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Amer. Zool., 33:29-39 (1993) from Physiological Insights the Stomatogastric Nervous Cellular System and Network of Lobsters Models of and Crabs1 Eve Marder,* Frank Laurence F. Abbott,* Irving R. Epstein,* Buchholtz,* KEPLER*4 B. AND THOMAS L. JORGE GOLOWASCH,*2 SCOTT HoOPER,f3 *Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254 and ^Department of Physiology and Biophysics, Mt. Sinai Medical School and Center for Neurobiology and Behavior, Columbia University P&S, New York, New York Synopsis. The stomatogastric nervous system of decapod crustaceans is an ideal system for the study of the processes the generation underlying of rhythmic movements by the nervous system. In this chapter we review recent work that uses mathematical simulations analyses and computer to understand: currents in controlling the activity 1) the role of individual on the activity of neurons, of and 2) the effects of electrical coupling neuronal oscillators. The aim of this review is to highlight, for the physwhat these studies have taught us about the organization and iologist, function of single cell and multicellular neuronal oscillators. components play in defining the activity of a system. On the other hand, this is an area where mathematical and computer models can provide substantial insights. The crustacean stomatogastric ganglion and generates 30 neurons, (STG) contains two motor the pyloric patterns, rhythm ?1 see) and the gastric rhythm (period (period 5-10 see). The pyloric rhythm consists of repeating bursts of action potentials in the motor neurons that innervate muscles and constrict the dilate that alternately pyloric valve. The pyloric rhythm depends on a bursting pacemaker for its rhythmicity neuron, the Anterior Burster (AB) neuron. The AB neuron is electrically coupled to the Pyloric Dilator (PD) neurons, which there? connections fore fire in bursts. Inhibitory from the pacemaker group (AB and PD neu? neurons (Lateral rons) cause the constrictor to Pyloric (LP) and Pyloric (PY) neurons) fire out of phase with the dilator neurons. In this article, we will describe our first attempts to construct models ofthe neurons in the STG. For each exam? and networks the form of the ple, rather than describe elsemodel in detail (these are described will describe the we where), neurobiological problem that led us to formulate the model, the salient features ofthe model, summarize neuand then focus on what experimental can learn from this enterprise. roscientists Introduction The reductionist to neurosci? approach ence has taught us to seek to understand the nervous to identify, system by attempting isolate, and analyze each of its components. At the level of cellular biophysics this has led to the study of single ionic currents and the second messenger systems that underlie At the level of sys? many of these processes. tems the reductionist neuroscience, has led us to attempt to identify approach the neurons in a given behavior involved and to define the connections among them. In both cellular and systems neuroscience, it is always easier to define the however, individual of a system, than it components is to understand what each identified com? be it a current or a neuron, conponent, tributes to the dynamic activity ofthe whole. conventional Indeed, electrophysiological and biophysical methods are almost totally to analyze the role individual inadequate 1 From the on Computational Symposium Approaches to Comparative Neurobiology presented at the Annual Meeting ofthe American Society of Zool? ogists, 27-30 December 1990, at San Antonio, Texas. 2 Present address: Department of Neurology, MGH, Boston, Massachusetts. 3 Present address: Department of Biological Sci? ences, Ohio University, Athens, Ohio. 4 Present address: Department of Biostatistics, North Carolina State University. \9 E. Marder 30 Cellular Models AB neuron The amplitude and frequency of the of the AB membrane oscillations potential in characteristic and neuron are modulated of both different ways by a large number and aminergic neurotransmitpeptidergic ters (Marder and Eisen, 1984&; Flamm and Harris-Warrick, 1986; Hooper and Marder, the AB neuron is a target 1987). Because sub? for so many different neuromodulatory for to determine, it is interesting stances, the modeach, the mechanism underlying of burst amplitude ulation and frequency and Flamm, (Harris-Warrick 1987). There are at least two different classes of general mechanisms by which such a large number of neuromodulatory substances could induce specific changes in AB neuron activity. First, the AB neuron's rhythmic oscillations could, in all cases, depend on