The pulse-coupled neural network (PCNN) model is a third-generation artificial neural network without training that uses the synchronous pulse bursts of neurons to process digital images, but the lack of in-depth theoretical research limits its extensive application. By analyzing the working mechanism of the PCNN, we present an expression for the fire-extinguishing time of neurons that fire in the second iteration and an expression for the firing time of neurons that extinguish in the second iteration. In addition, we find a phenomenon of the PCNN and name it mathematically coupled fire extinguishing. Based on the above analysis, we propose a new working mode for the PCNN, where the refiring of fire-extinguishing neurons is only allowed when all firing neurons are extinguished. We also work out the constraint conditions of the parameter settings under this mode. Furthermore, we analyze the relationship between the network parameters and mathematically coupled fire extinguishing, the coupling of neighboring neurons, and the convergence rate of the PCNN, respectively. In addition, we demonstrate the essential regularity of extinguished neuron in the PCNN and then propose an optimal parameter setting to achieve the best comprehensive performance of the PCNN.