Triadic interactions in the brain are general mechanisms by which a node, e.g. a neuron or a glia cell such as the astrocyte, can regulate directly the link, e.g. synapse between other two nodes. The regulation takes place in a familiar way by either depressing or facilitating synaptic transmission. Such interactions are ubiquitous in neural systems, accounting both for axo-axonic and tripartite synapses mediated by astrocytes, for instance, and have been related to neuronal and synaptic processes at different time-scales, including short- and long-term synaptic plasticity. In the field of network science, triadic interactions have been shown to produce complex spatio-temporal patterns of connectivity. Here, we investigate the emergent behavior of an in silico neural medium constituted by a population of leaky integrate-and-fire neurons with triadic interactions. We observe that, depending on relevant parameters defining triadic interactions, different activity patterns emerge. These include (i) a silent phase, (ii) a low-activity phase in which complex spatio-temporal patterns of low neuronal firing rate emerge that propagate through the medium, (iii) a high-activity phase characterized by complex spatio-temporal patterns of high neuronal firing rate that propagate through the neural medium as waves of high firing activity over a bulk of low activity neurons, and (iv) a pseudo-blinking phase in which the neural medium switches between high and low activity states. Here we analyze in depth the features of such patterns and relate our findings to the recently proposed model of triadic percolation.