In this paper, we present new findings in constructing and applications of artificial neural networks that use a biologically inspired spiking neuron model. The used model is a point neuron with the interaction between neurons described by postsynaptic potentials. The synaptic plasticity is achieved by using a temporal correlation learning rule, specified as a function of time difference between the firings of pre- and post-synaptic neurons. Using this rule we show how certain associations between neurons in a network of spiking neurons can be implemented. As an example we analyze the dynamic properties of networks of laterally connected spiking neurons and we show their capability to self-organize into topological maps in response to external stimulation. In another application we explore the capability networks of spiking neurons to solve graph algorithms by using temporal coding of distances in a given spatial configuration. The paper underlines the importance of temporal dimension in artificial neural network information processing.