Biomedical Engineering Reference
In-Depth Information
(a)
2 π
(b)
2 π
0
0
0
2
0
2
π
π
φ n
φ n
Fig. 9.6. Graphical solution to the fixed points of the second-iterate map (9.12):
the right-hand side of the equation is represented by the straight line , and the left-
hand side is represented by the curve . The parameter values are A =0 . 8and( a )
T =0 . 47, ( b ) T =0 . 53 (in units of the period of the oscillator 2 π/ω ). As the forcing
frequency is decreased, the main change that is observed in the curve is a global shift
upwards. These values of forcing period define the borders of the period-two region;
for T between 0 . 47 and 0 . 53, there are four intersections (solutions to (9.12)), but
only two of them represent stable solutions
at T ∼ 0 . 47 (Fig. 9.6a), four intersections for 0 . 47 <T < 0 . 53, again two in-
tersections at T ∼ 0 . 53 (Fig. 9.6b) and, finally, no intersections for T> 0 . 53.
Only two of the four intersections found for 0 . 47 <T < 0 . 53 represent stable
solutions. This behavior tells us that the values T
0 . 47 and T
0 . 53 are
the borders of the period-two tongue for A =0 . 8.
It is left for dynamicists to explain what is the minimal set of “geometric”
conditions that a one-dimensional map has to satisfy in order to show this
organization of temporal patterns. At the rough qualitative level that we are
interested in, nonlinear relaxation oscillators and excitable systems do fall
within the class.
What is the relationship between this abstract oscillator and the hornero
that we met in Sect. 9.2? The hornero is a suboscine. The brain structure in
the suboscines is less well understood than that of the oscine songbirds. For
this reason, it is a complex task to try to map the abstract oscillator that we
are using in this section to a particular structure in the brain. However, the
generation of repetitive syllables requires an oscillatory rhythm in order to
generate expirations, and therefore some population of neurons (which will
eventually feel somatosensory feedback) should show oscillatory activity. The
temporal structures described in the section on hornero duets could then be
produced if this neural substrate behaved as a nonlinear oscillator and could
be forced by auditory signals.
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