Andrey Shilnikov

andrey leonidovich shilnikov
  I received PhD in Dynamical Systems from University of Nizhny Novgorod (formerly Gorky) in 1990. The Andronov-Shilnikov school in Gorky had pioneered the qualitative theory of dynamical systems and bifurcations. I am very fortunate to be a member of the famous Gorky school.
I was a Royal Society postdoctoral fellow at DAMTP in Cambridge University (UK) and UC Berkeley. Prior to joining GSU in 2000, I held visiting positions at UC Berkeley, Georgia Institute of Technology, and Cornell University.

I hold a joint appointment at Neuroscience Institute and Department of Mathematics & Statistics. I am a faculty of Center for Nonlinear Science at Gatech, and a member of Center for Behavioral Neuroscience.
I currently serve on Editorial board of J. Mathematical Neuroscience J. Frontiers of Applied Mathematics and J. Discontinuity, Nonlinearity & Complexity.

Here are my complete CV as well research and teaching statements.

My original area of expertise is the theory of applied dynamical systems and global bifurcations. I study dynamics and their origin in diversely phenomenological systems and in exact models from life sciences. Of my special interest is a new emergent cross‐disciplinary field known as mathematical neuroscience. Its scopes include nonlinear models of individual neurons and networks. In‐depth analysis of such systems requires development of advanced mathematical tools paired with sophisticated computations. I derive models and create bifurcation toolkits for studying a stunning array of complex activities such as multistability of individual neurons and polyrhythmic bursting patterns discovered in multifunctional central pattern generators governing vital locomotor behaviors of animals and humans.

Deterministic chaotic dynamics, Lorenz and any strange attractors with underlying homo- and heteroclinic puzzles are always on my mind.

Prospective PhD candidates, MS and BS students interested in Mathematical Neuroscience, Dynamical Systems and Applied Mathematics: contact me. I have research positions funded by NSF, B&B and NI.

I seek outstanding PhD candidates in mathematical/computational neuroscience and/or applied dynamical systems. Apply through Neuroscience Institiute or Mathematics Department graduate programs.

The books that I have co-authored are available in English (1998, 2001), Russian (2003,2009) and Chinese (2011):

Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. Part I . World Sci. 1998
Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. Part II. World Sci. 2001

Шильников Л.П., Шильников А.Л., Тураев Д.В., Чуа Л., Методы качественной теории в нелинейной динамике. Часть 1. 2004
Шильников Л.П., Шильников А.Л., Тураев Д.В., Чуа Л., Методы качественной теории в нелинейной динамике. Часть 2. 2009

