB(C) values.p is the root mean square sound pressure and As can be seen in pressure . For souMediaRecorder API are averaged arithmetically and stored as a single value, which is supposed to represent the average sound level. This is erroneous, as sound levels are logarithmic values, which must be transformed to their energetic source values before they can be used for calculations [Erroneous calculation 1: The 30 amplitude values sampled by the app as retrieved by the Android ulations .Erroneous calculation 2: The first two measured values of the app often contained errors in the initial experiments. For multiple of the investigated devices, the first measured value was consistently 0, while the second value was often too low. Further experiments showed that these errors occur very frequently for measurements within the first 1000 ms after the start of the recording. These findings indicate that these first two values should be excluded from the calculations.AMR_NB [Unsuitable audio codec: The audio encoder AMR_NB  is used AMR_NB . Lower aLack of user transparency: The app does not indicate that the sound measurement is ongoing. The user could therefore interact with the mobile device in an unfavorable way, which might interfere with the measurement . For example, interacting with the touchscreen of the mobile device during the measurement resulted in an increase of the measured sound level by about 10 to 20 dB(C). Placing the device on a table led to values above 100 dB(C).As mentioned in SPL (or dB(C), respectively), and thus cannot be used for corresponding conclusions. However, the values could still be used to compare them relatively  and to investigate correlations with other data .To estimate the magnitude of the error due to the erroneous calculation, a worst case was simulated, for which 29 of the 30 measured amplitude values are used as input for the calculation that are rather small and one value that is rather high. We chose the amplitude values measured for 50 dB(C) and 80 dB(C) respectively, as these were the lowest and highest sound level values used in the experiments. The resulting dB value that would be calculated by the TYT app as well as the correct dB value are shown in soundlevel data from the TrackYourTinnitus database  is possible if the devices are calibrated correctly beforehand. However, there are several aspects that should be considered. The mobile application used to measure the sound level should be carefully revised regarding the following aspects:SPL in the experiments, which cannot be generalized for frequencies other than 1000 Hz. Furthermore, for environmental noise measurements usually the A-weighting filter is used to better reflect the hearing of the human ear. Sixth, the output of the mobile application is a peak value and not an effective value as measured by the SLM. These values should not be compared directly, but were nevertheless used to simulate the behavior of the TYT app. Seventh, it is assumed that the Android API used to retrieve the amplitude values behaves the same on each Android version, since the experiments were performed with the maximum version that was officially supported by the acquired devices , might have distorted the results. Second, the measuring distance of 30 cm from the sound source was chosen for spatial reasons. It was not verified whether a greater distance would lead to more accurate measurement results. Third, the measurements were limited to levels between 50 and 80 dB(C). Values below or above these limits cannot be verified. Fourth, the measurements were performed with a single sinus signal (pure tone) sound source at 1000 Hz. Generalizations for other sound signals and different frequencies might not be accurate. In addition, pure tones might lead to room modes and standing waves that could have distorted the results, which was not considered in the experiments. Fifth, in this context, dB(C) values as measured by the SLM are treated as dBices see . This asIn this work, an experiment was described with the objective to make a large data set of environmental sound measurements captured with smartphones and stored in the TrackYourTinnitus (TYT) database usable and comparable to enable meaningful interpretations in the context of tinnitus research. To this end, the existing data were analyzed to find the device models that contributed the most data entries. Four of these device models were then acquired for the experiments and equipped with a mobile app that mimics the environmental sound measurement of the TYT Android app. For the actual experiments, a sound signal was generated, the volume was adjusted to different sound levels using a professional calibrated sound level meter (SLM), and the values captured by the source code of the app on the Android devices were recorded. The results indicate that the amplitude values retrieved by the devices behave similarly except for a constant offset. Furthermore, equations derived from the results with a logarithmic regression analysis can be used to transform the values in the TYT database to  comparable dB values. However, there are several limitations to the experiments due to the code of the TYT app and the experimental setup.Since the experiments within the scope of this work were only conducted for a number of selected Android device models, in future work, more device models should be considered. This includes both Android as well as iOS device models. For the latter, there are far fewer different models, which are all produced by a single manufacturer, which simplifies the process. Once the values retrieved by the system APIs of the different device models and operation system versions are known, respective equations can be derived and used for any future measurements of the same models. Alternatively, along with the recommendations in In conclusion, it has been shown that measuring sound levels with mobile devices is possible and feasible for healthcare purposes, but there are many challenges to ensuring that the measured values are accurate, comparable, and interpretable and thus more future work towards the interpretation of mobile crowdsensing data should be conducted.


