* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Modeling the Gastric Mill Central Pattern Generator of the Lobster
Survey
Document related concepts
Biochemical switches in the cell cycle wikipedia , lookup
Extracellular matrix wikipedia , lookup
Endomembrane system wikipedia , lookup
Tissue engineering wikipedia , lookup
Cytokinesis wikipedia , lookup
Cell growth wikipedia , lookup
Cellular differentiation wikipedia , lookup
Cell culture wikipedia , lookup
Cell encapsulation wikipedia , lookup
Chemical synapse wikipedia , lookup
Organ-on-a-chip wikipedia , lookup
Transcript
JOURNALOF NEUROPHYSIOLOGY Vol. 70, No. 3, September 1993. Printed in U.S.A. Modeling the Gastric Mill Central Pattern Generator of the Lobster With a Relaxation-Oscillator Network PETER F. ROWAT AND ALLEN I. SELVERSTON Department of Biology, University of California, San Diego, La Jolla, California SUMMARY AND CONCLUSIONS 1. The gastric mill central pattern generator (CPG) controls the chewing movements of teeth in the gastric mill of the lobster. This CPG has been extensively studied, but the precise mechanism underlying pattern generation is not well understood. The goal of this research was to develop a simplified model that captures the principle, biologically significant features of this CPG. We introduce a simplified neuron model that embodies approximations of well-known membrane currents, and is able to reproduce several global characteristics of gastric mill neurons. A network built with these neurons, using graded synaptic transmission and having the synaptic connections of the biological circuit, is sufficient to explain much of the network’s behavior. 2. The cell model is a generalization and extension of the Van der Pol relaxation oscillator equations. It is described by two differential equations, one for current conservation and one for slow current activation. The model has a fast current that may, by adjusting one parameter, have a region of negative resistance in its current-voltage (1-v) curve. It also has a slow current with a single gain parameter that can be regarded as the combination of slow inward and outward currents. 3. For suitable values of the fast current parameter and the slow current parameter, the isolated model neuron exhibits several different behaviors: plateau potentials, postinhibitory rebound, postburst hyperpolarization, and endogenous oscillations. When the slow current is separated into inward and outward fractions with separately adjustable gain parameters, the model neuron can fire tonically, be quiescent, or generate spontaneous voltage oscillations with varying amounts of depolarization or hyperpolarization. 4. The most common form of synaptic interaction in the gastric CPG is reciprocal inhibition. A pair of identical model cells, connected with reciprocal inhibition, oscillates in antiphase if either the isolated cells are endogenous oscillators, or they are quiescent without plateau potentials, or they have plateau potentials but the synaptic strengths are below a critical level. If the isolated cells have widely differing frequencies (or would have if the cells were made to oscillate by adjusting the fast currents), reciprocal inhibition entrains the cells to oscillate with the same frequency but with phases that are advanced or retarded relative to the phases seen when the cells have the same frequency. The frequency of the entrained pair of cells lies between the frequencies of the original cells. The relative phases can also be modified by using very unequal synaptic strengths. 5. A reduced network model was used to study the coordination between the lateral and medial subsets and the effect of deleting a cell from the circuit. The results of killing Int 1 in the model had effects similar to killing Intl in the biological circuit. This suggests that Int 1 accomplishes the coordination of the two subsets by indirectly altering the effective strengths of synapses between them. 6. A network of cells with all the known connections was also studied. It was found that the network would oscillate and produce an approximately biologically correct output pattern over a wide 1030 92093-0322 range of synaptic strengths. This remained true when the individual cells were adjusted to be oscillators or to be quiescent. Random changes in parameter values of up to 40% had little effect on the overall pattern. The pattern of phase relationships remained approximately constant when the model frequency was varied. The phase lag between the lateral and medial subsets of the gastric network could be obtained by incorporating known slow synapses and by adjusting a slow current parameter. If cells are killed sequentially in the model, the network continues to generate a pattern so long as at least one pair of reciprocal inhibitory cells remains. Changes in the relative phases of slow-wave activity can be obtained by changing the gains of the slow currents. 7. A cell model that has a fast current with an N-shaped 1-V curve, and slow inward and outward currents with linear steadystate 1-V curves, captures important characteristic properties of gastric neurons, and a network model built by connecting these cells with instantaneous graded synaptic transmission captures important features of small CPGs. This simple cell model is an abstraction that delineates a basic mechanism common to all gastric cells and provides a foundation on which to build more comprehensive models of the gastric mill network. INTRODUCTION Central pattern generators (CPGs) produce rhythmic motor outputs that underly repetitive behaviors such as chewing, swimming, or walking. Many invertebrate CPGs have very small numbers of neurons and have been intensively studied (Kristan 1980; Selverston and Moulins 1987). The gastric mill CPG controls striated muscles operating three teeth that comprise the gastric mill of the lobster (Harris-Warrick et al. 1992). All but two of the 13 neurons forming the gastric mill CPG lie in the stomatogastric ganglion (STG), a network of -30 neurons located on the dorsal surface of the stomach of the lobster. Most of the nongastric cells in the STG belong to the pyloric CPG. This controls the pyloric region of the stomach and operates at a higher frequency. A great deal is now known about the gastric mill CPG (Selverston and Moulins 1987). All 13 cells are identifiable. Many connections are known (Mulloney and Selverston 1974a,b; Selverston and Mulloney 1974) and the results of many experimental perturbations, such as intracellular current pulse injections and the removal of cells from the network, have been examined (Selverston et al. 1983). Despite this, the known neuronal interactions are too complex to easily explain how the overall dynamics of the system produces the multiphase gastric pattern. To help in understanding the network mechanisms, a rigorous modeling study was undertaken. The gastric mill network provides an ideal system with which to attempt a theoretical account of 0022-3077/93 $2.00 Copyright 0 1993 The American Physiological Society MODEL OF GASTRIC MILL CPG 1031 LPGI DG AM GM INTI 110mv 5s Inhibitory i Functional Inhibitory A Excitatory i Functional Excitatory l - Elect rotonic E J 5mV 1 s. FIG. 1. A: network diagram for the gastric mill central pattern generator (CPG). Circles represent cells. Overlapping circles represent multiple cells that are regarded as identical for modeling purposes. All but the 2 E cells are in the stomatogastric ganglion (STG). The key to the types of connections in the network appears below the diagram. A functional inhibitory or excitatory connection means that not all the tests for monosynapticity, as described in Mulloney (1987), have been carried out on the connection in question. All cells except Intl and the E cells are motor neurons. B: simultaneous intracellular membrane-potential recordings of all 7 types of gastric cells in the STG during ongoing activity in an in vitro preparation (recording courtesy of H.-G. Heinzel). Slow waves are clearly seen. The STG preparation was attached to 3 other ganglia providing neuromodulatory input necessary for the maintenance of the rhythm. In each cells’ recording, high-frequency spikes superimposed on the upper parts of the slow waves are action potentials. Signals with frequency between that of the spikes and the slow waves, particularly prominent in the LPG, DG, AM, and Int 1 traces, are due to inputs from the pyloric network, another CPG contained in the STG. The small spike amplitude in these recordings is due to the microelectrode placement being in the soma, at a large electrical distance from the spike initiation zone. C: intracellular recordings from a GM and an E cell (modified from Selverston et al. 1976). AM, anterior median; DG, dorsal gastric; GM, gastric mill; LG, lateral gastric; MG, median gastric; LPG, lateral posterior gastric; Int 1, interneuron 1; E, excitatory. its principles of operation. There have been several previous modeling studies (Friesen and Stent 1978; Selverston et al. 1976; Thompson 1982; Tsung et al. 1990; Warshaw and Hartline 1976) of this system. In these studies there were many free parameters, few of which could be given unambiguous physiological interpretations, and the models were not tested with perturbations similar to those possible experimentally (Selverston et al. 1983). Thus a clear picture of the basic principles and mechanisms underlying pattern generation has not emerged. The cells of the gastric mill network, together with the currently known interconnections, are shown in Fig. 1~. The key to cell name abbreviations is given in the legend for this figure. There are four identical GM cells, each one connected to the others with electrical junctions, and each playing the same functional role in pattern generation, as far as is known (Selverston and Mulloney 1974). Because intracellular recordings from individual GM cells are identical, these cells were modeled as a single unit. The two lateral posterior gastric (LPG) cells are connected with an electrical junction and were, for the same reasons, modeled as a single unit. The two E cells are not in the STG. There is one in each of the bilaterally paired commissural ganglia of the stomatogastric nervous system (Russell 1976; Selverston et al. 1976). The E cells are inhibited by interneuron 1 (Int l), and they excite the GM, lateral gastric (LG), median gastric (MG), and LPG cells (Fig. IA). The E cells fire in bursts that are out of phase with Int 1 when the gastric rhythm is active. They fire tonically if not receiving phasic inhibition from Intl. Although their role is not clear, the E cells may be regarded as components of the gastric mill CPG (Russell 1976). The two E cells are functionally identical and were also modeled as a single unit. We included the E cells in most of our models of the complete gastric network. Thus there are seven different types of gastric mill cell within the stomatogastric ganglion, which together with the E cells make a total of eight different cell types. Simultaneous intracellular recordings of all seven types in the STG are shown in Fig. 1B (Heinzel and Selverston 1988). Simultaneous intracellular recordings of a GM and an E cell are shown in Fig. 1C (Selverston et al. 1976). The gastric mill CPG generates a pattern only when nonphasic, neuromodulatory, inputs from other ganglia in the stomatogastric nervous system are present, or when specific neuromodulatory substances have been experimentally applied to the STG. The stomatogastric nerve (Stn) is the sole input tract from these other ganglia. When these inputs are removed, by cutting or blocking the Stn, no pattern is generated. We will refer to these different states as cycling and noncycling states of the gastric mill CPG. 1032 I? F. ROWAT AND The gastric mill itself consists of one medial tooth and two lateral teeth, together with a number of ancillary ossicles and surrounding musculature (see Selverston et al. 1976). Functionally the GM cells cause the medial tooth to protract in a power stroke, whereas the dorsal gastric (DG) and anterior median (AM) cells cause it to retract. The LG and MG cells cause the lateral teeth to come together in a cut or squeeze movement and are opened by the LPG cells. The alternate firing of GM with DG and AM, and of LG and MG with LPG, is seen in the traces of Fig. 1B. Note that there is a small but distinct phase lag between the outputs of the cells controlling the lateral teeth (namely LG, MG, and LPG) and the outputs of cells controlling the medial tooth (GM, DG, and AM). For example, the bursts of spikes from the GM cell lag behind the LG and MG bursts, whereas the DG and AM bursts lag behind the rather poorly defined LPG bursts. The E cells are synchronous with the GM cells (Fig. 1C) and thus serve to increase the spiking frequency of the GM bursts. The E cells therefore modify the force exerted by the medial tooth and provide a means for CNS and sensory modulation of medial tooth action. Cell properties Much is already known about the properties of gastric mill cells and their synapses (Selverston and Moulins 1987). All cells, except for the interneurons Int 1 and the E cells, are motorneurons having two roles: pattern generation and muscle activation. The bursts of spikes from each motorneuron activate gastric mill muscles and are not believed to contribute significantly to the phase-setting mechanisms of pattern generation. Evidence for this is provided by Elson and Selverston (1992), who showed that the isolated gastric mill network, when activated by the presence of pilocarpine, produces slow-wave oscillations that continue with approximately unchanged phase relationships when spikes are suppressed by the application of tetrodotoxin (TTX). In the closely related pyloric CPG, Raper (1979) and Anderson and Barker (198 1) showed that the suppression of spikes by TTX has little effect on an ongoing pyloric rhythm. Also in the pyloric CPG, Graubard et al. (1983) showed that graded and spike-mediated synaptic transmission occur simultaneously between synaptically coupled pyloric cells. For the purpose of this work therefore, we assume that the contribution of network connectivity to pattern generation is primarily dependent on graded synaptic transmission and that spike-mediated synaptic transmission makes no significant contribution to the overall pattern-generating process. The E cells are located in separate ganglia and cannot communicate with the gastric mill cells in the STG by graded synaptic transmission. To simplify the initial modeling studies, it was assumed that the overall contribution to pattern generation of the excitatory postsynaptic currents caused by E cell spikes is equivalent to the postsynaptic effect of excitatory graded transmission from the E cells. Under normal experimental conditions the patternforming properties of the gastric mill network appear to be widely distributed. Several cells exhibit plateau potentials and postinhibitory rebound. Thus intrinsic cellular proper- A. I. SELVERSTON ties appear to play an important role in pattern formation (Russell and Hartline 1984; Selverston et al. 1976). No single cell has been shown to be a strong endogenous b urster. In a