J. exp. Bid. (1976), 64, 385-404 385With gfigttret

iPrinUd in Great Britain

ANALYSIS OF BURST ACTIVITY OF THE BUCCALGANGLION OF APLYSIA DEPILANS

BY R. M. ROSE

Department of Physiology, University of Bristol

(Received 8 August 1975)

SUMMARY

1. Certain nonlinear properties of molluscan neurones suggest that net-work activity could be described in relation to nonlinear relaxationoscillators.

2. A specific example is fitted by the Van der Pol equation with anexponentially decaying damping coefficient, and changes in the associatedenergy and sensitivity are considered.

3. More precise details of the interaction pattern are determined by analy-sis of a single cycle of burst activity, and it is shown that there are twostates of activity, depending on the balance of presumed inhibitorycomponents.

4. Previous results are discussed in relation to an overall decayingoscillation.

5. Changes in subcomponents of a cycle are described for decaying se-quences and the constancy of synaptic interaction is demonstrated. Theresults are briefly discussed in relation to catastrophe theory and learning.

INTRODUCTION

It is now established that decaying oscillatory activity can be initiated in certainprimitive nerve networks which are accessible to detailed study with microelectrodes.In the pedal network of Tritonia gilberti (Dorsett, Willows & Hoyle, 1969) this kindof activity occurs during the swimming escape response, and in the buccal ganglionof Aplysia depilans (Rose, 1972) decaying cyclical activity has been demonstratedduring induced feeding. In discussing modifications to this activity, it is importantto distinguish between modification of the interaction pattern within a single cyclecompared with changes in the number of cycles. We might expect that in relativelysimple systems, characterized by few subphases per cycle, experimental procedureswhich modify the activity would produce significant alterations in the number ofcycles. The buccal ganglia of Pleurobranchea appear to belong to this category, for ithas recently been shown (Mpitsos & Davis, 1974), that in this case the number ofcycles of decaying activity is markedly increased following several weeks classicalconditioning of the feeding response. By contrast, in more complex systems, charac-terized by a greater number of subphases per cycle, the major modifications couldoccur with respect to the interaction pattern itself, and the number of cycles mightnot be altered significantly. From a neurophysiological point of view a more complex

R. M. ROSE

Nl

R2

N

Fig. i. Cubic representation of network structure. The stimulus (N») excites one foce of thecube (network Ni), which projects to the opposite face (network Na). Excitatory neurones solidcircles, inhibitory neurones open circles, (a) Simple representation, with two extremearrangements in which the stimulus projects on to an excitatory face or an inhibitory face.(6) Representation of a circuit discussed by Rashevsky (1072) in relation to epilepticattacks. This is simply the left-hand circuit of (a) if we make three assumptions: that theneurones in Ni are totalty interconnected by excitatory connexions; that neurones in N2 arenot connected to each other; and that activity projects excitatorily from Ni to N2 along theedges of the cube and is reflected back inhibitorily from each neurone of Na to the entireface Ni. The network Nz may be a system of stretch receptors embedded in a muscle system(M). Ri, R2, recording planes between the shaded faces of the cube. By interconnecting twosuch networks to form a hypercube, it becomes possible to analyse complex switching effectsin burst networks.

system might be more convenient to investigate, since it may provide a sensitivemeasure of these changes. In this paper it will be shown that in the buccal ganglionof A. depilans the intrinsic balance within the network is of greatest significance.The components of this interaction will be determined, and the activity patterns willbe related to certain theoretical predictions of interactions in large networks.

As an initial generalization we assume that a given neural network is composedof inhibitory and excitatory elements (neurones), and the activity level of the net-work represents the percentage of excitatory activity. We may also expect that theinformation transmitted from one network to another, or to an effector system(muscle system) reflects in some way this activity level. Furthermore a given network

Burst activity of buccal ganglion of A. depilans 387

p not a homogeneous population of identical elements. As a first approximation tothe structural organization we must consider that the neurones (random thresholdelements) are organized into random threshold element networks (RATENS), to usethe network terminology of Amari (1974). Functionally distinct component networkshave been shown to exist in the buccal ganglion of Navanax (Levitan, Tauc & Segun-do, 1970; Spira & Bennett, 1972), where there is a network having weak electrotonicconnexions; and in the buccal ganglion of Aplysia californica (Gardner, 1971), wherethere is common synaptic input to one identified subpopulation. In order toorganize the neurophysiological and theoretical discussion we assume for simplicitythat there are four functionally distinct subpopulations arranged as shown in Fig. 1 aon one face of a cube (network Ni). The network is activated by sensory input, eitherdirectly or indirectly via another network, and we designate this input N#. Theactivity levels of the network Ni are projected to the muscle effector system M, andthe resulting sequence of contractions activate peripheral stretch receptors embeddedin the muscle system (network N2), as Kater & Fraser Rowell (1973) have found inthe buccal mass of Heliosoma. The activity of this receptor network N2 is in turnfed back either excitatorily or inhibitorily to Ni . Two simple configurations are shownin Fig. 1 a.

