Page 9 - Cardiac Electrophysiology | A Modeling and Imaging Approach

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g g Na Na Na g • • m m • h• h (2.4)
g = g g=
Na
Na Na
Where m is an activation gate, h an inactivation gate and ḡ Na is the maximum membrane
conductance for the Na ion species (when all gates are open). I current activation (increase of
+
Na
conductance) is simulated accurately with three identical m gates that transition from 0 (closed)
to 1 (open) upon V depolarization. The time dependence of m follows first-order kinetics:
m

dm dm = α • ( −1 m )− β • m

dt (2.5)
dt

Where m and (1-m) are the gate open and closed probabilities, respectively; α and ß are
opening and closing transition rates that depend on V . Because the transitions of each gate are
m
assumed independent of the other gates, the probability that all m gates are open is m . Upon
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depolarization, these gates open very rapidly, generating a large inward I that generates the rapid
Na
upstroke of the action potential. The voltage clamp recordings of Hodgkin and Huxley indicated
that I decreases shortly after activation (a process termed inactivation). This process is
Na
represented by the parameter h, which is also governed by a first-order differential equation similar
to equation (2.5). At hyperpolarized V , h is fully open; it closes when the membrane is depolarized
m
to cause a monoexponential decrease of I during voltage-clamp protocols, as observed
Na
experimentally. Based on the independent gating assumption, h transitions are independent of m
and the open probability for I is m h. For the axon, a similar formalism was used to compute a
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Na
potassium current I . The combination of inward I and outward I (together with a leakage
K Na K
current I ) was sufficient to reproduce the morphology of the action potential under a variety of
L
conditions.

2.2 Computing the Action Potential from Ion Channels Kinetic Properties

In the Hodgkin Huxley scheme, I in equation (2.1) is the macroscopic current through an
ion
ensemble of many ion channels. The gating variables are model parameters that do not
correspond to specific kinetic properties of the ion channel, nor to its molecular structure. In fact,

at the time when Hodgkin and Huxley formulated the model, the mechanism of voltage and time
dependent gating was not known and ion-channel proteins were not identified as the pathway
for transmembrane flow of ions. Based on current knowledge of ion channels and their structure/
function properties, it is apparent that the Hodgkin-Huxley formalism is limited in its ability to

describe various aspects of channel behavior. The gating parameters do not represent specific
kinetic states of the channel and the independent gating assumption precludes dependence of a
given transition on occupancy of the different states of the channel (for example, sodium channel