Advertisement

Computational and Mathematical Models of Neurons

  • Shinji Doi
  • Junko Inoue
  • Zhenxing Pan
Part of the A First Course in “In Silico Medicine” book series (FCISM, volume 2)

Abstract

What are models? The HH equations (2.3) are often called a physiological model, whereas the models appeared in the following sections are simplified models or abstract models. However, there is no model in which any simplifications or abstractions have not been made. Of course, many features of real neurons are ignored even in the HH equations. All models have their applicability and limits to describe natural phenomena. Therefore, all types of models whatever simplified or physiological, have their own values to model real phenomena. Starting with the BVP or FHN model which is a simplification of the HH equations, this chapter proceeds to several neuronal models with higher abstractions which are useful to tract some essential features of neurons.

Keywords

Hopf Bifurcation Bifurcation Diagram Neuron Model Bifurcation Curve 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. Aihara K, Matsumoto G, Ikegaya Y (1984) Periodic and non-periodic responses of a periodically forced Hodgkin–Huxley oscillator. J Theor Biol 109:249–269MathSciNetCrossRefGoogle Scholar
  2. Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144:333–340MathSciNetCrossRefGoogle Scholar
  3. Alexander JC, Doedel EJ, Othmer JC (1990) On the resonance structure in a forced excitable system. SIAM J Appl Math 50:1373–1418MathSciNetMATHCrossRefGoogle Scholar
  4. Arnold L (1995) Random dynamical systems. In: Johnson R (ed) Dynamical systems. Lecture Notes in Mathematics, vol 1609. Springer, Berlin, pp 1–43CrossRefGoogle Scholar
  5. Braaksma B (1993) Critical phenomena in dynamical systems of van der Pol type. Thesis, Rijksuniversiteit Utrecht, UtrechtGoogle Scholar
  6. Bulsara AR, Elston TC, Doering CR, Lowen SB, Lindenberg K (1996) Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys Rev E 53:3958–3969CrossRefGoogle Scholar
  7. Buonocore A, Nobile AG, Ricciardi LM (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv Appl Prob 19:784–800MathSciNetMATHCrossRefGoogle Scholar
  8. Caianiello ER (1961) Outline of a theory of thought-processes and thinking machines. J Theor Biol 2:204–235MathSciNetCrossRefGoogle Scholar
  9. Chhikara RS, Folks JL (1988) The inverse Gaussian distribution: theory, methodology, and applications. M. Dekker, New YorkGoogle Scholar
  10. Clay JR (1976) A stochastic analysis of the graded excitatory response of nerve membrane. J Theor Biol 59:141–158CrossRefGoogle Scholar
  11. Doi S (1993) On periodic orbits of trapezoid maps. Adv Appl Math 14:184–199MathSciNetMATHCrossRefGoogle Scholar
  12. Doi S, Inoue J, Kumagai S (1998) Spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally-forced van der Pol Oscillator with additive noise. J Stat Phys 90:1107–1127MathSciNetMATHCrossRefGoogle Scholar
  13. Doi S, Inoue J, Sato S, Smith CE (1999) Bifurcation analysis of neuronal excitability and oscillations. In: Poznanski R (ed) Modeling in the neurosciences: from ionic channels to neural networks, chap 16. Harwood, Newark, NJ, pp 443–473Google Scholar
  14. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophy J 1:445–466CrossRefGoogle Scholar
  15. Gardiner CW (1983) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, BerlinMATHCrossRefGoogle Scholar
  16. Gerstein GL, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4:41–68CrossRefGoogle Scholar
  17. Glass L, Mackey MC (1979) A simple model for phase locking of biological oscillators. J Math Biol 7:339–352MathSciNetMATHCrossRefGoogle Scholar
  18. Glass L, Mackey MC (1988) From clocks to chaos, the rhythms of life. Princeton University Press, PrincetonMATHGoogle Scholar
  19. Glass L, Sun J (1994) Periodic forcing of a limit-cycle oscillator: fixed points, Arnold tongues, and the global organization of bifurcations. Phys Rev E 50:5077–5084CrossRefGoogle Scholar
  20. Grasman J, Jansen MJW (1979) Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J Math Biol 7:171–197MathSciNetMATHCrossRefGoogle Scholar
  21. Guckenheimer J (1975) Isochrons and phaseless sets. J Math Biol 1:259–273MathSciNetMATHCrossRefGoogle Scholar
  22. Guckenheimer J (1986) Multiple bifurcation problems for chemical reactors. Physica D 20:1–20MathSciNetMATHCrossRefGoogle Scholar
  23. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, BerlinGoogle Scholar
  24. Guevara MR, Glass L (1982) Phase locking, period-doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J Math Biol 14:1–23MathSciNetMATHCrossRefGoogle Scholar
  25. Hadeler KP, an der Heiden U, Schumacher K (1976) Generation of the nervous impulse and periodic oscillations. Biol Cybern 23:211–218MATHGoogle Scholar
  26. Hassard B (1978) Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J Theor Biol 71:401–420MathSciNetCrossRefGoogle Scholar
  27. Hata M (1982) Dynamics of Caianiello’s equation. J Math Kyoto Univ 22(1):155–173MathSciNetMATHGoogle Scholar
  28. Hoppensteadt FC, Keener JP (1982) Phase locking of biological clocks. J Math Biol 15:339–349MathSciNetMATHCrossRefGoogle Scholar
  29. Inoue J, Doi S (2007) Sensitive dependence of the coefficient of variation of interspike intervals on the lower boundary of membrane potential for the leaky integrate-and-fire neuron model. Biosystems 87:49–57CrossRefGoogle Scholar
  30. Kawato M (1981) Transient and steady phase response curves of limit cycle oscillators. J Math Biol 12:13–30MathSciNetMATHCrossRefGoogle Scholar
  31. Kawato M, Suzuki R (1978) Biological oscillators can be stopped. Topological study of a phase response curve. Biol Cybern 30:241–248MathSciNetMATHCrossRefGoogle Scholar
  32. Keener JP, Glass L (1984) Global bifurcations of a periodically forced nonlinear oscillator. J Math Biol 21:175–190MathSciNetMATHCrossRefGoogle Scholar
  33. Keener JP, Hoppensteadt FC, Rinzel J (1981) Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J Appl Math 41:503–517MathSciNetCrossRefGoogle Scholar
  34. Kepler TB, Marder E, Abbott LF (1990) The effect of electrical coupling on the frequency of model neuronal oscillators. Science 248:83–85CrossRefGoogle Scholar
  35. Konig P, Engel AK, Singer W (1996) Integrator or coincidence detector? The role of the cortical neuron revisited. Trends Neurosci 19:130–137CrossRefGoogle Scholar
  36. Koper MTM (1995) Bifurcation of mixed-mode oscillations in a three-variable autonomous Van der Pol–Duffing model with a cross-shaped phase diagram. Physica D 80:72–94MathSciNetMATHCrossRefGoogle Scholar
  37. Lasota A, Mackey MC (1994) Chaos, fractals, and noise: stochastic aspects of dynamics. Springer, BerlinMATHGoogle Scholar
  38. Leonov NN (1959) Map of the line on to itself. Radiofisica 2:942–956Google Scholar
  39. Matsumoto G, Aihara K, Ichikawa M, Tasaki A (1984) Periodic and nonperiodic responses of membrane potentials in squid giant axons during sinusoidal current stimulation. J Theor Neurobiol 3:1–14Google Scholar
  40. Meunier C (1992) Two and three-dimensional reductions of the Hodgkin–Huxley system: separation of time scales and bifurcation schemes. Biol Cybern 67:461–468MATHCrossRefGoogle Scholar
  41. Mira C (1987) Chaotic dynamics. World Scientific, SingaporeMATHGoogle Scholar
  42. Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50:1645–1662MathSciNetMATHCrossRefGoogle Scholar
  43. Nagumo J, Sato S (1972) On a response characteristic of a mathematical neuron model. Kybernetik 10:155–164MATHCrossRefGoogle Scholar
