{"title": "Mapping Between Neural and Physical Activities of the Lobster Gastric Mill", "book": "Advances in Neural Information Processing Systems", "page_first": 913, "page_last": 920, "abstract": "", "full_text": "Mapping Between Neural and Physical \nActivities of the Lobster Gastric Mill \n\nKenji Doya \n\nMary E. T. Boyle \n\nAllen I. Selverston \n\nDepartment of Biology \n\nUniversity of California, San Diego \n\nLa Jolla, CA 92093-0322 \n\nAbstract \n\nA computer model of the musculoskeletal system of the lobster \ngastric mill was constructed in order to provide a behavioral in(cid:173)\nterpretation of the rhythmic patterns obtained from isolated stom(cid:173)\natogastric ganglion. The model was based on Hill's muscle model \nand quasi-static approximation of the skeletal dynamics and could \nsimulate the change of chewing patterns by the effect of neuromod(cid:173)\nulators. \n\n1 THE STOMATOGASTRIC NERVOUS SYSTEM \n\nThe crustacean stomatogastric ganglion (STG) is a circuit of 30 neurons that con(cid:173)\ntrols rhythmic movement of the foregut. It is one of the best elucidated neural \ncircuits. All the neurons and the synaptic connections between them are identi(cid:173)\nfied and the effects of neuromodulators on the oscillation patterns and neuronal \ncharacteristics have been extensively studied (Selverston and Moulins 1987, H arris(cid:173)\nWarrick et al. 1992). However, STG's function as a controller of ingestive behavior \nis not fully understood in part because of our poor understanding of the controlled \nobject: the musculoskeletal dynamics of the foregut. We constructed a mathemat(cid:173)\nical model of the gastric mill, three teeth in the stomach, in order to predict motor \npatterns from the neural oscillation patterns which are recorded from the isolated \nganglion. \n\nThe animal we used was the Californian spiny lobster (Panulirus interruptus), which \n\n913 \n\n\f914 \n\nDoya, Boyle, and Selverston \n\n(a) \n\nmedial tooth __ - -\n\nesophagus \n\n\u2022 Inhibitory \n6 Excitatory \n\n\u2022 Functional Inhibitory \n, FWlCionai Excitalory \n\nflexible endoscope \n\nJVV\\... Electronic \n\nFigure 1: The lobster stomatogastric system. \n(objects are not to scale). (b) The gastric circuit. \n\n(a) Cross section of the foregut \n\nis available locally. The stomatogastric nervous system controls four parts of the \nforegut: esophagus, cardiac sac (stomach), gastric mill, and pylorus (entrance to \nthe intestine) (Figure l.a). The gastric mill is composed of one medial tooth and \ntwo lateral teeth. These grind large chunks of foods (mollusks, algae, crabs, sea \nurchins, etc.) into smaller pieces and mix them with digestive fluids. The chewing \nperiod ranges from 5 to 10 seconds. Several different chewing patterns have been \nanalyzed using an endoscope (Heinzel 1988a, Boyle et al. 1990). Figure 2 shows \ntwo of the typical chewing patterns: \"cut and grind\" and \"cut and squeeze\" . \n\nThe STG is located in the opthalmic artery which runs from the heart to brain over \nthe dorsal surface of the stomach. When it is taken out with two other ganglia (the \nesophageal ganglion and the commissural ganglion), it can still generate rhythmic \nmotor outputs. This isolated preparation is ideal for studying the mechanism of \nrhythmic pattern generation by a neural circuit. \nFrom pairwise stimulus and \nresponse of the neurons, the map of synaptic connections has been established. \nFigure 1 (b) shows a subset of the STG circuit which controls the motion of the \ngastric mill. It consists of 11 neurons of 7 types. GM and DG neurons control the \nmedial tooth and LPG, MG, and LG neurons control the lateral teeth. A question of \ninterest is how this simple neural network is utilized to control the various movement \npatterns of the gastric mill, which is a fairly complex musculoskeletal system. \n\nThe oscillation pattern of the isolated ganglion can be modulated by perfusing