We construct a minimal but biologically plausible spiking neuron model operating in discrete time, combining multiplicative spike-timing-dependent plasticity (WSTDP), divisive normalization of synaptic integration, homeostatic threshold adaptation, and a one-step refractory period. We show that this normalization admits a biologically plausible dendritic implementation in which each binary junction operates using only locally available information. Assembling excitatory-inhibitory pairs of such neurons into a two-dimensional recurrent network and applying periodic localized stimulation, we find that the network spontaneously gives rise to stable, self-propagating wave packets with the properties of dissipative solitons: they maintain a stable spatial profile, propagate at constant speed, and annihilate upon frontal collision. Their emergence requires a geometric asymmetry between excitatory and inhibitory connection radii, and initial inhibitory synapses stronger than excitatory ones. WSTDP engraves the direction of propagation into the synaptic weight profile, so that the network learns by itself to sustain propagation in one direction while suppressing the reverse. When two sources are active simultaneously, the resulting waves annihilate upon collision, defining a semi-persistent boundary whose position encodes the relative phase and frequency of the two sources. These results provide a minimal computational framework for studying the emergence of cortical traveling waves, activity zone delimitation, and spatial memory from local plasticity rules alone.

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