Lotka-Volterra Recurrent Neural Networks Research Articles

Attractors coexistence and representation research of Lotka–Volterra recurrent neural networks by matrix analysis theory

Attractor neural network models have been extensively studied in maintaining brain working memory state, spatial representation, and other brain processing mechanisms. The coexistence of multiple attractors in nonlinear neural network is particularly worth studying. The attractor is a stable state in the dynamic system, and all adjacent states will eventually be pulled to the attractor over time. Attractors can be regarded as the expression of data in low-dimensional state space, and different kinds of attractors can be used to process different information. The coexistence of multiple attractors means that a neural network can store multiple patterns, that is to say different memory mechanisms appear at the same time. Investigating the coexistence and competition of attractors within Lotka–Volterra(LV) recurrent neural networks presents a pivotal avenue for understanding the dynamical underpinnings of complex systems, particularly in the realm of computational neuroscience and ecological modeling. This paper studies the coexistence of different types of attractors in asymmetric Lotka–Volterra recurrent neural networks. Through the weight matrix block and matrix decomposition theory, we obtain the explicit expressions of continuous attractors and discrete attractors. Since neurons have multiple grouping methods, each weight matrix block satisfying certain conditions corresponds to different attractors. Different from traditional symmetric networks, the eigenvalues of the asymmetric weight matrices may be complex, which may correspond to ring attractors. Our research shows that attractors can coexist under certain conditions, depending on the weight matrix and external input. Moreover, through simulation experiments, we show that external input will cause the competition of attractors. Finally, we provide several simulations of attractor coexistence to illustrate our theory.

Dynamic Behavior of a Class of Delayed Lotka–Volterra Recurrent Neural Networks on Time Scales

Dynamic Behavior of a Class of Delayed Lotka–Volterra Recurrent Neural Networks on Time Scales

Stepanov stability for delayed Lotka–Volterra recurrent neural networks on time scales

This paper studies the existence, uniqueness, asymptotic and -globally exponential stability of Stepanov-like pseudo almost automorphic solution to Lotka–Volterra recurrent neural networks (LVRNNs) with time-varying delays on time scales by applying D-matrix theory, Banach fixed point theorem, constructing a suitable Lyapunov functional and the theory of time scales calculations. An example and simulations are given to prove the effectiveness of main theoretical results.

The pseudo almost periodic solutions of the new class of Lotka–Volterra recurrent neural networks with mixed delays

This paper is concerned with the dynamics and oscillations of a new class of Lotka-Volterra recurrent neutral networks. We obtain results on the existence and uniqueness of the pseudo almost periodic solution under some sufficient and proper conditions. In addition, the asymptotic and exponential stability of the pseudo almost periodic solutions are investigated. Finally, two numerical examples with their simulations are presented to support our theoretical results.

Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays

Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays

Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays

Abstract: By means of Mawhin’s continuation theorem of coincidence degree theory and Lyapunov function, some simple sufficient conditions are obtained for the existence and stability of a unique positive almost periodic solution of a delayed Lotka–Volterra recurrent neural networks. To a certain extent, the work in this paper corrects the defect of a recent paper. Finally, an example and simulations are given to illustrate the feasibility and effectiveness of the main result.

Nearest convex hull classification by using Lotka–Volterra recurrent neural networks

Distance-based classifiers have been applied to many multi-class classification problems. The nearest convex hull classifier (NCHC) is a useful distance-based classifier. It assigns a test sample to the class that has the closest convex hull. This paper proposes a new algorithm to implement the NCHC. Considering an alternative interpretation of NCHC, the distance from the test sample to the convex hull of the training data in a certain class can be thought of as the reconstruction error. We propose an algorithm that uses neural networks to implement the NCHC. Our experimental results show that NCHC using Lotka–Volterra recurrent neural networks outperforms other classifiers in a whole.

Almost Periodicity in a Harvesting Lotka-Volterra Recurrent Neural Networks with Time-Varying Delays

By using the theory of exponential dichotomy and Banach fixed point theorem, this paper is concerned with the problem of the existence and uniqueness of positive almost periodic solution in a delayed Lotka-Volterra recurrent neural networks with harvesting terms. To a certain extent, our work in this paper corrects some result in recent years. Finally, an example is given to illustrate the feasibility and effectiveness of the main result. Keywords—positive almost periodic solution, Lotka-Volterra, neural networks, Banach fixed point theorem, harvesting

Periodic Solution of Impulsive Lotka-Volterra Recurrent Neural Networks with Delays

In this paper, periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays is studied. Using the continuation theorem of coincidence degree theory and analysis techniques, we establish criteria for the existence of periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays.

Using competitive layer model implemented by Lotka–Volterra recurrent neural networks for detecting brain activated regions from fMRI data

The competitive layer model (CLM) implemented by the Lotka–Volterra recurrent neural networks (LV RNNs) is prominently characterized by its capability of binding neurons with similar feature into the same layer by competing among neurons at different layers in a column. This paper proposes to use the CLM of the LV RNN for detecting brain activated regions from the fMRI data. The correlated voxels from brain fMRI data can be obtained, and the clusters from fMRI time series can be uncovered. Experiments on synthetic and real fMRI data demonstrate the effectiveness of binding activated voxels into the ‘active’ layers of the CLM. The activated voxels can be detected more accurately than some existing methods by the proposed method.

