12.3 Mathematical Model for the Central Pattern Generator 429
implications, or rather assumptions, of this are that the limit cycles γ
j
, j = 1, 2,... ,N
of the isolated oscillators will only be slightly perturbed by the coupling effects (recall
Section 9.5). So, it is still appropriate to use the oscillator equation form (12.13) and
(12.14) involving the phase θ and deviations r
j
from the limit cycle γ
j
, but now we
have to include an extra small coupling term in the equation. (In fact we shall make
even further simplifying assumptions based on what has been observed experimentally,
but it is instructive to proceed a little further with the current line since it is the basis for
a rigourous justification of the assumptions we make later.) The set of N equations we
have to study is then
dr
j
dt
= f
j1
(r
j
,θ
j
) + g
j1
(r
1
,... ,r
N
,θ
1
,... ,θ
N
, c),
dθ
j
dt
= ω
j
+ f
j2
(r
j
,θ
j
) + g
j2
(r
1
,... ,r
N
,θ
1
,... ,θ
N
, c), j = 1, 2,... ,N.
(12.20)
Experimentally it has been observed that the individual oscillators when uncoupled,
by severing and thus isolating them from their neighbours, have different frequencies
ω
j
and hence different periods T
j
= 2π/ω
j
. A crucially important point to keep in
mind is that when the segmental oscillators are coupled they still perform limit cycle
oscillations. So, even when coupled we can still characterise them in terms of their phase
θ
j
. Since fictive swimming is a reflection of phase coupling we need only consider a
phase coupling model for the system (12.19). So, instead of studying the system (12.19)
perturbed about r
j
= 0 we can consider a system of phase coupled equations of the form
dθ
j
dt
= ω
j
+h
j
(θ
1
,... ,θ
N
, c), j = 1,... ,N, (12.21)
where h
j
includes the (weak) coupling effect of all the other oscillators. Equations
(12.21) do not involve the amplitudes of the oscillators. The problem of weak cou-
pling in a population of oscillators has been studied in some depth, for example, by
Neu (1979, 1980), Rand and Holmes (1980), Ermentrout (1981) and in the book by
Guckenheimer and Holmes (1983). Carrying out a perturbation of (12.19) about r
j
= 0
eventually results in a phase coupled system of equations (12.21) (see Chapter 9). So,
there is a mathematical, as well as biological, justification for considering the simpler
model (12.21).
Since we assume small perturbations from the individual limit cycle oscillators to
come from the coupling, it is reasonable to consider a linear coupling model where the
effect of the jth oscillator on the ith one is simply proportional to x
j
. In this situation
the coupled oscillator system is of the form (12.9) perturbed by linear terms; namely,
dx
i
dt
= f
i
(x
i
) +
N
j=1
j=i
A
ij
x
j
, (12.22)
where A
ij
are matrices of the coupling coefficients.