volume | PIER Journals

Abstract

Electric fields have been widely used for the treatment of neurological diseases, using techniques such as non-invasive brain stimulation. An electric current controls cell excitability by imposing voltage changes across the cell membrane. At the same time, the presence of the cell itself causes a re-distribution of the local electric field. Computation of the electric field distribution at a single cell microscopic level is essential in understanding the mechanism of electric stimulation. In addition, the impact of the cellular biophysical properties on the field distribution in the vicinity of the cell should also be addressed. In this paper, we have begun by first computing the field distribution around and within a spherical model cell. The electric fields in the three regions differed by several orders of magnitude. The field intensity in the extracellular space was of the same order as that of the externally applied field, while in the membrane, it was calculated to be several thousand times greater than the applied field. In contrast, the field intensity inside the cell was greatly attenuated to approximately 1/133th of the applied field. We then performed a detailed analysis on the dependency of the local field distribution on both the electrical properties (i.e., conductivity, dielectricity), and the geometrical properties (i.e., size, membrane thickness) of the target cell. Variations of these parameters caused significant changes to the amplitude and direction of the electric field around a single cell. The biophysical mechanisms of such observations and their experimental implications are discussed. These results highlight the significance of considering cellular properties during the electric stimulation of neuronal tissues.

1. Gross, R. E. and A. M. Lozano, "Advances in neurostimulation for movement disorders," Neurol. Res., Vol. 22, 247-58, Apr. 2000.

2. Ridding, M. C. and U. Ziemann, "Determinants of the induction of cortical plasticity by non-invasive brain stimulation in healthy subjects," J. Physiol, Vol. 588, 2291-2304, Jul. 1, 2010.
doi:10.1113/jphysiol.2010.190314

3. Chi, R. P. and A. W. Snyder, "Facilitate insight by non-invasive brain stimulation," PLoS One, Vol. 6, e16655, 2011.
doi:10.1371/journal.pone.0016655

4. Fedorov, A., Y. Chibisova, A. Szymaszek, M. Alexandrov, C. Gall, and B. A. Sabel, "Non-invasive alternating current stimulation induces recovery from stroke," Restor. Neurol. Neurosci., Vol. 28, 825-833, 2010.

5. Lee, D. C. and W. M. Grill, "Polarization of a spherical cell in a nonuniform extracellular electric field," Ann. Biomed. Eng., Vol. 33, 603-615, May 2005.
doi:10.1007/s10439-005-2397-3

6. Kotnik, T. and D. Miklavcic, "Analytical description of transmembrane voltage induced by electric fields on spheroidal cells," Biophys. J., Vol. 79, 670-679, Aug. 2000.
doi:10.1016/S0006-3495(00)76325-9

7. Fricke, H., "The electric permittivity of a dilute suspension of membrane-covered ellipsoids," J. Appl. Phys., Vol. 24, 644-646, 1953.
doi:10.1063/1.1721343

8. Schwan, H. P., "Electrical properties of tissue and cell suspensions," Adv. Biol. Med. Phys., Vol. 5, 147-209, 1957.

9. Kotnik, T., F. Bobanovic, and D. Miklavcic, "Sensitivity of transmembrane voltage induced by applied electric fields --- A theoretical analysis," Bioelectrochem. Bioenerg., Vol. 43, 285-291, 1997.
doi:10.1016/S0302-4598(97)00023-8

10. DeBruin, K. A. and W. Krassowska, "Modeling electroporation in a single cell. I. Effects of field strength and rest potential," Biophys. J., Vol. 77, 1213-1224, Sep. 1999.
doi:10.1016/S0006-3495(99)76973-0

11. DeBruin, K. A. and W. Krassowska, "Modeling electroporation in a single cell. II. Effects of ionic concentrations," Biophys. J., Vol. 77, 1225-1233, Sep. 1999.
doi:10.1016/S0006-3495(99)76974-2

12. Ye, H., M. Cotic, and P. L. Carlen, "Transmembrane potential induced in a spherical cell model under low-frequency magnetic stimulation," J. Neural. Eng., Vol. 4, 283-293, Sep. 2007.
doi:10.1088/1741-2560/4/3/014