few preparations, however, the DG cell has been found to have the capability for endogenous bursting when subjected to injected current or other nonphasic input (Hartline and Russell 1984). This may be a side effect of the cellular properties of DG that are necessary for the overall pattern-generating process. When the gastric mill is cycling, it is possible to remove several cells from the network experimentally, including DG, with little degradation of the remaining rhythm (M. Wadepuhl, unpublished data). Hence the rhythm is not driven by a single oscillatory cell but must emerge from the interplay of intrinsic cellular properties and network connectivity. Synaptic properties Neurons linked by reciprocal inhibition have often been proposed as the basis for the production of alternate bursting in antagonistic pairs of muscles (Brown 19 14; McDougall 1903). A reciprocal inhibitory pair, i.e., two single cells linked with reciprocal inhibition, occurs in many invertebrate CPGs (Benjamin 1980; Calabrese et al. 1989; Getting 19 8 1, 19 8 3) and frequently innervates antagonistic muscles (Robertson and Pearson 1985). Perkel and Mulloney (1974) using a spike-based model of synaptic interaction, showed that postinhibitory rebound in each cell of an inhibitory pair was sufficient to cause oscillatory behavior. Satterlie (1985) provided evidence that the swimming pattern in Clione is based on reciprocal inhibition between two nonoscillatory cells with postinhibitory rebound. Wang and Rinzel ( 1992) studied a simplified, conductance-based model of reciprocal inhibition. Each cell had a leak current and a simplified T-type Ca2+ current that activated immediately but inactivated slowly. They found that there were two distinct modes of operation: “release” of inhibition by the inhibiting cell or “escape” from inhibition by the inhibited cell. A necessary requirement for oscillation in an inhibitory pair is some time-dependent mechanism to gradually remove inhibition from one cell so that it can burst. Physiologically, this may take the form of spike adaptation or disinhibition by a hyperpolarization-activated inward current in the presynaptic cell (Angstadt and Calabrese 1989). In the gastric mill CPG it can be seen from the circuit and traces in Fig. 1 that there are several pairs of neurons, connected by reciprocal inhibitory synapses, that fire approximately out of phase with one another. There are four inhibitory pairs: Int l/LG, Int l/MG, MG/LPG, and LG/DG. The LG/MG cells are connected with reciprocal inhibition but also share an electrical junction and have always been observed to fire approximately together (Mulloney and Selverston 1974a). Consequently, for the purposes of this paper, the LG/MG interaction is considered to be dominated by the electrical junction rather than by the inhibitory synaptic connections. The MG/LPG inhibitory pair innervates antagonistic muscles that move the lateral teeth. The cells Intl, MG, and LPG form a three-element chain with each pair connected by reciprocal inhibition. Therefore, because MG alternates with Int 1 and with LPG, syn- MODEL OF GASTRIC chronization of Int 1 and LPG is expected, as can be seen in Fig. 1B. Thus it is clear that the synaptic arrangement of pairs of reciprocal inhibitory synapses plays an important role in the GM CPG. There are two synapses in the network for which the postsynaptic response is very slow. When Intl is depolarized, the excitatory synapses from Intl to DG and from Intl to AM produce a response in the postsynaptic cells that is very slow and takes on the order of 400 ms to reach its peak (Figs. 8 and 9 in Selverston and Mulloney 1974; Selverston et al. 1976, 1983). Modeling To investigate the essential mechanisms underlying pattern generation by mathematical modeling, one has to first address the question of what level of modeling to use. Ideally the model should be a simplification of the numerous biophysical events taking place, whereas at the same time it should be consistent with the experimental data. One has to find a balance between simplicity, which implies using the smallest possible number of parameters in a model, and faithfulness to the experimental data, which may well require an essentially unlimited number of parameters. Many sophisticated and complex models of single neurons have been used. Multi-compartmental models (Edwards and Mulloney 1984; Perkel and Mulloney 1978) and single compartment models with large numbers of voltageand time-dependent ionic currents (Rose and Hindmarsh 1989a-c; Yamada et al. 1989) have been studied. The former are ideal for modeling the effect of cell geometry on network function, whereas the latter are ideal for studying the detailed interplay of specific types of membrane currents in endogenous oscillations. We considered both classes of model to be more complex than needed for models of a small network in which the goal is to elucidate general principles of operation. This is in part because the complexity of a network model constructed with such model neurons would be too great and in part because membrane currents in gastric mill cells are not characterized in sufficient detail to permit the construction of cell models that include accurate representations of ionic currents. We did not use a cell model based on the Hodgkin-Huxley equations (Hodgkin and Huxley 1952) because these equations can produce behavior much more complex than necessary (Carpenter 1979), and because they model action potentials, which do not play a major role in gastric mill pattern generation. The basic requirement was for a model of slow-wave dynamics, not a model for spikes which in the gastric mill ride on the crests of the slow waves. An even more crucial reason for not using a conductance-based model is that there are no data available for membrane currents in the gastric mill CPG. There are, however, data available on the membrane currents in cells of the pyloric CPG (Gola and Selverston 198 1; Golowasch and Marder 1992; Graubard and Hartline 199 1). The Golowasch and Marder (1992) data were incorporated in a biophysical model of the LP cell in the pyloric network (Buchholtz et al. 1992). If one hvpothesizes that cells in the same ganglion MILL CPG 1033 share similar sets of ionic conductances, then the latter model may be adaptable for subsequent modeling studies of the gastric mill. An important question of principle is whether the gastric mill CPG is driven by network interactions, by intrinsic cellular properties, or some combination of the two. Earlier work shed some light on this matter. Using an abstract formulation, Tsung et al. ( 1990) modeled the gastric mill CPG with a network of identical neurons. Each neuron was represented as a piece of passive membrane, and the input to a cell was a weighted sum of the presynaptic potentials. Thus the pattern-generating mechanism had, of necessity, to reside in the connection strengths. After parameter adjustment by means of back-propagation (Williams and Zipser 1989), this model captured the approximate phase relations between the slow membrane-potential waves seen in Fig. 1B but ceased oscillations when any one cell was removed from the network. Another problem was that two units connected with reciprocal inhibition would not oscillate. Thus a pair of reciprocal inhibitory cells, an important constituent of the gastric mill network, could not be modeled. The failure of this network model tells us that the pattern-generating mechanism cannot reside in the connections alone but must also depend on intrinsic cellular mechanisms. Therefore a more sophisticated model was required. Our goal was to develop a model that was both computationally tractable and physiologically plausible, in that it had to be consistent with the most significant features of the biological data. Specifically, the model had to satisfy the following requirements. I) Every model element (every variable and parameter) had to have a clear and plausible physiological interpretation. 2) It should be easy, by adjusting physiologically meaningful parameters, to cause a model cell to display characteristic properties of gastric cells such as plateau potentials and postinhibitory rebound. 3) It should be possible to obtain appropriate behavior from small networks of model cells such as the reciprocal inhibitory pair and other subnetworks of the gastric mill CPG. 4) At the network level, the responses of the complete model to perturbations should correspond to the response of gastric mill preparations to experimental perturbations. Other desirable features, not addressed in this paper, were that the model should provide a useful approximation for a wide variety of complex biophysical models and that the model should be mathematically tractable. Generalized nonlinear oscillators have been applied to neurophysiology by many authors (e.g., Pavlidis and Pinsker 1977). The application of nonlinear oscillators to CPGs in particular has been studied by Cohen et al. (1982) Kopell(1988), Friesen and Stent (1978), Stein (1977), and many others. To achieve simplicity, and thus derive general properties of networks of oscillators, the nonlinear oscillator has often been simplified to a point moving with constant speed on the circumference of a circle. However, the relation between the circular motion of the point and physiological phenomena such as membrane currents has been deliberately left undefined. To satisfy 1) above, we considered that another approach was necessary. 1034 P. F. ROWAT A AND C / /- linear Of = 0.9 FIG. 2. Simulated current-voltage (I-V) curves for the proposed cell model. A and B: I-Vcurves for the fast current when of = 2.0 (A) and of = 0.9 (B). C steady-state I- Vcurve for the slow current for B, = 3. This curve can be regarded as the combination of 2 parts: an I- Curve for an inward current and an I-V curve for an outward current. D: steady-state slow current I- V curve when the inward and outward parts have different gains (Oin = 3, lout = 0 The relaxation oscillator has a venerable history. It was studied in the 19th century by Lord Rayleigh, analyzed further by Van der Pol(l926), used by him and Van der Mark (1928) to model the heart, and claimed by the physiologist A. V. Hill (1933) to be the oscillator “with which alone we are concerned in physiology.” The essential feature of a relaxation oscillator is that it consists of two opposing forces acting on different time scales. A generalized CPG model consisting of a network of relaxation oscillators has been studied (Bay and Hemami 1987), but its parameters cannot be interpreted easily in physiological terms. Again, to satisfy 1) above, some other model was needed. We used an extension of the relaxation oscillator. METHODS Cell model The model for an isolated cell was adapted from the generalization by Lienard (1928) of Van der Pol’s relaxation oscillator (1926). It is written as two equations dV 7,-&+F(v)+q=O (1) dq 7”dt = -4 + q,(V) (2) where F(V) is given by F(V) = V - Af tanh [(af/Af) V] (3) and qco (V) is given by q,(V) = %V An alternative definition 4*(V) (4) is also used = I (TinV for V-c 0 GoutI/ for V> 0 Here V represents the membrane potential, 7, represents membrane time constant, F(V) represents the current-voltage (5) the (I- A. I. SELVERSTON V) curve of an instantaneous, voltage-dependent current (Fig. 2, A and B), and q represents a slow current with time constant 7, and steady-state I-V curve q, ( V) (Fig. 2, C and 0). F( V), the “fast I- V curve, ” is N-shaped. Equation 3 allows the degree of N-shape and the width of the N-shape to be adjusted independently of one another by means of the parameters of and Af. of adjusts the degree of N-shape in Ifast without changing the distance between the asymptotic values for F(V), V t A, (Fig. 2, A and B), whereas A, adjusts the width of the N-shape without affecting the degree of N-shape. Af scales the fast I-V curve without changing its slope at the origin. The dashed lines in Fig. 2A are the asymptotes for F( V) when V is large. A, measures one-half the distance between the asymptotes along the V-axis. 1 - gf is the slope of the 1-V curve at the origin. When of = 0, F(V) = V so the fast current is purely ohmic. When 0 < cf 5 1, F(V) has a point of inflexion at V = 0 (Fig. 2B), and when of > 1, F(V) is N-shaped with a region of negative slope (negative resistance) around the origin. In the latter case, it crosses the V-axis at three points: on the left of the origin, at 0, and on the right of the origin (Fig. 2A). qm (V) was usually defined by Eq. 4 (Fig. 2C), so the slow current could be regarded as the combination of a slowly activating linear outward current for V > 0 and a slowly activating linear inward current for V < 0, each with the same gain us. Sometimes qco (V) was split into inward (V < 0) and outward (V > 0) parts with separate gains oin and tout (Eq. 5 and Fig. 20). 7, always has a value smaller than the slow current time constant 7,. Eqs. 1 and 2 become a Lienard equation if the term -q in the second equation is removed, of > 1, and CT,> cf - 1. Network connections The current passing through an electrical junction was modeled as a fixed conductance times the difference in membrane potentials. The current through the rectifying electrical junction between GM and LPG is computed as for an electrical junction when V& > VLpG, otherwise it is zero. The model of graded chemical synaptic transmission followed Katz and Miledi ( 1967). Because synaptic transmission time in the gastric mill CPG is about three orders of magnitude smaller than the period of the slow waves (Mulloney and Selverston 1974a,b; Selverston and Mulloney 1974), synaptic delay was ignored. Let VV be the maximum postsynaptic conductance at a synapse between a presynaptic cell with membrane potential VP,, and a postsynaptic cell with potential Vpost. The proportion of postsynaptic channels open at any time due to the binding of transmitter released in graded manner from the presynaptic terminal was assumed to be an instantaneous, O-l valued, sigmoid function .fof the presynaptic membrane potential. Hence the postsynaptic conductance is wf( Y,,), and the post-synaptic current S at a synapse with maximum conductance VV > 0 is given by s = wf( v,re><vpost - E,ost) (6) where f( V,,) = ( 1 + e-4Vpre)-1 and Epost is the synaptic reversal potential. The sigmoid function/( Vi,,,) has unit slope at V = 0. We sometimes refer to the maximum synaptic conductance I/t/as the synaptic weightor connectionstrength.For an inhibitory synapse this expression must always be positive (outward current), hence the value of E’i.,ostmust be below the normal lowest value of the postsynaptic membrane potential. Similarly, at an excitatory synapse Epost must be above the normal highest value of the postsynaptic membrane potential. At a slow synapse, the slow response of the postsynaptic membrane potential to a change in the presynaptic potential was assumed to be mediated by a slow process that intervenes between the binding of transmitter to receptors and the opening of the MODEL channel conductances. At a slow synapse the postsynaptic ductance G is given by dG -= rd dt and the postsynaptic -G OF GASTRIC con- (7) + wf( v,,> s = (-3&OS* - Epost) (8) 7d is the time constant for the slow change in postsynaptic conductance in response to a change in the presynaptic membrane potential, V,,,. Model summary The model can be summarized in a physiological form as follows. The current equation for a single cell embedded in a network is obtained from Eq. 1 by adding terms &, lej, and Iinj dV+r fast +I slow + Isyn + Iej = Iinj (9) where V is membrane potential, 7, is membrane time constant, and 1, is injected current. Ifast = F(V) is the fast current, defined by Eq. 3. ISlOw = q is the slow current, which activates by Eq. 2 with time constant 7,, to the steady-state value qm(V) given by Eq. 4 or 5. Isyn is the sum of all postsynaptic currents, where the current S from one synapse is given by Eq. 6, or if a slow synapse, by Eqs. 7 and 8. 