  Most of animal and human behavior occurs on time scales much longer than the
response times of individual neurons. In many cases it is plausible that these
long time scales emerge from the recurrent dynamics of electrical activity in
networks of neurons. In linear models, time scales are set by the eigenvalues
of a dynamical matrix whose elements measure the strengths of synaptic
connections between neurons. It is not clear to what extent these matrix
elements need to be tuned in order to generate long time scales; in some cases,
one needs not just a single long time scale but a whole range. Starting from
the simplest case of random symmetric connections, we combine maximum entropy
and random matrix theory methods to construct ensembles of networks, exploring
the constraints required for long time scales to become generic. We argue that
a single long time scale can emerge generically from realistic constraints, but
a full spectrum of slow modes requires more tuning. Langevin dynamics that will
generate patterns of synaptic connections drawn from these ensembles involve a
combination of Hebbian learning and activity-dependent synaptic scaling.
