Martin Wechselberger

Martin Wechselberger

Professor

Publications

Books
  • Book Cover
    M. Wechselberger, Geometric singular perturbation theory beyond the standard form, Frontiers in Applied Dynamical Systems: Reviews and Tutorials 6, Springer, Cham, 2020.
    DOI: 10.1007/978-3-030-36399-4
Book Chapters
  1. R. Bertram, J. Tabak, W. Teka, T. Vo, M. Wechselberger, V. Kirk, and J. Sneyd, "Mathematical analysis of complex cellular activity", Frontiers in Applied Dynamical Systems: Reviews and Tutorials 1, Springer, Cham, 2015.
    DOI: 10.1007/978-3-319-18114-1
  2. M. Wechselberger, J. Mitry, and J. Rinzel, "Canard theory and excitability", in Nonautonomous dynamical systems in the life sciences, Lecture Notes in Math. 2102, Springer, Cham, 89–132, 2013.
    DOI: 10.1007/978-3-319-03080-7_3
  3. M. Brøns, M. Krupa, and M. Wechselberger, "Mixed mode oscillations due to the generalized canard phenomenon", in Bifurcation theory and spatio-temporal pattern formation, Fields Inst. Commun. 49, Amer. Math. Soc., Providence, RI, 39–63, 2006.
    DOI: 10.1090/fic/049/03
Journal Articles
  1. T. Miller, A. K. Y. Tam, R. Marangell, M. Wechselberger, and B. H. Bradshaw-Hajek, "Shock selection in reaction–diffusion equations with partially negative diffusivity using nonlinear regularisation", Phys. D 474, Paper No. 134561, 2025.
    DOI: 10.1016/j.physd.2025.134561
  2. B. H. Bradshaw-Hajek, I. Lizarraga, R. Marangell, and M. Wechselberger, "A geometric singular perturbation analysis of shock selection rules in composite regularized reaction-nonlinear diffusion models", SIAM J. Appl. Dyn. Syst. 23(3), 2099–2137, 2024.
    DOI: 10.1137/23M1591803
  3. T. Miller, A. K. Y. Tam, R. Marangell, M. Wechselberger, and B. H. Bradshaw-Hajek, "Analytic shock-fronted solutions to a reaction–diffusion equation with negative diffusivity", Stud. Appl. Math. 153(1), Paper No. e12685, 2024.
    DOI: 10.1111/sapm.12685
  4. D. Abelson, et al., "Mucus clears from the trachea in a helix: a new twist to understanding airway diseases", Thorax 79(7), 607–614, 2024.
    DOI: 10.1136/thorax-2023-221052
  5. S. Jelbart, K. U. Kristiansen, P. Szmolyan, and M. Wechselberger, "Singularly perturbed oscillators with exponential nonlinearities", J. Dynam. Differential Equations 34(3), 1823–1875, 2022.
    DOI: 10.1007/s10884-021-10041-1
  6. E. W. Hester, G. M. Vasil, and M. Wechselberger, "On the stability of shocks in isothermal black hole accretion discs", Mon. Not. R. Astron. Soc. 512(4), 5771–5781, 2022.
    DOI: 10.1093/mnras/stac731
  7. S. Jelbart, N. Pages, V. Kirk, J. Sneyd, and M. Wechselberger, "Process-oriented geometric singular perturbation theory and calcium dynamics", SIAM J. Appl. Dyn. Syst. 21(2), 982–1029, 2022.
    DOI: 10.1137/21M1412402
  8. S. Jelbart, K. U. Kristiansen, and M. Wechselberger, "Singularly perturbed boundary-equilibrium bifurcations", Nonlinearity 34(11), 7371–7414, 2021.
    DOI: 10.1088/1361-6544/ac23b8
  9. I. Lizarraga, B. Rink, and M. Wechselberger, "Multiple timescales and the parametrisation method in geometric singular perturbation theory", Nonlinearity 34(6), 4163–4201, 2021.
    DOI: 10.1088/1361-6544/ac04bf
  10. S. Jelbart, K. U. Kristiansen, and M. Wechselberger, "Singularly perturbed boundary-focus bifurcations", J. Differential Equations 296, 412–492, 2021.
    DOI: 10.1016/j.jde.2021.06.008
  11. Y. Li, P. van Heijster, M. J. Simpson, and M. Wechselberger, "Shock-fronted travelling waves in a reaction–diffusion model with nonlinear forward–backward–forward diffusion", Phys. D 423, Paper No. 132916, 2021.
    DOI: 10.1016/j.physd.2021.132916
  12. D. Avitabile, M. Desroches, R. Veltz, and M. Wechselberger, "Local theory for spatio-temporal canards and delayed bifurcations", SIAM J. Math. Anal. 52(6), 5703–5747, 2020.
    DOI: 10.1137/19M1306610
  13. I. Lizarraga, R. Marangell, and M. Wechselberger, "Slow unfoldings of contact singularities in singularly perturbed systems beyond the standard form", J. Nonlinear Sci. 30(6), 3161–3198, 2020.
    DOI: 10.1007/s00332-020-09647-4
  14. K. E. Harley, et al., "(In)stability of travelling waves in a model of haptotaxis", SIAM J. Appl. Math. 80(4), 1629–1653, 2020.
    DOI: 10.1137/19M1259705
  15. S. Jelbart and M. Wechselberger, "Two-stroke relaxation oscillators", Nonlinearity 33(5), 2364–2408, 2020.
    DOI: 10.1088/1361-6544/ab6a77
  16. I. Lizarraga and M. Wechselberger, "Computational singular perturbation method for nonstandard slow-fast systems", SIAM J. Appl. Dyn. Syst. 19(2), 994–1028, 2020.
    DOI: 10.1137/19M1242677
  17. P. Carter, E. Knobloch, and M. Wechselberger, "Transonic canards and stellar wind", Nonlinearity 30(3), 1006–1033, 2017.
    DOI: 10.1088/1361-6544/aa5743
  18. J. Mitry and M. Wechselberger, "Folded saddles and faux canards", SIAM J. Appl. Dyn. Syst. 16(1), 546–596, 2017.
    DOI: 10.1137/15M1045065
  19. M. G. Hayes, T. J. Kaper, P. Szmolyan, and M. Wechselberger, "Geometric desingularization of degenerate singularities in the presence of fast rotation", Indag. Math. (N.S.) 27(5), 1184–1203, 2016.
    DOI: 10.1016/j.indag.2015.11.005
  20. S. Boie, V. Kirk, J. Sneyd, and M. Wechselberger, "Effects of quasi-steady-state reduction on biophysical models with oscillations", J. Theoret. Biol. 393, 16–31, 2016.
    DOI: 10.1016/j.jtbi.2015.12.011
  21. G. Ullah, et al., "The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization", PLoS Comput. Biol. 11(8), Paper No. e1004414, 2015.
    DOI: 10.1371/journal.pcbi.1004414
  22. K. Roberts, J. E. Rubin, and M. Wechselberger, "Averaging, folded singularities, and torus canards: explaining transitions between bursting and spiking in a coupled neuron model", SIAM J. Appl. Dyn. Syst. 14(4), 1808–1844, 2015.
    DOI: 10.1137/140981770
  23. T. Vo and M. Wechselberger, "Canards of folded saddle-node type I", SIAM J. Math. Anal. 47(4), 3235–3283, 2015.
    DOI: 10.1137/140965818
  24. P. De Maesschalck and M. Wechselberger, "Neural excitability and singular bifurcations", J. Math. Neurosci. 5, Art. 16, 2015.
    DOI: 10.1186/s13408-015-0029-2
  25. A. Roberts, E. Widiasih, M. Wechselberger, and C. K. R. T. Jones, "Mixed mode oscillations in a conceptual climate model", Phys. D 292/293, 70–83, 2015.
    DOI: 10.1016/j.physd.2014.11.003
  26. K. Harley, et al., "Novel solutions for a model of wound healing angiogenesis", Nonlinearity 27(12), 2975–3003, 2014.
    DOI: 10.1088/0951-7715/27/12/2975
  27. X. Lin and M. Wechselberger, "Transonic evaporation waves in a spherically symmetric nozzle", SIAM J. Math. Anal. 46(2), 1472–1504, 2014.
    DOI: 10.1137/120875363
  28. T. Vo, J. Tabak, R. Bertram, and M. Wechselberger, "A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells", J. Comput. Neurosci. 36(2), 259–278, 2014.
    DOI: 10.1007/s10827-013-0470-8
  29. K. Harley, et al., "Existence of traveling wave solutions for a model of tumor invasion", SIAM J. Appl. Dyn. Syst. 13(1), 366–396, 2014.
    DOI: 10.1137/130923129
  30. J. Mitry, M. McCarthy, N. Kopell, and M. Wechselberger, "Excitable neurons, firing threshold manifolds and canards", J. Math. Neurosci. 3, Art. 12, 32, 2013.
    DOI: 10.1186/2190-8567-3-12
  31. T. Vo, R. Bertram, and M. Wechselberger, "Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting", SIAM J. Appl. Dyn. Syst. 12(2), 789–830, 2013.
    DOI: 10.1137/120892842
  32. M. Desroches, et al., "Mixed-mode oscillations with multiple time scales", SIAM Rev. 54(2), 211–288, 2012.
    DOI: 10.1137/100791233
  33. T. Vo, R. Bertram, and M. Wechselberger, "Bifurcations of canard-induced mixed mode oscillations in a pituitary lactotroph model", Discrete Contin. Dyn. Syst. 32(8), 2879–2912, 2012.
