May  2011, 10(3): 995-1009. doi: 10.3934/cpaa.2011.10.995

Neuronal coding of pacemaker neurons -- A random dynamical systems approach

1. 

Institut für Analysis, TU Dresden, 01219 Dresden, Germany

Received  March 2009 Revised  September 2009 Published  December 2010

The behaviour of neurons under the influence of periodic external input has been modelled very successfully by circle maps. The aim of this note is to extend certain aspects of this analysis to a much more general class of forcing processes. We apply results on the fibred rotation number of randomly forced circle maps to show the uniqueness of the asymptotic firing frequency of ergodically forced pacemaker neurons. In particular, this allows to treat the forced leaky integrate-and-fire model, which serves as a paradigm example.
Citation: T. Jäger. Neuronal coding of pacemaker neurons -- A random dynamical systems approach. Communications on Pure & Applied Analysis, 2011, 10 (3) : 995-1009. doi: 10.3934/cpaa.2011.10.995
References:
[1]

L. Lapicque, Quantitative investigations of electrical nerve excitation treated as polarization,, Biol. Cybern., 97 (2007), 341.   Google Scholar

[2]

N. Brunel and M. C. W. van Rossum, Lapicque's 1907 paper: from frogs to integrate-and-fire,, Biol. Cybern., 97 (2007), 337.  doi: doi:10.1007/s00422-007-0190-0.  Google Scholar

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L. F. Abbott, Lapicque's introduction of the integrate-and-fire model neuron (1907),, Brain Res. Bull., 50 (1999), 303.  doi: doi:10.1016/S0361-9230(99)00161-6.  Google Scholar

[4]

W. Gerstner and W. M. Kistler, "Spiking Neuron Models: Single Neurons, Populations, Plasticity,", Cambridge University Press, (2002).   Google Scholar

[5]

D. H. Perkel, J. H. Schulman, T. H. Bullock, G. P. Moore and J. P. Segundo, Pacemaker neurons: Effects of regularly spaced synaptic input,, Science, 145 (1964), 61.  doi: doi:10.1126/science.145.3627.61.  Google Scholar

[6]

B. W. Knight, The relationship between the fiding rate of a single neuron and the level of activity in a population of neurons,, J. Gen. Physiol., 59 (1972), 767.  doi: doi:10.1085/jgp.59.6.767.  Google Scholar

[7]

R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173.  doi: doi:10.1016/S0006-3495(65)86709-1.  Google Scholar

[8]

B. W. Knight, Dynamics of encoding in a population of neurons,, J. Gen. Physiol., 59 (1972), 734.  doi: doi:10.1085/jgp.59.6.734.  Google Scholar

[9]

J. P. Segundo, Pacemaker synaptic interactions: Modelled locking and paradoxical features,, Biol. Cybern., 35 (1979), 55.  doi: doi:10.1007/BF01845844.  Google Scholar

[10]

L. Glass, Cardiac arrhythmias and circle maps - a classical problem,, Chaos, 1 (1991), 13.  doi: doi:10.1063/1.165810.  Google Scholar

[11]

V. I. Arnold, Cardiac arrythmias and circle mappings,, Chaos, 1 (1991), 20.   Google Scholar

[12]

Source: http://commons.wikimedia.org., GNU Free Documentation License., The \emph{Limulus} picture is a drawing by Ernst Haeckel (, (): 1899.   Google Scholar

[13]

K. Pakdaman, Periodically forced leaky integrate-and-fire model,, Phys. Rev. E, 63 (2001).  doi: doi:10.1103/PhysRevE.63.041907.  Google Scholar

[14]

R. Brette, Dynamics of one-dimensional spiking neuron models,, J. Math. Biol., 48 (2004), 38.  doi: doi:10.1007/s00285-003-0223-9.  Google Scholar

[15]

A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input.,, Biol. Cybern., 95 (2006), 1.  doi: doi:10.1007/s00422-006-0068-6.  Google Scholar

[16]

P. Lansky and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models,, Biol. Cybern., 99 (2008), 253.  doi: doi:10.1007/s00422-008-0237-x.  Google Scholar

[17]

R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?,, J. Theor. Biol., 257 (2009), 90.  doi: doi:10.1016/j.jtbi.2008.11.004.  Google Scholar

[18]

A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties,, Biol. Cybern., 95 (2006), 97.  doi: doi:10.1007/s00422-006-0082-8.  Google Scholar

[19]

L. Arnold, "Random Dynamical Systems,", Springer, (1998).   Google Scholar

[20]

W. Li and K. Lu, Rotation numbers for random dynamical systems on the circle,, Trans. Am. Math. Soc., 360 (2008), 5509.  doi: doi:10.1090/S0002-9947-08-04619-9.  Google Scholar

[21]

K. Bjerklöv and T. Jäger, Rotation numbers for quasiperiodically forced circle maps - Mode-locking vs strict monotonicity,, J. Am. Math. Soc., 22 (2009), 353.  doi: doi:10.1090/S0894-0347-08-00627-9.  Google Scholar

