We assume that motion of each oscillator along its limit cycle can be rewritten in terms of a phase variable. Thus an oscillator's membrane potential is periodic with period T and follows the function where is the phase of the oscillator, j=1,2 and V is the voltage component of the limit cycle trajectory. In the absence of coupling the dynamics is given simply as where is an arbitrary phase-shift. Now consider the effect of small coupling. A brief, weak synaptic current to cell i from activity in cell j will cause a phase shift in cell i
where is the infinitesimal phase-response function, and the minus sign converts excitatory current to positive phase shift. The synaptic current is given by
where the postsynaptic gating variable in cell i is activated by the presynaptic voltage , is the reversal potential for the synapse, and is the strength of the synaptic coupling. The gating variable could be represented by a so-called (event-triggered) alpha function introduced by Rall (cf. chapter 2 of this volume). Alternatively, it could obey a voltage-gated differential equation.
In the method of averaging we simply ``add up'' all the phase-shifts due to the synaptic perturbations and average them over one cycle of the oscillation. Thus, after averaging, one finds that the coupled system satisfies:
where H is a T-periodic ``averaged'' interaction function, given by
The key to these models is the computation of H (see  and ).
In Figure 14A, we show the function along with the synaptic gating variable over one cycle for exactly the same parameters as in Figure 12B. Here is an ``alpha'' function with a 5 msec time constant. Note the similarity (except for scale) of the excitatory PRC and the infinitesimal PRC, As with the PRC, is mainly positive showing that the predominant effect of depolarizing perturbations is to advance the phase or equivalently speed up the oscillator. In only a very small interval of time can the phase be delayed, and this is a general property of membranes which become oscillatory through a saddle-node bifurcation ().
This has been set up so that if you integrate it you will obtain
exactly one cycle with the peak of the oscillation at t=0.
Integrate the equation. Plot the synaptic alpha function, s
as a function of time. Now compute the adjoint as follows. Choose
(nUmerics) (Averaging) (in older versions of XPP, use (Adjoint)
instead of (Averaging)) Choose (New adjoint) and let it compute. Escape
to the main menu and plot V versus t as XPP replaces
the columns with the adjoint. Compare this with the PRC computed earlier. They are close up
to a scaling factor. Now compute the H function. Choose
(nUmerics) (Averaging) (Make H). Type in
Figure 14B shows the function defined in equation (31) for the alpha function shown in Figure 14A and for mV. We can use this function along with equations (29-30) to determine the stable phase-locked patterns for this coupled system. Let denote the phase difference between the two oscillators. From equations (29-30) we see that satisfies:
Here is just the odd part of the function H. Since the coupling is weak, the higher order terms, are ignored. Equation 32 is just a first order equation. Phase-locked states are those for which does not change, that is, they are roots of the function and they are stable fixed points if Since any odd periodic function has at least two zeros, and there will always exist phase-locked states. However, these may not be stable, and there may be others. Synchronous solutions () imply that both membranes fire together. Anti-phase solutions () are exactly one half cycle apart. Figure 14b shows the function and from this we see that there are 4 distinct fixed points: (i) the synchronous (precisely in-phase) solution, (ii) the anti-phase solution, and (iii) a pair of phase-shifted solutions at msec. Both the synchronous and anti-phase solutions are unstable but the phase-shifted solution is stable. Thus, if two of these oscillators are coupled with weak excitatory coupling and the parameters chosen as above, they will phase-lock with a phase-shift of about of the period. Although the classical view is that mutual excitation leads to perfect synchrony, computations with a variety of neuronal models suggest that this is not generally the case.
This type of analysis is easily extended to systems where the oscillators are not exactly identical, when coupling is not symmetric, and when there are many more oscillators. The behavior of such phase-models and the forms of the interaction functions, H, are the topics of current research.