In the framework of the Spike Response Model (SRM) the state of a neuron is described by a single variable $u$ which we interpret as the membrane potential. In the absence of input, the variable $u$ is at its resting value, $u_{\rm rest}$. A short current pulse will perturb $u$ and it takes some time before $u$ returns to rest (Fig. 6.11). The function $\kappa(s)$ describes the time course of the voltage response to a short current pulse at time $s=0$. Because the subthreshold behavior of the membrane potential is taken as linear, the voltage response $h$ to an arbitrary time-dependent stimulating current $I^{\rm ext}(t^{\prime})$ is given by the integral $h(t)=\int_{0}^{\infty}\kappa(s)\,I^{\rm ext}(t-s)\,{\text{d}}s$.

Spike firing is defined by a threshold process. If the membrane potential reaches the threshold $\vartheta$, an output spike is triggered. The form of the action potential and the after-potential is described by a function $\eta$. Let us suppose that the neuron has fired some earlier spikes at times $t^{(f)}<t$. The evolution of $u$ is given by

The sum runs over all past firing times $t^{(f)}$ with $f=1,2,3,\dots$ of the neuron under consideration. Introducing the spike train $S(t)=\sum_{f}\delta(t-t^{(f)})$, Eq. (6.27) can be also written as a convolution

In contrast to the leaky integrate-and-fire neuron discussed in Chapter 1 the threshold $\vartheta$ is not fixed, but time-dependent

Firing occurs whenever the membrane potential $u$ reaches the dynamic threshold $\vartheta(t)$ from below

Dynamic thresholds can be directly measured in experiments (162; 35; 338) and are a standard feature of phenomenological neuron models.

So far Eq. (6.27) in combination with the threshold condition (6.30) defines a mathematical model. Can we give a biological interpretation of the terms?

The kernel $\kappa(s)$ is the linear response of the membrane potential to an input current. It describes the time course of a deviation of the membrane potential from its resting value that is caused by a short current pulse (“impulse response”).

The kernel $\eta$ describes the standard form of an action potential of neuron $i$ including the negative overshoot which typically follows a spike (the spike afterpotential). Graphically speaking, a contribution $\eta$ is ‘pasted in’ each time the membrane potential reaches the threshold $\vartheta$ (Fig. 6.12A). Since the form of the spike is always the same, the exact time course of the action potential carries no information. What matters is whether there is the event ‘spike’ or not. The event is fully characterized by the firing time $t^{(f)}$.

In a simplified model, the form of the action potential may therefore be neglected as long as we keep track of the firing times $t^{(f)}$. The kernel $\eta$ describes then simply the ‘reset’ of the membrane potential to a lower value after the spike at $t^{(f)}$ just like in the integrate-and-fire model

with a parameter $\eta_{0}>0$. The spike after-potential decays back to zero with a recovery time constant $\tau_{\rm recov}$. The leaky integrate-and-fire model is in fact a special case of the SRM, with parameter $\eta_{0}=(\vartheta-u_{r})$ and $\tau_{\rm recov}=\tau_{m}$.

In this section, we show that the leaky integrate-and-fire neuron with adaptation defined above in Eq. (6.7) and (6.8) is a special case of the Spike Response Model. Let us recall that the leaky integrate-and-fire model follows the equation of a linear circuit with resistance $R$ and capacity $C$

where $\tau_{m}=RC$ is the time constant, $E_{0}$ the leak reversal potential, $w_{k}$ are adaptation variables, and $I_{i}$ is the input current to neuron $i$. At each firing time

the voltage is reset to a value $u_{r}$. At the same time, the adaptation variables are increased by an amount $b_{k}$

The equations of the adaptive leaky integrate-and-fire model, Eqs. (6.35) and  (6.37) can be classified as linear differential equations. However, because of the reset of the membrane potential after firing, the integration is not completely trivial. In fact, there are two different ways of proceeding with the integration. The first method is to treat the reset after each firing as a new initial condition - this is the procedure typically chosen for a numerical integration of the model. Here we follow a different path and describe the reset as a current pulse. As we will see, the result enables a mapping of the leaky integrate-and-fire model to the SRM.

Let us consider a short current pulse $I_{i}^{\rm out}=-q\,\delta(t)$ applied to the $RC$ circuit. It removes a charge $q$ from the capacitor $C$ and lowers the potential by an amount $\Delta u=-q/C$. Thus, a reset of the membrane potential from a value of $u=\vartheta$ to a new value $u=u_{r}$ corresponds to an ‘output’ current pulse which removes a charge $q=C\,(\vartheta-u_{r})$. The reset takes place every time when the neuron fires. The total reset current is therefore

where the sum runs over all firing times $t_{i}^{(f)}$. We add the output current (6.38) on the right-hand side of (6.35),

Since Eqs. (6.39) and (6.40) define a system of linear equations, we can integrate each term separately and superimpose the result at the end. To perform the integration, we proceed in three steps. First, we shift the voltage so as to set the equilibrium potential to zero. Second, we calculate the eigenvalues and eigenvectors of the ‘free’ equations in the absence of input (and therefore no spikes). If there are $K$ adaptation variables, we have a total of $K+1$ eigenvalues which we label as $\lambda_{1},\lambda_{2}\dots$. The associated eigenvectors are $\mbox{\boldmath\(e\)}_{k}$ with components $(e_{k0},e_{k1},\dots,e_{kK})^{T}$. Third, we express the response to an impulse $\Delta u=1$ in the voltage (no perturbation in the adaptation variables) in terms of the $K+1$ eigenvectors: $(1,0,0,...,0)^{T}=\sum_{k=0}^{K}\beta_{k}\mbox{\boldmath\(e\)}_{k}$. Finally, we express the pulse caused by a reset of voltage and adaptation variables in terms of the eigenvectors $(-\vartheta+u_{r},b_{1},b_{2},...,b_{K})^{T}=\sum_{k=0}^{K}\gamma_{k}\mbox{\boldmath\(e\)}_{k}$.

