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Pettersson, Irina
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Jerez-Hanckes, C., Martínez, I. A., Pettersson, I. & Rybalko, V. (2021). Multiscale Analysis of Myelinated Axons. In: SEMA SIMAI Springer Series: (pp. 17-35). Springer
Öppna denna publikation i ny flik eller fönster >>Multiscale Analysis of Myelinated Axons
2021 (Engelska)Ingår i: SEMA SIMAI Springer Series, Springer , 2021, s. 17-35Kapitel i bok, del av antologi (Refereegranskat)
Abstract [en]

We consider a three-dimensional model for a myelinated neuron, which includes Hodgkin–Huxley ordinary differential equations to represent membrane dynamics at Ranvier nodes (unmyelinated areas). Assuming a periodic microstructure with alternating myelinated and unmyelinated parts, we use homogenization methods to derive a one-dimensional nonlinear cable equation describing the potential propagation along the neuron. Since the resistivity of intracellular and extracellular domains is much smaller than the myelin resistivity, we assume this last one to be a perfect insulator and impose homogeneous Neumann boundary conditions on the myelin boundary. In contrast to the case when the conductivity of the myelin is nonzero, no additional terms appear in the one-dimensional limit equation, and the model geometry affects the limit solution implicitly through an auxiliary cell problem used to compute the effective coefficient. We present numerical examples revealing the forecasted dependence of the effective coefficient on the size of the Ranvier node

Ort, förlag, år, upplaga, sidor
Springer, 2021
Nationell ämneskategori
Annan naturvetenskap
Identifikatorer
urn:nbn:se:hig:diva-39972 (URN)10.1007/978-3-030-62030-1_2 (DOI)2-s2.0-85100969030 (Scopus ID)
Tillgänglig från: 2022-09-22 Skapad: 2022-09-22 Senast uppdaterad: 2022-09-22Bibliografiskt granskad
Jerez-Hanckes, C., Pettersson, I. & Rybalko, V. (2020). Derivation of cable equation by multiscale analysis for a model of myelinated axons. Discrete and continuous dynamical systems. Series B, 25(3), 815-839
Öppna denna publikation i ny flik eller fönster >>Derivation of cable equation by multiscale analysis for a model of myelinated axons
2020 (Engelska)Ingår i: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 25, nr 3, s. 815-839Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order ε, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to ε which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model. 

Ort, förlag, år, upplaga, sidor
American Institute of Mathematical Sciences, 2020
Nyckelord
Cellular electrophysiology, Hodgkin-Huxley model, Homogenization, Multiscale modeling, Nonlinear cable equation
Nationell ämneskategori
Neurovetenskaper
Identifikatorer
urn:nbn:se:hig:diva-31374 (URN)10.3934/dcdsb.2019191 (DOI)000501609800001 ()2-s2.0-85076437746 (Scopus ID)
Anmärkning

Funding:

Swedish Foundation for International Cooperation in Research and Higher Education STINT (IB 2017-7370)

Chile Fondecyt Regular (1171491)

Tillgänglig från: 2020-01-07 Skapad: 2020-01-07 Senast uppdaterad: 2021-03-03Bibliografiskt granskad
Viirman, O. & Pettersson, I. (2019). What to do when there is no formula? Navigating between less and more familiar routines for determining velocity in a calculus task for engineering students.. In: J. Monaghan, E. Nardi and T. Dreyfus (Ed.), Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC: . Paper presented at Calculus in Upper Secondary and Beginning University Mathematics, Kristiansand, Norway, 6-8 August 2019 (pp. 167-170).
Öppna denna publikation i ny flik eller fönster >>What to do when there is no formula? Navigating between less and more familiar routines for determining velocity in a calculus task for engineering students.
2019 (Engelska)Ingår i: Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC / [ed] J. Monaghan, E. Nardi and T. Dreyfus, 2019, s. 167-170Konferensbidrag, Publicerat paper (Refereegranskat)
Nationell ämneskategori
Didaktik
Identifikatorer
urn:nbn:se:hig:diva-30813 (URN)
Konferens
Calculus in Upper Secondary and Beginning University Mathematics, Kristiansand, Norway, 6-8 August 2019
Tillgänglig från: 2019-10-18 Skapad: 2019-10-18 Senast uppdaterad: 2019-10-22Bibliografiskt granskad
Viirman, O., Pettersson, I., Björklund, J. & Boustedt, J. (2018). Programming in mathematics teacher education: A collaborative teaching approach. In: : . Paper presented at INDRUM 2018: Second conference of the International Network for Didactic Research in University Mathematics, University of Agder, Norway, 5-7 April 2018 (pp. 464-465).
Öppna denna publikation i ny flik eller fönster >>Programming in mathematics teacher education: A collaborative teaching approach
2018 (Engelska)Konferensbidrag, Muntlig presentation med publicerat abstract (Refereegranskat)
Nyckelord
novel approaches to teaching, teaching and learning of mathematics in other fields, team teaching, algorithmic thinking, programming.
Nationell ämneskategori
Didaktik
Forskningsämne
Innovativt lärande
Identifikatorer
urn:nbn:se:hig:diva-30812 (URN)
Konferens
INDRUM 2018: Second conference of the International Network for Didactic Research in University Mathematics, University of Agder, Norway, 5-7 April 2018
Tillgänglig från: 2019-10-18 Skapad: 2019-10-18 Senast uppdaterad: 2024-05-21Bibliografiskt granskad
Pettersson, I. & Piatnitski, A. (2018). Stationary convection-diffusion equation in an infinite cylinder. Journal of Differential Equations, 264(7), 4456-4487
Öppna denna publikation i ny flik eller fönster >>Stationary convection-diffusion equation in an infinite cylinder
2018 (Engelska)Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, nr 7, s. 4456-4487Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

