Linear noise approximation for oscillations in a stochastic inhibitory network with delay

Grégory Dumont, Georg Franz Josef Northoff, André Longtin

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm. © Published by the American Physical Society.
Original languageEnglish
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume90
Issue number1
DOIs
Publication statusPublished - 2014
Externally publishedYes

Fingerprint

Oscillation
oscillations
Approximation
approximation
Network Dynamics
Delay Differential Equations
neurons
Power Spectrum
Conduction
power spectra
Neuron
Fluctuations
neurology
conduction
rhythm
spiking
Neuroscience
Hilbert Transform
Stochasticity
Multiplicative Noise

Keywords

  • Computation theory
  • Differential equations
  • Neural networks
  • Power spectrum
  • Stochastic systems
  • Additive Gaussian white noise
  • Delay differential equations
  • Infinite network size limit
  • Intrinsic randomness
  • Linear noise approximation
  • Population activities
  • Probabilistic descriptions
  • Theoretical expression
  • Low noise amplifiers
  • biological model
  • cytology
  • nerve cell
  • nerve cell inhibition
  • nerve cell network
  • physiology
  • statistical model
  • statistics
  • Linear Models
  • Models, Neurological
  • Nerve Net
  • Neural Inhibition
  • Neurons
  • Stochastic Processes

Cite this

@article{85250b359f1f49c8b8bd182025ed07af,
title = "Linear noise approximation for oscillations in a stochastic inhibitory network with delay",
abstract = "Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian {"}delay{"} case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm. {\circledC} Published by the American Physical Society.",
keywords = "Computation theory, Differential equations, Neural networks, Power spectrum, Stochastic systems, Additive Gaussian white noise, Delay differential equations, Infinite network size limit, Intrinsic randomness, Linear noise approximation, Population activities, Probabilistic descriptions, Theoretical expression, Low noise amplifiers, biological model, cytology, nerve cell, nerve cell inhibition, nerve cell network, physiology, statistical model, statistics, Linear Models, Models, Neurological, Nerve Net, Neural Inhibition, Neurons, Stochastic Processes",
author = "Gr{\'e}gory Dumont and Northoff, {Georg Franz Josef} and Andr{\'e} Longtin",
note = "Cited By :2 Export Date: 11 May 2016 CODEN: PLEEE Funding Details: HDRF, Hope for Depression Research Foundation Funding Details: NSERC, Hope for Depression Research Foundation References: Abbott, L., Rajan, K., Sompolinksy, H., (2011) The Dynamic Brain: An Exploration of Neuronal Variability and Its Functional Significance, , (Oxford University Press, New York); Churchland, M.M., Abbott, L.F., (2012) Nat. Neurosci., 15, p. 1472. , NANEFN 1097-6256 10.1038/nn.3247; Faisal, A.A., Selen, L.P.J., Wolpert, D.M., Noise in the nervous system (2008) Nature Reviews Neuroscience, 9 (4), pp. 292-303. , DOI 10.1038/nrn2258, PII NRN2258; Lindner, B., Doiron, B., Longtin, A., Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback (2005) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 72 (6), p. 061919. , http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix= oai_apsmeta_2&set=journal:PRE:71, DOI 10.1103/PhysRevE.72.061919; Zillmer, R., Brunel, N., Hansel, D., (2009) Phys. Rev. e, 79, p. 031909. , PLEEE8 1539-3755 10.1103/PhysRevE.79.031909; Wallace, E., Benayoun, M., Van Drongelen, W., Cowan, J.D., (2011) Plos One, 6, pp. e14804. , 1932-6203 10.1371/journal.pone.0014804; Bressloff, P.C., (2009) SIAM J. Appl. Math., 70, p. 1488. , SMJMAP 0036-1399 10.1137/090756971; McKane, A.J., Newman, T.J., Predator-prey cycles from resonant amplification of demographic stochasticity (2005) Physical Review Letters, 94 (21), pp. 1-4. , http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix= oai_apsmeta_2&set=journal:PRL:94, DOI 10.1103/PhysRevLett.94.218102, 218102; Alonso, D., McKane, A.J., Pascual, M., Stochastic amplification in epidemics (2007) Journal of the Royal Society Interface, 4 (14), pp. 575-582. , DOI 10.1098/rsif.2006.0192; Kukjin Kang, M.S., Henrie, J.A., Shapley, R., (2010) J. Comput. Neurosci., 29, p. 495. , JCNEFR 0929-5313 10.1007/s10827-009-0190-2; Buzs{\'a}ki, G., Wang, X.-J., (2012) Annu. Rev. Neurosci., 35, p. 203. , ARNSD5 0147-006X 10.1146/annurev-neuro-062111-150444; Dumont, G., Henry, J., (2013) J. Math. Biol., 67, p. 453. , JMBLAJ 0303-6812 10.1007/s00285-012-0554-5; Dumont, G., Henry, J., (2013) Bull. Math. Biol., 75, p. 629. , BMTBAP 0092-8240 10.1007/s11538-013-9823-8; C{\'a}ceres, M.J., Carrillo, J.A., Perthame, B., (2011) J. Math. Neurosci., 1, p. 8567. , 10.1186/2190-8567-1-7; Brunel, N., Hakim, V., (1999) Neural Computat., 11, p. 1621. , NEUCEB 0899-7667 10.1162/089976699300016179; Stiefel, K.M., Englitz, B., Sejnowski, T.J., (2013) Proc. Natl. Acad. Sci. (USA), 110, p. 7886. , PNASA6 0027-8424 10.1073/pnas.1305219110; Erneux, T., Applied Delay Differential Equations (2009) Surveys and Tutorials in the Applied Mathematical Sciences, 3. , Springer, Berlin; Klosek, M.M., Kuske, R., Multiscale analysis of stochastic delay differential equations (2005) Multiscale Modeling and Simulation, 3 (3), pp. 706-729. , DOI 10.1137/030601375; Longtin, A., Milton, J., Bos, J., Mackey, M., (1990) Phys. Rev. A, 41, p. 6992. , PLRAAN 1050-2947 10.1103/PhysRevA.41.6992; Ermentrout, B., Terman, D., (2010) Mathematical Foundations of Neuroscience, , (Springer, Berlin); E. Wallace, arXiv:1004.4280Wallace, E., Gillespie, D.T., Sanft, K.R., Petzold, L.R., (unpublished)Buice, M.A., Cowan, J.D., Field-theoretic approach to fluctuation effects in neural networks (2007) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 75 (5), p. 051919. , http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org: PhysRevE.75.051919&metadataPrefix=oai_apsmeta_2, DOI 10.1103/PhysRevE.75.051919; Benayoun, M., Cowan, J.D., Van Drongelen, W., Wallace, E., (2010) Plos Computat. Biol., 6, pp. e1000846. , 1553-7358 10.1371/journal.pcbi.1000846; Bressloff, P.C., Lai, Y.M., (2011) J. Math. Neurosci., 1, p. 2. , 2190-8567 10.1186/2190-8567-1-2; Swadlow, H.A., Waxman, S.G., (2012) Scholarpedia, 7, p. 1451. , 1941-6016 10.4249/scholarpedia.1451; Boashash, B., (1992) Proc. IEEE, 80, p. 540. , IEEPAD 0018-9219 10.1109/5.135378; Nikolić, D., Fries, P., Singer, W., (2013) Trends Cogn. Sci., 17, p. 54. , 1364-6613 10.1016/j.tics.2012.12.003; Bressloff, P.C., (2010) Phys. Rev. e, 82, p. 051903. , PLEEE8 1539-3755 10.1103/PhysRevE.82.051903; Buice, M.A., Chow, C.C., (2013) Plos Computat. Biol., 9, pp. e1002872. , 1553-7358 10.1371/journal.pcbi.1002872; Bressloff, P.C., Newby, J.M., (2013) SIAM J. Appl. Dyn. Syst., 12, p. 1394. , 10.1137/120898978; Tsimring, L.S., Pikovsky, A., (2001) Phys. Rev. Lett., 87, p. 250602. , PRLTAO 0031-9007 10.1103/PhysRevLett.87.250602; Gillespie, D.T., Stochastic simulation of chemical kinetics (2007) Annual Review of Physical Chemistry, 58, pp. 35-55. , DOI 10.1146/annurev.physchem.58.032806.104637; Gillespie, D.T., (2000) J. Chem. Phys., 113, p. 297. , JCPSA6 0021-9606 10.1063/1.481811; Gillespie, D.T., Approximate accelerated stochastic simulation of chemically reacting systems (2001) Journal of Chemical Physics, 115 (4), pp. 1716-1733. , DOI 10.1063/1.1378322",
year = "2014",
doi = "10.1103/PhysRevE.90.012702",
language = "English",
volume = "90",
journal = "Physical Review E",
issn = "2470-0045",
publisher = "American Physical Society",
number = "1",

