Complex Rhythms - The Physics of Birdsong - page 131

This is a universal feature of nonlinear oscillators known as the Farey sum
rule . There is nothing special about neurons. Any excitable system would
show such a structure, as well as any nonlinear oscillator. For example, two
subpopulations of interconnected excitatory and inhibitory neurons might
have excitatory behavior as a whole. By this we mean that the activity
of the subpopulation (defined as the average number of spikes in some
time window for the subpopulation) can behave as an excitable system
[Hoppensteadt and Izhikevich 1997]. For this reason, under a periodic forc-
ing, a whole neural nucleus could display this complex rhythmic structure.
In summary, nonlinear systems are capable of generating complex and ap-
pealing rhythms. However, these temporal patterns are seriously constrained
by a mathematical structure. What about the beautiful rhythms present in
birdsong?
9.2 Duets
9.2.1 Hornero Duets
The South American hornero ( Furnarius rufus ) is a suboscine bird, widely
known for its nest, which is made of mud and looks like an oven. Beyond
their architectural skills, male and female horneros engage in highly struc-
tured duets. A sonogram of a typical duet is displayed in Fig. 9.1a, where
the continuous traces represent notes. The male starts singing at a note pro-
duction rate of approximately 6 Hz [Amador 2004], and in a few seconds it
increases the note production rate by about 200 percent. The duration of
the notes decreases. The female shows a large diversity: it may sing with an
increasing, decreasing or nonmonotonically varying note production rate, as
can be seen in Fig. 9.2. At the begining of the song (which can last up to 10
seconds), the female is capable of following the male, singing a note each time
the male does in a “one-by-one” fashion. However, after a while the female
seems to lose synchrony. Nevertheless, the timing has a nontrivial structure,
with features characteristic of nonlinear forced oscillators such as the one
that we discussed in the previous section.
In [Laje and Mindlin 2003], sonograms of hornero duets were computed
from field recordings, and only the fundamental frequencies were displayed,
as shown in Fig. 9.1a. The time intervals in which two notes are present are
those in which the two duetting birds are vocalizing simultaneously. In order
to characterize the duet, we define a coincidence as an event in which the
maximum of a male note occurs within a time interval in which the female is
vocalizing a note. We then define a number that describes the locking between
the male and female voices. This number r ap is the quotient of two integers
p/q ,where q stands for the number of male notes between consecutive male-
female coincidences, and p is the number of female notes between consecutive
male-female coincidences.