1ej is the sum of all electrical junction currents where the current in cell A due to an electrical junction with conductance G between cells A and B is G( VA - VB). Simulation system An interactive simulation system called “The Preparation” was written and used for running all models. The Preparation continually integrates a model and simultaneously displays model output graphically, while also allowing any parameters of a model to be changed, without having to stop the model. Integration was done by the LSODA software package developed by Lawrence Livermore Laboratories for integrating large sets of differential equations with automatic method switching when the system of equations becomes stiff (Petzold and Hindmarsh 1987). Phase diagrams The phase diagram is a mathematical device used to deduce qualitative properties of solutions of a system of differential equations without actually solving the system explicitly. The phase diagram can be used with a system having any number of differential equations but will be described here for a two-equation system and applied to the system defined by Eqs. 1 and 2. We will refer to this system as MC. Figure 3 shows seven phase diagrams. The state of MC is completely specified by the pair of values (V,q). This is represented as a point ( V,q) in the phaseplane, a two-dimensional coordinate system with V on the horizontal axis and q on the vertical axis. As the state changes over time, the phasepoint [ V(t), q(t)] describes a curve in the phase plane. This curve is also referred to as a trajectory or phasepath. An oscillatory solution with all transients fully decayed will describe a closed curve that does not intersect itself. This is called a limit cycle. A limit cycle is attracting if, after a small perturbation that moves the phase point off the limit cycle, the phase point returns to the limit cycle. Experimentally, only attracting limit cycles are seen. A and F of Fig. 3 have limit cycles. A solution of the system can be viewed as a point moving on a trajectory in such a manner that the direction and speed of movement CPG at each point of the trajectory dV is given by dt’ 1035 dq . The V-nukline is the curve on which dtdV = 0, and the q-nulldt cline is the curve on which dt&I = 0. Every part of Fig. 3 includes and q-nullclines. current is then rm dt MILL Nullclines scribing the movement V- are useful because they assist in de- dV =0 dt on the V-nullcline, any trajectory of MC must cross the V-nullcline vertically. Similarly, any trajectory must cross the q-nullcline horizontally. These statements can be checked in every part of Fig. 3 except E. At any point of intersection of the V and q nullclines, dV dq -E-E 0, so it is an equilibrium point (e.p.; see, e.g., Fig. 3E. dt dt the points L, H, and the origin, and D, the starred points and the origin). The stability of each e.p. can be found by examining the stability of the linear aproximation to MC at the e.p. Experimentally, only stable e.p.s are seen. In Fig. 3E, points L and H are stable e.p.s, whereas the origin is unstable. The origin is an unstable e.p. in A and D-G, but it is stable in B and C (where the V-nullcline crosses the V-axis with 0 or negative slope). For the system MC, the V-nullcline is given by the equation q = -F(V), and the q-nullcline is given by q = q, (V), or q = a,V. Thus the V-nullcline is obtained by reflecting the fast I-V curve in the Vaxis, and the q-nullcline is identical with the slow current’s steadystate I-V curve. These two facts make it particularly easy to relate the I- V curves of our model cell to nullclines in its phase diagram. Compare Fig. 2, A and C, with the nullclines in Fig. 3A. For instance, injection of a steady depolarizing current causes the fast I- V curve to move downward, whereas the V-nullcline moves upward. When a system is described by two differential equations with time constants that are significantly different, as in the case here where 7, < 7,, then the motion of the phase point can be described approximately 2forz of the phase point in time. Because - dV. as follows. Eq. I for dt is the fast equation is the slow equation. and Eq. Away from the fast, or V-, nullcline, the changes in the slow variable q are so slow in comparison with the rate of change of the fast variable V that for all intents and purposes the phase point is controlled by the fast equation, with the slow variable essentially constant. Thus the phase point describes a trajectory that is roughly parallel to the fast variable axis, which means that in our model it is parallel to the V-axis. Once on or very close to the fast nullcline, however, the movement due to the fast equation is negligible, because, by definition of the fast nullcline, the right-hand side of the fast equation is zero or very close to zero. Instead, the slow equation becomes dominant and moves the phase point along the fast nullcline. For example, in Fig. 3, A and F, the roughly horizontal segments of the limit cycle labeled 1 and 3 are fast movements caused by the fast equation. The diagonal segments of the limit cycle, labeled 2 and 4, are slow movements in which the phase point stays close to the fast nullcline while being moved in the vertical (q- or slow) direction by the slow equation. The fast movements 1 and 3 in the phase diagrams correspond to the rapid rise and fall in the potential traces in Figs. 3, A and F. The slow movements 2 and 4 in the phase diagram correspond to the depolarized part and the hyperpolarized part, respectively, of the potential traces. Thus by sketching the nullclines, and taking into account the presence of widely different time scales in the two equations, one can obtain a qualitative description of the trajectories of the state of the model cell MC and hence a description of the form of the membrane-potential trace. When two nullclines are roughly parallel and close together, the area between the nullclines is referred to as a narrowchannel.If the P. F. ROWAT 1036 P , w (7 l q nullcline AND A. I. SELVERSTON Q . I I PBH - V FIG. 3. Each box describes a different condition of the model cell. Each box contains a phase portrait, corresponding traces for membrane potential as a function of time and for applied current if present, and the settings of the parameters of and us. In the phase portraits, trajectories are shown by consecutive dots drawn at equal-length time intervals. Thus the larger the dot spacing, the faster the speed of movement of the phase point. The very thick dashed lines are the V-nullclines, the thinner dashed lines are the q-nullclines. In the B-D and G phase diagrams, a large dot with a surrounding circle marks the starting point of a trajectory. V, membrane potential; q, slow current. In all boxes A, = 1. Except for G, 7, = 0.1666,~~ = 5. A: cf = 2, cs = 3. An attracting limit cycle in the phase diagram produces potential traces similar to those often seen in cells LG and MG in the gastric mill CPG. A, B, and C form a series in which cf is reduced to 0, or equivalently the degree of N-shape goes to 0, whereas A, D, and E form a series in which cs is reduced to 0. F and G are 2 cases in which a split slow current is used. B: of = 1. The cell is quiescent but nearly an oscillator, as shown by the slow spiral of the phase point into the stable equilibrium point (e.p.) at the origin (v, q) = (0,O); the membrane-potential trace shows a damped oscillation. C of = 0. The model cell is quiescent and exhibits only postinhibitory rebound (PIR) and postburst hyperpolarization (PBH). The steadystate points marked by dot-circles on the q-nullcline arise from the initial hyperpolarizing and depolarizing currents. D: this case exhibits PIR, PBH, and plateau potentials but does not oscillate. It is intermediate between A and E, with os = 0.5. There are two stable e.p.s, starred, and an unstable e.p. at the origin. Top pair of traces shows PIR with the cell coming to rest at H, bottom pair of traces shows PBH with the cell stabilizing at the lower plateau potential L. The last parts of the trajectories, showing the slow approaches to H and L, have been omitted. E: cell exhibits pure plateau potentials. The slow current q = 0, so the phase point lies on the V-axis. The phase diagram is effectively 1 dimensional with stable e.p.s at L and H and an unstable e.p. at the origin. F: a split slow current was used, with the gain of the inward part less than the gain of the outward part. Thus the traversal of the lefthand downward section of the limit cycle is slower than the traversal of the righthand upward part. This difference in speeds on different parts of the limit cycle corresponds to the difference in the lengths of the hyperpolarized and depolarized portions of the potential trace. G: inward slow gain is 0, resulting in a single stable e.p. at L. A current pulse (arrowhead) moved the phase point from L to a positive potential (dotted circle). The final approach to L is very slow and stays very close to the fast I/-nullcline; this is not visible in the figure. MODEL OF GASTRIC FIG. MILL CPG 1037 5. (continued) phasepoint enters a narrow channel it is forced to move very exceeds the threshold value of = 1 + T,/T,. Because T,/T, is assumed to be small, of is - 1.0,, the threshold for the slow slowly becauseit is closeto two lines,with dt = 0 on oneline and current gain CT~,can be computed from of, T,, and 7,. If the &I = 0 on the other. time scales 7,, TV,of the two equations are such that 7, is at dt least an order of magnitude smaller than T,, then the wavedV RESULTS Our overall strategy was to start by modeling a single cell and then more complex circuits. We present results for the single-cell model followed by an inhibitory pair of cells, then a complete network model, a reduced four-cell network, and finally a complete network model incorporating slow synapses. Properties of a single cell When the fast current 1-V curve F(V) is sufficiently Nshaped (Fig. 24 and the gain (TVof the slow current is above a certain threshold value 8, > 0 (Fig. 34, the model cell oscillates (Fig. 34. F(V) is sufficiently N-shaped when cf F= l M ,\ z 0 0” 0 10 Slow current gain 20 FIG. 4. Plot of the oscillation frequency of a model cell as a function of the slow current gain cS. 1038 P. F. ROWAT AND form is that of a relaxation oscillator. Often, the intracellular slow waves generated by gastric mill cells LG, DG, or MG are remarkably similar to relaxation oscillator waveforms, as illustrated by Fig. 5, I and J. If F( V) is deformed into a non-N-shaped curve by setting cf 5 1 (Fig. 2B), then the model will not oscillate (Fig. 3B; also Fig. 3C). Thus the N-shaped feature of F( V) is necessary for the model cell to oscillate. The frequency of oscillation of the model cell is primarily determined by the setting of the gain us of the slow current. A plot of frequency u against slow current gain as is shown in Fig. 4. This dia.gram shows how a particular cell model suddenly changed from the quiescent state to endogenous oscillations as the parameter gs passed through a threshold value 8, = 0.49. The amplitude of oscillation of the cell is primarily determined by the setting of the parameter A, in Eq. 3, because this determines the width of the N-shape in the fast current and hence the approximate maximum and minimum values of the membrane-potential excursions. If the cell is oscillatory, changing the value of Af changes the oscillation amplitude but has no significant effect on the oscillation frequency. This is because when Af is changed, the speed of the phase point changes with it, so that although the limit cycle has larger dimensions, the overall time for one cycle remains approximately the same. Almost all statements about the qualitative behavior of the cell model can be verified and made more precise by mathematical analysis of the system MC (Guckenheimer and Holmes 1983). The steady-state I- Vcurve of the model cell is the sum of the N-shaped curve F(V) of the fast current and the steadystate 1-V curve of the slow current. The resultant summed 1- Vcurve has a region of negative resistance only when cf > landl-~f+~,<O,thusonlywhen~,<~f-l.When~,> cf - 1, the steady-state I-V curve has no region of negative resistance. If, however, the current is measured in voltage clamp immediately after the voltage has been stepped from a holding potential Vh, the instantaneous 1-V curve has a region of negative resistance. This is because the slow current has not had time to change significantly, and the resulting 1-V curve is simply F( V), shifted up or down by the slow current flowing at the holding potential Vh. Thus the instantaneous I-V curve will always have a region of negative resistance when cf > 1. 