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 (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

  • 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

  • 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]
  • 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, No. 12-(23pp). [More Information]
  • Wechselberger, M., Pettet, G. (2010). Folds, canards and shocks in advection-reaction-diffusion models. Nonlinearity, 23, 1949-1969.
  • 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: an interdisciplinary journal of nonlinear science, 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.
  • 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.
  • 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.
  • 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: an interdisciplinary journal of nonlinear science, 18(1), 015105-1-015105-12.
  • 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 (Print), 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.

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]
  • 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, No. 12-(23pp). [More Information]

2010

  • Wechselberger, M., Pettet, G. (2010). Folds, canards and shocks in advection-reaction-diffusion models. Nonlinearity, 23, 1949-1969.
  • 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: an interdisciplinary journal of nonlinear science, 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.
  • 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.
  • 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.

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: an interdisciplinary journal of nonlinear science, 18(1), 015105-1-015105-12.

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 (Print), 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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