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Pattern formation of a predator-prey model with the cost of anti-predator behaviors
A multi-base harmonic balance method applied to Hodgkin-Huxley model
Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France |
Our aim is to propose a new robust and manageable technique, called multi-base harmonic balance method, to detect and characterize the periodic solutions of a nonlinear dynamical system. Our case test is the Hodgkin-Huxley model, one of the most realistic neuronal models in literature. This system, depending on the value of the external stimuli current, exhibits periodic solutions, both stable and unstable.
References:
[1] |
U. Asher, J. Christiansen and R. D. Russell,
Collocation software for boundary-value ODEs, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209-222.
doi: 10.1145/355945.355950. |
[2] |
U. Asher, R. Mattheij and R. D. Russell,
Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, 1995.
doi: 10.1137/1.9781611971231. |
[3] |
G. Bader and U. Asher,
A new basis implementation for a mixed order boundary value ODE solver, SIAM Journal on Scientific and Statistical Computing, 8 (1987), 483-500.
doi: 10.1137/0908047. |
[4] |
M. Basso, R. Genesio and A. Tesi,
A frequency method for predicting limit cycle bifurcations, Nonlinear Dynamics, 13 (1997), 339-360.
doi: 10.1023/A:1008298205786. |
[5] |
F. Bonani and M. Gilli,
Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46 (1999), 881-890.
doi: 10.1109/81.780370. |
[6] |
T. H. Bullock, M. V. L. Bennett, D. Johnston, R. Josephson, E. Marder and R. D. Fields, The neuron doctrine, Redux, Science, 310 (1999), 791-793,2005. Google Scholar |
[7] |
T. Chan and H. B. Keller,
Arc-length continuation and multigrid techniques for nonlinear elliptic eigenvalue problems, SIAM Journal on Scientific and Statistical Computing, 3 (1982), 173-194.
doi: 10.1137/0903012. |
[8] |
E. Doedel, H. B. Keller and J. P. Kernevez,
Numerical analysis and control of bifurcation problems (I): Bifurcation in finite dimensions, International journal of bifurcation and chaos, 1 (1991), 493-520.
doi: 10.1142/S0218127491000397. |
[9] |
S. Doi, S. Nabetani and S. Kumagai,
Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes, Biological cybernetics, 85 (2001), 51-64.
doi: 10.1007/PL00007996. |
[10] |
M. Farkas,
Periodic Motions, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4757-4211-4. |
[11] |
M. Glickstein,
Golgi and Cajal: The neuron doctrine and the 100th anniversary of the 1906 Nobel Prize, Current Biology, 16 (2006), R147-R151.
doi: 10.1016/j.cub.2006.02.053. |
[12] |
D. Gottlieb and S. Orszag,
Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. |
[13] |
J. Guckenheimer and R. A. Oliva,
Chaos in the Hodgkin-Huxley Model, SIAM Journal on Applied Dynamical Systems, 1 (2002), 105-114.
doi: 10.1137/S1111111101394040. |
[14] |
J. Guckenheimer and J. S. Labouriau, Bifurcation of the Hodgkin and Huxley equations: A new twist, Bulletin of Mathematical Biology, 55 (1993), 937-952. Google Scholar |
[15] |
B. Hassard,
Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon, Journal of Theoretical Biology, 71 (1978), 401-420.
doi: 10.1016/0022-5193(78)90168-6. |
[16] |
J. S. Hestheaven, S. Gottlieb and D. Gottlieb,
Spectral Methods for Time-Dependent Problems, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618352. |
[17] |
A. L. Hodgkin,
The local electric changes associated with repetitive action in a non-medullated axon, The Journal of physiology, 107 (1948), 165-181.
doi: 10.1113/jphysiol.1948.sp004260. |
[18] |
A. L. Hodgkin and A. F. Huxley,
Propagation of electrical signals along giant nerve fibres, Proceedings of the Royal Society of London. Series B, Biological Sciences, 140 (1952), 177-183.
doi: 10.1098/rspb.1952.0054. |
[19] |
E. M. Izhikevich,
Dynamical Systems in Neuroscience, MIT press, 2007. |
[20] |
S. Karkar, B. Cochelin and C. Vergez, A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems, Journal of Sound and Vibration, 333 (2004), 2554-2567. Google Scholar |
[21] |
J. P. Keener and J. Sneyd,
Mathematical Physiology, Springer, 1998. |
[22] |
J. Kierzenka and L. F. Shampine,
A BVP solver based on residual control and the Matlab PSE, ACM Transactions on Mathematical Software (TOMS), 27 (2001), 299-316.
doi: 10.1145/502800.502801. |
[23] |
K. S. Kundert, J. K. White and A. Sangiovanni-Vicentelli,
Steady-state Methods for Simulating Analog and Microwave Circuits, Kluwer Academic Publishers Boston, 1990.
