A general rigorous analytic framework for computing the transmembrane potential shift resulting from the nonlinear voltage-current membrane relationship in response to wideband stochastic electromagnetic radiation is outlined, based on Volterra functional series. The special case of an insulated cylindrical cell with Hodgkin-Huxley membrane in an infinite homogeneous medium is worked out in detail, for the simplest case where the applied electric is normal to the cell axis, and independent from the axial coordinate. Representative computational results for a zero-average stationary band-limited white Gaussian incident field are illustrated and briefly discussed.
2. Foster, K. R. and H. P. Schwan, "Dielectric properties of tissues: A review," Handbook of Biological E®ects of Electromagnetic Radiation, C. Polk and E. Postow (eds.), CRC Press, Boca Raton (FL), 1995.
3. Barnes, F. S. and C. L. Hu, "Model of some non-thermal effects of radio and microwave fields in biological membranes," IEEE Trans. Microw. Theory Tech., Vol. 25, No. 9, 742-746, 1977.
4. Berkovitz, G. C. and F. S. Barnes, "The effects of nonlinear membrane capacity on the interaction of microwave and radio frequency fields with biological materials," IEEE Trans. Microw. Theory Tech., Vol. 27, No. 2, 204-207, 1979.
5. Chang, S. K. W., "Electric-field-induced volume and membrane ionic permeability changes of red blood cells," EEE Trans. Biomed. Eng., Vol. 40, No. 10, 1054-1059, 1993.
6. Wachtel, H., "Firing-pattern changes and trans-membrane currents produced by ELF fields in pacemaker neuron," 18th Annual Meeting, Hanford Life Sciences Symposium, WA (USA), Richland, Oct. 16-18, 1978.
7. Postow, E. and M. L. Swicord, "Modulated fields and window effects," Handbook of Biological Effects of Electromagnetic Fields, C. Polk and E. Postow (eds.), CRC Press, Boca Raton, FL, 1995.
8. Franceschetti, G. and I. M. Pinto, "Cell membrane nonlinear response to an applied electromagnetic field," IEEE Trans. Microw. Theory Tech., Vol. 32, No. 7, 653-658, 1984.
9. Franceschetti, G. and I. M. Pinto, "Who is who in nonlinear electromagnetics," Electromagnetics, Vol. 11, 281-305, 1991.
10. Casaleggio, A., L. Marconi, G. Morgavi, S. Ridella, and C. Rolando, "Current flow in a cell with a non-linear membrane stimulated by an electric field," Bioelectrochemistry and Bioenergetics, Vol. 14, No. 1-3, 13-21, 1985.
11. Marconi, L., G. Morgavi, S. Ridella, and C. Rolando, "Non-linear ionic fluxes in an electrically exposed cell," Bioelectrochemistry and Bioenergetics, Vol. 16, No. 1, 89-98, 1986.
12. Bisceglia, B., G. Franceschetti, I. M. Pinto, and M. R. Scarfi, "Volterra series solution of Hodgkin-Huxley equation," Atti V RiNEM, 1-5, Saint Vincent (IT), 1984.
13. Kistler, W., W. Gerstner, and J. L. Van Hemmen, "Reduction of the Hodgkin Huxley equation to a single-variable threshold model," Neural Comput., Vol. 9, No. 5, 1015-1045, 1997.
14. Hodgkin, A. L. and A. F. Huxley, "A quantitative description of membrane current and its application to conduction and excitation in nerves," J. Physiol., Vol. 117, No. 4, 500-544, 1952.
15. De Vita, A., B. Bisceglia, R. P. Croce, and I. M. Pinto, "An analytic model for the response of an excitable cell with a nonlinear Hodgkin-Huxley membrane radiated by a stochastic electromagnetic field ," Proc. 2008 URSI General Assembly, paper K01p5, 2008.
16. Middleton, D., "Statistical physical models of urban radio noise environments --- Part I: Foundations," IEEE Transactions on Electromagn. Compatib., Vol. 14, 38-56, 1972.
17. Middleton, D., "Man-made noise in urban environments and transportation systems: Models and measurements," IEEE Transactions on Electromagn. Compatib., Vol. 21, 1232-1241, 1973.
18. Middleton, D., "Statistical-physical models of electromagnetic interference ," IEEE Transactions on Electromagn. Compatib., Vol. 19, 106-127, 1977.
19. De Felice, L. J., Introduction to Membrane Noise, Plenum Press, New York, 1981.
20. Guttman, R., L. Feldman, and H. Lecar, "Squid axon membrane response to white noise stimulation," Biophys. J., Vol. 14, No. 12, 941-955, 1974.
21. Horikawa, Y., "Noise effects on spike propagation in the stochastic Hodgkin-Huxley models," J. Biological Cybernetics, Vol. 66, No. 1, 190-196, 1991.
22. Tanaka, H. and K. Aihara, "Analysis of the Hodgkin-Huxley equations with noise: The effects of noise on chaotic neurodynamics ," Artificial Life and Robotics, Vol. 8, No. 2, 190-196, 2004.
23. Luchian, T., B. Bancia, C. Pavel, and G. Popa, "Biomembrane excitability studied within a wideband frequency of an interacting exogenous electric field ," Electromagnetic Biology and Medicine, Vol. 21, 287-291, 2002.
24. McDonnell, M. D. and D. Abbott, "What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology," PLoS Comput. Biol., Vol. 5, No. 5, e1000348, 2009.
25. Collins, J., C. C. Chow, A. C. Capela, and T. T. Imhoff, "Aperiodic stochastic resonance," Physical Review E, Vol. 54, 5575-5584, 1996.
26. Paffi, A., M. Liberti, F. Apollonio, M. Gianni, and G. D'Inzeo, "Modeling electromagnetic fields detectability in a HH-like neuronal system: Stochastic resonance and window behavior," Biol. Cybernetics, Vol. 94, 118-127, 2006.
