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

BMC Neuroscience200910 (Suppl 1) :P250

  • Published:


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


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
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
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].



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

Authors’ Affiliations

Quantum Physics Division, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan, Poland


  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.PubMedView ArticleGoogle Scholar
  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.View ArticleGoogle Scholar
  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.View ArticleGoogle Scholar


© Borkowski; licensee BioMed Central Ltd. 2009

This article is published under license to BioMed Central Ltd.