Journal of Computational Neuroscience

, Volume 31, Issue 3, pp 647–666 | Cite as

Stochastic amplification of calcium-activated potassium currents in Ca2+ microdomains

  • David Arthur StanleyEmail author
  • Berj L. Bardakjian
  • Mark L. Spano
  • William L. Ditto


Small conductance (SK) calcium-activated potassium channels are found in many tissues throughout the body and open in response to elevations in intracellular calcium. In hippocampal neurons, SK channels are spatially co-localized with L-Type calcium channels. Due to the restriction of calcium transients into microdomains, only a limited number of L-Type Ca2+ channels can activate SK and, thus, stochastic gating becomes relevant. Using a stochastic model with calcium microdomains, we predict that intracellular Ca2+ fluctuations resulting from Ca2+ channel gating can increase SK2 subthreshold activity by 1–2 orders of magnitude. This effectively reduces the value of the Hill coefficient. To explain the underlying mechanism, we show how short, high-amplitude calcium pulses associated with stochastic gating of calcium channels are much more effective at activating SK2 channels than the steady calcium signal produced by a deterministic simulation. This stochastic amplification results from two factors: first, a supralinear rise in the SK2 channel’s steady-state activation curve at low calcium levels and, second, a momentary reduction in the channel’s time constant during the calcium pulse, causing the channel to approach its steady-state activation value much faster than it decays. Stochastic amplification can potentially explain subthreshold SK2 activation in unified models of both sub- and suprathreshold regimes. Furthermore, we expect it to be a general phenomenon relevant to many proteins that are activated nonlinearly by stochastic ligand release.


Stochastic Calcium-activated potassium SK2 Microdomain Noise Hill coefficient Subthreshold 



The authors wish to thank Dr. Berj Bardakjian, Dr. Avrama Blackwell, Behnam Kia, Ernest Ho, and Pengpeng Cao for valuable discussion. We are grateful to Dr. Neil V. Marrion and Dr. Pankaj Sah for providing elaboration on their published experimental results, which were essential for this paper. We also thank Janet Stanley for proofreading the manuscript and Kerstin Menne for making her GENESIS code available. The authors also wish to acknowledge ONR and NSERC for providing funding for this work.


  1. Avery, R., & Johnston, D. (1996). Multiple channel types contribute to the low-voltage-activated calcium current in hippocampal CA3 pyramidal neurons. Journal of Neuroscience, 16(18), 5567.PubMedGoogle Scholar
  2. Bekkers, J. (2000). Distribution of slow AHP channels on hippocampal CA1 pyramidal neurons. Journal of Neurophysiology, 83(3), 1756.PubMedGoogle Scholar
  3. Belle, M., Diekman, C., Forger, D., & Piggins, H. (2009). Daily electrical silencing in the mammalian circadian clock. Science, 326(5950), 281.PubMedCrossRefGoogle Scholar
  4. Berridge, M. (2006). Calcium microdomains: organization and function. Cell Calcium, 40(5–6), 405–412.PubMedCrossRefGoogle Scholar
  5. Berridge, M., & Galione, A. (1988). Cytosolic calcium oscillators. The FASEB Journal, 2(15), 3074.PubMedGoogle Scholar
  6. Bhalla, U. (2004). Signaling in small subcellular volumes. II. Stochastic and diffusion effects on synaptic network properties. Biophysical Journal, 87(2), 745–753.PubMedCrossRefGoogle Scholar
  7. Blatz, A., & Magleby, K. (1987). Calcium-activated potassium channels. Trends in Neurosciences, 10(11), 463–467.CrossRefGoogle Scholar
  8. Bleasel, A., & Pettigrew, A. (1992). Development and properties of spontaneous oscillations of the membrane potential in inferior olivary neurons in the rat. Developmental Brain Research, 65(1), 43–50.PubMedCrossRefGoogle Scholar