}

TY - JOUR

T1 - Linear noise approximation for oscillations in a stochastic inhibitory network with delay

AU - Dumont, Grégory

AU - Northoff, Georg Franz Josef

AU - Longtin, André

N1 - Cited By :2 Export Date: 11 May 2016 CODEN: PLEEE Funding Details: HDRF, Hope for Depression Research Foundation Funding Details: NSERC, Hope for Depression Research Foundation References: Abbott, L., Rajan, K., Sompolinksy, H., (2011) The Dynamic Brain: An Exploration of Neuronal Variability and Its Functional Significance, , (Oxford University Press, New York); Churchland, M.M., Abbott, L.F., (2012) Nat. Neurosci., 15, p. 1472. , NANEFN 1097-6256 10.1038/nn.3247; Faisal, A.A., Selen, L.P.J., Wolpert, D.M., Noise in the nervous system (2008) Nature Reviews Neuroscience, 9 (4), pp. 292-303. , DOI 10.1038/nrn2258, PII NRN2258; Lindner, B., Doiron, B., Longtin, A., Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback (2005) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 72 (6), p. 061919. , http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix= oai_apsmeta_2&set=journal:PRE:71, DOI 10.1103/PhysRevE.72.061919; Zillmer, R., Brunel, N., Hansel, D., (2009) Phys. Rev. e, 79, p. 031909. , PLEEE8 1539-3755 10.1103/PhysRevE.79.031909; Wallace, E., Benayoun, M., Van Drongelen, W., Cowan, J.D., (2011) Plos One, 6, pp. e14804. , 1932-6203 10.1371/journal.pone.0014804; Bressloff, P.C., (2009) SIAM J. Appl. Math., 70, p. 1488. , SMJMAP 0036-1399 10.1137/090756971; McKane, A.J., Newman, T.J., Predator-prey cycles from resonant amplification of demographic stochasticity (2005) Physical Review Letters, 94 (21), pp. 1-4. , http://oai.aps.org/oai/?verb=ListRecords&metadataPrefix= oai_apsmeta_2&set=journal:PRL:94, DOI 10.1103/PhysRevLett.94.218102, 218102; Alonso, D., McKane, A.J., Pascual, M., Stochastic amplification in epidemics (2007) Journal of the Royal Society Interface, 4 (14), pp. 575-582. , DOI 10.1098/rsif.2006.0192; Kukjin Kang, M.S., Henrie, J.A., Shapley, R., (2010) J. Comput. Neurosci., 29, p. 495. , JCNEFR 0929-5313 10.1007/s10827-009-0190-2; Buzsáki, G., Wang, X.-J., (2012) Annu. Rev. Neurosci., 35, p. 203. , ARNSD5 0147-006X 10.1146/annurev-neuro-062111-150444; Dumont, G., Henry, J., (2013) J. Math. Biol., 67, p. 453. , JMBLAJ 0303-6812 10.1007/s00285-012-0554-5; Dumont, G., Henry, J., (2013) Bull. Math. Biol., 75, p. 629. , BMTBAP 0092-8240 10.1007/s11538-013-9823-8; Cáceres, M.J., Carrillo, J.A., Perthame, B., (2011) J. Math. Neurosci., 1, p. 8567. , 10.1186/2190-8567-1-7; Brunel, N., Hakim, V., (1999) Neural Computat., 11, p. 1621. , NEUCEB 0899-7667 10.1162/089976699300016179; Stiefel, K.M., Englitz, B., Sejnowski, T.J., (2013) Proc. Natl. Acad. Sci. (USA), 110, p. 7886. , PNASA6 0027-8424 10.1073/pnas.1305219110; Erneux, T., Applied Delay Differential Equations (2009) Surveys and Tutorials in the Applied Mathematical Sciences, 3. , Springer, Berlin; Klosek, M.M., Kuske, R., Multiscale analysis of stochastic delay differential equations (2005) Multiscale Modeling and Simulation, 3 (3), pp. 706-729. , DOI 10.1137/030601375; Longtin, A., Milton, J., Bos, J., Mackey, M., (1990) Phys. Rev. A, 