Dynamics of a continuous-valued discrete-time Hopfield neural network with synaptic depression

A continuous-valued discrete-time Hopfield neural network with synaptic depression (CDHSD) is constructed. We prove that the fixed point of CDHSD is the same as that of a network without synaptic depression and with an activation function determined by the parameters of the synaptic depression. We analyze the stability of the equilibrium, and then give a sufficient condition for the existence of a unique equilibrium of CDHSD. Numerical analysis shows that the attractor of CDHSD might be an equilibrium, a periodic orbit or a nonperiodic orbit depending on its parameter values and initial conditions. A weak external input of the network contributes to the genesis of nonperiodic dynamics of the network. If the value of parameter ɛɛ, which is the steepness parameter of the activation function f(x)=1/(1+exp(-x/ɛ))f(x)=1/(1+exp(-x/ɛ)), is large enough or small enough, nonperiodic dynamics of CDHSD does not appear. It is also shown that nonperiodic dynamics is likely to emerge with intermediate strength of synaptic depression.