1-V curves obtained by measuring current after a time interval sufficiently short compared with the time constant for the slow current will also have a region of negative resistance. The time interval \ must be less than 7,log, (.,+7-O-,). If, in the endogenously oscillating model cell, the parameter gf is reduced so that F(V) is no longer sufficiently Nshaped, but (T, remains positive, the model cell does not oscillate but exhibits postinhibitory rebound (PIR) and postburst hyperpolarization (PBH; Fig. 3, B and C). It does not have plateau potentials. If instead, F(V) remains N-shaped but (T, is sufficiently reduced (Fig. 3, D and E), the model cell does not oscillate but does exhibit plateau potentials. Provided gs > 0, it will in addition display PIR and PBH, as shown in Fig. 30. The top pair of (V, i) traces is a case where a hyperpolarizing A. I. SELVERSTON current was stepped to zero; the V trace rebounds too high then settles down to the high plateau potential H. Similarly when a depolarizing current was stepped to zero, the Vtrace rebounded too low then slowly settled up to the low potential L. In Fig. 3E the depolarizing pulse was adjusted to a critical size barely sufficient to move the phase point past the unstable e.p. at the origin, similarly for the hyperpolarizing pulse. The existence of the unstable e.p. can be seen as a brief pause in the rise (fall) of the membrane potential to H (L)* Thus according to this model, if an isolated cell oscillates, then it necessarily also displays plateau potentials and PIR/ PBH. Conversely, if a cell displays plateau potentials, then it must have a fast current with a region of negative slope, even if it does not oscillate endogenously. A slow current may be present but must not be of sufficient magnitude to cause oscillations. If a cell displays PIR/PBH, then there must be slow inward and outward currents present, even if the fast current does not have an 1- Ycurve with a region of negative resistance. Thus one would expect that there would be certain neuromodulatory conditions such that both the fast current has a region of negative resistance and the slow current is of sufficient magnitude to cause endogenous oscillations. The diagrams of traces from the model cell do not have a time scale bar, because the time scale is essentially arbitrary. This is because when the time constants 7, and 7, are both scaled by a constant factor k, Eqs. 1 and 2 are invariant, except that time t is also scaled by k. Thus the qualitative behavior of the model cell does not change when the time constants are changed, provided their ratio 7,:7, is fixed. As explained in METHODS, the difference in time constants, but not their absolute values, is important to the correct functioning of the model cell. When the ratio 7,:7, becomes 1: 1, the oscillations cease. However, this value for the 7,:7, ratio is not physiological. In all model cells considered here it is 1: 10 or greater. When the slow current is separated into two fractions, a slow inward and a slow outward current, the parameter us, which controls the gain of the original slow current, is replaced by two parameters, gin and gout, which control, respectively, the gains of the new slow inward and outward currents. By adjusting the ratio of gin to gout, the waveform of the cell can be modified to have larger depolarized or hyperpolarized parts (Fig. 3E). In Fig. 5, physiological examples of plateau potentials, PIR and PBH, and endogenous oscillations, are compared with model output that demonstrates these properties. The physiological plateau potential in Fig. 5A is simulated by the model traces in Fig. 5C. The first two current pulses in Fig. 5n shows that the pulses must exceed a threshold to trigger a plateau, and the third pulse shows that a small hyperpolarizing pulse can terminate a plateau. The model traces in Fig. 5C simulates this behavior ( 1st pulse and last 2 pulses). Note, however, that the plateau in Fig. 5A continues to rise slowly after initiation, a feature that cannot be captured by our model. Figure 5B shows a self-terminating plateau potential, simulated by the model traces in Fig. 5D. These were obtained by having a split slow current with zero slow inward gain (gin = 0, coUt > 0), and N-shaped fast MODEL A OF GASTRIC v -2 nA 3 nA i A nn . .. ... .. .... .. A J /\ ioff I CPG 1039 1-V curve (a, > 1). Note the presence of slowly decaying PBH in B, which is also present in the model potential trace in D. Figure 5, E and F, shows physiological PIR and PBH. Figure 5, G and H, shows simulations of these effects, where the model must have of < 1 (no N-shape) and g’s> 0. Figure 51 shows a physiological trace from gastric cell LG, which is simulated well by the model trace shown in Fig. 5J. The latter trace was obtained by having a split slow current with injected current. tin ’ gout and a constant depolarizing These properties of gastric cells can be understood in the model in terms of movements in the (V, q) phase plane. Figure 3 illustrates how plateau potentials and PIR are phenomena necessarily associated with an oscillatory cell. For example, plateau potentials as in Fig. 5C can be interpreted as movements in the phase diagram of Fig. 3E. There are two stable e.p.s at L and H and an unstable e.p. at the origin. If the membrane potential is originally at L and a current pulse is applied that moves the phase point just to the right of the origin, then the phase point will continue to move further right to the high e.p. at H. If the pulse does not move the phase point as far as the origin then the potential falls back to L. This occurred for the first two pulses in Fig. 5C. The third pulse moved the phase point just to the right of B 2 MILL I I A q I i off I FIG. 5. Comparison of simulated cell behavior with physiological traces from gastric mill cells under various neuromodulatory conditions. In the model output traces, Vis membrane potential, i is injected current. Thick dashed lines have been drawn to emphasize differences in membrane-potential values. Thinner vertical bars to right of model traces (C, D, G, H, and J) are 1 unit in length. A and B show physiological plateau potentials; C and D show model behavior that simulates A and B. A: gastric mill cell given small current pulses at the arrowheads. The 1st pulse failed to elicit a plateau but the 2nd, slightly larger pulse succeeded (3 nA instead of 2 nA). The plateau was terminated by a small hyperpolarizing pulse. This was a DG cell, Stn cut, Ba2+ substituted for Ca2+, 10e7 M tetrodotoxin (TTX) in bath; 300-ms current pulses at arrowheads. Calibration: 5 s, 20 mV. Data from Russell and Hartline (1984, Fig. 1 lA2). B: another gastric cell, given a single depolarizing pulse, produces a plateau potential that terminates endogenously. Note the small after hyperpolarization. This was an MG cell, Stn cut, IO-Hz continuous Stn stimulation, -4 nA offset current. Calibration: 1 s, 20 mV, 10 nA. Data from Russell and Hartline (1984, Fig. 8B4). C: model cell was given a succession of depolarizing or hyperpolarizing current pulses. Model parameters used: slow current gain of = 1.8. The other parameters were Af = O’s = 0 and N-shape parameter values of these latter parameters do not 1.o, 7, = 0.2,7, = 4. The particular affect the production of plateau potentials. D: in a series of small current pulses of increasing amplitude applied to the cell model, the 3rd elicited a self-terminating plateau. The parameters used were split slow current, (T,,~ = 3.0, gin = 0, of = 1.8, 7, = 0.5, r, = 5. E and F: PIR and PBH in gastric mill cells. The timing dots in E and Fare at 1-s intervals. E: top trace is the membrane potential of an LPG cell when steady hyperpolarizing current was suddenly shut off at the arrowhead. The trace is not shown during the hyperpolarizing current (off scale). Discontinuous bottom trace is the current record. Calibration bar: 10 mV, 10 nA. Data from Selverston et al. (1976, Fig. 6 1). F: top trace is the membrane potential of an LPG cell when steady depolarizing current was suddenly shut off (arrowhead). Bottom trace is the current record. Calibration bar: 10 mV. Data from Selverston et al. (1976, Fig. 59b). G: PIR in the model cell when hyperpolarizing current was shut off. Top trace is model potential, bottom trace is model injected current. H: PBH in the model cell when depolarizing current was shut off. The 0 arrow in G and H shows the 0 current level. Critical parameters used for G and H: CT,= 2, cf = 0. I: membrane potential trace of an LG cell in a cycling gastric mill CPG. Calibration: 5 s, 10 mV. Data from Heinzel and Selverston ( 1988, Fig. 1OA). J: membrane-potential trace from an isolated model cell, with parameters of = 1.3, oout = 1.2, Gin = 1.O, 7, = 0.06, r, = 4.68, iniected current 0.133. 1040 P. F. ROWAT AND A. I. SELVERSTON the unstable e.p. at the origin; consequently, the phase point continued to move right until it reached H, where its motion stopped. The fourth pulse, equal and opposite to the third, moved the phase point back to L. The fifth and sixth pulses were of exactly the right size to move the phase point from L to H, and from H to L, without any further movement. The seventh pulse was a little too large because it moved the phase point to the right of H; subsequently, it moved back left to H. Similarly, the eighth pulse was a little bigger than needed to move the phase point back to L. If the fast nullcline were moved up slightly by, e.g., injecting inward current, the unstable central e.p. would be closer to the new low e.p. and further from the new high e.p. In this case the pulse required to move the potential up from the low e.p. to the high e.p. would be smaller than the pulse required to move the membrane potential down from the high e.p. to the low e.p. The transition from plateau potentials to oscillations appears in Fig. 3 as the sequence from E to D to A. This sequence is created by changing the slow current gain us from 0 to 0.5 to a value, 3, that is greater than the threshold 0,. Geometrically, in Fig. 3, the q-nullcline is being rotated anticlockwise about the origin. The transition from PIR/ PBH to endogenous oscillations, shown in the Fig. 3 sequence C to B to A, corresponds to the increase of the fast current parameter cf from zero to a value greater than one. This transition corresponds to the transition in phase diagrams in Fig. 3, from C to B to A, as cf is increased from 0 to 2. Geometrically, the Knullcline is being changed from a linear to an N-shaped form. When the fast and slow nullclines are both linear, as occurs with cf = 0 in Fig. 3C, the cell exhibits PIR and PBH. Here the trajectory follows a rapidly decreasing spiral. When cf is increased, the spiral becomes more open, as in Fig. 3B with gf = 1, and then becomes a limit cycle as in A. In Fig. 3F the split slow current has smaller inward gain gin, and therefore the movement along the fast nullcline at part 4 of the limit cycle is slower than the movement along part 2; hence the hyperpolarized portion is longer than the depolarized portion, as seen in the trace in Fig. 3F. If gin is considerably less than 8,, for instance gin = 0 in Fig. 3G, then there is exactly one stable e.p., at L. If a current pulse moves the phase point to the right of the central unstable e.p., the phase point follows a trajectory very close to that of a regular oscillation except that it comes to rest slowly at L: a self-terminating plateau potential (Fig. 3G). For this case the ratio of the membrane time constant and slow current time constant was changed from 1:30 to 1:200. When this ratio is made very large the rate at which the phase point moves up or down the fast nullcline is much smaller then the phase point’s horizontal movements. Consequently, the downward slope of part 1 of the potential trace in Fig. 3G is smaller. The trace in Fig. 3G is qualitatively the same as the trace in Fig. 5D. The slow nullcline need not touch the fast nullcline at the origin as in this example; any curve that swings down from upper right and crosses the fast nullcline at a point L on its left branch will suffice. Figure 3 outlines some of the different behaviors that can be obtained from this cell model and how they can be understood by using phase diagrams. Clearly many other be- haviors can be obtained by using, for example, slow nullclines with extra bends or even dips and bumps that could be obtained in a cell with enough slow currents. Reciprocal inhibitory pair When two identical cells that are endogenous oscillators are connected with reciprocal inhibition of equal strength, the two cells oscillate exactly out of phase with each other. The waveforms of the cells in the reciprocal inhibitory pair are similar to the waveform of the isolated cells, but with increased amplitude due to lower minimum potential caused by the inhibition and higher maximum potential due to PIR when the inhibition is suddenly reduced (Fig. 6, A and B). When the of parameters for both cells are set below one so that the cells are quiescent in isolation and do not display plateau potentials but still exhibit PIR/PBH, the reciprocal inhibitory pair still oscillates in antiphase. If the of parameters are reduced further, to zero, the inhibitory pair still oscillates (Fig. 6, C and D), provided the connecting strength IV is large enough. This is in agreement with the observation by Satterlie ( 1985) that PIR together with synaptic inhibition is sufficient to cause an inhibitory pair to oscillate. It also confirms, and extends to graded transmission, the spike-based simulations of Perkel and Mulloney (1974). As the value of cTfis reduced below one, the synaptic strength Wrequired for the inhibitory pair to oscillate with the same amplitude increases as a linear function of Us In one model inhibitory pair, the minimum value of IV required for oscillations to occur was found to be IV = 0.7 - 0.6a, for 0 < cTf< 1. Two quiescent model cells, when connected with strong enough synapses for joint oscillations to occur, may remain quiescent for a short period before oscillatory behavior begins. In this case the model is sitting at an unstable e.p. inside a limit cycle, but due to instability, the model will not remain at this point because the small errors arising in any numerical integration procedure ensure that the system will eventually burst into activity. A partial explanation of the oscillation of the inhibitory pair, when neither cell is an oscillator, can be given with reference to its phase diagram as follows. The reciprocal inhibitory pair whose waveforms are shown in Fig. 6D is analyzed in Fig. 7. When of = 0 the fast current in each cell is purely linear as in Fig. 7A. When two cells each having cf = 0 are connected by reciprocal inhibition, the inhibitory postsynaptic currents effectively deform the linear fast currents in each cell into N-shapes, as shown in Fig. 7, B and C, thus allowing each cell to oscillate as a nearly two-dimensional system with the limit cycle shown in the two-dimensional phase diagram of Fig. 7E. Because the reciprocal inhibitory pair is described by four differential equations, its phase diagram is four dimensional. The Vi-nullcline in Fig. 7E is a projection onto two dimensions of the true Vi-nullcline in the four-dimensional phase diagram. The true Vi-nullcline, which, from the current conservation equation is defined by q1 = -0 6) - wf(w 5 - Epos*)9is a three-dimensional surface in ( VI, V&ql,q2) space. The reverse N-shape is obtained as the VI- V2 trajectory of Fig. 70 is traversed. In Fig. 700 the traiectory has been de- MODEL OF GASTRIC MILL CPG 1041 5050 t i I I I I BB ??i *L -1 ViPL 8 4 period C 1 2 65:35 E ‘I LL El1 *J G 1 -------. 