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4.1 Integrate-and-fire model

In this section, we give an overview of integrate-and-fire models. The leaky integrate-and-fire neuron introduced in Section 4.1.1 is probably the best-known example of a formal spiking neuron model. Generalizations of the leaky integrate-and-fire model include the nonlinear integrate-and-fire model that is discussed in Section 4.1.2. All integrate-and-fire neurons can either be stimulated by external current or by synaptic input from presynaptic neurons. Standard formulations of synaptic input are given in Section 4.1.3.

4.1.1 Leaky Integrate-and-Fire Model

Figure 4.1: Schematic diagram of the integrate-and-fire model. The basic circuit is the module inside the dashed circle on the right-hand side. A current I(t) charges the RC circuit. The voltage u(t) across the capacitance (points) is compared to a threshold $ \vartheta$. If u(t) = $ \vartheta$ at time ti(f) an output pulse $ \delta$(t - ti(f)) is generated. Left part: A presynaptic spike $ \delta$(t - tj(f)) is low-pass filtered at the synapse and generates an input current pulse $ \alpha$(t - tj(f)).
\centerline{ \includegraphics[width=100mm]{I-Fnew.eps}}

The basic circuit of an integrate-and-fire model consists of a capacitor C in parallel with a resistor R driven by a current I(t); see Fig. 4.1. The driving current can be split into two components, I(t) = IR + IC. The first component is the resistive current IR which passes through the linear resistor R. It can be calculated from Ohm's law as IR = u/R where u is the voltage across the resistor. The second component IC 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 IC = C du/dt. Thus

I(t) = $\displaystyle {u(t)\over R}$ + C $\displaystyle {{\text{d}}u\over {\text{d}}t}$ . (4.2)

We multiply (4.2) by R and introduce the time constant $ \tau_{m}^{}$ = R C of the `leaky integrator'. This yields the standard form

$\displaystyle \tau_{m}^{}$ $\displaystyle {{\text{d}}u\over {\text{d}}t}$ = - u(t) + R I(t) . (4.3)

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

In integrate-and-fire models the form of an action potential is not described explicitly. Spikes are formal events characterized by a `firing time' t(f). The firing time t(f) is defined by a threshold criterion

t(f) :     u(t(f)) = $\displaystyle \vartheta$ . (4.4)

Immediately after t(f), the potential is reset to a new value ur < $ \vartheta$,

$\displaystyle \lim_{{t\to t^{(f)}; t>t^{(f)}}}^{}$u(t) = ur . (4.5)

For t > t(f) the dynamics is again given by (4.3) until the next threshold crossing occurs. The combination of leaky integration (4.3) and reset (4.5) defines the basic integrate-and-fire model (Stein, 1967b). We note that, since the membrane potential is never above threshold, the threshold condition (4.1) reduces to the criterion (4.4), i.e., the condition on the slope du/dt can be dropped.

In its general version, the leaky integrate-and-fire neuron may also incorporate an absolute refractory period, in which case we proceed as follows. If u reaches the threshold at time t = t(f), we interrupt the dynamics (4.3) during an absolute refractory time $ \Delta^{{\rm abs}}_{}$ and restart the integration at time t(f) + $ \Delta^{{\rm abs}}_{}$ with the new initial condition ur. Example: Constant stimulation and firing rates

Before we continue with the definition of the integrate-and-fire model and its variants, let us study a simple example. Suppose that the integrate-and-fire neuron defined by (4.3)-(4.5) is stimulated by a constant input current I(t) = I0. For the sake of simplicity we take the reset potential to be ur = 0.

As a first step, let us calculate the time course of the membrane potential. We assume that a spike has occurred at t = t(1). The trajectory of the membrane potential can be found by integrating (4.3) with the initial condition u(t(1)) = ur = 0. The solution is

u(t) = R I0$\displaystyle \left[\vphantom{ 1 - \exp\left(-{t-t^{(1)} \over \tau_m} \right) }\right.$1 - exp$\displaystyle \left(\vphantom{-{t-t^{(1)} \over \tau_m} }\right.$ - $\displaystyle {t-t^{(1)} \over \tau_m}$$\displaystyle \left.\vphantom{-{t-t^{(1)} \over \tau_m} }\right)$$\displaystyle \left.\vphantom{ 1 - \exp\left(-{t-t^{(1)} \over \tau_m} \right) }\right]$ . (4.6)

The membrane potential (4.6) approaches for t$ \to$$ \infty$ the asymptotic value u($ \infty$) = R I0. For R I0 < $ \vartheta$ no further spike can occur. For R I0 > $ \vartheta$, the membrane potential reaches the threshold $ \vartheta$ at time t(2), which can be found from the threshold condition u(t(2)) = $ \vartheta$ or

$\displaystyle \vartheta$ = R I0$\displaystyle \left[\vphantom{1 - \exp\left(-{t^{(2)}-t^{(1)} \over \tau_m}\right) }\right.$1 - exp$\displaystyle \left(\vphantom{-{t^{(2)}-t^{(1)} \over \tau_m}}\right.$ - $\displaystyle {t^{(2)}-t^{(1)} \over \tau_m}$$\displaystyle \left.\vphantom{-{t^{(2)}-t^{(1)} \over \tau_m}}\right)$$\displaystyle \left.\vphantom{1 - \exp\left(-{t^{(2)}-t^{(1)} \over \tau_m}\right) }\right]$ . (4.7)

Solving (4.7) for the time interval T = t(2) - t(1) yields

T = $\displaystyle \tau_{m}^{}$ ln$\displaystyle {R\, I_0 \over R\, I_0 - \vartheta}$ . (4.8)

After the spike at t(2) the membrane potential is again reset to ur = 0 and the integration process starts again. If the stimulus I0 remains constant, the following spike will occur after another interval of duration T. We conclude that for a constant input current I0, the integrate-and-fire neuron fires regularly with period T given by (4.8). For a neuron with absolute refractory period the firing period T' is given by T' = T + $ \Delta^{{\rm abs}}_{}$ with T defined by Eq. (4.8). In other words, the interspike interval is longer by an amount $ \Delta^{{\rm abs}}_{}$ compared to that of a neuron without absolute refractory period.

Figure 4.2: A. Time course of the membrane potential of an integrate-and-fire neuron driven by constant input current I0 = 1.5. The voltage u(t) is normalized by the value of the threshold $ \vartheta$ = 1. B. Gain function. The firing rate $ \nu$ of an integrate-and-fire neuron without (solid line) and with absolute refractoriness of $ \delta_{{\rm abs}}^{}$ = 4 ms (dashed line) as a function of a constant driving current I0. Current units are normalized so that the onset of repetitive firing is at I$\scriptstyle \theta$ = 1. Other parameters are R = 1, $ \tau_{m}^{}$ = 10ms, and ur = 0.
{\bf A}
... B}

The mean firing rate of a noiseless neuron is defined as $ \nu$ = 1/T. The firing rate of an integrate-and-fire model with absolute refractory period $ \Delta^{{\rm abs}}_{}$ stimulated by a current I0 is therefore

$\displaystyle \nu$ = $\displaystyle \left[\vphantom{\Delta^{\rm abs}+ \tau_m \, {\rm ln} {R\, I_0 \over R\, I _0 - \vartheta} }\right.$$\displaystyle \Delta^{{\rm abs}}_{}$ + $\displaystyle \tau_{m}^{}$ ln$\displaystyle {R\, I_0 \over R\, I_0 - \vartheta}$$\displaystyle \left.\vphantom{\Delta^{\rm abs}+ \tau_m \, {\rm ln} {R\, I_0 \over R\, I _0 - \vartheta} }\right]^{{-1}}_{}$ . (4.9)

In Fig. 4.2B the firing rate is plotted as a function of the constant input I0 for neurons with and without absolute refractory period. Example: Time-dependent stimulus I(t)

The results of the preceding example can be generalized to arbitrary stimulation conditions and an arbitrary reset value ur < $ \vartheta$. Let us suppose that a spike has occurred at $ \hat{{t}}$. For t > $ \hat{{t}}$ the stimulating current is I(t). The value ur will be treated as an initial condition for the integration of (4.3), i.e.,

u(t) = ur exp$\displaystyle \left(\vphantom{-{t-\hat{t}\over \tau_m}}\right.$ - $\displaystyle {t-\hat{t}\over \tau_m}$$\displaystyle \left.\vphantom{-{t-\hat{t}\over \tau_m}}\right)$ + $\displaystyle {1\over C}$$\displaystyle \int_{0}^{{t-\hat{t}}}$exp$\displaystyle \left(\vphantom{ -{s\over \tau_m}}\right.$ - $\displaystyle {s\over \tau_m}$$\displaystyle \left.\vphantom{ -{s\over \tau_m}}\right)$ I(t - s) ds . (4.10)

This expression describes the membrane potential for t > $ \hat{{t}}$ and is valid up to the moment of the next threshold crossing. If u(t) = $ \vartheta$, the membrane potential is reset to ur and integration restarts; see Fig. 4.3.