俄罗斯数学教材选译 2011
非线性动力学定性理论方法(第二卷) 2011

Recent papers

2017
1. Bondorenko V., and Shilnikov AL. Bursting dynamics in normal and failing hearts, Scientific Reports by Nature, 7, 5927, 2017 doi:10.1038/s41598-017-05198-z [pdf]
2. Lodi M., Shilnikov AL, and Storace M. Design of synthetic central pattern generators producing desired quadruped gaits. IEEE Transactions on Circuits and Systems I: Regular Papers, 2017 [pdf]
3. Pusuluri K., Pikovsky A., and Shilnikov AL. Unraveling the Chaos-land and its organization in the Rabinovich System. in Challenges in Complexity: Dynamics, Patterns,
and Cognition, Springer series “Nonlinear Systems and Complexity” 2017 [pdf]
4. Lodi M., Shilnikov AL, and, Storace M. CEPAGE: a toolbox for Central Pattern Generator analysis. 2017 IEEE International Symposium on Circuits and Systems, 2017 [pdf]
2016
1. Knapper D. Schwabedal J. and Shilnikov AL. Qualitative and Quantitative Stability Analysis of Penta-rhythmic Circuits. Nonlinearity, 2016 [pdf]
2. Shilnikov AL, and Maurer AP. The Art of grid fields: Geometry of neuronal time. J. Frontiers in Neural Circuits. 2016 [pdf]
3. Nagornov R., Osipov G., Komarov M., Pikovsky A., and Shilnikov AL. Mixed mode synchronization and network bursting of neurons with post-inhibitory rebound. J. Communications in Nonlinear Science and Numerical Simulation 36, 175-191, 2016 [pdf]
2015
1. Barrio R., Rodriguez M. and Shilnikov AL. Mechanism of quasi-periodic lag jitter in bursting rhythms by a neuronal network, European Physics Letters 12(3), 38002, 2015 10.1209/0295-5075/112/38002 [pdf]
2. Krishnan G., Filatov G., Shilnikov AL., and Bazhenov M. Electrogenic properties of the Na+/K+ ATPase controls transitions between normal and pathological brain states. J. Neurophysiology, 113: 3356-3374, 2015, doi:10.1152/jn.00460.2014 [pdf]
3. Alacam D. and Shilnikov AL. Making a swim central pattern generator out of latent parabolic bursters. Bifurcations and Chaos 25(7), 1540003, 2015, doi: 10.1142/s0218127415400039 [pdf]
4. Wojcik J. and Shilnikov AL. Voltage interval mappings for an elliptic burster, a referred chapter in "Nonlinear Dynamics: New Directions," Springer. 2015, ISBN 978-3-319-09866-1 [pdf]
5. Xing T., Wojcik J., Zaks M. and Shilnikov AL. Fractal Parameter Space of Lorenz-like Attractors: A Hierarchical Approach. in "Chaos, Information Processing and Paradoxical Games: The legacy of J.S. Nicolis.” World Scientific Publishing, 2015, ISBN 978-981-4602-12-9 [pdf]
2014
1. Shilnikov AL. and Turaev DV. Editorial: Leonid Pavlovich Shilnikov. Bifurcations and Chaos, 4(8), 2014 [pdf] 
2. Afraimovich VS., Gonchenko SV., Lerman LM., Shilnikov AL. and Turaev DV. Scientific heritage of L.P. Shilnikov. Part 1. Regular and Chaotic Dynamics. 19(4), 435-460. 2014 [pdf]
3. Wojcik J., Clewley R., Schwabedal J. and Shilnikov AL. Key bifurcations of bursting polyrhythms in 3-cell central pattern generators. PLoS ONE 9(4): e92918. doi:10.1371/journal.pone.0092918 [pdf]
4. Xing T., Barrio R. and Shilnikov AL. Symbolic quest into homoclinic chaos. Bifurcations and Chaos, 4(8), 2014 [pdf]
5. Schwabedal JTC., Neiman AB. and Shilnikov AL. Robust design of polyrhythmic neural circuits. Phys. Review E, 002700, 2014 [pdf]
6. Shilnikov LP. Shilnikov AL. and Turaev DV., Showcase of Blue Sky Catastrophes, Bifurcations and Chaos, 4(8), 2014 [pdf]
7. Barri, R. Martinez MA, Serrano S. and Shilnikov AL. Micro-chaotic and macro-chaotic structures in the Hindmarsh-Rose model of bursting neurons. Chaos, 24(2):023128, 2014 [pdf]
8. Xing T., Wojcik J., Barrio R. and Shilnikov AL. Symbolic toolkit for chaos exploration, book chapter in "International Conference on Theory and Application in Nonlinear Dynamics" (ICAND 2012). Springer series Understanding complex systems, 2014. ISBN: 978-3-319-02924-5 [pdf]
