Control of multistability in ring circuits of oscillators

Transcription

1 Biol. Cybern. 80, 87±102 (1999) Control of multistability in ring circuits of oscillators C.C. Canavier 1, D.A. Baxter 2, J.W. Clark 3, J.H. Byrne 2 1 Department of Psychology, University of New Orleans, New Orleans, LA 70148, USA 2 Department of Neurobiology and Anatomy and the W.M. Keck Center for the Neurobiology of Learning and Memory, University of Texas Medical School, Houston, TX 77225, USA 3 Department of Electrical and Computer Engineering, Rice University, Houston, TX , USA Received: 29 May 1998 / Accepted in revised form: 18 September 1998 Abstract. The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of su cient complexity, are capable of multistability, that is, of generating di erent ring patterns distinguished by the phasic relationships between the ring in each circuit element (neuron). Moreover, a shift in ring pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the ring pattern. The rst step was to visualize the solution space of the ring circuit, including the attractive basins for each stable ring pattern; this was possible using the relative phase of N 1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the e ect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N 1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Correspondence to: C.C. Canavier ( canavie@uno.edu, Tel.: Fax: Introduction A common and productive approach (Selverston 1980) to studying neural networks is to identify the neurons that compose them, and then to describe the intrinsic electrophysiological properties of the neurons and the type of synaptic coupling that exists between them. However, neurons are an example of nonlinear oscillators, and it has been observed that much of the behavior of coupled nonlinear oscillators is independent of at least some of the details of the properties of the individual neurons and of the coupling, and instead characteristic of the general architecture of the network (Canavier et al. 1997; Collins and Stewart 1993, 1994; Kopell and Ermentrout 1988a). The speci c neural networks in which we are interested are central pattern generators (CPG), and we hypothesize that the essential dynamics of some biological CPGs can be captured by a model consisting of N neurons connected in a ring structure. Previously, only coupling that was brief and pulsatile in nature was thought to fall within the purview of phase-response curve (PRC) theory (Kopell 1988). In this study, we extend our previous e orts (Canavier et al. 1997) to apply PRC methods to coupling that is not pulsatile, but rather more spread out within each cycle. We then add to the theory of CPGs (Kopell 1988) by identifying general principles that specify how to use PRCs to determine the entire solution structure for unidirectional ring circuits, and outline the possible forms the solution structure can take. However, the details of the individual oscillators and their connections determine the speci c PRCs that are expressed, which in turn determine which of the possible forms the solution structure actually takes. We, therefore, examine particular examples of these circuits to show how the general principles can be applied to particular networks of oscillators. CPGs are believed to mediate certain rhythmic, stereotyped behaviors, such as walking, ying, swimming, chewing, feeding, and scratching (Pearson 1993). The

2 88 patterns of neural activity that underlie these behaviors are internally generated (i.e., they do not require sensory feedback) by interconnected neural assemblages designated unit burst generators (Grillner 1975). In this study, a unit burst generator is represented by a single oscillator that is a reduced model of a physiological bursting neuron (see Methods). Physiologically, it is often the case that the same CPG circuitry is capable of generating distinct behaviors, such as walking, trotting, and galloping, or forward swimming and backward swimming. Two mechanisms for altering a CPG so that it produces a di erent pattern have been widely suggested: (1) changing the pattern of connectivity and (2) changing the intrinsic properties of the oscillators themselves via neuromodulation (Harris-Warrick and Marder 1991; Harris-Warrick et al. 1997). Collins and Richmond (1994) proposed a third mechanism, that of altering the driving signal to the CPG, and some of our previous work (Canavier et al. 1997) supports the feasibility of this third mechanism. However, in some cases yet another mechanism for changing output patterns may apply: switching between competing stable patterns using only a transient perturbation. We call this mechanism the control of multistability. We have chosen to focus on ring circuits initially due to their relative simplicity and the frequency of their usage as models of biological oscillators (Collins and Stewart 1994; Ermentrout 1985; Pasemann 1995). We focus speci cally on unidirectional, phasically (synaptically) connected rings, and on 1:1 entrainment modes in this study. The central hypothesis underlying this work is that the complete structure of the solution space of the unidirectional ring circuit can be predicted from the PRC if the solution space is investigated in a coordinate system composed of the relative phases of the individual oscillators. The two assumptions we make regarding the circuit are (1) that the phasic inputs (bursts) received in the circuit by each component oscillator from its respective presynaptic oscillator have substantially the same e ect as a phasic input from an uncoupled oscillator would have on another uncoupled oscillator, and (2) that the e ects of such inputs e ectively die out by the time the next input is received. Mathematically, the latter assumption implies that the oscillator has returned to its characteristic limit cycle oscillation (see Methods) by the time the next input is received. In a stable mode with all oscillators entrained at the same frequency, the elapsed time between inputs is equivalent to one trip around the oscillator's limit cycle. This assumption is less restrictive than the assumption that the e ects die out by the time of the next burst (Dror et al. 1999). In order to predict the entire solution structure of the unidirectional ring circuit, three things must be determined in this sequence: (1) the patterns the circuit can generate in a repetitive manner, (2) the stability of each possible pattern with respect to perturbations, and (3) the particular set of conditions that lead to the expression of each distinct stable pattern (its basin of attraction). A given ring circuit can frequently produce multiple output patterns distinguished by the phasic relationships of the component oscillators, entrained frequency, or both. We have already shown how PRCs can be used to predict what patterns could be exhibited by a unidirectional ring circuit (Canavier et al. 1997). The mere existence of a solution is not su cient for it to be observable in practice due to the issue of stability. Mathematically, a solution is stable if the system returns to this solution after su ciently small perturbations. A rule of thumb for determining some stable patterns was given in Canavier et al. (1997), and Dror et al. (1999) addressed stability in a unidirectional ring circuit more rigorously. However, when there is more than one stable pattern, the pattern currently exhibited by a system depends on the history of the system, including its initialization. Thus, even stability is not a guarantee that a solution will be easily observable. A quantity related to the probability that a given solution will be observed in practice is the relative stability of a solution. There are at least two ways to gauge the relative stability of a solution. One is the speed with which the solution reestablishes itself after a perturbation, and the other is the size of the perturbation that a solution can withstand. If a solution restablishes itself very slowly after a perturbation, the chances are increased that yet another perturbation will be received before the solution is reestablished, decreasing the chance that it will eventually be reestablished. This type of relative stability in terms of speed of convergence is related to the slope of the PRC curves as presented in Dror et al. (1999). Alternatively, some solutions may have such a small set of initial conditions that lead to them, that is, such small basins of attraction, that it is unlikely that the system will converge to those solutions. The present study addresses relative stability in terms of the size of the perturbation a solution can withstand, which is, of course, related to the size of the basin of attraction. The minimum perturbation that will precipitate a transition to another solution is very much dependent upon the phase at which the perturbation is applied, and, as we shall show in this study, upon the geometry of the attractive basin of the solution. This study uses PRCs for the rst time to predict the basins of attraction of the stable patterns, as well as the perturbations that would be e ective in inducing a transition between patterns, thus providing a scheme for controlling such transitions. This control scheme identi es the appropriate switch signal for each desired transition between patterns composed of inputs to some or all of the component oscillators, in which each input has a characteristic sign, intensity, duration, and timing. In this study, we systematically identi ed the required switch signals for circuits with two, three, and four oscillators. Our method was as follows. First, the attractive basin of each pattern was delineated using a circuit emulator (DeFranceschi 1995) based on PRC theory (see Methods). This enabled the prediction of the amount of resetting within a cycle that was required for each oscillator. Next, the duration of the switch signal was arbitrarily xed at the duration of an intrinsic burst, and the spike frequency was not allowed to vary within the

