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Volumn 63, Issue 1, 2001, Pages

Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients: Application to postural sway

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EID: 0007920742     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.011905     Document Type: Article
Times cited : (22)

References (82)
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    • A. K. Aziz Van Nostrand Reinhold, New York
    • H. P. McKean, Jr., in Lectures in Differential Equations, edited by A. K. Aziz (Van Nostrand Reinhold, New York, 1969), Vol. II, pp. 177–193.
    • (1969) Lectures in Differential Equations , vol.II , pp. 177-193
    • McKean, H.P.1
  • 35
    • 0033831890 scopus 로고    scopus 로고
    • Three pertinent issues concerning the modeling of brain activity: nonlinearities, time scales and neural underpinnings
    • A. Daffertshofer, T. D. Frank, C. E. Peper, and P. J. Beek, Behav. Brain Sci. 23, 400 (2000). Three pertinent issues concerning the modeling of brain activity: nonlinearities, time scales and neural underpinnings.
    • (2000) Behav. Brain Sci. , vol.23 , pp. 400
    • Daffertshofer, A.1    Frank, T.D.2    Peper, C.E.3    Beek, P.J.4
  • 51
    • 85035249572 scopus 로고    scopus 로고
    • More precisely, these solutions are stable with respect to perturbations that again yield Gaussian shaped probability densities. The approach, however, fails to address stability with respect to arbitrary perturbations. Strictly speaking, this analysis provides sufficient conditions for the instability of stationary solutions and necessary conditions for the stability
    • More precisely, these solutions are stable with respect to perturbations that again yield Gaussian shaped probability densities. The approach, however, fails to address stability with respect to arbitrary perturbations. Strictly speaking, this analysis provides sufficient conditions for the instability of stationary solutions and necessary conditions for the stability.
  • 55
    • 85035306275 scopus 로고    scopus 로고
    • fact, this stationary value represents a stable fixed point: let us rewrite Eq. (40) as the gradient dynamics of an overdamped motion, (Formula presented) with (Formula presented) The integral (Formula presented) is a monotonically increasing function that scales faster than (Formula presented) Therefore, the gradient dynamics has a unique and globally stable fixed point
    • In fact, this stationary value represents a stable fixed point: let us rewrite Eq. (40) as the gradient dynamics of an overdamped motion, (Formula presented) with (Formula presented) The integral (Formula presented) is a monotonically increasing function that scales faster than (Formula presented) Therefore, the gradient dynamics has a unique and globally stable fixed point.
  • 76
    • 85035285915 scopus 로고    scopus 로고
    • Note that Eq. (53) is solved by Gaussian time-dependent probability densities whose BGS entropies can be expressed in terms of their variances. The BGS entropies, in turn, can be related by Fitts law to neural processing times
    • Note that Eq. (53) is solved by Gaussian time-dependent probability densities whose BGS entropies can be expressed in terms of their variances. The BGS entropies, in turn, can be related by Fitts law to neural processing times.
  • 79
    • 85035265390 scopus 로고    scopus 로고
    • W. Feller, An introduction to Probability Theory and its Applications, 2nd ed. (Wiley, New York, 1971)
    • W. Feller, An introduction to Probability Theory and its Applications, 2nd ed. (Wiley, New York, 1971).
  • 81
    • 85035299050 scopus 로고    scopus 로고
    • We have (Formula presented) with (Formula presented) and (Formula presented) for all (Formula presented) Since (Formula presented) for (Formula presented) we find that (Formula presented) and (Formula presented) Consequently, the Lindeberg condition holds for (Formula presented) Roughly speaking, if we can neglect the asymptotic parts of (Formula presented) then we can also neglect the asymptotic parts of the corresponding squeezed normalized probability density (Formula presented) with (Formula presented) e.g., consider (Formula presented)
    • We have (Formula presented) with (Formula presented) and (Formula presented) for all (Formula presented) Since (Formula presented) for (Formula presented) we find that (Formula presented) and (Formula presented) Consequently, the Lindeberg condition holds for (Formula presented) Roughly speaking, if we can neglect the asymptotic parts of (Formula presented) then we can also neglect the asymptotic parts of the corresponding squeezed normalized probability density (Formula presented) with (Formula presented) e.g., consider (Formula presented)


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