Spontaneous spiking in an autaptic Hodgkin-Huxley setup

The effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop. The loop is based on the so-called autapse phenomenon in which dendrites establish connections not only to neighboring cells but also to its own axon. The biophysical modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing such a built in delayed feedback. The fluctuations stem from intrinsic channel noise, being caused by the stochastic nature of the gating dynamics of ion channels. The influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval. The delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution, exhibiting a delay-induced reduction in the spontaneous spiking activity at characteristic frequencies. Moreover, a specific frequency-locking mechanism is detected for the mean interspike interval.

Figure 1

(Color online) Sketch of an autapse: sketch of a neuronal cell exhibiting a self-delayed feedback mechanism.

Figure 2

The phase diagram $ϵ\text{−}\tau$ for repetitive firing within the Hodgkin-Huxley model containing the delay coupling strength $ϵ$ is depicted vs the delay time $\tau$. The two black solid lines indicate the boundaries of critical coupling strength $ϵ$ versus the delay time $\tau$ at which the behavior changes from the rest state to a repetitive firing. The gray areas give the parameter regime for which repetitive firing is observed.

Figure 3

Simulated spike trains: (a) membrane potential $V\left(t\right)$ from the stochastic Hodgkin-Huxley model in the absence of feedback, i.e., with $ϵ=0$ versus time $t$; (b) $V\left(t\right)$ versus time $t$ in the presence of finite feedback of strength $ϵ=0.07\text{mS}/{\text{cm}}^2$ exhibiting repetitive spiking for a chosen delay time $\tau =35\text{ms}$. The number of potassium ion channels is set to $N_{\text{K}}=300$ and that of the sodium ion channels is set to $N_{\text{Na}}=1000$. The spontaneous, i.e., noise-induced, action potentials exhibit a burstinglike behavior in case of the delay coupling (b). In (a) we indicated with the double-arrow line a typical occurring noise-induced interspike interval $T_i$.

Figure 4

(Color online) Interspike interval histograms (ISIHs) versus the interspike interval $T$ of the stochastic Hodgkin-Huxley model with a delay coupling mechanism at different coupling strengths $ϵ$, as indicated in the figure in units of $\text{mS}/{\text{cm}}^2$. The bin width for $T$ has been chosen throughout this work at 0.2 ms. The chosen number of potassium ion channels is $N_{\text{K}}=150$, while the number of sodium ion channels is set at $N_{\text{Na}}=500$. The delay time $\tau$ is set at 35 ms.

Figure 5

(Color online) ISIH vs interspike duration $T$ for different noise levels corresponding to different numbers of sodium and potassium channels as indicated. The chosen delay time is $\tau =35\text{ms}$; the coupling strength is set at $ϵ=0.03\text{mS}/{\text{cm}}^2<{ϵ}_c^{+}\left(\tau \right)$ and $N_{\text{Na}}=\frac{10}{3}{N}_{\text{K}}$.

Figure 6

(Color online) ISIH for different values of the suprathreshold coupling strength $ϵ>{ϵ}_c^{+}\left(\tau \right)$: the distribution of the interspike intervals is depicted for fixed delay time $\tau =35\text{ms}$; the numbers of ion channels are $N_{\text{K}}=150$ and $N_{\text{Na}}=500$. The values of the coupling strengths indicated in the plot are in units of $\text{mS}/{\text{cm}}^2$.

Figure 7

(Color online) Average interspike interval $⟨T⟩$ as a function of the delay time $\tau$: the mean interspike interval $⟨T⟩$ [cf. Eq. (6)] is plotted versus the delay time $\tau$ for the following parameters: the used ion channel numbers are $N_{\text{K}}=150$ and $N_{\text{Na}}=500$, and the coupling strengths are set at $ϵ=0.1\text{mS}/{\text{cm}}^2$ (black dashed line), $ϵ=0.2\text{mS}/{\text{cm}}^2$ (red dotted line), and $ϵ=0.4\text{mS}/{\text{cm}}^2$ (blue solid line).

Figure 8

The inverse of the number of spikes that fit into the delay-time interval $\kappa$ is plotted as a function of the delay time $\tau$ for $N_{\text{K}}=150$ and $N_{\text{K}}=500$ and a delay coupling strength of $ϵ=0.4\text{mS}/{\text{cm}}^2$.

Figure 9

(Color online) Comparison of the stochastic neuronal dynamics Hodgkin-Huxley model including feedback [Eqs. (1, 2, 3a, 3b, 3c, 3d, 3e, 3f, 4, 5a, 5b, 5c)] with the dynamics of the feedback assisted Kuramoto-type phase oscillator model [Eq. (8)]. The mean interspike interval (red dashed lines) is the result of the numerical integration of the stochastic Hodgkin-Huxley model with delayed feedback; there is quantitatively good agreement between periods of oscillations as obtained from a phase oscillator dynamics of the Kuramoto-type delayed feedback model (solid black line). Regions with multiple solutions in the phase model correspond to discontinuous jumps between different types of multiplets in the Hodgkin-Huxley model. The parameters for the neuronal modeling are chosen as $N_{\text{K}}=150$, $N_{\text{Na}}=500$, and $ϵ=0.4\text{mS}/{\text{cm}}^2$; those of the phase oscillator are given by oscillator frequency $\omega =2\pi /15.5{\text{ms}}^{-1}$ and feedback strength $A=1/15.5{\text{ms}}^{-1}$.