13. Gimsa, J. and D. Wachner, "Analytical description of the transmembrane voltage induced on arbitrarily oriented ellipsoidal and cylindrical cells," Biophys. J., Vol. 81, 1888-1896, Oct. 2001.
doi:10.1016/S0006-3495(01)75840-7

14. Kotnik, T. and D. Miklavcic, "Theoretical evaluation of voltage inducement on internal membranes of biological cells exposed to electric fields," Biophys. J., Vol. 90, 480-491, Jan. 15, 2006.
doi:10.1529/biophysj.105.070771

15. Pavlin, M., N. Pavselj, and D. Miklavcic, "Dependence of induced transmembrane potential on cell density, arrangement, and cell position inside a cell system," IEEE Trans. Biomed. Eng., Vol. 49, 605-612, Jun. 2002.
doi:10.1109/TBME.2002.1001975

16. Valic, B., M. Golzio, M. Pavlin, A. Schatz, C. Faurie, B. Gabriel, J. Teissie, M. P. Rols, and D. Miklavcic, "Effect of electric field induced transmembrane potential on spheroidal cells: Theory and experiment," Eur. Biophys. J., Vol. 32, 519-528, Sep. 2003.
doi:10.1007/s00249-003-0296-9

17. Farkas, D. L., R. Korenstein, and S. Malkin, "Electrophotoluminescence and the electrical properties of the photosynthetic membrane. I. Initial kinetics and the charging capacitance of the membrane," Biophys. J., Vol. 45, 363-373, Feb. 1984.
doi:10.1016/S0006-3495(84)84160-0

18. Jerry, R. A., A. S. Popel, and W. E. Brownell, "Potential distribution for a spheroidal cell having a conductive membrane in an electric field," IEEE Trans. Biomed. Eng., Vol. 43, 970-972, Sep. 1996.
doi:10.1109/10.532132

19. Wachner, D., M. Simeonova, and J. Gimsa, "Estimating the subcellular absorption of electric field energy: Equations for an ellipsoidal single shell model," Bioelectrochemistry, Vol. 56, 211-213, May 15, 2002.
doi:10.1016/S1567-5394(02)00020-8

20. Faber, D. S. and H. Korn, "Field effects trigger post-anodal rebound excitation in vertebrate CNS," Nature, Vol. 305, 802-804, Oct. 27--Nov. 2, 1983.
doi:10.1038/305802a0

21. Dudek, F. E., T. Yasumura, and J. E. Rash, "`Non-synaptic' mechanisms in seizures and epileptogenesis," Cell Biol. Int., Vol. 22, 793-805, Nov. 1998.
doi:10.1006/cbir.1999.0397

22. Rattay, F., "Analysis of the electrical excitation of CNS neurons," IEEE Trans. Biomed. Eng., Vol. 45, 766-772, Jun. 1998.
doi:10.1109/10.678611

23. McIntyre, C. C., W. M. Grill, D. L. Sherman, and N. V. Thakor, "Cellular effects of deep brain stimulation: Model-based analysis of activation and inhibition," J. Neurophysiol., Vol. 91, 1457-1469, Apr. 2004.
doi:10.1152/jn.00989.2003

24. Gimsa, U., U. Schreiber, B. Habel, J. Flehr, U. van Rienen, and J. Gimsa, "Matching geometry and stimulation parameters of electrodes for deep brain stimulation experiments --- Numerical considerations," J. Neurosci. Methods, Vol. 150, 212-227, Jan. 30, 2006.
doi:10.1016/j.jneumeth.2005.06.013

25. Sukharev, S. I., V. A. Klenchin, S. M. Serov, L. V. Chernomordik, and A. Chizmadzhev Yu, "Electroporation and electrophoretic DNA transfer into cells. The effect of DNA interaction with electropores," Biophys. J., Vol. 63, 1320-1327, Nov. 1992.
doi:10.1016/S0006-3495(92)81709-5

26. Kinosita, Jr., K. and T. Y. Tsong, "Voltage-induced pore formation and hemolysis of human erythrocytes," Biochim. Biophys. Acta, Vol. 471, 227-242, Dec. 1, 1977.