We now consider the later work of Rashevsky (1972, 1973, 1974) which was directedtowards the problem of the neurophysiological basis of epilepsy. Fig. 1 b is a cubicre-presentation of a particular circuit which he discussed (Rashevsky, 1972), as maybe seen by identifying the appropriate interconnexions I-VI. A very simple represen-tation of this complex circuit is given by the left-hand figure of ia with threeassumptions detailed in the legend. For such a circuit, Rashevsky showed that focalactivity produced by supra-threshold excitation of one subnet of Ni will spread fromthe focal point according to the ratio of the time factors a* and b* which characterizeV and VI. If b* £> a* no spread will occur, whereas under other conditions a focalepileptic fit can be produced, and the activity once initiated will take some time tosubside. The network N2 was not identified as a peripheral network, however. Thesystem which we are discussing here is of this general type, and for this reason wewill try to consider the relationship between previous theoretical findings on net-works in discussing the temporal dynamics. In relation to Fig. 1 it should be pointedout that extracellular recordings were made from the cerebro-buccal connectives andthe buccal nerves, and this corresponds to recording the activity flowing between thenetworks N*-Ni and N1-N2 in the recording planes indicated by the arrows Riand R2. The direction of flow is not specified, and the recorded bursts should be re-garded as samples of the activity of each subnetwork, being the output of specificneurones embedded in the given subpopulation.

METHODS

Details of the preparation and recording techniques have been given in an earlierpublication (Rose, 1972). The recordings discussed in this paper were made as partof the same series of experiments as the earlier paper, but here more emphasis hasbeen placed on the analysis of interactions. The recordings were made on ultravioletrecording paper, each sequence being several feet in length, which allowed detailed

388 R. M. ROSE

measurement. Interspike interval measurements were made by hand and the results arapresented as lines fitted to the points by eye, and representing changes in frequency^(pps) to a reasonably high degree of accuracy. Numerical integrations were carriedout by a Runge-Kutta-Merson method (Merson, 1957) with variable step size, givinga high degree of accuracy in the calculation of rapid energy changes.

THEORY

It is now established that many Molluscan neurones have a nonlinearity in thecurrent-voltage (I-V) relationship, referred to as 'anomalous rectification' (Kandel& Tauc, 1966). From voltage clamp analyses of bursting neurones in A. califarnica(Wachtel & Wilson, 1971) and in A. fasciata (Gola, 1974), it also appears that thedegree of nonlinearity is further exaggerated in these bursting neurones and thecurve becomes cubic. Gola (1974) has found that during a slow-ramp voltage clampthere is a hysteresis effect, in which the cubic curve is less pronounced during the fallof the ramp than during the rise. Wilson & Wachtel (1962) also found that the degreeof nonlinearity is temperature-sensitive. From a dynamic point of view this exaggera-tion of existing nonlinearity which underlies burst production may be consideredmathematically to be an example of a bifurcation phenomenon as discussed by Zeeman(1972) in relation to Thorn's catastrophe theory (Thom, 1972). That is, the deforma-tion of a linear characteristic to an S-shaped cubic results in the establishment of aclosed limit cycle in the phase plane representation of the oscillator. Such nonlinearoscillatory processes may be thought of as falling into two categories given by thefollowing systems of first-order differential equations (d.e.s), in which the functionsXi are independent (autonomous) or dependent on time:

dxjdt = X(xi), i = 1, 2, ..., n, (1)

dxjdt = X(Xi, t), i = 1, 2, .... n. (2)

For t = 2, the oscillation may be represented as the opposition of two componentprocesses. The interaction of these processes may be treated by means of the con-tinuous or the discontinuous theories of nonlinear mechanics (Minorsky, 1947;Andronov, Vitt & Khaikin, 1966). In the continuous theory we consider the Lie"nardd.e. to be the most general form:

x+ef(x)x+g(x) = o. (3)

Dividing equation (3) by e, followed by integration with respect to time gives

Setting y = ^—-— (Lie"nard transformation), we obtain two subsystJ e ems:

x= —

The function/(x) characterizes the damping coefficient, and has the form

/(*) =

Burst activity of buccal ganglion of A. depilans 389

L

Fig. 2. Intracellukr recordings from an unidentified neurone of the buccal ganglion ofA. depilans, showing fast jump to a maintained plateau level, and fast return, (a) Followingthe initial jump there is a slowly damped movement of the membrane potential to themaintained level, particularly evident in the burst on the left. (6) In the same cell the plateauis cut short, presumably because the initial downswing of the slow damping movement islarger in magnitude than in (a). The high-frequency action potentials are transmitted to abuccal nerve (extracellular recording, lower beam). The membrane potentials in (6) are verysimilar to heart potentials in Eriochar (Tazaki, 1971) and to paroxysmal discharges of epilepticfoci (Spencer & Kandel, 1969). Note that it is possible that there is a dual recurrent feedbacksystem in this ganglion such as has been suggested for epileptic foci (Spencer & Kandel,1969), whereby the prolonged depolarizing potential is produced by positive feedback, andthis is terminated by recurrent inhibition. Cell 1 (see Fig. 4) could discharge as in (6),analogous to the centre of the focus, and the remaining cells would play the role of the inhibitorycells surrounding the focus. Vertical bar = 50 mV, horizontal bar = 2 8.