3 89 burst, whereas the sign and intensity of the synapse(s) through which the switch signal was delivered were adjusted so that the required phase resetting for each oscillator was attainable. Finally, the timing of each input was selected in order to produce the desired amount of phase resetting. We do not suggest that biological oscillators utilize our algorithm, only that biological organisms may implement some strategy to produce results similar to ours. 2 Methods 2.1 Component oscillators and the ring circuit In this paper, a general technique for predicting the attractive basins for competing patterns exhibited by a unidirectional ring circuit is presented. A particular model of an oscillator (Butera et al. 1995) is used as an example to illustrate the technique. In this reduced model of a bursting neuron, a single current is responsible for burst initiation and termination, due to its voltage activation and calcium inactivation. The four variables in the reduced model are the slow activation variable for the current (s), the internal calcium concentration ( CaŠ i ), membrane potential (v), and spike frequency (q). Action potentials are not modeled explicitly; instead, the variable q (a function of the distance from the spike threshold in the phase plane containing the slow variables s and CaŠ i ) represents the instantaneous spike frequency, which is positive during a simulated burst. For details of the model and an explanation of the parameter values given in the gure legends, see Butera et al. (1995). We chose this model for two reasons. First, it is based on a physiological bursting neuron, hence the parameters have meaning in terms of the biophysics and biochemistry of bursting neurons, which facilitates an analogy with real neurons and enables a better intuitive understanding of parameter manipulations. Second, the level of complexity of this model, which derives from the presence of two variables that are slow in comparison with the other two as well as the large number of parameters associated with each variable, enables one to generate PRCs with the desired characteristics more easily than for a simpler model (such as a Van der Pol oscillator). The individual oscillators are connected in a ring pattern by synapses in which the postsynaptic current is proportional to the presynaptic spike frequency, hence the scale factor for the synaptic strength (g syn ) is given in ls/hz. This current is ohmic with a reversal potential at E syn and no kinetics: I syn ˆ g syn q pre v post E syn. For details of the implementation of the ring connections into the oscillator model, see Canavier et al. (1997). The four simultaneous nonlinear di erential equations per oscillator were solved using a fth-order implicit Runge-Kutta method (Hairer and Wanner 1991) to determine which patterns were generated by the ring circuits and how perturbations induced switching between patterns. 2.2 Modi ed phase-response curve method In classical PRC theory (Kopell 1988; Murray 1989; Pavlides 1973), a brief perturbation is applied to an uncoupled oscillator at various phases in the burst, and the e ect on the period of the current cycle is tabulated. However, here we attempt to predict the behavior of the circuit from the response of the uncoupled oscillator; therefore, we tried to make the perturbation with which the PRC was generated resemble as closely as possible the input that the oscillator would receive in the circuit. Thus, the perturbation chosen to generate the PRCs was a single burst from an identical oscillator acting via a synapse identical to those employed in the circuit (see Fig. 1). In some cases, such as the one shown in Fig. 1, the long duration of the input stimulus did not require a di erent treatment than classical PRC theory. If a stimulus were given repetitively at t s, then the driven oscillation would have an entrained period (P e ), which could be predicted by the e ect of a single stimulus on the rst burst following the stimulus. However, as a result of the long perturbation, sometimes the rst burst following a perturbation occurred before the perturbation had ended (see Fig. 2). This necessitated a deviation from classical PRC theory in that a single PRC was no longer su cient. For example, in order for subsequent stimuli in a train to be delivered when the oscillator is at the same phase as it was when it received the rst stimulus, the latency of the stimulus after the beginning of the burst must now be t s;e rather than t s;i because of the resetting e ect that occurs after the rst burst Fig. 1A,B. Classical phase response curve theory (applied to a long pulse). A The free running oscillation exhibited by a single uncoupled oscillator (solid line) is shown here, with the spike threshold indicated by the dotted line and the intrinsic period (P i ) indicated by the interval between the start of successive bursts. B An identical presynaptic oscillator is coupled to the oscillator (see inset) for a single presynaptic burst (dot-dash line) at time t s, and the perturbed period (P e ) is measured. P e is equal to the entrained period produced by a stimulus applied at stimulus time (t s ) after the beginning of a burst. This process is repeated for various values of t s between 0 and P i.onecan then plot the change in period (F / ) versus the phase of the applied perturbation / in order to generate a PRC. In this and subsequent circuit diagrams, an inhibitory synapse is indicated by a solid circle and an excitatory one by an open triangle