doi: 10.1007/978-1-4757-2081-5. |
[24] |
Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory, Springer, 1998. |
[25] |
V. Lanza, M. Bonnin and M. Gilli,
On the application of the describing function technique to the bifurcation analysis of nonlinear systems, IEEE, Trans. Circuits Systems II Express Briefs, 54 (2007), 343-347.
doi: 10.1109/TCSII.2006.890406. |
[26] |
V. Lanza, L. Ponta, M. Bonnin and F. Corinto, Multiple attractors and bifurcations in hard oscillators driven by constant inputs, International Journal of Bifurcation and Chaos, 22 (2012), 1250267, 16pp.
doi: 10.1142/S0218127412502677. |
[27] |
A. I. Mees,
Dynamics of Feedback Systems, Wiley Ltd., Chirchester, 1981. |
[28] |
R. E. Mickens,
Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods, World Scientific, 2010.
doi: 10.1142/9789814291668. |
[29] |
N. Minorsky, Nonlinear Oscillations, Krieger, Huntington, New York, 1974. Google Scholar |
[30] |
C. Piccardi,
Bifurcation analysis via harmonic balance in periodic systems with feedback structure, International Journal of Control, 62 (1995), 1507-1515.
doi: 10.1080/00207179508921611. |
[31] |
S. Ramon and Y. Cajal, Textura del Sistema Nervioso del Hombre y de los Vertebrados, Imprenta y Librería de Nicolás Moya, Madrid, 1899. Google Scholar |
[32] |
J. Rinzel and R. N. Miller,
Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations, Mathematical Biosciences, 49 (1980), 27-59.
doi: 10.1016/0025-5564(80)90109-1. |
[33] |
A. Scott,
Neuroscience: A mathematical Primer, Springer, 2002. |
[34] |
L. F. Shampine, J. Kierzenka and M. W. Reichelt, Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, Tutorial notes, 49 (2000), 437-448. Google Scholar |
[35] |
H. C. Tuckwell,
Introduction to Theoretical Neurobiology: Volume 1, Linear Cable Theory and Dendritic Structure, Cambridge University Press, 1988. |
[36] |
M. Urabe,
Galerkin's procedure for nonlinear periodic systems, Archive for Rational Mechanics and Analysis, 20 (1965), 120-152.
doi: 10.1007/BF00284614. |
[37] |
X. Wang and J. Rinzel, Oscillatory and bursting properties of neurons, in The handbook of brain theory and neural networks, MIT Press, (1998), 686–691. Google Scholar |
[38] |
A. Zygmund,
Trigonometric Series, Cambridge University Press, 2002. |
show all references
References:
[1] |
U. Asher, J. Christiansen and R. D. Russell,
Collocation software for boundary-value ODEs, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209-222.
doi: 10.1145/355945.355950. |
[2] |
U. Asher, R. Mattheij and R. D. Russell,
Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, 1995.
doi: 10.1137/1.9781611971231. |
[3] |
G. Bader and U. Asher,
A new basis implementation for a mixed order boundary value ODE solver, SIAM Journal on Scientific and Statistical Computing, 8 (1987), 483-500.
doi: 10.1137/0908047. |
[4] |
M. Basso, R. Genesio and A. Tesi,
A frequency method for predicting limit cycle bifurcations, Nonlinear Dynamics, 13 (1997), 339-360.
doi: 10.1023/A:1008298205786. |
[5] |
F. Bonani and M. Gilli,
Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46 (1999), 881-890.
doi: 10.1109/81.780370. |
[6] |
T. H. Bullock, M. V. L. Bennett, D. Johnston, R. Josephson, E. Marder and R. D. Fields, The neuron doctrine, Redux, Science, 310 (1999), 791-793,2005. Google Scholar |
[7] |
T. Chan and H. B. Keller,
Arc-length continuation and multigrid techniques for nonlinear elliptic eigenvalue problems, SIAM Journal on Scientific and Statistical Computing, 3 (1982), 173-194.
doi: 10.1137/0903012. |
[8] |
E. Doedel, H. B. Keller and J. P. Kernevez,
Numerical analysis and control of bifurcation problems (I): Bifurcation in finite dimensions, International journal of bifurcation and chaos, 1 (1991), 493-520.
doi: 10.1142/S0218127491000397. |
[9] |
S. Doi, S. Nabetani and S. Kumagai,
Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes, Biological cybernetics, 85 (2001), 51-64.
doi: 10.1007/PL00007996. |
[10] |
M. Farkas,
Periodic Motions, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4757-4211-4. |
[11] |
M. Glickstein,
Golgi and Cajal: The neuron doctrine and the 100th anniversary of the 1906 Nobel Prize, Current Biology, 16 (2006), R147-R151.
