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Ambrosio B, Aziz-Alaoui MA, Mondal A, Mondal A, Sharma SK, Upadhyay RK. Non-Trivial Dynamics in the FizHugh-Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions Systems. BIOLOGY 2023; 12:918. [PMID: 37508349 PMCID: PMC10376066 DOI: 10.3390/biology12070918] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2023] [Revised: 06/16/2023] [Accepted: 06/19/2023] [Indexed: 07/30/2023]
Abstract
This article focuses on the qualitative analysis of complex dynamics arising in a few mathematical models in neuroscience context. We first discuss the dynamics arising in the three-dimensional FitzHugh-Rinzel (FHR) model and then illustrate those arising in a class of non-homogeneous FitzHugh-Nagumo (Nh-FHN) reaction-diffusion systems. FHR and Nh-FHN models can be used to generate relevant complex dynamics and wave-propagation phenomena in neuroscience context. Such complex dynamics include canards, mixed-mode oscillations (MMOs), Hopf-bifurcations and their spatially extended counterpart. Our article highlights original methods to characterize these complex dynamics and how they emerge in ordinary differential equations and spatially extended models.
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Affiliation(s)
- Benjamin Ambrosio
- UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Normandie University, 76600 Le Havre, France
- The Hudson School of Mathematics, New York, NY 10001, USA
| | - M A Aziz-Alaoui
- UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Normandie University, 76600 Le Havre, France
| | - Argha Mondal
- Department of Mathematics, Sidho-Kanho-Birsha University, Purulia 723104, India
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
| | - Arnab Mondal
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Sanjeev K Sharma
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Ranjit Kumar Upadhyay
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
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Voliotis M, Plain Z, Li XF, McArdle CA, O’Byrne KT, Tsaneva‐Atanasova K. Mathematical models in GnRH research. J Neuroendocrinol 2022; 34:e13085. [PMID: 35080068 PMCID: PMC9285519 DOI: 10.1111/jne.13085] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 08/25/2021] [Revised: 11/18/2021] [Accepted: 12/16/2021] [Indexed: 12/05/2022]
Abstract
Mathematical modelling is an indispensable tool in modern biosciences, enabling quantitative analysis and integration of biological data, transparent formulation of our understanding of complex biological systems, and efficient experimental design based on model predictions. This review article provides an overview of the impact that mathematical models had on GnRH research. Indeed, over the last 20 years mathematical modelling has been used to describe and explore the physiology of the GnRH neuron, the mechanisms underlying GnRH pulsatile secretion, and GnRH signalling to the pituitary. Importantly, these models have contributed to GnRH research via novel hypotheses and predictions regarding the bursting behaviour of the GnRH neuron, the role of kisspeptin neurons in the emergence of pulsatile GnRH dynamics, and the decoding of GnRH signals by biochemical signalling networks. We envisage that with the advent of novel experimental technologies, mathematical modelling will have an even greater role to play in our endeavour to understand the complex spatiotemporal dynamics underlying the reproductive neuroendocrine system.
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Affiliation(s)
- Margaritis Voliotis
- Department of Mathematics and Living Systems InstituteCollege of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK
| | - Zoe Plain
- Department of Mathematics and Living Systems InstituteCollege of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK
| | - Xiao Feng Li
- Department of Women and Children’s HealthSchool of Life Course SciencesKing’s College LondonLondonUK
| | - Craig A. McArdle
- Laboratories for Integrative Neuroscience and EndocrinologySchool of Clinical SciencesUniversity of BristolBristolUK
| | - Kevin T. O’Byrne
- Department of Women and Children’s HealthSchool of Life Course SciencesKing’s College LondonLondonUK
| | - Krasimira Tsaneva‐Atanasova
- Department of Mathematics and Living Systems InstituteCollege of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK
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MacGregor DJ, Leng G. Network and Population Function in Neuroendocrine Systems. SYSTEMS MEDICINE 2021. [DOI: 10.1016/b978-0-12-801238-3.11371-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022] Open
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Clément F, Crépieux P, Yvinec R, Monniaux D. Mathematical modeling approaches of cellular endocrinology within the hypothalamo-pituitary-gonadal axis. Mol Cell Endocrinol 2020; 518:110877. [PMID: 32569857 DOI: 10.1016/j.mce.2020.110877] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Revised: 05/13/2020] [Accepted: 05/19/2020] [Indexed: 01/26/2023]
Abstract
The reproductive neuroendocrine axis, or hypothalamo-pituitary-gonadal (HPG) axis, is a paragon of complex biological system involving numerous cell types, spread over several anatomical levels communicating through entangled endocrine feedback loops. The HPG axis exhibits remarkable dynamic behaviors on multiple time and space scales, which are an inexhaustible source of studies for mathematical and computational biology. In this review, we will describe a variety of modeling approaches of the HPG axis from a cellular endocrinology viewpoint. We will in particular investigate the questions raised by some of the most striking features of the HPG axis: (i) the pulsatile secretion of hypothalamic and pituitary hormones, and its counterpart, the cell signaling induced by frequency-encoded hormonal signals, and (ii) the dual, gametogenic and glandular function of the gonads, which relies on the tight control of the somatic cell populations ensuring the proper maturation and timely release of the germ cells.
