Multimodal transition and stochastic antiresonance in squid giant axons

The experimental data of Takahashi et al. [Physica D 43, 318 (1990)], on the response of squid giant axons stimulated by periodic sequence of short current pulses is interpreted within the Hodgkin-Huxley model. The minimum of the firing rate as a function of the stimulus amplitude $I_0$ in the high-frequency regime is due to the multimodal transition. Below this singular point only odd multiples of the driving period remain and the system is sensitive to noise. The coefficient of variation has a maximum and the firing rate has a minimum as a function of the noise intensity, which is an indication of the stochastic coherence antiresonance. The model calculations reproduce the frequency of occurrence of the most common modes in the vicinity of the transition. A linear relation of output frequency vs $I_0$ above the transition is also confirmed.

Figure 1

The average firing rate $f_o/f_i$ as a function of the stimulus amplitude $I$ from the work of Takahashi et al. [12]. $I_t$ is the minimum current threshold obtained in the range $T_i=2.5\text{ms}$ to $T_i=6.5\text{ms}$. The measurements were carried out at $T_i=3.8\text{ms}$.

Figure 2

The calculated average firing rate at $T_i=7\text{ms}$ without noise.

Figure 3

The bifurcation diagram in the $T_i\text{−}{I}_0$ plane, showing the main mode-locked states in the model without noise. The unmarked intrusion in the upper left corner is the 5:1 state. The bottom part of the figure is occupied by the silent state. In the firing part of the diagram there are two solutions below the dotted line (see, e.g., the tip of the 3:1 and 4:1 states). Here, the limit cycle coexists with the fixed point. The average firing rate is a continuous function of $I_0$ at the excitation threshold between 4 and 5 ms and between 7.2 and 9 ms. The average firing rate is a continuous function of $I_0$. Full squares show the location of the minima of the firing rate. The borders of states below $T_i=4.5\text{ms}$ are shown in an approximate form. The detailed picture is less regular and somewhat more complex.

Figure 4

The relative frequency of occurrence of low-order even (upper) and odd (lower) modes for the parameter set of Fig. 2. Numbers $2,3,\dots$ indicate modes $2:1,3:1,\dots$, respectively. The vertical line marks the position of the minimum of the firing rate. Odd modes disappear below approximately $I_0=17.5\mu \text{A}/{\text{cm}}^2$.

Figure 5

The linear relation of the firing rate vs the stimulus amplitude above the multimodal transition point at $T_i=4\text{ms}$. These are experimental results of Takahashi et al. [12].

Figure 6

Calculated average firing rate vs stimulus amplitude above the multimodal transition for three values of $T_i$. The current pulse width is 0.6 ms.

Figure 7

Sensitivity of the $f_o/f_i=0.4$ plateau from Fig. 2 to noise. The location of the minimum of the firing rate remains almost unchanged up to $D\simeq {10}^{-2}$.

Figure 8

The relative frequency of occurrence of low-order even (upper) and odd (lower) modes for the parameter set of Fig. 2 and in the presence of small Gaussian noise, $D=0.001$.

Figure 9

The firing rate (upper) and the coefficient of variation (lower) as functions of the noise intensity at $T_i=7\text{ms}$. The middle curve in the upper diagram was obtained for $I_0=15\mu \text{A}/{\text{cm}}^2$. At $D=0$ all three curves start in the 3:1 mode. The maximum of the coefficient of variation as a function of the noise intensity is a property of the stochastic coherence antiresonance. The maximum of $C_V$ and the minimum of the firing rate occur at different noise levels. The irregularity of the $I_0=14\mu \text{A}/{\text{cm}}^2$ curve is a consequence of proximity to the excitation threshold.