The Integrate-and-Fire Model
The integrate-and-fire neuron is one of the simplest models of a neuron’s electrical properties
and probably the most commonly used in the field of neuroscience. The essence of the model is
to divide the voltage changes of the neuron into two parts:
1) Below threshold, it is assumed that the membrane behaves passively (i.e. has no
voltage-dependent ion channels) and acts as a leaky capacitor whose voltage, in the absence of
injected current, decays (or “leaks”) to a resting level E
L
(short for “E
Leak
”).
2) When the voltage reaches the action potential threshold (due to injected currents
charging up the membrane), the model assumes that the voltage spikes immediately to a level
V
spike
and is then immediately reset to a hyperpolarized level V
reset
. There is no explicit
modeling of the ion channel kinetics responsible for this spiking. Rather, it is simply assumed (a
reasonable assumption…) that once the cell reaches its threshold it will rapidly produce an action
potential and reset itself. The reason we can get away with this assumption is that we don’t
really care about the exact shape of the action potential: since all action potentials sent down the
axon are to a good approximation identical, the only informative feature of a neuron’s spiking is
the times at which the action potentials occur.
I. Mathematics of the integrate-and-fire neuron
Consider a neuron modeled as a leaky capacitor with membrane resistance R
m
, time
constant τ
m
=R
m
C
m
(where C
m
is the membrane’s capacitance), and resting potential E
L
. Below
the action potential threshold, the equation for the voltage of this cell when it receives current
injection I
e
is:
mL
dV
EVRI
dt
τ
=−+
me
(1)
Exercise: When the current injection I
e
is constant over time, verify
(i) that the solution to this equation is:
(
)
0
() ( ( ) )exp ( )/
Lme Lme om
Vt E RI Vt E RI t t
τ
=+ + −− −− (2)
where the constant t
0
is any reference time. (Do this by inserting the solution on both sides of
equation (1). Recall that
()
() ()
()
ft ft
dd
ee
dt dt
=
ft
e
).
(ii) that when t=t
0
, the left and right sides of equation (2) agree (and both equal what value?), and
(iii) that when t → ∞, V(t) → .
Lm
VERI
∞
≡+
Setting t
0
equal to the current time in a computer simulation and t equal to the time a
single time-step Δt later gives the one-time-step update rule we will use for simulating the
integrate-and-fire neuron (and which we will use more generally for simulating any equation of
the form of equation (1) above, e.g. in your homework for the spike-rate-adaptation conductance
if we substitute g
sra
for V and set E
L
=I
e
=0):
(
)
()exp
Lme Lme m
V E RI V E RI t/
τ
→+ +−− −Δ ,
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