The variable $u_{i}$ describes the momentary value of the membrane potential of neuron $i$. In the absence of any input, the potential is at its resting value $u_{\rm rest}$. If an experimentalist injects a current $I(t)$ into the neuron, or if the neuron receives synaptic input from other neurons, the potential $u_{i}$ will be deflected from its resting value.

In order to arrive at an equation that links the momentary voltage $u_{i}(t)-u_{\rm rest}$ to the input current $I(t)$, we use elementary laws from the theory of electricity. A neuron is surrounded by a cell membrane, which is a rather good insulator. If a short current pulse $I(t)$ is injected into the neuron, the additional electrical charge $q=\int I(t^{\prime})dt^{\prime}$ has to go somewhere: it will charge the cell membrane (Fig. 1.6A). The cell membrane therefore acts like a capacitor of capacity $C$. Because the insulator is not perfect, the charge will, over time, slowly leak through the cell membrane. The cell membrane can therefore be characterized by a finite leak resistance $R$.

The basic electrical circuit representing a leaky integrate-and-fire model consists of a capacitor $C$ in parallel with a resistor $R$ driven by a current $I(t)$; see Fig. 1.6.

If the driving current $I(t)$ vanishes, the voltage across the capacitor is given by the battery voltage $u_{\rm rest}$. For a biological explanation of the battery we refer the reader to the next chapter. Here we have simply inserted the battery ‘by hand’ into the circuit so as to account for the resting potential of the cell (Fig. 1.6A).

In order to analyze the circuit, we use the law of current conservation and split the driving current into two components,

The first component is the resistive current $I_{R}$ which passes through the linear resistor $R$. It can be calculated from Ohm’s law as $I_{R}=u_{R}/R$ where $u_{R}=u-u_{\rm rest}$ is the voltage across the resistor. The second component $I_{C}$ charges the capacitor $C$. From the definition of the capacity as $C=q/u$ (where $q$ is the charge and $u$ the voltage), we find a capacitive current $I_{C}={\text{d}}q/{\text{d}}t=C\,{\text{d}}u/{\text{d}}t$. Thus

We multiply Eq. (1.4) by $R$ and introduce the time constant $\tau_{m}=R\,C$ of the ‘leaky integrator’. This yields the standard form

We refer to $u$ as the membrane potential and to $\tau_{m}$ as the membrane time constant of the neuron.

From the mathematical point of view, Eq. (1.5) is a linear differential equation. From the point of view of an electrical engineer, it is the equation of a leaky integrator or $RC$-circuit where resistor $R$ and capacitor $C$ are arranged in parallel. From the point of view of the neuroscientist, Eq. (1.5) is called the equation of a passive membrane.

What is the solution of Eq. (1.5)? We suppose that, for whatever reason, at time $t=0$ the membrane potential takes a value $u_{\rm rest}+\Delta u$. For $t>0$ the input vanishes $I(t)=0$. Intuitively we expect that, if we wait long enough, the membrane potential relaxes to its resting value $u_{\rm rest}$. Indeed, the solution of the differential equation with initial condition $u(t_{0})=u_{\rm rest}+\Delta u$ is

Thus, in the absence of input, the membrane potential decays exponentially to its resting value. The membrane time constant $\tau_{m}=RC$ is the characteristic time of the decay. For a typical neuron it is in the range of 10ms, and hence rather long compared to the duration of a spike which is on the order of 1ms.

The validity of the solution (1.6) can be checked by taking the derivative on both sides of the equation. Since it is the solution in the absence of input, it is sometimes called the ‘free’ solution.

Before we continue with the definition of the integrate-and-fire model and its variants, let us study the dynamics of the passive membrane defined by Eq. (1.5) in a simple example. Suppose that the passive membrane is stimulated by a constant input current $I(t)=I_{0}$ which starts at $t=0$ and ends at time $t=\Delta$. For the sake of simplicity we assume that the membrane potential at time $t=0$ is at its resting value $u(0)=u_{\rm rest}$.

