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June  2018, 15(3): 807-825. doi: 10.3934/mbe.2018036

A multi-base harmonic balance method applied to Hodgkin-Huxley model

Aymen Balti , Valentina Lanza , and  Moulay Aziz-Alaoui

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

* Corresponding author: V. Lanza.

Received  November 23, 2016 Revised  October 09, 2017 Published  December 2017

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.

Keywords: Periodic solutions, harmonic balance method, Hodgkin-Huxley model.
Mathematics Subject Classification: Primary: 34C25; Secondary: 65L10.
Citation: Aymen Balti, Valentina Lanza, Moulay Aziz-Alaoui. A multi-base harmonic balance method applied to Hodgkin-Huxley model. Mathematical Biosciences & Engineering, 2018, 15 (3) : 807-825. doi: 10.3934/mbe.2018036
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. 

[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. 

[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. 

[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.

[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.

[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. 

[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.

[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. 

[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. 

[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. 

[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.

[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.

[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. 

[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.

[38]

A. Zygmund, Trigonometric Series, Cambridge University Press, 2002.

Figure 1.  Bifurcation diagram of the HH model, showing the stable (solid line) and unstable (dotted line) branches of the periodic solutions of HH model. For each periodic solution the minimum and the maximum values of the potential $V$ over one period are represented. Depending on the values of $I$, two regions with different dynamical behaviors can be identified.
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Figure 2.  Zoom for $I \in~[0, ~I_{2}]$ of Fig 1. HH model exhibits one equilibrium point, one stable limit cycle (solid line) and up to 3 unstable ones (dotted lines).
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Figure 3.  (a) The stable periodic solution detected by the HB method for $I =6.25$ exhibits a sort of Gibbs phenomenon. (b) Zoom showing the small oscillations, sign of a non accurate approximation of the limit cycle, despite the exploitation of 50 harmonics.
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Figure 4.  (a) Stable and (b) unstable periodic solutions for different values of $I$, in a neighborhood of (a) $I_1$ and (b) $I_2$, respectively.
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Figure 5.  Time series of (a) the stable periodic solution for $I = 152.2500$ and (b) the unstable periodic solution for $I = 9.71889$.
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Figure 6.  Stable (solid line) and unstable (dashed line) limit cycles near the first saddle-node of cycles bifurcation, for (a) $I =6.2649$ both solutions are almost coincident, and for (b) $I =6.2716$.
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Figure 7.  Projection of two unstable limit cycles on the $(V, n)$ plane for (a) $I = 7.92198548\lesssim I_3$ and (b) $I = I_3 = 7.92198549$.
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Figure 8.  Projection of two unstable limit cycles on the $(V, n)$ plane for (a) $I = I_4 = 7.84654752$ and (b) $I = 7.84654876\lesssim I_4$.
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Figure 9.  (a)-(b) Floquet multipliers for the stable limit cycles and unstable limit cycles, respectively, associated to the first saddle node of cycles bifurcation for $I\in [6.2792, 6.7872]$. As $I$ increases, in (a) the multiplier $\mu_4$ starts from the value +1 and then enters in the unit circle, while in (b) the multiplier $\mu_4$ starts to the value +1 and becomes bigger and bigger. (c)-(d) Floquet multipliers for the two unstable limit cycles associated to the second saddle node of cycles bifurcation for $I\in [7.921985465, 7.921985491]$. Here, in both cases, the third multiplier is outside the unit circle (this makes the limit cycle unstable) and is not shown, since it takes very high values with respect to the others. As in the previous case, as $I$ decreases, the multiplier $\mu_4$ starts from the value +1 and either (c) enters in the unit circle, or (d) takes higher and higher values. (e)-(f) Floquet multipliers for the two unstable limit cycles associated to the third saddle node of cycles bifurcation for $I\in [7.846557778, 7.846616827]$. Also in this case, for both limit cycles, the third multiplier is not represented. As $I$ increases, the multiplier $\mu_4$ starts from the value +1 and either (e) escapes from, or (f) enters in the unit circle.
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Figure 10.  Floquet multipliers near the period-doubling bifurcation for different values of $I\in [7.92197743, 7.92197799]$. By decreasing $I$, the multiplier $\mu_4$ crosses the unit cycle through $-1$.
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Table 1.  By decreasing the value of I, the multipliers $\mu_4$ decreases, crosses the value -1 for $I =7.92197768$ and enters into the unit circle.
I $\mu_1$ $\mu_2$ $\mu_3$ $\mu_4$
7.92197799 1.000 0.000 -2940.687 -1.041
7.92197793 1.000 -0.000 -2964.042 -1.033
7.92197787 1.000 0.000 -2987.386 -1.025
7.92197781 1.000 0.000 -3010.719 -1.017
7.92197775 1.000 -0.000 -3034.042 -1.009
7.92197768 1.000 0.000 -3057.354 -1.001
7.92197762 1.000 -0.000 -3080.655 -0.993
7.92197756 1.000 0.000 -3103.946 -0.986
7.92197750 1.000 0.000 -3127.225 -0.978
7.92197743 1.000 0.000 -3150.494 -0.9713
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I $\mu_1$ $\mu_2$ $\mu_3$ $\mu_4$
7.92197799 1.000 0.000 -2940.687 -1.041
7.92197793 1.000 -0.000 -2964.042 -1.033
7.92197787 1.000 0.000 -2987.386 -1.025
7.92197781 1.000 0.000 -3010.719 -1.017
7.92197775 1.000 -0.000 -3034.042 -1.009
7.92197768 1.000 0.000 -3057.354 -1.001
7.92197762 1.000 -0.000 -3080.655 -0.993
7.92197756 1.000 0.000 -3103.946 -0.986
7.92197750 1.000 0.000 -3127.225 -0.978
7.92197743 1.000 0.000 -3150.494 -0.9713
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