Subsections

• • 5.7.0.1 Example: Comparison of diffusion model and Arrhenius&Current escape rate

5.7 From diffusive noise to escape noise

In the subthreshold regime, the integrate-and-fire model with stochastic input (diffusive noise) can be mapped approximately onto an escape noise model with a certain escape rate f (Plesser and Gerstner, 2000). In this section, we motivate the mapping and the choice of f.

In the absence of a threshold, the membrane potential of an integrate-and-fire model has a Gaussian probability distribution, around the noise-free reference trajectory u₀(t). If we take the threshold into account, the probability density at u = $\vartheta$ of the exact solution vanishes, since the threshold acts as an absorbing boundary; see Eq. (5.92). Nevertheless, in a phenomenological model, we can approximate the probability density near u = $\vartheta$ by the `free' distribution (i.e., without the threshold)

$$\big\{ \vartheta \Delta \big\} \propto \Delta \left\{\vphantom{ - {[u_0(t)- \vartheta ]^2 \over 2 \langle \Delta u^2(t) \rangle } }\right. {[u_0(t)- \vartheta ]^2 \over 2 \langle \Delta u^2(t) \rangle} \left.\vphantom{ - {[u_0(t)- \vartheta ]^2 \over 2 \langle \Delta u^2(t) \rangle } }\right\}$$ (5.114)

where u₀(t) is the noise-free reference trajectory. The idea is illustrated in Fig. 5.21.

We have seen in Eq. (5.97) that the variance $\langle$$\Delta$u²(t)$\rangle$ of the free distribution rapidly approaches a constant value $\sigma^{2}_{}$/2. We therefore replace the time dependent variance 2$\langle$$\Delta$u(t)²$\rangle$ by its stationary value $\sigma^{2}_{}$. The right-hand side of Eq. (5.114) is then a function of the noise-free reference trajectory only. In order to transform the left-hand side of Eq. (5.114) into an escape rate, we divide both sides by $\Delta$t. The firing intensity is thus

$$\vartheta {c_1 \over \tau_m} \left\{\vphantom{ - {[u_0(t)-\vartheta]^2 \over \sigma^2} }\right. {[u_0(t)-\vartheta]^2 \over \sigma^2} \left.\vphantom{ - {[u_0(t)-\vartheta]^2 \over \sigma^2} }\right\}$$ (5.115)

The factor in front of the exponential has been split into a constant parameter c₁ > 0 and the time constant $\tau_{m}^{}$ of the neuron in order to show that the escape rate has units of one over time. Equation (5.115) is the well-known Arrhenius formula for escape across a barrier of height ($\vartheta$ - u₀)² in the presence of thermal energy $\sigma^{2}_{}$ (van Kampen, 1992).

Figure 5.21: The distribution of the membrane potential around the noise free reference trajectory u₀(t) is given by p(u, t). At t = t₀, where the reference trajectory has a discontinuity, the distribution of the membrane potential is shifted instantaneously across the threshold. The probability of firing at t₀ is given by the shaded surface under the distribution.

Let us now suppose that the neuron receives, at t = t₀, an input current pulse which causes a jump of the membrane trajectory by an amount $\Delta$u > 0; see Fig. (5.21). In this case the Gaussian distribution of membrane potentials is shifted instantaneously across the threshold so that there is a nonzero probability that the neuron fires exactly at t₀. To say it differently, the firing intensity $\rho$(t) = f[u₀(t) - $\vartheta$] has a $\delta$ peak at t = t₀. The escape rate of Eq. (5.115), however, cannot reproduce this $\delta$ peak. More generally, whenever the noise free reference trajectory increases with slope $\dot{{u}}_{0}^{}$ > 0, we expect an increase of the instantaneous rate proportional to $\dot{{u}}_{0}^{}$, because the tail of the Gaussian distribution drifts across the threshold; cf. Eq. (5.111). In order to take the drift into account, we generalize Eq. (5.115) and study

