A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks


Spiking neural networks are widely applied to simulate cortical dynamics in the brain and are regarded as the next generation of machine learning. The classical Hodgkin–Huxley (HH) neuron is the foundation of all spiking neural models. In numerical simulation, however, the stiffness of the nonlinear HH equations during an action potential (a spike) period prohibits the use of large time steps for numerical integration. Outside of this stiff period, the HH equations can be efficiently simulated with a relatively large time step. In this work, we present an efficient and accurate offline–online combined method that stops evolving the HH equations during an action potential period, uses a pre-computed (offline) high-resolution data set to determine the voltage value during the spike, and restarts the time evolution of the HH equations after the stiff period using reset values interpolated from the offline data set. Our method allows for time steps an order of magnitude larger than those used in the standard Runge–Kutta (RK) method, while accurately capturing dynamical properties of HH neurons. In addition, this offline–online method robustly achieves a maximum of a tenfold decrease in computation time as compared to RK methods, a result that is independent of network size.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16


  1. 1.

    Aihara, K.: Chaotic oscillations and bifurcations in squid giant axons. In: Chaos, pp. 257–269 (1986)

  2. 2.

    Beierlein, M., Gibson, J.R., Connors, B.W.: A network of electrically coupled interneurons drives synchronized inhibition in neocortex. Nat. Neurosci. 3(9), 904–910 (2000)

    Google Scholar 

  3. 3.

    Börgers, C., Nectow, A.R.: Exponential time differencing for Hodgkin–Huxley-like ODEs. SIAM J. Sci. Comput. 35(3), B623–B643 (2013)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Connors, B.W., Long, M.A.: Electrical synapses in the mammalian brain. Annu. Rev. Neurosci. 27, 393–418 (2004)

    Google Scholar 

  5. 5.

    Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430–455 (2002)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Crodelle, J., Zhou, D., Kovacic, G., Cai, D.: A role for electrotonic coupling between cortical pyramidal cells. Front. Comput. Neurosci. 13, 33 (2019)

    Google Scholar 

  7. 7.

    Dayan, P., Abbott, L.: Theoretical neuroscience: computational and mathematical modeling of neural systems. J. Cogn. Neurosci. 15(1), 154–155 (2003)

    Google Scholar 

  8. 8.

    Dayan, P., Abbott, L.F.: Theoretical Neuroscience, vol. 806. MIT Press, Cambridge (2001)

    Google Scholar 

  9. 9.

    Ding, L., Hou, C.: Stabilizing control of hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term. Nonlinear Dyn. 60(1–2), 131–139 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Galarreta, M., Hestrin, S.: A network of fast-spiking cells in the neocortex connected by electrical synapses. Nature 402(6757), 72–75 (1999)

    Google Scholar 

  11. 11.

    Gerstner, W., Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. 12.

    Gu, H., Pan, B.: A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn. 81(4), 2107–2126 (2015)

    MathSciNet  Google Scholar 

  13. 13.

    Guckenheimer, J., Oliva, R.A.: Chaos in the Hodgkin–Huxley model. SIAM J. Appl. Dyn. Syst. 1(1), 105–114 (2002)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Hansel, D., Mato, G., Meunier, C., Neltner, L.: On numerical simulations of integrate-and-fire neural networks. Neural Comput. 10(2), 467–483 (1998)

    Google Scholar 

  15. 15.

    Hansel, D., Sompolinsky, H.: Chaos and synchrony in a model of a hypercolumn in visual cortex. J. Comput. Neurosci. 3(1), 7–34 (1996)

    Google Scholar 

  16. 16.

    Hassard, B.: Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J. Theor. Biol. 71(3), 401–420 (1978)

    MathSciNet  Google Scholar 

  17. 17.

    Hertz, J., Prügel-Bennett, A.: Learning short synfire chains by self-organization. Netw. Comput. Neural Syst. 7(2), 357–363 (1996)

    MATH  Google Scholar 

  18. 18.