Dislocation of joints in the human body is a common occurrence, with the upper limb being commonly involved in most dislocations. Despite the common occurrence of upper limb dislocations, sternoclavicular dislocation is rare, comprising just 1% of all dislocations seen. This is attributed to a combination of restricted range of movements and strong ligaments around it. A thorough search of literature revealed a handful of cases of sternoclavicular dislocation. We report a case of a 35-year-old firefighter who presented to us an hour after his injury. He was diagnosed with anterior dislocation of the left sternoclavicular joint and was successfully managed by the closed reduction maneuver. We present this case to highlight the rarity of this unusual dislocation and also shed light on its etiopathogenesis and current management trends. Due to the presence of strong ligaments around it, sternoclavicular joint (SCJ) dislocation often requires a massive amount of force to cause it. SCJ dislocation is a rare injury that is classified into anterior and posterior dislocations. Anterior SCJ dislocations account for almost 3% of all shoulder girdle injuries but only 1% of all joint dislocations . A rapidA 35-year-old man presented to our emergency department (ED) with complaints of pain and swelling in the upper portion of his left chest and left hand. A firefighter by profession, he had sustained a fall from approximately a height of 4 meters while on active duty. Almost immediately, he complained of severe pain and swelling. He was brought to the hospital an hour after his fall and was conscious and oriented with no neurovascular deficit. On examination, there was localized tenderness along with a prominent palpable bump just lateral to the sternum on the left side Figure . We suspA radiograph of the left hand revealed a displaced transverse fracture of the shaft of the first metacarpal of the left hand. There was no other osseous deformity seen. After thorough discussion with the relatives, a decision to reduce the SCJ dislocation under sedation was made. Under general anesthesia, the patient was lying supine with a bolster firmly placed between his shoulders. The upper limb was then abducted to 90 degrees with traction applied to it in neutral flexion and direct pressure applied over the sternal end of the clavicle. Successful reduction of the SCJ dislocation was confirmed clinically as well as under fluoroscopy. The first metacarpal shaft fracture was fixed with open reduction and internal fixation using a lag screw (10 mm x 2 mm). The CLW was thoroughly debrided. A polysling was applied to the left upper limb for 4-6 weeks, and the patient was called for fortnightly follow-up. At four-week follow-up, the patient was fine with the maintenance of the reduction in alignment and negligible tenderness at the injury site Figure .The shoulder comprises four joints: the glenohumeral joint, the scapulothoracic joint, the SCJ, and the acromioclavicular joint . The SCJPatients with anterior dislocations of the SCJ clinically present with a painful lump just lateral to the sternum. Utmost precaution needs to be taken to ascertain whether this is a true SCJ dislocation or a fracture of the sternal clavicle. If the patient is under 25 then the possibility of a physeal injury should also be considered . The diaThere has been an ongoing debate about whether anterior dislocation warrants reduction. This is based on the theory that persistent anterior clavicular prominence causes no significant functional difficulties, and that recurrent, and even irreducible, anterior dislocations are usually tolerated without significant sequelae or complications . HoweverThis case report describes a case of an anterior SCJ dislocation with no other osseous injury. SCJ dislocations are rare and should be investigated properly. Though the diagnosis of anterior SCJ dislocation is obvious with its clinical finding, a high degree of suspicion and astute clinical acumen are also needed since it can be missed in the setting of polytrauma. If SCJ dislocation presents within eight to 10 days of injury, it can be reduced successfully. Open reduction should be carried out for unsuccessful cases of relocation or for re-dislocation cases.