2 -------_ 44:56 III I i ii F E21 i H : I i i FIG. 6. Potential traces from pairs of reciprocal inhibitory model neurons. A: 2 isolated identical oscillatory cells, by chance not out-of-phase. of = 1.8, c’s= 3.0,~~ = 0.333, 7, = 5.0. Scale bar extends from + 1 to - 1, centered on 0. B: same cells as in A, linked by reciprocal inhibition with the synaptic weights set to 0.2. Plot to the right compares the periods and the voltage ranges of the oscillations in A and B. This has the same scale as the traces, on both axes. C: 2 isolated cells each with of = 0. No activity is present. The other parameters in each cell were of = 0, (T,= 3.0,~~ = 0.333,7, = 5.0. D: same cells as in C, linked by reciprocal inhibition. The synaptic connection strengths were 0.8. E: 2 isolated cells each with different periods. In both, cf = 1.8,~~ = 0.333,~~ = 5.0. In the top cell, (T,= 10.0, whereas the bottom cell had os = 1.5, with ratio of periods ~3. F: same 2 cells as in E, linked by reciprocal inhibition, with synaptic strengths set to 0.6. The cells have entrained to a common period between the periods of the original cells and have phases different from the phases of the symmetrical inhibitory pair. Using the points of maximum potential to define the phases, the cell that was slowest originally has retarded phase (65% instead of 50%), and the cell that was fastest originally has advanced phase (35% instead of 50%). The plot to the right compares the periods and voltage excursions of the oscillations before (El and E2) and after (F) introducing the connections. G: 2 isolated cells that exhibit plateau potentials and PIR/PBH as in Figs. 4E and 5E: of = 3, us = 1.15 in cell 1, us = 1.11 incell2, 7, = 0.1, 7, = 2.0. I-? same 2 cells as in G, linked by setting the weights to 0.06. The asymmetry in the relative phase is due to small differences in the slow currents. Dashed lines linking the traces in G and H show that the opposing plateaus of the cells in the inhibitory pair are approximately the same as when the cells are isolated. formed by stretching along a diagonal line from lower left to top right, thus showing the asymmetry between the depolarized and hyperpolarized parts of the waveforms. The reciprocal inhibition moved the V1-V2 trajectory down and left from the origin (D). In Fig. 7E the dotted trajectory is the projection of the limit cycle of the reciprocal inhibitory pair onto the V,-q, plane. Note the similarity between this trajectory and the trajectory of a single endogenous oscillatory cell in Fig. 3A. The points of steep increase and decrease in the membrane potential in the traces of Fig. 6D are not caused by endogenous plateau potentials but by postinhibitory rebound when the inhibition from the other cell is sud- denly removed. Thus for this parameter setting our model reciprocal inhibitory pair has behavior similar to the release mode of operation of the Wang and Rinzel ( 1992) model of an reciprocal inhibitory pair. When two nonoscillatory cells that display plateau potentials and PIR/PBH (Fig. 30) are connected with reciprocal inhibition, the inhibitory pair will oscillate, provided the connecting weights are small enough (Fig. 6, G and k7). The cells begin to relax to their plateau potentials as in Fig. 30. Suddenly, the cell at the high potential loses stability due to the hyperpolarizing postsynaptic current, and the cells switch polarities. If the connecting weights are too large, the 1042 P. F. ROWAT AND A. I. SELVERSTON fast1 2 I v2 synl L K c Vl fast1 +synl FqLJ I I /w ql nullcline I I 91 Vl nullcline 7. Phase diagram for 1 cell of a pair of silent cells linked by reciprocal inhibition. of = 0 in both cells so there is no N-shape in their fast currents. A: fast current in cell 1, labeled fast 1, as a function of vi. B: synaptic current syn 1 in cell 1 during 1 oscillation cycle, plotted as a function of vi. The synaptic current as vi increases differs from the synaptic current as V, decreases because of different values for I$; hence the double lines shown. C: the total fast current in cell 1 is the sum of the currents in A and B. D: plot of vi vs. I$ over 1 limit cycle. When vi is low, & is high, so the synaptic current syn 1 in B is high. DD: data of D have been distorted to show detail. Twice as many points have been used. (If the points were plotted on a rubber sheet, and the sheet then uniformly stretched in a northeast-southwest direction, the resulting data point positions would be similar to those plotted in DD). E: total fast current in cell 1 is plotted as if it were the nullcline for I’i . The q,-nullcline and the qi- I$ phaseplot are also drawn. In each ofA-E the axes meet at (O,O),and the endpoints of the axes are at + 1 or - 1. In B and C the vertical axis extends from 0 to 1. FIG. third, when each cell in isolation is quiescent but exhibits plateau potentials, the inhibitory connections ensure that the cells will always be in plateau potentials of opposite polarities. If the connections are too strong, then one cell becomes dominant in a high plateau. When the connections are weak enough, the cell in the lower plateau is able to destabilize the cell in the high plateau, and the polarities switch. For an asymmetric reciprocal inhibitory pair, the notation ~1.772 means that the interval between the peak of ceU i and the following peak of cell 2 is ~1% of the period of the pair, and the interval between this peak of cell 2 and the next peak of cell I is ~2% of the period of the pair, where p2 + pl = 100. This notation has been used in Fig. 6. When two oscillatory model cells with widely differing endogenous frequencies are connected with reciprocal inhibition, they immediately entrain to a common frequency. The entrained frequency lies between the two endogenous frequencies, and the entrained cells are no longer exactly out of phase with each other (Fig. 6, E and F). When compared with the phases of an inhibitory pair with identical cells, the cell that originally has the slower frequency comes to a maximum with a slight phase lag, whereas the cell that originally has the faster frequency comes to a maximum with a slight phase advance. This effect is more pronounced when the connections are weak than when they are strong. By this means it is possible to set phase differences between cells of up to 15% (Fig. 68). Because the main determinant of the endogenous frequency of a cell is the gain of the slow current of the cell, one can state this effect slightly differently: by adjusting the ratios of the gains of the slow currents in different cells, one can obtain phase differences between cells of up to 15%. When the weights of the inhibitory connections are not identical but are still of the same order of magnitude, there is no significant difference in phase from the case of identical weights. However, when the weights differ by more than a factor of 10 (e.g., by a factor of 20), there is a small change in phase. If cell 2 receives strong inhibition from cell I but cell 1 receives only weak inhibition from cell 2, then cell 2 will lag behind cell I by >50%, and cell 1 will lag behind cell 2 by 40%. Complete net work model cell at the high potential cannot be destabilized, and the pair does not oscillate. The dashed horizontal lines in Fig. 6 between the traces, Gl and HI, and between the traces G2 and H2, bring out the fact that the plateau potentials in the reciprocal inhibitory pair, in this case, are approximately the same as the plateau potentials in the isolated cells. Thus there are, even for this simple model of an inhibitory pair, three distinct mechanisms of oscillation. In the first, when both cells are endogenous oscillators, essentially any nonzero connection strengths, however small, suffice to set the two cells exactly out of phase with each other. In the second, when both cells are quiescent in isolation but show PIR and PBH, the synaptic currents play an essential role in the oscillation mechanisms, and the connection strengths are required to be above a minimum value. In the We constructed a network model of the complete gastric mill network that included all the connections in Fig. IA. In view of the large number of independently adjustable parameters, we had assumed that it would be necessary to use a parameter adjustment algorithm, such as was presented in Rowat and Selverston (199 l), to find sets of parameter values that would cause the model to generate reasonably correct output patterns. However, the situation turned out to be somewhat different. The network model, with all the cells set as endogenous oscillators, all the excitatory and inhibitory synaptic strengths set to the same small value, and all the electrical junctions set to a small value, produced approximately correct output patterns. such as appear in Fig. 8A. Similar MODEL OF GASTRIC I LG MG GM LPG DG AM lntl LG MG GM LPG DG AM lntl C MG LPG, AM GM,LG I A phases II DG, Intl II --TI II I B pluses MG GM LG LPG DG* AAi IC -I -i II - lntl IC FIG. 8. Pattern generation in the model network, with all connections present. A: each cell is an endogenous oscillator. B: each cell would be quiescent if isolated. Dotted lines connect the maximum points in each trace, to show the phase relationships. In B, the cf value for all the cells has been reduced so that each cell would be quiescent if isolated. Each cell has PIR/PBH but does not have plateau potentials. The phase of MG has been advanced by increasing the gain of the slow current, gs, in MG. This is seen from the slight rightward tilt in the dotted line connecting the maxima of MG and LG. All the excitatory weights had the same value, and with 2 exceptions, all the inhibitory weights had the same, slightly larger value. Exceptions were for the LPG cell: inhibitory weights from DG and AM were reduced by an order of magnitude. All the electrical junctions had the same strength. C: comparisons of the phases and amplitudes of the traces in A and B. Vertical scale is the same in A-C. A horizontal line represents 1 complete cycle. There is a vertical line for each cell, connecting the maximum and minimum values of the cell’s membrane potential, with 0 aligned on the cycle line. Some of the phase lines overlap, e.g., LPG and AM, so the individual amplitude excursions cannot be seen. IC labels a dashed line drawn at the relative period of an isolated cell, with symmetrical extent indicating its maximum and minimum. MILL CPG approximately correct patterns were obtained when the parameter values were varied over relatively wide ranges (Fig. 11). By “approximately correct pattern” we mean that LG and MG are in phase and are out of phase with Intl and LPG, AM and DG are in phase together and out of phase with GM, and that AM and DG are in phase with Intl. Because there was a large volume of parameter space containing suitable sets of parameter values, the problem was transformed from one of using a parameter adjustment algorithm to find an element in a small volume of parameter space, into a somewhat different problem of understanding the phaselocking properties of networks of relaxation oscillators. The network model was then altered by making each cell quiescent, with each cell’s parameter a,just under one (Fig. 3, B and C). The cells did not have plateau potentials but still had PIR/PBH. This model also produced approximately correct patterns except for the LPG cell, whose output was greatly reduced in amplitude and sometimes exhibited phase walk-through, i.e., LPG did not have the same frequency as the other cells of the network, so that the phase of LPG with respect to the other cells was always changing (Ermentrout and Rinzel 1984). The phase of LPG always advanced with respect to the remaining cells of the network, because the frequency of oscillation of the complete network was slower than the frequency of individual cells (or potential frequency of an individual cell if it were quiescent in isolation). This behavior is not entirely unforeseen, because LPG receives antiphase inhibitory input from LG and MG as well as in-phase inhibitory input from DG and AM. Because inhibitory input was received at all phases of the oscillatory cycle, LPG was being subjected to a hyperpolarizing current, whose amplitude of variation was too small to produce entrainment. It is known that LPG has only weak connections back to other cells of the gastric mill network and has little effect on the overall pattern-generating process. What was needed was synaptic input with large enough amplitude of variation to entrain LPG. Therefore we reduced the strength of the connections from DG and AM to LPG by a factor of 10, so that the LG and MG connections were the dominant factor in determining the phase of LPG. With this change, the model produced approximately correct output patterns. The output of the model in these two cases is shown in Fig. 8, A and B. In A, each cell is an endogenous oscillator, whereas in B, each cell would be quiescent in isolation. The parameter values used in B are given in Table 1. The small phase advance of the MG signal over LG, corresponding to the data of Fig. 1B, was set by increasing the gain of the slow current in the MG model. It can be seen that the output is approximately correct in both cases except that the characteristic phase lag between the cells controlling the lateral and medial teeth is not present. Figure 8C shows that the phases are essentially the same in both cases. The phase lines extend further below the cycle line than above, showing that the traces are similar to the traces of an isolated cell, except for being shifted down a small amount by the reciprocal inhibition. In A, the oscillators were slowed down, whereas in B, the frequency of the network oscillations is 1044 TABLE P. F. ROWAT 1. Parametersfor first network model(Fig. 8) LG Network connections LG LPG DG AM GM Intl E Cellular parameters *r?l 9 Af 0s *S AND A. I. SELVERSTON MG LPG in 0.06 ej 0.02 in 0.06 ej 0.02 in 0.06 in 0.06 ex 0.05 in 0.06 ej 0.02 in 0.06 DG AM in 0.06 in 0.06 GM Intl E in 0.06 ex 0.05 in 0.06 ex 0.05 ej 0.02 in 0.06 in 0.06 ej 0.02 in 0.06 in 0.006 in 0.006 d+ 0.003 ex 0.05 ex 0.05 in 0.06 in 0.06 ej 0.02 in 0.06 in 0.06 ex 0.02 ej 0.02 ex 0.05 d- 0.003 in 0.06 in 0.06 0.066 1.004 1.0 3.0 10.0 0.066 1.004 1.0 5.0 10.0 0.066 0.066 0.066 0.066 0.066 0.066 1.004 1.0 1.004 1.0 1.004 1.0 1.004 1.0 1.004 1.0 1.004 1.0 3.0 10.0 3.0 3.0 3.0 3.0 3.0 10.0 10.0 10.0 10.0 10.0 Synaptic reversal potentials: E,,, = -4.0 (all inhibitory synapses), ,!Zpst= 4.0 (all excitatory synapses). LG, lateral gastric; MG, median gastric; LPG, lateral posterior gastric; DG, dorsal gastric; AM, anterior median; GM, gastric mill; Int 1, interneuron 1; E, excitatory. Explanation of network connection entries: “(type) (value)” in column A, row B, means the connection from cell A to cell B is of type (type) with strength (value), where (type) is one of the following: ex, excitatory; in, inhibitory, ej, electrical junction; d+, source for rectifying gap junction (diode); d-, sink for diode. faster than the frequency that an isolated cell would have, if it were made to oscillate by setting of > 1. Reduced four-cell network model In the network model just discussed, it became apparent that the phase changes obtained by adjusting the gains of the slow currents were insufficient to duplicate the phase lag observed between the cells that control the lateral teeth (LG, MG, and LPG) and the cells that control the medial tooth (DG, AM, and GM; Fig. 84. However, the excitatory synapses from Intl to DG and to AM produce a very slow postsynaptic response. Also, it has been observed that one result of killing cell Int 1 is for the lateral teeth cells and the medial tooth cells to continue to oscillate but with widely different frequencies (H.