Are connections in the human brain directed? Do connections in our brain have a direction, or do they behave equally in both directions? I am especially interested in the structures responsible for memory.
There are two kinds of neuron-to-neuron connections in the brain.  The most well known is the chemical synapse, which is unidirectional.  Despite this, two neurons can each synapse on each other with a chemical synapse, and both types of connection are observable in brain networks (unidirectional and bidirectional)

The second kind of connection is a gap-junction: electrical coupling.  In this case, it can be bidirectional or unidirectional, but all in the same synapse (whereas a bidirectional chemical synapse requires two synapses).

Typically, the chemical synapses are associated with memory.  There have been synapses observed that behave as engrams.