# Alexey Kuznetsov

Associate Professor, Mathematical Sciences

## Education

| |

6/96 – 7/99 | PhD equivalent in Physics and Mathematics |

6/94 – 5/96 | M.S. in Physics with Honors |

6/90 – 5/94 | B.S. in Physics |

## Teaching Assignments

**Current**

**Past**

- Fall 12 Ordinary Differential Equations
- Fall 12 Analytic Geometry and Calculus II
- Spring 12 Ordinary Differential Equations
- Spring 12 Multidimensional Mathematics
- Fall 11 Ordinary Differential Equations
- Fall 11 Calculus For Technology II
- Spring 11 Analytic Geometry and Calculus II
- Spring 11 Ordinary Differential Equations
- Fall 10 Qualitative Theory of Differential Equations
- Spring 2010 Ordinary Differential Equations
- Spring 2010 Calculus for Technology II
- Fall 2009 Qualitative Theory of Differential Equations.
- Fall 2009 Ordinary Differential Equations.
- Spring 2009 Multidamentional Mathematics
- Spring 2009 Calculus for Technology II
- Fall 2008 Qualitative Theory of Differential Equations.
- Fall 2008 Calculus for Technology I
- Spring 2008 Calculus for Life Sciences I
- Spring 2008 Calculus for Technology II
- Fall 2007 Qualitative Theory of Differential Equations.
- Fall 2007 Calculus for Technology I.
- Spring 2007 Calculus for Technology I.
- Fall 2006 Calculus for Technology I.
- Fall 2006 Qualitative Theory of Differential Equations.
- Spring 2006 Calculus for Technology I.
- Fall 2005 Calculus for Technology I.
- Fall 2004 Linear Algebra CAS MA242 (at Boston University
- Fall 2003 Calculus I CAS MA123 (at Boston University)

## Current Research

My central professional goal is to establish interdisciplinary research that combines experimental and theoretical biology with mathematical and statistical modeling. I am an applied mathematician by training. My dissertation addressed phenomena of synchronization and pattern formation in active, oscillatory networks using qualitative theory of differential equations and computational modeling. In addition to mathematics, I learned biophysics and molecular biology during my postdoctoral training. This cross-disciplinary background has proven to be a significant advantage in seeking explanations to biological problems via mathematical and computational modeling. In the future, I want to connect different levels of biological research, from molecular and cellular to system and behavioral, focusing on medical and pharmaceutical problems and further developing my experimental and medical collaborations.Ongiong projects

### Synaptic integration and firing patterns of the midbrain dopaminergic neuron

One of my main research projects is study of firing properties of the midbrain dopaminergic (DA) neuron and signal processing performed by this cell. The work has been initiated in collaboration with an experimentalist Charles Wilson (UTSA, Biology Department) and Nancy Kopell (BU, Mathematics Department) (Kuznetsov et al., J Neurophys 2006). The central dopamine system has been suggested to be involved with many behavioral and cognitive tasks. Dopamine neurotransmission has been found responsible for addictive behavior and is impaired in psychiatric disorders. However, a combination of experimental and modeling studies has not given a consistent picture of the DA neuron input-output relation (see Kuznetsov et al., Scholarpedia 2007 for discussion). Therefore, the role of the dopaminergic neuron and the role of dopamine itself are topics for active investigations for neuroscientists, pharmacists and physicians. More about this project### Oscillatory regulatory networks: synchrony and pattern formation

Another major field of my research interest is self-sustained oscillatory dynamics of protein expression. Most well known robust oscillatory circuits in molecular biology are the circadian clock and the cell cycle engine. Recently, other examples of such circuits have been discovered in many other regulatory processes, such as apoptosis, metabolism and morphogenesis. A question addressed in many recent studies is how specific are these oscillatory circuits and their elements. Experimentally this problem is investigated in the Applied Biodynamics Laboratory headed by James Collins at the Biomedical Engineering Department, BU. In close collaboration with an experimentalist Mads Kaern from the above laboratory, I have constructed a robust, hysteresis-based genetic relaxation oscillator and provided a theoretical analysis of the conditions necessary for single-cell oscillations (Kuznetsov et al., SIAM Appl. Math. 2004). Based on my analysis, I have proposed a modification in the architecture of the gene regulatory network that has substantially increased robustness of the oscillations. More about this project## Select Publications

