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Next: Summary Up: Phase-resetting and Phase-locking Previous: Phase-response curves

Averaging and weak coupling

The general behavior of coupled neural oscillators is very difficult to analyze, however limiting cases can be treated ([22]). We will describe one method, the method of averaging, that has been used successfully to study the dynamics of two or more neural oscillators that are weakly coupled (e.g., Hansel [16] and Ermentrout and Kopell [9]). In this limit the coupling is sufficiently weak that each oscillator's trajectory remains close to its intrinsic limit cycle. The primary effect of the coupling is to perturb the relative phase between the oscillators much as we described above. However, since the perturbation per cycle is small (with weak coupling) the net effect occurs only over many cycles and the per cycle effect is seen as averaged. For illustration, we summarize the use of averaging to describe the phase-locking properties of two identical Morris-Lecar oscillators when coupled with identical mutually excitatory synapses. Detailed derivations of the equations can be found in the above mentioned papers.

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 [9] and [22]).

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 ([8]).

Figure 14

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


when asked for the Coupling for V and type in 0 for the rest. After a few seconds, the bell will ring and the H will be computed. Escape to the main menu and plot V which is replaced by the interaction function. Then plot w which is replaced by the odd part of the interaction function. Change Vsyn to make the synapse inhibitory, say -75mv. What are the stable phaselocked points? Make the synapse very fast by setting alpha=1 Recompute the orbit by choosing (Init conds) (Last) a few times from the main menu. Compute the adjoint again. Compute the interaction functions for excitatory Vsyn=0 and inhibitory Vsyn=-75 synapses and determine the stable phaselocked states. Change alpha to 20 and do all this again.

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.

next up previous
Next: Summary Up: Phase-resetting and Phase-locking Previous: Phase-response curves

Bard Ermentrout
Mon Jul 29 17:47:46 EDT 1996