A classification of the Lie"nard equation may be given by analysing the behaviour ofsolutions of a 3-parameter (a, e, c) system (Minorsky, 1947), a simple and extensivelyanalysed case being the Van der Pol equation (Van der Pol, 1926); f{x) = 1 — *2

(i.e. a = — c = 1, e = o). The Lie"nard d.e. is in fact a generalization of the Van derPol equation. The constant parameter e may be thought of as an energy exchangeterm. Asymmetry is produced by introducing a constant forcing term z giving theinhomogeneous form

X-e (i-x*)z x+x = z. (6)

Qualitatively the values of the constant parameters e and z govern the period of theoscillation as follows. For small values of e <̂ 1, the oscillation is nearly sinusoidaland changes in z mainly affect the amplitude but not the period. For larger values ofe §> 1, the oscillation is a more rectangular relaxation oscillation with rapid changesalternating with periods of slow variation. Analytically, linear approximations maybe applied for e <̂ 1, whereas piecewise approximations have to be used for e > 1(Minorsky, 1947; Andronov, Vitt & Khaikin, 1966). In Fig. 2, examples of intra-cellularly recorded burst activity from an unidentified cell in the buccal ganglion ofA. depilans are shown, and it is clear that a maintained depolarizing plateau is acommon feature. This fact, together with the finding that there is a primary changein burst period for decaying sequences, suggests that we should try large values ofe > 1. An example of extracellularly recorded burst activity of the buccal nerves ofA. depilans during induced feeding activity is shown in Fig. 3. This activity was

39° R- M. ROSE

initiated in the isolated buccal ganglion-buccal mass preparation using food as a,natural stimulus (Rose, 1972). There appears to be an alternation between two main1

groups of bursting neurones, the alternating periods being defined as TAI and rBt

with subscripts denoting cycle number (TCt = TA( + TB{, i = 1, 2, ...n). Theseperiods have a behavioural correlate of forward (rAt) and backward (TB{) movementof the radula of the Aplysia buccal mass. The points tt can be clearly denned as thoseinstants at which activity switches between the rAi group and the rBi group. For thisparticular sequence there is an exponential decay in the burst periods of both rAt andT 5 4 periods. The maintained ratio between TA+ and rBi period durations suggeststhat the alternating sequence could be fitted by solutions of equation (6) with anexponentially decaying damping coefficient e(t) = A exp (— tjr) and z constant. Inpractice the values r — A = 30, z = 07 give a good fit to cycles TC^—TC^ butgreater physiological relevance is attained by considering

e(t) =

giving a rapid rise and slow exponential fall in e(t). Such a waveform might relate eitherto direct sensory inflow or to the long-lasting postsynaptic waveforms characteristicof some Aplysia neurones (Parnas, Armstrong & Strumwasser, 1974). We are thereforesuggesting that the variable parameter e(t) is related to input to the network, and x(t)is the resulting activity level of the whole network. Details of the fitting of the oscillationto burst period are given in the appendix.

In Fig. 3 the oscillation initially has a value e = 1, and the exponential form of thedamping coefficient is introduced at time *„. The initial condition, e = 1, could beregarded as describing the potentiality of the system to oscillate. Alternatively, we mighthave introduced threshold properties. In this respect, Karreman (1949) has consideredthe behaviour of the Van der Pol equation with/(«) = (x—p) (x — q) and also withf(x) = (x—i) (x+i) (x— 3) (* + 3) = ( a ^ - i o ^ + g). In the first case the coefficientof*, and therefore the damping of the motion, is positive for X < p, but negative forp < x < q, and again positive for x > q. Therefore if the initial conditions are set suchthat x < p, the oscillation is always positively damped and dies out. By contrast the Vander Pol equation does not show this threshold effect, the damping being negative between+1 and — 1. For the second case, the frequency will depend on the amplitude of theoscillation. Modifications of this type have not been introduced since they would com-plicate rather than add to the existing model.

Here we may also note that the opposite case, of increasing e, has been treated byDewan (1964) in relation to epileptic after-discharges. As e increases, there is a transitionfrom a sinusoidal (tonic stage) to a relaxation oscillation (clonic stage) with a loweringof frequency. These alterations may also be directly incorporated into the Van der Polequation by appropriate modification of the damping term.

For this to be a plausible model of the underlying physiological process, it wouldprobably have been more realistic to consider the oscillation as having arisen fromstrong coupling of two relaxation oscillations. An alternative analysis might have beento apply the discontinuous theory in which the second derivative term is multiplied bya small constant parameter /i in each of two coupled oscillations. It is then found thateach second-order oscillation reduces to a first-order process by ignoring the fiX term,and the two coupled oscillators may again be considered an oscillator of second order

Burst activity of buccal ganglion of A. depilans 391

(*)

(a) 0-4.1

•X,' »|

(c)

I •••••••••.

•E,

20 s

'5 U

Fig. 3. Single oscillator description of a decaying burst sequence. Recording shows a 4-cyclesequence in which burst period durations rAt and T.B, decay exponentially on cycles 3-4.The network is in state A (see Fig. 5 A), and only unit 6 can be measured, (a) Inter-spikeinterval changes (b{, s) for unit 6, lines are best-fitting polynomials to the points. Crossesindicate approximate centre of gravity of the shaded sections. (6) Solution of the Van der Polequation (equation (6)) with exponentially decaying damping coefficient

(A - 29. 28, 7, = 20-5, Tt = 10).

(c) Associated energy change.

rather than of fourth order. For instance, the symmetrical fourth-order Abraham-Bloch oscillator reduces essentially to the second order Van der Pol equation(Minorsky, 1947). We are not interested here in such a detailed justification, butinstead will retain the basic simplicity of the Van der Pol equation in regard to para-meter changes and look at two associated properties.

In discussions of properties of large networks, two statistical parameters are usuallyconsidered to govern the activity level (Amari, 1974), namely the threshold distribu-tion and the distribution of synaptic weights. Wilson & Cowan (1972) assumed con-stant average synaptic weights, but a subpopulation response function S(x) charac-terized by a distribution function D{a) of individual neural thresholds.

rx(.t)S(x) = D{&) dff.

Jo

392 R. M. ROSE

If D(a) has a unimodal distribution, then the response function will assume a sigdmoid form. A characteristic feature of the Van der Pol equation is that periods of slowvariation alternate with rapid changes. It is at the points t{ of rapid change that thethreshold distribution becomes significant. In particular the threshold distributionwill determine the degree of synchronization of groups of neurones. An analogousfunction associated with the Van der Pol equation is the associated energy given bymultiplication by £ followed by integration with respect to time:

E(t) =

Numerical solutions show a progressive decrease in the magnitude of this energyfunction which appears as a series of near-instantaneous pulses located around thepoints tt. The period asymmetry is reflected in the appearance of two different ampli-tudes of pulses which alternate, and the ratio between these amplitudes shows a diver-gence near the end of the sequence. This differentiation of pulse amplitudes couldpossibly be related to the extinction of one oscillatory process by another.

Another associated property is given by applying the Lie"nard transformation. Forthis particular case we have:

x =

y = -The variable y may be associated with sensitivity changes. In Fitzhugh's pheno-

menological model (Fitzhugh, 1962) of the oscillatory processes which generate theaction potential, an additional term ( — a + by) is added to the numerator of the expres-sion for^. Dynamic changes are then represented on the x-y plane, the x andy nullclinesbeing obtained by setting the derivatives equal to zero. The * nullcline is an N-shapedcubic, while the y nullcline is a straight line whose slope varies with time. As the slopeof the y nullcline increases, the phase point moves slowly along the x nullcline, untila threshold point is reached, when an instantaneous jump occurs to the descendingopposite branch of the x nullcline. The slow movement down this branch is terminatedwhen a second threshold point is reached, and another rapid jump occurs, completinga closed limit cycle. The fast jumps occur at the points to ti+1 in our model, and theirmagnitude may be regarded as reflecting the degree of synchronicity of burst neurones(see later). The slow increases in sensitivity determine the relaxation phase, and mightbe expected to be associated with a decrease in the inhibitory component of the network(increasing excitability). That this may well be the case is shown by comparison ofsensitivity changes (y) with inter-spike interval changes of the burst labelled ' 6' inFig. 3. A noticeable feature is that the inter-spike interval changes for this burst changein a characteristic fashion on each period rBit an initial rapid fall being followed by aslow rise followed by a second fall in frequency. Such a change in frequency has alsobeen calculated for neurones in the Limulus eye preparation (Hartline, Ratliff & Miller,1961) during illumination of neighbouring, mutually inhibitory units.

As a first approximation to the physiological mechanism it might therefore be thecase that the strongly synchronized units characteristic of the rfi^ period (Rose, 1972)do in fact inhibit neurones of the rAt period and that the observed damped oscillatorywaveform in units 6 inter-spike interval change results from a mutual inhibitory

Burst activity of buccal ganglion of A. depilans 393

hiechanism within neurones of the TA^ group. The inhibition TB^ -*• TAt might result infebound excitation of the T ^ neurones at the end of the TB{ period, followed by mutualinhibition of one neurone by another in the excited iAi group. On this basis the heightof the energy impulse which in turn reflects the kinetic energy (JC2 > ** near thepoints ti), could indeed be a measure of the degree of synchronicity of the rBt group,the cells of which are possibly coupled positively through weak electrotonic connexionsas in Navanax (Spira & Bennett, 1972). At the end of the TB{ period this kinetic energyis transferred to the near-synchronous rebound of the rAt group, and the balance ofmutual inhibition within this group would determine the TA+ period duration. Thereforethe degree of positive coupling within the rBt group would determine the duration ofthe TB{ period, and also the strength and balance of negative coupling within the rAt

group would determine the TA^ period duration. The associated energy could bethought of as being determined by the efficacity of synaptic transmission within eachburst group, and the resulting fall as a fatigue in this transmission process. To returnto basic assumptions, we can therefore state that there may be two main subpopulations,one containing largely excitatory neurones and the other group mainly inhibitoryneurones. The activity of these two groups alternates.

RESULTS

The first approximation model described the overall decay process for a completesequence, and relied on a simple bi-stable mechanism for each cycle. Analysis of furtherexamples shows that the interactions are more complicated.

Detailed interaction mechanism (Fig. 4)

From previous work (Rose, 1972) it is known that there are two main types of burstactivity, previously called regular and sequential activity. The simpler regular activitymay be associated with a lower level of activation, and the cycle period is relativelyconstant. By contrast the sequential activity is of the type just described in whichan intense activation decays fairly rapidly over 1-2 min. In order to understand alter-native types of interaction, we therefore take an example of the simpler regular burstactivity and subdivide one cycle into the maximum number of subcomponents insteadof the minimum number as in the first approximation. The results of this analysis areshown in Fig. 4. The most important finding is that there are three subdivisions insteadof two. The oscillator may now be thought of as bi-stable-monostable instead of simplybi-stable.