-G. Heinzel, unpublished data; YI= 2). This result is illustrated in Fig. 9, A and B. It appears, therefore, that Intl serves to coordinate the sets of cells controlling the lateral and medial teeth. To determine whether the model was sufficiently adaptable to generate simultaneously the lateral-medial phase lag and Heinzel’s Intl killing data, with the addition of slow processes to the Intl to DG and AM synapses, a reduced four-cell network was used. This consisted of one model cell to represent the LG/MG pair of cells that normally oscillate together with no more than - 5% phase difference, one cell to represent the DG/AM pair of cells that also usually oscillate together with no more than -5% phase difference, one cell corresponding to Intl, and one cell corresponding to LPG. The GM cells were ignored because, in general, they respond passively to inhibition from the DG/AM cells and have little influence on the remainder of the network (Selverston et al. 1976). The network diagram of Fig. 9 includes three inhibitory pairs: Intl ++ LG/MG, LG/MG * LPG, and LG/MG * DG/AM. Because none of the gastric mill cells, except perhaps DG, are known to be endogenous oscillators, the cf parameter was set less than one in each of the cells Int 1, LG/MG, and LPG so that none would oscillate when isolated but would still exhibit PIR/PBH. The connection strengths were set strong enough that the inhibitory pairs would oscillate. DG/AM was made to be an endogenous oscillator with frequency three to four times slower than the frequency of the inhibitory pairs in the network, by setting of larger than one and the gain of the slow current to be about one-third less than the slow current gain in the other three cells. The reciprocal inhibition between DG/AM and LG/MG had to be weak enough that when Intl was killed, DG/AM would not be entrained 1 to 1 to the inhibitory pair LG/MG * LPG, but when Intl was present, DG/AM would be entrained 1 to 1 to the lateral teeth cells. The slow synapse was modeled by the use of Eqs. 7 and 8 given above, with the use of a time constant of about onehalf that of the slow currents. The combination of the delayed synapse and the slower endogenous frequency of DG/ AM sufficed to set the phase lag of DG/AM when Intl is present. An example is shown in Fig. 9C. When Intl was removed from the network by setting all the pre- and postsynaptic conductances involving Intl to zero, the inhibitory pair LG/MG ++ LPG continued to oscillate, at a slightly higher frequency than before, whereas the DG/AM cell now had a frequency three times smaller (Fig. 9D). The inhibitory pair was phase locked (3:l) to DG/AM by the weak connections. Note also that in the model DG/AM trace there is a small secondary peak between the primary bursts, which may be considered to correspond to the short secondary bursts in the real AM trace between the long primary AM bursts. All parameter values used in this reduced model are shown in Table 2. Thus the network model was capable of generating the lateral-medial delay and, simultaneously, providing a mechanism for the coordination of the lateral and medial sets of cells by Int 1. MODEL A LPG W; OF GASTRIC MILL CPG 1045 Control ii I 10mV 5s B 0 . Int 1 killed LG- ?--- LPG - * b -wwt e LG/MG D LPG DG/AM lntl 2.0 2.0 2*o I 1 2.0 9. Comparison of removing the real Int 1 cell from the physiological GM circuit, with removing a model Int 1 cell from a model network, using a reduced model gastric network. A: control traces from the GM. B: GM traces from same preparation after cell Int 1 was killed. Traces in A and B are on the same time and voltage scales. Unpublished data courtesy of H.-G. Heinzel. The Intl trace in A, and the MG and GM traces in B, are intracellular recordings. All others are nerve recordings showing spike trains only. In A, all cells are phase locked. Note the phase lag between Int 1 and DG, AM. In B, cells LG, LPG, and MG are phase locked at 1 frequency (O), whereas cells DG, AM, and GM are phase locked at a much slower frequency (Cl). Inset: reduced network used; solid circles are inhibitory synapses, the solid triangle is an excitatory synapse. C and D: output from model network. In C, the phase lag between Int 1 and DG/AM (vertical dotted lines) was obtained by the combined effect of a slow synapse between Intl and DG/AM and a much reduced slow current gain parameter in the DG/AM cell. In D, Int 1 was “killed” in the model, and DG/AM now has a frequency 3 times slower than LG/MG and LPG. FIG. Complete network model incorporating slow synapses To obtain a new complete network model that exhibited the correct lateral-medial delay, as well as correct phase relationships among the other cells, we combined the parameter sets from the reduced four-cell model and the previous complete network model as follows. In the new complete network, the Int 1+DG, Int 1-+AM synapses were adjusted to be slow. The parameter set found for the reduced network of Fig. 9 was extended to the new complete network, as follows: First, the parameter values in the reduced network were inserted into the complete network. If a connection such as LG/MG to Int 1 in the reduced network had value X, then the connections LG to Intl and MG to Intl were both given the value l/2x in the complete network. Thus the effect of LG and MG on Intl in the complete network would be approximately the same as the effect of the single-cell LG/MG on Int 1 in the reduced network, provided LG and MG are approximately synchronous in the complete network. If a connection such as Int 1 to LG/MG had value y in the reduced model, then the connections Intl to LG and Intl to MC were both given the value y in the complete network. Thus the effect of Int 1 on LG and on MG would be the same as the effect of Int 1 on the single-cell LG/MG in the reduced model. Initially, all other connection strengths in the complete network were zero. Then all these connection strengths were simultaneously slowly increased, in the same proportions as had been established in 1046 TABLE P. F. ROWAT 2. Parameters for reduced four-cell network (Fig. 9) LG/MG Network LG/MG LPG DG/AM Intl Cellular 7rt.l ?f Af gs 7s AND LPG DG/AM in 0.33 in 0.02 Int 1 connections in 0.66 in 0.02 in 0.2 ex 0.3 rd 0.25 in 0.4 parameters 0.125 1.0 1.0 1.0 1.25 0.125 1.0 1.0 1.0 1.25 0.125 1.4 1.0 0.36 1.25 Synaptic reversal potentials: Epost = -4.0 (all inhibitory synapses), = 4.0 (all excitatory synapses). rd, time constant for slow excitatory apse. For other abbreviations and explanations, see Table 1. 0.125 1.0 1.0 1.0 1.25 Ei,,,t syn- the previous complete network model, to the same order of magnitude as the strengths already established in the current complete network model. The idea was that by this means the correct overall pattern would still be generated. When we did this and ran the new complete model, there was considerable distortion of the waveforms because of the larger synaptic currents; these in turn were caused by the larger numbers of synapses per cell in the complete network than in the reduced network. Therefore we reduced all the parameter values by an order of magnitude. A further constraint was that the relative connection strengths in the final set of parameters should be consistent with the relative functional connection strengths given by Russell ( 1985b). This has been satisfied. However, we have not yet been able to do this and, simultaneously, retain the property, exhibited by the reduced circuit, of being able to duplicate Heinzel’s data on removal of Int 1. The output from the resulting model is shown dotted in Fig. 10, and the parameters used are given in Table 3. For comparison we took a set of seven simultaneous intracellular recordings made by H.-G. Heinzel and passed them through a smoothing filter to remove spikes. In Fig. A. I. SELVERSTON 10 the smoothed biological traces and the model traces have been overlaid so that the mutual fit can be seen. The model traces have been individually vertically scaled so that each model trace has the same amplitude as its corresponding biological trace. The model LG and DG traces have the best fit, and the model Intl trace has the worst fit. The length of the depolarized part of the model MG trace is not as long as the depolarized part of the biological MG trace. The interburst potential of the model LPG trace is not sufficiently depolarized and should show a more constant gradient leading to the next burst. The interburst potential of the model AM trace needs to descend more slowly from the previous burst. The model GM trace should ascend faster and descend more slowly. The bursts of the model Intl trace should be more prolonged and drop more steeply, and the interburst hyperpolarization should be shorter and decrease faster when the burst begins. The need for individual vertical scaling factors (rightmost scalebars in Fig. 10) in matching the model and biological traces could have been removed by adjusting the amplitude of oscillation of the individual cells. This is done by changing the value of the A, parameter for individual cells. The only significant effect of a change in A, is a change in the amplitude of the cell’s traces, not a change in its frequency (or its potential frequency if not oscillatory). The network effect of changing the A, value in a cell is to change, by Eq. 6, the amplitudes of synaptic currents in that cell and in cells that are postsynaptic to it. Thus it was expected that changing the amplitude of oscillation of a single cell would have little effect on the network phase relationships. This was confirmed in the model: such adjustments caused insignificant changes in the phase relationships and the slowwave shapes. The A, parameter is approximately linearly related to the saturated amplitude of the fast inward current (such as a persistent Na current, or some combination of several fast inward currents), which, when summed with a fixed leak current, gives rise to an overall N-shaped fast 1-V curve. If we assume a unique solution for the A, factors, their relative LG FIG. 10. Simultaneous intracellular recordings (courtesy of H.-G. Heinzel) were passed through a filter to remove spikes. A set of model traces was horizontally aligned with the biological traces, then each trace was individually vertically scaled to obtain the match shown. The 1st set of vertical bars on the right shows the vertical scale of the biological traces, whereas the 2nd set shows the relative sizes of the vertical scaling factors used for the model traces. I MODEL TABLE MG LPG in 0.01 ej 0.0066 LPG DG in ej in in AM ex 0.10 GM in 0.01 ej 0.0033 in 0.06 Synaptic and 2. MILL CPG 1047 DG AM in 0.02 in 0.06 GM Int 1 E connections MG Intl E Cellular 7, ajAf OS 7s GASTRIC 3. Parameters for network model matching Heinzel’s traces (Fig. 10) LG Network LG OF 0.01 0.0066 0.06 0.02 in 0.06 ex 0.05 in 0.06 ex 0.05 ej 0.0033 in 0.06 in 0.06 in 0.02 in 0.006 in 0.006 ej 0.06 ej 0.0033 d+ 0.003 ej 0.06 in 0.06 in 0.06 ex 0.02 ex 0.10 Td 5.0 ex 0.10 ?-d 5.0 in 0.01 in 0.06 ex 0.05 d- 0.003 in 0.06 in 0.06 parameters 0.2 1.004 1.0 3.0 10.0 reversal potentials: Epost = -4.0 0.2 1.004 1.0 5.0 10.0 (all inhibitory 0.2 1.004 1.0 3.0 10.0 synapses), 0.2 1.004 1.0 1.0 10.0 Epost = 4.0 (all excitatory sizes should predict the relative magnitudes of fast inward currents in different cells. However, the recording site for the membrane potential of an STG cell is the soma, which is electrically distant from the slow-wave generation site in the neurites. Thus the amplitude of membrane-potential recordings depends on the electrical distance to the neurites as well as the actual amplitudes. The electrical distance is not known and probably varies from cell to cell; consequently, the A, parameters cannot be directly related to the amplitudes of fast currents. Adjusting the asymptotic values of the fast current (V A,) is not the same as ad hoc scaling of the output after the model has been run. Scaling A, affects synaptic strengths, but scaling output does not. To test the robustness of the model, all of the parameters except A, had their values randomly changed by as much as 40%. No significant change in the overall output pattern was found, as shown in Fig. 11. The largest difference in phase occurred for GM, whose phase relative to MC changed by 14%. When each individual cell was made to oscillate with a randomly assigned frequency, and then suddenly the known connections were introduced, as in Fig. 12, the network rapidly entrained to a common period that was close to the average of the isolated periods. In addition, the pattern was approximately correct. The period of the entrained oscillators is approximately the average of the periods of the isolated oscillators. The average period of the isolated cells is 6.19, the period of the entrained network is 6.15, and entrainment of all the cells to a common frequency is complete within about 2 cycles of the new frequency. When the frequency of all the cells in the network was uniformly changed by simultaneously decreasing the gain of the slow current by 50% in all the cells, the phase relationships remained essentially the same, as shown in Fig. 13. The pattern frequency decreased by 67%. 0.2 1.004 1.0 3.0 10.0 synapses). 0.2 1.004 1.0 3.0 10.0 For abbreviations 0.2 1.004 1.0 3.0 10.0 and explanations, 0.2 1.004 1.0 3.0 10.0 see Tables 1 To determine the degree to which the overall computation was distributed, a succession of cells was removed from the network (Fig. 14). This was accomplished by setting all of a cell’s pre- and postsynaptic conductances to zero, and by setting cf = 0. This completely disconnects the cell from the remainder of the network and changes its phase diagram from one similar to Fig. 3B to one similar to Fig. 3C. Consequently, after removal, the cell’s membrane potential smoothly settled to zero with no apparent damped oscillations. In Fig. 14 neurons were sequentially removed from the gastric mill network starting with MC (Fig. 14A). MC is known to play a key role in the formation of the gastric mill rhythm (Selverston et al. 1983) and is part of the kernel pattern-generating circuit under muscarinic modulation (Elson and Selverston 1992). Although there was no change in phase relationships, there was an increase in the amplitude of the slow wave for both LG and GM and a decrease in the amplitude for LPG. The LPG cell makes only weak connection to the MC cell, and its removal had no effect on the ongoing rhythm (Fig. 14B). Removal of the AM cell (Fig. 14C) produced a small decrease in the amplitude of the GM neurons but had very little effect on the DC cell to which it is electrically connected. Removal of the GM cell (Fig. 140), which is connected to the remaining cells only by an electrical connection, had no noticeable effect. Int 1, on the other hand, made connections to all three of the remaining cells, and its removal therefore had significant effects on the remaining pattern (Fig. 14E). The amplitudes of LG and DC were reduced considerably almost immediately, and their plateau properties disappeared so that their waveform was markedly altered. The E cell, whose only input comes from Int 1, did not stop immediately but showed a damped oscillation (Fig. 14, E and F). This sequence of simulated cell kills left only the LG and DC cells, which are connected to each other by reciprocal inhibitory 1048 P. F. ROWAT AND synapses. Removing LG therefore (Fig. 14F) removed the inhibition to DG, which on its own gradually stopped oscillating and became quiescent. Note that at the right-hand side of Fig. 14F there are still two cells alive but now quiescent: DG and E. Because there are no connections between them, they are isolated cells that become quiescent, as in Fig. 3B. The network continued to oscillate so long as at least one inhibitory pair remained. In the example shown, the LG/DG inhibitory pair continued to oscillate after five other cells had been removed. The behavior of