Between Adjacent Unmyelinated Parallel Fibers

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers Roger C. Barr and Robert Plonsey Departments of Biomedical Engineering, Pediatrics, and Cell Biology, Duke University, Durham, North Carolina 27706 ABSTRACT The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial intersitital potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm - 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-p,m radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function. INTRODUCTION Extracellular potentials cellular potentials in most in vitro studies. Small extracellular potentials allowed them to equate transmembrane The existence of extracellular potentials of significant potentials to intracellular ones and to compute mem- magnitude has been recognized and utilized from the brane currents as the second spatial derivatives of earliest investigations in electrophysiology. The pres- transmembrane potentials. This pattern has been fol- ence of such potentials arising from cardiac muscle lowed in most analyses of experimental studies until the made as possible such formative studies those of Lewis present because the structure of the experimental envi- and Rothschild on sequences, as (22) cardiac excitation ronment, which often involves a large conducting vol- well as the investigation of the intracellular-extracellu- ume surrounding an active fiber, makes the assumption lar voltage and current relationships by Lorente de No of near-zero extracellular potentials a good one. (24) and others in nerves. In 1940 Katz and Schmitt (18) Paradoxically, it is well known that substantial intersti- showed electrical interactions in vitro in a crab nerve preparation, and in 1941, Arvanitaki (1) defined the tial potentials exist in vivo, especially in cardiac muscle. Vander Ark and Reynolds (47) described potentials "ephapse" as the locus of close vicinity of two active with magnitudes in excess of 50 mV within the ventricu- membranes, and called transmission across such a site "ephaptic." lar wall. Spach and Barr (41) reported voltages that were Nevertheless, the development of the penetrating tens of millivolts in potential maps of the canine heart. microelectrode by Ling and Gerhard (23), which al- Kleber and Riegger (19) showed large extracellular potentials (equal to or greater than intracellular poten- lowed investigators to measure intracellular potentials tials) in rabbit papillary muscle under carefully con- directly, greatly diminished the need to use extracellular trolled conditions, and Taccardi et al. (44) potentials as indirect measures of transmembrane events. showed Thereafter, Hodgkin and Huxley (16), in their study of dramatic changes in observed extracellular waveforms of large magnitude based entirely on changes in the extra- the nerve membrane, took advantage of the fact that cellular conducting paths. Studying the extracellular potentials are small in comparison to intra- spinal motoneurons of the cat, Nelson (28) showed that facilitation of motoneurons can occur as a result of activation of other Address correspondence to Dr. Barr. motoneurons. Recently, an extensive review of electrical 1164 0006-3495/92/05/1164/120006-3495/92/05/1 164/12 $2.00 Biophys.I JJ. © Biophysical Society Volume 61 May 1992 1 164-1 175 field effects and their relevance to central neural net- Plonsey and Barr (2, 29) showed that the current flow works has been presented by Faber and Korn (10), patterns within the muclse would have multiple mem- especially as related to the mauthner cell. brane transitions (rather than a simple local circuit with In part because of the papers of Spach et al. (e.g., only two) and that, under extreme conditions, the shape reference 42) on the importance of cellular discontinui- of the excitation wavefront would deviate markedly from ties to propagation, the effects of all aspects of structure its expected elliptical shape. Wikswo and co-workers on propagation have received more systematic scrutiny (40, 48) used bidomain analysis to show that stimulation in recent years. Nonetheless, the effects of substantial of cardiac tissue can produce a virtual cathode, a interstitial or extracellular potentials on propagation is a dog-bone pattern of response to stimulation that arises topic that has received much less attention, presumably from the anisotropic interstitial and intracellular conduc- because of the recognized importance of syncytial con- tivity. nections of the intracellular spaces of adjacent cells Using bidomain analysis, Plonsey and Barr (30) ar- along with the technical difficulty of studying interstitial gued on theoretical grounds that interstitial potentials interactions. This technical difficulty arises experimen- rose quite rapidly with movement from the surface into tally because the greatest effects occur when cells or the tissue, so that substantial interstitial potentials could fibers are closely packed, producing the largest intersti- be expected within a few cell diameters of the surface. tial potentials but allowing the least access to electrodes. Henriquez and Plonsey (14, 15) went on to show for Specifically focused experimental design thereby is re- cylindrical bundles that the traditional concept of a quired, as in the work of Knisley et al. (20). In mathemat- plane wave of excitation traveling down a bundle or ical and numerical models, extracellular and interstitial block of tissue in contact with an external bathing studies also have been few. (Exceptions have included solution could not be supported and that concave bidomain models [below], and significant recent studies wavefronts were to be expected, a subject also explored such as Halter and Clark for myelinated nerve [13] and by Roth (37). Leon and Roberge [21] showing velocity changes.) With These papers also show examples of a major shortcom- models, the technical difficulty has been that mathemat- ing of the bidomain models: the scale of some of the ical analysis is simpler and numerical analysis is shorter emerging findings is the same as that of the discretiza- when extracellular potentials are assumed to be zero. tion of the tissue formed by its cellular nature, so that the macroscopic averaging that underlies the bidomain Bidomain formulation may not be justified. The bidomain model models probably best portrays real tissue when it is extensive The one area of analysis that routinely has included and healthy, since in these circumstances the interstitial interstitial potentials has been "bidomain" models, in potentials are largest and the uniformity assumed by the which the averaged properties of intracellular and inter- bidomain is greatest. In contrast, if one considers small stitial "domains" have been incorporated. Anticipated numbers of active fibers, examines behavior when fibers somewhat by Schmitt (39) in 1969 with his description of are more loosely coupled, or seeks the potential field at a interpenetrating domains, Tung (46), Miller and Ge- cellular (microscopic) level, a more sophisticated treat- selowitz (11, 26), and others took the viewpoint that, in ment is necessary. The goal of this paper is to investigate spite of the actual discrete structure, the electrical interstitial and extracellular effects so as to see how events within cardiac muscle could be examined by excitation is affected by each, but to avoid the averaging regarding the tissue as a continuum (syncytium). Both assumptions of the bidomain. intracellular space and interstitial space were consid- ered continuous and described by the same coordinates, Model of two fibers separated everywhere by the membrane. The same basic framework has been used, for example, by Muler and A central difficulty in conceiving a plan to investigate the Markin (27) in a mathematical analysis, by Roberts et al. effect of interstitial potentials on propagation has been (36) in the ventricle, by Eisenberg et al. (9) as related to the complexity of the experimental preparation or the the lens of the eye, and by Roth (37) for cardiac muscle. model, which is the basis of a simulation. The problem The results

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