27. Somiari, S., J. Glasspool-Malone, J. J. Drabick, R. A. Gilbert, R. Heller, M. J. Jaroszeski, and R. W. Malone, "Theory and in vivo application of electroporative gene delivery," Mol. Ther., Vol. 2, 178-187, Sep. 2000.
doi:10.1006/mthe.2000.0124

28. Yousif, N., R. Bayford, and X. Liu, "Revealing the biophysical mechanism for configuring electrode contacts in deep brain stimulation," Sixteenth Annual Computational Neuroscience Meeting: CNS*2007, P143 Toronto, Canada, 2007.

29. Mossop, B. J., R. C. Barr, J. W. Henshaw, and F. Yuan, "Electric fields around and within single cells during electroporation --- a model study," Ann. Biomed. Eng., Vol. 35, 1264-1275, Jul. 2007.
doi:10.1007/s10439-007-9282-1

30. Mossop, B. J., R. C. Barr, D. A. Zaharoff, and F. Yuan, "Electric fields within cells as a function of membrane resistivity --- A model study," IEEE Trans. Nanobioscience, Vol. 3, 225-231, Sep. 2004.
doi:10.1109/TNB.2004.833703

31. Bryant, G. and J. Wolfe, "Electromechanical stresses produced in the plasma membranes of suspended cells by applied electric fields," J. Membr. Biol., Vol. 96, 129-139, 1987.
doi:10.1007/BF01869239

32. Pucihar, G., T. Kotnik, B. Valic, and D. Miklavcic, "Numerical determination of transmembrane voltage induced on irregularly shaped cells," Ann.Biomed. Eng., Vol. 34, 642-652, Apr. 2006.
doi:10.1007/s10439-005-9076-2

33. Vigmond, E. J., J. L. Perez Velazquez, T. A. Valiante, B. L. Bardakjian, and P. L. Carlen, "Mechanisms of electrical coupling between pyramidal cells," J. Neurophysiol, Vol. 78, 3107-3116, Dec. 1997.

34. Loew, L. M., "Voltage-sensitive dyes: Measurement of membrane potentials induced by DC and AC electric fields," Bioelectromagnetics, Vol. 1, 179-189, 1992.
doi:10.1002/bem.2250130717

35. Ghai, R. S., M. Bikson, and D. M. Durand, "Effects of applied electric fields on low-calcium epileptiform activity in the CA1 region of rat hippocampal slices," J. Neurophysiol, Vol. 84, 274-280, Jul. 2000.

36. Grill, Jr., W. M., "Modeling the effects of electric fields on nerve fibers: Influence of tissue electrical properties," IEEE Trans. Biomed. Eng., Vol. 46, 918-928, Aug. 1999.
doi:10.1109/10.775401

37. Sotiropoulos, S. N. and P. N. Steinmetz, "Assessing the direct effects of deep brain stimulation using embedded axon models," J. Neural. Eng., Vol. 4, 107-119, Jun. 2007.
doi:10.1088/1741-2560/4/2/011

38. Gehl, J., "Electroporation: Theory and methods, perspectives for drug delivery, gene therapy and research," Acta Physiol. Scand., Vol. 177, 437-447, Apr. 2003.
doi:10.1046/j.1365-201X.2003.01093.x

39. Ehrenberg, B., D. L. Farkas, E. N. Fluhler, Z. Lojewska, and L. M. Loew, "Membrane potential induced by external electric field pulses can be followed with a potentiometric dye," Biophys. J., Vol. 51, 833-837, May 1987.
doi:10.1016/S0006-3495(87)83410-0

40. Teruel, M. N. and T. Meyer, "Electroporation-induced formation of individual calcium entry sites in the cell body and processes of adherent cells," Biophys. J., Vol. 73, 1785-1796, Oct. 1997.
doi:10.1016/S0006-3495(97)78209-2

41. Holsheimer, J., "Electrical conductivity of the hippocampal CA1 layers and application to current-source-density analysis," Exp. Brain Res., Vol. 67, 402-410, 1987.
doi:10.1007/BF00248560

42. Tyner, K. M., R. Kopelman, and M. A. Philbert, "Nanosized voltmeter enables cellular-wide electric field mapping," Biophys. J., Vol. 93, 1163-1174, Aug. 15, 2007.
doi:10.1529/biophysj.106.092452

43. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.

Dr. Hui Ye

Dr. Marija Cotic

Dr. Michael G. Fehlings

Dr. Peter L. Carlen