An improved mathematical model of artificial neuron with active generation of action potentials was developed, and the behavior of an artificial network consisting of thousands of described neurons was investigated. The passive part of the neuron consists of soma and asymmetric dendritic branches that provide multi-stability. The active component of the neuron is described with the help of a simplified non-linear neuron model with mechanisms for the spiking generation. The presented model reproduces all types of electrical generations of known biological neurons, e.g. neocortical. The model combines the biological similarity of the Hodgkin-Huxley type dynamics and the computational efficiency of integrative-spiking neurons. It is shown that switching between different modes of generation is possible under the condition of structural three-stability of the neuron in common. A neural network consisting of multi-stable neurons is capable of generating synchronous regular spikes if all neurons in the network are in a similar electrical state. In the case where a part of the neurons at non-similar stable condition, the network generates asynchronous regular spikes, without adding any synaptic plasticity mechanisms or modulating stimulation processes. The obtained model can be used for studying the features of real-time data processing by artificial neural networks, which can be used for such modern tasks as recognition and classification of biophysical signal patterns or for the development of elements of artificial intelligence.