Professor Martin Wechselberger

F07 - Carslaw Building
The University of Sydney

Telephone 9351 3860
Fax 9351 4534

Website Personal web page

Research interests

Martin Wechselberger is a member of the

Research Interests

  • Dynamical systems, ordinary and partial differential equations
  • Geometric singular perturbation theory
  • Dynamics of rhythms and patterns, nonlinear waves and stability
  • Mathematical Physiology

Teaching and supervision

Timetable

International links

New Zealand

(Department of Mathematics, University of Auckland)

Marsden Fund (2016-2018): `Surprisingly slow dynamics in calcium models: where are the slow time scales', V. Kirk, J. Sneyd, H. Osinga, M. Wechselberger.

Marsden Fund (2012-2014): `Applications of multiscale excitable systems to calcium dynamics and neuroscience', V. Kirk, J. Sneyd, H. Osinga, M. Wechselberger.

Marsden Fund (2008-2011): `Designing experiments using mathematics: nonlinear dynamics and calcium oscillations', V. Kirk, J. Sneyd, H. Osinga, M. Wechselberger.


United States

(Mathematical Biosciences Institute, Ohio State University)

Co-organizer current topic workshop (March 22-26, 2011): New Developments in Dynamical Systems Arising from the Biosciences


Selected grants

2013

  • Mathematics of non-communicable diseases: understanding failure of cell signalling; Wechselberger M, Krauskopf B, Sherman A, Bogacz R, Osinga H, Kirk V, Tsaneva-Atanasova K, Randall A, Sneyd J; DVC International/IPDF Grant.

2012

  • Geometric methods in mathematicalphysiology; Wechselberger M; Australian Research Council (ARC)/Future Fellowships (FT).

2011

  • A geometric theory for travelling waves in advection-reaction-diffusion models; Wechselberger M, Pettet G, Jones C; Australian Research Council (ARC)/Discovery Projects (DP).

2010

  • Diagnostic tools for mixed mode oscillations; Wechselberger M, Guckenheimer J; DVC Research/International Visiting Research Fellowship (IVRF).

2008

  • Canards and Rifurcations of Spatid-Temporal Patterns; Wechselberger M, Knobloch E, Knobloch E; DVC Research/International Visiting Research Fellowship (IVRF).

2007

  • Multiple Time Scales Analysis of Physiological Rhythms; Wechselberger M; University of Sydney/Bridging Support.

Selected publications

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Book Chapters

  • Bertram, R., Tabak, J., Teka, W., Vo, T., Wechselberger, M. (2015). Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells. In R. Bertram, J. Tabak, W. Teka, T. Vo, M. Wechselberger, V. Kirk, J. Sneyd (Eds.), Mathematical Analysis of Complex Cellular Activity, (pp. 1-52). Cham: Springer. [More Information]
  • Wechselberger, M., Mitry, J., Rinzel, J. (2013). Canard Theory and Excitability. In Peter E Kloeden, Christian Potzsche (Eds.), Nonautonomous Dynamical Systems in the Life Sciences, (pp. 89-132). Cham, Switzerland: Springer.

