I received my 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 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 and J. Discontinuity, Nonlinearity & Complexity.
Here is my complete CV for reviews.
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.
This research is currently funded through DMS-1009591 Multistability and bifurcations for polyrhythmic Central Pattern Generators from NSF,
as well as B&B grants, and a grant 14.740.11.0919 Rhythm formation mechanisms in neuronal networks in application to adaptive bio-robotics from Russian Education and Science Ministry.
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 and B&B.
The books that I have co-authored
are available in English, Russian and Chinese:
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. Appendix C [pdf]
Russian edition of Part I. R&C D, 2004 [pdf]
Russian edition of Part II. R&C D, 2009
Chinese Edition of Methods of Qualitative Theory in Nonlinear Dynamics. Part I, 2011 俄罗斯数学教材选译
Chinese Edition of Methods of Qualitative Theory in Nonlinear Dynamics. Part II, 2011 非线性动力学定性理论方法(第二卷)
Xing T, Wojcik J, Barrio R and Shilnikov A. Symbolic toolkit for chaos exploration. in "Theory and Applications in Nonlinear Dynamics," Springer, 2013 [pdf]
R. Barrio, F. Blesa, S. Serrano, T. Xing and A. Shilnikov, Homoclinic spirals: theory and numeric. “Dynamical Systems: 100 years after Poincaré,” Lecture Notes in Mathematics and Statistics, Springer, 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 (24 pages) 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 [pdf] DOI 10.1007/s11071-011-0046-y
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] see link
Shilnikov LP. Shilnikov AL and Turaev DV, Showcase of Blue Sky Catastrophes, a referred chapter in "Nonlinear Dynamics: New Directions," Springer Series "Nonlinear Physical Science," 2012 [pdf]
Shilnikov A., Shilnikov L. 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]
Wojcik J. and Shilnikov A.L. Voltage interval mappings for an elliptic burster, a referred chapter in in "Nonlinear Dynamics: New Directions," Springer Series "Nonlinear Physical Science," 2012 [pdf]
Shilnikov AL and Shilnikov LP, Torus bifurcations in a slow-fast system with a saddle-focus, to be submitted, 2012
Shilnikov AL, Shilnikov LP and Turaev DV, Homoclinic Bifurcations, Scholarpedia, submitted, 2012
Shilnikov AL, Shilnikov LP and Turaev DV, Homoclinic Orbits, Scholarpedia, submitted, 2012
Shilnikov AL, Routes to Bustling, Scholarpedia, submitted, 2012
Shilnikov AL, Multistability and Polyrhythmicity in neurodynamics, Scholarpedia, submitted, 2012
Neiman A. Dierkes K, Lindner B and Shilnikov A. Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model od sensory hair cells, J. Mathematical Neuroscience, 1:11 2011 [pdf] doi:10.1186/2190-8567-1-11
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 [pdf] doi: 10.1103/PhysRevE.84.035201
Wojcik J., Clewley R, and Shilnikov A., Order parameter for bursting polyrhythms in multifunctional central pattern generators. Physics Review E 83, 056209-6, 2011 [pdf] DOI: 10.1103/PhysRevE.83.056209
Wojcik J. and Shilnikov A.L. 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
Hu X, Youker J., Wojcik J, Clewley R and Shilnikov A, 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]
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]
Ukrainian translation of the paper by S.Kravchuk
Malaschenko T, Shilnikov A and Cymbalyuk G. Bistability of bursting and silence regimes in a model of a leech heart interneuron, Physics Review E 84, 041910, 2011 [pdf]
Malaschenko T, Shilnikov A 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]
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.
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]
Коломиец M.Л и Шильников А.Л. Методы качественной теории для модели Хиндмарш-Роуз. Нелинейная Динамика, Т. 6, №2, с. 1–30, 2010 [pdf]
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.
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
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
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.
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.
Shilnikov L.P. and Shilnikov A., Shilnikov Saddle-Node, Scholarpedia, 2008,3(4):4789.
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 [pdf] [gzip.ps]. Virtual Journal of Biological Physics, 3(7), 2007.
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]
Shilnikov L.P. and Shilnikov A., Shilnikov Bifurcation, Scholarpedia, 2007, 2(8):1891.
Shilnikov A.L. and Turaev D., Blue Sky Catastrophe, Scholarpedia, 2006,
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
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]
Chua, L.O, Turaev, D.V. and Shilnikov, A.L., Editorial, Bifurcations and Chaos 15(11), 3509-3534,
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]
Cymbaluyk G. and Shilnikov A.L., Co-existent tonic spiking modes in a
leech neuron model, J. Computational Neuroscience 18 (3), 269-282, 2005
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]
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,
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]
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]