f_i_curve.tex
Computation Neuroscience – 3 integrate and fire
Figure 1: A cartoon to help you remember that there is more sodium outside the cell and more
potassium inside.
Introduction
This note is about the dynamics of a single neuron, it will cover one of the simplest such
models: the leaky integrate and fire model.
Electrical properties of a neuron
The potential inside a neuron is lower than the potential on the outside; this difference is
created by ion pumps, small molecular machines that use energy to pump ions across the
membrane seperating the inside and outside of the cell. One typical ion pump is Na+/K+-
ATPase (Sodium-potassium adenosine triphosphatase); this uses energy in the form of ATP,
the energy carrying molecule in the body, and through each cycle it moves three sodium ions
out of the cell and two potassium ions into the cell. Since both sodium and potassium ions
have a charge of plus one, this leads to a net loss of one atomic charge to the inside of the
cell lowering its potential. It also creates an excess of sodium outside the cell and an excess
of potassium inside it. We will return to these chemical imbalances later. The potential
difference across the membrane is called the membrane potential. At rest a typical value of
the membrane potential is EL = −70mV. It is useful to remember that the excessive sodium is
outside the cell and potassium inside; I think of islands which are surrounded by salty water,
as in Fig. 1.
Spikes
So the summary version of what happens in neuons is that synapses cause a small increase
or decrease in the voltage; excitatory synapses cause an increase, inhibitory synapses a
decrease. This drives the internal voltage dynamics of the cell, these dynamics are what we
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Computation Neuroscience – 3 integrate and fire
will learn about here. If the voltage exceeds a threshold, say VT = −55 mV there is a nonlinear
cascade which produces a spike or action potential, a spike in voltage 1-2 ms wide which
rises above 0 mV before, in the usual description, falling to a reset value of VR = −65 mV, the
cell then remains unable to produce another spike for a refractory period which may last
about 5 ms. We will examine how spikes are formed later, this involves the nonlinear dynamics
of ion channels in the membrane; first though we will consider the integrate and fire model
which ignores the details of how spikes are produced and simplifies the voltage dynamics.
The bucket-like equation for neurons
We will now try to extend the bucket-like equation we looked at before so that it applies to
neurons. First off we replace h, the height of the water, by V the voltage in the cell and C
will be replaced by Cm, the capacitance of the membrane, the amount of electrical charge that
can be stored at the membrane is CmV . The amount of electrical charge is the analogue of
the volume of water. Thus, voltage is like height, charge is like the amount of water.
The leak is a bit more complicated, because of the chemical gradients, that is the effects of
the differing levels of ions inside and outside the cell along and their propensity to diffuse, the
voltage at which there is no leaking of charge is not zero, it is EL = −70mV, roughly. This is
an important aspect of how neurons behave, and one we will encounter again looking at the
Hodgkin-Huxley equation: you might at first expect that if the voltage inside the cell was,
say, -60 mV then even if there was a high conductivity for potassium at the membrane, the
potassium ions would stay in the cell: they are positive ions after all and so a negative voltage
means the electrical force is attracting them to the inside of the cell. However, this isn’t quite
what happens, there is a high concentration of potassium inside the cell and because of the
random motion of particles associated with temperature, these have a tendency to diffuse, that
is to increase the entropy of the situation by spreading out. It takes a force to counteract this.
This is the reversal potential, EL, the voltage required for zero current even if there is some
conductivity. It turns out that the normal Ohm’s law applies around the reversal potential so
that the current out of the cell is proportional to V − EL.
G is now Gm, a conductance, measuring the porousness of the membrane to the flow of
ions, in other words, it gives the constant of proportionality for the leak current: the leak
current out of the cell is Gm(V − EL). We actually divide across by the conductance, and
write Rm = 1/Gm, the resistance. Finally, we write τm = Cm/Gm to get
τm
dV
dt
= EL − V +RmI (1)
I might end up being synaptic input, but traditionally we write the equation to match the
in vivo experiment where I is an injected current from an electrode, so we write Ie, ‘e’ for
electrode. τm is a time constant, using the notation of dimensional analysis we have [τm] = T .
To check this note that the units of capacitance are charge per voltage: [Cm] = QV
−1, the units
of resistance is voltage per current [Rm] = V I
−1 and current is charge per time, [I] = QT−1
so [CmRm] = T , time.
The equation above leaves out the possibility that there are other non-linear changes in
the currents through the membrane as V changes. This is a problem since there are other
non-linear changes in the currents through the membrane as V changes. The equation above
leaves these out, in fact, the nonlinear effects are strongest for values of V near where a spike
is produced, so one approach is to use the linear equation unless V reaches a threshold value
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References
and then add a spike ‘by hand’. This has the effect of changing the voltage to a reset value,
this mimics what happens in the neuron, or in the Hodgkin Huxley model which we will look
at next and which includes the full non-linear dynamics which makes the spike. Anyway, in
summary
• V satisfies
τm
dV
dt
= EL − V +RmIe (2)
• If V ≥ VT a spike is recorded and the voltage is set to a reset value VR.
The reset value, the voltage after the spike is often set equal to the leak potential. This is the
leaky integrate and fire model, a surprisingly old model first introduced in [1]. It lacks lots
of the details important in the dynamics of neurons, but is useful and often used for modeling
the behavior of large neuronal networks or for exploring ideas about neuronal computation in
a relatively straight-forward setting.
This model is easy to solve; if Ie is constant we have already solved it above up to messing
around with constants:
V (t) = EL +RmIe + [V (0)− EL −RmIe]e
−t/τm (3)
If Ie is not constant it may still be possible to solve the equation, but in any case the equation
can be solved numerically on a computer. An example in given in Fig. 2.
One thing to notice is that there are no spikes for low values of the current. Looking at the
equation
τm
dV
dt
= EL − V +RmIe (4)
so the equilibrium value for constant Ie, the value where V stops changing, is
V̄ = EL +RmIe (5)
Now if this value V̄ > VT then as the neuron voltage increased towards its equilibrium value, V̄ ,
it would reach the threshold, VT , and spike. Hence, if V̄ > VT the neuron will spike repeatedly.
However if V̄ < VT then the neuron will not spike for that input because it will never reach
threshold. We won’t do it here1, but, in fact, since we can solve the equations for constant Ie
we can work out the f−I curve, the relationship between the firing rate and the input current.
It is plotted in Fig. 3.
References
[1] Lapicque, L. (1907). Recherches quantitatives sur l’excitation lectrique des nerfs traite
comme une polarisation. J. Physiol. Pathol. Gen, 9:620–635.
1
This calculation is described in note on the f-I curve.pdf
3
References
-70
-60
-50
-40
-30
-20
-10
0
10
20
0 20 40 60 80 100
RmIe = 16 mV
RmIe = 12 mV
t (ms)
V (mV)
Figure 2: An integrate and fire neuron with different inputs. For RmI = 12mV the voltage
relaxes towards the equilibrium value V = EL + RmIe = −58 mV. It never reaches
the threshold value of VT = −55mV. For RmI = 16 mV the voltage reaches threshold
and so there is a spike; the spike is added by hand, in this case by setting V to 20
mV for one time step. The voltage is then reset. Here τm = 10 ms.
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References
0
10
20
30
40
50
60
70
80
0 5 10 15 20
fi
ri
n
g
ra
te
in
H
z
RmIe
Figure 3: The firing rate, that is spikes per second, for the integrate and fire neuron with
different constant inputs with τm = 10 ms, VT = −55 mV and both the leak and
reset given by −70 mV. Notice how there is no firing until a threshold is reached and
after that the firing increases very quickly.
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