Journals

  • Boie, S., Kirk, V., Sneyd, J., Wechselberger, M. (2016). Effects of quasi-steady state reduction on biophysical models with oscillations. Journal of Theoretical Biology, 393, 16-31. [More Information]
  • Roberts, K., Rubin, J., Wechselberger, M. (2015). Averaging, foIded singularities and torus canards: explaining transitions between bursting and spiking in a coupled neuron model. SIAM Journal on Applied Dynamical Systems, 14(4), 1808-1844. [More Information]
  • Vo, T., Wechselberger, M. (2015). Canards of folded saddle-node type I. SIAM Journal On Mathematical Analysis, 47(4), 3235-3283. [More Information]
  • Roberts, A., Widiasih, E., Wechselberger, M., Jones, C. (2015). Mixed mode oscillations in a conceptual climate model. Physica D: Nonlinear Phenomena, 292-293, 70-83. [More Information]
  • De Maesschalck, P., Wechselberger, M. (2015). Neural Excitability and Singular Bifurcations. Journal of Mathematical Neuroscience, 5(16), 1-32. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2015). Numerical computation of an Evans function for travelling waves. Mathematical Biosciences, 266, 36-51. [More Information]
  • Ullah, G., Wei, Y., Dahlem, M., Wechselberger, M., Schiff, S. (2015). The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization. PLoS Computational Biology, 11(8), e1004414-25 pages. [More Information]
  • Vo, T., Tabak, J., Bertram, R., Wechselberger, M. (2014). A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. Journal of Computational Neuroscience, 36(2), 259-278. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2014). Existence of Traveling Wave Solutions for a Model of Tumor Invasion. SIAM Journal on Applied Dynamical Systems, 13(1), 366-396. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2014). Novel solutions for a model of wound healing angiogenesis. Nonlinearity, 27(12), 2975-3003. [More Information]
  • Lin, X., Wechselberger, M. (2014). Transonic evaporation waves in a spherically symmetric nozzle. SIAM Journal On Mathematical Analysis, 46(2), 1472-1504. [More Information]
  • Vo, T., Tabak, J., Bertram, R., Wechselberger, M. (2013). Canard- and Hopf-induced bursting in pituitary cells. Gazette of the Australian Mathematical Society, 40(3), 199-201.
  • Mitry, J., McCarthy, M., Kopell, N., Wechselberger, M. (2013). Excitable Neurons, Firing Threshold Manifolds and Canards. Journal of Mathematical Neuroscience, 3(12), 1-32. [More Information]
  • Vo, T., Bertram, R., Wechselberger, M. (2013). Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting. SIAM Journal on Applied Dynamical Systems, 12(2), 789-830. [More Information]
  • Wechselberger, M. (2012). A Propos de Canards. Transactions of the American Mathematical Society, 364(6), 3289-3309. [More Information]
  • Vo, T., Bertram, R., Wechselberger, M. (2012). Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model. Discrete and Continuous Dynamical Systems - Series A, 32(8), 2879-2912. [More Information]
  • Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H., Wechselberger, M. (2012). Mixed-Mode Oscillations with Multiple Time Scales. SIAM Review, 54(2), 211-288. [More Information]
  • Zhang, W., Kirk, V., Sneyd, J., Wechselberger, M. (2011). Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. Journal of Mathematical Neuroscience, 1(9), 1-22. [More Information]
  • Harvey, E., Kirk, V., Wechselberger, M., Sneyd, J. (2011). Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics. Journal of Nonlinear Science, 21, 639-683. [More Information]