The most interesting component of this cycle is the positive coupling between units1 and 5, for burst 1 appears to rise slowly at first, and then at a very rapid rate indicativeof a positive feedback mechanism. Here we should note that Kater & Frazer-Rowell(1973) have shown that in Heliosoma there are two clusters of cells embedded in themusculature of the buccal mass, which appear to operate as mechanoreceptors, andsend axons in the dorso-buccal nerve. During the retraction phase, these receptorsfeedback positively (E.P.S.P.s) to retractor motoneurones, and negatively (I.P.S.P.s)to protractor motoneurones. It appears that the presumed positive feedback loop 1-5 isthe equivalent of this system in A. depilans, particularly when we consider that units 1and s both occur in nerve B4, which supplies the same region of the buccal mass asthe dorso-buccal nerve in Heliotoma. Also the fact that a number of bursts fire

R. M. ROSE

- 1

Fig. 4. Simple building-block for complex sequences, showing substructure of one cycle ofactivity, (a) Four-channel extracellular recording. At the onset of the TA{ period (£<) units 5,6and 7 are excited as a result of rebound from inhibition. At t{+1 6 switches off and there ispositive feedback 5-1. At the onset of the TB( period (*<+i) the synchronized group fires, ofwhich 2 is one component. At f', unit 4, which may be a giant cell, inhibits unit a. (6) Frequency( + 6 , pps) plotted against time for units 1-7. Burst 1 and 5 (nerve B4) have been plottedupside down to emphasize the inhibitory balance between 5 and 6 (shaded), and the rapidswitch of activity between 1 and 2. Periods a-d, bi-stable phase. Period «, relaxation phase.(e) Activity level (solid squares) compared with activity level (solid line) calculated theoreti-cally for a large network (Amari, 1969). Waveforms have been added together and normalizedto ± 1 (see text).

synchronously with 2 suggests that electrotonic coupling is involved in the high level ofactivity of this component, which may be analogous with the electrotonically couplednetwork described by Spira & Bennett (1972) in Navanax.

Associated with bursts 1 and 2 are two bursts (3, 4) which are of significance in con-trolling the 'on' and 'off' times of the bi-stable pair. Burst 3 does not occur in thisparticular recording but fires usually in the phase labelled' a' (see later). Burst 4 fires inthe phase 'd ' and is a low-frequency large-amplitude burst; initial comparisons withprevious work on Archidoris (Rose, 1971) suggest that this may be a giant cell, but thisconclusion is only tentative. The remaining bursts (5, 6) determine the mono-stable orrelaxation phase. In Fig. 4, burst 6 has been plotted in the positive direction comparedwith burst 5 which is plotted in the negative direction (i.e. upside down). When plottedin this way the inter-spike interval changes appear to confirm the idea that there is amutually inhibitory balance between 5 and 6. Furthermore, at the onset of the TAt

period, burst 6 fires at a frequency higher than burst 5, but as the TA( period progressesthis balance shifts in favour of burst 5. Burst 1 appears to be excited by burst 5 with atime lag of several seconds. Consequently when the balance has shifted sufficiently from

Burst activity of buccal ganglion of A. depilans 395

16 to 5 such that 6 terminates, burst 5 becomes excited, possibly the result of reboundWrom inhibition by 6. This brief excitatory wave in 5 triggers the bi-stable component

1-2.

This recording i9 particularly valuable in that it shows how the first approximationmodel relates to the more detailed mechanism. The first approximation fitted theperiods rAf and TBit whereas we now identify 3 subdivisions of TA{ and 2 subdivisionsof TB{, 5 subperiods in total. We could therefore have proposed an intermediate third-order model by describing the cycle 1-2-6 as a blocking oscillator (Andronov, Vitt &Khaikin, 1966) - that is two phases of positive feedback followed by a third slow relaxa-tion phase. In reality we see that the model must be of at least sixth order to accom-modate the balance between 5 and 6 and the controlling bursts 3/4. The implicationfrom the analysis at this stage is that there is a dynamic balance within the networksuch that the activity of any one subcomponent is related to the activity of a later com-ponent. The interaction 1-2 appears to be similar to a model proposed by Camerer(1974). In this model, positive self-feedback to unit 1 results in a high level of activitywhich is terminated by reflex inhibition from 2, which in turn has positive self-feed-back. The resulting bi-stable action may constitute the ' bite' in the Aplysia case - thatis a rapid movement of the radula teeth to grasp food, followed by ingestion of food.The relationship between the burst activity sequence and the muscle contractionsequence will have to be resolved by cinematography or myographic recording if thepreparation is of significance. In this preliminary paper a detailed mathematicaldescription of the interaction will not be presented, since the object is only to attemptto demonstrate the feasibility of linking theoretical and experimental models. Thislink is made possible by comparing the waveform calculated by Amari (1969) for theactivity level of a network consisting of two coupled RATENs of inhibitory andexcitatory types, with the activity of the Aplysia network averaged for a given instant(Fig. 4c). There are probably several hundred cells in the Aplysia buccal ganglion, ofwhich we are looking at the activity of six. We may therefore assume that this sample ofsix units possibly represents the activity levels in six subpopulations of the ganglion,and that in this system it might be possible to give a neurophysiological basis to thistheoretical model. However, it must be realized that Amari's model has a statisticalbasis that cannot be claimed for our system, which has a limited number of cells. Itmay nevertheless be instructive to draw some comparisons.

