Skip to main content

2016 | OriginalPaper | Buchkapitel

3. Population and Subpopulation Models

verfasst von : Priscilla E. Greenwood, Lawrence M. Ward

Erschienen in: Stochastic Neuron Models

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We have seen stochastic neuron firing models that have inherent frequencies in their subthreshold dynamics, e.g., for the Morris-Lecar neuron [28]. This frequency shows up at the population level. If we record the firings of several model neurons over a period of time, the firings of each single neuron follow an inherent frequency, but often skipping many repeats of that frequency. The skipping phenomenon is a result of the tendency of the neurons to fire on their subthreshold quasicycles, a stochastic facilitation phenomenon explained in Sections 1.​2 and 2.​5 When the firings of the several neurons, driven by the same or partially common noise, are added together as a function of time, we obtain a function that oscillates at the common inherent frequency of the family.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
3.
Zurück zum Zitat Greenwood, P.E., McDonnell, M.D., Ward, L.M.: Dynamics of gamma bursts in local field potentials. Neural Comput. 27, 74–103 (2015)CrossRef Greenwood, P.E., McDonnell, M.D., Ward, L.M.: Dynamics of gamma bursts in local field potentials. Neural Comput. 27, 74–103 (2015)CrossRef
4.
Zurück zum Zitat Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11, 1621–1671 (1999)CrossRef Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11, 1621–1671 (1999)CrossRef
18.
21.
Zurück zum Zitat Izhikevich, E.M., Edelman, G.M.: Large-scale models of mammalian thalamocortical systems. Proc. Natl. Acad. Sci. U. S. A. 105(9), 3593–3598 (2008)CrossRef Izhikevich, E.M., Edelman, G.M.: Large-scale models of mammalian thalamocortical systems. Proc. Natl. Acad. Sci. U. S. A. 105(9), 3593–3598 (2008)CrossRef
28.
Zurück zum Zitat Ditlevsen, S., Greenwood, P.E.: The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67, 239–259 (2013)MathSciNetCrossRefMATH Ditlevsen, S., Greenwood, P.E.: The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67, 239–259 (2013)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Baxendale, P.H., Greenwood, P.E.: Sustained oscillations for density dependent Markov processes. J. Math. Biol. 63, 433–457 (2011)MathSciNetCrossRefMATH Baxendale, P.H., Greenwood, P.E.: Sustained oscillations for density dependent Markov processes. J. Math. Biol. 63, 433–457 (2011)MathSciNetCrossRefMATH
31.
Zurück zum Zitat Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer, Berlin (1990)MATH Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer, Berlin (1990)MATH
40.
Zurück zum Zitat Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972)CrossRef Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972)CrossRef
41.
Zurück zum Zitat Kang, K., Shelley, M., Henrie, J.A., Shapley, R.: LFP spectral peaks in V1 cortex: network resonance and cortico-cortical feedback. J. Comput. Neurosci. 29, 495–507 (2010)CrossRef Kang, K., Shelley, M., Henrie, J.A., Shapley, R.: LFP spectral peaks in V1 cortex: network resonance and cortico-cortical feedback. J. Comput. Neurosci. 29, 495–507 (2010)CrossRef
42.
Zurück zum Zitat Ray, S., Maunsell, J.H.R.: Different origins of gamma rhythm and high-gamma activity in macaque visual cortex. PLOS Biol. 9(4), 1000610 (2011)CrossRef Ray, S., Maunsell, J.H.R.: Different origins of gamma rhythm and high-gamma activity in macaque visual cortex. PLOS Biol. 9(4), 1000610 (2011)CrossRef
43.
Zurück zum Zitat McKane, A.J., Biancalani, T., Rogers, T.: Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond. Bull. Math. Biol. 76, 895–921 (2014)MathSciNetCrossRefMATH McKane, A.J., Biancalani, T., Rogers, T.: Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond. Bull. Math. Biol. 76, 895–921 (2014)MathSciNetCrossRefMATH
44.