    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500 (1952)

    Google Scholar 

  19. 19.

    Ikegaya, Y., Sasaki, T., Ishikawa, D., Honma, N., Tao, K., Takahashi, N., Minamisawa, G., Ujita, S., Matsuki, N.: Interpyramid spike transmission stabilizes the sparseness of recurrent network activity. Cerebral Cortex 23(2), 293–304 (2012)

    Google Scholar 

  20. 20.

    Ito, S., Hansen, M.E., Heiland, R., Lumsdaine, A., Litke, A.M., Beggs, J.M.: Extending transfer entropy improves identification of effective connectivity in a spiking cortical network model. PLoS ONE 6(11), e27431 (2011)

    Google Scholar 

  21. 21.

    Koch, C., Segev, I.: Methods in Neuronal Modeling: From Ions to Networks. MIT Press, Cambridge (1998)

    Google Scholar 

  22. 22.

    Kopell, N., Ermentrout, B.: Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proc. Natl. Acad. Sci. 101(43), 15482–15487 (2004)

    Google Scholar 

  23. 23.

    Mainen, Z.F., Sejnowski, T.J.: Influence of dendritic structure on firing pattern in model neocortical neurons. Nature 382(6589), 363 (1996)

    Google Scholar 

  24. 24.

    Mattia, M., Del Giudice, P.: Efficient event-driven simulation of large networks of spiking neurons and dynamical synapses. Neural Comput. 12(10), 2305–2329 (2000)

    Google Scholar 

  25. 25.

    McLaughlin, D., Shapley, R., Shelley, M., Wielaard, D.J.: A neuronal network model of macaque primary visual cortex (v1): orientation selectivity and dynamics in the input layer 4c\(\alpha \). Proc. Natl. Acad. Sci. 97(14), 8087–8092 (2000)

    Google Scholar 

  26. 26.

    Monteforte, M., Wolf, F.: Dynamic flux tubes form reservoirs of stability in neuronal circuits. Phys. Rev. X 2(4), 041007 (2012)

    Google Scholar 

  27. 27.

    Nie, Q., Wan, F.Y., Zhang, Y.T., Liu, X.F.: Compact integration factor methods in high spatial dimensions. J. Comput. Phys. 227(10), 5238–5255 (2008)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19(2), 197–231 (1968)

    MathSciNet  Google Scholar 

  29. 29.

    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  30. 30.

    Parker, T.S., Chua, L.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (2012)

    Google Scholar 

  31. 31.

    Perkel, D.H., Gerstein, G.L., Moore, G.P.: Neuronal spike trains and stochastic point processes: II. Simultaneous spike trains. Biophys. J. 7(4), 419–440 (1967)

    Google Scholar 

  32. 32.

    Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y., Markram, H., Destexhe, A.: Minimal Hodgkin–Huxley type models for different classes of cortical and thalamic neurons. Biol. Cybern. 99(4), 427–441 (2008)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Quinn, C.J., Coleman, T.P., Kiyavash, N., Hatsopoulos, N.G.: Estimating the directed information to infer causal relationships in ensemble neural spike train recordings. J. Comput. Neurosci. 30(1), 17–44 (2011)

    MathSciNet  Google Scholar 

  34. 34.

    Rangan, A.V., Cai, D.: Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks. J. Comput. Neurosci. 22(1), 81–100 (2007)

    MathSciNet  Google Scholar 

  35. 35.

    Revel, J., Karnovsky, M.: Hexagonal array of subunits in intercellular junctions of the mouse heart and liver. J. Cell Biol. 33(3), C7 (1967)

    Google Scholar 

  36. 36.

    Rinzel, J., Ermentrout, G.B.: Analysis of neural excitability and oscillations. Methods Neuronal Model. 2, 251–292 (1998)

    Google Scholar 

  37. 37.

    Shelley, M.J., Tao, L.: Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks. J. Comput. Neurosci. 11(2), 111–119 (2001)

    Google Scholar 

  38. 38.

