Page 71 - Cardiac Electrophysiology | A Modeling and Imaging Approach

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3. PROPAGATION OF THE CARDIAC

ACTION POTENTIAL

3.1. Basic Biophysical Principles

The fundamental biophysical principles of action potential propagation in multicellular
cardiac tissue are best explained in a 1-dimensional strand of cells. Propagation is accomplished
by axial current flow from a depolarized cell to its less depolarized neighbors through specialized

intercellular resistive pathways called gap junctions. During conduction, the membrane potential
V is a function of both time (t) and space (x) and the following reaction-diffusion equation holds:
m

∂V  a   ∂ V 
2
C m + I =    •   m  (3.1)


m
∂t ion  2r i   ∂X 2 

Where a is the fiber radius and r its axial resistance per unit length. The left side of
i
the equation is the total transmembrane current, the sum of the capacitive component C m ∂ V m

and the ionic component I . The right side is the gain or loss of axial current as it flows down t ∂
ion
the fiber. This equation is simply a mathematical statement of the conservation principle that any
gain or loss of axial current is accounted for by current crossing the membrane. Note that under
2
space-clamp conditions V does not vary in space, ∂ V /∂X 2 = 0 and equation (3.1) is reduced to
m m
equation (2.1) that describes the non-propagated action potential.

In an isolated single cell, dV / dt is proportional to I (equation 2.1) because the entire
m ion
charge contribution from I is used to discharge the local membrane capacitance C . Therefore,
ion m
I determines the rate of V depolarization, dV / dt, in a single cell. This property identifies
ion m m
dV / dt as a measure of the transmembrane current flow; its maximum (dV / dt , the steepest
m m max
portion of the action potential upstroke) coincides in time with peak I . In the multicellular tissue
Na
(equation 3.1) this proportionality is lost because the charge generated by I is divided between
ionic
the local membrane and the axial current. Peak I occurs later than dV / dt which is no longer
Na m max
an accurate marker of local activation.

Equation (3.1) represents cardiac tissue as a continuous electrical syncytium, because r in
i
this equation is an average resistance that lumps together the cytoplasmic and gap junctional
resistances. In this simplified model (borrowed from the nerve axon) the velocity of conduction,
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θ is proportional to the square root of dV / dt and (independently) inversely proportional to the
m max
square root of r . Importantly, in this continuous model dV / dt depends only on membrane
i m max