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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[24]

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

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[24]

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

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901

[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835

[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315

[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857

[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263

[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977

[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[13]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430

[14]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[15]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635

[16]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129

[17]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473

[18]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

[19]

João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641

[20]

Antonio Giorgilli, Stefano Marmi. Convergence radius in the Poincaré-Siegel problem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 601-621. doi: 10.3934/dcdss.2010.3.601

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]