  9. Bowden, S., Fletcher, S., Loane, D., & Marrion, N. (2001). Somatic colocalization of rat SK1 and D class (Cav 1.2) L-type calcium channels in rat CA1 hippocampal pyramidal neurons. Journal of Neuroscience, 21(20), 175.Google Scholar
  10. Bower, J., Beeman, D., & Hucka, M. (2002). The GENESIS simulation system. In M. A. Arbib (Ed.), The handbook of brain theory and neural networks (pp. 475–478, 2nd ed.). Cambridge: MIT Press.Google Scholar
  11. Bower, J., Beeman, D., & Wylde, A. (1998). The book of GENESIS: Exploring realistic neural models with the GEneral NEural SImulation System (2nd ed.). New York: Springer.Google Scholar
  12. Bruce, I. (2009). Evaluation of stochastic differential equation approximation of ion channel gating models. Annals of Biomedical Engineering, 37(4), 824–838.PubMedCrossRefGoogle Scholar
  13. Catterall, W., Perez-Reyes, E., Snutch, T., & Striessnig, J. (2005). International union of pharmacology. XLVIII. Nomenclature and structure-function relationships of voltage-gated calcium channels. Pharmacological Reviews, 57(4), 411.PubMedCrossRefGoogle Scholar
  14. Catterall, W., Perez-Reyes, E., Snutch, T., & Striessnig, J. (2010). Voltage-gated calcium channels: Cav1.3. Last modified on 2010-07-01. Accessed on 13 October 2010. IUPHAR database (IUPHAR-DB) URL
  15. Chay, T., & Keizer, J. (1983). Minimal model for membrane oscillations in the pancreatic beta-cell. Biophysical Journal, 42(2), 181–189.PubMedCrossRefGoogle Scholar
  16. Choi, S., Yu, E., Kim, D., Urbano, F., Makarenko, V., Shin, H., et al. (2010). Subthreshold membrane potential oscillations in inferior olive neurons are dynamically regulated by P/Q-and T-type calcium channels: A study in mutant mice. The Journal of Physiology, 588(16), 3031.PubMedCrossRefGoogle Scholar
  17. Chorev, E., Yarom, Y., & Lampl, I. (2007). Rhythmic episodes of subthreshold membrane potential oscillations in the rat inferior olive nuclei in vivo. Journal of Neuroscience, 27(19), 5043.PubMedCrossRefGoogle Scholar
  18. Chow, C., & White, J. (1996). Spontaneous action potentials due to channel fluctuations. Biophysical Journal, 71(6), 3013–3021.PubMedCrossRefGoogle Scholar
  19. Cingolani, L., Gymnopoulos, M., Boccaccio, A., Stocker, M., & Pedarzani, P. (2002). Developmental regulation of small-conductance Ca2+-activated K+ channel expression and function in rat Purkinje neurons. Journal of Neuroscience, 22(11), 4456.PubMedGoogle Scholar
  20. De Schutter, E,, & Smolen, P. (1998). Calcium dynamics in large neuronal models. In C. Koch, & I. Segev (Eds.), Methods in neuronal modeling: From ions to networks (pp. 211-215, 2nd ed., chap. 6). Cambridge: MIT Press.Google Scholar
  21. Diba, K., Lester, H., & Koch, C. (2004). Intrinsic noise in cultured hippocampal neurons: Experiment and modeling. Journal of Neuroscience, 24(43), 9723.PubMedCrossRefGoogle Scholar
  22. Dyhrfjeld-Johnsen, J., Maier, J., Schubert, D., Staiger, J., Luhmann, H., Stephan, K., et al. (2005). CoCoDat: A database system for organizing and selecting quantitative data on single neurons and neuronal microcircuitry. Journal of Neuroscience Methods, 141(2), 291–308.PubMedCrossRefGoogle Scholar
  23. Eliasmith, C., & Anderson, C. (2004). Neural engineering: Computation, representation, and dynamics in neurobiological systems. Cambridge: MIT Press.Google Scholar
  24. Ellis, L., Mehaffey, W., Harvey-Girard, E., Turner, R., Maler, L., & Dunn, R. (2007). SK channels provide a novel mechanism for the control of frequency tuning in electrosensory neurons. Journal of Neuroscience, 27(35), 9491.PubMedCrossRefGoogle Scholar
  25. Fakler, B., & Adelman, J. (2008). Control of KCa channels by calcium nano/microdomains. Neuron, 59(6), 873–881.PubMedCrossRefGoogle Scholar
