MATH 3104: THE MEMBRANE EQUATION AND
INTEGRATE-AND-FIRE MODELS
A/Prof Geoffrey Goodhill, Semester 1, 2009
Neurons as electric circuits
Neuronal membranes have both capacitance (opposite charges can build up on either side of the
membrane), and resistance (ion channels allow only a limited flow of ions through them). The simplest
model of a neuronal membrane is therefore as an RC-circuit, as shown in Fig. 1.
Neuronal capacitance
In general we have
Q = CVm
where Q is the charge that can build up, Vm is the membrane potential, and C is the capacitance of the
membrane. The specific capacitance Cm is the capacitance per unit area. For neuronal membranes
Cm ≈ 1µF/cm2 .
A change in voltage causes a capacitive current IC to flow:
IC =
dVm
dQ
=C
dt
dt
Neuronal resistance
Ohm’s Law states that
I=
V
R
Or in our case
Vm − Vrest
R
where IR is the resistive current. The resistance of a neural membrane is usually in the range of
gigaohms (GΩ). Thus the conductance (1/R) is usually measured in picosiemens (pS).
IR =
The Membrane Equation
If we inject current Iinj , Kirchoff’s current law says that
IC + IR = Iinj
That is,
dVm Vm − Vrest
+
= Iinj
dt
R
If we now define the membrane time constant τ = RC (units of seconds) we can write
dVm
= −Vm + Vrest + RIinj
τ
dt
C
Solving the membrane equation
The solution depends on whether Iinj = Iinj (t), and on the initial conditions.
Assume that at time t = 0, Vm = Vrest and Iinj = I0 . Then we have
Vm = RI0 1 − e−t/τ + Vrest
Note that
τ = RC = Rm Cm
that is τ is independent of the size of the neuron. For neurons τ is generally in the range of 1 − 10
ms. The effect of injecting various levels of current is shown in Fig. 2A.
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The integrate-and-fire model
In reality whenever Vm ≥ Vthreshold a neuron emits a spike. We discuss shortly how this occurs
in terms of changes in the permeability of certain ion channels. Here we consider a much simpler
approach, the “integrate-and-fire” model. We use the passive membrane equation, and just assert
that whenever Vm ≥ Vthreshold a spike occurs, and that Vm is then instantaneously reset to Vreset.
Since this is obviously highly nonlinear, in general the membrane equation will then need to be solved
numerically. However, for a constant injected current, it is still possible to calculate analytically the
time between spikes. The solution of the membrane equation when Iinj is constant and Vm (0) =
Vreset is:
Vm = Vrest + RIinj + (Vreset − Vrest − RIinj ) e−t/τ
We would now like to know the time tisi when Vm reaches Vthreshold
Vthreshold = Vrest + RIinj + (Vreset − Vrest − RIinj ) e−tisi /τ
Solving this for tisi gives
tisi = τ ln
RIinj + Vrest − Vreset
RIinj + Vrest − Vthreshold
From the interspike-interval tisi we can define a firing rate r = 1/tisi
1
r=
τ ln
RIinj +Vrest −Vreset
RIinj +Vrest −Vthreshold
It is left as an exercise to show that when Iinj is sufficiently large one can use ln(1 + z) ≈ z for small
z to prove that
Vrest − Vthreshold + RIinj
r≈
τ (Vthreshold − Vreset)
i.e. the firing rate grows linearly with Iinj .
Spike rate adaptation
The above analysis so far ignores the fact that neurons are not linear: their spike rates cannot increase indefinitely. A computationally cheap way to model this is to include an extra current in the
model, which varies with time:
τ
dVm (t)
= Vrest − Vm (t) − rm gsra (t)(Vm (t) − VK ) + RIinj
dt
Since it’s a K + current it will hyperpolarize the neuron, slowing spiking. Assume that this conductance
decays exponentially to zero with time constant τsra
τsra
dgsra
= −gsra
dt
Whenever the neuron fires a spike we increase gsra by ∆gsra . This produces spike trains which match
quite well with real spike trains, as shown in Fig. 3.
Summary
• Neuronal membranes can be modelled as simple electrical circuits with resistance and capacitance.
• Integrate-and-fire models are a relatively easy way to generate realistic-looking spike trains.
Recommended reading
Dayan, P. & Abbott, L.F. (2001). Theoretical Neuroscience. MIT Press, Cambridge, MA (pp 153-166).
Koch, C. (1999). Biophysics of Computation. Oxford University Press (pp 5-12).
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Figure 1: Neurons can be represented as electrical circuits with resistance and capacitance (pictures describing the same thing are shown from two different sources). The battery represents the
equilibrium (resting) potential we calculated in the previous lecture.
A
B
Figure 2: A Evolution of the membrane potential in a simple RC neuron when a current step of
different amplitudes I0 is turned on at t = 0 and turned off at 100 msec. Paraneters: Vrest = −70mV,
R = 100MΩ, C = 100pF, τ = 10ms, I0 = −0.1, 0.1, 0.2, 0.3 nA. B In reality neurons produce a spike
once their membrane potential reaches a certain threshold.
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Figure 3: A Comparison of interspike interval firing rates as a function of injected current for an
integrate-and-fire model and a cortical neuron measured in vivo. The line gives risi for a model
neuron with τm = 30ms, EL = Vreset = −65mV, Vth = −50mV, and Rm = 90MΩ. The data points are
from a cat visual cortex neuron. The filled circles show the inverse of the interspike interval for the first
two spikes fired, and the open circles show the steady interspike interval firing rate after spike-rate
adaptation. B A recording of the firing of a cortical neuron under constant current injection, showing
spike-rate adaptation. C Membrane voltage and spikes for an integrate-and-fire model with an added
current, with rm ∆gs ra = 0.06, τrsa = 100ms, and EK = −70mV.
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