41, p. 6992. , PLRAAN 1050-2947 10.1103/PhysRevA.41.6992; Ermentrout, B., Terman, D., (2010) Mathematical Foundations of Neuroscience, , (Springer, Berlin); E. Wallace, arXiv:1004.4280Wallace, E., Gillespie, D.T., Sanft, K.R., Petzold, L.R., (unpublished)Buice, M.A., Cowan, J.D., Field-theoretic approach to fluctuation effects in neural networks (2007) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 75 (5), p. 051919. , http://oai.aps.org/oai?verb=GetRecord&Identifier=oai:aps.org: PhysRevE.75.051919&metadataPrefix=oai_apsmeta_2, DOI 10.1103/PhysRevE.75.051919; Benayoun, M., Cowan, J.D., Van Drongelen, W., Wallace, E., (2010) Plos Computat. Biol., 6, pp. e1000846. , 1553-7358 10.1371/journal.pcbi.1000846; Bressloff, P.C., Lai, Y.M., (2011) J. Math. Neurosci., 1, p. 2. , 2190-8567 10.1186/2190-8567-1-2; Swadlow, H.A., Waxman, S.G., (2012) Scholarpedia, 7, p. 1451. , 1941-6016 10.4249/scholarpedia.1451; Boashash, B., (1992) Proc. IEEE, 80, p. 540. , IEEPAD 0018-9219 10.1109/5.135378; Nikolić, D., Fries, P., Singer, W., (2013) Trends Cogn. Sci., 17, p. 54. , 1364-6613 10.1016/j.tics.2012.12.003; Bressloff, P.C., (2010) Phys. Rev. e, 82, p. 051903. , PLEEE8 1539-3755 10.1103/PhysRevE.82.051903; Buice, M.A., Chow, C.C., (2013) Plos Computat. Biol., 9, pp. e1002872. , 1553-7358 10.1371/journal.pcbi.1002872; Bressloff, P.C., Newby, J.M., (2013) SIAM J. Appl. Dyn. Syst., 12, p. 1394. , 10.1137/120898978; Tsimring, L.S., Pikovsky, A., (2001) Phys. Rev. Lett., 87, p. 250602. , PRLTAO 0031-9007 10.1103/PhysRevLett.87.250602; Gillespie, D.T., Stochastic simulation of chemical kinetics (2007) Annual Review of Physical Chemistry, 58, pp. 35-55. , DOI 10.1146/annurev.physchem.58.032806.104637; Gillespie, D.T., (2000) J. Chem. Phys., 113, p. 297. , JCPSA6 0021-9606 10.1063/1.481811; Gillespie, D.T., Approximate accelerated stochastic simulation of chemically reacting systems (2001) Journal of Chemical Physics, 115 (4), pp. 1716-1733. , DOI 10.1063/1.1378322

PY - 2014

Y1 - 2014

N2 - Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm. © Published by the American Physical Society.

AB - Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm. © Published by the American Physical Society.

KW - Computation theory

KW - Differential equations

KW - Neural networks

KW - Power spectrum

KW - Stochastic systems

KW - Additive Gaussian white noise

KW - Delay differential equations

KW - Infinite network size limit

KW - Intrinsic randomness

KW - Linear noise approximation

KW - Population activities

KW - Probabilistic descriptions

KW - Theoretical expression

KW - Low noise amplifiers

KW - biological model

KW - cytology

KW - nerve cell

KW - nerve cell inhibition

KW - nerve cell network

KW - physiology

KW - statistical model

KW - statistics

KW - Linear Models

KW - Models, Neurological

KW - Nerve Net

KW - Neural Inhibition

KW - Neurons

KW - Stochastic Processes

U2 - 10.1103/PhysRevE.90.012702

DO - 10.1103/PhysRevE.90.012702

M3 - Article

C2 - 25122330

VL - 90

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 1

ER -