scholarly journals Avalanches in a short-memory excitable network

2021 ◽  
Vol 53 (3) ◽  
pp. 609-648
Author(s):  
Reza Rastegar ◽  
Alexander Roitershtein
Keyword(s):  
Dynamical System ◽  
Epidemic Model ◽  
Branching Process ◽  
Rates Of Convergence ◽  
Large Network ◽  
Exact Relation ◽  
Short Memory ◽  
Rigorous Bounds ◽  
The One ◽  
Deterministic Dynamical System

AbstractWe study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced by Larremore et al. (Phys. Rev. E, 2012) and is mathematically equivalent to an endemic variation of the Reed–Frost epidemic model introduced by Longini (Math. Biosci., 1980). Two types of heuristic approximation are frequently used for models of this type in applications: a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

scholarly journals Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions

2014 ◽  
Vol 46 (01) ◽  
pp. 241-255 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Epidemic Model ◽  
Branching Process ◽  
Endemic Equilibrium ◽  
Infectious Period ◽  
Sis Epidemic Model ◽  
Branching Process With Immigration ◽  
Sis Epidemic ◽  
Sis Epidemics ◽  
Surprising Observation

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

scholarly journals Human Creativity and Consciousness: Unintended Consequences of the Brain’s Extraordinary Energy Efficiency?

2020 ◽  
Author(s):  
Tim Palmer
Keyword(s):  
Energy Efficiency ◽  
Human Brain ◽  
Quantum Physics ◽  
Unintended Consequences ◽  
Large Network ◽  
Active Neuron ◽  
Available Energy ◽  
Human Creativity ◽  
The One ◽  
The Brain

It is proposed that both human creativity and human consciousness are (unintended) consequences of the human brain’s extraordinary energy efficiency. The topics of creativity and consciousness are treated separately, though have a common sub-structure. It is argued that creativity arises from a synergy between two cognitive modes of the human brain (which broadly coincide with Kahneman’s Systems 1 and 2). In the first, available energy is spread across a relatively large network of neurons. As such, the amount of energy per active neuron is so small that the operation of such neurons is susceptible to thermal (ultimately quantum decoherent) noise. In the second, available energy is focussed on a small enough subset of neurons to guarantee a deterministic operation. An illustration of how this synergy can lead to creativity with implications for computing in silicon are discussed. Starting with a discussion of the concept of free will, the notion of consciousness is defined in terms of an awareness of what are perceived to be nearby counterfactual worlds in state space. It is argued that such awareness arises from an interplay between our memories on the one hand, and quantum physical mechanisms (where, unlike in classical physics, nearby counterfactual worlds play an indispensable dynamical role) in the ion channels of neural networks. As with the brain’s susceptibility to noise, it is argued that in situations where quantum physics plays a role in the brain, it does so for reasons of energy efficiency. As an illustration of this definition of consciousness, a novel proposal is outlined as to why quantum entanglement appears so counter-intuitive.

On the Rates of Convergence of Chlodovsky–Durrmeyer Operators and their Bézier Variant

2009 ◽  
Vol 16 (4) ◽  
pp. 693-704
Author(s):  
Harun Karsli ◽  
Paulina Pych-Taberska
Keyword(s):  
Recent Result ◽  
Pointwise Convergence ◽  
Rates Of Convergence ◽  
Durrmeyer Operators ◽  
The One ◽  
Special Case ◽  
Appl Math ◽  
Bézier Variant

Abstract We consider the Bézier variant of Chlodovsky–Durrmeyer operators 𝐷𝑛,α for functions 𝑓 measurable and locally bounded on the interval [0,∞). By using the Chanturia modulus of variation we estimate the rate of pointwise convergence of (𝐷𝑛,α 𝑓) (𝑥) at those 𝑥 > 0 at which the one-sided limits 𝑓(𝑥+), 𝑓(𝑥–) exist. In the special case α = 1 the recent result of [Ibikli, Karsli, J. Inequal. Pure Appl. Math. 6: 12, 2005] concerning the Chlodovsky–Durrmeyer operators 𝐷𝑛 is essentially improved and extended to more general classes of functions.

A coupling of finite particle systems

1993 ◽  
Vol 30 (01) ◽  
pp. 258-262 ◽  
Author(s):  
T. S. Mountford
Keyword(s):  
Invariant Measure ◽  
Branching Process ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Initial Configuration ◽  
Limit Measure ◽  
One Dimensional ◽  
Interacting Particle ◽  
The One ◽  
Finite Particle

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

scholarly journals Spatial long memory

2019 ◽  
Vol 3 (1) ◽  
pp. 243-256
Author(s):  
Peter M. Robinson
Keyword(s):  
Time Series ◽  
Statistical Modeling ◽  
Spatial Data ◽  
Long Memory ◽  
The Other ◽  
Future Prospects ◽  
Memory Time ◽  
Short Memory ◽  
The One ◽  
Relevant Work

AbstractWe discuss developments and future prospects for statistical modeling and inference for spatial data that have long memory. While a number of contributons have been made, the literature is relatively small and scattered, compared to the literatures on long memory time series on the one hand, and spatial data with short memory on the other. Thus, over several topics, our discussions frequently begin by surveying relevant work in these areas that might be extended in a long memory spatial setting.

scholarly journals Human Creativity and Consciousness: Unintended Consequences of the Brain’s Extraordinary Energy Efficiency?

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 281
Author(s):  
Tim Palmer
Keyword(s):  
Energy Efficiency ◽  
Quantum Physics ◽  
Unintended Consequences ◽  
Large Network ◽  
Available Energy ◽  
Human Creativity ◽  
Creative Computing ◽  
The One ◽  
Definition Of ◽  
Susceptibility To Noise

It is proposed that both human creativity and human consciousness are (unintended) consequences of the human brain’s extraordinary energy efficiency. The topics of creativity and consciousness are treated separately, though have a common sub-structure. It is argued that creativity arises from a synergy between two cognitive modes of the human brain (which broadly coincide with Kahneman’s Systems 1 and 2). In the first, available energy is spread across a relatively large network of neurons, many of which are small enough to be susceptible to thermal (ultimately quantum decoherent) noise. In the second, available energy is focussed on a smaller subset of larger neurons whose action is deterministic. Possible implications for creative computing in silicon are discussed. Starting with a discussion of the concept of free will, the notion of consciousness is defined in terms of an awareness of what are perceived to be nearby counterfactual worlds in state space. It is argued that such awareness arises from an interplay between memories on the one hand, and quantum physical mechanisms (where, unlike in classical physics, nearby counterfactual worlds play an indispensable dynamical role) in the ion channels of neural networks, on the other. As with the brain’s susceptibility to noise, it is argued that in situations where quantum physics plays a role in the brain, it does so for reasons of energy efficiency. As an illustration of this definition of consciousness, a novel proposal is outlined as to why quantum entanglement appears to be so counter-intuitive.

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Harry Dankowicz ◽  
Frank Schilder
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
The Other ◽  
Extended Formulation ◽  
Hybrid Dynamical System ◽  
Families Of Periodic Orbits ◽  
Continuation Problem ◽  
The One ◽  
Additional Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

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