Switch of encoding characteristics in single neurons by subthreshold and suprathreshold stimuli

Spike-triggered analysis is a statistical method used to elucidate encoding properties in neural systems by estimating the statistical structure of input stimulus preceding spikes. A recent numerical study suggested that the profile of the spike-triggered average (STA) changes depending on whether the mean input stimuli are subthreshold or suprathreshold. Here we analytically verify the difference between subthreshold STA and suprathreshold STA by using the spike response model (SRM). We show by moment expansion that the suprathreshold STA is proportional to the first derivative of the response kernel, and that the subthreshold STA is expressed by a linear combination of the response kernel and its first derivative. We verify whether the analytical results obtained from the SRM can be applied to a multicompartment model with Hodgkin-Huxley type dynamics.

Figure 1

(Color online) Time course of membrane potential in the SRM (thick solid line) and that in the threshold crossing model (thin solid line): (a) the case of subthreshold mean stimuli and (b) the case of suprathreshold mean stimuli. The mean trajectory of the SRM and that of the threshold crossing model are indicated by the dashed line (time varying) and dash dotted line (time constant), respectively. The firing threshold $\theta$ is shown by the dotted line.

Figure 2

(Color online) (A) Type I response kernel $\kappa \left(t\right)$ of SRM $\left({\tau }_0=1\right)$. (B) Suprathreshold STA obtained in numerical simulation (crosses) and the suprathreshold STA derived from the response kernel of the SRM as the derivative of the kernel ${\kappa }^{\prime }\left(t\right)$ (solid line). Parameters are set as $\sigma =0.01,\theta =0.99$. (C) Subthreshold STA obtained in numerical simulation (crosses). The solid line in C is a line fitted to the numerically obtained subthreshold STA by a linear combination of the response kernel and its first derivative $c_1\kappa \left(\tau \right)+c_2{\kappa }^{\prime }\left(\tau \right)$. Coefficients of the linear combination are determined by the least-squares method ($c_1=0.4156$ and $c_2=0.3274$). Parameters are set as $\sigma =0.1,\theta =1.1$.

Figure 3

(Color online) (A) Type II response kernel $\kappa \left(t\right)$ of the SRM $\left({\tau }_0=1,{\tau }_1=2\right)$. (B) Suprathreshold STA obtained in numerical simulation (crosses) and the suprathreshold STA derived from the kernel of the SRM as the derivative of the kernel ${\kappa }^{\prime }\left(t\right)$ (solid line). Parameters are set as $\sigma =0.01,\theta =0.55$. (C) Subthreshold STA obtained in numerical simulation (crosses) and a curve fitted to the numerically obtained subthreshold STA by the linear combination of the response kernel and its first derivative (solid line). Coefficients are determined by the least square method ($c_1=0.6456$ and $c_2=0.32$). Parameters are set as $\sigma =0.12,\theta =0.6$.

Figure 4

Dependence of the STA on the firing threshold $\theta$ in numerical simulations using the SRM. The numerically obtained STAs for different firing thresholds are fitted by a linear combination of the response kernel and its derivative and the dependence of the ratio between the coefficients $c_1/c_2$ on the threshold $\theta$ is shown. The fitting curve is calculated using the least-squares method. The type I kernel is used, and the noise amplitude is set at $\sigma =0.01$. The suprathreshold (subthreshold) regime corresponds to $\theta <1\left(\theta >1\right)$.

Figure 5

(Color online) Dependence of the suprathreshold STA of the SRM on noise amplitude $\left(\sigma \right)$. The numerically obtained suprathreshold STA (symbols) is fitted well by the theoretically derived curve for the small amplitudes of noise. This shows that the dependence of STA on noise amplitude derived in our analysis is valid for weak noise. The STA for each noise level is indicated by the following symbols: $\sigma =0.0025$ (plus, $+$), 0.005 (cross, $\times$), 0.0075 (triangle, △), 0.01 (square, ◻), 0.015 (circle, ○). The type I kernel is used, and the firing threshold is set at $\theta =0.99$.

Figure 6

(Color online) (A) Schematic diagram of a multicompartment model with Hodgkin-Huxley type dynamics. The external inputs are given to a dendritic compartment. (B) Impulse response (corresponding to response kernel) obtained from numerical simulation. (C) Suprathreshold STA derived analytically as a first-order derivative of the response kernel shown in B (solid line) and the suprathreshold STA obtained directly by numerical simulations using the multicompartment model (crosses). (D) Subthreshold STA fitted using a linear combination of the response kernel shown in B and its derivative (solid line) to the numerically obtained subthreshold STA (crosses).