the same ensemble of conductances, with the changes induced substances resulting by neuromodulatory in the from only quantitative alterations of these conductances. Second, expression AB neuron could arise from oscillations different sets of conductances, qualitatively each set induced by different neuromodu? lators. The ultimate answer to this question ofthe ionic requires the complete description currents found in the AB neuron, as well as an analysis ofthe mechanisms by which each influences the AB substance modulatory neuron. Although we are far from having to model the these data, a first attempt activity ofthe AB neuron gives us some new insights into this problem. and Flamm (1987) com? Harris-Warrick pared the AB neuron bursts obtained in the seroto? of three different amines, presence and dopamine. These nin, octopamine, workers noted that under a given set of TTX failed to conditions experimental but abolish bursts, dopamine-enhanced bursts. serotonin-enhanced suppressed From these and other observations, Harristhat Warrick and Flamm (1987) suggested there might be an essential difference in the mechanism nature of the burst generating found in the presence of these different amines. Partly motivated by these data, ET AL. an Epstein and Marder (1990) constructed burster. This ad hoc model of an AB-like model was not based on specific data from but is studies of AB neurons, biophysical model an isopotential, conductance-based five ionic currents found in that contains almost all central nervous system neurons. This "AB" model consists of a HodgkinNa+ current, a Huxley type TTX-sensitive K+ current, a voltagedelayed-rectifier Ca++ current, a Ca++-dependent dependent K+ current, current. and a Cl" leakage that showed and Marder Epstein (1990) in the values ofthe maximal convariations ofthe Na+, Ca++ and Cl" leakage ductances bursts with significantly currents produced different waveforms (Fig. IA, B) and cur? In one form of the properties. rent-voltage is model (Fig. IA), the Ca++ conductance was sim? low, and when TTX application ulated by turning off the Na+ conductance, In was suppressed (not shown). bursting another form ofthe model in which the Ca++ is higher, bursting persists in conductance of the Na+ current (Fig. 1C). the absence The currents active during bursting in these forms of the model are shown in Figure 1D, E. not specifically This model, although data from the AB based on real biophysical neuron, teaches us several things. First, even a relatively modest change in the balance of of a neuron can produce conductances behavior under current different markedly the two forms clamp conditions. Although differof the model respond qualitatively of TTX application, ently to the simulation this effect is produced by a relatively minor Ca++ and leakage difference in the maximal in the two forms ofthe model. conductances are involved, and No new conductances these changes are well within the kinds of effects produced by modulatory substances in many tissues. Thus, the apparent quali? of the bio? tative difference in the behavior logical AB neuron in the presence of dopa? in the mine (in which bursting continues of and serotonin presence TTX) (in which is suppressed bursting by TTX) (HarrisWarrick and Flamm, 1987) may result from a modest quantitative difference in the Ca++ current which participates in bursting in both STG Models 31 A 50 ?B0K -50 [ J 1000 2000 3000 mS 1500 mS 1500 mS Fig. 1. Multiple forms of a burster. A. Epstein and Marder (1990) model with maximal conductances (mmho/ cm2): Na+ = 100; Ca++ = 0.08; CL = 0.11. B. Epstein and Marder (1990) with maximal conductances (mmho/ cm2): Na+ = 8; Ca++ = 0.12; CL = 0.18. All other parameters are the same for the two models. C. Conductances as in (B) with the Na+ conductance set to 0. D. Currents flowing during burst shown in (A). E. Currents flowing during burst of (B). Adapted from Figs. 2, 3, 4, and 8 of Epstein and Marder (1990). that bursting cases, rather than indicating in the two amines occurs by qualitatively different mechanisms. for this Support comes from recent work of interpretation Johnson et al (1992) who showed that serotonin-activated bursting persists in the pres? ence of TTX when the temperature is elevated. This if occur the may higher increases the Ca++ current, for temperature In summary, a relatively minor example. modification of the ratio of the maximal conductances of the currents involved in burst generation can markedly influence the dynamic activity of the neuron. The work of Epstein and Marder (1990) makes another In this point. interesting model the only Na+ current is the early inward TTX-sensitive