the subnetwork remaining after a number of cells has been removed does not depend on the order in which the removals are carried out. Equivalently, every subnetwork has at most one stable mode of oscillation, in the parameter ranges used. A. I. SELVERSTON A RANDOM FREQUENCIES CONNECTIONS ON E .i .i .; . .i .i LG MG GM LPG DG AM DISCUSSION We have shown that a network of model neurons, each with an N-shaped fast current and slow inward and outward currents, together with graded synaptic transmission, captures several important aspects of pattern generation in the gastric mill CPG. In particular, the network produced approximately correct phase relationships for the in vitro gastric mill rhythm; it continued to generate an approximately correct pattern even when no constituent cell was an endogenous oscillator; when cells were serially removed A PARAMETER LG MG GM LPG DG AM lntl 2.0 LG, E GM B LPG, AM I MG E LG 2.0 DG Phases before I II I GM Inll, AM. In!1 Phases LPG. PERIODS B E LG MG GM LPGDG AM lntl - I; I.I. 1. I, I I 1; f! 2.0 I Entrained Ii 1; Average II FIG. 12. Entrainment of oscillators with randomized frequencies to a common rhythm. All cells were assigned a random frequency by randomly setting the parameter us in the range 1 S-6 for each cell. A: to the left of the vertical dashed line, the cells are isolated. At the dashed line all the network connections were introduced. B: bar chart of the isolated periods. The vertical line with long dashes indicates the entrained period, the shortdashed line the average period. RANDOMIZATION E MG lntl after DG FIG. 11. Randomization of parameters. A: at the vertical dashed line, the connection and cellular parameters were given new random values within 40% of their original values. In most cases, as illustrated here, there is no major change in intercellular phase relationships. The period did not change. Dotted lines join corresponding maximum points in each cycle, thus making clear the phase relationships. Similar dotted lines are used in Figs. 12 and 13. All the model cells were endogenous oscillators. B: comparison of phases before and after randomization, using same conventions as in Fig. 8C. from the network it continued to oscillate and produced approximately correct phase relationships in the remaining cells; and, ignoring spikes, the network produced slow-wave shapes that corresponded approximately to the shapes of the slow waves in physiological intracellular recordings of membrane potential. Small changes in phase relationships were due to differences in gains of the slow currents, whereas larger phase differences required other mechanisms such as a synapse with a delayed activation time, as in the case of the phase lag between the cells controlling the lateral teeth and the cell controlling the medial tooth. The most critical point in the development of this model was the realization that a simple generalization of the wellknown relaxation oscillator of Van der Pol could be given a physiologically correct interpretation, whereas, at the same time, several characteristic properties of stomatogastric neurons, such as plateau potentials, PIR, and endogenous oscillations, flowed naturally from this generalization. Additionally, the waveforms produced by this model cell compared favorably with the typical waveforms of several gastric mill neurons. Further supporting evidence was the fact that an N-shaped fast current is necessary for an isolated model cell to oscillate, consistent with the finding of Wilson and Wachtel(1974) that bursting neurons require a region of negative resistance in their fast 1-V curve. The model makes MODEL 50 % DECREASE IN SLOW CURRENT OF GASTRIC GAIN LPG DG : : : : : : : : AM lntl 2.0 LG, GM MG E LPG Pluses before AM, lnl 1 I B I MG E. LG.MG I 2.0 DG : : AM.LPG.lnll.DG FIG. 13. Constancy of phase relationships when the frequency changes. A: at the vertical dashed line, the slow current gain oS of each cell was reduced by 50%, with insignificant change in phase relationships. All the model cells were endogenous oscillators. B: comparison of phases before and after the frequency change. Dotted line on the “Phases after” cycle line is the original period, relative to the period after the frequency change. precise the statement that if a cell exhibits plateau potentials in one modulatory condition, and PIR and/or PBH in another condition, then the cell will be an endogenous burster in a modulatory condition, which allows the expression of plateau potentials and of sufficiently strong PIR. Physiologically, an N-shaped fast current could arise, for example, from a combination of a fixed leak conductance and a fast, persistent, sodium current. This is the case in neocortical neurons (Stafstrom et al. 1985). The slow current can be regarded as the combination of a slow depolarization-activated, noninactivating, outward current such as a calcium-activated potassium current, and a slow hyperpolarization-activated, non-inactivating, inward current similar to the & current in heart cells (DiFrancesco and Noble 1985) and leech heart interneurons (Angstadt and Calabrese 1989), or the sag current found in the LP cell in the pyloric CPG (Golowasch and Marder 1992). A pair of identical model cells, when connected with reciprocal graded synaptic inhibition, automatically, and very stably, oscillated exactly out of phase with one another. This behavior is seen in reciprocally inhibited pairs in the gastric mill CPG and other CPGs (Calabrese et al. 1989; Satterlie 1985) in the spike-based modeling study of Perkel and Mulloney (1974) and in the model of Wang and Rinzel (1992). Our model reciprocal inhibitory pair can oscillate in three distinct modes. In the first the connections merely serve a phase-setting function, because both cells are endogenous oscillators. In the second the inhibitory connections contribute in an essential way to the oscillations of MILL CPG 1049 the system. The cells are not endogenous oscillators and do not have plateau potentials. In the third the inhibitory connections contribute in an essential way, different from the preceding, to the oscillations of the system. Both cells have plateau potentials but do not have large enough slow PIR/ PBH currents for them to be endogenous oscillators. Provided the connections are not too strong, the system continually switches between a high plateau-low plateau configuration and a low plateau-high plateau configuration. If the connections are too strong, the system locks up and cannot change configurations. When a chain of three model cells is constructed with reciprocal inhibition between the first and second pairs of cells, the two end cells are in synchrony and out of phase with the center cell. The three cells Int l-MG-LPG form a chain in the gastric mill network (Fig. lA), and, as predicted by the model, gastric mill recordings (Fig. 1B) show that the traces of Intl and LPG are synchronous and out of phase with the trace of MG. By extension of this idea, if six cells are made into a ring with reciprocal inhibition, then the cells numbered 0, 2, and 4 will be in synchrony and out of phase with the synchronous cells numbered 1, 3, and 5. A network of this form and with these phase relationships occurs in the leech heartbeat timing oscillator (Calabrese et al. 1989). A pair of model cells with different frequencies (or different potential frequencies if they are not endogenous oscillators) are entrained by reciprocal inhibition to a frequency lying between the two original frequencies and with small changes in relative phase from the case of two identical cells. An example of two cells of widely differing endogenous frequencies entraining to a common frequency occurs in the isolated gastric mill under the effect of pilocarpine (Elson and Selverston 1992, Fig(s). 3 and 8). Here the inhibitory pair consists of the cells LG and DG; when DG is hyperpolarized, LG slows down, but when LG is hyperpolarized, DG speeds up. Thus the reciprocal inhibition between LG and DG serves to entrain two apparently oscillatory cells to a common frequency. Because frequency of a model cell is controlled primarily by the slow current gain parameter u,, in a network of cells with many reciprocal inhibitory connections, the relative phases of different cells can be adjusted over a small range by adjusting the relative values of the intracellular g’sparameters. These parameters correspond to maximal slow current conductances. In our model the small phase advance of MG over LG was set by increasing the slow current gain in MG. This is a testable prediction for the gastric mill network. Small phase delays can also be set in a model reciprocal inhibitory pair by using synaptic weights that differ by more than a factor of 10. Essentially, this mechanism is hypothesized by Williams (1992) to set the intersegmental phase lag in the spinal cord of the lamprey. This suggests an alternative mechanism, namely pairs of cells linked by reciprocal inhibitory synapses of very unequal strengths, for establishing phase relations in the gastric mill CPG. Getting (1989) drew up a list of potential building blocks of network operation. This network model demonstrates how, when combined appropriately, the cellular mechanisms of PIR, PBH, and plateau potentials; the synaptic 1050 P. F. ROWAT a MG AND LPG KILLED LG MG MG 1 LPG DG AM B KILLED LG LPG A. I. SELVERSTON /\ DG /\ AM GM GM lntl lntl E E C AM //\ D KILLED LG LG MG MG GM KILLED LPG DG AM GM GM lntl lntl E E E Intl w w7 F KILLED LG KILLED + LG LG MG MG LPG LPG DG DG AM AM GM GM lntl lntl E 14. Changes in the output removal of MG. B: removal of LPG. details. FIG. \ \ E pattern of the simulated network following sequential removal of single neurons. A: C: removal of AM. D: removal of GM. E: removal of Int 1. F: removal of LG. See text for mechanisms of instantaneous and slow graded transmission; and the connectivity mechanisms of reciprocal inhibition and electrical junction all contribute to correct network operation. With the use of a model such as the present one, the relative importance of different mechanisms to overall network performance can be computed. The model presented here can be related to other wellknown simplifying cell models. It is similar to the FitzhughNagumo model (Fitzhugh 196 1; Nagumo et al. 1962) in that both are defined by two differential equations with widely differing time constants. However, the form of our currents is more closely comparable with the physiological currents believed to underly slow-wave generation in crustacean bursting neurons. It is not surprising that there is a difference between the forms of the currents, because the Fitzhugh-Nagumo equations were intended as a simplification of the Hodgkin-Huxley equations for action potentials in the squid axon, not as a model of slow waves in bursting neurons. It is also comparable with the two-equation Hindmarsh-Rose model (1982), but again the form of our currents is very different from theirs. They use the narrow channel property in the phase diagram, in which two nullclines are very close together and thus causes the phase point to move very slowly in the channel between them, as the basis for very large and variable interspike intervals. Kepler et al. ( 1990) and Abbott et al. ( 199 1) used simplified cell models to study, respectively, control of burst frequency and control of burst duration in the pyloric network in the STG. Abbott et al. used a two-equation cell model whose only major difference from ours is the inclusion of an explicit voltage-dependent membrane conductance. Their equations appear more complex, but in fact the dynamics of their model cell is very similar to ours. The cell model used by Kepler et al. is almost identical to ours; their N-shaped fast current has an 1-V curve composed of three linear sections, whereas ours is curvilinear. Wang and Rin- MODEL OF GASTRIC zel(1992) used a simplified, conductance-based cell model in their study of the reciprocal inhibitory pair. Their cell model has a linear leak (fast) current and a slowly inactivating current (“pi? current) that models a T-type calcium current. Our model has a linear fast current when the parameter of = 0, as in the reciprocal inhibitory pair illustrated in Fig. 6D. However, their pir current cannot be directly compared with our slow current. For instance, its steady-state 1-v curve is bell-shaped around -8 1 to -65 mV, below the nominal rest potential, whereas our slow current steady-state 1-V curve is purely linear both above and below our model’s rest potential. Our model is more specific than any considered by Kopell and Ermentrout (1990) in their models of the lamprey spinal cord; their general theorems on phase locking in chains of coupled oscillators apply to chains of our model cells. In summary, we have constructed a simplifying, biologically constrained network model of the gastric mill CPG, which shows that only a few basic mechanisms are sufficient to produce the relatively complex patterns characteristic of the cycling gastric mill. Further evidence to support the validity of this network model is the fact that it automatically produces approximately correct gastric mill output patterns over a wide range of parameter values. Thus it is a very robust model of the gastric mill mechanism. By this we mean that, because the network model captures a wide range of characteristic properties of the gastric mill network, at the cellular, subnetwork, and network levels, and over a broad spectrum of parameter values, then it is probable that any future, more detailed model of the gastric mill network should be reducible to this one. Our network model therefore contains some fundamental mechanisms underlying the dynamics of the gastric mill CPG. Problem ofneuromodulation As we have shown, our model is robust. On the one hand this is advantageous, in that it produces approximately correct gastric mill output patterns over a relatively wide range of parameter values. Thus the network pattern generation mechanism is not sensitive to many parameters and is not disrupted by small changes in synaptic strengths or in the intrinsic properties of neurons. On the other hand it may be disadvantageous because the gastric mill pattern is highly modulatable (Elson and Selverston 1992; Heinzel and Selverston 1988; Turrigiano and Selverston 1989, 1990). To take the simplest example, when the STG is dissected out and disconnected from the other ganglia in the stomatogastric nervous system, the gastric mill CPG does not produce a pattern. Some cells fire tonically (Int 1, LPG, and GM), two are quiescent (LG and MG), and others burst irregularly (DG and AM) (Selverston and Moulins 1987). The current model can generate this behavior but requires drastic changes in the parameter set. It is unlikely that changes of this sort are an adequate reflection of the neuromodulator-induced difference between the noncycling state of the isolated gastric mill CPG and the cycling state that we have modeled here. This brings into question the usefulness and validity of our model. MILL CPG 1051 The situation can be viewed as follows. There are many dynamically changing variables in the gastric mill CPG, many more than are represented in our