Nyckelord
Convection–diffusion equation, Infinite cylinder, Stabilization at infinity, Effective drift
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27148 (URN)10.1016/j.jde.2017.12.015 (DOI)
Tillgänglig från: 2018-06-25 Skapad: 2018-06-25 Senast uppdaterad: 2018-06-25Bibliografiskt granskad
Pettersson, I. (2017). Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary. Differential Equations & Applications, 9(3), 393-412
Öppna denna publikation i ny flik eller fönster >>Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary
2017 (Engelska)Ingår i: Differential Equations & Applications, ISSN 1847-120X, Vol. 9, nr 3, s. 393-412Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

The aim of this paper is to adapt the notion of two-scale convergence in Lp to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients and a p-Laplacian stated in thin cylinders with locally periodic rapidly varying thickness.

Nyckelord
Two-scale convergence, singular measure, homogenization, thin domain with varying thickness, oscillating boundary, dimension reduction, locally periodic operators, p-Laplacian
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27149 (URN)10.7153/dea-2017-09-28 (DOI)
Tillgänglig från: 2018-06-25 Skapad: 2018-06-25 Senast uppdaterad: 2018-06-25Bibliografiskt granskad
Chechkina, A., Pankratova, I. & Pettersson, K. (2015). Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator. Asymptotic Analysis, 93(1-2), 141-160
Öppna denna publikation i ny flik eller fönster >>Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator
2015 (Engelska)Ingår i: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 93, nr 1-2, s. 141-160Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27150 (URN)10.3233/ASY-151291 (DOI)
Tillgänglig från: 2018-06-25 Skapad: 2018-06-25 Senast uppdaterad: 2018-06-25Bibliografiskt granskad
Pankratova, I. & Pettersson, K. (2015). Spectral asymptotics for an elliptic operator in a locally periodic perforated domain. Applicable Analysis, 94(6), 1207-1234
Öppna denna publikation i ny flik eller fönster >>Spectral asymptotics for an elliptic operator in a locally periodic perforated domain
2015 (Engelska)Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, nr 6, s. 1207-1234Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.

Nyckelord
homogenization, spectral problem, localization of eigenfunctions, locally periodic perforated domain, dimension reduction
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27151 (URN)10.1080/00036811.2014.924110 (DOI)
Tillgänglig från: 2018-06-25 Skapad: 2018-06-25 Senast uppdaterad: 2018-06-25Bibliografiskt granskad
Chiadò Piat, V., Pankratova, I. & Piatnitski, A. (2013). Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions. SIAM Journal on Mathematical Analysis, 45(3), 1302-1327
Öppna denna publikation i ny flik eller fönster >>Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions
2013 (Engelska)Ingår i: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 45, nr 3, s. 1302-1327Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of localization of the eigenfunctions. Moreover, the limit behavior of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive estimates for the rate of convergence. 

Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27152 (URN)10.1137/120868724 (DOI)
Tillgänglig från: 2018-06-25 Skapad: 2018-06-25 Senast uppdaterad: 2018-06-25Bibliografiskt granskad
Allaire, G., Pankratova, I. & Piatnitski, A. (2012). Homogenization and concentration for a diffusion equation with large convection in a bounded domain. Journal of Functional Analysis, 262(1), 300-330
Öppna denna publikation i ny flik eller fönster >>Homogenization and concentration for a diffusion equation with large convection in a bounded domain
2012 (Engelska)Ingår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 262, nr 1, s. 300-330Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.

Nyckelord
Homogenization, Convection–diffusion, Localization
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:hig:diva-27154 (URN)10.1016/j.jfa.2011.09.014 (DOI)
Tillgänglig från: 2018-06-21 Skapad: 2018-06-21 Senast uppdaterad: 2018-06-21Bibliografiskt granskad
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