  26. Fettiplace, R., & Fuchs, P. (1999). Mechanisms of hair cell tuning. Annual Review of Physiology, 61(1), 809–834.PubMedCrossRefGoogle Scholar
  27. Fisher, J., Kowalik, L., & Hudspeth, A. (2011). Imaging electrical resonance in hair cells. Proceedings of the National Academy of Sciences, 108(4), 1651.CrossRefGoogle Scholar
  28. Fuchs, P., Nagai, T., & Evans, M. (1988). Electrical tuning in hair cells isolated from the chick cochlea. Journal of Neuroscience, 8(7), 2460.PubMedGoogle Scholar
  29. Gillespie, D. (1977). Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25), 2340–2361.CrossRefGoogle Scholar
  30. Gleeson, P., Crook, S., Cannon, R., Hines, M., Billings, G., Farinella, M., et al. (2010). NeuroML: A language for describing data driven models of neurons and networks with a high degree of biological detail. PLoS Computational Biology, 6(6), e1000815.CrossRefGoogle Scholar
  31. Gleeson, P., Steuber, V., & Silver, R. (2007). neuroConstruct: A tool for modeling networks of neurons in 3D space. Neuron, 54(2), 219–235.PubMedCrossRefGoogle Scholar
  32. Hallworth, N., Wilson, C., & Bevan, M. (2003). Apamin-sensitive small conductance calcium-activated potassium channels, through their selective coupling to voltage-gated calcium channels, are critical determinants of the precision, pace, and pattern of action potential generation in rat subthalamic nucleus neurons in vitro. Journal of Neuroscience, 23(20), 7525.PubMedGoogle Scholar
  33. Hell, J., Westenbroek, R., Warner, C., Ahlijanian, M., Prystay, W., Gilbert, M., et al. (1993). Identification and differential subcellular localization of the neuronal class C and class D L-type calcium channel alpha 1 subunits. The Journal of Cell Biology, 123(4), 949.PubMedCrossRefGoogle Scholar
  34. Helton, T., Xu, W., & Lipscombe, D. (2005). Neuronal L-type calcium channels open quickly and are inhibited slowly. Journal of Neuroscience, 25(44), 10247.PubMedCrossRefGoogle Scholar
  35. Hille, B. (2001). Ion channels of excitable membranes (3rd ed.). Sinauer Sunderland, MA.Google Scholar
  36. Hirschberg, B., Maylie, J., Adelman, J., & Marrion, N. (1998). Gating of recombinant small-conductance Ca-activated K+ channels by calcium. The Journal of General Physiology, 111(4), 565.PubMedCrossRefGoogle Scholar
  37. Hirschberg, B., Maylie, J., Adelman, J., & Marrion, N. (1999). Gating properties of single SK channels in hippocampal CA1 pyramidal neurons. Biophysical Journal, 77(4), 1905–1913.PubMedCrossRefGoogle Scholar
  38. Hodgkin, A., & Huxley, A. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500.PubMedGoogle Scholar
  39. Hudspeth, A., & Lewis, R. (1988). A model for electrical resonance and frequency tuning in saccular hair cells of the bull-frog, Rana catesbeiana. The Journal of Physiology, 400(1), 275.PubMedGoogle Scholar
  40. Jacobson, D., Mendez, F., Thompson, M., Torres, J., Cochet, O., & Philipson, L. (2010). Calcium-activated and voltage-gated potassium channels of the pancreatic islet impart distinct and complementary roles during secretagogue induced electrical responses. The Journal of Physiology, 588(18), 3525–3537.PubMedCrossRefGoogle Scholar
  41. Jaffe, D., Ross, W., Lisman, J., Lasser-Ross, N., Miyakawa, H., & Johnston, D. (1994). A model for dendritic Ca2+ accumulation in hippocampal pyramidal neurons based on fluorescence imaging measurements. Journal of Neurophysiology, 71(3), 1065.PubMedGoogle Scholar
  42. Kang, Y., & Kitai, S. (1993). Calcium spike underlying rhythmic firing in dopaminergic neurons of the rat substantia nigra. Neuroscience Research, 18(3), 195–207.PubMedCrossRefGoogle Scholar