model or will ever be represented in any future model. Any model is an approximation to reality. In particular, our model is a gross approximation to the true dynamics of the gastric mill, but it happens to be quite adequate for the neuromodulatory condition that induces the cycling state we have studied. In other neuromodulatory conditions, which induce qualitatively different behaviors, the approximations embedded in our model break down, and other aspects of the true gastric mill dynamics must be included to adequately model the induced behaviors. To gain a better understanding of the generation of multiple, qualitatively different behaviors by the single gastric mill network, a more complex model will ultimately be required. This model will have many more differential equations than our model and will have a large parameter space (e.g., several kinetic parameters for each channel type in each neuron). This model will have many qualitatively different behaviors, each associated with a different, welldefined region of its high-dimensional parameter space. Each qualitatively different model behavior will correspond to an experimentally observed gastric mill behavior. In each region of the high-dimensional parameter space, a simpler, low-dimensional model will be definable that 1) provides an approximate description of the observed behavior under the neuromodulatory conditions defined by this region and 2) whose behavior can be shown to be a reasonable approximation of the behavior of the complex model in this region of parameter space. A change in neuromodulatory condition is represented as a movement between regions in the high-dimensional parameter space. The motion in parameter space causes a change in qualitative behavior of the complex model that in turn corresponds to a transition from one approximating model to another. The mathematical investigation of this topic is the concern of bifurcation theory. The model presented in this paper is part of the picture just sketched, in that it satisfies 1): it approximately describes the behavior observed in the most well-studied neuromodulatory state, that which is obtained when the STG is left connected to the other ganglia in the stomatogastric nervous system. The more complex model for which it is an approximation is unknown. We thank J. E. Lewis and R. L. Calabrese for comments on the manuscript, I-Teh Hsieh for assistance with the computer system, and H.-G. Heinzel for the data used in figures 1,9, and 10. We also thank the referees for their detailed and perceptive comments. This work was supported by Office of Naval Research Grant NOOO149 l-J-1 720, National Institutes of Mental Health Grant NH-46899, and National Science Foundation Grant IBN-9 1227 12. Address for reprint requests: P. F. Rowat, Biology Dept. 0322, University of California at San Diego, 9500 Gilman Dr., La Jolla, CA 920930322. Received 10 December 1992; accepted in final form 13 April 1993. REFERENCES ABBOTT, control L. F., MARDER, E., AND HOOPER, S. L. Oscillating networks: of burst duration by electrically coupled neurons. Neural Computation 3: 487-497, 199 1. ANDERSON, W. W. AND BARKER, D. L. Synaptic mechanisms that gener- 1052 P. F. ROWAT AND ate network oscillations in the absence of discrete postsynaptic potentials. J. Exp. Zool. 216: 187-191, 1981. ANGSTADT, J. D. AND CALABRESE, R. L. A hyperpolarization-activated inward current in heart interneurons of the medicinal leech. J. Neurosci. 9: 2846-2857, 1989. AYERS, J. L. AND SELVERSTON, A. I. Synaptic perturbation and entrainment of gastric mill rhythm of the spiny lobster. J. Neurophysiol 5 1: 113-125, 1984. BAY, J. S. AND HEMAMI, H. Modeling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans. Biomed. Eng. BME-34: 297-306, 1987. BENJAMIN, P. R. AND ROSE, R. M. Interneuronal circuitry underlying feeding in gastropod molluscs. Trends Neurosci. 3: 272-274, 1980. BROWN, T. G. On the nature of the fundamental activity of the nervous centers together with an analysis of the conditioning of rhythmic activity in progression and a theory of the evolution of function in the nervous system. J. Physiol. Land. 48: 18-46, 19 14. BUCHHOLTZ, F., GOLOWASCH, J., EPSTEIN, I.R., ANDMARDER, E.Mathematical model of an identified stomatogastric ganglion neuron. J. Neurophysiol. 67: 332-340, 1992. CALABRESE, R.L., ANGSTADT, J.D., ANDARBAS, A.A.Aneuraloscillator based on reciprocal inhibition. In: Perspectives in Neural Systems and Behavior, edited by T. J. Carew and D. Kelly. New York: Liss, 1989, p. 33-50. CALABRESE, R. L. AND DE SCHUTTER, E. Motor-pattern-generating networks in invertebrates: modeling our way toward understanding. Trends Neurosci. 15: 439-445, 1992. CARPENTER, G. A. Bursting phenomena in excitable membranes. SIAlM J. Appl. Math. 36: 334-372, 1979. COHEN, A. H., HOLMES, P. J., AND RAND, R. H. The nature ofthe coupling between segmental oscillators of the lamprey spinal generator for locomotion: a mathematical model. J. Math. Biol. 13: 345-369, 1982. DIFRANCESCO, D. AND NOBLE, D. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos. Trans. R. Sot. Lond. B Biol. Sci. 307: 353-398, 1985. EDWARDS, D. H. AND MULLONEY, B. Compartmental models of electronic structure and synaptic integration in an identified neurone. J. Physiol. Lond. 348: 89-l 13, 1984. ELSON, R. C. AND SELVERSTON, A. I. Mechanisms of gastric rhythm generation in the isolated stomatogastric ganglion of spiny lobsters: bursting pacemaker potentials, synaptic interactions and muscarinic modulation. J. Neurophysiol. 68: 890-907, 1992. ERMENTROUT, G. B. AND RINZEL, J. Beyond a pacemaker’s entrainment limit: phase walk-through. Am. J. Physiol. 246 (Regulatory Integrative Comp. Physiol. 15): R 102-R 106, 1984. FITZHUGH, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1: 445-466, 196 1. FRIESEN, W. 0. AND STENT, G. S. Neural circuits for generating rhythmic movements. Annu. Rev. Biophys. Bioeng. 7: 37-6 1, 1978. GETTING, P. A. Mechanisms of pattern generation underlying swimming in Tritonia. I. Network formed by monosynaptic connections. J. Neurophysiol. 46: 65-79, 198 1. GETTING, P. A. Mechanisms of pattern generation underlying swimming in Tritonia. II. Network reconstruction. J. Neurophysiol. 49: 10 171035, 1983. GETTING, P. A. Emerging principles governing the operation of neural networks. Annu. Rev. Neurosci. 12: 185-204, 1989. GOLA, M. AND SELVERSTON, A. I. Ionic requirements for bursting activity in lobster stomatogastric neurons. J. Comp. Physiol. 145: 19 l-207, 1981. GOLOWASCH, J. AND MARDER, E. Ionic currents of the lateral pyloric neuron of the stomatogastric ganglion of the crab. J. Neurophysiol. 67: 3 18331, 1992. GRAUBARD, K. AND HARTLINE, D. K. Voltage clamp analysis of intact stomatogastric neurons. Brain Res. 557: 24 l-254, 199 1. GRAUBARD, K., RAPER, J. A., AND HARTLINE, D. K. Graded synaptic transmission between identified spiking neurons. J. Neurophysiol. 50: 508-521, 1983. GUCKENHEIMER, J. AND HOLMES, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983. HARRIS-WARRICK, H. W., MARDER, E., SELVERSTON, A. I., AND MouLINS, M. (Editors) Dynamic Biological Networks: The Stomatogastric Nervous System. Cambridge, MA: MIT Press, 1992. A. I. SELVERSTON H ARTLINE, D. K. AND RUSSELL, D. F. Endogenous burst capability neuron of the gastric mill pattern generator of the spiny lobster in a Panu- lirus interruptus. J. Neurobiol. 15: 345-364, 1984. HEINZEL, Effects H. AND SELVERSTON, A. I. Gastric mill activity in the lobster. III. of proctolin on the isolated central pattern generator. J. Neurophysiol. 59: 566-585, 1988. HILL, A. V. Wave transmission as the basis of nerve activity. Cold Spring Harbor Symp. @ant. Biol. 1: 146- 15 1, 1933. HINDMARSH, J. L. AND ROSE, R. M. A model of the nerve impulse using two first-order differential equations. Nature Land. 296: 162- 164, 1982. HODGKIN, A. L. AND HUXLEY, A. F. A quantitive description of membrane current and its application to conduction and excitation in nerve. J. Physiol. Lond. 117: 500-544, 1952. KATZ, B. AND MILEDI, R. A study of synaptic transmission in the absence of nerve impulses. J. Physiol. Land. 192: 407-436, 1967. KEPLER, T. B., MARDER, E., AND ABBOTT, L. F. The effect of electrical coupling on the frequency of model neuronal oscillators. Science Wash. DC 248: 83-85, 1990. KOPELL, N. Toward a theory of modelling central pattern generators. In: Neural Control of Rhythmic Movements in Vertebrates, edited by A. H. Cohen, S. Rossignol, and S. Grillner. New York: Wiley, 1988, p. 369413. KOPELL, N. AND ERMENTROUT, G. B. Phase transitions and other phenomena in chains of coupled oscillators. SIAM J. Appl. Math. 50: 10 141052, 1990. KRISTAN, W. B., JR. Generation of rhythmic motor patterns. In: Infirmation Processing in the Nervous System, edited by H. M. Pinsker and W. D. Willis, Jr. New York: Raven, 1980, p. 24 l-26 1. LI~NARD, A. Etude des oscillations entretenues. Revue G&&ale de Z’Electricitd 23, pp. 90 l-9 12, 946-954, 1928. MAYNARD, D. M. AND WALTON, K. D. Effects of maintained depolarization of presynaptic neurons on inhibitory transmission in lobster neuropil. J. Clomp. Physiol. 97: 2 15-243, 1975. MCDOUGALL, W. The nature of the inhibitory processes within the nervous system. Brain 26: 153-19 1, 1903. MULLONEY, B. Neural circuits. In: The Crustacean Stomatogastric System, edited by A. I. Selverston and M. Moulins. New York: SpringerVerlag, 1987, p. 57-75. MULLONEY, B. AND SELVERSTON, A. I. Organization of the stomatogastric ganglion of the spiny lobster. I. Neurons driving the lateral teeth. J. Comp. Physiol. 9 1: l-32, 1974a. MULLONEY, B. AND SELVERSTON, A. I. Organization of the stomatogastric ganglion of the spiny lobster. III. Coordination of the two subsets of the gastric system. J. Comp. Physiol. 9 1: 53-78, 1974b. NAGUMO, J., ARIMOTO,S.,ANDYOSHIZAWA, S.Anactivepulsetransmission line simulating nerve axon. Proc. Inst. Radio Eng. 50: 206 l-2070, 1962. PAVLIDIS, T. AND PINSKER, H. M. Oscillator theory and neurophysiology. Federation Proc. 36: 2033-2035, 1977. PERKEL, D. H. AND MULLONEY, B. Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science Wash. DC 185: 181-183, 1974. PERKEL, D. H. AND MULLONEY, B. Electrotonic properties of neurons: the steady-state compartmental model. J. Neurophysiol. 4 1: 62 l-639, 1978. PETZOLD, L. R. AND HINDMARSH, A. C. LSODA: Livermore Solver for Ordinary D&&erential Equations, with Automatic Method Switching for Stirand Nonst#Problems. Livermore, CA 9450: Computing and Mathematics Division, l-3 16, Lawrence Livermore National Laboratory, 1987. RAPER, J. A. Nonimpulse-mediated synaptic transmission during the generation of a cyclic motor program. Science Wash. DC 205: 304-306, 1979. ROBERTSON, R. M. AND PEARSON, K. G. Neural circuits in the flight system of the locust. J. Neurophysiol. 53: 110-128, 1985. ROSE, R. M. AND HINDMARSH, J. L. The assembly of ionic currents in a thalamic neuron. I. The three dimensional model. Proc. R. Sot. Lond. B Biol. Sci. 237: 267-288, 1989a. ROSE, R. M. AND HINDMARSH, J. L. The assembly of ionic currents in a thalamic neuron. II. The stability and state diagrams. Proc. R. Sot. Lond. B Biol. Sci. 237: 289-3 12, 1989b. ROSE, R. M. AND HINDMARSH, J. L. The assembly of ionic currents in a thalamic neuron. III. The seven-dimensional model. Proc. R. Sot. Land. B Biol. Sci. 237: 3 13-334, 1989~. MODEL OF GASTRIC ROWAT, P. F. AND SELVERSTON, A. I. Learning algorithms for oscillatory networks with gap junctions and membrane currents. Network 2: 17-4 1, 1991. RUSSELL, D. F. Rhythmic excitatory inputs to the lobster stomatogastric ganglion. Brain Res. 101: 582-588, 1976. RUSSELL, D. F. Pattern and reset analysis of the gastric mill rhythm in a spiny lobster, Panulirus interruptus. J. Exp. Biol. 114: 7 l-98, 1985a. RUSSELL, D. F. Neural basis of teeth coordination during gastric rhythms in spiny lobsters, Panulirus interruptus. J. Exp. Biol. 114: 99-l 19, 1985b. RIJSSELL,D. F. AND HARTLINE, D. K. Synaptic regulation of cellular properties and burst oscillations of neurons in gastric mill system of spiny lobsters, Panulirus interruptus. J. Neurophysiol. 52: 54-73, 1984. SATTERLIE, R. A. Reciprocal inhibition and postinhibitory rebound produce reverberation in a locomotor pattern generator. Science Wash. DC 229: 402-404, 1985. SELVERSTON,A.I., MILLER, J-P., ANDWADEPUHL, M.Cooperativemechanisms for the production of rhythmic movements. In: Neural Origin of Rhythmic Movements, edited by A. Roberts and B. L. Roberts. Cambridge, UK: Cambridge Univ. Press, 1983, p. 55-87. SELVERSTON,A. I. AND MOULINS, M. The Crustacean Stomatogastric System. New York: Springer-Verlag, 1987. SELVERSTON,A. I. AND MULLONEY, B. Organization of the stomatogastric ganglion of the spiny lobster. II. Neurons driving the medial tooth. J. Comp. Physiol. 9 1: 33-5 1, 1974. SELVERSTON,A.I., RUSSELL, D.F., MILLER, J.P., ANDKING, D.G.The stomatogastric nervous system: structure and function of a small neural network. Prog. Neurobiol. 7: 2 15-290, 1976. STAFSTROM, C. E., SCHWINDT, P. C., CHUBB, M. C., AND CRILL, W. E. Properties of persistent sodium conductance and calcium conductance of layer V neurons from cat sensorimeotor cortex in vitro. J. Neurophysiol. 53: 153-170, 1985. STEIN, P. S. G. Application of the mathematics of coupled oscillator systems to the analysis of the neural control of locomotion. Federation Proc. 36: 2056-2059, 1977. MILL CPG 1053 THOMPSON, R. S. A model for basic pattern generating mechanisms in the lobster stomatogastric ganglion. Biol. Cybern. 43: 7 l-78, 1982. TSUNG, F.-S., COTTRELL,G.,AND SELVERSTON, A.I. Someexperiments on learning stable network oscillations. In: Proceedings of the International Joint Conference on Neural Networks. San Diego, CA: IEEE Neural Networks Council, 1990, p. 169- 174. TURRIGIANO, G. G. AND SELVERSTON,A. I. Cholecystokinin-like peptide is a modulator of a crustacean central pattern generator. J. Neurosci. 9: 2486-250 1, 1989. TURRIGIANO, G. G. AND SELVERSTON, A. I. A cholecystokinin-like hormone activates a feeding-related neural circuit in lobster. Nature Land. 344: 866-868, 1990. VAN DER POL, B. On “relaxation-oscillations.” Philos. Mag. 2: 978-992, 1926. VAN DER POL, B. AND VAN DER MARK, J. The heartbeat considered asa relaxation oscillator, and an electrical model of the heart. Philos. Mag. 7: 763-775, 1928. WANG, X.-J. AND RINZEL, J. Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Computation 4: 84-97, 1992. WARSHAW, H. S. AND HARTLINE, D. K. Simulation of network activity in stomatogastric ganglion of the spiny lobster, Panulirus. Brain Res. 110: 259-272, 1976. WILLIAMS, R. J. AND ZIPSER, D. A learning algorithm for continually running fully recurrent neural networks. Neural Computation 1: 270280, 1989. WILLIAMS, T. L. Phase coupling by synaptic spread in chains of coupled neuronal oscillators. Science Wash. DC 258: 662-665, 1992. WILSON, W. A. AND WACHTEL, H. Negative resistance characteristic essential for the maintenance of slow oscillations in bursting neurons. Science Wash. DC 186: 932-934, 1974. YAMADA, W. M., KOCH, C., AND ADAMS, P. R. Multiple channels and calcium dynamics. In: Methods in Neuronal Modeling: From Synapses to Networks, edited by C. Koch and I. Segev. Cambridge, MA: MIT Press, 1989, p. 97-l 34.