it \nwith of several neuromodulators, e.g. proctolin, octopamine (Heinzel and Selver(cid:173)\nston 1988), CCK (Turrigiano 1990), and pilocarpine (Elson and Selverston 1992). \nHowever, the behavioral interpretation of these different activity patterns is not \nwell understood. The gastric mill is composed of 7 ossicles (small bones) which is \nloosely suspended by more than 20 muscles and connective tissues. That makes it is \nvery difficult to intuitively estimate the effect of the change of neural firing patterns \nin terms of the teeth movement. Therefore we, decided to construct a quantitative \nmodel of the musculoskeletal system of the gastric mill. \n\n\fMapping Between Neural and Physical Activities of the Lobster Gastric Mill \n\n915 \n\n(a) \n\n(b) \n\nFigure 2: Typical chewing patterns of the gastric mill. (a) cut and grind. (b) cut \nand squeeze. \n\n2 PHYSIOLOGICAL EXPERIMENTS \n\nIn order to design a model and determine its parameters, we performed anatomical \nand physiological experiments described below. \n\nAnatomical experiments: The carapace and the skin above the stomach mill \nwas removed to expose a dorsal view of the ossicles and the muscles which control \nthe gastric mill. Usually, the gastric mill was quiescent without any stimuli. The \npositions of the ossicles and the lengths of the muscles at the resting state was \nmeasured. After the behavioral experiments mentioned below, the gastric mill was \ntaken out and the size of the ossicles and the positions of the attachment points of \nthe muscles were measured. \n\nBehavioral experiments: With the carapace removed and the gastric mill ex(cid:173)\nposed, one video camera was used to record the movement of the ossicles and the \nmuscles. Another video camera attached to a flexible endoscope was used to record \nthe motion of the teeth from inside the stomach. In the resting state, muscles were \nstimulated by a wire electrode to determine the behavioral effects. In order to in(cid:173)\nduce chewing, neuromodulators such as proctolin and pilocarpine were injected into \nthe artery in which STG is located. \n\nSingle muscle experiments: The gm!, the largest of the gastric mill muscles, \nwas used to estimate the parameters of the muscle model mentioned below. It was \nremoved without disrupting the carapace or ossicle attachment points and fixed to a \ntension measurement apparatus. The nerve fiber aln that innervates gmt was stim(cid:173)\nulated using a suction electrode. The time course of isometric tension was recorded \nat different muscle lengths and stimulus frequencies. The parameters obtained from \nthe gmt muscle experiment were applied to other muscles by considering their rel(cid:173)\native length and thickness. \n\n\f916 \n\nDoya, Boyle, and Selverston \n\n(a) \n\ncontraction element (CE) \n\nserial elasticity (SE) \n\nparallel elasticity (PE) \n\n(c) \n\nfO \nfmax \n\n(d) \n\nfs \n\no \n\no \n\nleo \n\nFigure 3: The Hill-based muscle model. \n\n3 MODELING THE MUSCULOSKELETAL SYSTEM \n\n3.1 MUSCULAR DYNAMICS \n\nThere are many ways to model muscles. In the simplest models, the tension or the \nlength of a muscle is regarded as an instantaneous function of the spike frequency \nof the motor nerve. In some engineering approaches, a muscle is considered as \na spring whose resting length and stiffness are modulated by the nervous input \n(Hogan 1984). Since these models are a linear static approximation of the nonlinear \ndynamical characteristics of muscles, their parameters must be changed to simulate \ndifferent motor tasks (Winters90). Molecular models (Zahalak 1990), which are \nbased on the binding mechanisms of actin and myosin fibers, can explain the widest \nrange of muscular characteristics found in physiological experiments. However, \nthese complex models have many parameters which are difficult to estimate. \nThe model we employed was a nonlinear macroscopic model based on A. V. Hill's \nformulation (Hill 1938, Winters 1990). The model is composed of a contractile \nelement (CE), a serial elasticity (SE), and a parallel elasticity (PE) (Figure 3.a). \nThis model is based on empirical data about nonlinear characteristics of muscles \nand its parameters can be determined by physiological experiments. \nThe output force Ie of the CE is a function of its length Ie and its contraction speed \nVe = -dle/dt (Figure 3.b) \n\nVe ?:: 0 (contraction), \nVe < 0 (extension), \n\n(1) \n\nwhere 10 is the isometric output force (at Ve = 0) and Vo is the maximal contraction \nvelocity. The parameters of the I-v curve were a = 0.25 and f3 = 0.3. The isometric \nforce 10 was given as the function of CE length Ie and the activation level a(t) of \n\n\fMapping Between Neural and Physical Activities of the Lobster Gastric Mill \n\n917 \n\nthe muscle (Figure 3.c) \n\nfo(l\"a(t)) = { ~m.Z!~, (f.;)' (f.; - r) a(t) \n\no < Ie < 'Y, \notherwise, \n\nwhere leo is the resting length of the CE and 'Y = 1.5. \nThe SE was modeled as an exponential spring (Figure 3.d) \n\nI${I$) = { okl(exp[k2l'~~'Q] -1) 1$ ~ 1$0, \n1$ < 1$0, \n\n(2) \n\n(3) \n\nwhere 1$ is the output force, 1$0 is the resting length, and kl and k2 are stiffness \nparameters. The PE was supposed to have the same exponential elasticity (3). \nIn the simulations, the CE length Ie was taken as the state variable. The total \nmuscle length 1m = Ie + 1$ is given by the skeletal model and the muscle activation \na(t) is given by the the activation dynamics described below. The SE length is \nIe and then the output force I${I$) = Ie + Ip = 1m is given \ngiven from 1$ = 1m -\nby (3). The contraction velocity Ve = -~ is derived from the inverse of (1) at \nIe = I${I$) - Ip(le) and then integrated to update the CE length Ie. \nThe activation level a(t) of a muscle is determined by the free calcium concentration \nin muscle fibers. Since we don't have enough data about the calcium dynamics in \nmuscle cells, the activation dynamics was crudely approximated by the following \nequations. \n\nda(t) \n\nTa-;{t = -a(t) + e(t), \n\nand \n\nde(t) \n\nTe~ = -e(t) + n(t)2, \n\n(4) \n\nwhere n(t) is the normalized firing frequency of the nerve input and e(t) is the elec(cid:173)\ntric activity of the muscle fibers. The nonlinearity in the nervous input represents \nstrong facilitation of the postsynaptic potential (Govind and Lingle 1987). \n\nWe incorporated seven of the gastric mill muscles: gml, gm2, gm3a, gm3c, gm4, \ngm6b, and gm9a (Maynard and Dando 1974). The muscles gml, gm2, gm3a, and \ngm3c are extrinsic muscles that have one end attached to the carapace and gm4, \ngm6b, and gm9a are intrinsic muscles both ends of which are attached of the ossicles. \nThree connective tissues were also incorporated and regarded as muscles without \ncontraction elements. See Figure 4 for the attachment of these muscles and tissues \nto the ossicles. \n\n3.2 SKELETAL DYNAMICS \n\nThe medial tooth was modeled as three rigid pieces PI, P2 and P3 . PI is the base of \nthe medial tooth. P2 is the main body of the medial tooth. P3 forms the cusp and \nthe V-shaped lever on the dorsal side. The lateral tooth was modeled as two rigid \npieces P4 and Ps. P4 is a L-shaped plate with a cusp at the angle and is connected \nto P3 at the dorsal end. Ps is a rod that is connected to P4 near the root of the \ncusp (Figure 4). \nWe assumed that the motion is symmetric with respect to the midline. Therefore \nthe motion of the medial tooth was two-dimensional and only the left one of the \n\n\f918 \n\nDoya, Boyle, and Se1verslon \n\ngm3c \n\n~':-;--...J......... gm9a \n\nz \n\ny \n\n30 \n\nFigure 4: The design of the gastric mill model. Ossicle PI stands for the ossicles \nI and II, P2 for VII, P3 for VI, P4 for III, IV, and V, Ps for XIV in the standard \ndescription by Maynard and Dando (1974). \n\ntwo lateral teeth was considered. The coordinate system was taken so that x-axis \npoints to the left, y-axis backward, and z-axis upward. The rotation angles of the \nossicles around x, y, and z axes ware represented as 0,
