Journal of Computational Neuroscience

, Volume 35, Issue 1, pp 87–108 | Cite as

Bifurcations of large networks of two-dimensional integrate and fire neurons

  • Wilten Nicola
  • Sue Ann Campbell


Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by Touboul (SIAM J Appl Math 68(4):1045–1079, 2008). However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.


Bifurcation theory Mean field Large networks Bursting Population density methods Integrate and fire 



This work benefitted from the support of the Natural Sciences and Engineering Research Council of Canada and the Ontario Graduate Scholarship program. The authors would like to thank F. Skinner for useful discussions. The authors would also like to thank the reviewers for their suggestions which improved the manuscript.


  1. Abbott, L.F., & van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Learning and Memory, 48(2), 1483–1490.Google Scholar
  2. Apfaltrer, F., Ly, C., Tranchina, D. (2006). Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods. Network: Computation in Neural Systems, 17(4), 373–418.CrossRefGoogle Scholar
  3. di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P., Nordmark, A., Tost, G., Piiroinen, P. (2008). Bifurcations in non-smooth dynamical systems. SIAM Review, 50(4), 629–701.CrossRefGoogle Scholar
  4. Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5), 3637–3642.PubMedCrossRefGoogle Scholar
  5. Casti, A., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B.W., Victor, J., Sirovich, L. (2002). A population study of integrate-and-fire-or-burst neurons. Neural Computation, 14(5), 957–986.PubMedCrossRefGoogle Scholar
  6. Destexhe, A., Mainen, Z., Sejnowski, T. (1998). Kinetic models of synaptic transmission. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling: From synapses to networks (chap. 1). Cambridge, MA: MIT Press.Google Scholar
  7. Dhooge, A., Govaerts, W., Kuznetsov, Y.A. (2003). MatCont: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29, 141–164.CrossRefGoogle Scholar
  8. Dur-e-Ahmad, M., Nicola, W., Campbell, S.A., Skinner, F. (2012). Network bursting using experimentally constrained single compartment CA3 hippocampal neuron models with adaptation. Journal of Computational Neuroscience, 33(1), 21–40.PubMedCrossRefGoogle Scholar
  9. Ermentrout, G.B., & Terman, D.H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer.CrossRefGoogle Scholar
  10. Fitzhugh, R. (1952). Impulses and phsyiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466.CrossRefGoogle Scholar
  11. Gerstner, W., & Kistler, W. (2002). Spiking Neuron Models. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  12. Hines, M.L., Morse, T., Migliore, M., Carnevale, N.T., Shepherd, G.M. (2004). ModelDB: a database to support computational neuroscience. Journal of Computational Neuroscience, 17(1), 7–11.PubMedCrossRefGoogle Scholar
  13. Hemond, P., Epstein, D., Boley, A., Migliore, M., Ascoli, G., Jaffe, D. (2008). Distinct classes of pyramidal cells exhibit mutually exclusive firing patterns in hippocampal area CA3b. Hippocampus, 18(4), 411–424.PubMedCrossRefGoogle Scholar
  14. Ho, E.C., Zhang, L., Skinner, F.K. (2009). Hippocampus, 19(2), 152–165.PubMedCrossRefGoogle Scholar
  15. Izhikevich, E. (2003). Simple model of spiking neurons. Neural Networks, IEEE Transactions, 14(6), 1569–1572.CrossRefGoogle Scholar
  16. Knight, B.W. (2000). Dynamics of encoding in neuron populations: some general mathematical features. Neural Computation, 12, 473–518.PubMedCrossRefGoogle Scholar
  17. La, Camera, G., Rauch, A., Luscher, H.R., Senn, W., Fusi, S. (2004). Minimal models of adapted neuronal response to in-vivo like input currents. Neural Computation, 16, 2101–2124.PubMedCrossRefGoogle Scholar
  18. La Camera, G., Giugliano, M., Senn, W., Fusi, S. (2008). The response of cortical neurons to in vivo-like input current: theory and experiment. Biological Cybernetics, 99, 279–301.PubMedCrossRefGoogle Scholar
  19. Ly, C., & Tranchina, D. (2007). A critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Computation, 19(8), 2032–2092.PubMedCrossRefGoogle Scholar
  20. Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., Wu, C. (2004). Interneurons of the neocortical inhibitory system. Nature Reviews: Neuroscience, 5(10), 793–807.PubMedCrossRefGoogle Scholar
  21. MATLAB (2012). Version 7.10.0 (R2012a). The MathWorks Inc. Massachusetts: Natick.Google Scholar
  22. Naud, R., Marcille, N., Clopath, C., Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99, 335–347.PubMedCrossRefGoogle Scholar
  23. Nesse, W., Borisyuk, A., Bressloff, P. (2008). Fluctuation-driven rhythmogenesis in an excitatory neuronal network with slow adaptation. Journal of Computational Neuroscience, 25, 317–333. doi: 10.1007/s10827-008-0081-y.PubMedCrossRefGoogle Scholar
  24. Nykamp, D., & Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. Journal of Computational Neuroscience, 8, 19–50.PubMedCrossRefGoogle Scholar
  25. Omurtag, A., Knight, B.W., Sirovich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8, 51–63.PubMedCrossRefGoogle Scholar
  26. Sirovich, L., Omurtag, A., Knight, B.W. (2000). Dynamics of neuronal populations: the equilibrium solution. SIAM Journal on Applied Mathematics, 60(6), 2009–2028.CrossRefGoogle Scholar
  27. Sirovich, L., Omurtag, A., Lubliner, K. (2006). Dynamics of neural populations: stability and synchrony. Network: Computation in Neural Systems, 17, 3–29.CrossRefGoogle Scholar
  28. Strogatz, S., & Mirollo, R.E. (1991). Stability of incoherence in a population of coupled oscillators. Journal of Statistical Physics, 63, 613–635.CrossRefGoogle Scholar
  29. Tikhonov, A. (1952). Systems of differential equations containing small parameters in the derivatives (in Russian). Matematicheskii Sbornik (NS), 31(73), 575–586.Google Scholar
  30. Touboul, J. (2008). Bifurcation analysis of a general class ofnonlinear integrate-and-fire neurons. SIAM Journal on Applied Mathematics, 68(4), 1045–1079.CrossRefGoogle Scholar
  31. Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Computation in Neural Systems, 4(3), 259–284.CrossRefGoogle Scholar
  32. van Vreeswijk, C. (1996). Partial synchronization in populations of pulse-coupled oscillators. Physical Review E, 54, 5522–5537. doi:  10.1103/PhysRevE.54.5522.CrossRefGoogle Scholar
  33. van Vreeswijk, C., & Hansel, D. (2001). Patterns of synchrony in neural networks with spike adaptation. Neural Computation, 13(5), 959–992.PubMedCrossRefGoogle Scholar
  34. van Vreeswijk, C., Abbott, L.F., Ermentrout, G.B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313–321.PubMedCrossRefGoogle Scholar
  35. Vladimirski, B.B., Tabak, J., O’Donovan, M.J., Rinzel, J. (2008). Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. Journal of Computational Neuroscience, 25, 39–63.PubMedCrossRefGoogle Scholar
  36. Wu, Y., Lu, W., Lin, W., Leng, G., Feng, J. (2012). Bifurcations of emergent bursting in a neuronal network. PLoS ONE, 7(6), e38402.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations