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Input space bifurcation manifolds of recurrent neural networks

Robert Haschke, Jochen J. Steil
2005 Neurocomputing  
We derive analytical expressions of local codimension-1 bifurcations for a fully connected, additive, discrete-time recurrent neural network (RNN), where we regard the external inputs as bifurcation parameters  ...  The complexity of the bifurcation diagrams obtained increases exponentially with the number of neurons.  ...  Acknowledgements: This work was supported by the DFG grants GK-231.  ... 
doi:10.1016/j.neucom.2004.11.030 fatcat:kft6ielzpvgotp537xo7pyld6u

Interpreting Recurrent Neural Networks Behaviour via Excitable Network Attractors

Andrea Ceni, Peter Ashwin, Lorenzo Livi
2019 Cognitive Computation  
Simulations conducted on a controlled benchmark task confirm the relevance of these attractors for interpreting the behaviour of recurrent neural networks, at least for tasks that involve learning a finite  ...  of a neural network while solving tasks.  ...  Acknowledgements We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.  ... 
doi:10.1007/s12559-019-09634-2 fatcat:k4l5iqd6znfknl2hiogbtfzcuy

Dynamics of Fuzzy-Rough Cognitive Networks

István Á. Harmati
2021 Symmetry  
It will be shown that their mathematical properties highly depend on the size of the network, i.e., there are structural differences between the long-term behaviour of FRCN models of different size, which  ...  Although there are many applications on fuzzy cognitive maps and recently for FRCNS, only a very limited number of studies discuss the theoretical issues of these models.  ...  Finally, we provide a geometrical reasoning of the structure of fixed points. Consider two fixed points of type FP 1 , i.e., they have one high and N − 1 low coordinates.  ... 
doi:10.3390/sym13050881 fatcat:ghwsxjghsffe3psnlnxnnzv5hu

Conditions for wave trains in spiking neural networks

Johanna Senk, Karolína Korvasová, Jannis Schuecker, Espen Hagen, Tom Tetzlaff, Markus Diesmann, Moritz Helias
2020 Physical Review Research  
Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations  ...  We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability.  ...  While the neural-field model describes neural activity as a quantity that is continuous in space and time, the spiking model assumes a network of recurrently connected spiking model neurons in discrete  ... 
doi:10.1103/physrevresearch.2.023174 fatcat:olzbii5ov5f3vbkmvkfpjnzhcu

Chaotic Dynamical Behavior of Recurrent Neural Network

A. Zerroug, L. Terrissa, A. Faure
2013 Annual Review of Chaos Theory, Bifurcations and Dynamical Systems  
In order to reduce the degree of complexity of this work, we have considered in this paper a fully connected neural network of two discrete neurons.  ...  On account of their role played in the fundamental biological rhythms and by considering their potential use in information processing, the dynamical properties of an artificial neural network are particularly  ...  the origin is not a stable fixed point.  ... 
doaj:6f7b4f9cf53042259da17f8ddc5bedff fatcat:saxk2j2lmvfi3j5z5pvqvkrpmy

Conditions for wave trains in spiking neural networks [article]

Johanna Senk, Karolína Korvasová, Jannis Schuecker, Espen Hagen, Tom Tetzlaff, Markus Diesmann, Moritz Helias
2019 arXiv   pre-print
Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations  ...  We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability.  ...  " of the RWTH Aachen University, and the Research Council of Norway (NFR) through COBRA (grant No 250128).  ... 
arXiv:1801.06046v2 fatcat:yic5qssqqrf5nhpowdw5nbb6cu

Chaotic Dynamics in Iterated Map Neural Networks with Piecewise Linear Activation Function [article]

Sitabhra Sinha
1999 arXiv   pre-print
The paper examines the discrete-time dynamics of neuron models (of excitatory and inhibitory types) with piecewise linear activation functions, which are connected in a network.  ...  The properties of a pair of neurons (one excitatory and the other inhibitory) connected with each other, is studied in detail.  ...  Figure 6 . 6 Bifurcation diagram for k = k ′ = 1 at a = 4.0. Fig. Fig. 6 shows the bifurcation structure of the map for a = 4. For b/a < 0.25, the fixed point Z * 2 is stable.  ... 
arXiv:chao-dyn/9903009v1 fatcat:sfhabotfbvf2dlsezgco2is33a

Structure and Dynamics of Random Recurrent Neural Networks

Hugues Berry, Mathias Quoy
2006 Adaptive Behavior  
It is possible to store information in these networks through hebbian learning. Eventually, learning "destroys" the dynamics and leads to a fixed point attractor.  ...  We investigate here the structural change in the networks through learning, and show a "small-world" effect.  ...  Model A random recurrent neural network is a set of N fully connected neurons.  ... 
doi:10.1177/105971230601400204 fatcat:g3pxnp7jdnawvjnzx36b3u55mi

Characterization of periodic attractors in neural ring networks

Frank Pasemann
1995 Neural Networks  
The paper presents a discussion of parameterized discrete dynamics of neural ring networks. For specific parameter domains stable periodic orbits coexist.  ...  The dynamical effects of inhibitory connections are analysed, and a characterization of attractors in terms of their "firing pattern" is presented.  ...  Neural n-ring networks The standard additive nonlinear neuron model is chosen, i.e. the activity a i of unit i is given by the sum over the weighted outputs o j of units j connected to unit i plus a bias  ... 
doi:10.1016/0893-6080(94)00085-z fatcat:jpy45f5vsngrrmm7rlbsolf5mi

Chaotic dynamics and the geometry of the error surface in neural networks

Y.J. Choie, S. Kim, C.N. Lee
1992 Physica D : Non-linear phenomena  
Ahlers We have observed transient periodic and chaotic oscillations in the learning process of a class of multi-layered neural networks called perceptrons.  ...  This illustrates that transient dynamics can be used to extract information on the geometry of the error surface in neural networks. 0167-2789/92/$05.00 0 1992 -Elsevier Science Publishers B.V.  ...  We would also like to thank referees for helpful remarks and correcting several mistakes in the original manuscript.  ... 
doi:10.1016/0167-2789(92)90191-o fatcat:v3b2clrsfzbqbiq6kiwpoqt2hy

From neuron to neural networks dynamics

B. Cessac, M. Samuelides
2007 The European Physical Journal Special Topics  
We discuss some models reducing the Hodgkin-Huxley model to a two dimensional dynamical system, keeping one of the main feature of the neuron: its excitability.  ...  A last section is devoted to a detailed example of recurrent model where we go in deep in the analysis of the dynamics and discuss the effect of learning on the neuron dynamics.  ...  Obviously, representing the synaptic connections between two neurons by an edge between two nodes is certainly a very rough way of sketching a neural network structure.  ... 
doi:10.1140/epjst/e2007-00058-2 fatcat:ajxl6csjdzbrng3jrouf4b6swa

From Neuron to Neural Networks dynamics [article]

B. Cessac, M. Samuelides
2006 arXiv   pre-print
We discuss some models reducing the Hodgkin-Huxley model to a two dimensional dynamical system, keeping one of the main feature of the neuron: its excitability.  ...  A last section is devoted to a detailed example of recurrent model where we go in deep in the analysis of the dynamics and discuss the effect of learning on the neuron dynamics.  ...  Obviously, representing the synaptic connections between two neurons by an edge between two nodes is certainly a very rough way of sketching a neural network structure.  ... 
arXiv:nlin/0609038v1 fatcat:del6kbqslza2torwpzi6rxveva

Biologically Plausible Artificial Neural Networks [chapter]

Joao Luis Garcia Rosa
2013 Artificial Neural Networks - Architectures and Applications  
For some configurations of m points, a straight line is able to separate them in two classes (figures 3 and 4).  ...  We may add a fourth type: a network which considers populations of neurons instead of individual ones and the existence of chaotic oscillations, perceived by electroencephalogram (EEG) analysis.  ...  Author details João Luís Garcia Rosa Bioinspired Computing Laboratory (BioCom), Department of Computer Science, University of São Paulo at São Carlos, Brazil References  ... 
doi:10.5772/54177 fatcat:xeavo23zinczve6n6ovdkf226q

Attractive Periodic Sets in Discrete-Time Recurrent Networks (with Emphasis on Fixed-Point Stability and Bifurcations in Two-Neuron Networks)

Peter Tiňo, Bill G. Horne, C. Lee Giles
2001 Neural Computation  
We perform a detailed xed-point analysis of two-unit recurrent neural networks with sigmoid-shaped transfer functions.  ...  Finally, for an N-neuron recurrent network, we give lower bounds on the rate of convergence of attractive periodic points toward the saturation values of neuron activations, as the absolute values of connection  ...  Acknowledgments This work was supported by grants VEGA 2/6018/99 and VEGA 1/7611/20 from the Slovak Grant Agency for Scienti c Research (VEGA) and the Austrian Science Fund (FWF) SFB010.  ... 
doi:10.1162/08997660152002898 pmid:11387050 fatcat:riuiczchvnbzrbqnmycyu2w3ui

A Very Small Chaotic Neural Net

C. Lourenco
2006 The 2006 IEEE International Joint Conference on Neural Network Proceedings  
Those networks contain a moderate to large number of units connected in a spatial arrangement providing instances of so-called Cellular Neural Networks.  ...  Here we aim to find a minimal network of realistic neurons already featuring a chaotic regime.  ...  ACKNOWLEDGMENT The author acknowledges the partial support of Fundação para a Ciência e a Tecnologia and EU FEDER via the Center for Logic and Computation and the project ConT-Comp (POCTI/MAT/45978/2002  ... 
doi:10.1109/ijcnn.2006.246993 dblp:conf/ijcnn/Lourenco06 fatcat:pv2c2uzrlzagjo2lwtcjslmxwu
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