Searching for emergent long time scales without fine tuning

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 connections between neurons. It is not clear to what extent connection strengths need to be tuned in order to generate sufficiently long time scales; in some cases, one needs not just a single long time scale but a whole range. For a system with random symmetric connections, random matrix theory allows us to show that imposing a global stability constraint is sufficient to generate a diverging density of arbitrarily slow modes. But as soon as the detection mechanism for stability is set to be biologically plausible, these modes disappear for all system sizes. We will give a progress report on the more realistic, and challenging, case of asymmetric interactions.