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  • 1.
    Gyllingberg, Linnéa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Mathematical models of biological interactions2021Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Mathematical models are used to describe and analyse different types of biological interactions.  From self-propelled particle models capturing the collective motion of fish schools to models in mathematical neuroscience describing the interactions between neurons to individual-based models of ecological interactions. A question that arises for all such models is how we scale from one level to another. How do we scale from fish interactions to the movement of the school of fish? How do we scale from neuronal interactions to the functioning of the brain?  How do we scale from animal competition to population dynamics? It is approaches to this question that we study in this thesis for two different systems. 

    In paper I,  we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. The first is based on local interactions of the individuals and approximates the individual-based model well for small dispersal distances. The second approximates the long-range interactions of the individual-based model. Both approximations incorporate the demographic stochasticity from the individual-based model and show that dispersal stabilizes the population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the classical mean field approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Taken together, our results deepen the understanding of spatial population dynamics and introduces new approximate analytical descriptions.

    In paper II,  we propose a model of social burst and glide motion in pairs of fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion.  Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike self-propelled particle models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of from a deterministic interaction.  We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity. 

    List of papers
    1. Finding analytical approximations for discrete, stochastic, individual-based models of ecology
    Open this publication in new window or tab >>Finding analytical approximations for discrete, stochastic, individual-based models of ecology
    2023 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 365Article in journal (Refereed) Published
    Abstract [en]

    Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

    Place, publisher, year, edition, pages
    Elsevier, 2023
    National Category
    Computational Mathematics Other Mathematics Probability Theory and Statistics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-455245 (URN)10.1016/j.mbs.2023.109084 (DOI)001103942100001 ()
    Available from: 2021-10-05 Created: 2021-10-05 Last updated: 2024-02-21Bibliographically approved
    2. Using neuronal models to capture burst and glide motion and leadership in fish
    Open this publication in new window or tab >>Using neuronal models to capture burst and glide motion and leadership in fish
    (English)Manuscript (preprint) (Other academic)
    National Category
    Other Mathematics Computational Mathematics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-455553 (URN)
    Available from: 2021-10-08 Created: 2021-10-08 Last updated: 2021-10-12
  • 2.
    Gyllingberg, Linnéa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    The Art of Modelling Oscillations and Feedback across Biological Scales2024Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers in the field of mathematical biology. All papers aim to advance our understanding of biological systems through the development and application of innovative mathematical models. These models cover a diverse range of biological scales, from the nuclei of unicellular organisms to the collective behaviours of animal populations, showcasing the broad applicability and potential of mathematical approaches in biology. While the first three papers study mathematical models of very different applications and at various scales, all models contribute to the understanding of how oscillations and/or feedback mechanisms on the individual level give rise to complex emergent patterns on the collective level. In Paper I, we propose a mathematical model of basal cognition, inspired by the true slime mould, Physarum polycephalum. The model demonstrates how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner. In Paper II, we propose a model of social burst-and-glide motion in pairs of swimming fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. In Paper III, we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. Paper IV takes a reflective turn, advocating from a philosophical perspective the importance of developing new mathematical models in the face of current scientific challenges.

    List of papers
    1. A minimal model of cognition based on oscillatory and reinforcement processes
    Open this publication in new window or tab >>A minimal model of cognition based on oscillatory and reinforcement processes
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential approach is to start by identifying models of basal cognition, which give an abstract representation of a range organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner. We first show that our model connects resources in an efficient manner when the environment is constant. We then show that in an oscillatory environment our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We show that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. We identify connections between our model and basal cognition in biological systems and slime moulds, in particular, showing how oscillatory and problem-solving properties of these systems are captured by our model.

    National Category
    Other Mathematics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-523413 (URN)
    Available from: 2024-02-18 Created: 2024-02-18 Last updated: 2024-04-15Bibliographically approved
    2. Using neuronal models to capture burst-and-glide motion and leadership in fish
    Open this publication in new window or tab >>Using neuronal models to capture burst-and-glide motion and leadership in fish
    2023 (English)In: Journal of the Royal Society Interface, ISSN 1742-5689, E-ISSN 1742-5662, Vol. 20, no 204Article in journal (Refereed) Published
    Abstract [en]

    While mathematical models, in particular self-propelled particle models, capture many properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the use of intermittent locomotion, often referred to as burst-and-glide, by many species. In this paper, we propose a model of social burst-and-glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We first show that our model can capture the motion of a single fish swimming down a channel. Extending to a two-fish model, where visual stimulus of a neighbour affects the internal burst or glide state of the fish, we observe a rich set of dynamics found in many species. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and tit-for-tat turn taking. Moreover, unlike previous studies where a randomness is required for leadership switching to occur, we show that this can instead be the result of deterministic interactions. We give several empirically testable predictions for how bursting fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.

