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Volumn 66, Issue 5, 2002, Pages 19-

Population dynamics of interacting spiking neurons

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; PERTURBATION TECHNIQUES; RANDOM PROCESSES; SPURIOUS SIGNAL NOISE; TRANSFER FUNCTIONS;

EID: 85036133906     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.051917     Document Type: Article
Times cited : (224)

References (50)
  • 1
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    • A more complete historical account of the subject can be found, for example, in 7
    • A more complete historical account of the subject can be found, for example, in 7.
  • 11
    • 85036158015 scopus 로고    scopus 로고
    • B. W. Knight, D. Manin, and L. Sirovich, in Proceedings of Symposium on Robotics and Cybernetics, Lille-France, July 9-12, edited by E. C. Gerf (1996)
    • B. W. Knight, D. Manin, and L. Sirovich, in Proceedings of Symposium on Robotics and Cybernetics, Lille-France, July 9-12, edited by E. C. Gerf (1996).
  • 18
    • 85036169844 scopus 로고    scopus 로고
    • Conditions favoring independence are sparse connectivity, a random distribution of synaptic efficacies which can compensate for sparsness when connectivity is high or, in general, a suitable source of quenched randomness affecting the communication among the neurons
    • Conditions favoring independence are sparse connectivity, a random distribution of synaptic efficacies which can compensate for sparsness when connectivity is high or, in general, a suitable source of quenched randomness affecting the communication among the neurons.
  • 19
    • 85036160352 scopus 로고    scopus 로고
    • H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1984)
    • H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1984).
  • 20
    • 85036409267 scopus 로고    scopus 로고
    • We remark that in deriving the above Fokker-Planck equation in the general case (2.2) of a multiplicative noise, the condition (Formula presented) plays a crucial role, since if the latter condition is not fulfilled, a further, “noise-induced” drift appears in the A coefficient of the Fokker-Planck equation
    • We remark that in deriving the above Fokker-Planck equation in the general case (2.2) of a multiplicative noise, the condition (Formula presented) plays a crucial role, since if the latter condition is not fulfilled, a further, “noise-induced” drift appears in the A coefficient of the Fokker-Planck equation.
  • 24
    • 85036278722 scopus 로고    scopus 로고
    • The ratio between the standard deviation and the mean of the interspike intervals
    • The ratio between the standard deviation and the mean of the interspike intervals.
  • 25
    • 85036281017 scopus 로고    scopus 로고
    • The irrelevance of the detailed statistics of the interspike intervals for the collective dynamics of the network has been recognized in a different context in 50
    • The irrelevance of the detailed statistics of the interspike intervals for the collective dynamics of the network has been recognized in a different context in 50.
  • 26
    • 85036385382 scopus 로고    scopus 로고
    • Ref. 19 it is suggested (for a slightly different expansion strategy) that this is indeed the case when the terms A and B in the Fokker-Planck equation are rational functions of (Formula presented) (see 11 for an explicit example)
    • In Ref. 19 it is suggested (for a slightly different expansion strategy) that this is indeed the case when the terms A and B in the Fokker-Planck equation are rational functions of (Formula presented) (see 11 for an explicit example).
  • 27
    • 85036428520 scopus 로고    scopus 로고
    • Not to be individually considered as probability currents, since the (Formula presented)’s are not individually probability density functions
    • Not to be individually considered as probability currents, since the (Formula presented)’s are not individually probability density functions.
  • 28
    • 85036189170 scopus 로고    scopus 로고
    • As noted by several authors, the above definition of (Formula presented) is free from the difficulties that arise in the definition based on taking averages of single spike trains over small time intervals
    • As noted by several authors, the above definition of (Formula presented) is free from the difficulties that arise in the definition based on taking averages of single spike trains over small time intervals.
  • 29
    • 85036433132 scopus 로고    scopus 로고
    • Indeed, the (Formula presented) term is finite in the potentially dangerous limit (Formula presented) and the singularities possibly arising from the term in the numerator involving (Formula presented) are eliminated by the analogous term in the denominator
    • Indeed, the (Formula presented) term is finite in the potentially dangerous limit (Formula presented) and the singularities possibly arising from the term in the numerator involving (Formula presented) are eliminated by the analogous term in the denominator.
  • 30
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    • While the stability condition (Formula presented) appears superficially consistent with general and standard arguments, it applies to a special case (see below for other examples). Besides, we argue in Sec. IV that simple arguments appropriate for a first-order dynamics lead to wrong conclusions in the case under study