4 90 F 1 / ˆ P 1 P i =P i F 2 / ˆ P 2 P i =P i 1 2 where F 1 / constitutes the PRC curve for the e ect on the rst burst, and F 2 / the e ect on the second burst. Other useful de nitions are that F / ˆF 1 / F 2 /, t s;e ˆ P i / F 2 /, P e ˆ P i 1 F /, P 1 ˆ P i 1 F 1 /, and P 2 ˆ P i 1 F 2 /. Due to causality, if / is between 0 and 1, F 1 / cannot be less than / 1. Otherwise, an input could advance the next burst to a point in time before the input was received. Therefore, as / increases toward 1, F 1 / cannot approach a negative value. If, for example, F 1 0 is negative, then F 1 / cannot be a periodic function of / with a period of 1, because it will be discontinuous at 1. However, under our assumptions, F / is a continuous periodic function, thus enabling the use of the emulator algorithm described below. Fig. 2A,B. Modi ed phase-response curve theory. A The free running oscillation for two intrinsic periods is shown. B The predicted entrained period (P e ) is rede ned as the amount of time required to return to the same conditions (associated with a particular phase /,or point on the limit cycle, see Fig. 3) that prevailed immediately prior to the stimulus applied at time t s;i. The e ect of the stimulus on the timing of the rst burst is tabulated as F 1 /, whereas that on the second burst is tabulated as F 2 /. The total e ect on the entrained period F / is the sum of F 1 / and F 2 /. t s;e is the time elapsed between the rst burst following the stimulus and the return to prestimulus phase. This elapsed time is equivalent to the ring interval under conditions of entrainment by a stimulus repeatedly given at phase /. The main e ect of including F 2 / is to make the PRC continuous at t s;i ˆ 0andt s;i ˆ P e, which in turn allows the emulator to make correct predictions, particularly with respect to the synchronous modes following the initial perturbation. Also, in order to calculate P e for a repeated stimulus at phase /, the PRC generated by the alteration in the rst burst is not su cient; the e ect of the second burst must be known as well. Therefore, it was clearly necessary to tabulate the e ect on the second burst following the perturbation as well the e ect on the rst burst, so that two PRCs (one for the rst burst and one for the second) were generated. The driven oscillator (solid lines), with the indicated intrinsic period (P i ), was initialized at the beginning of a burst, which we de ned as the point at which the spike frequency q became positive. The driven oscillator was then integrated until the start of the perturbation (t s;i ), at which time the connection from the driving oscillator (dot-dash line) was turned on for the duration of a burst in the driving oscillator. The time between the beginning of the burst before the start of the perturbation and the burst after start of the perturbation is P 1. The time elapsed between the beginning of the rst and second bursts following the start of the perturbation is P 2. The results were tabulated in terms of /, which for purposes of PRC generation is equal to t s;i =P i : 2.3 Mapping the ring circuit dynamics onto the emulator Classical PRC theory assumes that each oscillator can be reduced to a single variable, the absolute phase h n, which in an unperturbed oscillator is equivalent to the elapsed time normalized by the intrinsic period. The phase space associated with the single oscillator, which normally would have the four dimensions mentioned above (s, CaŠ i, v, andq), can be reduced in the case of limit cycle dynamics to a line (a single dimension) from 0 to 1 with the position along this line indicated by h n. An oscillation generated by an example uncoupled oscillator is shown in Fig. 3A, and its corresponding limit cycle (Fig. 3B) is plotted for convenient viewing in the plane of the two slowest state variables of the model, with the phase indicated by the numbers alongside the cycle. If an input is received at any given phase, the phase is either advanced along the limit cycle, which shortens the period since a portion of the limit cycle is skipped, or delayed by being reset to an earlier phase, which lengthens the period because the trajectory must then traverse a portion of the limit cycle twice. In our extension of PRC theory to the ring circuit, each oscillator is assumed to follow its own limit cycle independently, with the only e ect of the coupling being to reset the phase of the postsynaptic oscillator along the limit cycle at the time of a burst in the presynaptic oscillator (with the caveat that sometimes the e ect on two bursts must be taken into account). Therefore, all the information needed to predict the future activity of a ring circuit resides in the PRCs. Rather than integrating all of the equations required to produce the limit cycle, we can merely assume that the phase of each oscillator will increment linearly in time, except when the phase is reset by an input. Phase resetting is assumed to occur instantaneously. The dynamics of the ring circuit under these assumptions can be reduced to a single dimension per oscillator. For ease of visualization, the dimensionality of the system can be reduced by one more dimension when we

5 91 Fig. 3A±E. Mapping the ring circuit dynamics onto the emulator. A Uncoupled oscillator model output. The waveform produced by an example free running uncoupled oscillator (solid line) is indexed by its absolute phase. Dotted line shows the threshold for the beginning of a burst (phase = 0). B Phase along the limit cycle. The limit cycle shown for the uncoupled oscillator in A is plotted in the plane of the two slowest state variables of the model oscillator, and each point is associated with a unique set of four state variables. The numbers alongside the cycle indicate the elapsed time normalized by the intrinsic period, where one trip around the limit cycle occurs during one period of the waveform in A. C Ring circuit model output. This is an example of how the waveform is modi ed as a result of coupling oscillators into the ring circuit. The waveforms are coded by oscillator as shown in D. As described in the text, this system can be analysed in terms of the phases of oscillators 2, 3 and 4 relative to oscillator 1, sampled once per cycle as shown at the arrow-marked trajectory point. D Circuit diagram. This particular example is for a ring of four oscillators coupled with inhibitory connections as shown. E Emulator output. The output of the emulator is coded by oscillator as shown in D. The phases increment linearly in time except when an input causes an instantaneous reset. The trajectories in the phase space of the emulator are mapped by taking the phases relative to oscillator 1 once per cycle when the absolute phase of oscillator 1 is zero, as shown at the trajectory point (arrow) consider that it is the relative phases of the oscillators that are signi cant in di erentiating between patterns exhibited by the ring circuit. In other words, if a particular pattern is shifted in time, it remains the same pattern. Therefore, the absolute phase along one of the limit cycles (designated the reference oscillator) can be eliminated as a dimension, provided that the absolute phase of the remaining oscillators is transformed into the phase relative to the reference oscillator. In this paper, the reference oscillator will be the number 1 oscillator in all cases. The trajectory of the system dynamics can then be plotted by taking a point composed of the relative phase of N 1 oscillators at the time the reference oscillator begins a burst ± actually when its absolute phase is zero, simply for ease of computing the relative phases. This technique allows the mapping of the attractive basins of the various patterns generated by the ring circuit. If one were to attempt this technique with the full model, computationally one is faced with a 4N dimensional system of nonlinear di erential equations, whereas the emulator is a much simpler system in which entire cycles around the limit cycle can be determined with just a few algebraic calculations of an iterated mapping. Moreover, in the full system, phase resetting does not occur instantaneously, and hence the phase in terms of the limit cycle is not clearly de ned following perturbations. This can be seen in Fig. 3C, in which the waveforms of the coupled oscillators in an example ring circuit (Fig. 3D) are quite di erent from that of the uncoupled oscillators (Fig. 3A), indicating that the trajectory is not on the limit cycle at all times, which is hardly surprising given the duration of the perturbations. By contrast, the output of the emulator is shown in Fig. 3E. The absolute phases follow a sawtooth waveform as they are linearly incremented, except at the instantaneous resets due to their own ring or the ring of their presynaptic oscillators. The arrows indicate the points at which the phases are sampled for trajectory points used to delineate basins of attraction. The simpli cation made possible by the mapping of the ring

