IIMAS - FENOMEC
UNAM
Resumen:
Standing bursts are three-spatial
scale generalization of standing pulses, in the same way as
bursts are three-timescale generalization of impulses (or
“spikes”). Standing pulses are generated by local excitation
/ lateral inhibition mechanism. At the biological level,
they underlie morphogenesis and various cognitive phenomena,
like direction sensitivity in the visual cortex and visual
working memory. Standing bursts are hypothetized by the
author to be generated by the same mechanism and to underly
similar biological functions simultaneously at multiple
spatial scales and organization levels.
The classical mathematical construction of standing pulses
relies on geometrical singular perturbation techniques
[Jones and Rubin, 1998] and, in particular, the Exchange
Lemma [Jones and Kopell, 1994]. Generalizing the same
construction to standing bursts is not trivial, because the
presence of three spatial scales leads to two distinct
singular limits that correspond to different phases of the
standing burst. The switch between the two limits along the
standing burst orbit goes beyond the standard theory. The
existence of standing bursts was shown numerically but
mathematically only conjectured in [Franci and Sepulchre
2016].
In this talk, I will provide a short introduction to the
Exchange Lemma and its application to the geometric
construction of standing pulses. I will then present ongoing
work toward a proof of the existence of standing bursts. I
will finish by summarizing the open questions and future
perspectives.
Informes: coloquiomym@gmail.com, o al 5622-3564.