  • Teka, W., Tabak, J., Vo, T., Wechselberger, M., Bertram, R. (2011). The dynamics underlying pseudo-plateau bursting in a pituitary cell model. Journal of Mathematical Neuroscience, 1(12), 1-23. [More Information]
  • Wechselberger, M., Pettet, G. (2010). Folds, canards and shocks in advection-reaction-diffusion models. Nonlinearity, 23, 1949-1969. [More Information]
  • Krupa, M., Wechselberger, M. (2010). Local analysis near a folded saddle-node singularity. Journal of Differential Equations, 248(12), 2841-2888. [More Information]
  • Vo, T., Bertram, R., Tabak, J., Wechselberger, M. (2010). Mixed mode oscillations as a mechanism for pseudo-plateau bursting. Journal of Computational Neuroscience, 28(3), 443-458. [More Information]
  • Harvey, E., Kirk, V., Osinga, H., Sneyd, J., Wechselberger, M. (2010). Understanding anomalous delays in a model of intracellular calcium dynamics. Chaos, 20(4), 045104-1-045104-19. [More Information]
  • Wechselberger, M., Weckesser, W. (2009). Bifurcations of mixed-mode oscillations in a stellate cell model. Physica D: Nonlinear Phenomena, 238(16), 1598-1614. [More Information]
  • Ermentrout, B., Wechselberger, M. (2009). Canards, Clusters, and Synchronization in a Weakley Coupled Interneuron Model. SIAM Journal on Applied Dynamical Systems, 8(1), 253-278. [More Information]
  • Wechselberger, M., Weckesser, W. (2009). Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete and Continuous Dynamical Systems - Series S, 2(4), 829-850. [More Information]
  • Rostein, H., Wechselberger, M., Kopell, N. (2008). Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II Stellate cell model. SIAM Journal on Applied Dynamical Systems, 7(4), 1582-1611. [More Information]
  • Beck, M., Jones, C., Schaeffer, D., Wechselberger, M. (2008). Electrical Waves in a One-Dimensional Model of Cardiac Tissue. SIAM Journal on Applied Dynamical Systems, 7(4), 1558-1581. [More Information]
  • Rubin, J., Wechselberger, M. (2008). The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos, 18(1), 015105-1-015105-12. [More Information]
  • Wechselberger, M. (2007). Canards. Scholarpedia, 2(4), 1356-1-1356-7.
  • Rubin, J., Wechselberger, M. (2007). Giant Squid - Hidden Canard: the 3D Geometry of the Hodgkin Huxley Model. Biological Cybernetics, 97(1), 5-32. [More Information]
  • Guckenheimer, J., Wechselberger, M., Young, L. (2006). Chaotic attractors of relaxation oscillators. Nonlinearity, 19(3), 701-720.
  • Wechselberger, M., Wright, C., Bishop, G., Boulant, J. (2006). Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity. American Journal of Physiology: Regulatory, Integrative and Comparative Physiology, 291(3), R518-R529.
  • Brons, M., Krupa, M., Wechselberger, M. (2006). Mixed mode oscillations due to the generalized canard phenomenon. Fields Institute Communications, 49, 39-63.
  • Wechselberger, M. (2005). Existence and bifurcation of canards in R^3 in the case of a folded node. SIAM Journal on Applied Dynamical Systems, 4(1), 101-139.
  • Best, J., Borisyuk, A., Rubin, J., Terman, D., Wechselberger, M. (2005). The dynamic range of bursting in a model respiratory pacemaker network. SIAM Journal on Applied Dynamical Systems, 4(4), 1107-1139.
  • Szmolyan, P., Wechselberger, M. (2004). Relaxation oscillations in R^3. Journal of Differential Equations, 200(1), 69-104. [More Information]
  • Wechselberger, M. (2002). Extending Melnikov theory to invariant manifolds on non-compact domains. Dynamical Systems-An International Journal, 17(3), 215-233.
  • Szmolyan, P., Wechselberger, M. (2001). Canards in R^3. Journal of Differential Equations, 177(2), 419-453.