6 92 circuit dynamics onto the emulator allows a visualization of the system that could not be achieved otherwise. 2.4 Using modi ed PRC theory to predict stable patterns Two previous studies addressed the existence and stability, respectively, of patterns in which all of the oscillators in the ring are entrained at the same frequency (Canavier et al. 1997; Dror et al. 1999). Only a limited number of patterns can be produced in a repetitive fashion; prior to settling into such a pattern (i.e., during transients), the phasic relationships between the oscillators are continuously changing. Thus, the hallmark of a pattern, or mode, is that the phasic relationships between pre and postsynaptic oscillators (/ n ) are constant, where / post ˆ h post h pre. In this study, the / n are determined when h pre =0 or at the time the presynaptic oscillator begins a burst. The PRC can be used to predict the values of / n that comprise a solution mode. The two criteria for existence of a mode, as developed in the previous study (Canavier et al. 1997), are formulated here for identical and identically connected oscillators entrained at a common frequency. In the terminology of the present study, these criteria are that (1) the sum of the ring intervals (t s;e ) between oscillators must add up to the common entrained period, P e, or a multiple thereof, and (2) each presynaptic oscillator must produce the same amount of phase advance or phase delay in its postsynaptic oscillator. Although the model used as an example in this study does not explicitly model spikes, the oscillator is considered to be ring when the state variable representing ring frequency is positive, and hence the term ring interval is applied to the interval t s;e, even in the absence of explicit spikes. Using the de nitions of t s;e and P e given in the section on the modi ed phase response curve method, and factoring out P i gives the rst criterion as: X N nˆ1 / n F 2 / n Š ˆ j 1 F / n Š 3 The quantity on the lefthand side is the normalized sum of the ring intervals, and the quantity on the righthand side is a multiple of the normalized entrained period. The second criterion is that F / 1 ˆF / 2 ˆˆF / n 4 That is, if the / n are not equal, then they must at least have equal values of F / n in order to be entrained at the same frequency. Only a limited number of sets of values of / n can satisfy both existence criteria for a given circuit, and numerical methods can be employed to examine the PRCs in order to identify all such sets. The relative phases (with respect to a reference oscillator) in a mode that satis es the existence criteria can be calculated as follows: h 1 ˆ 0 5 h n ˆ Xn / j F 2 / j Š 6 jˆ2 This conversion allows the modes that meet the existence criteria to be plotted as xed points in the N 1 dimensional space of the relative phases of the oscillators. The h n are equivalent to the time elapsed between the beginning of a burst in the reference oscillator and the beginning of a burst in oscillator n, normalized by the intrinsic period. If a particular h n exceeds 1 F / n, as it may in modes with j > 1, then a multiple of 1 F / n should be subtracted from h n until it is less than 1 F / n. It can easily be shown that h N ˆ 1 F 1 / 1 / 1. Thus, if modes are encountered in which F 1 / > /, the axes of the phase space will be extended beyond 1 to the maximum value of h n. There is a subtle distinction between when / n and h n are sampled, as explained in the section on the emulator algorithm. The advantage of using h n over / n as coordinates of the state space is that only N 1 h n need be plotted since h 1 is zero, even during transients. The / n can only be reduced to N 1 within a mode that meets the existence criteria, in which case, N 1 / uniquely determines the remaining value of /, using the rst existence criterion. The two previous studies each used somewhat different terminology than the current study. In addition, the underlying assumptions di ered slightly in each of the studies. In order to clarify the di erences, the equation embodying the rst criterion for existence has been recast for each of the previous studies using the terminology as de ned in the current paper. The rst study (Canavier et al. 1997) used the criterion: X N nˆ1 / n Šˆj 1 F / n Š 7 hence the addition of F 2 / n to the lefthand side of the equation in the current study increases the accuracy of the predictions. The second study focused on stability and ignored F 2 / n entirely because the stability analysis is much more tractable if one assumes that an oscillator returns to the limit cycle by the next time it res (begins a burst), rather than, as in the current study, by the time it receives its next input. Therefore, the second study (Dror et al. 1999) used the criterion: X N nˆ1 / n Šˆj 1 F 1 / n Š Emulator algorithm We assumed that the two types of PRCs (/; F 1 / and /; F 2 / ) generated by the method described above together contained all the information necesssary to describe the dynamics of the circuit. The emulator code had access to two les containing the pairs /; F 1 / and

7 93 /; F 2 /, respectively, generated as described above at numerous sequential values of /, with the capability to interpolate between points if necessary. The emulator kept track of the time elapsed and of the absolute phase of each oscillator h n, normalized in terms of P i to between 0 and 1. When the phase of an oscillator reached 1, it was instantaneously reset to 0, which was considered the beginning of a burst. Phase in this context is interpreted not just in terms of time elapsed since the previous burst, but rather in terms of position along the limit cycle associated with the oscillation. The phase of an oscillator is advanced along the limit cycle with the passage of time, but may also be instantaneously reset by being moved forward (advanced) or backward (delayed) along the limit cycle as a result of an input perturbation. The emulator was initialized with arbitrary values of the phase of each oscillator. At each step, the emulator determined which oscillator would re next by checking for the maximum phase. Then the phase of each oscillator was increased by the amount required to increase the maximum phase to 1. The oscillator at phase 1 is tabulated as having red, and its phase is reset to 0. The postsynaptic oscillator to the one that has just red has its phase reset as follows: h post newš ˆh post oldš F 1 h post oldš 9 The absolute phase (h post oldš) of the postsynaptic oscillator is equal to the relative phase (/ post ) between the post and presynaptic oscillators since the presynaptic oscillator has just been reset to 0 (h post h pre ˆ / post ). The / n that describe a pattern are de ned prior to correction for F 2 /. The emulator must also remember the quantity F 2 h post in order to subtract it from the phase of the postsynaptic oscillator after the next time it bursts (when it becomes the presynaptic oscillator and is reset to zero), in order to properly account for the e ects of a perturbation on the second burst after an input perturbation: 1 h pre newš ˆ0 F 2 rememberedš 11 The h that describe a pattern are de ned after correction for F 2 /, which, unlike the /, enables them to be equated exactly with the observed ring intervals. Under certain conditions, h may go negative, which has no meaning in terms of the limit cycle, but serves as a `placeholder' to lengthen the period. In fact, in Fig. 3E the phases are slightly negative immediately following the F 2 rememberedš adjustment after the reset to zero, although it is not obvious due to the limits of the graphical resolution. The e ective output of the emulator was the phase of all of the oscillators but one, the 1 A mathematically equivalent formulation is _h i t ˆf d t t n;i 1 F 1 h i t n;i 1 d t t n;i F 2 h i t n 1;i 1 10 where f is the intrinsic frequency, t n is the time of the most recent spike in the indexed oscillator, t n 1 is the time of the previous one, and i 1 is the index of the presynaptic oscillator. reference oscillator (h 1 ), at the time that the reference oscillator began a burst. This output was used to map the attractive basins of various competing patterns and make predictions about how to induce transitions between patterns. The order of the emulator is actually 2N because of the F 2 rememberedš array, but in practice, the e ect of F 2 is small and can be left out of the phase space visualization techniques for which only N 1 dimensions are required. The inclusion of this array is crucial, however, in order for the patterns generated by the emulator to match those generated by the full circuit. 3 Results The methods described above extend classical PRC theory, particularly with respect to application to a ring circuit of oscillators. While these methods in theory provide a way to map out the basins of attractions of such circuits a priori, in order to determine the usefulness of these methods, it was necessary to test them on circuits composed of model oscillators. Below, examples are given for the application of these novel methods to circuits composed of two, three, and four model oscillators. These circuits are not designed to correspond to particular biological circuits, but rather to illustrate the potential application of the PRC methods. The example circuits were chosen because they exibited two or more stable patterns characteristic of ring circuits of that size. For example, the two-oscillator circuit exhibits both synchronous and alternating modes. The di erent PRCs were produced by varying the parameters of the model in Butera et al. (1995) and the synaptic parameters E syn and g syn, as noted in the gure legends. The most sensitive parameter is E syn, the reversal potential of the synapse, because PRCs generated using excitatory versus inhibitory inputs have different characters. Other sensitive parameters include the strength of the synaptic connection and any parameter that a ects the kinetics of the slow variables. Excitatory connections were chosen for the circuits with two and three oscillators because it is much easier to produce synchrony using excitation. Inhibitory connections were chosen for the circuit composed of four oscillators because a variety of patterns could be produced with such a circuit (not including synchrony, however). An important assumption that was made while investigating perturbations that could induce transitions between patterns is that the perturbations to each oscillator were restricted to those that could be examined using PRC theory. That is, the system must return approximately to the limit cycle within approximately one period so that PRC methods could be applied to the transition-inducing perturbations. In practice, this restriction was implemented by requiring the duration of the pertubation to be no longer than the intrinsic bursts exhibited by the oscillators. The PRC for the perturbation was not broken down into two components because the additional complexity was not found to be necessary to predict the induction of transitions.

