Stability of Quasi-Periodic Orbits in Recurrent Neural Networks

Abstract

A simple discrete recurrent neural network model is considered. The local stability is analyzed with the associated characteristic model. In order to study the dynamic behavior of the quasi-periodic orbit, it is necessary to determine the Neimark-Sacker bifurcation. In the case of two neurons, one necessary condition that yields the Neimark-Sacker bifurcation is found. In addition to this, the stability and direction of the Neimark-Sacker bifurcation are determined by applying normal form theory and the center manifold theorem. An example is given and a numerical simulation is performed to illustrate the results. The phase-locking phenomena are analyzed for certain experimental results with Arnold Tongues in a particular weight configuration.

Hopf Bifurcation in a Chaotic Associative Memory

“…• All these studies [5,9,10,11,12,13,14,15,16,17,18,19,20] related the bifurcation parameter to time delay, while in our case the bifurcation parameter is a damping parameter of the system. Besides, in these studies the Hopf Bifurcation occurs in the input/output space of phase, while in our case this occurs in the internal state space of phases;…”

“…The study of the changes of dynamical behaviors of NLS as function of variation of the parameter values, includes the study of Bifurcation [3]. There are studies in which local bifurcations were verified in general RNNs [4,5,6,7], in Associative Memories (AM) and in Bidirectional Associative Memories (BAM) [5,8,9,10,11,12,13,14,15,16,17,18,19,20].…”

“…For neural networks, the typical goals were to prove the existence and stability of Hopf Bifurcation for continuous systems [9,12,15], and for discrete systems [4,5,10,16,18], deemed Neimark-Sacker Bifurcation [3]. More recently, several studies with Neural Networks in which the bifurcation parameter μ ∈ R 2 have been presented and show the occurrence of other types of bifurcations such as: Hopf-Hopf bifurcation, when there are two pairs of purely complex eigenvalues [20]; Zero-Hopf Bifurcation, when there is the presence of a zero eigenvalue and a pair of purely complex eigenvalues [21]; Bautin Bifurcation, when there is a pair of purely imaginary eigenvalues and the first Lyapunov coefficient for the Hopf Bifurcation when it converges to zero [14]; or Bogdanov-Takens Bifurcation, when a null eigenvalue has a multiplicity of two [6,22,23].…”

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

“…For example, the most prominent are gamma (30-100 Hz) oscillations in the cortex and theta (4-8 Hz) oscillations in the hippocampus (Llinas 1988). Recently, Marichal et al (2010) considered the dynamic behavior of a quasi-periodic orbit obtained by the Neimark-Sacker bifurcation in a simple discrete recurrent neural network model. It is well known that time delay is an inevitable factor in the signal transmission between biological neurons or electronic-model-neurons.…”

Analysis of multiple quasi-periodic orbits in recurrent neural networks

“…In particular, the R 2 bifurcation is a resonance represented by the rotational number 1:2. In fact, for neural network (1), the numerical simulations on Arnold tongues [10] conclude that the most important Arnold tongue is associated with a 1:2 rotational number, that is, the most frequent resonance is 1:2. Additionally, in R 2 resonance the most interesting dynamical behavior is the presence of multiple quasiperiodic orbits under some conditions of normal form coefficients (see Section 4).…”