Hodgkin-Huxley, current. Although this current activates and inactivates rapidly, enough current remains at the relatively membrane hyperpolarized of the slow oscillations for it to potentials a in role burst play significant generation {e.g., Fig. 1E). It has often been assumed in the analysis of burst generation (Benson and Adams, 1987, 1989), that the TTX-sensi? tive Na+ current which participates in burst- ing is a different current than that respon? sible for rapid action potentials. Our results suggest that in certain neurons the fast Na+ current may play both roles. LP neuron the ad hoc model of the AB Although neuron brought us insight into several phys? we wished to construct iological processes, models that were based on the actual con? ductances measured from a real STG neu? ron. Therefore Golowasch and Marder undertook a study ofthe conduc? (1992a) tances found in the Lateral Pyloric (LP) neu? ron ofthe crab, Cancer borealis. Each STG has a single LP neuron which is also subject to a host of neuromodulatory influences and (Hooper and Marder, 1987; Nusbaum and 1989a, Marder, 1988, b; Golowasch Marder, 1991*). two-electrode volt? Using conventional Golowasch and Marder age clamp methods, characterized (1992a) many ofthe major conductances voltage- and time-dependent in the LP neuron. The LP neuron shows three outward a delayed-rectifier currents, K+ current (ID), a fast transient K+ current 32 E. Marder -38 ? Fig. 2. Proctolin responses ofthe biological and model LP neurons. A. Intracellular recording from an LP neu? ron isolated from presynaptic inputs from the pyloric network by application of 105 M picrotoxin and a sucrose block on the input nerve. Proctolin was applied at the upward arrow from a puffer pipette (0.5 see puff). Note the slight depolarization and the increase in firing frequency. B. Response of the model LP neuron to a simulated proctolin puff (at the arrow). Once again, note the slight depolarization and the increase in firing frequency. Time tics are 1 see. K+ current (IoCa)(IA), and a Ca++-activated The inward currents displayed by the LP neuron include a hyperpolarization activated slow inward current (IH), a voltageCa++ current (ICa), and a Hodgdependent Na+ current type TTX-sensitive kin-Huxley and Marder, To (INa) (Golowasch 1992a). construct a model of the LP neuron, each of these currents was fit with equations of the general form of the Hodgkin-Huxley et al, 1992), and then equations (Buchholtz a model neuron was constructed by comand bining these with a leakage conductance The activity of the model LP capacitance. neuron was compared to the activity ofthe LP neuron a series by simulating biological of current and voltage-clamp experiments to those performed with the real similar et al, 1992; Golowasch neuron (Buchholtz etal, 1992). One of the comparisons the between model neuron and real neuron can be seen in Figure 2, which compares the response of the biological and model neurons to the of the peptide application proctolin (the current was simulated from bio? proctolin Golowasch and measurements, physical et al. et al, Marder, 1992&; Golowasch 1992). Proctolin is a peptide known to have impor? tant modulatory actions on the pyloric and Marder, rhythm (Hooper 1987; Nusbaum and Marder, 1989a, b). Figure 2A shows the response ofthe biological neuron to a pufFof proctolin (arrow). Note the slight and the sharp increase in the depolarization, of the LP neuron action poten? frequency tials. In Figure 2B, the same experiment was simulated by turning on (arrow) the proc? tolin current in the model neuron. Note in firing frequency and again the increase the slight depolarization of the membrane The similarity in the responses of potential. the model and the real neurons is gratifying, but does not in itself provide any insight into the roles of each ofthe membrane cur? rents in controlling membrane excitability. Insight into the roles of each of the con? in controlling ductances the activity of the neuron can be obtained the by examining currents during ongoplots ofthe individual ing rhythmic activity (Fig. 3). In the absence of the proctolin current, note that the outward current due to the activity ofthe Ca++K+ current activated contributes signifi? of the action cantly to the repolarization while the delayed rectifier con? potential, tributes less. considerably Interestingly, when the proctolin current is turned on, and the neuron is thus slightly depolarized, the relative contributions ofthe Ca++-activated K+ current and the delayed rectifier to the of the action repolarization potentials changes. Thus even a small current that produces only a modest change in membrane have effects on potential