The response to the reset pulses yields the kernel $\eta$ while the response to voltage pulses yields the filter $\kappa(s)$ of the SRM

with kernels

As usual, $\Theta(x)$ denotes the Heaviside step function.

The models discussed in this chapter are point neurons, i.e., models that do not take into account the spatial structure of a real neuron. In Chapter 3 we have already seen that the electrical properties of dendritic trees can be described by compartmental models. In this section, we want to show that neurons with a linear dendritic tree and a voltage threshold for spike firing at the soma can be mapped to the Spike Response Model.

We study an integrate-and-fire model with a passive dendritic tree described by $n$ compartments. Membrane resistance, core resistance, and capacity of compartment $\mu$ are denoted by $R_{\text{T}}^{\mu}$, $R_{\text{L}}^{\mu}$, and $C^{\mu}$, respectively. The longitudinal core resistance between compartment $\mu$ and a neighboring compartment $\nu$ is $r^{\mu\nu}=(R_{\text{L}}^{\mu}+R_{\text{L}}^{\nu})/2$; cf. Fig. 3.8. Compartment $\mu=1$ represents the soma and is equipped with a simple mechanism for spike generation, i.e., with a threshold criterion as in the standard integrate-and-fire model. The remaining dendritic compartments ($2\leq\mu\leq n$) are passive.

Each compartment $1\leq\mu\leq n$ of neuron $i$ may receive input $I_{i}^{\mu}(t)$ from presynaptic neurons. As a result of spike generation, there is an additional reset current $\Omega_{i}(t)$ at the soma. The membrane potential $V_{i}^{\mu}$ of compartment $\mu$ is given by

where the sum runs over all neighbors of compartment $\mu$. The Kronecker symbol $\delta^{\mu\nu}$ equals unity if the upper indices are equal; otherwise, it is zero. The subscript $i$ is the index of the neuron; the upper indices $\mu$ or $\nu$ refer to compartments. Below we will identify the somatic voltage $V_{i}^{1}$ with the potential $u_{i}$ of the Spike Response Model.

Equation (6.45) is a system of linear differential equations if the external input current is independent of the membrane potential. The solution of Eq. (6.45) can thus be formulated by means of Green’s functions $G_{i}^{\mu\nu}(s)$ that describe the impact of an current pulse injected in compartment $\nu$ on the membrane potential of compartment $\mu$. The solution is of the form

Explicit expressions for the Green’s function $G_{i}^{\mu\nu}(s)$ for arbitrary geometry have been derived by Abbott et al. (1) and Bressloff and Taylor (66).

We consider a network made up of a set of neurons described by Eq. (6.45) and a simple threshold criterion for generating spikes. We assume that each spike $t_{j}^{(f)}$ of a presynaptic neuron $j$ evokes, for $t>t_{j}^{(f)}$, a synaptic current pulse $\alpha(t-t_{j}^{(f)})$ into the postsynaptic neuron $i$. The actual amplitude of the current pulse depends on the strength $W_{ij}$ of the synapse that connects neuron $j$ to neuron $i$. The total input to compartment $\mu$ of neuron $i$ is thus

Here, $\Gamma_{i}^{\mu}$ denotes the set of all neurons that have a synapse with compartment $\mu$ of neuron $i$. The firing times of neuron $j$ are denoted by $t_{j}^{(f)}$.

In the following we assume that spikes are generated at the soma in the manner of the integrate-and-fire model. That is to say, a spike is triggered as soon as the somatic membrane potential reaches the firing threshold, $\vartheta$. After each spike the somatic membrane potential is reset to $V_{i}^{1}=u_{r}<\vartheta$. This is equivalent to a current pulse

so that the overall current due to the firing of action potentials at the soma of neuron $i$ amounts to

We will refer to equations (6.46)–(6.49) together with the threshold criterion for generating spikes as the multi-compartment integrate-and-fire model.

Using the above specializations for the synaptic input current and the somatic reset current the membrane potential (6.46) of compartment $\mu$ in neuron $i$ can be rewritten as

with

The kernel $\epsilon_{i}^{\mu\nu}(s)$ describes the effect of a presynaptic action potential arriving at compartment $\nu$ on the membrane potential of compartment $\mu$. Similarly, $\eta_{i}^{\mu}(s)$ describes the response of compartment $\mu$ to an action potential generated at the soma.

The triggering of action potentials depends on the somatic membrane potential only. We define $u_{i}=V_{i}^{1}$, $\eta_{i}(s)=\eta_{i}^{1}(s)$ and, for $j\in\Gamma_{i}^{\nu}$, we set $\epsilon_{ij}=\epsilon_{i}^{1\nu}$. This yields the equation of the SRM