Conferences

  • Wechselberger, M. (2005). PoincarĂ© maps for relaxation oscillations in R^3 - invariant manifolds, canards and turning points. EQUADIFF 2003, Hackensack, NJ: World Scientific Publishing.

2016

  • Boie, S., Kirk, V., Sneyd, J., Wechselberger, M. (2016). Effects of quasi-steady state reduction on biophysical models with oscillations. Journal of Theoretical Biology, 393, 16-31. [More Information]

2015

  • Roberts, K., Rubin, J., Wechselberger, M. (2015). Averaging, foIded singularities and torus canards: explaining transitions between bursting and spiking in a coupled neuron model. SIAM Journal on Applied Dynamical Systems, 14(4), 1808-1844. [More Information]
  • Vo, T., Wechselberger, M. (2015). Canards of folded saddle-node type I. SIAM Journal On Mathematical Analysis, 47(4), 3235-3283. [More Information]
  • Bertram, R., Tabak, J., Teka, W., Vo, T., Wechselberger, M. (2015). Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells. In R. Bertram, J. Tabak, W. Teka, T. Vo, M. Wechselberger, V. Kirk, J. Sneyd (Eds.), Mathematical Analysis of Complex Cellular Activity, (pp. 1-52). Cham: Springer. [More Information]
  • Roberts, A., Widiasih, E., Wechselberger, M., Jones, C. (2015). Mixed mode oscillations in a conceptual climate model. Physica D: Nonlinear Phenomena, 292-293, 70-83. [More Information]
  • De Maesschalck, P., Wechselberger, M. (2015). Neural Excitability and Singular Bifurcations. Journal of Mathematical Neuroscience, 5(16), 1-32. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2015). Numerical computation of an Evans function for travelling waves. Mathematical Biosciences, 266, 36-51. [More Information]
  • Ullah, G., Wei, Y., Dahlem, M., Wechselberger, M., Schiff, S. (2015). The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization. PLoS Computational Biology, 11(8), e1004414-25 pages. [More Information]

2014

  • Vo, T., Tabak, J., Bertram, R., Wechselberger, M. (2014). A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. Journal of Computational Neuroscience, 36(2), 259-278. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2014). Existence of Traveling Wave Solutions for a Model of Tumor Invasion. SIAM Journal on Applied Dynamical Systems, 13(1), 366-396. [More Information]
  • Harley, K., van Heijster, P., Marangell, R., Pettet, G., Wechselberger, M. (2014). Novel solutions for a model of wound healing angiogenesis. Nonlinearity, 27(12), 2975-3003. [More Information]
  • Lin, X., Wechselberger, M. (2014). Transonic evaporation waves in a spherically symmetric nozzle. SIAM Journal On Mathematical Analysis, 46(2), 1472-1504. [More Information]

2013

  • Wechselberger, M., Mitry, J., Rinzel, J. (2013). Canard Theory and Excitability. In Peter E Kloeden, Christian Potzsche (Eds.), Nonautonomous Dynamical Systems in the Life Sciences, (pp. 89-132). Cham, Switzerland: Springer.
  • Vo, T., Tabak, J., Bertram, R., Wechselberger, M. (2013). Canard- and Hopf-induced bursting in pituitary cells. Gazette of the Australian Mathematical Society, 40(3), 199-201.
  • Mitry, J., McCarthy, M., Kopell, N., Wechselberger, M. (2013). Excitable Neurons, Firing Threshold Manifolds and Canards. Journal of Mathematical Neuroscience, 3(12), 1-32. [More Information]
  • Vo, T., Bertram, R., Wechselberger, M. (2013). Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting. SIAM Journal on Applied Dynamical Systems, 12(2), 789-830. [More Information]

2012

  • Wechselberger, M. (2012). A Propos de Canards. Transactions of the American Mathematical Society, 364(6), 3289-3309. [More Information]
  • Vo, T., Bertram, R., Wechselberger, M. (2012). Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model. Discrete and Continuous Dynamical Systems - Series A, 32(8), 2879-2912. [More Information]
  • Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H., Wechselberger, M. (2012). Mixed-Mode Oscillations with Multiple Time Scales. SIAM Review, 54(2), 211-288. [More Information]

2011

  • Zhang, W., Kirk, V., Sneyd, J., Wechselberger, M. (2011). Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. Journal of Mathematical Neuroscience, 1(9), 1-22. [More Information]
  • Harvey, E., Kirk, V., Wechselberger, M., Sneyd, J. (2011). Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics. Journal of Nonlinear Science, 21, 639-683. [More Information]
  • Teka, W., Tabak, J., Vo, T., Wechselberger, M., Bertram, R. (2011). The dynamics underlying pseudo-plateau bursting in a pituitary cell model. Journal of Mathematical Neuroscience, 1(12), 1-23. [More Information]