As a first step, let us calculate the time course of the membrane potential. The trajectory of the membrane potential can be found by integrating (1.5) with the initial condition $u(0)=u_{\rm rest}$. The solution for $0<t<\Delta$ is

If the input current never stopped, the membrane potential (1.7) would approach for $t\to\infty$ the asymptotic value $u(\infty)=u_{\rm rest}+R\,I_{0}$. We can understand this result by looking at the electrical diagram of the $RC$-circuit in Fig. 1.6. Once a steady state is reached, the charge on the capacitor no longer changes. All input current must then flow through the resistor. The steady-state voltage at the resistor is therefore $RI_{0}$ so that the total membrane voltage is $u_{\rm rest}+RI_{0}$.

Throughout this book, the term ‘firing time’ refers to the moment when a given neuron emits an action potential $t^{(f)}$. The firing time $t^{(f)}$ in the leaky integrate-and-fire model is defined by a threshold criterion

The form of the spike is not described explicitly. Rather, the firing time is noted and immediately after $t^{(f)}$ the potential is reset to a new value $u_{r}<\vartheta$,

For $t>t^{(f)}$ the dynamics is again given by (1.5) until the next threshold crossing occurs. The combination of leaky integration (1.5) and reset (1.13) defines the leaky integrate-and-fire model (495). The voltage trajectory of a leaky integrate-and-fire model driven by a constant current $I_{0}$ is shown in Fig. 1.9.

For the firing times of neuron $i$ we write $t_{i}^{(f)}$ where $f=1,2,\dots$ is the label of the spike. Formally, we may denote the spike train of a neuron $i$ as the sequence of firing times

where $\delta(x)$ is the Dirac $\delta$ function introduced before, with $\delta(x)=0$ for $x\neq 0$ and $\int_{-\infty}^{\infty}\delta(x){\text{d}}x=1$. Spikes are thus reduced to points in time (Fig. 1.8). We remind the reader that the $\delta$-function is a mathematical object that needs to be inserted into an integral in order to give meaningful results.

We study a leaky integrate-and-fire model which is driven by an arbitrary time-dependent input current $I(t)$; cf. Fig. 1.9B. The firing threshold has a value $\vartheta$ and after firing the potential is reset to a value $u_{r}<\vartheta$.

In the absence of a threshold, the linear differential equation (1.5) has a solution

where $I(t)$ is an arbitrary input current and $\tau_{m}=RC$ is the membrane time constant. We assume here that the input current is defined for a long time back into the past: $t\to-\infty$ so that we do not have to worry about the initial condition. A sinusoidal current $I(t)=I_{0}\sin(\omega\,t)$ or a step current pulse, $I(t)=I_{0}\Theta(t)$ where $\Theta$ denotes the Heaviside step function with $\Theta(t)=0$ for $t\leq 0$ and $\Theta(t)=1$ for $t>0$, are two examples of a time-dependent current, but the solution, Eq. (1.15), is also valid for every other time-dependent input current.

So far our leaky integrator does not have a threshold. What happens to the solution Eq. (1.15), if we add a threshold $\vartheta$? Each time the membrane potential hits the threshold, the variable $u$ is reset from $\vartheta$ to $u_{r}$. In the electrical circuit diagram, the reset of the potential corresponds to removing a charge $q_{r}=C\,(\vartheta-u_{r})$ from the capacitor (Fig. 1.6) or, equivalently, adding a negative charge $-q_{r}$ onto the capacitor. Therefore, the reset corresponds to a short current pulse $I_{r}=-q_{r}\,\delta(t-t^{(f)})$ at the moment of the firing $t^{(f)}$. Indeed, it is not unusual to say that a neuron ‘discharges’ instead of ‘fires’. Since the reset happens each time the neuron fires, the reset current is

where $S(t)$ denotes the spike train, defined in Eq. (1.14).

The short current pulse corresponding to the ‘discharging’ is treated mathematically just like any other time-dependent input current. The total current $I(t)+I_{r}(t)$, consisting of the stimulating current and the reset current, is inserted into the solution (1.15) to give the final result

where the firing times $t^{(f)}$ are defined by the threshold condition

Note that with our definition of the Dirac $\delta$-function in Eq. (1.9), the discharging reset follows immediately after the threshold crossing, so that the natural sequence of events – first firing, then reset – is respected.

Eq. (1.17) looks rather complicated. It has, however, a simple explanation. In Sect. 1.3.2 we have seen that a short input pulse at time $t^{\prime}$ causes at time $t$ a response of the membrane proportional to $\exp\left(-{t-t^{\prime}\over\tau_{m}}\right)$, sometimes called the impulse response function or Green’s function; cf. Eq. (1.11). The second term on the right-hand side of Eq. (1.17) is the effect of the discharging current pulses at the moment of the reset.