In principle we could derive a waveform for the activity level in a number ofdifferent ways, but the method chosen is to plot the activity of all cells negatively,except for units 1-5 in periods a-b. The reason for this choice is clear if we claim thatactivity in 1-5 in period a-b is purely excitatory, and is followed by near-instantaneoussynchronization of unit 2 with the other cells with which it is associated. Immediatelyfollowing this synchronization at the point tit unit 2 may be subjected to inhibition,or desynchronization, and inhibitory activity is predominant until the next phase ofpositive feedback. Amari (1969) found that a purely excitatory network would oscillatewith a short period, whereas in an inhibitory-excitatory network the period is greatlyprolonged (Fig. 4c). This prolonged period of oscillation may be equivalent to theintroduction of a slow phase of desynchronization (c, d) followed by another slowphase of mutual inhibition (e) in the present case.

26 ixa 64

396 R. M. ROSE

6>5

Fig. 5. Schematic illustration of two states of activation of the network. State 'A' (6 > 5):the oscillation is of the type shown in Fig. 4. State ' B ' (5 > 6): unit 5 is dominant at theonset of the TA( period, and the point t'+j is displaced forwards. A new equilibrium is set upbetween a and 5.

Two states of activity (Fig. 5)Having established that there is an underlying pattern of interaction, we now

demonstrate that there are two extreme forms of this activity depending upon thebalance existing between units 5 and 6. These two states are shown schematically inFig. 5. In state A, unit 6 fires at a higher frequency than unit 5, which we write as6 > 5. The output pattern is exactly of the type discussed in the previous section(Fig. 4), with burst 5 cut off early in the cycle, and remaining subthreshold until thetime t*+1 when there follows a period of rebound from inhibition in unit 5, and a newcycle is initiated. State B represents the inverse case, with 5 > 6. Here burst 5 firesat higher frequency than burst 6 at the onset of the TA^ period, so that unit 6 cuts offearly, the time t\+1 being advanced to an early point in the TA{ period. In both state Aand state B, unit 1 follows activity in unit 5. A further feature of state B activity is thatfollowing the point f'+1 a new equilibrium is set up between bursts 2 and 5, for thetime period (t\+1—*<+i), the overall cycle period being largely unaltered comparedwith state A activity. These are in fact two forms of state B. In the simplest case(Figs. 6, 9), a new cycle is re-initiated at the point t\+1. In the more extreme case, asshown in Fig. 5 B, a new equilibrium is set up between 2 and 5. In the case where thisnew equilibrium is established, another burst, unit 8 (Figs. 7, 8) also fires.

The recognition of these two states of activity has two important consequences.Firstly, it enables us to understand the basis of the highly labile patterns producedby this ganglion (Rose, 1972); secondly, it is interesting in regard to the initial com-ments which were made concerning an intrinsic balance in the network. In at least

RB3

Burst activity of buccal ganglion of A. depilans

—--6

397

Fig. 6. Single oscillator description of the results given in an earlier paper (Rose, 1972,Figs. 13 and 14). Bursts have been renumbered as follows (earlier notation in parentheses):1 (04), a (fiy), 3 (a,), 4 (fit), 5 (a,), 6 (otj). Frequency plotted on a logarithmic scale againsttime. Bursts 1, 3 and 5 have been inverted and vertical lines have been added at the pointst{ (i =• i , . . . , 6) to emphasize the switching of activity between 1 and a. As activity declines in1, a, 5 and 6 there is a tendency for 3 and 4 to fire at slightly higher frequency. Burst 4 seemsto inhibit burst 2, and there is a presumed mutual inhibitory balance between 5 and 6(shaded), which controls the duration of the TA( period.

one theory of learning (Wilson & Cowan, 1972), it is considered that a network maygenerate different stable levels of activity, and that learning involves a switchingprocess from one stable position to another. The analogy with oscillatory processesis the existence of multiple limit cycles, with activity switching from one stableoscillatory state to another.

Decaying cyclic activity

A simple version of the dynamic changes during a decaying 3-cycle sequence maybe obtained by applying the model to an earlier description (Rose, 1972), as shownin Fig. 6. The bi-stable component 1-2 (ai-/?i) declines in activity on successivecycles, and simultaneously the subcomponents 3-4 (a6-p2) increase in amplitude.The presumed inhibitory groups 5-6 (0.2-0.4) again show damped oscillatory changesand the domination of unit 5 over unit 6 decreases progressively on successive cyclesresulting in a lengthening of the TAI, period on the third cycle. The sequence isterminated as both this process and the decay process go to completion on the fourthcycle. In this preparation three or four such sequences were initiated, and theyappeared remarkably similar, indicating a degree of permanence in the synapticinteraction. In order to obtain a better estimate of this permanence of the dynamicbalance, we take another example in which the activation is more intense, and com-pare two 3-cycle sequences.

26-2

B4R

B2R.

B3R

B2L

B4L

}-]— T B I -A I

-rA2- -rB2•2 rA3-|

T C I - I rC2-

rB3

TC3-

-rA4-

Fig. 7. Component changes in a decaying three-cycle sequence. Frequency (pps) plottedagainst time. Burst changes have been separated to show the growth of 3-4-6-9 (middletrace, B3R, see/I*1), and the simultaneous decline in units 1-2-5 (BCC particularly unit 5, upperand lower traces, and ./?**). The network is initially in state B (5 > 6), but this balancechanges until 6 > s, as the overall activity decays.

