Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 35

that their concentration provides a reasonable indica-
tion of the average level of neural activity over the
recent time period. This makes them useful for caus-
ing other things to happen in the neuron.

As we said, this net current affects the membrane po-
tential because the movement of charges decreases the
net charge difference, which causes the potential in the
first place. The following equation updates the mem-
brane potential V m ( v_m in the simulator) in our model
based on the previous membrane potential and net cur-
rent:

2.4.5 Putting It All Together: Integration

Having covered all the major ions and channels affect-
ing neural processing, we are now in a position to put
them all together into one equation that reflects the neu-
ral integration of information. The result of this will be
an equation for updating the membrane potential, which
is denoted by the variable V m ( V for voltage, and m
for membrane). All we have to do is use Ohm's law to
compute the current for each ion channel, and then add
all these currents together. For each type of ion chan-
nel (generically denoted by the subscript c for now), we
need to know three things: (1) its equilibrium potential
as given above, E c , (2) the fraction of the total number
of channels for that ion that are open at the present time,

(2.7)

The time constant 0 <dt vm < 1 ( dt_vm in the sim-
ulator) slows the potential change, capturing the corre-
sponding slowing of this change in a neuron (primarily
as a result of the capacitance of the cell membrane, but
the details of this are not particularly relevant here be-
yond the fact that they slow down changes).
In understanding the behavior of neurons, it is use-
ful to think of increasing membrane potential as result-
ing from positive current (i.e., “excitation”). However,
equation 2.7 shows that according to the laws of elec-
tricity, increasing membrane potential results from neg-
ative current. To match the more intuitively appealing
relationship between potential and current, we simply
change the sign of the current in our model ( I net ￿ =

), and (3) the maximum conductance that would re-
sult if all the channels were open (i.e., how many ions
the channels let through), g c . The product of g c (t) and

then gives us the total conductance.
The current for the channel is then given by the dif-
fusion corrected Ohm's law described above, which is
just the total conductance (fraction open times maxi-
mum conductance) times the net potential (difference
between the membrane potential at the present time
( V m (t) ) and the equilibrium potential):

) and add it to the previous membrane potential
instead of subtracting it:

(2.5)

The three basic channels that do most of the activa-
tion work in the neuron are: (a) the excitatory synaptic
input channel activated by glutamate and passing the
Na + ion (subscript e ), (b) the inhibitory synaptic in-
put channel activated by GABA and passing the Cl ￿
ion (subscript i ), and (c) the leak channel that is always
open and passing the K + ion (subscript l ). The total or
net current for these three channels is:

(2.8)

Equation 2.8 is of course mathematically equivalent to
equation 2.7, but it captures the more intuitive relation-
ship between potential and current. From now on, we
will just use I net ( I_net in the simulator) to refer to

net ￿ to simplify our notation.
Equation 2.8 provides the means for integrating all
the inputs into a neuron, which show up here as the dif-
ferent values of the conductances for the different ion

(2.6)