Skip to main content
Help us understand how you use our websites. Take part in our 30 minute study now.

Resonant response of a Hodgkin-Huxley neuron to a spike train input


Experiments show that neurons have a tendency to respond to signals tuned to a resonant frequency [1]. In order to understand the general properties of a resonant response of a neuron, we study the silent Hodgkin-Huxley neuron driven by periodic input. The current arriving through the synapse consists of a set of spikes I p (t) ~ gsyn ∑(t/τ) exp(-t/τ) C(t) (V a -V syn ), where g syn is the synapse conductivity, τ is the time constant associated with the synapse conduction, Va is the maximum membrane potential and Vsyn is the reversal potential of the synapse.


See Figures 1 and 2.

Figure 1

The phase diagram for typical HH model parameters [2]in the limit of small synaptic conductivity. There is a well-pronounced minimum at T i = 17.5 ms. The resonant nature of the response can be seen also at multiples of this value, at T i ≈ 34 ms and T i ≈ 50 ms. Near the resonance the system has the tendency to mode locking with high values of k, where k = T o /T i is the ratio of the output ISI to the input ISI. For example near the main resonance frequency we find narrow regions with k = 5, 6 or 9. Areas with bistable solutions are shown in grey. We expect the resonance at T i = 17.5 ms to survive in the presence of noise.

Figure 2

In the limit of small T i the distinction between the firing spikes and subthreshold oscillations disappears and the output signal decreases to 0 for sufficiently large g syn . Broken line in the figure indicates a transition to nonfiring behavior. In the area below this transition the amplitude of the spikes gradually increases. Solid lines are borders of the mode-locked states with different values of k. Properties of this model are similar to the HH model with a sinusoidal driving current at intermediate values of input ISI Ti = 5–12 ms. However the results in both the high and the low frequency regime are qualitatively different. In the case of a sinusoidal input there is only one resonance frequency and reported values of k are lower [3].


  1. 1.

    Hutcheon B, Yarom Y: Resonance, oscillation and the intrinsic frequency preference of neurons. Trends Neurosci. 2000, 23: 216-222. 10.1016/S0166-2236(00)01547-2.

    CAS  PubMed  Article  Google Scholar 

  2. 2.

    Hasegawa H: Responses of a Hodgkin-Huxley neuron to various types of spike-train inputs. Phys Rev E. 2000, 61: 718-726. 10.1103/PhysRevE.61.718.

    CAS  Article  Google Scholar 

  3. 3.

    Lee SG, Kim S: Bifurcation analysis of mode-locking structure in a Hodgkin-Huxley neuron under sinusoidal current. Phys Rev E. 2006, 73: 041924-10.1103/PhysRevE.73.041924.

    Article  Google Scholar 

Download references


Part of the numerical computation was performed in the Computer Center of the Tri-city Academic Computer Network in Gdansk, Poland.

Author information



Corresponding author

Correspondence to Lech S Borkowski.

Rights and permissions

Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Borkowski, L.S. Resonant response of a Hodgkin-Huxley neuron to a spike train input. BMC Neurosci 10, P250 (2009).

Download citation


  • Animal Model
  • Membrane Potential
  • Time Constant
  • Resonant Frequency
  • General Property