Collective Effects in Arrays of Cilia and Rotational Motors - Cell Motility - page 218

namic interactions, this dynamical state is only stable for periodic boundary condi-
tions and its dispersion relationship depends on the details of the angular dependence
of the interciliar interactions. This non-universal nature of our results shows that
one has to be very careful in drawing general conclusions from studies of specific
ciliar beating patterns.

8.3.4 Synchronization of Beating Frequencies

If two neighboring cilia have different frequencies ω 1 and ω 2 , hydrodynamic inter-
actions can, in principle, lead to synchronization of the frequencies, i.e., to a syn-
chronized movement of the two cilia with the same frequency ω . For example, for
cilia with a specific beating pattern [47], this phenomenon has been shown to occur
under suitable conditions.
However, this is not an universal feature. For example, for the general hydrody-
namic interaction introduced above, the two cilia obey

ϕ 1 + ϕ 2 J ( ϕ 1 ,ϕ 2 )= ω 1 , (8.31)

ϕ 2 + ϕ 1 J ( ϕ 2 ,ϕ 1 )= ω 2 . (8.32)

Here, synchronization only occurs for non-symmetric interactions J ( x, y ), i.e.,
J ( x, y ) = J ( y, x ). This can be seen by making the ansatz ϕ 1 = φ ( t ), ϕ 2 = φ ( t )+ δ ,
where δ is a time-independent constant. Then, Equations (8.31) and (8.32) yield

ω 1 (1 + J ( φ + δ, φ )) = ω 2 (1 + J ( φ, φ + δ )) . (8.33)

For symmetric interactions J ( x, y )= J ( y, x ) (as one has, e.g., in the case of the
monociliar beating introduced in Section 8.3.1) there is no synchronized solution.
There is an interesting connection to general synchronization phenomena occur-
ring, for example, in arrays of coupled oscillators. One of the best studied models
in this context is the Kuramoto model (for a review see [48]). The Kuramoto model
consists of N coupled oscillators with natural frequencies ω i whose motion is de-
scribed by the dynamical variable θ i . The equation of motion is given by

N

d θ i
d t

= ω i + K ij sin( θ j − θ i ) , (8.34)

j =1

where additionally the coupling-matrix K ij has to be specified. The simplest choice
is an identical coupling between all oscillators:

K
N .

K ij ≡ (8.35)

The basic properties of the Kuramoto model can be summarized as follows: For
N 1thereisacriticalcouplingstrength K ∗ above which (i.e., K>K ∗ )the
oscillator become synchronized to their average phase θ i ≈ ψ .For K<K ∗ the
oscillators behave incoherently. One can also easily see that in the Kuramoto model,
two oscillators synchronize (i.e., ϕ 1 = ϕ 2 )for ω 1 = ω 2 . However, for two cilia with
general interaction J ( ϕ 1 ,ϕ 2 ), this is not the case. Thus, it remains a challenge to
develop an understanding of what the essential difference between Kuramoto's and
ciliar systems is.