8 Two oscillators The simplest possible ring is one composed of two oscillators as shown in Fig. 4A. This circuit was capable of producing both a synchronous and an alternating mode, as shown in Fig. 4D1 and D2. In order to determine what type of perturbation to apply to this system to switch between modes, a PRC analysis was performed as described in Methods, using a single intrinsic burst as the perturbation to an uncoupled oscillator. Figure 4B shows a plot of F / versus / that was used to predict the existence of both stable and unstable modes as described in the Methods. Three possible modes were identi ed. As described previously (Canavier et al. 1997; Dror et al. 1999), positive slopes of the PRC at / between 0 and 1 usually correlate with stability, whereas negative slopes are unstable. Hence the synchronous mode (0 or 1, open circles) shown in Fig. 4D1 and the alternating mode (0:52P i, solid circle) shown in Fig. 4D2 are stable. A third mode, characterized by alternating values of / of 0.39 and 0.62 (asterisks) is not stable due to the negative slope at / ˆ 0:62. The two-oscillator circuit can be reduced to a single variable as described in Methods. If we sample h 2 only at the time a burst is initiated in the reference oscillator (i.e., h 1 ˆ 0), h 2 is not only the absolute phase of oscillator two, but also the relative phase at the sampling point. The ring interval is also equal to h 2, and in a mode that meets the existence criteria, this reliably repeated interval can be calculated as / 2 F 2 / 2. F 2 / is relatively small in this example, so h 2 and / 2 are approximately equal. In the 1D phase space of this circuit (Fig. 4C), the unstable mode forms the boundary between the two stable modes. Therefore, when the phase of the reference oscillator (h 1 ) is 0, if the phase of the other oscillator (h 2 ) is between 0.39 and 0.62 the alternating mode will result, otherwise the synchronous mode will result. The asterisks at both 0.39 and 0.62 correspond to rotational variants of the same mode. In order to switch between the modes, a switch signal was added to the circuit in Fig. 5A using an inhibitory synapse. This switch signal was used to generate the PRC given in Fig. 5B. The timing of the switch signal was determined simply by identifying a time at which the phase advance or delay met the minimum criteria to produce a transition. The number of cycles required for a switch to be e ected depends on the rate at which the transients die out, which is related to the magnitude of the slopes of the PRC associated with the eventual mode, and on how close the perturbation is to the exact perturbation required. Since the slope of the PRC is steeper at 0.52 than at 0, the system will converge more rapidly toward the alternating solution. The emulator described in the methods con rmed the existence and location of these boundaries. The boundaries imply that in the synchronous mode, h 2 must either be delayed by more than 0.39 or advanced by more than 0.38 in order to produce a transition. In Fig. 5C1, an oscillator received a switch signal when its phase was 0.93, producing a delay of P i, which was su cient to induce a Fig. 4A±D. Ring with two oscillators can produce both synchronous and antisynchronous patterns. A Circuit diagram. Two reciprocally connected oscillators with identical excitatory connections. Parameters were as in Butera et al. (1995) Figs. 5A and 6A, except that I stim ˆ)0.5 na and the time constant associated with the activation variable s was scaled to half its value at all potentials. The synapses have a reversal potential E syn of )30 mv and a conductance of g syn of ls=hz. B Phase-response curve. This PRC used a single presynaptic burst as the perturbation. Solid line shows F /, and dotted line shows the contribution of F 2 /. The synchronous mode (intervals equal to 0, indicated by the open circles) and the alternating mode (intervals equal to 0:52P i, solid circle) are stable, whereas a mode characterized by alternating intervals of 0:39P i and 0:62P i (asterisks) is unstable. C Emulator phase diagram. In the 1D space of the emulator, the boundaries between the two modes are indicated on a line with the arrows giving the direction of ow, where 1 is equivalent to0.d This circuit is capable of exhibiting a synchronous mode (D1) or an alternating mode (D2). Dotted line in both D1 and D2 indicates the threshold for the beginning of a burst, about )42 mv shift. On the other hand, in the alternating mode h 2 = 0.52, and must be delayed by more than 0.13 (0.52 ) 0.39) or advanced by more than 0.10 (0.62 ) 0.52) in order to induce a transition. In Fig. 5C2, an oscillator received a switch signal when its phase was 0.94, producing a delay of P i, which was well in excess of that required to induce a shift. While smaller shifts are also e ective, they take longer to settle into the synchronous mode. The phase of the oscillator at the time of the switch signal was calculated by dividing the time elapsed since the beginning of the previous burst by P i.if an input had been received in the interim, the phase was adjusted by the appropriate amount of advance or delay. Since a larger phase shift is required to go from the synchronous mode to the alternating mode than vice versa, in one sense the synchronous mode, in this instance, is more robust to perturbations. 3.2 Three oscillators A ring circuit composed of three oscillators was constructed as shown in Fig. 6A in order to produce the three distinct patterns shown in Fig. 6C. The PRC produced by a component oscillator in response to a burst from an identical oscillator coupled via a synapse