### Papers in reviewed journals

- A.Y. Kuznetsova, A.S. Kuznetsov, R.Deth, Working memory modulation by D4 dopamine receptor-mediated phospholipid methylation. submitted to J. Comp. Neuroscience. 2008.
- A.S. Kuznetsov, L. Rubchinsky, N. Kopell, C. Wilson, Models of Midbrain Dopaminergic Neurons. Scholarpedia, pp.23551, 2007.
- L. Rubchinsky, A. Kuznetsov, V. Wheelock, K. Sigvardt, Tremor. Scholarpedia, pp. 22658, 2007.
- A.S. Kuznetsov, C. Wilson, N. Kopell, Transient high-frequency firing in a coupled-oscillator model of the mesencephalic dopaminergic neuron. J. Neurophysiology. 95:932-947, 2006.
- A.S. Kuznetsov, G.L. Forty, V.D. Shalfeev, Analysis of Pattern Formation in a Chain of Coupled Bistable Elements. Int. J. Bifurcation and Chaos 16(3):741-747, 2006.
- A.S. Kuznetsov, V. D. Shalfeev, L. S. Tsimring. Regularization of dynamics in an ensemble of nondiffusively coupled chaotic elements. Phys. Rev. E 72, p. 046209 2005.
- A.S. Kuznetsov, M. Kaern, N. Kopell, Synchrony in a Population of Hysteresis-based Genetic Oscillators. SIAM Applied Math., V. 65, N 2, pp. 392-425, 2004.
- A.S. Kuznetsov, J. Kurths, Stable Heteroclinic Cycles for ensembles of chaotic oscillators, Phys. Rev. E 66, p. 026201, 2002.
- Kozlov A.K., Sushchik M.M., Molkov Ya.I., Kuznetsov A.S., Bistable phase synchronization and chaos in a system of coupled Van der Pol-Duffing oscillators. Int. J. Bifurcation and Chaos, V. 9, N 12, pp. 2271-2277, 1999.
- A.S. Kuznetsov, V.D. Shalfeev, Analysis of regularization processes in ensemble of coupled chaotic elements. Radiophysics and Quantum Electronics, V. 41. N 12, pp. 1056-1060, 1998.
- V.D. Shalfeev, A.S. Kuznetsov, Controlling pattern formation in a CNN of Chua's circuits. Int. J. Bifurcation & Chaos, V. 6, N 11, pp. 2127-2144, 1996.
- A.S.Kuznetsov, V.D.Shalfeev, Structure formation in the array of coupled active elements. Radiophysics and Quantum Electronics, V. 38, N 3-4, pp. 110-115, 1995.
- Kozlov A.K., Sushchik M.M., Molkov Ya.I., Kuznetsov A.S., Phase synchronization bistability and chaos in system of two identical Van der Pol-Duffing oscillators. Applied Nonlinear Dynamics. V. 7, N 1, pp. 68-80, (in Russian with abstract in English), 1999.
- Molkov Ya.I., Sushchik M.M., Kuznetsov A.S., Kozlov A.K., Zakharov D.G., Dynamical model of locomotor-like movements of human evoked by muscle vibration. Applied Nonlinear Dynamics. V. 7, N 2-3, pp. 107-121 (in Russian with abstract in English), 1999.
- A.S.Kuznetsov, Dynamics of modified Chua's circuit. Bulletin of University of Nizhny Novgorod. Radiophysica, N 1, pp. 135-144 (in Russian), 1998.
- Molkov Ya.I., Sushchik M.M., Kuznetsov A.S., Kozlov A.K., Zakharov D.G., A dynamical model of locomotor-like movements evoked by muscle vibration in humans. Bulletin of University of Nizhny Novgorod. Radiophysics, N 1, pp. 63-88 (in Russian), 1998.
- Kozlov A.K., Sushchik M.M., Molkov Ya.I., Kuznetsov A.S., Symmetry breaking, multistability and chaos in the system of two coupled identical Van der Pol-Duffing oscillators. Bulletin of University of Nizhny Novgorod. Radiophysics, N 1, pp. 89-104 (in Russian), 1998.
- A.S.Kuznetsov, V.D.Shalfeev, Structure formation and regularization in the system of globally coupled oscillators. Bulletin of University of Nizhny Novgorod. Nonlinear dynamics - synchronization and chaos, V. 2, pp. 84-92. (in Russian), 1997.

