With the notion of phase defined, we now examine how a perturbation shifts the
phase of the oscillator. In Figure 12A , we show the voltage time course for
the Morris-Lecar system in the oscillating regime. At a fixed time, say t,
after the voltage peak, we apply a brief depolarizing current pulse. This
shifts the time of the next peak (Figure 12A) and this shift remains for all
time (the solid curve is the perturbed oscillation and the dashed is the
unperturbed - in this case the time for the next peak is shortened). If the
time of the next peak is shortened from the natural time, we say that the
stimulus has advanced the phase. If the time of the next peak is lengthened
then we have delayed the phase. Let 
denote the time of the next peak.
The phase shift is 
, and 
depends on the time t or the
phase 
at which the stimulus is applied. Thus, we can define a
phase shift 
. The graph of this
function is called the phase response curve (PRC) for the oscillator. If
is positive then the perturbation advances the phase and the
peak will occur sooner. On the other hand, if 
is negative then
the phase is delayed and the next peak will occur later. We can easily compute
this function numerically and the same idea can be used to analyze an
experimental system. Moreover, this curve can be used as a rough approximation
of how the oscillator will be affected by repeated perturbation (periodic
forcing) with the same current pulse. More complete descriptions and numerous
examples of phase models and PRC's can be found in [13,38].
Do exactly like you would do in Fig. 12 for a variety of different values of ph and manually compute the phase shift.
In Figure 12B, we show a typical PRC for the Morris-Lecar model computed for
both a depolarizing stimulus (solid line) and a hyperpolarizing stimulus (dashed
line). The stimulus consists of a current pulse of magnitude 480
applied for
msec at different times after the voltage peak. The time of the
next spike
is determined and this yields the PRC as above. The figure agrees with our
intuition; if the depolarizing stimulus comes while 
is increasing (i.e.
during the upstroke or slow depolarization of recovery), the peak will occur
earlier and we will see a phase-advance. If the stimulus occurs while 
is
decreasing (i.e. during the downstroke), there will be a delay. The opposite
occurs for hyperpolarizing stimuli. The curves show that it is difficult
to delay the onset of an action potential with a depolarizing stimulus or
advance it with a hyperpolarizing one. For different sets of parameters, these
curves may change. As we have seen in the previous section, it is sometimes
possible to completely stop the oscillation if a stimulus is given at the right
time. In this case, the PRC is no longer defined; nearby phases can then have
arbitrarily long latencies before firing.
We now show how this function can be used to analyze a periodically forced
oscillator. Suppose that every P time units a current pulse is applied to the
cell. Let 
denote the phase right before the time of the nth stimulus.
This
stimulus will either advance or delay the onset of the next peak depending on
the phase at which the stimulus occurs. In any case, the new phase after time P
and just before the next stimulus will be 
. To
understand this, first consider the case where there is no stimulus. Then after
time P the oscillator will advance 
in phase. But the stimulus advances
or delays the phase by an amount 
so that this amount
is just added to the
unperturbed phase. This results in an equation for the new phase just before
the next stimulus:
This difference equation can be solved numerically. Here we will consider the natural question of whether the periodic stimulus can entrain the voltage oscillation. That is, we ask whether there is a periodic solution to this forced neural oscillation. In general, a periodic solution is one for which there are M voltage spikes for N stimuli where M and N are positive integers. When such a solution exists, we have what is known as M:N phase-locking.
Finding 
phase-locked solutions is quite easy. We require the oscillator
to
undergo M oscillations per stimulus period. In terms of (24) this
means we seek a solution which satisfies
for some value of 
. For if such a solution exists and if the solution
is
stable (to be defined below), then if we start near 
, we can iterate
(24) and
end up back at 
. This 
is the locking phase just before the
next stimulus and since it
doesn't change from stimulus to stimulus, the resulting solution must be
periodic. Obviously, a necessary condition for a solution to (25) is
that
lie between the maximum and minimum of 
, i.e., we must
solve:
Having solved this, we need to determine the stability of the solution. For
equations of the form (24) a necessary and sufficient condition for
to
be a stable solution is that 
. Since
is periodic and
continuous there will in general be two solutions to (26) (see Figure
12B), but
only one of them will occur where 
has a negative slope, so that there
will be a unique stable solution. We must also worry about whether the negative
slope is too steep (i.e., more negative than -2); for small stimuli, this
will never be the case - stability is assured. When
(instability), very
complex behavior can occur such as chaos (e.g., see [13]). The case of
phase-locking where N > 1 is more difficult to explain so we will not consider
it here. It is clear that if the stimulus is weak, the magnitude of 
will
also be small so that 
must be small in order to achieve 
locking.
On
the other hand if the stimulus is too strong, then we must be concerned with the
stability of the locked solution. We note that in a sense equation (24)
is only valid for stimuli which are weak compared to the strength of
attraction of the limit cycle,
since for stronger stimuli it will take the solution more than a single
oscillation to return to points close to the original cycle.
The PRC in Figure 12B shows that, when the stimulus is
depolarizing, it is easier to advance
the Morris-Lecar oscillator and thus force it at a higher frequency
(
) than it is
to force it at a lower frequency (
). For
hyperpolarizing stimuli, we can more easily drive it at frequencies lower than
the natural
frequency. (The counter results are possible but for small ranges of
parameters; also, see [25].)
To illustrate these concepts, we have periodically stimulated the Morris-Lecar
model (natural period of 95 msec) with the same brief depolarizing current pulse
repeated every 76 msec time units. Figure 13 shows that the oscillation is
quickly entrained to the new higher frequency. Equation (26) allows us
to
predict the time after the voltage peak that the stimulus will occur for 
phase-locking. From the PRC we can see that 
corresponds to two values of 
, one stable (cross in Fig 12B) 
and the other unstable. Thus the locking time after the voltage peak,
i.e. when the stimulus occurs, is predicted from the PRC to be 
msec. This is exactly the shift observed in Figure 13.
Integrate this equation. Then use the (Graphic stuff) (Add curve) to add the auxiliary variable stim so that you can see the relative phase shift of the stimulus and the oscillator. Change the period per of the stimulus and see how it affects things. Try to make it 110. Integrate for a longer time. Why can't you lock? Change s0 to -480 (hyperpolarizing). Can you lock now? Now change the period of the stimulus back to 76 keeping the hyperpolarizing current. Can you lock now? Explain the results of these experiments in terms of the PRC.
The technique illustrated here is useful for analyzing the behavior of a single oscillator when forced with a short pulsatile stimulus. For more continuous types of forcing, such as an applied sinusoidal current, other techniques must be used. One such technique is the method of averaging which is applicable when the forcing is weak. Since periodic forcing is just a special case of coupling, we will only describe the latter.
