Page 115 - YORAM RUDY BOOK FINAL

P. 115
and integration by parts gives (considering that 1/r = 0 at ∞ )

∞ 2
Z
1 σ ∂ V ( ) 1
φ = i ∫ dA ∫ m dZ
dA
dZ
π
4 σ A − ∞ ∂ Z 2 r (4.17)
° O
o

In (4.16), the sources that generate the potential field ϕ are double-layers of strength
°
– ∂V / ∂Z situated on cross-sectional disks A along the fiber. In (4.17) the sources are single-layer
m
(monopolar) disks of strength ∂ V / ∂Z . Note that the sources in (4.16) and (4.17) are equivalent
2
2
m
in the sense that they generate the same external potential field ϕ . Importantly, these are not
0
the true biophysical sources (membrane ionic currents); in fact, they are located inside the cell
volume while the true sources are confined to the cell membrane. Never the less, these equivalent
source formulations provide a way to compute ϕ associated with an action potential V . Note
0
m
that V can be measured experimentally or simulated using models as in section 3. Note also that
m
V contributes to ϕ only where it varies (∂V / ∂Z and ∂ V / ∂Z are zero where V is constant).
2
2
0
m
m
m
m
This implies that spatial potential gradients (e.g., during depolarization or repolarization) are
detected by the ECG, but uniform V (at rest or during the plateau phase) is “silent”.
m

Figure 4.3. Geometry of a cylindrical fiber. ρ, Z are cylindrical coordinates. a is fiber radius. σ ,σ are
i
o
intracellular and extracellular conductivities, respectively.