### Preprints

- A.S. Kuznetsov, V.D. Shalfeev and G.L.Forti, Regularization of Dynamics in an Ensemble of Multicoupled Chaotic Elements. Dipartimento di Matematica, Universita' degli studi di Milano, Milano, Italia, 2000.
- A.S. Kuznetsov, V.D. Shalfeev and G.L.Forti, Stationary Spatial Patterns in a Chain of Bistable Elements with Unidirectional Couplings. Dipartimento di Matematica, Universita' degli studi di Milano, Milano, Italia, 2000.

### Papers in conference proceedings

- A.S. Kuznetsov, Chaotic Dynamics of Oscillators with Frequency Control, Proc. of the IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2000), Catania, Italy, 18-20 May, 2000, pp 113-116.
- A.S. Kuznetsov, Suppression of oscillations in a system of two nonlinearly coupled Chua's circuit. Proc. of the IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2000), Catania, Italy, 18-20 May, 2000, pp 259-262.
- A.S. Kuznetsov and V.D. Shalfeev, Stationary Spatial Patterns in CNN of Bistable Cells with Nonlinear Couplings. Proc. of the IEEE Workshop on Cellular Neural Networks and their Applications (CNNA2000), Catania, Italy, 23-25 May, 2000, pp. 413-418.
- A.S. Kuznetsov and V.D. Shalfeev, Dynamics of a CNN with Variable Number of Couplings. Proc. of the IEEE Workshop on Cellular Neural Networks and their Applications (CNNA2000), Catania, Italy, 23-25 May, 2000, pp. 419-424.
- M.M.Sushchik, A.S.Kuznetsov, A.K.Kozlov, and Ya.I.Molkov, Chaotic Synchronization of Long Chains by a Small Number of Couplings. Proc. of International Conference on Control of Oscillations and Chaos, St. Petersburg, Russia, 5-7 July, 2000, pp. 185-186.
- A.S. Kuznetsov and V.D. Shalfeev, Analysis of regularization of dynamics in a circular chain of bistable chaotic elements with variable number of couplings. Proc. of International Conference on Control of Oscillations and Chaos, St. Petersburg, Russia, 5-7 July, 2000, pp. 365-368.
- A.S. Kuznetsov and V.D. Shalfeev, Description of System Dynamics by Low-Dimensional Models. Proc. of International Symposium on Nonlinear Theory and its Applications (NOLTA2000), Dresden, Germany, Sep. 17-21, 2000, pp. 575-578.
- D.G.Zakharov, M.M.Sushchik and A.S. Kuznetsov, The Effect of Additive and Multiplicative Noise on the Dynamics of Locomotor Central Pattern Generator. Proc. of International Symposium on Nonlinear Theory and its Applications (NOLTA2000), Dresden, Germany, Sep. 17-21, 2000, pp. 661-664.
- A.K.Kozlov, M.M.Sushchik, Ya.I.Molkov, A.S.Kuznetsov, "Phase bistability and chaos in a system of two identical Van der Pol-Duffing oscillators" Proc. 1998 International Symposium on Nonlinear Theory and its Applications (NOLTA'98), Crans - Monana, Switzerland, 1998, V. 2, pp. 527-530.
- Ya.I.Molkov, M.M.Sushchik, A.S.Kuznetsov, A.K.Kozlov, D.G.Zakharov, "Dynamical model for locomotor - like movements of humans" Proc. 1998 International Symposium on Nonlinear Theory and its Applications (NOLTA'98), Crans - Monana, Switzerland, 1998, V. 3, pp. 1325-1328.
- A.K.Kozlov, M.M.Sushchik, Ya.I.Molkov, A.S.Kuznetsov, "Bistable phase synchronization and chaos in a system of coupled Van der Pol â€“ Duffing oscillators" Proc. of International Workshop on "Synchronization, Pattern Formation and Spatio - Temporal Chaos in Coupled Chaotic Oscillators", University of Santiago de Compostella, Santiago de Compostella, Spain, June 7-10, 1998, p. 20.
- V.D.Shalfeev, A.S.Kuznetsov "Structure formation and regularization in an array of globally coupled oscillators", Pros. of the third international specialist workshop on nonlinear dynamics of electronic systems NDES'97, Moscow 1997, pp. 347-351.
- V.D.Shalfeev, G.V.Osipov, A.S.Kuznetsov "Spatio-temporal dynamics in Chua's circuits networks". Pros. of the third international specialist workshop on nonlinear dynamics of electronic systems NDES'95, ed. M.P.Kennedy, University College Dublin, 1995, pp. 299-302.

+ Abstracts (the total number is approximately 25).

- phone: (317) 278-7460
- office: LC 224T
- e-mail: askuznet@iupui.edu

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