may pronounced the role of each current in shaping the activ? ity of the neuron, in ways that are impos? sible to see unless one has a model in which the activity of each current can be visualized during ongoing Reduction activity. of Complex detailed Models conductance-based and Although models allow the multicompartmental to see what each variable in a investigator model is doing at all times and to ask what of a model is providing to each component the output of the system, they also have Most signifi? several major disadvantages. variof dynamical cantly, as the number STG 50 r 30 10 "m -10-30-50; 0- 'm -200 -300 60 40 'o(Ca) 20 0 JUIMMUIMJL 60 40 ?d 20 0 L/WIAAAAAAAAJ^ 20 ?A 0 -300 60 40 20 0 60 40 20 0 20 'A Ca [Ca] 0 'Ca 10 -10 4 2^ 0 50 30 10 -10 -30 -50 0 ?Na -100 -200 -100 'Na ?o(Ca) 33 Models ?proc [Ca] 100 200 Time (ms) 300 400 o -20 4 -2 mAMMAJUV-. 3 1 100 200 Time (ms) 300 400 Fig. 3. Computer models allow the examination ofthe roles of individual membrane currents. Left: Activity of the membrane currents in the model LP neuron during spontaneous activity in the absence of the proctolin current. Right: Activity in the model LP neuron in the presence of the proctolin current, which depolarizes the neuron, and causes it to fire more rapidly. Comparison of left and right shows that the relative contributions of and id currents to spike repolarization have changed. Modified from Golowasch et al, 1992. i0(Ca) ables in a model increases, our ability to analyze the behavior ofthe model in formal analytic terms decreases. Therefore, the ideal situation is a model that retains the essential features of a full, realistic model, but is as simple as possible. Over the years a number of simplified neuron models have been used for simulations of neurons and neural net? works. However, in most cases, these sim? plified models have been ad hoc, and their bear little or no relation to the parameters ofthe neu? properties underlying biological rons that they are meant to represent. To remedy this situation, Kepler et al. a method to (1991, 1992) have developed do systematic reductions of realistic conductance-based models. Kepler et al {1991, 1992) start out with the full Hodgkin-Huxfor an action potential. ley equations Using a method of calculating "equivalent poten- tials" they are able to reduce the four Hodgto a model with only equations kin-Huxley two dynamic equations. Figure 4A comthe behavior ofthe full pares Hodgkin-Huxto that of the reduced model ley equations and illustrates that although the model has been reduced, this has been accomplished without the essential character? sacrificing istics of the neuron's electrical activity. To test further this method, Kepler et al. (1991, an "A current" (IA of Con1992) introduce nor and Stevens, 1971). Figure 4B compares the activity of a six equation model in which the activity of these three conductances is and a reduction of this fully simulated, model with only three dynamical variables. Once again, the reduced model behaves almost identically to the full model. The general purpose of developing this method of reducing full models is not com- et al. E. Marder 34 > 60 full (order 4) reduced (order 2) O rtCD a O o CD c cd *-. ?> a -90 0) 3 e 10 0 10 20 time - full - reduced (order 30 40 (mS) 6) (order 3) CD O r-tCT> a o KWWWJI W Ww~m ?-i ?1 0) v\ IM o > 100 200 time 300 400 500 (mS) Fig. 4. Comparison of reduced and full models. Top panel: Simulation of full Hodgkin-Huxley model (solid lines) for the squid axon in response to a release from tonic hyperpolarization. Superimposed is the simulation ofthe reduced Hodgkin-Huxley model (dashed lines) in response to the same stimulus paradigm. Note that the reduced model duplicates the behavior of the full model (solid and dashed lines almost perfectly superimpose). Bottom panel: Simultaneous plot ofthe full model (six differential equations) of Connor et al. (1977) (solid line) and the reduced model (three equations) (dashed lines). The bottom trace of this panel shows the same randomly fiuctuating current imposed on both models to probe their response to perturbation. Note that the reduced model and the full model track almost perfectly. Modified from Kepler et al. (1991). putational ease, as the increasing speed and size of computers are making simulations of even large numbers of complicated neu? rons tractable. Instead, we hope that these reduction will bring these mod? techniques els to an intellectually accessible point and allow us to identify which features of the models (and of the neurons they complete are responsible for different represent) of the neuron's and mod? aspects activity ulation. Given that these