  44. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line stimulating nerve axon. Proc Inst Radio Eng 50:2061–2070Google Scholar
  45. Nakano H, Saito T (2002) Basic dynamics from a pulse-coupled network of autonomous integrate-and-fire chaotic circuits. IEEE Trans Neural Netw 13:92–100CrossRefGoogle Scholar
  46. Nomura T, Sato S, Doi S, Segundo JP, Stiber MD (1994a) Global bifurcation structure of a Bonhoeffer van der Pol oscillator driven by periodic pulse trains. Comparison with data from an inhibitory synapse. Biol Cybern 72:55–67MATHGoogle Scholar
  47. Okuda M (1981) A new method of nonlinear analysis for threshold and shaping actions in transient state. Prog Theor Phys 66:90–100CrossRefGoogle Scholar
  48. Pakdaman K (2001) Periodically forced leaky integrate-and-fire model. Phys Rev E 63:041907CrossRefGoogle Scholar
  49. Rescigno R, Stein RB, Purple RL, Poppele RE (1970) A neuronal model for the discharge patterns produced by cyclic inputs. Bull Math Biophys 32:337–353MATHCrossRefGoogle Scholar
  50. Ricciardi LM (1977) Diffusion processes and related topics in biology. Springer, BerlinMATHCrossRefGoogle Scholar
  51. Ricciardi LM, Sato S (1988) First-passage-time density and moments of the Ornstein–Uhlenbeck process. J Appl Prob 25:43–57MathSciNetMATHCrossRefGoogle Scholar
  52. Rinzel J (1978) On repetitive activity in nerve. Fed Proc 37:2793–2802Google Scholar
  53. Scharstein H (1979) Input–output relationship of the leaky-integrator neuron model. J Math Biol 8:403–420MathSciNetMATHCrossRefGoogle Scholar
  54. Shadlen MN, Newsome WT (1994) Noise, neural codes and cortical organization. Curr Opin Neurobiol 4:569–579CrossRefGoogle Scholar
  55. Softky WR, Koch C (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J Neurosci 13:334–350Google Scholar
  56. Stein RB, French AS, Holden AV (1972) The frequency response, coherence, and information capacity of two neuronal models. Biophys J 12:295–322CrossRefGoogle Scholar
  57. Takahashi N, Hanyu Y, Musha T, Kubo R, Matsumoto G (1990) Global bifurcation structure in periodically stimulated giant axons of squid. Physica D 43:318–334MATHCrossRefGoogle Scholar
  58. Tateno T, Doi S, Sato S, Ricciardi LM (1995) Stochastic phase-lockings in a relaxation oscillator forced by a periodic input with additive noise: a first-passage-time approach. J Stat Phys 78:917–935MATHCrossRefGoogle Scholar
  59. Torikai H, Saito T (1999) Return map quantization from an integrate-and-fire model with two periodic inputs. IEICE Trans Fundam E82-A:1336–1343Google Scholar
  60. Tuckwell HC (1988) Introduction to theoretical neurobiology. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  61. van der Pol B (1926) On “relaxation-oscillations”. Phil Mag 2:978–992Google Scholar
  62. Winfree AT (1980) The geometry of biological time. Springer, New YorkMATHGoogle Scholar
  63. Xu J-X, Jiang J (1996) The global bifurcation characteristics of the forced van der Pol oscillator. Chaos Solitons Fractals 7:3–19MathSciNetMATHCrossRefGoogle Scholar
  64. Yellin E, Rabinovitch A (2003) Properties and features of asymmetric partial devil’s staircases deduced from piecewise linear maps. Phys Rev E 67:016202MathSciNetCrossRefGoogle Scholar
  65. Yu X, Lewis ER (1989) Studies with spike initiators: linearization by noise allows continuous signal modulation in neural networks. IEEE Trans Biomed Eng 36:36–43CrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Shinji Doi
    • 1
  • Junko Inoue
    • 2
  • Zhenxing Pan
    • 3
  1. 1.Graduate School of EngineeringKyoto UniversityKyoto-Daigaku Katsura, Nishikyo-kuJapan
  2. 2.Faculty of Human ScienceKyoto Koka Women’s UniversityJapan
  3. 3.Graduate School of EngineeringOsaka UniversityJapan

Personalised recommendations