Given the input $I(t^{\prime})$ for $t^{\prime}<t$ and the past firing times $t^{(f)}<t$, the membrane potential of a generalized integrate-and-fire model (e.g., the adaptive leaky ingrate-and-fire model) or a Spike Response Model (SRM) can be calculated from Eq. (6.7) or Eq. (6.27) respectively; cf. Chapter 6. For example, the value of the membrane potential of a SRM can be expressed as

where $I^{\rm det}$ is the known driving current (the superscript ‘det’ stands for deterministic) and $\kappa$ and $\eta$ are filters that describe the response of the membrane to an incoming pulse or an outgoing spike; see Chapter 6. In the deterministic model the next spike occurs when $u$ reaches the threshold $\vartheta$. In order to introduce some variability into the neuronal spike generator, we replace the strict threshold by a stochastic firing criterion. In the noisy threshold model, spikes can occur at any time with a probability density

that depends on the momentary distance between the (noiseless) membrane potential and the threshold; see Fig. 9.1. We can think of $f$ as an escape rate similar to the one encountered in models of chemical reactions (529). In the mathematical theory of point processes, the quantity $\rho$ is called a ‘stochastic intensity’. Since we use $\rho$ in the context of neuron models we will refer to it as a firing intensity.

The choice of the escape function $f$ in Eq. (9.2) is arbitrary. A reasonable condition is to require $f\to 0$ for $u\to-\infty$ so that the neuron does not fire if the membrane potential is far below threshold. A common choice is the exponential,

where $\beta$ and $\tau_{0}$ are parameters. For $\beta\to\infty$, the soft threshold turns into a sharp one so that we return to the noiseless model. Below we discuss some further choices of the escape function in Eq. (9.2).

The SRM of Eq. (9.1) together with the exponential escape rate of Eq. (9.3) is an example of a Generalized Linear Model. Applying the theory of Generalized Linear Models (cf. Chapter 10) to the SRM with escape noise enables a rapid extraction of model parameters from experimental data.

In slice experiments it was found (247) that the exponential escape rate of Eq. (9.3) provides an excellent fit to the spiking intensity of real neurons (Fig. 9.3). Moreover, the firing times of an AdEx model driven by a deterministic fluctuating current $I^{\rm det}(t)$ and a unknown white noise current $\xi(t)$ can also be well fitted by the Spike Response Model of Eq. (9.1) combined with an exponential escape rate as we will see below in Section 9.4.

Nevertheless, we may wonder whether Eq. (9.2) is a sufficiently general noise model. We have seen in Chapter 2 that the concept of a pure voltage threshold is questionable. More generally, the spike trigger process could, for example, also depend on the slope $\dot{u}=du/dt$ with which the ‘threshold’ is approached. In the noisy threshold model, we may therefore also consider an escape rate (or hazard) which depends not only on $u$ but also on its derivative $\dot{u}$

We will return to Eq. (9.4) in Section 9.4.

Note that the time-dependence of the firing intensity $\rho(t)$ on the right-hand side of Eq. (9.4) arises implicitly via the membrane potential $u(t)$. In an even more general model, we could in addition include an explicit time dependence, e.g., to account for a reduced spiking probability immediately after a spike at $t^{(f)}$. Instead of an explicit dependence, a slightly more convenient way to implement an additional time dependence is via a time-dependent threshold $\vartheta\longrightarrow\vartheta(t)$ which we have already encountered in Eq. (6.31). An even more general escape rate model therefore is

In Chapter 11 we will see that a Spike Response Model with escape noise and dynamic threshold can explain neuronal firing with a high degree of accuracy.

In discrete time, we consider the probability $P_{F}(u)$ of firing in a finite time step given the neurons has membrane potential $u$. It is bounded from above by one – despite the fact that the escape rate can be arbitrarily large. We start from a model in continuous time and discretize time as it is often done in simulations. In a straightforward discretization scheme, we would calculate the probability of firing during a time step $\Delta t$ as $\int_{t}^{t+\Delta t}\rho(t^{\prime}){\text{d}}t^{\prime}\approx\rho(t)\,\Delta t$. However, for $u\gg\vartheta$, the hazard $\rho(t)=f[u(t)-\vartheta]$ can take large values; see, e.g., Eq. (9.3). Thus $\Delta t$ must be taken extremely short so as to guarantee $\rho(t)\,\Delta t<1$.

In order to arrive at an improved discretization scheme, we calculate the probability that a neuron does not fire in a time step $\Delta t$. The probability $S(t)$ to survive for a time $t$ without firing decays according to $dS/dt=-\rho(t)\,S(t)$. Integration of the differential equation over a finite time $\Delta t$ yields an exponential factor $S(t)=\exp[-\int_{0}^{t}\rho(t^{\prime})\,{\text{d}}t^{\prime}]$; compare the discussion of the survivor function in Chapter 7 (Section 7.5). If the neuron does not survive, it must have fired. Therefore we arrive at a firing probability

Even if $f$ diverges for $u\to\infty$, the probability of firing remains bounded between zero and one. Aslo, we see that for small $\Delta t$, the probability $P_{F}$ scales as $f\Delta t$. We see from Fig. 9.5A that, for an exponential escape rate, an increase in the discretization $\Delta t$ mainly shifts the firing curve to the left while the form remains roughly the same. An increase of the noise level makes the curve flatter; cf. Fig. 9.5B.

Note that, because of refractoriness, it is impossible for integrate-and-fire models (and for real neurons) to fire more than one spike in a short time bin $\Delta t$. Refractoriness is implemented in the model of Eq. (9.1) by a significant reset of the membrane potential via the refractory kernel $\eta$.