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channels. For example, the amount of excitatory input
determines what fraction of the excitatory synaptic in-
put channels are open ( g e (t) ), and thus how much the
neuron's membrane potential is driven toward the ex-
citatory reversal potential ( +55mV ). However, to ac-
curately represent what happens in a real neuron, one
would need to apply equation 2.8 at every point along
the dendrites and cell body of the neuron, along with
additional equations that specify how the membrane po-
tential spreads along neighboring points of the neuron.
The details of how to do this are beyond the scope of
this text — see Johnston and Wu (1995) for a detailed
treatment, and Bower and Beeman (1994) or Hines and
Carnevale (1997) for software that implements such de-
tailed models.
To avoid the need to implement hundreds or thou-
sands of equations to simulate a single neuron, we will
take advantage of a useful simplifying approximation
that enables us to directly use equation 2.8 to compute
the membrane potential of a simulated neuron. This ap-
proximation is based on the fact that a large part of what
happens to the electrical signals as they propagate from
the dendrites to the cell body is that they get averaged
together (though see section 2.5.1 for notable excep-
tions to this generalization). Thus, we can just use the
average fraction of open channels of the various types
across the entire dendrite as a very crude but efficient
approximation of the total conductance of a particular
type of channel, and plug these numbers directly into a
single equation (equation 2.8) that summarizes the be-
havior of the entire neuron.
This approximation is the essence of the point neuron
approximation mentioned previously, where we have
effectively shrunk the entire spatial extent of the neu-
ron down to a single point, which can then be modeled
by a single equation. We see later that our computa-
tional model actually captures some important aspects
of the spatial structure of the dendrites in the way the
excitatory input conductance ( g e (t) ) is computed.
To illustrate how a point neuron would behave in the
face of different amounts of excitatory input, we can
plot the results of repeatedly using equation 2.8 to up-
date the membrane potential in response to fixed levels
of input as determined by specified values of the various
conductances. Figure 2.9 shows a graph of the net cur-
40−
−30−
30−
−35−
−40−
20−
−45−
10−
I_net
−50−
0−
−55−
g_e = .4
−10−
−60−
−20−
g_e = .2
−65−
−30−
V_m
−70−
−40−
0
5 0 5 0 5 0 5 0
cycles
Figure 2.9: Two traces of the computed net current ( I net )
and membrane potential ( V m ) updated by excitatory inputs at
time 10, one of total conductance ( g e (t)g e , labeled as g e )of
.4 and the other of .2.
rent and membrane potential (starting at 0 current with
the rest potential of ￿70mV ) responding to two differ-
ent inputs that come in the form of a g e (t) value of .4 or
.2 starting at time step 10 (with a g e of 1). The g l value
is a constant 2.8 (and g l (t) is always 1 because the leak
is always on), there is no inhibitory conductance, and
(you will run this example yourself in sec-
tion 2.6).
The important thing to note about this figure is that
the membrane potential becomes elevated in response
to an excitatory input, and that the level of elevation is
dependent on the strength (conductance) of the excita-
tion compared to the leak conductance (and other con-
ductances, if present). This membrane potential eleva-
tion then provides the basis for the neuron's subsequent
output (i.e., if it gets over the threshold, the neuron will
fire). Although the firing mechanism is not reflected in
the graph, the stronger of these inputs put our simulated
neuron just over its threshold for responding, which was
, while the weaker was clearly sub-threshold .
This figure also shows that the value of I net repre-
sents the amount of change in the value of V m . Thus,
represents the derivative of V m (i.e., I net is large
when V m is rising rapidly, and settles back toward 0 as
settles into a steady or equilibrium value).
2.4.6
The Equilibrium Membrane Potential
The notion of an equilibrium membrane potential is an
important concept that we will explore in more detail
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