In the sequences of Figs. 7 and 8, state B occurs. In Fig. 7, the equilibrium is almostas shown schematically in Fig. 5 B, but as the cycle progresses the initial state where5 > 6 (state B) changes continuously until 6 > 5 (state A). The results have beenarranged to compare these components where activity declines with components whereactivity increases (unit 1 again fires with burst 5 in nerve B4, but cannot be clearlydistinguished and has not been drawn in). The main shifts in this record can be clearlyunderstood if we compare the increasing activity in units 4-6-9 (nerve B3R, middletrace), with the decreasing activity of 1 and 5 (nerve B4R/L, upper/lower trace). Itshould also be noted that unit 2 declines as units 3, and particularly 4, increase. Therise in 3-4-6-9, and the simultaneous fall in 1-2-5 ^ more rapid than in the previousexample, and these changes are suggestive that the previously noted exponentialdecline in cycle period (Fig. 3) may be accounted for by exponential changes in theamplitude (measured as pps) of component bursts. The results have been arranged in

Burst activity of buccal ganglion of A. depilans 399

Fig. 8. Another three-cycle sequence initiated from the same preparation as that shown inFig. 7, and showing the degree of similarity in details of the frequency changes in the samepreparation.

the most convenient form to illustrate this point. The schematic form of Fig. 5 may berecovered by inverting unit 5 and displacing unit 2 to the upper set, and unit 3 to thelower set. As previously remarked, a new equilibrium is set up between 2 and 5 oncycle period TAZ. There appears to be a secondary burst structure (marked by dotsin three upper traces) which arises as burst 8 becomes active and units associated withpurst 2 begin to synchronize.

A second sequence, shown in Fig. 8, was initiated about 5 min after thatshown in Fig. 7. Comparison of the two shows that they are remarkably similar.Because the activity level is slightly lower, probably because of some fatigue of thepreparation, it is possible to observe that units 5 and possibly 6 extend into the rBi

period, giving some confirmation to the possibility that there is a near-continuousoscillation between 5 and 6 with the latter growing in amplitude on each successivecycle.

To summarize the findings so far we may state that the overall interaction patternshows two states of activity (i.e. of firing order) and that decreasing cycle periodappears to be accompanied by increasing amplitude of a major subcomponent. Thelast two papers written by Rashevsky (Rashevsky, 1973, 1974) referred to a mechanismof generation of epileptic discharges based on an inhibitory-excitatory biochemical

400

B4R

B3R

B2R

B3L

B2L

'o h

R. M. ROSEt2 tz tt ts t6 tj t.

20

20

-

-6

_ _

I

_

_1 6

/ 20

20 s

TC2 T C 4

Fig. 9. Component changes in a decaying four-cycle sequence. The pattern changes fromstate B (5 > 6) on the first two cycles, to state A (6 > 5) on the fourth cycle. There is asignificant feedback between 1 and 5 (upper trace, A4), and there are two separate increasesin unit 4.

interaction (2-factor theory) in which oscillations grew to a maximum amplitude andsuddenly cut off as a fatigue substance accumulated to a saturation point. It must bestated that there is a similarity between this mechanism and that discussed in thispaper, although the complete mechanism of the Aplysia burst sequence generationhas additional complicating factors, if the interpretation discussed here is indeedcorrect.

In Fig. 9, a further sequence is analysed in which the presumed inhibitory andexcitatory subcomponents have been separated. This sequence was published pre-viously (Rose, 1972, Fig. 11), but no analysis was given. The most interesting featureis that the network initially activated into state B passes through an 'equilibrium'state and then into state A. In state B (cycles 1 and 2), unit 5 > 6 as previously.On cycle 3 these bursts appear to be in equilibrium (6 ~ 5), and finally on cycle 4units 6 > 5 and state A occurs. The overall effect of this interaction is a progressivedecrease in amplitude of both the inhibitory (5-6) and excitatory (1-2) pair fromcycle 1 to cycle 3, followed by a growth of activity in both of these components oncycle 4. The activity then suddenly terminated as unit 6 attains its highest amplitude.Simultaneous with this change in state, a noticeable feature is that a switchover occursin the growth of unit 4 which grows in amplitude on cycles TCI and TCZ (state B

Burst activity of buccal ganglion of A., depilans 401

!ctivity); this growth process being repeated on cycles TC3 and TC4 (state A activity),t is clear also from this recording that the feedback 1-5 plays a predominant part,

and that the initial state of unit 1 may govern the resulting burst patterns.

DISCUSSION

There are several features of the activity patterns considered in this paper whichsuggest that we could relate the properties of such a network to recent discussions ofThorn's catastrophe theory. Zeeman (1972) has applied catastrophe theory to theheart and nerve action potential and also discussed the significance of the Van der Polequation. The essential feature is the recognition that there are three underlying' qualities' of these oscillatory phenomena, namely a jump from a resting equilibriumlevel to a new level, followed by a slow or fast return to the resting state. In Fig. 2we have seen an example of a fast jump and fast return of the membrane potential,and noted the similarity to bursts recorded intracellularly from the heart of the crabEriocheir. It is therefore interesting that Mayeri (1973) has shown that interactions inthe network of the cardiac ganglion of the lobster Homarus have phenomenologicalcharacteristics resembling nonlinear oscillators of the Van der Pol type. In the cardiacganglion it is thought (Mayeri, 1973) that one or more of four small cells generatea burst rhythm and impose this rhythm upon five large cells via excitatory synapticconnexions. The rhythm may be controlled by exciting the small cells with singleshocks at different phases of the burst cycle. In an earlier paper (Rose, 1972) theexperimental arrangement was also designed so that shocks could be initiated atdifferent phases of the burst cycle. However, in Aplysia we are dealing with a hetero-geneous system rather than a synchronized set of neurones, and we must thereforeregard the relaxation oscillator description as a first approximation to a higher-ordersystem, which is useful in organizing the analysis of changes rather than providinga qualitative description. In this system synchronized firing occurs in at least two setsof neurones and there are other interactions besides these. There are however twoproperties, discontinuity of frequency changes, and divergence between differentunits in decaying sequences, which suggest that catastrophe theory might be usefulin understanding overall changes. It may therefore be worth commenting at this stagethat we obtain three subsystems by considering the integrated form of the Lidnardequation in which there are three separate terms,