Zurück zum Zitat Wallace, E., Benayoun, M., van Dronglen, W., Cowan, J.D.: Emergent oscillations in networks of stochastic spiking neurons. PLoS One 6(5), 14804 (2011)CrossRef Wallace, E., Benayoun, M., van Dronglen, W., Cowan, J.D.: Emergent oscillations in networks of stochastic spiking neurons. PLoS One 6(5), 14804 (2011)CrossRef
45.
Zurück zum Zitat Borodin, A.N., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae. Probability and Its Applications, 2nd edn. Birkhauser, Basel (2002) Borodin, A.N., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae. Probability and Its Applications, 2nd edn. Birkhauser, Basel (2002)
46.
Zurück zum Zitat Vakeroudis, S.: On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a stable process. arXiv:1209.4027v1 (2012) Vakeroudis, S.: On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a stable process. arXiv:1209.4027v1 (2012)
47.
48.
Zurück zum Zitat Asllani, M., Biancalini, T., Fanelli, D., McKane, A.J.: The linear noise approximation for reaction-diffusion systems on networks. arXiv:1305.7318v1 (2013) Asllani, M., Biancalini, T., Fanelli, D., McKane, A.J.: The linear noise approximation for reaction-diffusion systems on networks. arXiv:1305.7318v1 (2013)
49.
Zurück zum Zitat Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1992)MATH Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1992)MATH
50.
Zurück zum Zitat Doiron, B., Lindner, B., Longtin, A., Maler, L., Bastian, J.: Oscillatory activity in electrosensory neurons increases with spatial correlation of the stochastic input stimulus. Phys. Rev. Lett. 93(4), 048101 (2004)CrossRef Doiron, B., Lindner, B., Longtin, A., Maler, L., Bastian, J.: Oscillatory activity in electrosensory neurons increases with spatial correlation of the stochastic input stimulus. Phys. Rev. Lett. 93(4), 048101 (2004)CrossRef
51.
Zurück zum Zitat Lindner, B., Doiron, B., Longtin, A.: Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback. Phys. Rev. E 72, 061919 (2005)MathSciNetCrossRef Lindner, B., Doiron, B., Longtin, A.: Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback. Phys. Rev. E 72, 061919 (2005)MathSciNetCrossRef
52.
Zurück zum Zitat Dumont, G., Northoff, G., Longtin, A.: Linear noise approximation for oscillations in a stochastic inhibitory network with delay. Phys. Rev. E 90, 012702 (2014)CrossRef Dumont, G., Northoff, G., Longtin, A.: Linear noise approximation for oscillations in a stochastic inhibitory network with delay. Phys. Rev. E 90, 012702 (2014)CrossRef
53.
Zurück zum Zitat Klosek, M.M., Kuske, R.: Multiscale analysis of stochastic differential equations. SIAM Multiscale Model. Simul. 3, 706–729 (2005)MathSciNetCrossRefMATH Klosek, M.M., Kuske, R.: Multiscale analysis of stochastic differential equations. SIAM Multiscale Model. Simul. 3, 706–729 (2005)MathSciNetCrossRefMATH
54.
Zurück zum Zitat Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, New York (1984)CrossRefMATH Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, New York (1984)CrossRefMATH
55.
Zurück zum Zitat Acebron, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)CrossRef Acebron, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)CrossRef
56.
Zurück zum Zitat Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143, 1–20 (2000)MathSciNetCrossRefMATH Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143, 1–20 (2000)MathSciNetCrossRefMATH
57.
Zurück zum Zitat Greenwood, P.E., McDonnell, M.D., Ward, L.M.: A Kuramoto coupling of quasi-cycle oscillators. arXiv:1511.04124v2 Greenwood, P.E., McDonnell, M.D., Ward, L.M.: A Kuramoto coupling of quasi-cycle oscillators. arXiv:1511.04124v2
58.
Zurück zum Zitat DeVille, R.E.L., Peskin, C.S., Spencer, J.H.: Dynamics of stochastic neural networks and the connection to random graph theory. Math. Model. Nat. Phenom. 5(2), 26–66 (2010)MathSciNetCrossRefMATH DeVille, R.E.L., Peskin, C.S., Spencer, J.H.: Dynamics of stochastic neural networks and the connection to random graph theory. Math. Model. Nat. Phenom. 5(2), 26–66 (2010)MathSciNetCrossRefMATH
Metadaten
Titel
Population and Subpopulation Models
verfasst von
Priscilla E. Greenwood
Lawrence M. Ward
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-26911-5_3