A new method based on the CLM of the LV RNN for brain MR image segmentation

A new method based on the competitive layer model (CLM) implemented by Lotka-Volterra recurrent neural networks (LV RNNs) is proposed for brain MR image segmentation. This method firstly divides an MR image into sub-images, and segments each sub-image by the CLM of the LV RNN to obtain a lot of 4-connected regions. Secondly, any two neighboring regions that are similar to each other are merged to form one region. Finally, all remaining regions are clustered by the RFCM into background, CSF, GM and WM. Compared with other three methods using numerical simulations, our method is shown to be more effective.

The almost periodic solution of Lotka–Volterra recurrent neural networks with delays

By the fixed-point theorem subject to different polyhedrons and some inequalities (e.g., the inequality resulted from quadratic programming), we obtain three theorems for the Lotka–Volterra recurrent neural networks having almost periodic coefficients and delays. One of the three theorems can only ensure the existence of an almost periodic solution, whose existence and uniqueness the other two theorems are about. By using Lyapunov function, the sufficient condition guaranteeing the global stability of the solution is presented. Furthermore, two numerical examples are employed to illustrate the feasibility and validity of the obtained criteria. Compared with known results, the networks model is novel, and the results are extended and improved.

Continuous Attractors of Lotka–Volterra Recurrent Neural Networks With Infinite Neurons

Continuous attractors of Lotka-Volterra recurrent neural networks (LV RNNs) with infinite neurons are studied in this brief. A continuous attractor is a collection of connected equilibria, and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks. The existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network. A continuous attractor can be stable or unstable. It is shown in this brief that a LV RNN can possess multiple continuous attractors if the synaptic connections and the external inputs are Gussian-like in shape. Moreover, both stable and unstable continuous attractors can coexist in a network. Explicit expressions of the continuous attractors are calculated. Simulations are employed to illustrate the theory.

Foundations of Implementing the Competitive Layer Model by Lotka–Volterra Recurrent Neural Networks

The competitive layer model (CLM) can be described by an optimization problem. The problem can be further formulated by an energy function, called the CLM energy function, in the subspace of nonnegative orthant. The set of minimum points of the CLM energy function forms the set of solutions of the CLM problem. Solving the CLM problem means to find out such solutions. Recurrent neural networks (RNNs) can be used to implement the CLM to solve the CLM problem. The key point is to make the set of minimum points of the CLM energy function just correspond to the set of stable attractors of the recurrent neural networks. This paper proposes to use Lotka-Volterra RNNs (LV RNNs) to implement the CLM. The contribution of this paper is to establish foundations of implementing the CLM by LV RNNs. The contribution mainly contains three parts. The first part is on the CLM energy function. Necessary and sufficient conditions for minimum points of the CLM energy function are established by detailed study. The second part is on the convergence of the proposed model of the LV RNNs. It is proven that interesting trajectories are convergent. The third part is the most important. It proves that the set of stable attractors of the proposed LV RNN just equals the set of minimum points of the CLM energy function in the nonnegative orthant. Thus, the LV RNNs can be used to solve the problem of the CLM. It is believed that by establishing such basic rigorous theories, more and interesting applications of the CLM can be found.

Global convergence of Lotka-Volterra recurrent neural networks with delays

Recurrent neural networks of the Lotka-Volterra model have been proven to possess characteristics which are desirable in some neural computations. A clear understanding of the dynamical properties of a recurrent neural network is necessary for efficient applications of the network. This paper studies the global convergence of general Lotka-Volterra recurrent neural networks with variable delays. The contributions of this paper are: 1) sufficient conditions are established for lower positive boundedness of the networks; 2) global exponential stability conditions are obtained for the networks. These conditions are totally independent of the variable delays which are therefore allowed to be uncertain; 3) novel Lyapunov functionals are constructed to establish delays dependent conditions for global asymptotic stability, and 4) simulation results and examples are provided to supplement and illustrate the theoretical contributions presented.

Erratum: Dynamical stability conditions for Lotka-Volterra recurrent neural networks with delays [Phys. Rev. E66, 011910 (2002)

Erratum: Dynamical stability conditions for Lotka-Volterra recurrent neural networks with delays [Phys. Rev. E<b>66</b>, 011910 (2002)

Dynamic stability conditions for Lotka-Volterra recurrent neural networks with delays.

The Lotka-Volterra model of neural networks, derived from the membrane dynamics of competing neurons, have found successful applications in many "winner-take-all" types of problems. This paper studies the dynamic stability properties of general Lotka-Volterra recurrent neural networks with delays. Conditions for nondivergence of the neural networks are derived. These conditions are based on local inhibition of networks, thereby allowing these networks to possess a multistability property. Multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process. Under these nondivergence conditions, a compact set that globally attracts all the trajectories of a network can be computed explicitly. If the connection weight matrix of a network is symmetric in some sense, and the delays of the network are in L2 space, we can prove that the network will have the property of complete stability.