Subsections

• 5.6.1 Sub- and superthreshold stimulation
  • 5.6.1.1 Example: Interval distribution in the superthreshold regime
  
  
• 5.6.2 Coefficient of variation C_V
  • 5.6.2.1 Example: Poisson neuron with absolute refractoriness

5.6 The subthreshold regime

One of the aims of noisy neuron models is to mimic the large variability of interspike intervals found, e.g., in vertebrate cortex. To arrive at broad interval distributions, it is not just sufficient to introduce noise into a neuron model. Apart from the noise level, other neuronal parameters such as the firing threshold or a bias current have to be tuned so as to make the neuron sensitive to noise. In this section we introduce a distinction between super- and subthreshold stimulation (Bugmann et al., 1997; König et al., 1996; Shadlen and Newsome, 1994; Troyer and Miller, 1997; Abeles, 1991). In Section 5.7 we will show that, in the subthreshold regime, there is a close relation between the two different noise models discussed above, viz., escape noise (cf. Section 5.3) and diffusive noise (cf. Section 5.5). Finally, in Section 5.8 we turn to the phenomenon of stochastic resonance and discuss signal transmission in the subthreshold regime.

5.6.1 Sub- and superthreshold stimulation

An arbitrary time-dependent stimulus I(t) is called subthreshold, if it generates a membrane potential that stays - in the absence of noise - below the firing threshold. Due to noise, however, even subthreshold stimuli can induce action potentials. Stimuli that induce spikes even in a noise-free neuron are called superthreshold.

The distinction between sub- and superthreshold stimuli has important consequences for the firing behavior of neurons in the presence of noise. To see why, let us consider an integrate-and-fire neuron with constant input I₀ for t > 0. Starting from u(t = 0) = u_r, the trajectory of the membrane potential is

$$\infty \left[\vphantom{ 1 - e^{-t/\tau_m} }\right. \tau_{m} \left.\vphantom{ 1 - e^{-t/\tau_m} }\right] \tau_{m}$$ (5.109)

In the absence of a threshold, the membrane potential approaches the value u_$\infty$ = R I₀ for t$\to$$\infty$. If we take the threshold $\vartheta$ into account, two cases may be distinguished. First, if u_$\infty$ < $\vartheta$ (subthreshold stimulation), the neuron does not fire at all. Second, if u_$\infty$ > $\vartheta$ (superthreshold stimulation), the neuron fires regularly. The interspike interval is s₀ derived from u₀(s₀) = $\vartheta$. Thus

$$\tau {u_\infty - u_r \over u_\infty - \vartheta}$$ (5.110)

Figure 5.19: Interval distribution P₀(t| 0) for superthreshold stimuli. The membrane potential distribution p(u, t) is shifted across the threshold and generates an interval distribution P₀(t| 0) (schematic figure).

We now add diffusive noise. In the superthreshold regime, noise has little influence, except that it broadens the interspike interval distribution. Thus, in the superthreshold regime, the spike train in the presence of diffusive noise, is simply a noisy version of the regular spike train of the noise-free neuron.

On the other hand, in the subthreshold regime, the spike train changes qualitatively, if noise is switched on; see (König et al., 1996) for a review. Stochastic background input turns the quiescent neuron into a spiking one. In the subthreshold regime, spikes are generated by the fluctuations of the membrane potential, rather than by its mean (Bugmann et al., 1997; Feng, 2001; Shadlen and Newsome, 1994; Troyer and Miller, 1997; Abeles, 1991). The interspike interval distribution is therefore broad; see Fig. 5.20.

