Ergodicity Coefficients Defined by Vector Norms

Ipsen, Ilse C. F.; Selee, Teresa M.

doi:10.1137/090752948

Cited by 64 publications

(66 citation statements)

References 69 publications

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“…(18)- (21) for finite values of K, since the trapezoidal rule and quadrature are used for the numerical integration). Since the discretized SPTO is a positive matrix, the product of discretized SPTOs is weakly ergodic [44,45]. (15) and (16)], and the SPTO P K, ,A i ,I i expresses the relationship between the density just before the ith impulse to that just before the (i + 1)th impulse: The weak ergodicity leads to the following property for any densities h and h : (29) where H n,n 0 = P K, ,A n ,I n P K, ,A n−1 ,I n−1 .…”

Section: E Contribution To the Current State From The Past Statesmentioning

confidence: 99%

Global dynamics of a stochastic neuronal oscillator

Yamanobe

2013

Phys. Rev. E

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Nonlinear oscillators have been used to model neurons that fire periodically in the absence of input. These oscillators, which are called neuronal oscillators, share some common response structures with other biological oscillations such as cardiac cells. In this study, we analyze the dependence of the global dynamics of an impulse-driven stochastic neuronal oscillator on the relaxation rate to the limit cycle, the strength of the intrinsic noise, and the impulsive input parameters. To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. Previously, we derived the Markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane. Here, we construct a Markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle. In both cases, the response of the stochastic neuronal oscillator to time-varying impulses is described by a product of Markov operators. Furthermore, we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density. Specifically, we analyze the dynamics of the number of spikes per unit time based on the properties of the Markov operators. Each Markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions. This allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator, which we show to be based on the dependence of the oscillator on past activity. Our analysis shows how the duration of the past neuronal activity depends on the relaxation rate, the noise strength, and the impulsive input parameters.

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Section: E Contribution To the Current State From The Past Statesmentioning

confidence: 99%

Global dynamics of a stochastic neuronal oscillator

Yamanobe

2013

Phys. Rev. E

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“…In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some sufficient conditions for weak and strong ergodicity of nonhomogeneous Markov processes are given and estimates of the rate of convergence are proved.…”

Section: Introductionmentioning

confidence: 99%

“…In the simplest case, when all factors in the products are identical to the same stochastic operator T , ergodicity corresponds to the investigation of iterations of T . The Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of such products (see [17] for review) . Therefore, we will define such a ergodicity coefficient of a positive mapping defined on nonassociative L 1 -space, and study its properties.…”

Section: Introductionmentioning

confidence: 99%

“…Many ergodic type theorems have been proved for Markov operators acting in nonassociative and noncommutative L p -spaces (see for example, [4, 5, 20, 21, 8, 18]) On the other hand, nonhomogeneous Markov processes with general state space have become a subject of interest due to their applications in many branches of mathematics and natural sciences. In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some…”

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confidence: 99%

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On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Mukhamedov

2013

J. Phys.: Conf. Ser.

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Abstract. In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on non-associative L 1 -spaces associated with semi-finite JBWalgebras. Such results extends the well-known classical ones to a non-associative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a necessary and sufficient conditions for such processes to satisfy the weak ergodicity. IntroductionIt is known (see [23]) that the investigations of asymptotical behavior of iterations of Markov operators on commutative L 1 -spaces are very important. On the other hand, these investigations are related with several notions of ergodicity of L 1 -contractions of measure spaces. To the investigation of such ergodic properties of Markov operators were devoted lots of papers (see for example, [7,23]). On the other hand, such kind of operators were studied in noncommutative settings. Since, the study of quantum dynamical systems has had an impetuous growth in the last years, in view of natural applications to various field of mathematics and physics. It is then of interest to understand among the various ergodic properties, which ones survive and are meaningful by passing from the classical to the quantum case. Due to noncommutativity, the latter situation is much more complicated than the former. The reader is referred e.g. to [2,15,16,18,29] for further details relative to some differences between the classical and the quantum situations. It is therefore natural to study the possible generalizations to quantum case of the various ergodic properties known for classical dynamical systems. One of the generalizations of noncommutative algebras is Jordan algebras. Note that Jordan Banach algebras [3,6] are a non-associative real analogue of von Neumann algebras. The existence of exceptional JBW -algebras does not allow one to use the ideas and methods from von Neumann algebras. The motivation of these investigations arose in quantum statistical mechanics and quantum field theory (see, [9]). Mostly, in those investigations homogeneous Markov processes were considered. Many ergodic type theorems have been proved for Markov operators acting in nonassociative and noncommutative L p -spaces (see for example, [4, 5, 20, 21, 8, 18]) On the other hand, nonhomogeneous Markov processes with general state space have become a subject of interest due to their applications in many branches of mathematics and natural sciences. In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some

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“…For more on Markov's work see [4]. The column and row variation appear in research literature under various names; see [1,Section 3.3].…”

Section: Definitionsmentioning

confidence: 99%