  A growing literature in computational neuroscience leverages gradient descent
and learning algorithms that approximate it to study synaptic plasticity in the
brain. However, the vast majority of this work ignores a critical underlying
assumption: the choice of distance for synaptic changes - i.e. the geometry of
synaptic plasticity. Gradient descent assumes that the distance is Euclidean,
but many other distances are possible, and there is no reason that biology
necessarily uses Euclidean geometry. Here, using the theoretical tools provided
by mirror descent, we show that the distribution of synaptic weights will
depend on the geometry of synaptic plasticity. We use these results to show
that experimentally-observed log-normal weight distributions found in several
brain areas are not consistent with standard gradient descent (i.e. a Euclidean
geometry), but rather with non-Euclidean distances. Finally, we show that it
should be possible to experimentally test for different synaptic geometries by
comparing synaptic weight distributions before and after learning. Overall, our
work shows that the current paradigm in theoretical work on synaptic plasticity
that assumes Euclidean synaptic geometry may be misguided and that it should be
possible to experimentally determine the true geometry of synaptic plasticity
in the brain.