Figure 4.3: Voltage u(t) of an integrate-and-fire model (top) driven by the input current I(t) shown at the bottom. The input I(t) consists of a superposition of four sinusoidal components at randomly chosen frequencies plus a positive bias current I0 = 1.2 which drives the membrane potential towards the threshold.
\end{center} \vspace{-5mm}

4.1.2 Nonlinear integrate-and-fire model

In a general nonlinear integrate-and-fire model, Eq. (4.3) is replaced by

$\displaystyle \tau$$\displaystyle {{\text{d}}\over {\text{d}}t}$u = F(u) + G(uI ; (4.11)

cf. Abbott and van Vreeswijk (1993). As before, the dynamics is stopped if u reaches the threshold $ \vartheta$ and reinitialized at u = ur. A comparison with Eq. (4.3) shows that G(u) can be interpreted as a voltage-dependent input resistance while - F(u)/(u - urest) corresponds to a voltage-dependent decay constant. A specific instance of a nonlinear integrate-and-fire model is the quadratic model (Feng, 2001; Hansel and Mato, 2001; Latham et al., 2000),

$\displaystyle \tau$$\displaystyle {{\text{d}}\over {\text{d}}t}$u = a0 (u - urest) (u - uc) + RI , (4.12)

with parameters a0 > 0 and uc > urest; cf. Fig. 4.4. For I = 0 and initial conditions u < uc, the voltage decays to the resting potential urest. For u > uc it increases so that an action potential is triggered. The parameter uc can therefore be interpreted as the critical voltage for spike initiation by a short current pulse. We will see in the next example that the quadratic integrate-and-fire model is closely related to the so-called $ \Theta$-neuron, a canonical type-I neuron model (Ermentrout, 1996; Latham et al., 2000).

Figure 4.4: Quadratic integrate-and-fire model. A. Without external current I = 0, the membrane potential relaxes for all initial condition u < uc to the resting potential urest. If the membrane potential is moved above uc, the potential increases further since du/dt > 0. The neuron is said to fire if u reaches the threshold $ \vartheta$ = - 40mV. B. A constant super-threshold current I is characterized by the fact that du/dt > 0 for all u. If u reaches the firing threshold of -40mV, it is reset to -80mV. This results in repetitive firing.
{\bf A}
{\bf B}
\end{minipage} Rescaling and standard forms (*)

It is always possible to rescale the variables so that threshold and membrane time constant are equal to unity and that the resting potential vanishes. Furthermore, there is no need to interpret the variable u as the membrane potential. For example, starting from the nonlinear integrate-and-fire model Eq. (4.11), we can introduce a new variable $ \tilde{{u}}$ by the transformation

u(t) $\displaystyle \longrightarrow$ $\displaystyle \tilde{{u}}$(t) = $\displaystyle \tau$ $\displaystyle \int_{0}^{{u(t)}}$$\displaystyle {{\text{d}}x \over G(x)}$ (4.13)

which is possible if G(x)$ \ne$ 0 for all x in the integration range. In terms of $ \tilde{{u}}$ we have a new nonlinear integrate-and-fire model of the form

$\displaystyle {{\text{d}}\tilde{u}\over {\text{d}}t}$ = $\displaystyle \gamma$($\displaystyle \tilde{{u}}$) + I(t) (4.14)

with $ \gamma$($ \tilde{{u}}$) = $ \tau$F(u)/G(u). In other words, a general integrate-and-fire model (4.11) can always be reduced to the standard form (4.14). By a completely analogous transformation, we could eliminate the function F in Eq. (4.11) and move all the dependence into a new voltage dependent G (Abbott and van Vreeswijk, 1993). Example: Relation to a canonical type I model (*)

In this section, we show that there is a close relation between the quadratic integrate-and-fire model (4.12) and the canonical type I phase model,

$\displaystyle {{\text{d}}\phi \over {\text{d}}t}$ = [1 - cos$\displaystyle \phi$]  + $\displaystyle \Delta$I [1 + cos$\displaystyle \phi$] ; (4.15)

cf. Section 3.2.4 (Strogatz, 1994; Ermentrout and Kopell, 1986; Ermentrout, 1996; Latham et al., 2000; Hoppensteadt and Izhikevich, 1997).

