Regular spiking in high-conductance states: The essential role of inhibition

Strong inhibitory input to neurons, which occurs in balanced states of neural networks, increases synaptic current fluctuations. This has led to the assumption that inhibition contributes to the high spike-firing irregularity observed in vivo. We used single compartment neuronal models with time-correlated (due to synaptic filtering) and state-dependent (due to reversal potentials) input to demonstrate that inhibitory input acts to decrease membrane potential fluctuations, a result that cannot be achieved with simplified neural input models. To clarify the effects on spike-firing regularity, we used models with different spike-firing adaptation mechanisms, and we observed that the addition of inhibition increased firing regularity in models with dynamic firing thresholds and decreased firing regularity if spike-firing adaptation was implemented through ionic currents or not at all. This fluctuation-stabilization mechanism provides an alternative perspective on the importance of strong inhibitory inputs observed in balanced states of neural networks, and it highlights the key roles of biologically plausible inputs and specific adaptation mechanisms in neuronal modeling.

Figure 1

(a) During a 2-s-long simulation, the intensity of inhibitory input increases from 0 to 20 kHz. The presynaptic spike trains are modeled as Poisson point processes. (b) The intensity of excitatory input is increased simultaneously with the inhibition in order to maintain the mean membrane potential constant. This increases fluctuations of the synaptic current but decreases fluctuations of the membrane potential. The orange lines signify the mean value $\pm$ standard deviation. (c),(d) The effect of membrane potential stabilization on firing regularity. The intensity of the inhibitory input follows the time course shown in (a), and the intensity of the excitatory activity is increased in order to maintain the steady-state postsynaptic firing rate at approximately 10 Hz (blue trace). The firing regularity (measured here by the Fano factor, orange trace) decreases with the addition of inhibition to the input in the model with spike-firing adaptation by M currents (c). However, the model with dynamic threshold (d) exhibits a clear increase in regularity with the added inhibitory input. The mean firing rate and Fano factor were calculated by a sliding window of length 100 ms from approximately ${10}^5$ trials.

Figure 2

Stabilization of the membrane potential. (a) Top panel: Membrane potential fluctuations as a function of the mean membrane potential for different values of $c$. The markers represent data obtained from simulations with different excitatory input intensities ${\lambda }_{\mathrm{e}}$. The full lines represent the effective time-constant approximation (ETA, Appendix pp1). Bottom panel: The standard deviation ${\sigma }_I$ of the total synaptic current $I_{\mathrm{syn}}=I_{\mathrm{e}}+I_{\mathrm{i}}$. Note that ${\sigma }_V$ decreases with growing $c$ even though ${\sigma }_I$ increases. (b) Overview of ${\sigma }_V$ (color) for all achievable $\langle V\rangle$ ($x$-axis) at given $c$ ($y$-axis). $C=1\mu \mathrm{F}/{\mathrm{cm}}^2$ and $g_{\mathrm{L}}=0.045\mathrm{mS}/{\mathrm{cm}}^2$, approximation of ${\sigma }_V$ computed from the ETA. Heatmaps for different values of $g_{\mathrm{L}}$ are provided in the supplementary Fig. 2 in [50] and for different values of $A_{\mathrm{i}}$ [Eq. (21)] in the supplementary Fig. 3 in [50].

Figure 3

The effect of membrane potential stabilization on spiking regularity in the GLIF models. (a)–(c) Dependence of the ${\mathrm{C}}_{\mathrm{V}}$ of ISIs on the postsynaptic firing rate for different values of $c$ (color-coded). Data points where ${\lambda }_{\mathrm{e}}>100\mathrm{kHz}$ are represented as filled points. In the LIF and AHP-LIF model, higher $c$ universally leads to higher ${\mathrm{C}}_{\mathrm{V}}$. In contrast, in the DT-LIF model, higher $c$ can lead to more regular spike trains, especially if the input intensities are high. If $V_{\infty }\left(c\right)\le \theta$ (or ${\theta }_0$ for the DT-LIF model, i.e., $c\ge 2.75$), the firing rate will eventually drop to 0. (d),(e) Contour plots with color-coded ${\mathrm{C}}_{\mathrm{V}}$, $c$ on the $y$-axis. If more than one input can produce the same PSFR with the same $c$, the lowest possible value of ${\mathrm{C}}_{\mathrm{V}}$ is color-coded, resulting in the discontinuity in (f). The data points were obtained from simulations with different input intensities ${\lambda }_{\mathrm{e}}$, ${\lambda }_{\mathrm{i}}$.

Figure 4

The effect of membrane potential stabilization on spiking regularity in the Hodgkin-Huxley models. (a)–(c) Dependence of the ${\mathrm{C}}_{\mathrm{V}}$ of ISIs on the firing rate for different values of $c$ (color-coded). Data points where ${\lambda }_{\mathrm{e}}>100\mathrm{kHz}$ are represented as filled points. In the subthreshold regimes, the output rate reaches its maximum and then starts dropping to zero. For the HH-DT model (c), the ${\mathrm{C}}_{\mathrm{V}}$ decreases at this point, whereas for the HH-0 (a) and HH-M (b) models, no clear improvement is observed. (d)–(f) Contour plots with color-coded ${\mathrm{C}}_{\mathrm{V}}$, $c$ on the $y$-axis. If more than one input can produce the same PSFR with the same $c$, the lowest possible value of ${\mathrm{C}}_{\mathrm{V}}$ is color-coded. Note that there is no discontinuity in (f), unlike in Fig. 3. The transition to the more regular states with growing $c$ is continuous, as is illustrated in supplementary Fig. 4 in [50]. The data points were obtained from simulations with different input intensities ${\lambda }_{\mathrm{e}}$, ${\lambda }_{\mathrm{i}}$.

Figure 5

Constant inhibition trajectories for the dynamic threshold models. In both the DT-LIF (a) and HH-DT (b) models, increasing the presynaptic inhibitory firing rate (color) is beneficial for the firing regularity (measured by ${\mathrm{C}}_{\mathrm{V}}$, $y$-axis) for a wide range of PSFRs ($x$-axis).

Figure 6

Membrane potential with conductance input without synaptic filtering. (a) Membrane potential fluctuations as a function of the mean membrane potential for different values of $c=\frac{{\lambda }_{\mathrm{i}}{b}_{\mathrm{i}}}{{\lambda }_{\mathrm{e}}{b}_{\mathrm{e}}}$ (color-coded), as calculated from Eqs. (42) and (43). The full line represents the limit ${lim}_{{\lambda }_{\mathrm{e}},{\lambda }_{\mathrm{i}}\to \infty }{\sigma }_{V,W}\left({\langle V\rangle }_W\right)$. The membrane potential is not stabilized at infinite inputs. Above certain depolarizations, inhibition increases membrane potential fluctuations, contrary to the case of conductance input with synaptic filtering. (b) Heatmap with color-coded standard deviation of the membrane potential. Parameters used were $b_{\mathrm{e}}=0.0045$, $b_{\mathrm{i}}=0.0150$.

Figure 7

Approximation of the PSFR and ${\mathrm{C}}_{\mathrm{V}}$ of LIF. Simulation results are color-coded. The full lines represent the approximations from Eqs. (B2) and (B3). The firing rate is approximated very well (a). ${\mathrm{C}}_{\mathrm{V}}$ is approximated well for very high PSFRs ($>1\mathrm{kHz}$) (b). This is due to the short input time constants (3 ms for excitatory, 10 ms for inhibitory). For the simulation, we used the time step $\mathrm{\Delta }t=0.1\mathrm{ms}$ if the expected PSFR was $<100\mathrm{Hz}$, and $\mathrm{\Delta }t\frac{{\mu }_{\mathrm{ISI}}}{10\mathrm{ms}}$ otherwise.

Figure 8

High-conductance limit of the AHP-LIF model. (a) Equilibrium potential (in mV) in the infinite-conductance limit [Eq. (B8)] for different values of $c$ (shown in blue). For high $c$, the value $V_{\infty }^{\mathrm{AHP}}\left(c\right)$ is very close to the threshold (dashed black line). The value of $V_{\infty }\left(c\right)$ (33), corresponding to the LIF model, is shown in orange for comparison. The M-current adaptation clearly pushes the equilibrium potential closer to the threshold, leading to bursting behavior. (b)–(d) Time-course of the membrane potential of LIF with M-current adaptation with $c=1.7$ for different values of input intensities. The membrane potential (shown in blue) follows closely $V_{\mathrm{ef}}^{\mathrm{AHP}}$ (shown in orange). When $V_{\mathrm{ef}}^{\mathrm{AHP}}>\theta$, the neuron is bursting; otherwise, the neuron is silent. With higher input intensities, the probability of $V_{\mathrm{ef}}^{\mathrm{AHP}}\le \theta$ drops, and the firing rate becomes increasingly more regular.