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Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model

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Abstract

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons.

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References

  • Acebron, J.A., Bulsarra, A.R., Rappel, W.-J., 2004. Noisy FitzHugh-Nagumo model: From single elements to globally coupled networks. Phys. Rev. E 69, 026202.

    Article  MathSciNet  Google Scholar 

  • Bahar, S., Neiman, A., Wilkens, L.A., Moss, F., 2002. Phase synchronization and stochastic resonance effects in the crayfish caudal photoreceptor. Phys. Rev. E 69, 050901(R).

    Google Scholar 

  • Bahar, S., Moss F., 2003. Stochastic phase synchronization in the crayfish mechanoreceptor/photoreceptor system. Chaos 13, 138–144.

    Article  PubMed  MathSciNet  Google Scholar 

  • Bal, T., Debay, D., Destexhe, A., 2000. Cortical feedback controls the frequency and synchrony of oscillations in the visual thalamus. J. Neurosci. 20, 7478–7488.

    PubMed  Google Scholar 

  • Berdichevsky, V., Gitterman, M., 1996. Stochastic resonance in a bistable piecewise potential: Analytical solution. J. Phys. A 29, L447—L452.

    Article  MathSciNet  Google Scholar 

  • Börgers, C., Epstein, S., Kopell, N., 2005. Background gamma rhythmicity and attention in cortical local circuits: A computational study. PNAS 102, 7002–7007.

    Article  PubMed  Google Scholar 

  • Braun, H.A., Huber, M.T., Dewald, M., Schafer, K., Voigt, K., 1998. Computer simulations of neuronal signal transduction: The role of nonlinear dynamics and noise. Int. J. Bif. Chaos 8, 881–889.

    Article  MATH  Google Scholar 

  • Bressloff, P.C., Coombes, S., 2000. A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators. SIAM J. Appl. Math. 60, 820–841.

    Article  MathSciNet  MATH  Google Scholar 

  • Brunel, N., Sergi, S., 1998. Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics. J. Theor. Biol. 195, 87–95.

    Article  PubMed  Google Scholar 

  • Brunel, N., Hakim, V., Richardson, M.J.E., 2003. Firing-rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance. Phys. Rev. E 67, 051916.

    Article  MathSciNet  Google Scholar 

  • Bulsarra, A.R., Elston, T.C., Doering, C.R., Lowen, S.B., Lindenberg, K., 1996. Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys. Rev. E 53, 3958–3969.

    Article  Google Scholar 

  • Chow, C.C., White, J.A., 1996. Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 3013–3020.

    Article  PubMed  Google Scholar 

  • Contreras, D., Destexhe, A., Sejnowski, T.J., Steriade, M., 1996. Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science 274, 771–774.

    Article  PubMed  Google Scholar 

  • Destexhe, A., Bal, T., McCormick, D.A., Sejnowski, T.J., 1996. Ionic mechanisms underlying synchronized oscillations and propagating waves in a model of ferret thalamic slices. J. Neurophysiol. 76, 2049–2070.

    PubMed  Google Scholar 

  • Freund, J.A., Schimansky-Geier, L., Hanggi, P., 2003. Frequency and phase synchronization in stochastic systems. Chaos 13, 225–238.

    Article  PubMed  MathSciNet  MATH  Google Scholar 

  • Hu, G., Ditzinger, T., Ning, C.Z., Haken, H., 1993. Stochastic resonance without external periodic force. PRL 71, 807–810.

    Article  Google Scholar 

  • Gluckman, B.J., Netoff, T.I., Neel, E.J., Ditto, W.L., Spano, M.L., Schiff, S.J., 1996. Stochastic resonance in a neuronal network from mammalian brain. Phys. Rev. Lett. 77, 4098–4101.

    Article  PubMed  Google Scholar 

  • Hauptmann, C., Kaiser, F., Eichwald, C., 1999. Signal transfer and stochastic resonance in coupled nonlinear systems. Int. J. Bif. Chaos 9, 1159–1167.