Let us denote by I$\scriptstyle \theta$ the minimal current necessary for repetitive firing of the quadratic integrate-and-fire neuron. With a suitable shift of the voltage scale and constant current I = I$\scriptstyle \theta$ + $ \Delta$I the equation of the quadratic neuron model can then be cast into the form

$\displaystyle {{\text{d}}u\over {\text{d}}t}$ = u2 + $\displaystyle \Delta$I . (4.16)

For $ \Delta$I > 0 the voltage increases until it reaches the firing threshold $ \vartheta$ $ \gg$ 1 where it is reset to a value ur $ \ll$ - 1. Note that the firing times are insensitive to the actual values of firing threshold and reset value because the solution of Eq. (4.16) grows faster than exponentially and diverges for finite time (hyperbolic growth). The difference in the firing times for a finite threshold of, say, $ \vartheta$ = 10 and $ \vartheta$ = 10 000 is thus negligible.

We want to show that the differential equation (4.16) can be transformed into the canonical phase model (4.15) by the transformation

u(t) = tan$\displaystyle \left(\vphantom{{\phi(t)\over 2}}\right.$$\displaystyle {\phi(t)\over 2}$$\displaystyle \left.\vphantom{{\phi(t)\over 2}}\right)$ . (4.17)

To do so, we take the derivative of (4.17) and use the differential equation (4.15) of the generic phase model. With help of the trigonometric relations dtan x/dx = 1/cos2(x) and 1 + cos x = cos2(x/2) we find
$\displaystyle {{\text{d}}u\over {\text{d}}t}$ = $\displaystyle {1\over \cos^2(\phi/2)}$ $\displaystyle {1\over 2}$ $\displaystyle {{\text{d}}\phi \over {\text{d}}t}$  
  = tan2($\displaystyle \phi$/2) + $\displaystyle \Delta$I = u2 + $\displaystyle \Delta$I . (4.18)

Thus Eq. (4.17) with $ \phi$(t) given by (4.15) is a solution to the differential equation of the quadratic integrate-and-fire neuron. The quadratic integrate-and-fire neuron is therefore (in the limit $ \vartheta$$ \to$$ \infty$ and ur$ \to$ - $ \infty$) equivalent to the generic type I neuron (4.15).

4.1.3 Stimulation by Synaptic Currents

So far we have considered an isolated neuron that is stimulated by an external current I(t). In a more realistic situation, the integrate-and-fire model is part of a larger network and the input current I(t) is generated by the activity of presynaptic neurons.

In the framework of the integrate-and-fire model, each presynaptic spike generates a postsynaptic current pulse. More precisely, if the presynaptic neuron j has fired a spike at tj(f), a postsynaptic neuron i `feels' a current with time course $ \alpha$(t - tj(f)). The total input current to neuron i is the sum over all current pulses,

Ii(t) = $\displaystyle \sum_{{j}}^{}$wij$\displaystyle \sum_{{f}}^{}$$\displaystyle \alpha$(t - tj(f)) . (4.19)

The factor wij is a measure of the efficacy of the synapse from neuron j to neuron i.

Though Eq. (4.19) is a reasonable model of synaptic interaction, reality is somewhat more complicated, because the amplitude of the postsynaptic current pulse depends on the actual value of the membrane potential ui. As we have seen in Chapter 2, each presynaptic action potential evokes a change in the conductance of the postsynaptic membrane with a certain time course g(t - t(f)). The postsynaptic current generated by a spike at time tj(f) is thus

$\displaystyle \alpha$(t - tj(f)) = - g(t - tj(f)$\displaystyle \left[\vphantom{u_i(t) - E_{\rm syn}}\right.$ui(t) - Esyn$\displaystyle \left.\vphantom{u_i(t) - E_{\rm syn}}\right]$ . (4.20)

The parameter Esyn is the reversal potential of the synapse.

The level of the reversal potential depends on the type of synapse. For excitatory synapses, Esyn is much larger than the resting potential. For a voltage ui(t) close to the resting potential, we have ui(t) < Esyn. Hence the current Ii induced by a presynaptic spike at an excitatory synapse is positive and increases the membrane potential 4.1. The higher the voltage, the smaller the amplitude of the input current. Note that a positive voltage ui > urest is itself the result of input spikes which have arrived at other excitatory synapses. Hence, there is a saturation of the postsynaptic current and the total input current is not just the sum of independent contributions. Nevertheless, since the reversal potential of excitatory synapses is usually significantly above the firing threshold, the factor [ui - Esyn] is almost constant and saturation can be neglected.