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Affiliation(s)
- Frédérique Clément
- Inria, Centre de Recherche Inria Saclay-Île-de-France, Palaiseau, France.
| | - Pascale Crépieux
- INRAE, UMR85, Unité Physiologie de la Reproduction et des Comportements, F-37380, Nouzilly, France; CNRS, UMR7247, F-37380, Nouzilly, France; Université de Tours, F-37041, Tours, France
| | - Romain Yvinec
- INRAE, UMR85, Unité Physiologie de la Reproduction et des Comportements, F-37380, Nouzilly, France; CNRS, UMR7247, F-37380, Nouzilly, France; Université de Tours, F-37041, Tours, France
| | - Danielle Monniaux
- INRAE, UMR85, Unité Physiologie de la Reproduction et des Comportements, F-37380, Nouzilly, France; CNRS, UMR7247, F-37380, Nouzilly, France; Université de Tours, F-37041, Tours, France
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Zavala E, Wedgwood KCA, Voliotis M, Tabak J, Spiga F, Lightman SL, Tsaneva-Atanasova K. Mathematical Modelling of Endocrine Systems. Trends Endocrinol Metab 2019; 30:244-257. [PMID: 30799185 PMCID: PMC6425086 DOI: 10.1016/j.tem.2019.01.008] [Citation(s) in RCA: 40] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/14/2018] [Revised: 01/23/2019] [Accepted: 01/25/2019] [Indexed: 12/12/2022]
Abstract
Hormone rhythms are ubiquitous and essential to sustain normal physiological functions. Combined mathematical modelling and experimental approaches have shown that these rhythms result from regulatory processes occurring at multiple levels of organisation and require continuous dynamic equilibration, particularly in response to stimuli. We review how such an interdisciplinary approach has been successfully applied to unravel complex regulatory mechanisms in the metabolic, stress, and reproductive axes. We discuss how this strategy is likely to be instrumental for making progress in emerging areas such as chronobiology and network physiology. Ultimately, we envisage that the insight provided by mathematical models could lead to novel experimental tools able to continuously adapt parameters to gradual physiological changes and the design of clinical interventions to restore normal endocrine function.
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Affiliation(s)
- Eder Zavala
- Living Systems Institute, University of Exeter, Exeter EX4 4QD, UK; EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QD, UK; Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter EX4 4QD, UK; College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK.
