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.
Similar content being viewed by others
References
Angelino E, Brenner MP (2007) Excitability constraints on voltage-gated sodium channels. PLoS Comput Biol 3(9): 1751–1760
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
Brette R (2004) Dynamics of one-dimensional spiking neuron models. J Math Biol 48(1): 38–56
Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J Neurophysiol 94: 3637–3642
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
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
Gerstner W, Kistler W (2002) Spiking neuron models. Cambridge University Press, Cambridge
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
Izhikevich E (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15(5): 1063–1070
Izhikevich E (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, Cambridge
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
Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation. J Physiol Pathol Gen 9: 620–635
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
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
Touboul J (2008) Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. SIAM Appl Math 68: 1045–1079
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-008-0267-4