M. Golubitsky, C. Postlethwaite, L-J. Shiau and Y. Zhang
The feed-forward chain as a filter amplifier motif
In: Coherent Behavior in Neuronal Networks.
(K. Josic, M. Matias, R. Romo, and J. Rubin, eds.)
Springer, 2009, 95-120.
Husdpeth, Magnasco, and collaborators have suggested that the auditory system
works by tuning a collection of hair cells near Hopf bifurcation, but each with
a different frequency. An incoming sound signal to the cochlea then resonates
most strongly with one of these hair cells, which then informs the auditory
neuronal system of the frequency of the incoming signal. In this paper we
discuss two mathematical issues. First, we describe how periodic forcing of
systems near a point of Hopf bifurcation is generally more complicated than
the description given in these auditory system models. Second, we discuss
how the periodic forcing of coupling identical systems whose internal dynamics
is each tuned near a point of Hopf bifurcation leads naturally to successive
amplification of the incoming signal. We call this coupled system a
feed-forward chain and suggest that it is a mathematical candidate for
a motif.