Spike-triggered averages for passive and resonant neurons receiving filtered excitatory and inhibitory synaptic drive

A path-integral approach is developed for the analysis of spike-triggered average quantities in neurons with voltage-gated subthreshold currents. Using a linearization procedure to reduce the models to the generalized integrate-and-fire form, analytical expressions are obtained in an experimentally relevant limit of fluctuation-driven firing. The influences of voltage-gated channels as well as excitatory and inhibitory synaptic filtering are shown to affect significantly the neuronal dynamics prior to the spike. Analytical forms are given for all relevant physiological quantities, such as the mean voltage triggered to the spike, mean current flowing through voltage-gated channels, and the mean excitatory and inhibitory conductance waveforms prior to a spike. The mathematical results are shown to be in good agreement with numerical simulations of the underlying nonlinear conductance-based models. The method promises to provide a useful analytical tool for the prediction and interpretation of the temporal structure of spike-triggered averages measured experimentally.

Figure 1

(Color online) Three different types of neuron model give rise to three qualitatively different spike-triggered averages. (A) Response of the model to a small current pulse causing a steady-state depolarization of about 0.5 mV. Three types of response are distinguished: passive decay, sag-rebound, and damped oscillations. (B) Sample voltage trajectories leading to an output spike in the simulations. [(C) and (E)] Comparison of numerical simulations (symbols) of the full conductance-based model to the low-rate theoretical predictions [solid lines, Eqs. (30, 31, 32, 34, 35, 36, B20, B21)]. (C) Spike-triggered average voltage. (D) Spike-triggered average synaptic conductances (circles: excitatory conductance; squares: inhibitory conductance). (E) Spike-triggered average values of the intrinsic membrane current (circles: $I_h$, $h$-current; squares: $I_{\text{NaP}}$, persistent sodium current). Parameters for the case of passive decay were rate of arrival of excitatory and inhibitory conductance pulses, ${\mathcal{R}}_e=4.17\text{kHz}$ and ${\mathcal{R}}_i=1.25\text{kHz}$; amplitude of conductance pulses, $c_e=3.6\text{nS}$ and $c_i=4.6\text{nS}$; firing rate $r=1.1\text{Hz}$. For the sag-rebound case, ${\mathcal{R}}_e=3.06\text{kHz}$, ${\mathcal{R}}_i=0.8\text{kHz}$, $c_e=3.27\text{nS}$, $c_i=6.26\text{nS}$, and firing rate $r=1.3\text{Hz}$. For damped oscillations, ${\mathcal{R}}_e=4.59\text{kHz}$, ${\mathcal{R}}_i=2.81\text{kHz}$, $c_e=0.76\text{nS}$, $c_i=0.80\text{nS}$, and firing rate $r=0.1\text{Hz}$.

Figure 2

(Color online) Spike-triggered averages in the simulations and in the theory. Numerical simulations of the simplified model (symbols) are compared to the low-rate theoretical predictions (solid lines). The model is defined by Eqs. (7, 8, 9, 10), and the theoretical predictions were calculated using Eqs. (30, 31, 32) for the passive case, and Eqs. (34, 35, 36, B20, B21) for the two other cases. (A) Sample voltage trajectories leading to an output spike in the simulations. (B) Spike-triggered average voltage. (C) Spike-triggered average synaptic conductances (circles: excitatory conductance; squares: inhibitory conductance). (D) Spike-triggered average values of the intrinsic membrane current (circles: $I_h$, $h$-current; squares: $I_{\text{NaP}}$, persistent sodium current). The parameters of the reduced model were derived from the full model in Fig. 1. For all three cases, ${\tau }_w=75\text{ms}$. For the case of passive decay, ${\tau }_v=6.56\text{ms}$, $\gamma =0$, ${\sigma }_x=3.65\text{mV}$, ${\sigma }_y=2.13\text{mV}$, and firing rate $r=1.0\text{Hz}$. For the sag-rebound case, ${\tau }_v=6.68\text{ms}$, $\gamma =0.62$, ${\sigma }_x=2.86\text{mV}$, ${\sigma }_y=2.41\text{mV}$, and firing rate $r=1.0\text{Hz}$. For damped oscillations, ${\tau }_v=39.02\text{ms}$, $\gamma =3.20$, ${\sigma }_x=4.67\text{mV}$, ${\sigma }_y=3.53\text{mV}$, and firing rate $r=0.2\text{Hz}$.

Figure 3

(Color online) Spike-triggered averages at very low firing rates. The theoretical predictions [solid lines, Eqs. (30, 31, 32, 34, 35, 36, B20, B21)] are compared with numerical simulations of the simplified model (symbols) for a firing rate of ${10}^{-4}\text{Hz}$. (A) Sample voltage trajectories leading to an output spike in the simulations. (B) Spike-triggered average voltage. (C) Spike-triggered average synaptic conductances (circles: excitatory conductance; squares: inhibitory conductance). (D) Spike-triggered average values of the intrinsic membrane current (circles: $I_h$, $h$-current; squares: $I_{\text{NaP}}$, persistent sodium current). The parameters are the same as in Fig. 2, except ${\sigma }_x=1.54\text{mV}$, ${\sigma }_y=1.37\text{mV}$ (decay), ${\sigma }_x=1.39\text{mV}$, ${\sigma }_y=1.31\text{mV}$ (sag-rebound), ${\sigma }_x=2.33\text{mV}$, and ${\sigma }_y=2.36\text{mV}$ (damped oscillations).

Figure 4

(Color online) Spike-triggered averages: effects of correlations between excitation and inhibition. The theoretical predictions of Appendix (solid lines) with numerical simulations of the simplified model (symbols) for a correlation coefficient $\rho =0.4$. (A) Sample voltage trajectories leading to an output spike. (B) Spike-triggered average voltage. (C) Spike-triggered average synaptic conductances (circles: excitatory conductance; squares: inhibitory conductance). (D) Spike-triggered average values of the intrinsic membrane current (circles: $I_h$, $h$-current; squares: $I_{\text{NaP}}$, persistent sodium current). The parameters are the same as in Fig. 2, except ${\sigma }_x=4.86\text{mV}$, ${\sigma }_y=3.33\text{mV}$, firing rate 0.7 Hz (decay), ${\sigma }_x=3.27\text{mV}$, ${\sigma }_y=2.90\text{mV}$, firing rate 0.1 Hz (sag-rebound), and ${\sigma }_x=5.84\text{mV}$, ${\sigma }_y=5.89\text{mV}$, firing rate 0.2 Hz (damped oscillations).