Interplay of subthreshold activity, time-delayed feedback, and noise on neuronal firing patterns

Feedback connections and noise are ubiquitous features of neuronal networks and affect in a determinant way the patterns of neural activity. Here we study how the subthreshold dynamics of a neuron interacts with time-delayed feedback and noise. We use a Hodgkin-Huxley-type model of a thermoreceptor neuron and assume the feedback to be linear, corresponding effectively to a recurrent electrical connection via gap junctions. This type of feedback can model electrical autapses, which connect the terminal fibers of a neuron’s axon with dendrites from the same neuron. Thus the delay in the feedback loop is due basically to the axonal propagation time. We chose model parameters for which the neuron displays, in the absence of feedback and noise, only subthreshold oscillations. These oscillations, however, take the neuron close to the firing threshold, such that small perturbations can drive it above the level for generation of action potentials. The resulting interplay between weak delayed feedback, noise, and the subthreshold intrinsic activity is nontrivial. For negative feedback, depending on the delay, the firing rate can be lower than in the noise-free situation. This is due to the fact that noise inhibits feedback-induced spikes by driving the neuronal oscillations away from the firing threshold. For positive feedback, there are regions of delay values where the noise-induced spikes are inhibited by the feedback; in this case, it is the feedback that drives the neuronal oscillations away from the threshold. Our study contributes to a better understanding of the role of electrical self-connections in the presence of noise and subthreshold activity.

Figure 1

(Color online) (a) Interspike intervals (normalized to $T_0$) and (b) firing rate (number of spikes per intrinsic oscillation period) vs the delay time for negative feedback strength, $\eta =-0.001\mathrm{mS}∕{\mathrm{cm}}^2$, in the absence of noise. Other parameters are $C_M=1\mu \mathrm{F}∕{\mathrm{cm}}^2$, $g_l=0.1\mathrm{mS}∕{\mathrm{cm}}^2$, $g_{Na}=1.5\mathrm{mS}∕{\mathrm{cm}}^2$, $g_K=2\mathrm{mS}∕{\mathrm{cm}}^2$, $g_{sd}=0.25\mathrm{mS}∕{\mathrm{cm}}^2$, $g_{sr}=0.4\mathrm{mS}∕{\mathrm{cm}}^2$, $V_l=-60\mathrm{mV}$, $V_{Na}=50\mathrm{mV}$, $V_K=-90\mathrm{mV}$, $V_{sd}=50\mathrm{mV}$, $V_{sr}=-90\mathrm{mV}$, $s_{Na}=0.25{\mathrm{mV}}^{-1}$, $s_K=0.25{\mathrm{mV}}^{-1}$, $s_{sd}=0.09{\mathrm{mV}}^{-1}$, $V_{0Na}=-25\mathrm{mV}$, $V_{0K}=-25\mathrm{mV}$, $V_{0sd}=-40\mathrm{mV}$, ${\tau }_K=2\mathrm{ms}$, ${\tau }_{sd}=10\mathrm{ms}$, ${\tau }_{sr}=20\mathrm{ms}$, $\theta =0.17$, $\mu =0.012$, $A_1=1.3$, $A_2=3.0$, and $T_c=25°\mathrm{C}$.

Figure 2

(Color online) (a) Detail of the ISI profile displayed in Fig. 1a. Time traces of the membrane potential for $\eta =-0.001$, $\tau ∕T_0=0.31$ (b), 0.5 (c), 0.68 (d), and 0.695 (e).

Figure 3

(Color online) (a) Same as Fig. 2, but for larger delay times. Notice the logarithmic vertical scale. Time traces of the membrane potential for $\tau ∕T_0=4.3$ (b), 4.5 (c), 4.7 (d), and 4.75 (e).

Figure 4

(Color online) (a),(c) Interspike intervals (normalized to $T_0$) and (b),(d) firing rate (number of spikes per intrinsic oscillation period) vs the delay time for negative feedback strength, $\eta =-0.001$, in the presence of noise: (a), (b) $D=0.005$ and (c), (d) $D=0.01$.

Figure 5

(Color online) (a),(c) Interspike intervals (normalized to $T_0$) and (b),(d) firing rate (number of spikes per intrinsic oscillation period) vs the delay time for positive feedback strength, $\eta =0.001$, in the presence of noise: (a), (b) $D=0.005$ and (c), (d) $D=0.01$.

Figure 6

(Color online) Membrane potential when noise is included in the simulations $(D=0.01)$. No feedback (a), negative feedback $(\eta =-0.001)$ (b),(c), and positive feedback $(\eta =0.001)$ (d),(e). The delay time is $\tau =T_0∕2$ (b),(d) and $\tau =T_0$ (c),(e).

Figure 7

(Color online) Probability density function of interspike intervals for fixed noise $(D=0.01)$ and varying feedback. (a) Without feedback, (b),(c) with positive feedback [$\eta =0.0001$ in (b), $\eta =0.0005$ in (c)] and a delay value such that the feedback decreases the firing rate $(\tau =T_0∕2)$. In (a) the inset displays in detail the first five peaks of the distribution.

Figure 8

(Color online) Probability density function of interspike intervals for fixed negative feedback $(\eta =-0.001)$ and varying noise. The delay value is such that there is an irregular alternation of spikes and subthreshold oscillations [$\tau =4.7T_0$; a spike train is shown in Fig. 3d]. Without noise (a) and $D=0.001$ (b), 0.005 (c), and 0.01 (d).

Figure 9

(Color online) Amplitude (a),(c) and frequency (c),(d) of the limit-cycle oscillator, Eq. (A1), with positive feedback, $\eta =0.1$ (a),(b) and with negative feedback, $\eta =-0.1$ (c),(d). ${\omega }_0=1$.