Is the weight of neural inputs in the human brain as central as it is for neurons in an artifical neural network?
The general biological term for this kind of mechanism is synaptic plasticity. Synapses are the biological structure that allow neurons to communicate via the exchange of neurotransmitters. The strength of the synapse (the effectiveness of the link between the neurons) can be modified over time.

One important kind of modification is long-term potentiation, first reported by Bliss and Lømo in 1973. They demonstrated that short term bursts of activity in a neural pathway can cause long term augmentation of the post-synaptic response. In artificial neural network terms, they were able to demonstrate something akin to an increase in the connection strength between two neurons.