    Article  Google Scholar 

  • Hindmarsh, J.L., Rose, R.M., 1984. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B Biol. Sci. 221, 87–102.

    Article  PubMed  Google Scholar 

  • Huguenard, J.R., McCormick, D.A., 1992. Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons. J. Neurophys. 68, 1373–1383.

    Google Scholar 

  • Hutcheon, B., Yarom, Y., 2000. Resonance, oscillation, and the intrinsic frequency preferences of neurons. TINS 23, 216–222.

    PubMed  Google Scholar 

  • Izhikevich, E.M., 2001. Resonate-and-fire neurons. Neural Netw. 14, 883–894.

    Article  PubMed  Google Scholar 

  • Kurrer, C., Schulten, K., 1995. Noise-induced neuronal oscillations. Phys. Rev. E 51, 6213–6218.

    Article  Google Scholar 

  • Kuske, R., Baer, S.M., 2002. Asymptotic analysis of noise sensitivity in a neuronal burster. Bull. Math. Bio 64, 447–481.

    Article  Google Scholar 

  • Lee, S.-G., Neiman, A., Kim, S., 1998. Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron. Phys. Rev. E 57, 3292–3297.

    Article  Google Scholar 

  • Lindner, J.F., Meadows, B.K., Ditto, W.L., Inchiosa, M.E., Bulsara, A.R., 1995. Array enhanced stochastic resonance and spatiotemporal synchronization. Phys. Rev. Lett. 75, 3–6.

    Article  PubMed  Google Scholar 

  • Lindner, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L., 2004. Effects of noise in excitable systems. Phys. Rep. 392, 321–424.

    Article  Google Scholar 

  • Liu, Y.-H., Wang, X.-W., 2001. Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron. J. Comp. Neurosci 10, 25–45.

    Article  Google Scholar 

  • Longtin, A., 1997. Autonomous stochastic resonance in bursting neurons. Phys. Rev. E 55, 868–876.

    Article  Google Scholar 

  • Longtin, A., 2000. Effect of noise on the tuning properties of excitable systems. Chaos Solitons Fractals 11, 1835–1848.

    Article  MATH  Google Scholar 

  • Massanes, S., Vicente, C., 1999. Nonadiabatic resonances in a noisy FitzHugh-Nagumo neuron model. Phys. Rev. E 59, 4490–4497.

    Article  Google Scholar 

  • McCormick, D.A., 1999. Spontaneous activity: Signal or noise? Science 285, 541–543.

    Article  PubMed  Google Scholar 

  • Mori, T., Kai, S., 2002. Noise-induced entrainment and stochastic resonance in human brain waves. Phys. Rev. Lett. 88, 218101.

    Article  PubMed  Google Scholar 

  • Neiman, A., Schimansky-Geier, L., Cornell-Bell, A., Moss, F., 1999. Noise-enhanced phase synchronization in excitable media. Phys. Rev. Lett. 83, 4896–4899.

    Article  Google Scholar 

  • Pei, X., Wilkens, L., Moss, F., 1996. Noise-mediated spike timing precision from aperiodic stimuli in an array of Hodgkin-Huxley-type neurons. Phys. Rev. Lett. 77, 4679–4682.

    Article  PubMed  Google Scholar 

  • Pikovsky, A., Kurths, J., 1997. Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775–778.

    Article  MathSciNet  MATH  Google Scholar 

  • Qian, M., Zhang, X.J., 2002. Rotation number, stochastic resonance, and synchronization of coupled systems without periodic driving. Phys. Rev. E 65, 031110.

    Article  MathSciNet  Google Scholar 

  • Rappel, W.-J., Karma, A., 1996. Noise-induced coherence in neural networks. Phys. Rev. Lett. 77, 3256–3259.

    Article  PubMed  Google Scholar 

  • Reinker, S., Puil, E., Miura, R.M., 2003. Resonances and noise in a stochastic Hindmarsh-Rose model of thalamic neurons. Bull. Math. Biol. 65, 641–663.