  43. Knopfel, T., Vranesic, I., Gahwiler, B., & Brown, D. (1990). Muscarinic and beta-adrenergic depression of the slow Ca2+-activated potassium conductance in hippocampal CA3 pyramidal cells is not mediated by a reduction of depolarization-induced cytosolic Ca2+ transients. Proceedings of the National Academy of Science, 87, 4083–4087.CrossRefGoogle Scholar
  44. Koschak, A., Obermair, G., Pivotto, F., Sinnegger-Brauns, M., Striessnig, J., & Pietrobon, D. (2007). Molecular nature of anomalous L-type calcium channels in mouse cerebellar granule cells. Journal of Neuroscience, 27(14), 3855.PubMedCrossRefGoogle Scholar
  45. Kuznetsova, A., Huertas, M., Kuznetsov, A., Paladini, C., & Canavier, C. (2010). Regulation of firing frequency in a computational model of a midbrain dopaminergic neuron. Journal of Computational Neuroscience, 28(3), 1–15.CrossRefGoogle Scholar
  46. Li, G., Nair, S., & Quirk, G. (2009). A biologically realistic network model of acquisition and extinction of conditioned fear associations in lateral amygdala neurons. Journal of Neurophysiology, 101(3), 1629.PubMedCrossRefGoogle Scholar
  47. Llinas, R., & Yarom, Y. (1986). Oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: An in vitro study. The Journal of Physiology, 376(1), 163.PubMedGoogle Scholar
  48. Losonczy, A., & Magee, J. (2006). Integrative properties of radial oblique dendrites in hippocampal CA1 pyramidal neurons. Neuron, 50(2), 291–307.PubMedCrossRefGoogle Scholar
  49. Lu, L., Zhang, Q., Timofeyev, V., Zhang, Z., Young, J., Shin, H., et al. (2007). Molecular coupling of a Ca2+-activated K+ channel to L-type Ca2+ channels via α-Actinin2. Circulation Research, 100(1), 112.PubMedCrossRefGoogle Scholar
  50. Mainen, Z., & Sejnowski, T. (1996). Influence of dendritic structure on firing pattern in model neocortical neurons. Nature, 382(6589), 363–366.PubMedCrossRefGoogle Scholar
  51. Mainen, Z., & Sejnowski, T. (1998). Modeling active dendritic processes in pyramidal neurons. In C. Koch, & I. Segev (Eds.), Methods in neuronal modeling: From ions to networks (pp. 171-210). Cambridge: MIT Press.Google Scholar
  52. Marder, E., Kopell, N., & Karen, S. (1999). How computation aids in understanding biological networks. In P. S. G. Stein, S. Grillner, A.I. Selverston, & D. G. Stuart (Eds.), Neurons, networks, and motor behavior (pp. 139-150, chap. 13). Cambridge: MIT Press.Google Scholar
  53. Marrion, N., & Tavalin, S. (1998). Selective activation of Ca2+-activated K+ channels by co-localized Ca2+ channels in hippocampal neurons. Nature, 395(6705), 900–905.PubMedCrossRefGoogle Scholar
  54. Migliore, M., Cook, E., Jaffe, D., Turner, D., & Johnston, D. (1995). Computer simulations of morphologically reconstructed CA3 hippocampal neurons. Journal of Neurophysiology, 73(3), 1157.PubMedGoogle Scholar
  55. Millership, J., Heard, C., Fearon, I., & Bruce, J. (2010). Differential regulation of calcium-activated potassium channels by dynamic intracellular calcium signals. Journal of Membrane Biology, 235(3), 191–210.Google Scholar
  56. Mino, H., Rubinstein, J., & White, J. (2002). Comparison of algorithms for the simulation of action potentials with stochastic sodium channels. Annals of Biomedical Engineering, 30(4), 578–587.PubMedCrossRefGoogle Scholar
  57. Navedo, M., Amberg, G., Westenbroek, R., Sinnegger-Brauns, M., Catterall, W., Striessnig, J., et al. (2007). Cav1.3 channels produce persistent calcium sparklets, but Cav1.2 channels are responsible for sparklets in mouse arterial smooth muscle. American Journal of Physiology—Heart and Circulatory Physiology, 293(3), H1359.CrossRefGoogle Scholar
  58. Nedergaard, S., Flatman, J., & Engberg, I. (1993). Nifedipine-and omega-conotoxin-sensitive Ca2+ conductances in guinea-pig substantia nigra pars compacta neurones. The Journal of Physiology, 466(1), 727.PubMedGoogle Scholar
