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Andrey Shilnikov

Professor    Faculty    ,

Ph.D. Mathematics, University of Nizhny Novgorod, Russia, 1990
M.S. Physics and Mathematics, University of Nizhny Novgorod, Russia, 1984


Dynamical Systems
Mathematical Neuroscience
Bifurcation Theory


I received PhD in Dynamical Systems from University of Nizhny Novgorod (formerly Gorky) in 1990. The Gorky school had pioneered the qualitative theory of dynamical systems and bifurcations. 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 andJ. Discontinuity, Nonlinearity & Complexity.

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.



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

Recent Papers

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]
Xing T, Barrio R and Shilnikov AL. Symbolic quest into homoclinic chaos. Bifurcations and Chaos, 4(8), 2014
Shilnikov LP. Shilnikov AL and Turaev DV, Showcase of Blue Sky Catastrophes, Bifurcations and Chaos, 4(8), 2014 [pdf]
Barri, R. Martinez MA, Serrano S. and Shilnikov AL. Micro-chaotic and macro-chaotic structures in the Hindmarsh-Rose model of bursting neurons. Chaos, 2014.
Jalil S, Allen D, Youker J and Shilnikov A. 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]
Barrio R, Shilnikov A., Shilnikov L. Kneadings, symbolic dynamics, and painting Lorenz chaos. a Tutorial. J. Bifurcations and Chaos, Vol. 22, No. 4, 1230016, 2012 [pdf]
Shilnikov A. 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 DOI 10.1007/s11071-011-0046-y [pdf]
Jalil S., Belykh I and Shilnikov A. Spikes matter in phase-locking of inhibitory bursting networks. Phys Review E 85, 036214, 2012, doi:10.1103/PhysRevE.85.036214 [pdf]
Neiman A. Dierkes K, Lindner B and Shilnikov A. Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells, J. Mathematical Neuroscience, 1:11 2011 doi:10.1186/2190-8567-1-11 [pdf]
Barrio R, Blesa F., Serrano S. and Shilnikov A. Global organization of spiral structures in parametric phase space of dissipative flows, Physics Review E84, 035201R, 2011 doi: 10.1103/PhysRevE.84.035201 [pdf]
Wojcik J., Clewley R, and Shilnikov A., Order parameter for bursting polyrhythms in multifunctional central pattern generators. Physics Review E 83, 056209-6, 2011 DOI: 10.1103/PhysRevE.83.056209 [pdf]
Wojcik J. and Shilnikov A.L. Voltage interval mappings for dynamics transitions in elliptic bursters, Physica D 240, 1164-1180, 2011 [pdf]
Barrio R and Shilnikov A. 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]
Jalil S., Belykh I., and Shilnikov A. Fast reciprocal inhibition can synchronize bursting neurons, Physics Review E 81(4), 045201-4, Rapid Communications, 2010 Virtual Journal of Biological Physics Research: biological networks.19(9), 2010. [pdf]

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, DOI 10.1007/s10827-009-0167-1 [pdf]
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 DOI: 10.1142/S0218127408021634 [pdf]
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 Virtual Journal of Biological Physics Research: biological networks. 16(7), 2008. [pdf]
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 DOI: 10.1103/PhysRevLett.101.078102. Virtual Journal of Biological Physics Research: biological networks, 16(4), 2008. [original_pdf] [published_pdf]
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]
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]
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]
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]
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]
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]
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 DOI: 10.1142/S0218127404010539 [pdf]
Shilnikov A.L. and Rulkov N., Subthreshold oscillations in a map-based neuron model, Physics Letters A 328, 177-184, 2004 [pdf]
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]

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