9. Wojcik J., Clewley R., and Shilnikov AL. The role of duty cycle in three cell central pattern generator. book chapter in "International Conference on Theory and Application in Nonlinear Dynamics" (ICAND 2012). Springer series Understanding complex systems, 2014. ISBN: 978-3-319-02924-5 [pdf]
2013
1. Jalil S., Allen D., Youker J. and Shilnikov AL. Toward robust phase-locking in Melibe swim central pattern generator model. J. Chaos, 23(4), focus issue "Rhythms and Dynamic Transitions in Neurological Disease," 2013 [pdf]
2. Barrio R, F. Blesa, S. Serrano, T. Xing and A. Shilnikov, Homoclinic spirals: theory and numerics. “Progress and Challenges in Dynamical Systems,” Springer Proceedings in Mathematics & Statistics, v. 54, 2013 [pdf]
2012
1. Barrio R., Shilnikov AL., Shilnikov LP. Kneadings, symbolic dynamics, and painting Lorenz chaos. a Tutorial. J. Bifurcations and Chaos, Vol. 22, No. 4, 1230016, 2012 [pdf]
2. Shilnikov AL. Complete dynamical analysis of an interneuron model. Invited referred review. Special Issue: Dynamics in Biology and Medicine. J. Nonlinear Dynamics, 68(3), 305-328, 2012 [pdf]  DOI 10.1007/s11071-011-0046-y
3 Jalil S., Belykh I. and Shilnikov AL. Spikes matter in phase-locking of inhibitory bursting networks. Phys Review E 85, 036214, 2012, doi:10.1103/PhysRevE.85.036214 [pdf] see link
4. Shilnikov AL., Shilnikov LP. and Barrio R., Symbolic dynamics and spiral structures due to the saddle-focus bifurcations, a referred chapter in “Chaos, CNN, Memristors and Beyond”, 2012 [pdf]
2011
1. Neiman A., Dierkes K., Lindner B. and Shilnikov AL. Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells, J. Mathematical Neuroscience, 1:11 2011 [pdf] doi:10.1186/2190-8567-1-11
2. Barrio R, Blesa F., Serrano S. and Shilnikov AL. Global organization of spiral structures in parametric phase space of dissipative flows, Physics Review E84, 035201R, 2011 [pdf] doi: 10.1103/PhysRevE.84.035201
3. Wojcik J., Clewley R, and Shilnikov AL. Order parameter for bursting polyrhythms in multifunctional central pattern generators.  Physics Review E 83, 056209-6, 2011 [pdf] DOI: 10.1103/PhysRevE.83.056209
4. Wojcik J. and Shilnikov AL. Voltage interval mappings for dynamics transitions in elliptic bursters, Physica D 240, 1164-1180, 2011 [pdf] http://dx.doi.org/10.1016/j.physd.2011.04.003
5. Hu X, Youker J., Wojcik J., Clewley R. and Shilnikov AL. Phase and exact models for multifunctional central pattern generators, Proc. the 4th Dynamical Systems and Control Conference, Arlington, VA, Oct 31-Nov 2, 2011 [pdf]
6. Barrio R. and Shilnikov AL. Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model, J Mathematical Neuroscience. 1:6, 2011. doi:10.1186/2190-8567-1-6 [pdf]   Ukrainian translation of the paper by S.Kravchuk
7. Malaschenko T., Shilnikov AL. and Cymbalyuk G. Bistability of bursting and silence regimes in a model of a leech heart interneuron, Physics Review E 84, 041910, 2011 [pdf]
8. Malaschenko T., Shilnikov AL. and Cymbalyuk G. Six Types of Multistability in a Neuronal Model Based on Slow Calcium Current. PLOS ONE 6(7): e21782. doi:10.1371/journal.pone.0021782. 2011. [pdf]
2010
1. Jalil S., Belykh I., and Shilnikov A. Fast reciprocal inhibition can synchronize bursting neurons, Physics Review E 81(4), 045201-4, Rapid Communications, 2010 [pdf] Virtual Journal of Biological Physics Research: biological networks.19(9), 2010.
2. Belykh I., Jalil S., and Shilnikov A. Burst-duration mechanism of in-phase bursting in inhibitory networks. Regular & Chaotic Dynamics, 15(2-3), 148-160, 2010 [pdf]
3. Коломиец M.Л и Шильников А.Л. Методы качественной теории для модели Хиндмарш-Роуз. Нелинейная Динамика, Т. 6, №2, с. 1–30, 2010 [pdf]
