128 J. W. WOODBURY AND W . E. CRILL 



unspecified "junctionar' transmission. Their evidence is all indirect, and little 

 of it critical in the sense that it does not sharply distinguish between the possi- 

 bihties. The only evidence presented by these investigators which is appar- 

 ently not compatible with local circuit propagation is the reversible blocking 

 of conduction in heart by a solution made three times isotonic with sucrose. 

 This effect is probably explained by the electron-micrographic observation 

 that the intercalated disks separate in this solution (S. Scheyer, personal 

 communication). Also, the spread of applied currents is simultaneously 

 greatly reduced or abolished (Crill, unpublished experiments). 



The principal question raised by these experiments is not what is the 

 mechanism of spread, but what factors give rise to the short space constant. 

 There is at present insufficient knowledge of the tissue to permit the synthesis 

 of an accurate equivalent electrical circuit. Thus, the first step in evaluating 

 the data is to assume an equivalent circuit and then to see if the measured 

 properties are approximately predicted by the model. The simplest model is 

 that the atrium consists electrically of a single planar cell of practically in- 

 finite extent, bounded by two parallel membranes 15 /x apart. The cell is 

 bathed by thick layers of extracellular fluid of negligible resistance. This 

 model is the two-dimensional generalization of the equivalent circuit of a 

 nerve fiber in a large volume of fluid. The equation describing the voltage as a 

 function of distance and time in both models is A^V'^e — e ^ t'e where 

 A and t are the space and time constants, V-e is the Laplacian of membrane 

 voltage, £, and e = de/dt. For the punctate appfication of current the appro- 

 priate form for V-e is in polar co-ordinates. If e is a function of /• only, i.e. 

 no angular variation in the properties of the tissue, the steady-state solution 

 for which e = Oatr=ooisa zero-order Bessel function of the second 

 kind with imaginary argument. As would be supposed from the geometry 

 of the system, this Bessel function falls off rapidly with distance near the 

 origin, where the membrane area available to current penetration is increasing 

 rapidly. 



An attempt was made to fit data of the type shown in Fig. 2 to the appro- 

 priate Bessel function. The fit of the data to the theoretical curve is modera- 

 tely good and considerably better than the fit to an exponential function. A 

 reasonably good approximation to the space constant in the fiber direction 

 is 130 /x. The method of fitting the data was strictly trial-and-error. It appeared 

 that a better fit could have been obtained with additional trials. However, for 

 preliminary analysis the possible improvement in accuracy was not considered 

 worth the extra effort. 



An idea of the shortness of this space constant can be obtained by com- 

 paring it with the theoretical space constant of the model. Assuming that the 

 specific membrane resistance (Rm) is 1000 il cm- the spacing, S, between the 

 parallel membranes is 10 /x and the internal specific resistivity, p., is 100 

 12 cm, then the space constant is A = ^ '[/?,„/(/> • /•(•)] = 1200 /u, where 