    Shinomoto, S., Kim, H., Shimokawa, T., Matsuno, N., Funahashi, S., Shima, K., Fujita, I., Tamura, H., Doi, T., Kawano, K., et al.: Relating neuronal firing patterns to functional differentiation of cerebral cortex. PLoS Comput. Biol. 5(7), e1000433 (2009)

    MathSciNet  Google Scholar 

  39. 39.

    Shlens, J., Field, G.D., Gauthier, J.L., Grivich, M.I., Petrusca, D., Sher, A., Litke, A.M., Chichilnisky, E.: The structure of multi-neuron firing patterns in primate retina. J. Neurosci. 26(32), 8254–8266 (2006)

    Google Scholar 

  40. 40.

    Somers, D.C., Nelson, S.B., Sur, M.: An emergent model of orientation selectivity in cat visual cortical simple cells. J. Neurosci. 15(8), 5448–5465 (1995)

    MATH  Google Scholar 

  41. 41.

    Song, S., Sjöström, P.J., Reigl, M., Nelson, S., Chklovskii, D.B.: Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol. 3(3), e68 (2005)

    Google Scholar 

  42. 42.

    Sun, Y., Zhou, D., Rangan, A.V., Cai, D.: Library-based numerical reduction of the Hodgkin–Huxley neuron for network simulation. J. Comput. Neurosci. 27(3), 369–390 (2009)

    MathSciNet  Google Scholar 

  43. 43.

    Sussillo, D., Abbott, L.F.: Generating coherent patterns of activity from chaotic neural networks. Neuron 63(4), 544–557 (2009)

    Google Scholar 

  44. 44.

    Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, New York (2002)

    Google Scholar 

  45. 45.

    Tian, Z.Q.K., Zhou, D.: Exponential time differencing algorithm for pulse-coupled Hodgkin–Huxley neuronal networks. arXiv preprint arXiv:1910.08724 (2019)

  46. 46.

    Wang, Y., Barakat, A., Zhou, H.: Electrotonic coupling between pyramidal neurons in the neocortex. PLoS ONE 5(4), e10253 (2010)

    Google Scholar 

  47. 47.

    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D: Nonlinear Phenom. 16(3), 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Xu, Z.Q.J., Bi, G., Zhou, D., Cai, D.: A dynamical state underlying the second order maximum entropy principle in neuronal networks. Commun. Math. Sci. 15(3), 665–692 (2017)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Zhou, D., Rangan, A.V., Sun, Y., Cai, D.: Network-induced chaos in integrate-and-fire neuronal ensembles. Phys. Rev. E 80(3), 031918 (2009)

    Google Scholar 

  50. 50.

    Zhou, D., Sun, Y., Rangan, A.V., Cai, D.: Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type. J. Comput. Neurosci. 28(2), 229–245 (2010)

    MathSciNet  Google Scholar 

  51. 51.

    Zhou, D., Xiao, Y., Zhang, Y., Xu, Z., Cai, D.: Granger causality network reconstruction of conductance-based integrate-and-fire neuronal systems. PLoS ONE 9(2), e87636 (2014)

    Google Scholar 

Download references


This work was supported by National Key R&D Program of China (2019YFA0709503), NSFC-11671259, NSFC-11722107, SJTU-UM Collaborative Research Program, and the Student Innovation Center at Shanghai Jiao Tong University (D.Z.); the NSF Mathematical Sciences PostDoctoral Research Fellowship (MSPRF) DMS-1703761 (J.C.). We dedicate this paper to our late professor David Cai.

Author information



Corresponding author

Correspondence to Douglas Zhou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tian, Z.K., Crodelle, J. & Zhou, D. A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks. J Sci Comput 84, 10 (2020). https://doi.org/10.1007/s10915-020-01261-6

Download citation


  • Fast algorithm
  • Offline–online method
  • Hodgkin–Huxley
  • Numerical simulation