5.6.1.1 Example: Interval distribution in the superthreshold regime

Figure 5.20: Integrate-and-fire neuron ( $\tau_{m}^{}$ = 10 ms) with superthreshold (left column) and subthreshold (right column) stimulation. A. Without noise, a neuron with superthreshold stimulus I_a fires regularly. Spikes are marked by vertical lines. The threshold is indicated by a horizontal line. The dashed line shows the evolution of the membrane potential in the absence of the threshold. B. The same neuron with subthreshold stimulation I_b does not fire. C. If we add stochastic excitatory and inhibitory spike input ( w_± = 0.05 at $\nu_{{\pm}}^{}$ = 1.6 kHz) to the constant input I_a, the membrane potential drifts away from the noise-free reference trajectory, but firing remains fairly regular. D. The same sequence of input spikes added to the subthreshold current I_b generates irregular spiking. E and F. Histogram of interspike intervals in C and D, respectively, as an estimator of the interval distribution P₀(s) in the super- and subthreshold regime. The mean interval $\langle$s$\rangle$ is 12ms (E) and 50ms (F); the C_V values are 0.30 and 0.63, respectively.

For small noise amplitude 0 < $\sigma$ $\ll$ u_$\infty$ - $\vartheta$, the interval distribution is centered at s₀. Its width can be estimated from the width of the fluctuations $\langle$$\Delta$u²_$\infty$$\rangle$ of the free membrane potential; cf. Eq. (5.99). Since the membrane potential crosses the threshold with slope u₀', there is a scaling factor u₀' = du₀(t)/dt evaluated at t = s₀; cf. Fig. 5.19. The interval distribution is therefore approximately given by a Gaussian with mean s₀ and width $\sigma$/$\sqrt{{2}}$ u₀' (Tuckwell, 1988), i.e.,

$${1\over \sqrt{\pi}} {u_0' \over \sigma} \left[\vphantom{ - { (u_0')^2\, (t - s_0)^2 \over \sigma^2 } }\right. {(u_0')^2\, (t - s_0)^2 \over \sigma^2} \left.\vphantom{ - { (u_0')^2\, (t - s_0)^2 \over \sigma^2 } }\right]$$ (5.111)

5.6.2 Coefficient of variation C_V

Figures 5.20e and 5.20f show that interval distributions in the super- and subthreshold regime look quite differently. To quantify the width of the interval distribution, neuroscientists often evaluate the coefficient of variation, short C_V, defined as the ratio of the variance and the mean squared,

$${\langle \Delta s^2 \rangle \over \langle s \rangle^2}$$ (5.112)

where $\langle$s$\rangle$ = $\int_{0}^{\infty}$P₀(s) ds and $\langle$$\Delta$s²$\rangle$ = $\int_{0}^{\infty}$s² P₀(s) ds - $\langle$s$\rangle^{2}_{}$. A Poisson distribution has a value of C_V = 1. A value of C_V > 1, implies that a given distribution is broader than a Poisson distribution with the same mean. If C_V < 1, then the spike train is more regular than that generated by a Poisson neuron of the same rate. A long refractory period and low noise level decrease the C_V value.

5.6.2.1 Example: Poisson neuron with absolute refractoriness

We study a Poisson neuron with absolute refractory period $\Delta^{{\rm abs}}_{}$. For t - $\hat{{t}}$ > $\Delta^{{\rm abs}}_{}$, the neuron is supposed to fire stochastically with rate r. The interval distribution is given in Eq. (5.20) with mean $\langle$s$\rangle$ = $\Delta^{{\rm abs}}_{}$ + 1/r and variance $\langle$$\Delta$s²$\rangle$ = 1/r². The coefficient of variation is therefore

$${\Delta^{\rm abs} \over \langle s \rangle}$$ (5.113)

Let us compare the C_V of Eq. (5.113) with that of a homogeneous Poisson process of the same mean rate $\nu$ = $\langle$s$\rangle^{{-1}}_{}$. As we have seen, a Poisson process has C_V = 1. A refractory period $\Delta^{{\rm abs}}_{}$ > 0 lowers the C_V, because a neuron with absolute refractoriness fires more regularly than a Poisson neuron. If we increase $\Delta^{{\rm abs}}_{}$, we must increase the instantaneous rate r in order to keep the same mean rate $\nu$, In the limit of $\Delta^{{\rm abs}}_{}$$\to$$\langle$s$\rangle$, the C_V approaches zero, since the only possible spike train is regular firing with period $\langle$s$\rangle$.