For inhibitory synapses, the reversal potential is close to the resting potential. An action potential arriving at an inhibitory synapse pulls the membrane potential towards the reversal potential Esyn. Thus, if the neuron is at rest, inhibitory input hardly has any effect on the membrane potential. If the membrane potential is instead considerably above the resting potential, then the same input has a strong inhibitory effect. This is sometimes described as silent inhibition: inhibition is only seen if the membrane potential is above the resting potential. Strong silent inhibition is also called `shunting' inhibition, because a significantly reduced resistance of the membrane potential forms a short circuit that literally shunts excitatory input the neuron might receive from other synapses. Example: Pulse-coupling and $ \alpha$-function

The time course of the postsynaptic current $ \alpha$(s) introduced in Eq. (4.19) can be defined in various ways. The simplest choice is a Dirac $ \delta$-pulse, $ \alpha$(s) = q $ \delta$(s), where q is the total charge that is injected in a postsynaptic neuron via a synapse with efficacy wij = 1. More realistically, the postsynaptic current $ \alpha$ should have a finite duration, e.g., as in the case of an exponential decay with time constant $ \tau_{s}^{}$,

$\displaystyle \alpha$(s) = $\displaystyle {q\over \tau_s}$exp$\displaystyle \left(\vphantom{ -{s\over \tau_s} }\right.$ - $\displaystyle {s\over \tau_s}$$\displaystyle \left.\vphantom{ -{s\over \tau_s} }\right)$ $\displaystyle \Theta$(s) . (4.21)

As usual, $ \Theta$ is the Heaviside step function with $ \Theta$(s) = 1 for s > 0 and $ \Theta$(s) = 0 else. Equation (4.21) is a simple way to account for the low-pass characteristics of synaptic transmission; cf. Fig. 4.1.

An even more sophisticated version of $ \alpha$ includes a finite rise time $ \tau_{r}^{}$ of the postsynaptic current and a transmission delay $ \Delta^{{\rm ax}}_{}$,

$\displaystyle \alpha$(s) = $\displaystyle {q \over \tau_s - \tau_r}$$\displaystyle \left[\vphantom{ \exp\left(-{s-\Delta^{\rm ax}\over \tau_s}\right) - \exp\left(-{s-\Delta^{\rm ax}\over \tau_r}\right)}\right.$exp$\displaystyle \left(\vphantom{-{s-\Delta^{\rm ax}\over \tau_s}}\right.$ - $\displaystyle {s-\Delta^{\rm ax}\over \tau_s}$$\displaystyle \left.\vphantom{-{s-\Delta^{\rm ax}\over \tau_s}}\right)$ - exp$\displaystyle \left(\vphantom{-{s-\Delta^{\rm ax}\over \tau_r}}\right.$ - $\displaystyle {s-\Delta^{\rm ax}\over \tau_r}$$\displaystyle \left.\vphantom{-{s-\Delta^{\rm ax}\over \tau_r}}\right)$$\displaystyle \left.\vphantom{ \exp\left(-{s-\Delta^{\rm ax}\over \tau_s}\right) - \exp\left(-{s-\Delta^{\rm ax}\over \tau_r}\right)}\right]$ $\displaystyle \Theta$(s - $\displaystyle \Delta^{{\rm ax}}_{}$) . (4.22)

In the limit of $ \tau_{r}^{}$$ \to$$ \tau_{s}^{}$, (4.22) yields

$\displaystyle \alpha$(s) = q $\displaystyle {s-\Delta^{\rm ax} \over \tau_s^2}$exp$\displaystyle \left(\vphantom{-{s-\Delta^{\rm ax}\over \tau_s}}\right.$ - $\displaystyle {s-\Delta^{\rm ax}\over \tau_s}$$\displaystyle \left.\vphantom{-{s-\Delta^{\rm ax}\over \tau_s}}\right)$ $\displaystyle \Theta$(s - $\displaystyle \Delta^{{\rm ax}}_{}$) . (4.23)

In the literature, a function of the form x exp(- x) such as (4.23) is often called an $ \alpha$-function. While this has motivated our choice of the symbol $ \alpha$ for the synaptic input current, $ \alpha$ may stand for any form of an input current pulse.

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Gerstner and Kistler
Spiking Neuron Models. Single Neurons, Populations, Plasticity
Cambridge University Press, 2002

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