9 95 Fig. 5A±C. Switch signal produces transitions between modes. A Circuit diagram. A transient switch signal has been added to the circuit and connected to only one oscillator. B Switch signal PRC. The switch signal is equivalent to a burst of constant frequency of equal duration to the intrinsic bursts of the oscillators in the circuit, but with a reversal of potential (E syn ˆ 100 mv) corresponding to an inhibitory connection. Since the burst was of constant frequency, the quantity g syn q pre was held constant at ls for the duration of the input perturbation. A PRC was generated using this signal, rather than a burst, as the perturbation. C Using this transient switch signal, transitions between the synchronous and alternating modes could be induced by applying the stimulus at a phase of about 0.9 when the phase of the oscillator indicated by the dashed lines was reset by about 0.5 in order to produce a speedy transition. Dotted line in both C1 (synchronous to alternating transition) and C2 (alternating to synchronous transition) indicates the threshold for the beginning of a burst, about 42 mv identical to the ones in the circuit is shown in Fig. 6B. There must be three values of / associated with a pattern that can be produced by a ring with three oscillators, but the / need not all be distinct. All values of / may be identical, in which case the mode is symmetrical, and there are no rotational variants. Alternatively, only two might be the same, in which case there are three possible rotational variants of the pattern; in each of the variants, the value of / occupies a di erent one of the three possible positions between the oscillators. In this example, no modes with three distinct / were encountered. As a rule of thumb, stable modes have a small positive slope on the PRC at each values of /. Unstable modes have all negative slopes. Saddles, which attract in some directions and repel in others, have at least one positive and one negative value. Analysis of the PRC indicated that the modes at / =0, 0.337, and were stable (open circles) and corresponded to the synchronous, counterclockwise (with respect to the ring), and clockwise modes, respectively. In these modes, there is only a single value of /, repeated three times. In addition to the three stable patterns, the analysis of the PRC predicted an unstable mode with one / of and two of 0.438, indicated by the two connected solid circles. This mode is an unstable repellor because the slopes associated with both of these / are negative. Finally, two saddles, corresponding to the pairs of points represented by asterisks, were also identi ed, each with one positive slope and one negative. The pairs of points that compose each saddle are connected by a horizontal dashed line: the upper set of asterisks is at (value of / repeated twice) and 0.437, and the lower set of asterisks is at and (repeated twice). In a circuit of three or more oscillators, a clear distinction needs to be made between the phases between adjacent oscillators around the ring, that is, the phase of the postsynaptic oscillator at the time the presynaptic oscillator begins a burst, which we shall designate /, and the phase of each oscillator at the time that the reference oscillator begins a burst, which we shall designate h. The PRC allows us to predict the N values of / associated with each mode. The / can be converted using simple formulae from Methods to the appropriate h. h 1 ˆ 0 h 2 ˆ / 2 F 2 / 2 h 3 ˆ / 2 F 2 / 2 / 3 F 2 / The symmetric modes, that is, those in which all three / are identical, translate into a single point in this plane. However, as stated above, those modes that have one / di erent from the other two can have this unique value of / equal to / 1, / 2,or/ 3, hence three distinct xed points are associated with the rotational variants of such a mode. The xed points have been transformed to the plane de ned by h 2 and h 3 in Fig. 7. The curves indicate the trajectories in this space from any given point. If a trajectory leaves the phase plane on the righthand side it reemerges on the left, and vice versa, and similarly for top and bottom. The three attractors are shown by open circles. All trajectories terminate on an attractor. The trajectories terminating on an attractor constitute its

10 96 Fig. 6A±C. Ring of three oscillators can produce three stable modes. A Circuit diagram. Parameters were as in Butera et al. (1995) except that I stim ˆ )0.6 na, I CaP = 15.0 na, Vol i ˆ 1:0 nl,g R =0.15lS, and the time constant associated with the activation variable s wasscaledtoa tenth of its value at all potentials. E syn was set to )25 mv and g syn to ls/hz. B Phase response curve. The PRC (F / ) for this oscillator with stable attractors indicated by open circles at 0 (synchronous), (counterclockwise), and (clockwise). Values of / associated with an unstable xed point are indicated by solid circles at and Pairs of values of / associated with two saddles (stable in some directions but not others) are indicated by asterisks at and and at and Since unstable xed points and saddles occur in pairs (one of the two ring intervals is used twice in each case), the pairs are connected by a dashed line to show which ones go together. Dotted line shows the contribution of F 2 /. C Stable oscillatory modes. This circuit was capable of ring synchronously (C1, 0 on the PRC) and of ring regularly spaced bursts with ring order that was either counterclockwise (C3, on the PRC) or clockwise (C2, on the PRC) with respect to the ring. Dotted line in C1, C2, andc3 indicates the threshold for the beginning of a burst, about )52 mv basin of attraction and are therefore color-coded by attractor. The open circle at 0; 0 corresponds to the synchronous mode (green lines), the open circle at 0:34; 0:67 corresponds to the counterclockwise mode (red lines), and the one at 0:64; 0:32 to the clockwise mode (blue lines). The entire ow can be mapped by the trajectories emanating from the three unstable xed points ( lled circles). The six saddle points (asterisks) attract trajectories from the unstable xed points, but then repel them toward the attractor. The xed points together de ne the ow, and an approximation of the ow can be made knowing only the xed points. Due to symmetry, xed points occur either singly or in sets of three, hence the triangles whose vertices consist of a set of unstable xed points or whose sides are drawn through a set of saddle points approximate the attractive basin of a xed point within. If modes with three distinct / were encountered, a symmetry based on six rotational variants rather than three would be expected. From the phase-plane plot of this circuit, we can deduce that perturbing a single oscillator will never induce a shift between the clockwise and counterclockwise modes because you cannot draw a horizontal (perturb oscillator 2 only) or a vertical line (perturb oscillator 3 only) from one attractor to the attractive basin of the other. One can also deduce from Fig. 7 how much the phase of oscillators 2 and 3 need to be reset in order to induce a transition between any two modes. Three such transitions are shown in Fig. 8. Usually, one oscillator needed to be advanced and the other delayed. Two different strategies were implemented for di erentially perturbing the two oscillators. The rst was to perturb both oscillators at the same time, but through di erent synaptic connections which resulted in di erent PRCs. The second was to use the same synaptic connections but to stagger the perturbations so that di erent points on the PRC curve predicted the resultant phase resetting. Figure 8A shows the rst strategy, a transition induced by the simultaneous application of the switch signal at two di erent excitatory synapses, with slightly di erent parameters that produce slightly di erent PRCs. At left, the circuit diagram is shown with a transient switch signal applied to two oscillators. A transition from the synchronous mode to the clockwise mode was desired, so Fig. 7 was examined to determine the optimum perturbation, given the PRCs in Fig. 8A1. The four corners of Fig. 7 are all nearly equivalent to the synchronous mode since normalized ring intervals of 0 and P e =P i are equivalent in a mode that meets the exis-

