Journal of Computational Neuroscience

, Volume 37, Issue 1, pp 161-180

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

  • Victor J. BarrancaAffiliated withMathematical Sciences Department, Rensselaer Polytechnic Institute Email author 
  • , Daniel C. JohnsonAffiliated withDivision of Applied Mathematics, Brown University
  • , Jennifer L. MoyherAffiliated withMathematical Sciences Department, Rensselaer Polytechnic Institute
  • , Joshua P. SauppeAffiliated withPhysics Department, University of Wisconsin-Madison
  • , Maxim S. ShkarayevAffiliated withDepartment of Physics and Astronomy, Iowa State University
  • , Gregor KovačičAffiliated withMathematical Sciences Department, Rensselaer Polytechnic Institute
  • , David CaiAffiliated withDepartment of Mathematics, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong UniversityCourant Institute of Mathematical Sciences and Center for Neural Science, New York University


In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.


Adaptation current Integrate-and-fire networks Bifurcations Numerical methods Efficient neuronal models