x+ je(t) xdt+ ((g(x)-z)dt = o.

Further manipulation of equations of this type could give a representation of thisnew system in terms of an elementary cusp catastrophe, although the problem ofobtaining the appropriate form which clearly separates the slow and fast processesof the Van der Pol equation from the additional slow process arising through theintroduction of e(t) is rather difficult. It is interesting that Amari (1969, 1974) hasdiscussed the properties of large networks in relation to catastrophe theory. Heuses a different approach, basing the argument on the properties of a threshold distri-bution and distribution of synaptic weights in a population of neurones. By couplingtwo networks together, one containing excitatory neurones and the other inhibitory

402 R. M. ROSE

neurones, he obtains the waveform shown in Fig. 4. In this paper an attempt has beeimade to give a neurophysiological basis to such oscillations.

We may approach the analysis of this activity in two ways. If we try to extract theessential quality of the interactions this will lead to a generalized description such asthat just discussed. However, it is obviously also necessary to express the detailedinteraction in terms of coupled first-order d.e.s. It is interesting that Rubio (1967,1971) has provided an interpretation of activity of the respiratory centre followingsimilar lines. Rubio (1967) discussed the respiratory centre as a second-order non-linear oscillator, while further physiological evidence made it possible to provide asimpler model expressed in terms of four coupled first-order equations of theVolterra type :

Ut = out+ki /(u<+Xi,) - ki+1 /(u,+1+*<+l i r).

Camerer (1974) has discussed similar equations. Before we can take this approachhowever, it will be necessary to provide more specific evidence of connectivity, toincorporate a mechanism of rebound from hyperpolarization, and to determine thefrequency-membrane potential dependency for the main cells.

Returning to the initial comments regarding interaction during a cycle, perhapsthe most important finding of this paper is that there is a high degree of stability ofthe interaction in a given preparation. In contrast there are relatively large differencesbetween different individuals. This suggests that the preparation might be sensitiveenough to investigate learning phenomena over a period of hours rather than weeksas in Pleurobranchea (Mpitsos & Davis, 1974). Furthermore, we can now investigatesensory feedback mechanisms (bursts 1-5, nerve B4) and synchronization (units 2and associated bursts, units 1-5-6) since we now understand the basic features of theinteraction pattern.

APPENDIX

With reference to the discontinuous limit cycle for the system

t = G(x,y),an expression for the period of self-oscillations may be derived for small values of ft:

T1 = T0+Afi*+Bfiln—\-Cfi+o{fii},/*

where A, B and C are numbers determined by values of the functions F (x, y) andG (x, y) on the discontinuous limit cycle. We now consider if we can arrive at a quan-titative expression of the period for the Van der Pol equation

For e > i (e constant), an approximate estimate of the period may be obtained byconsidering that x ~ ofor \x\ = 2 to \x\ = i and is large for most of the remainder.Setting x = o, we obtain

T = 2 l * " 1 ^ 1 - * " ) ^ = 2 e ( t - l n 2)J -1 X va /

Burst activity of buccal ganglion of A. depilans

jFor constant z on the R.H.S. this becomes:403

Tz -J Ldx

J_a * -*

-1

- 8

- 1

- a : : ) •

A more accurate estimate of the period has been given by Cartwright (1952), by con-sidering the equivalent energy, differentiated, and integrated forms of the Van derPol equation, and using a 'piecewise' analytic treatment. Dorodnitzin (see Minorsky,1947) obtained the same result (but to higher terms) using asymptotic expansions:

This suggests that an improved estimate of the period of constant z might be given by

= Ts+jeri.

Since e(t) is slowly varying compared with T, an approximated value for cyclesTC1-3 may be given by:

For a single exponential decay in e(t) suggested by the results of Fig. 3, an approxi-mate estimate of T would then be:

— 2— 2ZX

-t

- 1

- 3

The investigation of the Van der Pol equation for e(t), does in fact present quite aninteresting, though complicated mathematical problem in itself, but a detailedinvestigation is not justified in the present work, since we are only trying to representthe overall process of decay in a qualitative manner. It is however necessary to provideestimates for the constants A, rlt z, if our approximate expression is to be useful.Since

4 - 2

l<-3

/ i-21 7

for the sequence shown in Fig. 3 this suggests that

(Tz)rBil(Tz)TAi~ 17

if we consider e const, for cycles 2-3 (this assumption will lead to some error though).Evaluation of this ratio for different values of z suggests that z ~ 0-65 — 07. The

value of e(t) at different times in the sequence can then be estimated, giving roughvalues for A and TX. Finally solutions were obtained numerically and more accurateparameter adjustments were made by trial and error.

404 R. M. ROSE

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