11 97 Fig. 7. Phase plane for the three-oscillator circuit emulator. The x-axis gives the phase of oscillator 2 at the beginning of a burst in oscillator 1, and the y-axis gives the phase of oscillator 3 at the same time. These phases are not relative to the postsynaptic oscillator as in Fig. 6, but to a single reference oscillator, hence the xed point attractors identi ed in Fig. 6 have been transformed to the new coordinates. The three attractors are shown by open circles. The open circle at 0, 0 corresponds to the synchronous mode (green lines), the open circle at 0.34, 0.67 corresponds to the counterclockwise mode (red lines), and the one at 0.64, 0.32 to the clockwise mode (blue lines). The entire ow can be mapped by the trajectories emanating from the three unstable xed points (solid circles). The six saddle points (asterisks) attract trajectories from the unstable xed points, but then repel them toward the attractor. Dashed line in the lower right portion of the gure shows the shortest path between the synchronous mode and the clockwise mode (see text and Fig. 8A) tence criteria, and P e =P i is nearly 1 in this example. Hence, the shortest distance from the synchronous mode can be seen as a line drawn between the lower right corner at (1,0) and the asterisk at ( ). Thus, a minimum of an advance of to oscillator 2 and a delay of to oscillator 3 should produce a transition from the synchronous mode to the clockwise mode. At a phase of 0.5 (vertical dotted line), the input to oscillator 2 produces an advance of 0.302, whereas the input to oscillator 3 produces a delay of 0.279, which is very close to the minimum required perturbation. Applying this input at the time shown by the thick line in Fig. 8A2 was indeed su cient to produce a transition to the clockwise mode. The two PRCs that correspond to the two switch inputs in the top trace are very similar. They both have a sharp shift from phase delay to phase advance near midcycle. The illustration of the transition induced from the synchronous mode reveals the reason for the apparent discontinuity. A perturbation delivered at midcycle can either trigger a burst (see dashed waveform of oscillator 2 that exceeds threshold during and immediately after the perturbation) or not trigger a burst (see dotted waveform of oscillator 3 that does not exceed threshold during or immediately after the perturbation). In the rst case, a burst occurs before the expected time of the next burst in an unperturbed oscillator and is tabulated as an advance. In the second case, the subthreshold perturbation actually delays the onset of the next burst. Though the two perturbations were very similar, their e ects were drastically di erent, because they fell on opposite sides of the discontinuity. The predictions of the phase resetting required are very robust even though the shape, duration, and amplitude of the bursts in the circuit can di er greatly from the intrinsic bursts with which the predictions were generated. In the example of Fig. 8B, a similar strategy is applied, except that one inhibitory (solid circle) and one excitatory connection is used. Since the initial mode is not the synchonous mode, even though the switch signal is applied to both oscillators at the same time, the two perturbed oscillators do not have the same phase, so the PRCs are shifted appropriately to show the amount of reset expected. A transition from the counterclockwise to the clockwise mode is induced. Note that the PRC for the inhibitory synapse di ers qualitatively from that of the excitatory synapse: the discontinuity falls near the end of the cycle. In this case a stimulus just prior to a burst either keeps it from initiating within one intrinsic period (a delay) or it cuts the burst short, which advances the second burst following the stimulus. Therefore, this is usually also the region in which F 2 / becomes signi cant. Finally, in the example of Fig. 8C, rather than apply simultaneous switch inputs with potentially di erent

12 98 Fig. 8A±C. Strategies for switching between modes. A switch signal was added to oscillators 2 and 3. The switch signal is again equivalent to a burst of constant frequency of equal duration to the intrinsic bursts of the oscillators in the circuit. Since the burst was of constant frequency, the quantity g syn q pre was held constant for the duration of the input perturbation. In A, the switch signal for oscillator 2 had E syn =0mVandg syn q pre =0.03lS. The input to oscillator 3 in A and to oscillator 2 in B had E syn =20mVandg syn q pre =0.02lS. The input to oscillator 3 in B had E syn ˆ )120 mv and g syn q pre =0.01lS. Both inputs in C had E syn ˆ )120 mv and g syn q pre =0.02lS. The inset PRCs were generated using the corresponding switch signal rather than a burst. Horizontal dotted lines indicate the burst threshold, and thick horizontal lines indicate the timing of the perturbations. Vertical dotted lines connecting the PRCs in the insets indicate the timing of the perturbations and therefore the amount of resetting predicted for oscillators 2 and 3 PRCs to di erent oscillators, the second strategy is employed. In this second strategy, both switch synapses have the same PRC, but a delay is applied so that the switch signal is received at the two oscillators at di erent times. A quick transition from the clockwise mode to the synchronous mode is induced. This second strategy is more versatile because arbitrary relative phases of oscillators 2 and 3 can be chosen, whereas if the perturbations are simultaneous, the relative phases of oscillators 2 and 3 are limited to the constant phase di erence within the currently exhibited mode. 3.3 Four oscillators A four-oscillator circuit was constructed as shown in Fig. 9A, this time with inhibitory connections. In the four-oscillator circuit, a mapping can be made between each oscillator and the legs of a quadruped, enabling an analogy between the patterns exhibited by the circuit and quadrupedal gaits. This circuit, for example, was capable of exhibiting a counterclockwise (Fig. 9C1) and a clockwise (Fig. 9C3) mode, which are analogous to two kinds of walking gaits in quadrupeds, as well as a mode in which nonadjacent oscillators are in phase and in antiphase with the other pair. This latter gait is analogous to a bound (Fig. 9C2), in which forelegs are in phase and in antiphase with the hindlegs. For details of the mapping, see Canavier et al. (1997). The stable values of / (repeated four times) associated with the stable modes are indicated by open circles in Fig. 9B: 0.24, counterclockwise mode; 0.495, pairs in phase mode; and 0.92, clockwise mode. The synchronous mode is unstable because the slope is negative at /=0. Finally, three saddle modes, indicated by the pairs of asterisks, were also identi ed and associated with one

