Sawtooth effect: fact or fancy?

How does an electric stimulus affect the transmembrane potential in tissue far from the stimulating electrode?1 Plonsey and Barr2 proposed one of the  rst hypotheses for such “far- eld” stimulation, known as the “sawtooth effect.” Current  owing in the tissue encounters a periodic increase in resistance at the intercellular junctions coupling adjacent cardiac cells. As current approaches a junction, it leaves the intracellular space to take advantage of the relatively lowresistance extracellular space, thereby depolarizing the membrane. After passing the junction, current reenters the intracellular space, hyperpolarizing the membrane. This redistribution of current occurs at every intercellular junction, causing each cell to depolarize at one end and hyperpolarize at the other. Krassowska et al.3,4 developed this idea further, and others have used it to explain cardiac de brillation.5,6 Is the sawtooth hypothesis true? In the case of single isolated cells, it undoubtedly is.7 However, attempts to observe the sawtooth potential in intact tissue have failed.8,9 The article by Sharma and Tung10 in this issue of the Journal examines a system intermediate between a single cell and intact tissue: an isolated cell-pair. They chose an excellent system to study, because it is the simplest system that contains at least one intercellular junction. Isolated single cells are interesting but they lack any cell junctions, and the in uence of junctions is the very essence of the sawtooth effect. I have grown accustomed to describing the sawtooth effect as “each cell is depolarized at one end and hyperpolarized at the other,” so I was startled when Sharma and Tung demonstrated that this is not always correct. For two cells of different lengths, the transmembrane potential is discontinuous at the cell junction, but it may or may not change sign, depending on the junctional resistance. I wonder if this interesting result is applicable near the center of a long strand of cells, many length constants from the ends? One crucial parameter for any analysis of the sawtooth effect is the resistance of the intercellular junction. Sharma and Tung review several studies that suggest a junctional resistance of tens or even hundreds of M V . These values are higher than expected, given the reported macroscopic conductivity of the intracellular space. For example, Plonsey and Barr2 assumed a microscopic intracellular resistance per unit length of 1.6 3 108 V /cm (Sharma and Tung assume the same value) and then calculated the junctional resistance required to make the macroscopic intracellular conductivity match that measured by Clerc.11 Their result was 0.85 M V . How do we reconcile Clerc’s data with a junctional resistance of tens or hundreds of M V ? Simple calculations may not capture the structural complexity of cardiac tissue, and certainly Plonsey and Barr’s calculation is based on a simple model. Yet, regardless of the tissue structure, the intracellular current can hardly avoid passing through at least one intercellular junction every cell length. If we believe Clerc’s value for the effective intracellular conductivity, we can calculate an upper limit for the junctional resistance of about 1.6 M V .* I can think of ways to justify a junctional resistance lower than this, but I cannot image how the junctional resistance could be much higher, at least in healthy tissue. This issue is important, because the magnitude of the sawtooth effect increases with the junctional resistance (see Sharma and Tung’s Fig. 1C). If we overestimate this resistance, we overestimate the magnitude of the sawtooth effect. Although studies of cell-pairs are necessary and useful, the important question is what impact the sawtooth effect has on the heart. For instance, is the magnitude of the sawtooth effect large enough to be responsible for reentry induction or de brillation? We can set an upper limit for the sawtooth effect by assuming the cells are completely decoupled. In that case, the change in transmembrane potential from one end of the cell to the other equals the electric  eld strength times the cell length. If a 22 V/cm electric  eld stimulates a cell 120 mm long, the change in transmembrane potential from one end to the other should be 260 mV (130 mV depolarized on one side and 130 mV hyperpolarized on the other). Sharma and Tung present data from cell-pairs uncoupled by heptanol that are consistent with this estimate. Frazier et al.13 suggested that 5 V/cm is the minimum electric  eld strength necessary for reentry induction by formation of a critical point. An electric  eld of 5 V/cm corresponds to cells depolarized by at most 30 mV at one end and hyperpolarized by at most 30 mV at the other. Here my analysis becomes dif cult, because 30 mV is not a huge depolarization, but it is not tiny either. It corresponds roughly to the threshold depolarization needed to excite a resting membrane. Remember, 30 mV is an upper limit, found by assuming that the cell has no lateral connections to other cells and that the entire intracellular voltage drop occurs across the junctional resistance, with no voltage drop across the myoplasmic resistance. Juhlin and Pormann14 examined a more realistic model and found maximum depolarizations of 10 to 15 mV. Moreover, reentry induction and de brillation require excitation of relatively refractory tissue, where the threshold is high. Finally, excitation of a cell that is half depolarized and half hyperpolarized is a complicated process.4 Having just one end of the cell depolarized to threshold may not be suf cient to excite the entire cell. All things considered, I remain skeptical that the sawtooth effect alone can explain a 5 V/cm threshold for