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Pettersson, Irina

Open this publication in new window or tab >>Multiscale Analysis of Myelinated Axons### Jerez-Hanckes, Carlos

### Martínez, Isabel A.

### Pettersson, Irina

### Rybalko, Volodymyr

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##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2021
##### National Category

Other Natural Sciences
##### Identifiers

urn:nbn:se:hig:diva-39972 (URN)10.1007/978-3-030-62030-1_2 (DOI)2-s2.0-85100969030 (Scopus ID)
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Available from: 2022-09-22 Created: 2022-09-22 Last updated: 2022-09-22Bibliographically approved

Universidad Adolfo Ibáñez, Santiago, Chile.

Pontificia Universidad Católica de Chile, Santiago, Chile.

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.

Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine.

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

Open this publication in new window or tab >>Derivation of cable equation by multiscale analysis for a model of myelinated axons### Jerez-Hanckes, C.

### Pettersson, Irina

### Rybalko, V.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 25, no 3, p. 815-839Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Mathematical Sciences, 2020
##### Keywords

Cellular electrophysiology, Hodgkin-Huxley model, Homogenization, Multiscale modeling, Nonlinear cable equation
##### National Category

Neurosciences
##### Identifiers

urn:nbn:se:hig:diva-31374 (URN)10.3934/dcdsb.2019191 (DOI)000501609800001 ()2-s2.0-85076437746 (Scopus ID)
#####

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##### Note

Universidad Adolfo Ibáñez Diagonal Las Torres, Peñalolén, Santiago, 2700, Chile.

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.

B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky Ave., Kharkiv, 61103, Ukraine.

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.

Funding:

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

Chile Fondecyt Regular (1171491)

Available from: 2020-01-07 Created: 2020-01-07 Last updated: 2021-03-03Bibliographically approvedOpen this publication in new window or tab >>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.### Viirman, Olov

### Pettersson, Irina

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); 2019 (English)In: Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC / [ed] J. Monaghan, E. Nardi and T. Dreyfus, 2019, p. 167-170Conference paper, Published paper (Refereed)
##### National Category

Didactics
##### Identifiers

urn:nbn:se:hig:diva-30813 (URN)
##### Conference

Calculus in Upper Secondary and Beginning University Mathematics, Kristiansand, Norway, 6-8 August 2019
#####

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Available from: 2019-10-18 Created: 2019-10-18 Last updated: 2019-10-22Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.

Open this publication in new window or tab >>Programming in mathematics teacher education: A collaborative teaching approach### Viirman, Olov

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.### Pettersson, Irina

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.### Björklund, Johan

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.### Boustedt, Jonas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 2018 (English)Conference paper, Oral presentation with published abstract (Refereed)
##### Keywords

novel approaches to teaching, teaching and learning of mathematics in other fields, team teaching, algorithmic thinking, programming.
##### National Category

Didactics
##### Research subject

Innovative Learning
##### Identifiers

urn:nbn:se:hig:diva-30812 (URN)
##### Conference

INDRUM 2018: Second conference of the International Network for Didactic Research in University Mathematics, University of Agder, Norway, 5-7 April 2018
#####

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Available from: 2019-10-18 Created: 2019-10-18 Last updated: 2020-11-23Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Industrial Development, IT and Land Management, Computer science.

Open this publication in new window or tab >>Stationary convection-diffusion equation in an infinite cylinder### Pettersson, Irina

### Piatnitski, Andrey

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); 2018 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, no 7, p. 4456-4487Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Convection–diffusion equation, Infinite cylinder, Stabilization at infinity, Effective drift
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27148 (URN)10.1016/j.jde.2017.12.015 (DOI)
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Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved

The Arctic University of Norway, UiT, Norway.

The Arctic University of Norway, UiT, Norway; Institute for Information Transmission Problems of Russian Academy of Sciences, Russian Federation.

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.

Open this publication in new window or tab >>Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary### Pettersson, Irina

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Differential Equations & Applications, ISSN 1847-120X, Vol. 9, no 3, p. 393-412Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Two-scale convergence, singular measure, homogenization, thin domain with varying thickness, oscillating boundary, dimension reduction, locally periodic operators, p-Laplacian
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27149 (URN)10.7153/dea-2017-09-28 (DOI)
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Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved

UiT The Arctic University of Norway, Narvik, Norway.

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.

Open this publication in new window or tab >>Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator### Chechkina, Alexandra

### Pankratova, Iryna

### Pettersson, Klas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); 2015 (English)In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 93, no 1-2, p. 141-160Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27150 (URN)10.3233/ASY-151291 (DOI)
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Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved

Lomonosov Moscow State University, Moscow, Russia.

Narvik University College, Narvik, Norway.

Narvik University College, Narvik, Norway.

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 *j*th eigenfunction can be approximated by a rescaled function that is constructed in terms of the *j*th eigenfunction of fourth or second order effective operators with constant coefficients.

Open this publication in new window or tab >>Spectral asymptotics for an elliptic operator in a locally periodic perforated domain### Pankratova, Iryna

### Pettersson, Klas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 2015 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, no 6, p. 1207-1234Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

homogenization, spectral problem, localization of eigenfunctions, locally periodic perforated domain, dimension reduction
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27151 (URN)10.1080/00036811.2014.924110 (DOI)
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Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved

Department of Technology, Narvik University College, Narvik, Norway.

Department of Technology, Narvik University College, Narvik, Norway.

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 *j*th eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the *j*th eigenfunction of a one-dimensional harmonic oscillator operator.

Open this publication in new window or tab >>Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions### Chiadò Piat, V.

### Pankratova, Iryna

### Piatnitski, A.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); 2013 (English)In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 45, no 3, p. 1302-1327Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27152 (URN)10.1137/120868724 (DOI)
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Available from: 2018-06-25 Created: 2018-06-25 Last updated: 2018-06-25Bibliographically approved

Politecnico di Torino, Torino, Italy.

Narvik University College, Narvik, Norway.

Narvik University College, Narvik, Norway; P.N. Lebedev Physical Institute RAS, Moscow, Russian Federation.

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.

Open this publication in new window or tab >>Homogenization and concentration for a diffusion equation with large convection in a bounded domain### Allaire, G.

### Pankratova, Iryna

### Piatnitski, A.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); 2012 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 262, no 1, p. 300-330Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Homogenization, Convection–diffusion, Localization
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hig:diva-27154 (URN)10.1016/j.jfa.2011.09.014 (DOI)
#####

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#####

Available from: 2018-06-21 Created: 2018-06-21 Last updated: 2018-06-21Bibliographically approved

Ecole Polytechnique, Palaiseau Cedex, France.

Narvik University College, Narvik, Norway.

Lebedev Physical Institute RAS, Moscow, Russia.

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.