    Article  PubMed  Google Scholar 

  • Reinker, S., Puil, E., Miura, R.M., 2004. Membrane resonance and stochastic resonance modulate firing patterns of thalamocortical neurons. J. Comp. Neurosci. 16, 15–25.

    Article  Google Scholar 

  • Risken, H., 1989. The Fokker-Planck Equation, Springer, Berlin Heidelberg New York.

    MATH  Google Scholar 

  • Richardson, M.J.E., Brunel, N., Hakim, V., 2003. From subthreshold to firing-rate resonance. J. Neurophysiol. 89, 2538–2554.

    Article  PubMed  Google Scholar 

  • Shuai, J.-W., Durand, D.M., 1999. Phase synchronization in two coupled chaotic neurons. Phys. Lett. A 264, 289–297.

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, G.D., Cox, C.L., Sherman, S.M., Rinzel, J., 2000. Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model. J. Neurophysiol. 83, 588–610.

    PubMed  Google Scholar 

  • Stacey, W.C., Durand, D.M., 2000. Stochastic resonance improves signal detection in hippocampal CA1 neurons. J. Neurophysiol. 83, 1394–1402.

    PubMed  Google Scholar 

  • Steinmetz, P.N. Manwani, A., Koch, C., London, M., Segev, I., 2000. Subthreshold voltage noise due to channel fluctuations in active neuronal membranes. J. Comp. Neurosci. 9, 133–148.

    Article  Google Scholar 

  • Tateno, T. Jimbo, Y., 2000. Stochastic mode-locking for a noisy integrate-and-fire oscillator. Phys. Lett. A 271, 227–236.

    Article  MathSciNet  MATH  Google Scholar 

  • Tiesinga, P.H., Jose, J.V., 2000. Synchronous clusters in a noisy inhibitory neural network. J. Comp. Neurosci. 9, 49–65.

    Article  Google Scholar 

  • Tiesinga, P.H., Fellous, J.-M., Jose, J.V., Sejnowski, T.J., 2001. Computational model of carbachol-induced delta, theta, and gamma oscillations in the hippocampus. Hippocampus 11, 251–274.

    Article  PubMed  Google Scholar 

  • Wang, W., Wang, Z.D., 1997. Internal-noise-enhanced signal transduction in neuronal systems. Phys. Rev. E 55, 7379–7384.

    Article  Google Scholar 

  • Wang, Y., Wang, Z.D., 2000. Information coding via spontaneous oscillations in neural ensembles. Phys. Rev. E 62, 1063–1068.

    Article  Google Scholar 

  • Wang, Y., Chik, D.T.W., Wang, Z.D., 2000. Coherence resonance and noise-induced synchronization in globally coupled Hodgkin-Huxley neurons. Phys. Rev. E 61, 740–746.

    Article  Google Scholar 

  • White, J.A., Rubinstein, J.T., Kay, A.F., 2000. Channel noise in neurons. TINS 23, 131–137.

    PubMed  Google Scholar 

  • Zhou, C., Kurths, J., Hu, B., 2001. Array-enhanced coherence resonance: Nontrivial effects of heterogeneity and spatial independence of noise. Phys. Rev. Lett. 87, 098101.

    Article  PubMed  Google Scholar 

  • Zhou, C., Kurths, J., 2003. Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons. Chaos 13, 401–409.

    Article  PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Physics, FDM, University of Freiburg, Hermann-Herder-Str. 3, 79104, Freiburg, Germany

    Stefan Reinker

  2. Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada, V6T1Z2

    Yue-Xian Li & Rachel Kuske

Authors
  1. Stefan Reinker
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  2. Yue-Xian Li
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  3. Rachel Kuske
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Correspondence to Stefan Reinker.

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PACS 05.45.-a, 05.40.Ca, 87.18.Sn, 87.19

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Reinker, S., Li, YX. & Kuske, R. Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model. Bull. Math. Biol. 68, 1401–1427 (2006). https://doi.org/10.1007/s11538-006-9089-5

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  • DOI: https://doi.org/10.1007/s11538-006-9089-5

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