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

Introduction

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) ~ g_syn ∑(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, V_a is the maximum membrane potential and V_syn is the reversal potential of the synapse.

Results

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 T_i = 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].