    • While the stability condition (Formula presented) appears superficially consistent with general and standard arguments, it applies to a special case (see below for other examples). Besides, we argue in Sec. IV that simple arguments appropriate for a first-order dynamics lead to wrong conclusions in the case under study.
  • 32
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    • A pole near one of the eigenvalues of L, driving the instability of the network for small (Formula presented) (which our analysis proves to be a special, large (Formula presented) effect) had been predicted in Ref. 22 to be a general property, on the basis of what appears as an unwarranted condition imposed on the poles near (Formula presented)
    • A pole near one of the eigenvalues of L, driving the instability of the network for small (Formula presented) (which our analysis proves to be a special, large (Formula presented) effect) had been predicted in Ref. 22 to be a general property, on the basis of what appears as an unwarranted condition imposed on the poles near (Formula presented)
  • 33
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    • H. C. Tuckwell, Introduction to Theoretical Neurobiology (Cambridge University Press, Cambridge, England, 1988), Vol. 2
    • H. C. Tuckwell, Introduction to Theoretical Neurobiology (Cambridge University Press, Cambridge, England, 1988), Vol. 2.
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    • We mention for completeness that in the small-(Formula presented) limit, in the drift-dominated regime the eigenvalues (therefore, in most conditions, the diffusion poles) have a real part (Formula presented) such that as (Formula presented) the diffusion poles bring about a marginal stability. See also 6 and 22 for a discussion of the eigenvalues of L in the limit of small (Formula presented)
    • We mention for completeness that in the small-(Formula presented) limit, in the drift-dominated regime the eigenvalues (therefore, in most conditions, the diffusion poles) have a real part (Formula presented) such that as (Formula presented) the diffusion poles bring about a marginal stability. See also 6 and 22 for a discussion of the eigenvalues of L in the limit of small (Formula presented)
  • 36
    • 85036318510 scopus 로고    scopus 로고
    • For example, for the extreme case of full connectivity, with all synapses chosen as random variables drawn from the same probability distribution, the afferent currents to two generic neurons are determined by the convolution of the same realization of a stochastic point process (the common sequence of afferent spikes) with two independent realizations of the process generating the synaptic couplings
    • For example, for the extreme case of full connectivity, with all synapses chosen as random variables drawn from the same probability distribution, the afferent currents to two generic neurons are determined by the convolution of the same realization of a stochastic point process (the common sequence of afferent spikes) with two independent realizations of the process generating the synaptic couplings.
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    • P. Dayan and L. F. Abbott, Theoretical Neuroscience (MIT Press, Cambridge, MA, 2001)
    • P. Dayan and L. F. Abbott, Theoretical Neuroscience (MIT Press, Cambridge, MA, 2001).
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    • W. Gerstner and W. Kistler, Spiking Neuron Models (Cambridge University Press, Cambridge, England, 2002)
    • W. Gerstner and W. Kistler, Spiking Neuron Models (Cambridge University Press, Cambridge, England, 2002).
  • 42
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    • It is clear from the above discussion that the characteristic times of the population are not directly determined by single neuron properties such as the membrane time constant
    • It is clear from the above discussion that the characteristic times of the population are not directly determined by single neuron properties such as the membrane time constant.
  • 44
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    • P. Del Giudice and M. Mattia, in Neurocomputing–Computational Neuroscience: Trends in Research 2001, edited by J. Bower (Elsevier Science, Amsterdam, 2001), Vols. 38–40, pp. 1175–1180
    • P. Del Giudice and M. Mattia, in Neurocomputing–Computational Neuroscience: Trends in Research 2001, edited by J. Bower (Elsevier Science, Amsterdam, 2001), Vols. 38–40, pp. 1175–1180.
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    • A. Destexhe, Z. F. Mainen, and T. J. Sejnowski, in Methods in Neuronal Modeling, edited by C. Koch and I. Segev (MIT Press, Cambridge, MA, 1998), pp. 1–25
    • A. Destexhe, Z. F. Mainen, and T. J. Sejnowski, in Methods in Neuronal Modeling, edited by C. Koch and I. Segev (MIT Press, Cambridge, MA, 1998), pp. 1–25.
  • 49
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    • Previous attempts within the diffusion approximation are reported in 9 47 takes into account the higher-order contributions in the Kramers-Moyal expansion
    • Previous attempts within the diffusion approximation are reported in 9. 47 takes into account the higher-order contributions in the Kramers-Moyal expansion.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.