reductions are from realistic models, we have good biologically reason to hope that the associations we make between features of the reduced specific STG Models 35 of neuronal models and specific aspects with these activity will be correct. Moreover, from the models to guide us, it predictions should be relatively easy to test them in on real neurons. experiments Network Models Although recent years have seen an exploof the role of sion in our understanding and neurons in the substances modulatory control ofthe neuronal networks in the STG 1989; Harris-War(Marder and Nusbaum, rick and Marder, 1991), we are far from neu? the role each individual understanding in the STG connection ron and synaptic the motor patterns pro? plays in shaping duced by the STG. As part of an ongcing program to develop a network model ofthe in the networks central pattern generating the two celled STG, we began by modeling network formed by the electrically coupled the AB and AB and PD neurons. Although PD neurons burst synchronously during activity because they are ongoing rhythmic coupled, the AB and PD neurons electrically differ in terms of their membrane properties to inputs (Bal et al, 1988), their responses and the neu19846), (Marder and Eisen, that they release (Marder and rotransmitters Eisen, 1984a). We describe below data that show that the PD neuron shapes both the ofthe AB neu? frequency and the waveform that the ron oscillations, thus demonstrating for the pyloric rhythm is a net? pacemaker neu? work of several coupled electrically rons. The role ofelectrical control coupling in frequency con? that frequency The first intimation trol in the pyloric network results from an between the intrinsic frequency interaction of the AB neuron and network interactions came from the work of Hooper and Marder These authors noted that the iso? (1987). lated AB neuron in the presence of proctolin produced bursts at about 2Hz, while the full cycled at a fre? pyloric network in proctolin quency of about 1Hz. Thus an electrically coupled neuron could decrease the intrinsic of an oscillatory neu? pacemaker frequency ron (Hooper and Marder, 1987). this qualitative However, understanding, Fig. 5. Oscillator waveform is important in deter? mining the effect of an electrically coupled neuron on oscillator frequency. This figure shows the oscillator neuron's membrane potential for both cases described, in this case with coupling coefficient = 0. Dotted hor? izontal lines are shown for reference and correspond to the membrane potential of the hyperpolarized cell to which the cell was coupled in the example described in the text. obtained with physiological data alone, is only part ofthe story. Kepler et al. (1990) modeled the AB neuron using a Fitzhugh and then coupled the model model, (1961) AB neuron to a non-oscillatory PD neuron. In this simple model it is apparent that the effect ofthe non-oscillatory cou? electrically neuron on the wave? pled depends critically form of the oscillator (Fig. 5). When the neuron depolarizes oscillatory slowly and hyperpolarizes quickly (Fig. 5, top), an inac? tive, hyperpolarized, coupled electrically neuron lengthens the intrinsic period ofthe neuron. when the However, oscillatory neuron depolarizes oscillatory quickly and an hyperpolarizes slowly (Fig. 5, bottom), cou? inactive, hyperpolarized, electrically pled neuron shortens the intrinsic period of the oscillatory neuron (see Fig. 1 in Kepler et al, 1990). A qualitative for explanation this follows. The hyperpolarized, inactive neuron is effectively injecting outward cur? rent through the electrical junction into the oscillator the oscillator throughout cycle. When the oscillator depolarizes slowly and out? hyperpolarizes rapidly, the additional ward current from the coupled cell retards the depolarization ofthe oscillator more than it speeds up the repolarization of the oscil- E. Marder 36 (Fig. 5, lator, so the cycle period increases when the oscillator depotop). However, larizes rapidly and hyperpolarizes slowly, cell the outward current from the coupled but speeds up the slows the depolarization more (allowing the next burst repolarization to occur earlier). Therefore the cycle period ofthe calculation A quantitative decreases. on the fre? neuron effect of the coupled requires knowing quency of the oscillator between the neu? the coupling conductance and con? membrane the and potentials rons, neurons (Kepler ductances ofthe individual etal, 1990). of this finding for phys? The implications clear iology are several. First, it is becoming with conditional that neurons oscillatory in many brain are important properties the regions. We are starting to understand