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Dynamics and bifurcations of the adaptive exponential integrate-and-fire model

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Abstract

Recently, several two-dimensional spiking neuron models have been introduced, with the aim of reproducing the diversity of electrophysiological features displayed by real neurons while keeping a simple model, for simulation and analysis purposes. Among these models, the adaptive integrate-and-fire model is physiologically relevant in that its parameters can be easily related to physiological quantities. The interaction of the differential equations with the reset results in a rich and complex dynamical structure. We relate the subthreshold features of the model to the dynamical properties of the differential system and the spike patterns to the properties of a Poincaré map defined by the sequence of spikes. We find a complex bifurcation structure which has a direct interpretation in terms of spike trains. For some parameter values, spike patterns are chaotic.

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References

  • Angelino E, Brenner MP (2007) Excitability constraints on voltage-gated sodium channels. PLoS Comput Biol 3(9): 1751–1760

    Article  CAS  PubMed  Google Scholar 

  • Badel L, Lefort S, Brette R, Petersen C, Gerstner W, Richardson M (2008) Dynamic IV curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. J Neurophysiol 99(2): 656

    Article  PubMed  Google Scholar 

  • Brette R (2004) Dynamics of one-dimensional spiking neuron models. J Math Biol 48(1): 38–56

    Article  PubMed  Google Scholar 

  • Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J Neurophysiol 94: 3637–3642

    Article  PubMed  Google Scholar 

  • Clopath C, Jolivet R, Rauch A, Lüscher H, Gerstner W (2007) Predicting neuronal activity with simple models of the threshold type: Adaptive Exponential Integrate-and-Fire model with two compartments. Neurocomputing 70(10–2): 1668–1673

    Article  Google Scholar 

  • Fourcaud-Trocme N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci 23(37): 11,628

    CAS  Google Scholar 

  • Gerstner W, Kistler W (2002) Spiking neuron models. Cambridge University Press, Cambridge

    Google Scholar 

  • Goodman D, Brette R (2008) Brian: a simulator for spiking neural networks in Python. Front Neuroinformatics (in preparation)

  • Hille B (2001) Ion channels of excitable membranes. Sinauer Sunderland, Massachusetts

    Google Scholar 

  • Izhikevich E (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15(5): 1063–1070

    Article  PubMed  Google Scholar 

  • Izhikevich E (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, Cambridge

    Google Scholar 

  • Jolivet R, Kobayashi R, Rauch A, Naud R, Shinomoto S, Gerstner W (2008) A benchmark test for a quantitative assessment of simple neuron models. J Neurosci Meth 169(2): 417–424

    Article  Google Scholar 

  • Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation. J Physiol Pathol Gen 9: 620–635

    Google Scholar 

  • Markram H, Toledo-Rodriguez M, Wang Y, Gupta A, Silberberg G, Wu C (2004) Interneurons of the neocortical inhibitory system. Nat Rev Neurosci 5(10): 793–807

    Article  CAS  PubMed  Google Scholar 

  • Naud R, Macille N, Clopath C, Gerstner W (2008) Firing patterns in the adaptive exponential integrate-and-fire model. Biol Cybern (submitted)

  • Richardson MJ, Brunel N, Hakim V (2003) From subthreshold to firing-rate resonance. J Neurophysiol 89(5): 2538–2554

    Article  PubMed  Google Scholar 

  • Touboul J (2008) Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. SIAM Appl Math 68: 1045–1079

    Article  Google Scholar 

  • Touboul J, Brette R (2008) Spiking dynamics of bidimensional integrate-and-fire neurons (in preparation)

  • Wang XJ (1993) Genesis of bursting oscillations in the Hindmarsh–Rose model and homoclinicity to a chaotic saddle. Phys D 62: 263–274

    Article  Google Scholar 

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Correspondence to Romain Brette.

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Touboul, J., Brette, R. Dynamics and bifurcations of the adaptive exponential integrate-and-fire model. Biol Cybern 99, 319–334 (2008). https://doi.org/10.1007/s00422-008-0267-4

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  • DOI: https://doi.org/10.1007/s00422-008-0267-4

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