Abstract
The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.
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References
Bonhoeffer KF (1941) Über die Aktivierung von passiven Eisen in Salpetersäure. Z Elektrochem 47:147
Bonhoeffer KF (1948) Activation of passiv iron as a model for the excitation of nerve. J Gen Physiol 32:69
Bonhoeffer KF (1953) Modelle der Nervenerregung. Naturwissenschaften 40:301
Bonhoeffer KF, Langhammer G (1948) Über periodische Reaktionen. IV. Theorie der kathodischen Polarisation von Eisen in Salpetersäure. Z Elektrochem 52:67
Conti F, Wanke E (1975) Channel noise in nerve membranes and lipid bilayers. Qu Rev Biophys 8 (4):451
Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membranes. Biophys J 1:445
Gardiner CW (1983) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, Berlin Heidelberg New York (Springer Series in Synergetics, vol 13)
Guckenheim J, Holmes P (1983) Nonlinear oszillations, dynamical systems, and bifurcation of vector fields. Springer, Berlin Heidelberg New York (Appl Math Sci, vol 42)
Hennig R, Lømo T (1985) Firing patterns of motor units in normal rats. Nature 314:164
Holden AV (1976) Models of the stochastic activity of neurons. Springer, Berlin Heidelberg New York (Lecture Notes in Biomathematics, vol 12)
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500
Koester J (1985) Voltage-gated Channels and the generation of the action potential. In: Kandel ER, Schwartz JH (eds) Principles of neural science. Elsevier, New York Amsterdam Oxford
Lamm G, Schulten K (1981) Extended Brownian dynamics approach to diffusion-controlled processes. J Chem Phys 75:365
Lamm G, Schulten K (1983) Extended Brownian dynamics. II. Reactive, nonlinear diffusion. J Chem Phys 78:2713
Lecar H, Nossal R (1971 a) Theory of threshold fluctuations in nerves. I. Biophys J 11:1048
Lecar H, Nossal R (1971 b) Theory of threshold fluctuations in nerves. II. Biophys J 11:1068
Marsden J, McCracken M (1976) The Hopf-bifurcation and its applications. Springer, Berlin Heidelberg New York (Appl Math Sci, vol 19)
Ricciardi LM (1982) Diffusion approximations and computational problems for single neutrons' activity. In: Levin S (ed) Competition and cooperation in neural nets. Springer, Berlin Heidelberg New York (Lecture Notes in Biomathematics, vol 45)
Risken H (1985) The Fokker-Planck equation; methods of solution and applications. Springer, Berlin Heidelberg New York (Springer series in Synergetics, vol 18)
Treutlein H (1984) Die Lösungen der Fokker-Planck Gleichung zur Beschreibung nicht-linearer Reaktionen mit additivem Rauschen. Diplomarbeit, Technische Universität München
Treutlein H, Schulten K (1985) Noise induced limit cycles of the Bonhoeffer-van der Pol model of neural pulses. Ber Bunsenges Phys Chem 89:710
van der Pol B (1926) On relaxation oscillations. Philos Mag 2:978
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Treutlein, H., Schulten, K. Noise-induced neural impulses. Eur Biophys J 13, 355–365 (1986). https://doi.org/10.1007/BF00265671
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DOI: https://doi.org/10.1007/BF00265671