Qualitative Study Of Solutions For A Class Of Neural Network Models With Two Neurons

In this paper, we study the large-time behavior of solutions for the following delay neural network model of two neurons:x(t) = -μx(t) - af(x(t - r)) + bg(y(t - r))[x]⁺, y(t) = -μy(t) - ag(y(t - r)) + bf(x(t - r))[y]⁺,where μ > 0 is the internal decay rate; a, b > 0 are given constants and Ï„ > 0 is referred as the synaptic transmission delay; [x]⁺ is definited byAssuming the signal transmission functions f and g are given by the following functions (McCulloh-Pitts)type:In such a model, we consider not only the interaction of the two neurons, but also the message feedback of themselves, in particular, the model is considered the limit relations, it means that if only when the neuron is in the active state it can be influenced by the other neuron, that conforms to reality. In this paper we divide the values of different paramates in the model, in every different space we discuss the solutions (x(t),y(t))^T with different initial values, and according to (x(t) — σ₁)(y(t) —σ₂) = 0, we can get a series of time points t_n, so we can get the values of x(t), y(t) when t is t_n + r, then we can judge whether the solution is convergent or periodic.There are three chapters in this paper. In chapter 1 we mainly introduces the background, significance of the problem we investigate and its recent development, as well as some notations which will be used in our paper. In chapter 2 we discuss the convergence of solutions for the model when σ₁ ≠ σ₂. We prove that under most conditions the solutions of the model are convergent; For other conditions, we prove that the solutions of the model are also convergent in a special case. In chapter 3 we discuss the periodicity of the synchronous solutions for the model when σ₁ =σ₂ According to the argument of some conclusions in chapter 2, We can see that to this case,the solutions of the model are convergent under most...