  59. Ngo-Anh, T., Bloodgood, B., Lin, M., Sabatini, B., Maylie, J., & Adelman, J. (2005). SK channels and NMDA receptors form a Ca2+-mediated feedback loop in dendritic spines. Nature Neuroscience, 8(5), 642–649.PubMedCrossRefGoogle Scholar
  60. Ping, H., & Shepard, P. (1996). Apamin-sensitive Ca2+-activated K+ channels regulate pacemaker activity in nigral dopamine neurons. Neuroreport, 7(3), 809.PubMedCrossRefGoogle Scholar
  61. Regehr, W., Connor, J., & Tank, D. (1989). Optical imaging of calcium accumulation in hippocampal pyramidal cells during synaptic activation. Nature, 341, 533–536.PubMedCrossRefGoogle Scholar
  62. Sah, P., & Bekkers, J. (1996). Apical dendritic location of slow afterhyperpolarization current in hippocampal pyramidal neurons: Implications for the integration of long-term potentiation. Journal of Neuroscience, 16(15), 4537.PubMedGoogle Scholar
  63. Sah, P., & Clements, J. (1999). Photolytic manipulation of [Ca2+]i reveals slow kinetics of potassium channels underlying the afterhyperpolarization in pyramidal neurons. Journal of Neuroscience, 19(10), 3657.PubMedGoogle Scholar
  64. Sah, P., & Isaacson, J. (1995). Channels underlying the slow afterhyperpolarization in hippocampal pyramidal neurons: Neurotransmitters modulate the open probability. Neuron, 15(2), 435–441.PubMedCrossRefGoogle Scholar
  65. Sailer, C., Kaufmann, W., Marksteiner, J., & Knaus, H. (2004). Comparative immunohistochemical distribution of three small-conductance Ca2+-activated potassium channel subunits, SK1, SK2, and SK3 in mouse brain. Molecular and Cellular Neuroscience, 26(3), 458–469.PubMedCrossRefGoogle Scholar
  66. Schaefer, A., Larkum, M., Sakmann, B., & Roth, A. (2003). Coincidence detection in pyramidal neurons is tuned by their dendritic branching pattern. Journal of Neurophysiology, 89(6), 3143.PubMedCrossRefGoogle Scholar
  67. Schiller, J., Helmchen, F., & Sakmann, B. (1995). Spatial profile of dendritic calcium transients evoked by action potentials in rat neocortical pyramidal neurones. The Journal of Physiology, 487(Pt 3), 583.PubMedGoogle Scholar
  68. Sherman, A. (1996). Contributions of modeling to understanding stimulus-secretion coupling in pancreatic beta-cells. American Journal of Physiology—Endocrinology and Metabolism, 271(2), E362.Google Scholar
  69. Shuai, J., & Parker, I. (2005). Optical single-channel recording by imaging Ca2+ flux through individual ion channels: Theoretical considerations and limits to resolution. Cell Calcium, 37(4), 283–299.PubMedCrossRefGoogle Scholar
  70. Skupin, A., Kettenmann, H., Falcke, M. (2010). Calcium signals driven by single channel noise. PLoS Computational Biology, 6(8), 1183–1186.CrossRefGoogle Scholar
  71. Sourdet, V., Russier, M., Daoudal, G., Ankri, N., & Debanne, D. (2003). Long-term enhancement of neuronal excitability and temporal fidelity mediated by metabotropic glutamate receptor subtype 5. Journal of Neuroscience, 23(32), 10238.PubMedGoogle Scholar
  72. Stocker, M. (2004). Ca2+-activated K+ channels: Molecular determinants and function of the SK family. Nature Reviews Neuroscience, 5(10), 758–770.PubMedCrossRefGoogle Scholar
  73. Stocker, M., Hirzel, K., D’hoedt, D., & Pedarzani, P. (2004). Matching molecules to function: neuronal Ca2+-activated K+ channels and afterhyperpolarizations. Toxicon, 43(8), 933–949.PubMedCrossRefGoogle Scholar
  74. Stocker, M., Krause, M., & Pedarzani, P. (1999). An apamin-sensitive Ca2+-activated K+ current in hippocampal pyramidal neurons. Proceedings of the National Academy of Sciences of the United States of America, 96(8), 4662.PubMedCrossRefGoogle Scholar
  75. Stojilkovic, S., Zemkova, H., & Van Goor, F. (2005). Biophysical basis of pituitary cell type-specific Ca2+ signaling-secretion coupling. Trends in Endocrinology and Metabolism, 16(4), 152–159.PubMedCrossRefGoogle Scholar