2009
1. Шильников Л.П., Шильников А.Л., Тураев Д.В., Чуа Л. Mетоды качественной теории в нелинейной динамикe [pdf]
Russian Edition of Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. Part II.  World Scientific Pub., 2009
2. Channell P., Fuwape I., Neiman A., and Shilnikov A.L., Variability of bursting patterns in a neuronal model in the presence of noise, 2009, J. Computational Neuroscience, 27(3), 527-542, [pdf] DOI 10.1007/s10827-009-0167-1
2008
1. Shilnikov A. L. and Kolomiets M.L., Methods of the qualitative theory for the Hindmarsh-Rose model: a case study. Tutorial. Inter. Journal of Bifurcations and Chaos, 18 (8), 1-27, 2008 [pdf] DOI: 10.1142/S0218127408021634
2. Shilnikov A.L., Gordon R. and Belykh I.V., Polyrhythmic synchronization in bursting network motifs, J. Chaos, 18, 037120, 2008, DOI: 10.1063/1.2959850 [pdf] Virtual Journal of Biological Physics Research: biological networks. 16(7), 2008.
3. Belykh I.V. and Shilnikov, A.L., David vs. Goliath: when weak inhibition synchronizes strongly desynchronizing networks of bursting neurons, Phys. Rev. Letters 101, 078102, 2008 [original_pdf] [published_pdf] DOI: 10.1103/PhysRevLett.101.078102. Virtual Journal of Biological Physics Research: biological networks, 16(4), 2008.
4. Shilnikov L.P. and Shilnikov A., Shilnikov Saddle-Node, Scholarpedia, 2008,3(4):4789.
2007-
1. Channell P., Cymbalyuk G. and Shilnikov A. L., Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Letters 98, 134101, 2007; doi: 10.1103/PhysRevLett.98.134101. Virtual Journal of Biological Physics, 3(7), 2007. [pdf]
2. Channell P., Cymbalyuk, G. and Shilnikov, A. L., Applications of the Poincare mapping technique to analysis of neuronal dynamics, Neurocomputing, 70 (10-12), 2007; doi:10.1016/j.neucom.2006.10.091 [pdf]
3. Shilnikov L.P. and Shilnikov A., Shilnikov Bifurcation, Scholarpedia, 2007, 2(8):1891.
4. Shilnikov A.L. and Turaev D., Blue Sky Catastrophe, Scholarpedia, 2006, 2(8):1889.
2005
1. Shilnikov A.L. and Cymbalyuk G., Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe, Phys Rev Letters, 94, 048101, 2005 [pdf]
2. Shilnikov A.L., Shilnikov L.P. and Turaev D., Blue sky catastrophe in singularly perturbed systems. AMS Moscow Math. J., 5(1), 205-218,2005 [pdf]
3. Shilnikov A.L., Calabrese R. and Cymbalyuk G., Mechanism of bi-stability: tonic spiking and bursting in a neuron model, Phys Review E 71(5), 056214-046221, 2005 [pdf]
4. Chua, L.O, Turaev, D.V. and Shilnikov, A.L., Editorial, Bifurcations and Chaos 15(11), 3509-3534, 2005 [pdf]
5. Mira, C. and Shilnikov, A.L., Slow and fast dynamics generated by non-invertible plane maps, Bifurcations and Chaos 15(11), 3509-3534, 2005 [pdf]
6. Cymbaluyk G. and Shilnikov A.L., Co-existent tonic spiking modes in a leech neuron model, J. Computational Neuroscience 18 (3), 269-282, 2005 [pdf]
7. Shilnikov A.L, Calabrese R. and Cymbalyuk G., How a neuron model can demonstrate coexistence of tonic spiking and bursting? Neurocomputing 65-66, 869-875, 2005 [pdf]
2004-
1. Shilnikov A.L. and Cymbalyuk G., Homoclinic saddle-node orbit bifurcations en a route between tonic spiking and bursting in neuron models, Invited review. Regular & Chaotic Dynamics, 3(9), 281-297, 2004 [pdf]
2. Shilnikov A.L., Shilnikov L.P. and Turaev D., On some mathematical aspects of classical synchronization theory. a Tutorial. Inter. Journal of Bifurcations and Chaos 14(7) 2143-2160, 2004 [pdf] DOI: 10.1142/S0218127404010539
3. Shilnikov A.L. and Rulkov N., Subthreshold oscillations in a map-based neuron model, Physics Letters A 328, 177-184, 2004 [pdf]
4. Shilnikov A.L. and Rulkov N., Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity. Bifurcations and Chaos, 13(11), 3325-3340, 2003 [pdf]