Advertisement

Frequency Domain I: Bode Plots and Transfer Functions

  • John Milton
  • Toru Ohira
Chapter

Abstract

Up to now, we have focused on descriptions of the dynamics of biological systems in the time domain, that is, on how variables change as a function of time. However, for certain types of problems, it is often more convenient to analyze dynamics in the frequency domain. For example, we typically describe the heartbeat in terms of frequency, namely, the number of beats per minute, rather than in terms of its period, i.e., the time between successive beats. In this chapter, we focus on the response of linear dynamical systems to sinusoidal inputs in the frequency domain. Sinusoidal inputs are of interest in the laboratory, since they are easy to implement. The responses provide an estimate of the linearity of the dynamical system, and they can be used to approximate the differential equation that is consistent with the observed input–output relationships.

Keywords

Transfer Function Derivative Term Impulse Response Function Linear Dynamical System Sinusoidal Input 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. Abeles, E. Vaadia, and H. Bergman. Firing patterns of single units in the prefrontal cortex and neural network models. Network, 1:13–25, 1990.CrossRefGoogle Scholar
  2. 7.
    U. Alon, M. C. Surette, N. Barkai, and S. Leibler. Robustness in bacterial chemotaxis. Nature, 397:168–171, 1999.CrossRefGoogle Scholar
  3. 14.
    A. P. Arkin and J. Ross. Computational functions in biochemical reaction networks. Biophys. J., 67:560–578, 1994.CrossRefGoogle Scholar
  4. 37.
    H. C. Berg. E. coli in motion. Springer-Verlag, New York, 2004.Google Scholar
  5. 39.
    H. C. Berg and P. M. Tedesco. Transient response to chemotactic stimuli in Escherichia coli. Proc. Natl. Acad. Sci. USA, 72:3235–3239, 1975.Google Scholar
  6. 53.
    R. N. Bracewell. The Fourier transform and its applications, 2nd ed., revised. McGraw–Hill, San Francisco, 1986.Google Scholar
  7. 56.
    D. Bray. Bacterial chemotaxis and the question of gain. Proc. Natl. Acad. Sci. USA, 99:7–9, 2002.CrossRefGoogle Scholar
  8. 86.
    K. M. Chapman and R. S. Smith. A linear transfer function underlying impulse frequency modulation in a cockroach mechanoreceptor. Nature, 197:699–700, 1963.CrossRefGoogle Scholar
  9. 133.
    T. Drengstig, H. R. Ueda, and P. Ruoff. Predicting perfect adaptation motifs in reaction kinetic networks. J. Phys. Chem. B, 112:16752–16758, 2008.CrossRefGoogle Scholar
  10. 146.
    C. E. Epstein and J. Shotland. The bad truth about Laplace’s transform. SIAM Rev., 50:504–520, 2008.MathSciNetCrossRefMATHGoogle Scholar
  11. 173.
    A. S. French and P. H. Torkkeli. The power law of sensory adaptation: Simulation by a model of excitability in spider mechanoreceptor neurons. Ann. Biomed. Eng., 36:153–161, 2008.CrossRefGoogle Scholar
  12. 192.
    H. L. Gerber, R. P. Joshi, and C. C. Tseng. Using Bode plots to access intracellular coupling. IEEE Trans. Plasma Sci., 36:1659–1664, 2008.CrossRefGoogle Scholar
  13. 233.
    C. H. Hansen, R. G. Endres, and N. S. Wingreen. Chemotaxis in Escherichia coli: A molecular model for robust precise adaptation. PLoS Comput. Biol, 4:14–27, 2008.MathSciNetCrossRefGoogle Scholar
  14. 251.
    Q. He and Y. Lin. Molecular mechanism of light responses in Neurospora: From light-induced transcription to photoadaptaton. Genes Dev., 19:2888–2899, 2005.CrossRefGoogle Scholar
  15. 256.
    P. Hersen, M. N. McClean, L. Mahadevan, and S. Ramanathan. Signal processing by the HOG MAP kinase pathway. Proc. Natl. Acad. Sci. USA, 105:7165–7170, 2008.CrossRefGoogle Scholar
  16. 305.
    R. E. Kearney and I. W. Hunter. System identification of human joint dynamics. CRC Crit. Rev. Biomed. Eng., 18: 55–87, 1990.Google Scholar
  17. 319.
    Y. Kondoh and M. Hisada. Neuroanatomy of the terminal (sixth abdominal) ganglion of the crayfish, Procambarus clarkii (Girard). Cell Tissue Res., 243:273–288, 1986.CrossRefGoogle Scholar
  18. 336.
    I. Lestas, J. Paulsson, N. E. Ross, and G. Vinnicomb. Noise in gene regulatory networks. IEEE Trans. Automatic Control (Special Issue), 53:189–200, 2008.Google Scholar
  19. 338.
    A. Levchenko and P. A. Iglesias. Models of eukaryotic gradient sensing: application to chemotaxis and neutrophils. Biophys. J., 82 (1 Pt. 1):50–63, 2002.Google Scholar
  20. 362.
    J. Luo, J. Wang, T. M. Ma, and Z. Sun. Reverse engineering of bacterial chemotaxis pathway via frequency domain analysis. PLoS ONE, 5:e9182, 2010.CrossRefGoogle Scholar
  21. 383.
    P. J. Marin, A. J. Herrero, J. G. Milton, T. J. Hazell, and D. Garcia-Lopez. Whole-body vibration applied during upper body exercise improves performance. J. Strength Conditioning Res., 27:1807–1812, 2013.CrossRefGoogle Scholar
  22. 384.
    J. B. Marion. Classical dynamics of particles and systems, 2nd ed. Academic Press, New York, 1970.Google Scholar
  23. 396.
    B. A. Mello and Y. Tu. Perfect and near-perfect adaptation in a model of bacterial chemotaxis. Biophys. J., 84:2943–2956, 2003.CrossRefGoogle Scholar
  24. 397.
    B. A. Mello and Y. Tu. Effects of adaptation in maintaining high sensitivity over a wide range of backgrounds for Escherichia coli chemotaxis. Biophys. J., 92:2329–2337, 2007.CrossRefGoogle Scholar
  25. 401.
    J. T. Mettetal, D. Murray, C. Gómez-Uribe, and A. van Oudenaarden. The frequency dependence of osmo-regulation in Saccharomyces cerevisiae. Science, 319:482–484, 2008.Google Scholar
  26. 430.
    J. G. Milton, T. Ohira T, J. L. Cabrera, R. M. Fraiser, J. B. Gyorffy, F. K. Ruiz, M. A. Strauss, E. C. Balch, P. J. Marin, and J. L. Alexander. Balancing with vibration: A prelude for “drift and act” balance control. PLoS ONE, 4:e7427, 2009.Google Scholar
  27. 445.
    S. Mukherji and A. van Oudenaarden. Synthetic biology: understanding biological design from synthetic circuits. Nature Rev. Gen., 10:859–871, 2009.Google Scholar
  28. 454.
    A. M. Nakashima, M. J. Borland, and S. M. Abel. Measurement of noise and vibration in Canadian forces armored vehicles. Ind. Health, 45:318–327, 2007.CrossRefGoogle Scholar
  29. 508.
    J. W. S. Pringle and V. J. Wilson. The response of a sense organ to harmonic stimulus. J. Exp. Bio, 29:220–235, 1952Google Scholar
  30. 518.
    F. Ratliff, H. K. Hartline, and W. H. Miller. Spatial and temporal aspects of retinal inhibitory interaction. J. Opt. Soc. Am., 53:110–120, 1963.CrossRefGoogle Scholar
  31. 528.
    M. A. J. Roberts, E. August, A. Hamadeh, P. K. Maini, P. E. McSharry, J. P. Armitage, and A. Papchritodoulou. A model invalidation-based approach for elucidating biological signaling pathways, applied to the chemotaxis pathway in R. sphaeroides. BMC Systems Biol., 3:105, 2009.Google Scholar
  32. 551.
    J. Schwarzenbach and K. F. Gill. System modeling and control, 3rd ed. Halsted Press, New York, 1992.Google Scholar
  33. 564.
    Y-J. Shin, B. Hencey, S. M. Liplan, and X. Shen. Frequency domain analysis reveals external periodic fluctuations can generate sustained p53 oscillations. PLoS ONE, 6:e22852, 2011.Google Scholar
  34. 567.
    L. Simon. Drug-delivery systems for chemical, biomedical and pharmaceutical engineering. J. Wiley & Sons, New York, 2012.Google Scholar
  35. 585.
    L. Stark. Environmental clamping of biological systems: Pupil servomechanism. J. Opt. Soc. Amer. A, 52:925–930, 1962.CrossRefGoogle Scholar
  36. 586.
    L. Stark. Neurological control systems: Studies in bioengineering. Plenum Press, New York, 1968.Google Scholar
  37. 629.
    W. van Drongelen. Signal processing for neuroscientists: Introduction to the analysis of physiological signals. Academic Press, New York, 2007.Google Scholar
  38. 651.
    D. T. Westwick and R. E. Kearney. Identification of nonlinear physiological systems. Wiley–Interscience, New York, 2003.CrossRefGoogle Scholar
  39. 666.
    T-M. Yi, Y. Huang, M. I. Simon, and J. Doyle. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc. Natl. Acad. Sci. USA, 97:4649–4653, 2000.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • John Milton
    • 1
  • Toru Ohira
    • 2
  1. 1.W.M. Keck Science DepartmentThe Claremont CollegesClaremontUSA
  2. 2.Graduate School of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations