BIFURCATIONS AND OSCILLATORY BEHAVIOR IN A CLASS OF COMPETITIVE CELLULAR NEURAL NETWORKS

2000 ◽  
Vol 10 (06) ◽  
pp. 1267-1293 ◽  
Author(s):  
M. DI MARCO ◽  
A. TESI ◽  
M. FORTI
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Opposite Sign ◽  
Symmetric Matrix ◽  
General Theorem ◽  
Dynamical Behavior ◽  
Cellular Neural Networks ◽  
Oscillatory Behavior ◽  
Third Order ◽  
Small Perturbations

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some stationary state. In this paper it is shown that the interconnection symmetry, though ensuring complete stability, is not in the general case sufficient to guarantee that complete stability is robust with respect to sufficiently small perturbations of the interconnections. To this end, a class of third-order CNN's with competitive (inhibitory) interconnections between distinct neurons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This result is of obvious relevance in view of CNN's implementation, since perfect interconnection symmetry in unattainable in hardware (e.g. VLSI) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leonard model of the voting paradox, a special Volterra–Lotka model of three competing species. Finally, it is shown that the results in this paper can also be viewed as an extension of previous results by Zou and Nossek for a two-cell CNN with opposite-sign interconnections between distinct neurons. Such an extension has a significant interpretation in the framework of a general theorem by Smale for competitive dynamical systems.

ON THE MARGIN OF COMPLETE STABILITY FOR A CLASS OF CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (05) ◽  
pp. 1343-1361 ◽  
Author(s):  
MAURO DI MARCO ◽  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Neural Networks ◽  
Dynamical Behavior ◽  
Stability Margin ◽  
Cellular Neural Networks ◽  
Third Order ◽  
Dimensional Parameter ◽  
The Family ◽  
A New Technique ◽  
Two Parameters

In this paper, the dynamical behavior of a class of third-order competitive cellular neural networks (CNNs) depending on two parameters, is studied. The class contains a one-parameter family of symmetric CNNs, which are known to be completely stable. The main result is that it is a generic property within the family of symmetric CNNs that complete stability is robust with respect to (small) nonsymmetric perturbations of the neuron interconnections. The paper also gives an exact evaluation of the complete stability margin of each symmetric CNN via the characterization of the whole region in the two-dimensional parameter space where the CNNs turn out to be completely stable. The results are established by means of a new technique to investigate trajectory convergence of the considered class of CNNs in the nonsymmetric case.

GLOBAL CONSISTENCY OF DECISIONS AND CONVERGENCE OF COMPETITIVE CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (09) ◽  
pp. 3127-3150 ◽  
Author(s):  
M. DI MARCO ◽  
M. FORTI ◽  
M. GRAZZINI ◽  
P. NISTRI ◽  
L. PANCIONI
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Dynamical Behavior ◽  
Geometric Approach ◽  
Cellular Neural Networks ◽  
Convergence Properties ◽  
Competitive System ◽  
Decision Scheme ◽  
Extended Approach ◽  
Global Consistency

In a series of papers published in the seventies, Grossberg had developed a geometric approach for analyzing the global dynamical behavior and convergence properties of a class of competitive dynamical systems. The approach is based on the property that it is possible to associate a decision scheme with each competitive system in that class, and that global consistency of the decision scheme implies convergence of each solution toward some stationary state. In this paper, the Grossberg approach is extended to the class of competitive standard Cellular Neural Networks (CNNs), and it is used to investigate convergence under the hypothesis that the competitive CNN has a globally consistent decision scheme. The extension is nonobvious and requires to deal with the set-valued vector field describing the dynamics of the CNN output solutions. It is also stressed that the extended approach does not require the existence of a Lyapunov function, hence it is applicable to address convergence in the general case where the CNN neuron interconnections are not necessarily symmetric. By means of the extended approach, a number of classes of third-order nonsymmetric competitive CNNs are discovered, which have a globally consistent decision scheme and are convergent. Moreover, global consistency and convergence hold for interconnection parameters belonging to sets with non-empty interior, and thus they represent physically robust properties. The paper also shows that when the dimension is higher than three, there are fundamental differences between the convergence properties of competitive CNNs implied by a globally consistent decision scheme, and those of the class of competitive dynamical systems considered by Grossberg. These differences lead to the need to introduce a stronger notion of global consistency of decisions, with respect to that proposed by Grossberg, in order to guarantee convergence of competitive CNNs with more than three neurons.

HARMONIC BALANCE APPROACH TO PREDICT PERIOD-DOUBLING BIFURCATIONS IN NEARLY SYMMETRIC CNNs

2003 ◽  
Vol 12 (04) ◽  
pp. 435-459 ◽  
Author(s):  
MAURO DI MARCO ◽  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Neural Networks ◽  
Computer Simulations ◽  
Harmonic Balance ◽  
Recent Literature ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Third Order ◽  
Small Perturbations ◽  
Basic Issue ◽  
Period Doubling Bifurcations

This paper further investigates a basic issue that has received attention in the recent literature, namely, the robustness of complete stability of standard Cellular Neural Networks (CNNs) with respect to small perturbations of the nominal symmetric interconnections. More specifically, a class of third-order CNNs with a nominal symmetric interconnection matrix is considered, and the Harmonic Balance (HB) method is exploited for addressing the possible existence of period-doubling bifurcations, and complex dynamics, for small perturbations of the nominal interconnections. The main result is that there are indeed parameter sets close to symmetry for which period-doubling bifurcations are predicted by the HB method. Moreover, the predictions are found to be reliable and accurate by means of computer simulations.

scholarly journals On Smoother Attributions using Neural Stochastic Differential Equations

2021 ◽  
Author(s):  
Sumit Jha ◽  
Rickard Ewetz ◽  
Alvaro Velasquez ◽  
Susmit Jha
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Small Perturbations

Several methods have recently been developed for computing attributions of a neural network's prediction over the input features. However, these existing approaches for computing attributions are noisy and not robust to small perturbations of the input. This paper uses the recently identified connection between dynamical systems and residual neural networks to show that the attributions computed over neural stochastic differential equations (SDEs) are less noisy, visually sharper, and quantitatively more robust. Using dynamical systems theory, we theoretically analyze the robustness of these attributions. We also experimentally demonstrate the efficacy of our approach in providing smoother, visually sharper and quantitatively robust attributions by computing attributions for ImageNet images using ResNet-50, WideResNet-101 models and ResNeXt-101 models.

COMPLETE STABILITY FOR A CLASS OF CELLULAR NEURAL NETWORKS

2001 ◽  
Vol 11 (01) ◽  
pp. 169-177 ◽  
Author(s):  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Vector Field ◽  
Dynamical Systems ◽  
State Vector ◽  
Cellular Neural Networks ◽  
Output Function ◽  
Output Vector ◽  
Liapunov Functions ◽  
Lattice Dynamical Systems ◽  
Invariant Principle

This work investigates a class of lattice dynamical systems originated from cellular neural networks. In the vector field of this class, each component of the state vector and the output vector is related through a sigmoidal nonlinear output function. For two types of sigmoidal output functions, Liapunov functions have been constructed in the literatures. Complete stability has been studied for these systems using LaSalle's invariant principle on the Liapunov functions. The purpose of this presentation is two folds. The first one is to construct Liapunov functions for more general sigmoidal output functions. The second is to extend the interaction parameters into a more general class, using an approach by Fiedler and Gedeon. This presentation also emphasizes the complete stability when the equilibrium is not isolated for the standard cellular neural networks.

EMERGENT CHAOS SYNCHRONIZATION IN NONCHAOTIC CNNS

2008 ◽  
Vol 18 (05) ◽  
pp. 1337-1342 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QUAN YUAN
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Cellular Neural Networks ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable

It is shown that emergent chaos synchronization can take place in coupled nonchaotic unit dynamical systems. This is demonstrated by coupling two nonchaotic cellular neural networks, in which the couplings give rise to a synchronous chaotic dynamics and in the meanwhile the synchronous dynamics is globally asymptotically stable, thus chaos synchronization takes place under the suitable couplings.

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