| | - Kyle C A Wedgwood
- Living Systems Institute, University of Exeter, Exeter EX4 4QD, UK; EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QD, UK; Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter EX4 4QD, UK; College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
| | - Margaritis Voliotis
- Living Systems Institute, University of Exeter, Exeter EX4 4QD, UK; EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QD, UK; Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter EX4 4QD, UK; College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
| | - Joël Tabak
- Institute of Biomedical and Clinical Science, College of Medicine and Health, University of Exeter, Exeter EX4 4PS, UK
| | - Francesca Spiga
- Henry Wellcome Laboratories for Integrative Neuroscience and Endocrinology, University of Bristol, Bristol BS1 3NY, UK
| | - Stafford L Lightman
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QD, UK; Henry Wellcome Laboratories for Integrative Neuroscience and Endocrinology, University of Bristol, Bristol BS1 3NY, UK
| | - Krasimira Tsaneva-Atanasova
- Living Systems Institute, University of Exeter, Exeter EX4 4QD, UK; EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QD, UK; Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter EX4 4QD, UK; College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
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Leng G, MacGregor DJ. Models in neuroendocrinology. Math Biosci 2018; 305:29-41. [DOI: 10.1016/j.mbs.2018.07.008] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2018] [Revised: 07/20/2018] [Accepted: 07/24/2018] [Indexed: 12/18/2022]
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Kim LU, D’Orsogna MR, Chou T. Perturbing the Hypothalamic-Pituitary-Adrenal Axis: A Mathematical Model for Interpreting PTSD Assessment Tests. COMPUTATIONAL PSYCHIATRY (CAMBRIDGE, MASS.) 2018; 2:28-49. [PMID: 30090861 PMCID: PMC6067831 DOI: 10.1162/cpsy_a_00013] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/03/2017] [Accepted: 08/15/2017] [Indexed: 12/05/2022]
Abstract
We use a dynamical systems model of the hypothalamic-pituitary-adrenal (HPA) axis to understand the mechanisms underlying clinical protocols used to probe patient stress response. Specifically, we address dexamethasone (DEX) and ACTH challenge tests, which probe pituitary and adrenal gland responses, respectively. We show that some previously observed features and experimental responses can arise from a bistable mathematical model containing two steady-states, rather than relying on specific and permanent parameter changes due to physiological disruption. Moreover, we show that the timing of a perturbation relative to the intrinsic oscillation of the HPA axis can affect challenge test responses. Conventional mechanistic hypotheses supported and refuted by the challenge tests are reexamined by varying parameters in our mathematical model associated with these hypotheses. We show that (a) adrenal hyposensitivity can give rise to the responses seen in ACTH challenge tests and (b) enhanced cortisol-mediated suppression of the pituitary in subjects with PTSD is not necessary to explain the responses observed in DEX stress tests. We propose a new two-stage DEX/external stressor protocol to more clearly distinguish between the conventional hypothesis of enhanced suppression of the pituitary and bistable dynamics hypothesized in our model.
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Affiliation(s)
- Lae Un Kim
- Department of Biomathematics, University of California, Los Angeles, USA
| | | | - Tom Chou
- Department of Biomathematics, University of California, Los Angeles, USA
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Blanco W, Bertram R, Tabak J. The Effects of GABAergic Polarity Changes on Episodic Neural Network Activity in Developing Neural Systems. Front Comput Neurosci 2017; 11:88. [PMID: 29085291 PMCID: PMC5649201 DOI: 10.3389/fncom.2017.00088] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2017] [Accepted: 09/15/2017] [Indexed: 11/23/2022] Open
Abstract
Early in development, neural systems have primarily excitatory coupling, where even GABAergic synapses are excitatory. Many of these systems exhibit spontaneous episodes of activity that have been characterized through both experimental and computational studies. As development progress the neural system goes through many changes, including synaptic remodeling, intrinsic plasticity in the ion channel expression, and a transformation of GABAergic synapses from excitatory to inhibitory. What effect each of these, and other, changes have on the network behavior is hard to know from experimental studies since they all happen in parallel. One advantage of a computational approach is that one has the ability to study developmental changes in isolation. Here, we examine the effects of GABAergic synapse polarity change on the spontaneous activity of both a mean field and a neural network model that has both glutamatergic and GABAergic coupling, representative of a developing neural network. We find some intuitive behavioral changes as the GABAergic neurons go from excitatory to inhibitory, shared by both models, such as a decrease in the duration of episodes. We also find some paradoxical changes in the activity that are only present in the neural network model. In particular, we find that during early development the inter-episode durations become longer on average, while later in development they become shorter. In addressing this unexpected finding, we uncover a priming effect that is particularly important for a small subset of neurons, called the “intermediate neurons.” We characterize these neurons and demonstrate why they are crucial to episode initiation, and why the paradoxical behavioral change result from priming of these neurons. The study illustrates how even arguably the simplest of developmental changes that occurs in neural systems can present non-intuitive behaviors. It also makes predictions about neural network behavioral changes that occur during development that may be observable even in actual neural systems where these changes are convoluted with changes in synaptic connectivity and intrinsic neural plasticity.
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Affiliation(s)
- Wilfredo Blanco
- Department of Computer Science, State University of Rio Grande do Norte, Natal, Brazil.,Laboratory of Memory, Sleep and Dreams, Brain Institute, Federal University of Rio Grande do Norte, Natal, Brazil
| | - Richard Bertram
- Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, United States
| | - Joël Tabak
- Institute of Biomedical and Clinical Science, University of Exeter Medical School, Exeter, United Kingdom
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