$$\dot{{u}}_{0}^{} \left(\vphantom{ {c_1\over \tau_m} + {c_2 \over \sigma} [\dot{u}_0]_+ }\right. {c_1 \over \tau_m} {c_2 \over \sigma} \dot{{u}}_{0}^{} \left.\vphantom{ {c_1\over \tau_m} + {c_2 \over \sigma} [\dot{u}_0]_+ }\right) \left\{\vphantom{ - {[u_0(t)-\vartheta]^2 \over \sigma^2} }\right. {[u_0(t)-\vartheta]^2 \over \sigma^2} \left.\vphantom{ - {[u_0(t)-\vartheta]^2 \over \sigma^2} }\right\}$$ (5.116)

where $\dot{{u}}_{0}^{}$ = du₀/dt and [x]₊ = x for x > 0 and zero otherwise. We call Eq. (5.116) the Arrhenius&Current model (Plesser and Gerstner, 2000).

We emphasize that the right-hand side of Eq. (5.116) depends only on the dimensionless variable

$${u_0(t) - \vartheta \over \sigma}$$ (5.117)

and its derivative $\dot{{x}}$. Thus the amplitude of the fluctuations $\sigma$ define a `natural' voltage scale. The only relevant variable is the momentary distance of the noise-free trajectory from the threshold in units of the noise amplitude $\sigma$. A value of x = - 1 implies that the membrane potential is one $\sigma$ below threshold. A distance of u - $\vartheta$ = - 10mV at high noise (e.g., $\sigma$ = 10mV) is as effective in firing a cell as a distance of 1 mV at low noise ( $\sigma$ = 1mV).

5.7.0.1 Example: Comparison of diffusion model and Arrhenius&Current escape rate

To check the validity of the arguments that led to Eq. (5.116), let us compare the interval distribution generated by the diffusion model with that generated by the Arrhenius&Current escape model. We use the same input potential u₀(t) as in Fig. 5.18. We find that the interval distributions P_I^diff for the diffusive noise model and P_I^A&C for the Arrhenius&Current escape model are nearly identical; cf. Fig. (5.22). Thus the Arrhenius&Current escape model yields an excellent approximation to the diffusive noise model. We quantify the error of the approximation by the measure

$${\int_0^\infty \left[P^{\rm diff}_I(t\vert) - P_I^{\rm A\&C}(t\ve... ...{d}}t \over \int_0^\infty \left[P^{\rm diff}_I(t\vert) \right]^2\, {\text{d}}t}$$ (5.118)

For the example shown in Fig. 5.22 we find E = 0.026. Over a large set of subthreshold stimuli, the difference between the diffusive noise and the Arrhenius&Current model is typically in the range of E = 0.02; the best choice of parameters is c₁ $\approx$ 0.72 and c₂ $\approx$ $\pi^{{-1/2}}_{}$ (Plesser and Gerstner, 2000). The simple Arrhenius model of Eq. (5.115) or the sigmoidal model of Eq. (5.51) have errors which are larger by a factor of 3-5.

Even though the Arrhenius&Current model has been designed for sub-threshold stimuli, it also works remarkably well for super-threshold stimuli with typical errors around E = 0.04. An obvious shortcoming of the escape rate (5.116) is that the instantaneous rate decreases with u for u > $\vartheta$. The superthreshold behavior can be corrected if we replace the Gaussian exp(- x²) by 2 exp(- x²)/[1 + erf(- x)] (Herrmann and Gerstner, 2001a). The subthreshold behavior remains unchanged compared to Eq. (5.116) but the superthreshold behavior of the escape rate f becomes linear. With this new escape rate the typical error E in the super-threshold regime is as small as that in the subthreshold regime.

Figure 5.22: The interval distributions P_I(t| 0) for diffusive noise (solid line) and Arrhenius&Current escape noise (dashed line) are nearly identical. The input potential is the same as in Fig. 5.18. Taken from (Plesser and Gerstner, 2000).