27. Paffi, A., M. Liberti, F. Apollonio, M. Gianni, and G. D'Inzeo, "Effects of exogenous noise in a silent neuron model: Firing induction and EM signal detection," Proc. 28th IEEE-EMBS Ann. Intl. Conf., Vol. 1, 4183-4186, 2006.
28. Volterra, V., Theory of Functionals and of Integral and Integro-Differential Equations, , Dover, New York, 1959.
29. Schetzen, M., The Volterra and Wiener Theories of Nonlinear Systems, Wiley and Sons, New York, 1980.
30. Aidley, D. J., The Physiology of Excitable Cells, Cambridge University Press, UK, 2000.
31. FitzHugh, R., "Impulses and physiological states in theoretical models of nerve membrane ," Biophys. J., Vol. 1, No. 6, 445-466, 1961.
32. Izhikevich, E. M., "Simple model of spiking neurons," IEEE Trans. Neural Netw., Vol. 14, No. 6, 1569-1572, 2003.
33. Phillipson, P. E. and P. Schuster, "A Comparative study of the HH and Fitzhugh Nagumo models of neuron pulse propagation," Int. J. Bifurc. and Chaos, Vol. 15, 3851-3866, 2005.
34. Cain, C. A., "A theoretical basis for microwave and RF field effects on excitable cellular membranes ," IEEE Trans. Microw. Theory .
35. See, C. H., R. A. Abd-Alhameed, and P. S. Excell, "Computation of electromagnetic field in assemblages of biological cells using a modi¯ed ¯nite di®erence time domain scheme," IEEE Trans. Microw. Theory Tech., Vol. 55, No. 9, 1986-1994, 2007.
36. Fitzhugh, R. and J. Gen. Physiol., "Theoretical effects of temperature on threshold ," Theoretical effects of temperature on threshold in the Hodgkin-Huxley nerve model, Vol. 49, No. 5, 989-1005, 1966.
37. Bedrosian, E. and S. O. Rice, "The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and gaussian inputs," Proc. IEEE, Vol. 59, 1688-1707, 1971.
38. Isserlis, L., "On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables," Biometrika, Vol. 12, 134-139, 1918.
39. Stogryn, A., "Equations for calculating the dielectric constant of saline water," IEEE Trans. Microw. Theory Tech., Vol. 19, No. 8, 733-736, 1971.
40. Klein, L. A. and C. T. Swift, "An improved model for the dielectric constant of sea water at microwave frequencies," IEEE Trans. Antennas Propagat., Vol. 25, No. 1, 104-111, 1977.
41. Cowalkzuk, C. C., G. Yarwood, R. Blackwell, M. Priestner, Z. Sienkiewicz, S. Bou²er, I. Ahmed, R. Abd-Alhameed, P. Excell, V. Hodzic, C. Davis, R. Gammon, and Q. Balzano, "Absence of nonlinear responses in cells and tissues exposed to RF energy at mobile phone frequencies using a doubly resonant cavity," Bioelectromagnetics, Vol. 31, 556-565, 2010.
42. Davis, C. C. and Q. Balzano, "The brain is not a radio receiver for wireless phone signals: Human tissue does not demodulate a modulated radiofrequency carrier," Comp. Rend. Physique, Vol. 11, 585-591, 2011.
43. Nawarathna, D., J. R. Claycomb, G. Cardenas, J. Gardner, D. Warmflash, J. H. Miller, and W. R. Widger, "Harmonic generation by yeast cells in response to low-frequency electric fields," Physical Review E, Vol. 73, 0519141-0519146, 2006.
44. Lev, D., A. Puzenko, A. Manevitch, Z. Manevitch, L. Livshits, Y. Feldman, and A. Lewis, "D-glucose-induced second harmonic generation response in human erythrocytes," J. Phys. Chem., Vol. B113, 2513-2518, 2009.
45. Fishman, H. M. and H. R. Leuchtag, "Electrical noise in physics and biology," Curr. Topics Membr. Transp., Vol. 37, S. I. Helman and W. Van Driessche, Eds., Academic Press, 1990.
46. Weaver, J. C. and R. D. Astumian, "The response of cells to very weak electric fields. The thermal noise limit," Science, Vol. 247, No. 4941, 459-462, 1990.
47. Weaver, J. C. and R. D. Astumian, "Estimates for ELF effects: Noise-based thresholds and the number of experimental conditions required for empirical searches," Bioelectromagnetics Suppl., Vol. 1, 119-138, 1992.
48. Derksen, H. E. and A. A. Verveen, "Fluctuation of resting neural membrane potential," Science, Vol. 151, No. 3716, 1388-1389, 1966.
49. Bevan, S., R. Kullberg, and J. Rice, "An analysis of cell membrane noise," Ann. Statist., Vol. 7, No. 2, 237-257, 1979.
50. Eisenberg, R. S., M. Frank, and C. F. Stevens, "Membranes, Channels and Noise," Plenum Press, New York, 1984.
51. Bier, M., "How to evaluate the electric noise in a cell membrane," Acta Physica Polonica, Vol. 37, No. 5, 1409-1424, 2006.
52. Bukhari, M. S. H., J. H. Miller, and Z. H. Shah, "Intrinsic electromagnetic noise in living cells in vitro and its spectroscopy," J. Basic Appl. Sci., Vol. 5, 65-71, 2009.
53. Bukhari, M. S. H., J. H. Miller, and Z. H. Shah, "Intrinsic membrane noise in living cells and its coupling to external fields," Proc. 2nd International Conference on Computer Research and Development (ICCRD 2010), 540-544, IEEE Computer Society Press, 2010.