not only in the role of oscillatory processes but in of rhythmic movements, generation as well. order processing sensory higher the way in which electrical Understanding of these coupling can modify the frequency to is fundamental processes oscillatory are how processes oscillatory understanding of all kinds used in neural computations (Marder et al, 1992). Second, many modthe plateau modulate substances ulatory or change the phase of action potentials, shape of neuronal bursts. As our knowledge substances on of the effect of modulatory neurons progresses, conditional oscillatory to bear in mind that sub? it is important of an that change the waveform stances will a neuron produce change in oscillatory neuron frequency as well, if that oscillatory to other neurons. is electrically coupled that elec? Third, there is growing evidence are subject to themselves trical connections substances 1989). modulatory (Dowling, Thus, in a network in which the frequency of an oscillatory neuron is controlled through of coupling to other neurons, the frequency neuron may be influenced that oscillatory of the electrical coupling in by modulation the network. The role of electrical cycle modulation coupling in duty The PD neurons also change the character of the AB neuron burst. Figure 6B shows in which the membrane an experiment AB and frequency of an isolated potential et al. of neuron are manipulated by the injection current into the cell body. Note that the while the burst duration remains constant, if the interburst interval expands. However, is moved AB neuron membrane potential when the PD neurons are present, a com? is seen (Fig. relationship pletely different 6A). Here, as the frequency is decreased, the of the AB/PD burst duration pacemaker along with the interburst group increases and we see that the pacemaker interval, constant an approximately group maintains duty cycle (ratio ofthe duration ofthe oscil? lator burst to the cycle period). These data in Figure 6C. are summarized how the PD neurons To understand transform the AB neuron burst from one in to constant which burst duration remains one in which duty cycle remains constant, Abbott et al (1991) developed simple mod? els of the AB and PD neurons that retain of these neurons the essential properties when and then coupled them isolated, The AB neuron is rep? together electrically. the model resented as a simple oscillator; AB neuron when isolated maintains con? as current is injected stant burst duration AB neuron. (Fig. 6E), just like the biological as a neuron The PD neuron is represented more that can either oscillate significantly slowly than the AB neuron, or fire tonically. Most critically, the model PD neuron has a and inactivating current, slowly activating which operates on a time scale considerably the slower than the currents that control burst in the AB neuron. When the model AB and PD neurons are coupled electrically, we see that the coupled network now behaves as a constant duty cycle oscillator (Fig. 6D, F). This occurs because the slow current of the PD neuron oscillates around an average value as the AB-PD ensemble oscillates; this when value is unchanging the average increase of the current during the depolar? ized part of the AB-PD oscillation equals the decline ofthe current during the hyper? polarized part. This current acts as a duty because any time the duty cycle governor cycle changes, the average value of this cur? rent acts to compensate for this change until the ratio burst duration to interburst inter? val is restored to the original value (Abbott etal, 1991). This model is satisfying, since very simple STG 37 Models ^AB) ^b)^aaaa-^d) 7?> i ? i ' ? I ? I ' I ' I ' AB AAAAAAAA abAAAM MIAAAAAAAA^ LvuvuJ pdAAAA/ HOmV 1s PD PUA^l^U 1s ? 1 o (D ",0.8 ?AB-PD 0 Isolated AB J0.6 ? 0.4 ?^0.2 1 3 2 period (see) 1 2 period 3 (see) Fig. 6. Electrical coupling to the PD neurons changes the AB neuron from a constant burst duration to constant duty cycle pacemaker. A. Simultaneous intracellular recordings from the AB and PD neurons when the AB neuron was depolarized (top two traces), with no injected current (middle two traces), and with the AB neuron hyperpolarized (bottom two traces). B. Intracellular recording from isolated AB neuron. Top trace, depolarized; middle trace, no injected current; bottom trace, hyperpolarized. C. Plot of burst duration as a function of cycle period of data from the biological experiments. D. Model AB and PD neurons electrically coupled. Top two traces, depolarizing current added to the AB neuron; middle two traces, no injected current; bottom two traces, AB neuron hyperpolarized. E. Model AB neuron in the absence of