[22]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1997).   Google Scholar

[23]

M. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2,, Comment. Math. Helv., 58 (1983), 453.  doi: doi:10.1007/BF02564647.  Google Scholar

[24]

R. Johnson and J. Moser, The rotation numer for almost periodic potentials,, Commun. Math. Phys., 4 (1982), 403.  doi: doi:10.1007/BF01208484.  Google Scholar

show all references

References:
[1]

L. Lapicque, Quantitative investigations of electrical nerve excitation treated as polarization,, Biol. Cybern., 97 (2007), 341.   Google Scholar

[2]

N. Brunel and M. C. W. van Rossum, Lapicque's 1907 paper: from frogs to integrate-and-fire,, Biol. Cybern., 97 (2007), 337.  doi: doi:10.1007/s00422-007-0190-0.  Google Scholar

[3]

L. F. Abbott, Lapicque's introduction of the integrate-and-fire model neuron (1907),, Brain Res. Bull., 50 (1999), 303.  doi: doi:10.1016/S0361-9230(99)00161-6.  Google Scholar

[4]

W. Gerstner and W. M. Kistler, "Spiking Neuron Models: Single Neurons, Populations, Plasticity,", Cambridge University Press, (2002).   Google Scholar

[5]

D. H. Perkel, J. H. Schulman, T. H. Bullock, G. P. Moore and J. P. Segundo, Pacemaker neurons: Effects of regularly spaced synaptic input,, Science, 145 (1964), 61.  doi: doi:10.1126/science.145.3627.61.  Google Scholar

[6]

B. W. Knight, The relationship between the fiding rate of a single neuron and the level of activity in a population of neurons,, J. Gen. Physiol., 59 (1972), 767.  doi: doi:10.1085/jgp.59.6.767.  Google Scholar

[7]

R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173.  doi: doi:10.1016/S0006-3495(65)86709-1.  Google Scholar

[8]

B. W. Knight, Dynamics of encoding in a population of neurons,, J. Gen. Physiol., 59 (1972), 734.  doi: doi:10.1085/jgp.59.6.734.  Google Scholar

[9]

J. P. Segundo, Pacemaker synaptic interactions: Modelled locking and paradoxical features,, Biol. Cybern., 35 (1979), 55.  doi: doi:10.1007/BF01845844.  Google Scholar

[10]

L. Glass, Cardiac arrhythmias and circle maps - a classical problem,, Chaos, 1 (1991), 13.  doi: doi:10.1063/1.165810.  Google Scholar

[11]

V. I. Arnold, Cardiac arrythmias and circle mappings,, Chaos, 1 (1991), 20.   Google Scholar

[12]

Source: http://commons.wikimedia.org., GNU Free Documentation License., The \emph{Limulus} picture is a drawing by Ernst Haeckel (, (): 1899.   Google Scholar

[13]

K. Pakdaman, Periodically forced leaky integrate-and-fire model,, Phys. Rev. E, 63 (2001).  doi: doi:10.1103/PhysRevE.63.041907.  Google Scholar

[14]

R. Brette, Dynamics of one-dimensional spiking neuron models,, J. Math. Biol., 48 (2004), 38.  doi: doi:10.1007/s00285-003-0223-9.  Google Scholar

[15]

A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input.,, Biol. Cybern., 95 (2006), 1.  doi: doi:10.1007/s00422-006-0068-6.  Google Scholar

[16]

P. Lansky and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models,, Biol. Cybern., 99 (2008), 253.  doi: doi:10.1007/s00422-008-0237-x.  Google Scholar

[17]

R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?,, J. Theor. Biol., 257 (2009), 90.  doi: doi:10.1016/j.jtbi.2008.11.004.  Google Scholar

[18]

A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties,, Biol. Cybern., 95 (2006), 97.  doi: doi:10.1007/s00422-006-0082-8.  Google Scholar

[19]

L. Arnold, "Random Dynamical Systems,", Springer, (1998).   Google Scholar

[20]

W. Li and K. Lu, Rotation numbers for random dynamical systems on the circle,, Trans. Am. Math. Soc., 360 (2008), 5509.  doi: doi:10.1090/S0002-9947-08-04619-9.  Google Scholar

[21]

K. Bjerklöv and T. Jäger, Rotation numbers for quasiperiodically forced circle maps - Mode-locking vs strict monotonicity,, J. Am. Math. Soc., 22 (2009), 353.  doi: doi:10.1090/S0894-0347-08-00627-9.  Google Scholar

[22]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1997).   Google Scholar

[23]

M. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2,, Comment. Math. Helv., 58 (1983), 453.  doi: doi:10.1007/BF02564647.  Google Scholar

[24]

R. Johnson and J. Moser, The rotation numer for almost periodic potentials,, Commun. Math. Phys., 4 (1982), 403.  doi: doi:10.1007/BF01208484.  Google Scholar

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