  76. Strassberg, A., & DeFelice, L. (1993). Limitations of the Hodgkin–Huxley formalism: Effects of single channel kinetics on transmembrane voltage dynamics. Neural Computation, 5(6), 843–855.CrossRefGoogle Scholar
  77. Takahashi, Y., Jeong, S., Ogata, K., Goto, J., Hashida, H., Isahara, K., et al. (2003). Human skeletal muscle calcium channel α1S is expressed in the basal ganglia: Distinctive expression pattern among L-type Ca2+ channels. Neuroscience Research, 45(1), 129–137.PubMedCrossRefGoogle Scholar
  78. Thibault, O., & Landfield, P. (1996). Increase in single L-type calcium channels in hippocampal neurons during aging. Science, 272(5264), 1017.PubMedCrossRefGoogle Scholar
  79. Traub, R., Jefferys, J., Miles, R., Whittington, M., & Toth, K. (1994). A branching dendritic model of a rodent CA3 pyramidal neurone. The Journal of Physiology, 481(Pt 1), 79.PubMedGoogle Scholar
  80. Tzingounis, A., Heidenreich, M., Kharkovets, T., Spitzmaul, G., Jensen, H., Nicoll, R., et al. (2010). The KCNQ5 potassium channel mediates a component of the afterhyperpolarization current in mouse hippocampus. Proceedings of the National Academy of Sciences, 107(22), 10232.CrossRefGoogle Scholar
  81. Tzingounis, A., Kobayashi, M., Takamatsu, K., & Nicoll, R. (2007). Hippocalcin gates the calcium activation of the slow afterhyperpolarization in hippocampal pyramidal cells. Neuron, 53(4), 487–493.PubMedCrossRefGoogle Scholar
  82. Van Goor, F., Li, Y., & Stojilkovic, S. (2001). Paradoxical role of large-conductance calcium-activated K+ (BK) channels in controlling action potential-driven Ca2+ entry in anterior pituitary cells. Journal of Neuroscience, 21(16), 5902.PubMedGoogle Scholar
  83. Vergara, C., Latorre, R., Marrion, N., & Adelman, J. (1998). Calcium-activated potassium channels. Current Opinion in Neurobiology, 8(3), 321–329.PubMedCrossRefGoogle Scholar
  84. Von Wegner, F., & Fink, R. (2010). Stochastic simulation of calcium microdomains in the vicinity of an L-type calcium channel. European Biophysics Journal, 39(7), 1079–1088.CrossRefGoogle Scholar
  85. Warman, E., Durand, D., & Yuen, G. (1994). Reconstruction of hippocampal CA1 pyramidal cell electrophysiology by computer simulation. Journal of Neurophysiology, 71(6), 2033.PubMedGoogle Scholar
  86. Wei, A., Gutman, G., Aldrich, R., Chandy, K., Grissmer, S., & Wulff, H. (2005). International union of pharmacology. LII. Nomenclature and molecular relationships of calcium-activated potassium channels. Pharmacological Reviews, 57(4), 463.PubMedCrossRefGoogle Scholar
  87. Xia, X., Fakler, B., Rivard, A., Wayman, G., Johnson-Pais, T., Keen, J., et al. (1997). Mechanism of calcium gating in small-conductance calcium-activated potassium channels. Nature, 386, 167–170.CrossRefGoogle Scholar
  88. Yung, W., Häusser, M., & Jack, J. (1991). Electrophysiology of dopaminergic and non-dopaminergic neurones of the guinea-pig substantia nigra pars compacta in vitro. The Journal of Physiology, 436(1), 643.PubMedGoogle Scholar
  89. Zhang, M., Houamed, K., Kupershmidt, S., Roden, D., & Satin, L. (2005). Pharmacological properties and functional role of Kslow current in mouse pancreatic β-Cells. The Journal of General Physiology, 126(4), 353.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • David Arthur Stanley
    • 1
    Email author
  • Berj L. Bardakjian
    • 2
    • 3
  • Mark L. Spano
    • 1
  • William L. Ditto
    • 1
  1. 1.School of Biological and Health Systems EngineeringArizona State UniversityTempeUSA
  2. 2.Edward S. Rogers Sr. Department of Electrical and Computer EngineeringUniversity of TorontoTorontoCanada
  3. 3.Institute of Biomaterials and Biomedical EngineeringUniversity of TorontoTorontoCanada

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