doi: 10.1016/j.cub.2006.02.053. |
[12] |
D. Gottlieb and S. Orszag,
Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. |
[13] |
J. Guckenheimer and R. A. Oliva,
Chaos in the Hodgkin-Huxley Model, SIAM Journal on Applied Dynamical Systems, 1 (2002), 105-114.
doi: 10.1137/S1111111101394040. |
[14] |
J. Guckenheimer and J. S. Labouriau, Bifurcation of the Hodgkin and Huxley equations: A new twist, Bulletin of Mathematical Biology, 55 (1993), 937-952. Google Scholar |
[15] |
B. Hassard,
Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon, Journal of Theoretical Biology, 71 (1978), 401-420.
doi: 10.1016/0022-5193(78)90168-6. |
[16] |
J. S. Hestheaven, S. Gottlieb and D. Gottlieb,
Spectral Methods for Time-Dependent Problems, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618352. |
[17] |
A. L. Hodgkin,
The local electric changes associated with repetitive action in a non-medullated axon, The Journal of physiology, 107 (1948), 165-181.
doi: 10.1113/jphysiol.1948.sp004260. |
[18] |
A. L. Hodgkin and A. F. Huxley,
Propagation of electrical signals along giant nerve fibres, Proceedings of the Royal Society of London. Series B, Biological Sciences, 140 (1952), 177-183.
doi: 10.1098/rspb.1952.0054. |
[19] |
E. M. Izhikevich,
Dynamical Systems in Neuroscience, MIT press, 2007. |
[20] |
S. Karkar, B. Cochelin and C. Vergez, A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems, Journal of Sound and Vibration, 333 (2004), 2554-2567. Google Scholar |
[21] |
J. P. Keener and J. Sneyd,
Mathematical Physiology, Springer, 1998. |
[22] |
J. Kierzenka and L. F. Shampine,
A BVP solver based on residual control and the Matlab PSE, ACM Transactions on Mathematical Software (TOMS), 27 (2001), 299-316.
doi: 10.1145/502800.502801. |
[23] |
K. S. Kundert, J. K. White and A. Sangiovanni-Vicentelli,
Steady-state Methods for Simulating Analog and Microwave Circuits, Kluwer Academic Publishers Boston, 1990.
doi: 10.1007/978-1-4757-2081-5. |
[24] |
Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory, Springer, 1998. |
[25] |
V. Lanza, M. Bonnin and M. Gilli,
On the application of the describing function technique to the bifurcation analysis of nonlinear systems, IEEE, Trans. Circuits Systems II Express Briefs, 54 (2007), 343-347.
doi: 10.1109/TCSII.2006.890406. |
[26] |
V. Lanza, L. Ponta, M. Bonnin and F. Corinto, Multiple attractors and bifurcations in hard oscillators driven by constant inputs, International Journal of Bifurcation and Chaos, 22 (2012), 1250267, 16pp.
doi: 10.1142/S0218127412502677. |
[27] |
A. I. Mees,
Dynamics of Feedback Systems, Wiley Ltd., Chirchester, 1981. |
[28] |
R. E. Mickens,
Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods, World Scientific, 2010.
doi: 10.1142/9789814291668. |
[29] |
N. Minorsky, Nonlinear Oscillations, Krieger, Huntington, New York, 1974. Google Scholar |
[30] |
C. Piccardi,
Bifurcation analysis via harmonic balance in periodic systems with feedback structure, International Journal of Control, 62 (1995), 1507-1515.
doi: 10.1080/00207179508921611. |
[31] |
S. Ramon and Y. Cajal, Textura del Sistema Nervioso del Hombre y de los Vertebrados, Imprenta y Librería de Nicolás Moya, Madrid, 1899. Google Scholar |
[32] |
J. Rinzel and R. N. Miller,
Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations, Mathematical Biosciences, 49 (1980), 27-59.
doi: 10.1016/0025-5564(80)90109-1. |
[33] |
A. Scott,
Neuroscience: A mathematical Primer, Springer, 2002. |
[34] |
L. F. Shampine, J. Kierzenka and M. W. Reichelt, Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, Tutorial notes, 49 (2000), 437-448. Google Scholar |
[35] |
H. C. Tuckwell,
Introduction to Theoretical Neurobiology: Volume 1, Linear Cable Theory and Dendritic Structure, Cambridge University Press, 1988. |
[36] |
M. Urabe,
Galerkin's procedure for nonlinear periodic systems, Archive for Rational Mechanics and Analysis, 20 (1965), 120-152.
doi: 10.1007/BF00284614. |
[37] |
X. Wang and J. Rinzel, Oscillatory and bursting properties of neurons, in The handbook of brain theory and neural networks, MIT Press, (1998), 686–691. Google Scholar |
[38] |
A. Zygmund,
Trigonometric Series, Cambridge University Press, 2002. |









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