Abstract
There have been many research studies of various neuron models that typically take the form of ordinary nonlinear differential equations of several dimensions. The pattern of spiking is of great importance, because it is presumed that it codifies information transmitted by neurons. It is an actively studied problem to apply various neuron models to artificial neural networks (ANNs) for intelligent information processing in the field of nonlinear dynamics and brain research. One important aspects in this situation is the lack of universal discussion about the dynamical behaviors of various neuron models, although perturbation and bifurcation theories exist. Thus, we reveal that each model has its own potential function and active areas on the potential. Negative resistance is one of the active areas. This concept realizes the universal discussion of the dynamical behaviors of models, for example, bursting and spiking. On the other hand, Hopfield neural network is capable of solving combinatorial optimization problems and is a parallel-processing version of the gradient method. However, it has some drawbacks. One of its most serious drawbacks is that it frequently finds locally minimum solutions instead of global minima. The active areas of neuron models make the state of the network escape from local minima by their destabilization. In computer simulation and theoretical estimation, we have already shown that the ID network, which is active ANN for implementing associative memory and combinatorial optimization problems, can converge on optimal solutions only.
