Neurons, Models, and Invariants

Chapter
First Online:
Part of the Springer Series in Cognitive and Neural Systems book series (SSCNS, volume 1)

Abstract

The theory of dynamical systems is presented as an integrative theory, as it offers a consistent method to cross levels and scales. Behavior happens in recurrent loops where what happens next is what happened before plus how things work. Dynamical systems abstracts organisms into state variables, rules, and parameters. Analysis of structure and parameters can be done both analytically and computationally to address the invariances of a modeled system. Dynamical systems analysis reveals new facets of the phenomena modeled, and may effectively lead to powerful reductions. The Hodgkin–Huxley model of the action potential is presented as a prototypical example of dynamical systems analysis applied to neurons. The Hodgkin–Huxley model illustrates several interesting aspects of action potential generation, for example, how constancy in the action potentials appears despite variability in ion channel distribution. Effectively, dynamical systems modeling demonstrates how a phenomenon can be analyzed with respect to its parameters and structure, and helps solve the dichotomy between constancy and variability through mappings between parameter domains and dynamical varieties.

Keywords

Logical Operation Patch Clamp Recurrent Neural Network Neural Model Parameter Domain 
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.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Bernstein J (1910) Die Thermoströme des Muskels und die “Membrantheorie” der bioelektrischen Ströme. Pflügers Arch 131(10):589–600CrossRefGoogle Scholar
  2. 2.
    Bernstein J, Tschermak A (1902) Ueber die Beziehung der negativen Schwankung des Muskelstromes zur Arbeitsleistung des Muskels. Pflügers Arch 89(7):289–331CrossRefGoogle Scholar
  3. 3.
    Boeddeker N, Egelhaaf M (2005) A single control system for smooth and saccade-like pursuit in blowflies. J Exp Biol (208):1563–1572PubMedCrossRefGoogle Scholar
  4. 4.
    Braitenberg V, Schüz A (1998) Cortex: Statistics and geometry of neuronal connectivity. Springer, BerlinGoogle Scholar
  5. 5.
    Brazier M (1984) A history of neurophysiology in the 17th and 18th centuries: From concept to experiment. Lippincott Williams & Wilkins, New YorkGoogle Scholar
  6. 6.
    Cajal Santiago Ramon-y, Pasik P, Pasik T (2002) Texture of the nervous system of man and the vertebrates. Springer, New YorkGoogle Scholar
  7. 7.
    Cash S, Yuste R (1998) Input summation by cultured pyramidal neurons is linear and position-independent. J Neurosci 18(1):10–15PubMedGoogle Scholar
  8. 8.
    Cash S, Yuste R (1999) Linear summation of excitatory inputs by CA1 pyramidal neurons. Neuron 22:383–394PubMedCrossRefGoogle Scholar
  9. 9.
    Chow T, Li X (2000) Modeling of continuous time dynamical systems with input by recurrent neural networks. IEEE Trans Circuits Syst I Fundam Theory Appl 47(4):575–578CrossRefGoogle Scholar
  10. 10.
    Crick F (1996) The astonishing hypothesis: The scientific search for the soul. J Nerv Ment Dis 184(6):384CrossRefGoogle Scholar
  11. 11.
    Dayhoff J (2007) Computational properties of networks of synchronous groups of spiking neurons. Neural Comput 19(9):2433PubMedCrossRefGoogle Scholar
  12. 12.
    Funahashi K, Nakamura Y (1993) Approximation of dynamical systems by continuous time recurrent neural networks. Neural Netw 6(6):801–806CrossRefGoogle Scholar
  13. 13.
    Guckenheimer J, Labouriau I (1993) Bifurcation of the Hodgkin and Huxley equations: A new twist. Bull Math Biol 55(5):937–952Google Scholar
  14. 14.
    Hartree D, Womersley J (1937) A method for the numerical or mechanical solution of certain types of partial differential equations. Proc R Soc Lond A Math Phys Sci 161(906):353–366CrossRefGoogle Scholar
  15. 15.
    Herz A, Gollisch T, Machens C, Jaeger D (2006) Modeling single-neuron dynamics and computations: A balance of detail and abstraction. Science 314(5796):80–85PubMedCrossRefGoogle Scholar
  16. 16.
    Hodgkin A, Huxley A (1952) A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J Physiol 117:500–544PubMedGoogle Scholar
  17. 17.
    Iwasa K, Tasaki I, Gibbons R (1980) Swelling of nerve fibers associated with action potentials. Science 210(4467):338–339PubMedCrossRefGoogle Scholar
  18. 18.
    Izhikevich E (2007) Dynamical systems in neuroscience: The geometry of excitability and bursting. MIT, Cambridge, MAGoogle Scholar
  19. 19.
    Izhikevich EM (2004) Which model to use for cortical spiking neurons. IEEE Trans Neural Netw 15(5)Google Scholar
  20. 20.
    Jiang Y, Lee A, Chen J, Cadene M, Chait B, MacKinnon R (2002) Crystal structure and mechanism of a calcium-gated potassium channel. Nature 417:515–522PubMedCrossRefGoogle Scholar
  21. 21.
    Kaltenbrunner A, Gomez V, Lopez V (2007) Phase transition and hysteresis in an ensemble of stochastic spiking neurons. Neural Comput 19(11):3011–3050PubMedCrossRefGoogle Scholar
  22. 22.
    Kelso J (1995) Dynamic patterns: The self-organization of brain and behavior. MIT, Cambridge, MAGoogle Scholar
  23. 23.
    Koch C (1997) Computation and the single neuron. Nature 385:207–210PubMedCrossRefGoogle Scholar
  24. 24.
    Koch C, Poggio T, Torre V (1983) Nonlinear interactions in a dendritic tree: Localization, timing, and role in information processing. Proc Natl Acad Sci 80(9):2799–2802PubMedCrossRefGoogle Scholar
  25. 25.
    Konur S, Yuste R (2004) Imaging the motility of dendritic protrusions and axon terminals: Roles in axon sampling and synaptic competition. Mol Cell Neurosci 27(4):427–440PubMedCrossRefGoogle Scholar
  26. 26.
    Lapicque L (1907) Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarisation. J Physiol Pathol Gen 9:620–635Google Scholar
  27. 27.
    Lapicque L (1934) Neuro-muscular isochronism and chronological theory of curarization. J Physiol 81(1):113PubMedGoogle Scholar
  28. 28.
    Ling G (1992) A revolution in the physiology of the living cell. Krieger, Malabar, FloridaGoogle Scholar
  29. 29.
    Ling G (2001) Life at the cell and below-cell level: The hidden history of a fundamental revolution in biology. Pacific, New YorkGoogle Scholar
  30. 30.
    London M, Hausser M (2005) Dendritic computation. Annu Rev Neurosci 28:503–32PubMedCrossRefGoogle Scholar
  31. 31.
    Mel B, Schiller J (2004) On the fight between excitation and inhibition: Location is everything. Sci STKE 2004(250)Google Scholar
  32. 32.
    Neher E, Sakmann B, Steinbach J (1978) The extracellular patch clamp: A method for resolving currents through individual open channels in biological membranes. Pflügers Arch 375(2):219–228PubMedCrossRefGoogle Scholar
  33. 33.
    Pasemann F, Wennekers T (2000) Generalized and partial synchronization of coupled neural networks. Network 11(1):41–61PubMedCrossRefGoogle Scholar
  34. 34.
    Rinzel J, Keller J (1973) Traveling wave solutions of a nerve conduction equation. Biophys J 13(12):1313–1337PubMedCrossRefGoogle Scholar
  35. 35.
    Segev I, London M (2000) Untangling dendrites with quantitative models. Science 290(5492): 744–750PubMedCrossRefGoogle Scholar
  36. 36.
    Yellen G (2002) The voltage-gated potassium channels and their relatives. Nature 419(6902):35–42PubMedCrossRefGoogle Scholar
  37. 37.
    Yuste R, Denk W (1995) Dendritic spines as basic functional units of neuronal integration. Nature 375(6533):682–684PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Okinawa Institute of Science and TechnologyOkinawaJapan

Personalised recommendations