    DOI: 10.3934/dcds.2012.32.2879
  34. M. Wechselberger, "À propos de canards (Apropos canards)", Trans. Amer. Math. Soc. 364(6), 3289–3309, 2012.
    DOI: 10.1090/S0002-9947-2012-05575-9
  35. E. P. Harvey, V. Kirk, M. Wechselberger, and J. Sneyd, "Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics", J. Nonlinear Sci. 21(5), 639–683, 2011.
    DOI: 10.1007/s00332-011-9096-z
  36. E. Harvey, et al., "Understanding anomalous delays in a model of intracellular calcium dynamics", Chaos 20(4), 045104, 19, 2010.
    DOI: 10.1063/1.3523264
  37. M. Wechselberger and G. J. Pettet, "Folds, canards and shocks in advection-reaction-diffusion models", Nonlinearity 23(8), 1949–1969, 2010.
    DOI: 10.1088/0951-7715/23/8/008
  38. T. Vo, R. Bertram, J. Tabak, and M. Wechselberger, "Mixed mode oscillations as a mechanism for pseudo-plateau bursting", J. Comput. Neurosci. 28(3), 443–458, 2010.
    DOI: 10.1007/s10827-010-0226-7
  39. M. Krupa and M. Wechselberger, "Local analysis near a folded saddle-node singularity", J. Differential Equations 248(12), 2841–2888, 2010.
    DOI: 10.1016/j.jde.2010.02.006
  40. M. Wechselberger and W. Weckesser, "Homoclinic clusters and chaos associated with a folded node in a stellate cell model", Discrete Contin. Dyn. Syst. Ser. S 2(4), 829–850, 2009.
    DOI: 10.3934/dcdss.2009.2.829
  41. B. Ermentrout and M. Wechselberger, "Canards, clusters, and synchronization in a weakly coupled interneuron model", SIAM J. Appl. Dyn. Syst. 8(1), 253–278, 2009.
    DOI: 10.1137/080724010
  42. H. G. Rotstein, M. Wechselberger, and N. Kopell, "Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model", SIAM J. Appl. Dyn. Syst. 7(4), 1582–1611, 2008.
    DOI: 10.1137/070699093
  43. M. Beck, C. K. R. T. Jones, D. Schaeffer, and M. Wechselberger, "Electrical waves in a one-dimensional model of cardiac tissue", SIAM J. Appl. Dyn. Syst. 7(4), 1558–1581, 2008.
    DOI: 10.1137/070709980
  44. J. Rubin and M. Wechselberger, "The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales", Chaos 18(1), 015105, 12, 2008.
    DOI: 10.1063/1.2789564
  45. J. Rubin and M. Wechselberger, "Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model", Biol. Cybernet. 97(1), 5–32, 2007.
    DOI: 10.1007/s00422-007-0153-5
  46. J. Guckenheimer, M. Wechselberger, and L.-S. Young, "Chaotic attractors of relaxation oscillators", Nonlinearity 19(3), 701–720, 2006.
    DOI: 10.1088/0951-7715/19/3/009
  47. M. Wechselberger, et al., "Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity", Amer. J. Physiol. Regul. Integr. Comp. Physiol. 291(3), R518–R529, 2006.
    DOI: 10.1152/ajpregu.00039.2006
  48. J. Best, et al., "The dynamic range of bursting in a model respiratory pacemaker network", SIAM J. Appl. Dyn. Syst. 4(4), 1107–1139, 2005.
    DOI: 10.1137/050625540
  49. M. Wechselberger, "Existence and bifurcation of canards in R3 in the case of a folded node", SIAM J. Appl. Dyn. Syst. 4(1), 101–139, 2005.
    DOI: 10.1137/030601995
  50. P. Szmolyan and M. Wechselberger, "Relaxation oscillations in R3", J. Differential Equations 200(1), 69–104, 2004.
    DOI: 10.1016/j.jde.2003.09.010
  51. M. Wechselberger, "Extending Melnikov theory to invariant manifolds on non-compact domains", Dyn. Syst. 17(3), 215–233, 2002.
    DOI: 10.1080/14689360210136901
  52. P. Szmolyan and M. Wechselberger, "Canards in R3", J. Differential Equations 177(2), 419–453, 2001.
    DOI: 10.1006/jdeq.2001.4001
Editorials
  1. T. J. Kaper, B. Krauskopf, H. M. Osinga, and M. Wechselberger, "Preface: New developments in dynamical systems arising from the biosciences", Discrete Contin. Dyn. Syst. 32(8), i–iii, 2012.
    DOI: 10.3934/dcds.2012.32.8i
  2. P. de Maesschalck, F. Dumortier, and M. Wechselberger, "Preface: Special issue on bifurcation delay", Discrete Contin. Dyn. Syst. Ser. S 2(4), i–iii, 2009.
    DOI: 10.3934/dcdss.2009.2.4i