2010

  • Wechselberger, M., Pettet, G. (2010). Folds, canards and shocks in advection-reaction-diffusion models. Nonlinearity, 23, 1949-1969. [More Information]
  • Krupa, M., Wechselberger, M. (2010). Local analysis near a folded saddle-node singularity. Journal of Differential Equations, 248(12), 2841-2888. [More Information]
  • Vo, T., Bertram, R., Tabak, J., Wechselberger, M. (2010). Mixed mode oscillations as a mechanism for pseudo-plateau bursting. Journal of Computational Neuroscience, 28(3), 443-458. [More Information]
  • Harvey, E., Kirk, V., Osinga, H., Sneyd, J., Wechselberger, M. (2010). Understanding anomalous delays in a model of intracellular calcium dynamics. Chaos, 20(4), 045104-1-045104-19. [More Information]

2009

  • Wechselberger, M., Weckesser, W. (2009). Bifurcations of mixed-mode oscillations in a stellate cell model. Physica D: Nonlinear Phenomena, 238(16), 1598-1614. [More Information]
  • Ermentrout, B., Wechselberger, M. (2009). Canards, Clusters, and Synchronization in a Weakley Coupled Interneuron Model. SIAM Journal on Applied Dynamical Systems, 8(1), 253-278. [More Information]
  • Wechselberger, M., Weckesser, W. (2009). Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete and Continuous Dynamical Systems - Series S, 2(4), 829-850. [More Information]

2008

  • Rostein, H., Wechselberger, M., Kopell, N. (2008). Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II Stellate cell model. SIAM Journal on Applied Dynamical Systems, 7(4), 1582-1611. [More Information]
  • Beck, M., Jones, C., Schaeffer, D., Wechselberger, M. (2008). Electrical Waves in a One-Dimensional Model of Cardiac Tissue. SIAM Journal on Applied Dynamical Systems, 7(4), 1558-1581. [More Information]
  • Rubin, J., Wechselberger, M. (2008). The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos, 18(1), 015105-1-015105-12. [More Information]

2007

  • Wechselberger, M. (2007). Canards. Scholarpedia, 2(4), 1356-1-1356-7.
  • Rubin, J., Wechselberger, M. (2007). Giant Squid - Hidden Canard: the 3D Geometry of the Hodgkin Huxley Model. Biological Cybernetics, 97(1), 5-32. [More Information]

2006

  • Guckenheimer, J., Wechselberger, M., Young, L. (2006). Chaotic attractors of relaxation oscillators. Nonlinearity, 19(3), 701-720.
  • Wechselberger, M., Wright, C., Bishop, G., Boulant, J. (2006). Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity. American Journal of Physiology: Regulatory, Integrative and Comparative Physiology, 291(3), R518-R529.
  • Brons, M., Krupa, M., Wechselberger, M. (2006). Mixed mode oscillations due to the generalized canard phenomenon. Fields Institute Communications, 49, 39-63.

2005

  • Wechselberger, M. (2005). Existence and bifurcation of canards in R^3 in the case of a folded node. SIAM Journal on Applied Dynamical Systems, 4(1), 101-139.
  • Wechselberger, M. (2005). PoincarĂ© maps for relaxation oscillations in R^3 - invariant manifolds, canards and turning points. EQUADIFF 2003, Hackensack, NJ: World Scientific Publishing.
  • Best, J., Borisyuk, A., Rubin, J., Terman, D., Wechselberger, M. (2005). The dynamic range of bursting in a model respiratory pacemaker network. SIAM Journal on Applied Dynamical Systems, 4(4), 1107-1139.

2004

  • Szmolyan, P., Wechselberger, M. (2004). Relaxation oscillations in R^3. Journal of Differential Equations, 200(1), 69-104. [More Information]

2002

  • Wechselberger, M. (2002). Extending Melnikov theory to invariant manifolds on non-compact domains. Dynamical Systems-An International Journal, 17(3), 215-233.

2001

  • Szmolyan, P., Wechselberger, M. (2001). Canards in R^3. Journal of Differential Equations, 177(2), 419-453.

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