scholarly journals Hawkes processes with variable length memory and an infinite number of components

2017 ◽  
Vol 49 (1) ◽  
pp. 84-107 ◽  
Author(s):  
Pierre Hodara ◽  
Eva Löcherbach
Keyword(s):  
Infinite Number ◽  
Variable Length ◽  
Neural Nets ◽  
Interaction Graph ◽  
Hawkes Processes ◽  
Number Of Components ◽  
Stationary Version ◽  
Biological Neural Nets ◽  
Saturation Threshold ◽  
Type Decomposition

Abstract In this paper we propose a model for biological neural nets where the activity of the network is described by Hawkes processes having a variable length memory. The particularity in this paper is that we deal with an infinite number of components. We propose a graphical construction of the process and build, by means of a perfect simulation algorithm, a stationary version of the process. To implement this algorithm, we make use of a Kalikow-type decomposition technique. Two models are described in this paper. In the first model, we associate to each edge of the interaction graph a saturation threshold that controls the influence of a neuron on another. In the second model, we impose a structure on the interaction graph leading to a cascade of spike trains. Such structures, where neurons are divided into layers, can be found in the retina.

CLASSICAL LINK INVARIANTS AND THE HAWAIIAN EARRINGS SPACE

1992 ◽  
Vol 01 (04) ◽  
pp. 327-342
Author(s):  
TIM D. COCHRAN
Keyword(s):  
Infinite Number ◽  
Number Of Components ◽  
Link Invariants ◽  
Classical Link

We show that, in search of link invariants more discriminating than Milnor's [Formula: see text]-invariants, one is naturally led to consider seemingly pathological objects such as links with an infinite number of components and the join of an infinite number of circles (Hawaiian earrings space). We define an infinite homology boundary link, and show that any finite sublink of an infinite homology boundary link has vanishing Milnor's invariants. Moreover, all links known to have vanishing Milnor's invariants are finite sublinks of infinite homology boundary links. We show that the exterior of an infinite homology boundary link admits a map to the Hawaiian earrings space, and that this may be employed to get a factorization of K. E. Orr's omega-invariant through a rather simple space.

ATTENTION-LOCKED COMPUTATION WITH CHAOTIC NEURAL NETS

2004 ◽  
Vol 14 (02) ◽  
pp. 737-760 ◽  
Author(s):  
CARLOS LOURENÇO
Keyword(s):  
Neural Network ◽  
Information Processing ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Chaotic Dynamics ◽  
Neural Nets ◽  
Processing Capacity ◽  
Unstable Periodic Orbits ◽  
Pattern Processing ◽  
Static Pattern

We review a neural network model based on chaotic dynamics [Babloyantz & Lourenço, 1994, 1996] and provide a detailed discussion of its biological and computational relevance. Chaos can be viewed as a "reservoir" containing an infinite number of unstable periodic orbits. In our approach, the periodic orbits are used as coding devices. By considering a large enough number of them, one can in principle expand the information processing capacity of small or moderate-size networks. The system is most of the time in an undetermined state characterized by a chaotic attractor. Depending on the type of an external stimulus, the dynamics is stabilized into one of the available periodic orbits, and the system is then ready to process information. This corresponds to the system being driven into an "attentive" state. We show that, apart from static pattern processing, the model is capable of dealing with moving stimuli. We especially consider in this paper the case of transient visual stimuli, which has a clear biological relevance. The advantages of chaos over more regular regimes are discussed.

A Possible Infinite Number of Components in a Single Crystalline Phase: On the Isomorphism of Brivaracetam–Guest Molecules

2018 ◽  
Vol 18 (9) ◽  
pp. 4807-4810 ◽  
Author(s):  
Nicolas Couvrat ◽  
Morgane Sanselme ◽  
Yohann Cartigny ◽  
Frederic De Smet ◽  
Sandrine Rome ◽  
...  
Keyword(s):  
Infinite Number ◽  
Crystalline Phase ◽  
Guest Molecules ◽  
Single Crystalline ◽  
Number Of Components

scholarly journals Minimally Supported Frequency (MSF) d-Dilation Wavelets

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1284
Author(s):  
Aparna Vyas ◽  
Gibak Kim
Keyword(s):  
Infinite Number ◽  
Geometric Construction ◽  
The Other ◽  
Wavelet Sets ◽  
Number Of Components ◽  
Wavelet Set

In this paper, we provide a geometric construction of a symmetric 2n-interval minimally supported frequency (MSF) d-dilation wavelet set with d∈(1,∞) and characterize all symmetric d-dilation wavelet sets. We also provide two special kinds of symmetric d-dilation wavelet sets, one of which has 4m-intervals whereas the other has (4m+2)-intervals, for m∈N. In addition, we construct a family of d-dilation wavelet sets that has an infinite number of components.

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