PD neuron. Top, depolarized AB; middle, no injected current; bottom, AB hyperpolarized. F. Plot of burst duration as a function of cycle period for the model shown in D and E, above. neurons are suffi? models of the individual of for the phenomenon cient to account interest. In this case we are able to use caricatures of neurons to represent their essen? tial features and obtain the insight that the data can arise simply from the physiological we are constituent components. Specifically, the salient feature of the able to represent difference between AB and PD physiological "conAB neurons can generate neurons: ventional" high frequency bursts when iso? lated from the PD neurons, but the PD neu? with an rons generate only slow oscillations the AB from when isolated irregular period neurons (Bal et al, 1988). This model does not require that we know the nature of the in either the PD or the ionic conductances AB neurons, but only their general char- It should be stressed that this acteristics. kind of model cannot provide any insight of the ionic currents in into the identities the PD and AB neurons that underly these For this we will need to develop caricatures. conductance-based models, such as that for above. In that case the LP neuron discussed it will be possible to associate specific con? ductances with specific aspects ofthe behav? and the neurons ior of the physiological simple models. we do not know which ionic Although for the transforare responsible currents mation ofthe AB neuron burst by the electo the PD neuron, this has trical coupling for nonetheless implications important full The the pyloric rhythm. understanding of of the activity pyloric rhythm consists 38 five classes E. Marder of motor neurons that fire with Under stereotyped phase relationships. standard control conditions the full pyloric maintains fixed rhythm approximately phase relationships among its elements over a significant frequency range (Eisen and Marder, 1984). The ability ofthe pacemaker ensemble to maintain constant duty cycles at different frequencies explains at least par? fixed tially how the full network maintains at different frequencies. phase relationships To maintain fixed phase relationships, the neurons must all begin to fire later in the Because cycle as the cycle period increases. the pacemaker ensemble releases transmitter in a graded fashion during the burst, as the duration of the pacemaker burst the time during which inhibitory increases, transmitter is released is extended. This in turn will retard the onset of firing of the follower neurons inhibited by the pace? maker network. Thus the maintenance of constant neurons phase in these follower depends heavily on the ability of the pace? maker ensemble to maintain a constant duty is changed. cycle as frequency Conclusions We have used models of several kinds to represent the neurons in the pyloric network ofthe STG. Some of these models are based on real ionic conductances, others are caricatures of neurons, rather than realistic representations. in each case the However, model allowed us to articulate new and dif? ferent insights into the electrophysiological and biophysical the processes underlying of rhythmic movements. generation First, we have shown that major changes in neuronal can stern both from activity modest in existing quantitative changes conductances and from the induction of Models help us small, novel conductances. understand how even small currents or small changes in currents can, under certain cirin the cumstances, produce large changes activity of the neuron. Second, we have shown that whether elec? neurons to trically coupling non-oscillatory neurons or decreases increases oscillatory the oscillator on the depends frequency waveform of the oscillator and the mem? brane potential of the non-oscillator. Con- et al. the effect of electrical sequently, coupling, and of neuromodulatory changes in oscil? lator waveform, neuronal membrane poten? tial, and the strength of electrical coupling, can only be predicted if one has relatively detailed knowledge of the neurons and the circuitry involved. a simple mech? Third, we have proposed anism for the production of constant duty and cycles at different oscillator frequencies, argue that this helps maintain constant phase for the pyloric network as a relationships whole. While this proposal must be tested it illustrates how even physiologically, simple models can suggest solu? extremely tions to difficult neurobiological problems. ACKNOWLEDGMENTS This paper was prepared under the ausMH46742. Some of the pices of NIMH research was supported by NS 17813. 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