- 4.1.1 Leaky Integrate-and-Fire Model
- 4.1.1.1 Example: Constant stimulation and firing rates
- 4.1.1.2 Example: Time-dependent stimulus
*I*(*t*)

- 4.1.2 Nonlinear integrate-and-fire model

- 4.1.3 Stimulation by Synaptic Currents

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

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*) = *I*_{R} + *I*_{C}. 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* where *u* 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} = *C* d*u*/d*t*. Thus

We multiply (4.2) by

We refer to

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

Immediately after

For

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
and
restart the integration at time
*t*^{(f)} + with the new initial
condition *u*_{r}.

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*) = *I*_{0}. For the sake of simplicity we
take the reset potential to be *u*_{r} = 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)}) = *u*_{r} = 0. The solution is

The membrane potential (4.6) approaches for

Solving (4.7) for the time interval

After the spike at

The mean firing rate of a noiseless neuron is defined as = 1/*T*. The
firing rate of an integrate-and-fire model with absolute refractory period
stimulated by a current *I*_{0} is therefore

In Fig. 4.2B the firing rate is plotted as a function of the constant input

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

This expression describes the membrane potential for

4.1.2 Nonlinear integrate-and-fire model

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

cf. Abbott and van Vreeswijk (1993). As before, the dynamics is stopped if

with parameters

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 by the transformation

which is possible if

with () =

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,

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*_{} 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*_{} + *I* the
equation of the quadratic neuron model can then be cast into the form

For

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

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

= | |||

= | tan^{2}(/2) + I = u^{2} + I . |
(4.18) |

Thus Eq. (4.17) with (

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 *t*_{j}^{(f)}, a postsynaptic neuron *i* `feels' a
current with time course
(*t* - *t*_{j}^{(f)}). The total input current
to neuron *i* is the sum over all current pulses,

The factor

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 *u*_{i}. 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 *t*_{j}^{(f)} is thus

The parameter

The level of the reversal potential depends on the type of synapse. For
excitatory synapses,
*E*_{syn} is much larger than the resting potential.
For a voltage *u*_{i}(*t*) close to the resting potential, we have
*u*_{i}(*t*) < *E*_{syn}. Hence the current *I*_{i} 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
*u*_{i} > *u*_{rest} 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
[*u*_{i} - *E*_{syn}] 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
*E*_{syn}. 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.

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

As usual, is the Heaviside step function with (

An even more sophisticated version of includes a finite rise time of the postsynaptic current and a transmission delay ,

In the limit of , (4.22) yields

In the literature, a function of the form

Cambridge University Press, 2002

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