13 99 Fig. 9A±C. Ring of four oscillators can produce three stable modes. A A ring of four oscillators with inhibitory connections. Parameters were as in Butera et al. (1995) except that I stim =0.5nA,Vol i =4.5 nl, and K M;SI = mm. E syn was set to )75 mv and g syn to ls/hz. B The phase response curve (F / ) generated using an intrinsic burst as the stimulus. Dotted line shows the contribution of F 2 /. Solid circle at a phase of 0 indicates that the synchronous mode is unstable. open circles at 0.24, 0.5, and 0.91 correspond to the stable modes in C1, C2, and C3, respectively. Saddles are indicated by the three pairs of asterisks. The position of the rightmost open circle at the top of a narrow peak with a pair of saddle points immediately below it predicts that the attractive basin for this mode will be relatively small. C Stable oscillatory modes. C1 Counterclockwise mode (walk). C2 Pairs in phase mode (bound). C3 Clockwise mode (walk). Dotted line in C1, C2,andC3 indicates the threshold for the beginning of a burst, about )50 mv negative and one positive slope. These modes are: (repeated three times) and 0.973, (repeated twice), and (repeated twice), and 0.91 (repeated three times) and A qualitative prediction can be made that the basin of attraction of the mode at 0.92 will be small because it will be bounded by the modes associated with the asterisks at 0.91 and The xed points in terms of / can be transformed to h coordinates in a manner analogous to that given for the circuit of three oscillators. Saddles occur in groups of four or six, depending on whether one value of / is repeated three times (4) or if they each occur twice (6). The state space for this circuit consists of three dimensions (h 2, h 3, and h 4 as shown in the cube in Fig. 10). The basin for the counterclockwise ÔwalkÕ mode (red) appears to be roughly tetrahedral, as is the one for the clockwise ÔwalkÕ mode (blue). All trajectories that do not fall within those two basins comprise the largest basin of attraction, for the pairs-in-phase `bound' mode (green). If the trajectories do not converge on the attractors (open circles), then they exit the phase cube and reenter on the opposite face, indicating that the ring order of one of the oscillators with respect to the reference oscillator has changed. If no trajectories are shown emanating from a saddle, it is because the trajectories diverged away from the saddle through the face of the cube in a single cycle. Figure 10 allows one to discern what changes in h 2, h 3, and h 4 would be required to e ect a transition from a mode associated with one stable xed point to another. If a given resetting of these phases causes the resultant coordinates to lie in another basin of attraction, a transition will be e ected. As in the two- and threeoscillator cases, a switch signal was added to the circuit, as shown in Fig. 11A. Only a single strategy was explored in this example, that of using identical synaptic connections activated at di erent times. These synapses were inhibitory, and the resultant PRC is shown in Fig. 11B. C1, C2, C3, and C4 show transitions from the clockwise mode to the counterclockwise mode, from the counterclockwise mode back to the clockwise mode, from the clockwise mode to the pairs-in-phase mode, and from the pairs-in-phase mode to the counterclockwise mode, respectively. Interestingly, C1 and C2 show inverse transitions induced by the same switch signals. In general, di erent switch signals are required for inverse transitions. The transition in C4 applied the switch signal to only two oscillators, numbers 3 and 4. This is possible because there is a path in Fig. 10 from the xed point corresponding to the pairs-in-phase mode to the attractive basin on the counterclockwise mode along a plane in which h 2 is constant. 4 Discussion 4.1 Comparative strengths and limitations of the modi ed PRC method In the Introduction and Methods, we referred to classical PRC theory, in which the coupling studied is pulsatile in nature with a short duration, comparable to a delta function. The phase resetting due to coupling is instantaneous in classical PRC theory. We also described our modi ed version of PRC theory, in which the coupling is active for a signi cant fraction of the cycle, but is analyzed mathematically as if the resetting were instantaneous. A major limitation of PRC methods is that they generally assume one input per oscillator per cycle. In the modi ed method, the main di culty is that the phase of the oscillator is unde ned while the coupling is active (during a presynaptic burst). Thus, the mathematical treatment of simultaneous or overlapping inputs is problematic. Although phase resetting in the ring circuits that we have studied is most certainly not instantaneous, the practical constraint on the application of the modi ed PRC method is that the oscillator must have e ectively returned to the limit cycle by the time the next input is received. If this criterion is met, m:n entrainment or bidirectional coupling involving more than one input per oscillator per cycle can be handled by these methods. Evidence that supports this claim is provided by the observation

14 100 Fig. 10. Phase space for the four-oscillator emulator. The attractive basins for the counterclockwise mode, the pairs-in-phase mode, and the clockwise mode are shown in red, green, and blue, respectively, and the attractors are indicated by open circles. Representative trajectories emanating from the unstable xed point (solid circle) and the saddles (asterisks) are shown that in some transients produced by the emulator, particularly when converging on a synchronous mode, more than one input per cycle was received by an oscillator, and the emulator correctly predicted the trajectories exhibited by the ring circuit under these conditions. Furthermore, as shown in Canavier et al. (1997), the PRC techniques used herein can easily be generalized to nonidentical, nonidentically connected ring circuits. These results may also be easily generalized to rings with more than four oscillators, with the sole caveat that the resultant N 1 dimensional state space will not be as easily visualized. Another, perhaps more serious limitation of the modi ed PRC method is that the e ect of an input within a circuit is assumed to cause essentially the same amount of resetting in the postsynaptic oscillator as would an input from an uncoupled, free-running oscillator. This assumes that the waveform of the burst does not change due to the input it receives in the circuit in a way that a ects the phase resetting produced in the postsynaptic oscillator. In our experience with the example oscillators in this paper, the timing and sign of the input were the most critical parameters, with the amplitude and duration being less critical. It is the latter two that can vary as a result of the interaction within a circuit, and the variation was not very signi cant in this particular system. Another method has been proposed for the analysis of coupled nonlinear oscillators in terms of their phase, in addition to PRC theory. This alternate method is average phase di erence (APD) theory (Ermentrout and Kopell 1991; Kopell 1988; Kopell and Ermentrout 1988b). This analysis makes the critical assumption that under su ciently weak coupling, the relative phases of the oscillators vary so much more slowly than the absolute phases that the e ect of the coupling (as a function of the relative phase of the oscillators) can be averaged over the limit cycle. The net e ect of the coupling is then to uniformly increase or decrease the speed at which the limit cycle is traversed. Hence in APD theory, phase resetting does not occur instantaneously but rather uniformly and continuously. The advantage of the APD approach is that multiple inputs can be easily handled by simple summation. However, the key assumption that relative phase varies slowly compared to absolute phase would not seem to apply to the example system used in this paper. Although Figs. 7 and 10 do not explicitly show the passage of time, an examination of the data used to generate these gures revealed that frequently the relative phase varied 5% as fast as the absolute phase and in some cases it varied 20%±30% as fast. Furthermore, the relative phases of the oscillators vary rapidly and signi cantly within a single period of the oscillation due to the resetting that occurs within a cycle. Each method may be appropriate for certain applications, but close attention must be paid to whether the underlying assumptions for the chosen method are valid in the particular theoretical or experimental system under study. A minor limitation, in our view, that is common to both methods is that each