In order to interpret the last term on the right-hand side, we think of a stimulating current $I(t)$ as consisting of a rapid sequence of discrete and short current pulses. In discrete time, there would be a different current pulse in each time step. Because of the linearity of the differential equation, the effect of all these short current pulses can be added. When we return from discrete time to continuous time, the sum of the impulse response functions turns into the integral on the right-hand side of Eq. (1.17).

The leaky integrate-and-fire model is defined by the differential equation (1.5), i.e.,

combined with the reset condition

where $t^{(f)}$ are the firing times

As we have seen in the previous subsection, the linear equation can be integrated and yields the solution (1.17). It is convenient to rewrite the solution in the form

where we have introduced filters $\eta(s)=(u_{r}-\vartheta)\,\exp\left(-{s\over\tau_{m}}\right)$ and $\kappa(s)={1\over C}\exp\left(-{s\over\tau_{m}}\right)\,$. Interestingly, Eq. (1.22) is much more general than the leaky integrate-and-fire model, because the filters do not need to be exponentials but could have any arbitrary shape. The filter $\eta$ describes the reset of the membrane potential and, more generally, accounts for neuronal refractoriness. The filter $\kappa$ summarizes the linear electrical properties of the membrane. Eq. (1.22) in combination with the threshold condition (1.21) is the basis of the Spike Response Model and Generalized Linear Models, which will be discussed in Part II.

Formally, the complex Fourier transform of a real-valued function $f(t)$ with argument $t$ on the real line is

where $|\hat{f}(\omega)|$ and $\phi_{f}(\omega)$ are called amplitude and phase of the Fourier transform at frequency $\omega$. The mathematical condition for a well-defined Fourier transform is that the function $f$ be Lebesgue integrable with integral $\int_{-\infty}^{\infty}|f(t)|{\text{d}}t<\infty$. If $f$ is a function of time, then $\hat{f}(\omega)$ is a function of frequency. An inverse Fourier transform leads back from frequency-space to the original space, i.e., time.

For a linear system, the above definition gives rise to several convenient rules for Fourier-transformed equations. For example, let us consider the system

where $I(t)$ is a real-valued input (e.g., a current), $u(t)$ the real-valued system output (e.g., a voltage) and $\kappa$ a linear response filter, or kernel, with $\kappa(s)=0$ for $s<0$ because of causality. The convolution on the right-hand side of Eq. (1.24) turns after Fourier transformation into a simple multiplication, as shown by the following steps of calculation:

where we introduced in the last step the variable $t^{\prime}=t-s$ and used the definition (1.23) of the Fourier transform.

Similarly, the derivative $du/dt$ of a function $u(t)$ can be Fourier-transformed using the product rule of integration. The Fourier transform of the derivative of $u(t)$ is $i\omega\hat{u}(\omega)$.

While introduced here as a purely mathematical operation, it is often convenient to visualize the Fourier transform in the context of a physical system driven by a periodic input. Consider the linear system of Eq. (1.24) with an input

A short comment on the notation. If the input is a current, it should be real-valued, as opposed to a complex number. We therefore take $I_{0}$ as a real and positive number and focus on the real part of the complex equation (1.26) as our physical input. When we perform a calculation with complex numbers, we therefore implicitly assume that, at the very end, we only take the real part of solution. However, the calculation with complex numbers turns out to be convenient for the steps in between.

Inserting the periodic drive, Eq. (1.26), into Eq. (1.24) yields

Hence, if the input is periodic at frequency $\omega$ the output is so, too. The term in square brackets is the Fourier transform of the linear filter. We write $u(t)=u_{0}\,e^{i\phi_{\kappa}(\omega)}\,e^{i\omega t}$. The ratio between the amplitude of the output and that of the input is

The phase $\phi_{\kappa}(\omega)$ of the Fourier-transformed linear filter $\kappa$ corresponds to phase shift between input and output or, to say it differently, a delay $\Delta=\phi_{\kappa}/\omega=\phi_{\kappa}\,T/2\pi$ where $T$ is the period of the oscillation. Fourier transforms will play a role in the discussion of signal processing properties of connected networks of neurons in Part III of the book.