    Place, publisher, year, edition, pages
    The Royal Society, 2023
    Keywords
    collective behaviour, swimming dynamics, neuronal dynamics, dynamical systems, fish behaviour
    National Category
    Bioinformatics (Computational Biology)
    Identifiers
    urn:nbn:se:uu:diva-508872 (URN)10.1098/rsif.2023.0212 (DOI)001030842300005 ()37464800 (PubMedID)
    Funder
    Knut and Alice Wallenberg Foundation, 102 2013.0072EU, Horizon 2020, 101030688The Research Council of Norway, 262762
    Available from: 2023-08-11 Created: 2023-08-11 Last updated: 2024-02-21Bibliographically approved
    3. Finding analytical approximations for discrete, stochastic, individual-based models of ecology
    Open this publication in new window or tab >>Finding analytical approximations for discrete, stochastic, individual-based models of ecology
    2023 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 365Article in journal (Refereed) Published
    Abstract [en]

    Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

    Place, publisher, year, edition, pages
    Elsevier, 2023
    National Category
    Computational Mathematics Other Mathematics Probability Theory and Statistics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-455245 (URN)10.1016/j.mbs.2023.109084 (DOI)001103942100001 ()
    Available from: 2021-10-05 Created: 2021-10-05 Last updated: 2024-02-21Bibliographically approved
    4. The lost art of mathematical modelling
    Open this publication in new window or tab >>The lost art of mathematical modelling
    2023 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 362, article id 109033Article in journal (Refereed) Published
    Abstract [en]

    We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities - (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data - inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomenon can be modelled in an infinite number of different ways, through the adoption of a pluralistic approach, where we view a system from multiple, different points of view. We explain this pluralistic approach using fish locomotion as a case study and illustrate some of the pitfalls - universalism, creating models of models, etc. - that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling.

    Place, publisher, year, edition, pages
    Elsevier BV, 2023
    Keywords
    Mathematical biology, Hybrid models, Critical complexity, Machine learning, Equation-free approaches
    National Category
    Other Mathematics
    Identifiers
    urn:nbn:se:uu:diva-509274 (URN)10.1016/j.mbs.2023.109033 (DOI)001038884800001 ()37257641 (PubMedID)
    Available from: 2023-08-23 Created: 2023-08-23 Last updated: 2024-02-21Bibliographically approved
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  • 3.
    Gyllingberg, Linnéa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Using neuronal models to capture burst and glide motion and leadership in fishManuscript (preprint) (Other academic)
  • 4.
    Gyllingberg, Linnéa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Birhane, Abeba
    Mozilla Fdn, 2 Harrison St,Suite 175, San Francisco, CA 94105 USA..
    Sumpter, David J. T.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology.
    The lost art of mathematical modelling2023In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 362, article id 109033Article in journal (Refereed)
    Abstract [en]

    We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities - (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data - inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomenon can be modelled in an infinite number of different ways, through the adoption of a pluralistic approach, where we view a system from multiple, different points of view. We explain this pluralistic approach using fish locomotion as a case study and illustrate some of the pitfalls - universalism, creating models of models, etc. - that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling.

    Download full text (pdf)
    fulltext
  • 5.
    Gyllingberg, Linnéa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Sumpter, David J. T.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology.
    Brännström, Åke
    Finding analytical approximations for discrete, stochastic, individual-based models of ecology2023In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 365Article in journal (Refereed)
    Abstract [en]

    Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

    Download full text (pdf)
    fulltext
  • 6.
    Gyllingberg, Linnéa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Szorkovszky, Alex
    Univ Oslo, RITMO Ctr Interdisciplinary Studies Rhythm Time &, Oslo, Norway..
    Sumpter, David J. T.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology.
    Using neuronal models to capture burst-and-glide motion and leadership in fish2023In: Journal of the Royal Society Interface, ISSN 1742-5689, E-ISSN 1742-5662, Vol. 20, no 204Article in journal (Refereed)
    Abstract [en]

    While mathematical models, in particular self-propelled particle models, capture many properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the use of intermittent locomotion, often referred to as burst-and-glide, by many species. In this paper, we propose a model of social burst-and-glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We first show that our model can capture the motion of a single fish swimming down a channel. Extending to a two-fish model, where visual stimulus of a neighbour affects the internal burst or glide state of the fish, we observe a rich set of dynamics found in many species. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and tit-for-tat turn taking. Moreover, unlike previous studies where a randomness is required for leadership switching to occur, we show that this can instead be the result of deterministic interactions. We give several empirically testable predictions for how bursting fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.

    Download full text (pdf)
    fulltext
  • 7.
    Gyllingberg, Linnéa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Tian, Yu
    Sumpter, David J. T.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Artificial Intelligence. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control.
    A minimal model of cognition based on oscillatory and reinforcement processesManuscript (preprint) (Other academic)
    Abstract [en]

    Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential approach is to start by identifying models of basal cognition, which give an abstract representation of a range organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner. We first show that our model connects resources in an efficient manner when the environment is constant. We then show that in an oscillatory environment our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We show that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. We identify connections between our model and basal cognition in biological systems and slime moulds, in particular, showing how oscillatory and problem-solving properties of these systems are captured by our model.

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