586
14 Neural Networks and Brains
The ion active-transport channels are not included in the Hodgkin–Huxley
model, as only a small discharge of the membrane occurs on each pulse. Moreover,
the impulse drains the small charge in a millisecond or so, while the active-transport
channels take about 8 ms to pump the same amount of charge back.
Let’s see how the Hodgkin–Huxley neuron membrane operates. Including an
external source of current I into the membrane, charge conservation makes
I = g Na (V − V Na ) + g K (V − V K ) + g L (V − V L ) + C
dV
dt
,
(14.1)
where V is the potential difference across the membrane and each g i , called the
conductivity, is the inverse of the resistance in the respective channel. The constants
V i are set so that if the potential across the membrane matches V i , then the current in
that channel is zero. The passive nature of the leakage channel makes g L a constant.
However, the voltage-gated sodium and potassium channels will have conductivities
which depend on V itself. Here is where the non-linearity of the system arises.
By fitting the measured conductivity to particular models of the voltage-gated
channels, Hodgkin and Huxley discovered that the sodium channels have three
separate fast gates which must open before the sodium ions can pass, and another
slower-acting gate that closes to inactivate the channel. They found that the
potassium channels had four slower-acting gates to activate the potassium channel.
An external stimulus that increases the membrane potential from −70 mV to a
certain threshold value (about −45 mV), triggers a nerve impulse. Over the next half
a millisecond, the three sodium channel activation gates open, letting sodium ions
into the cell, raising the potential to about +30 mV. The slower-acting inactivating
sodium gate then close the sodium channel, while the four potassium gates in
each potassium channel open to release potassium from the cell. After another
half millisecond, the cell potential goes negative and the potassium gates close, but
acting relatively slowly, the potential goes below −70 mV, leading to a refractory
period of about two milliseconds, during which the membrane does not generate a
new impulse even when encountering the triggering stimulus.
Hodgkin and Huxley proceeded in their mathematical formulation as follows.
They let
m be the probability that one of the sodium activation gates is open,
h be the probability that the sodium channel inactivation gate is open,
and
n be the probability that one of the potassium activation gates is open.
The use of probabilities is justified by noting that even though any one channel
handles only a few ions at a time, there are many channels on the membrane to
average over.
Each of these probabilities varies as the voltage V varies. Combining the
probabilities gives
g Na = m
3 hg
max
Na
(14.2)
14 Neural Networks and Brains
The ion active-transport channels are not included in the Hodgkin–Huxley
model, as only a small discharge of the membrane occurs on each pulse. Moreover,
the impulse drains the small charge in a millisecond or so, while the active-transport
channels take about 8 ms to pump the same amount of charge back.
Let’s see how the Hodgkin–Huxley neuron membrane operates. Including an
external source of current I into the membrane, charge conservation makes
I = g Na (V − V Na ) + g K (V − V K ) + g L (V − V L ) + C
dV
dt
,
(14.1)
where V is the potential difference across the membrane and each g i , called the
conductivity, is the inverse of the resistance in the respective channel. The constants
V i are set so that if the potential across the membrane matches V i , then the current in
that channel is zero. The passive nature of the leakage channel makes g L a constant.
However, the voltage-gated sodium and potassium channels will have conductivities
which depend on V itself. Here is where the non-linearity of the system arises.
By fitting the measured conductivity to particular models of the voltage-gated
channels, Hodgkin and Huxley discovered that the sodium channels have three
separate fast gates which must open before the sodium ions can pass, and another
slower-acting gate that closes to inactivate the channel. They found that the
potassium channels had four slower-acting gates to activate the potassium channel.
An external stimulus that increases the membrane potential from −70 mV to a
certain threshold value (about −45 mV), triggers a nerve impulse. Over the next half
a millisecond, the three sodium channel activation gates open, letting sodium ions
into the cell, raising the potential to about +30 mV. The slower-acting inactivating
sodium gate then close the sodium channel, while the four potassium gates in
each potassium channel open to release potassium from the cell. After another
half millisecond, the cell potential goes negative and the potassium gates close, but
acting relatively slowly, the potential goes below −70 mV, leading to a refractory
period of about two milliseconds, during which the membrane does not generate a
new impulse even when encountering the triggering stimulus.
Hodgkin and Huxley proceeded in their mathematical formulation as follows.
They let
m be the probability that one of the sodium activation gates is open,
h be the probability that the sodium channel inactivation gate is open,
and
n be the probability that one of the potassium activation gates is open.
The use of probabilities is justified by noting that even though any one channel
handles only a few ions at a time, there are many channels on the membrane to
average over.
Each of these probabilities varies as the voltage V varies. Combining the
probabilities gives
g Na = m
3 hg
max
Na
(14.2)