s) respectively. P5 has only \ntwo degrees of rotation freedom since it is regarded as a rod. \nWe employed a quasi-static approximation. The configuration of the ossicles e \nwas determined by the static balance of force. Now let Lm and Fm be the vectors \nof the muscle lengths and forces. Then the balance of the generalized forces in the \ne space (force for translation and torque for rotation) is given by \n\n(6) \nwhere T m and Te represent the generalized forces from muscles and external loads. \nThe muscle force in the e space is given by \n\nTm(e, Fm) + Te = 0, \n\n(7) \nwhere J(e) = 8Lm/8e is the Jacobian matrix of the mapping e .- Lm determined \nby the ossicle kinematics and the muscle attachment. Since it is very difficult to \nobtain a closed form solution of (6), we used a gradient descent equation \n\nTm(e, Fm) = J(e)T Fm, \n\nde dt = -c:(Tm(e, Fm) + Te) = -c:(J(e)T Fm + Te) \n\n(8) \n\n\fMapping Between Neural and Physical Activities of the Lobster Gastric Mill \n\n919 \n\n(a) t=O. \n\nt=2. \n\nt=4. \n\nt=6. \n\n(b) \n\nt=O. \n\nt=1.5 \n\nt=3. \n\nt=4.5 \n\nFigure 5: Chewing patterns predicted from oscillation patterns of isolated STG . (a) \nspontaneous pattern. (b) proctolin induced pattern. \n\nto find the approximate solution of 0(t). This is equivalent to assuming a viscosity \nterm c- 1d0ldt in the motion equation. \n\n4 SIMULATION RESULTS \n\nThe musculoskeletal model is a 17-th order differential equation system and was \nintegrated by Runge-Kutta method with a time step 1ms. Figure 5 shows examples \nof motion patterns predicted by the model. The motoneuron output of spontaneous \noscillation of the isolated ganglion was used in (a) and the output under the effect \nof proctolin was used in (b). It has been reported in previous behavioral studies \n(Heinzel 1988b) that the dose of proctolin typically evokes \"cut and grind\" chewing \npattern. The trajectory (b) predicted from the proctolin induced rhythm has a \nlarger forward movement of the medial tooth while the lateral teeth are closed, \nwhich qualitatively agrees with the behavioral data. \n\n5 DISCUSSION \n\nThe motor pattern generated by the model is considerably different from the chew(cid:173)\ning patterns observed in the intact animal using an endoscope. This is partly \nbecause of crude assumptions in model construction and errors in parameter esti(cid:173)\nmation. However, this difference may also be due to the lack of sensory feedback in \nthe isolated preparation. The future subject of this project is to refine the model so \nthat we can reliably predict the motion from the neural outputs and to combine it \nwith models of the gastric network (Rowat and Selverston, submitted) and sensory \nreceptors. This will enable us to study how a biological control system integrates \ncentral pattern generation and sensory feedback. \n\n\f920 \n\nDoya, Boyle, and Selverston \n\nAcknowledgements \n\nWe thank Mike Beauchamp for the gml muscle data. This work was supported by \nthe grant from Office of Naval Research NOOOI4-91-J-1720. \n\nReferences \n\nBoyle, M. E. T., Turrigiano, G . G., and Selverston, A. 1. 1990. An endoscopic anal(cid:173)\n\nysis of gastric mill movements produced by the peptide cholecystokinin. Society \nfor Neuroscience Abstracts 16, 724. \n\nElson, R. C. and Selverston, A. 1. 1992. Mechanisms of gastric rhythm generation \nin the isolated stomatogastric ganglion of spiny lobsters: Bursting pacemaker \npotentials, synaptic interactions and muscarinic modulation. Journal of Neuro(cid:173)\nphysiology 68, 890-907. \n\nGovind, C. K. and Lingle, C. J. 1987. Neuromuscular organization and pharmacol(cid:173)\n\nogy. In Selverston, A. 1. and Moulins, M., editors, The Crustacean Stomatogastric \nSystem, pages 31-48. Springer-Verlag, Berlin. \n\nHarris-Warrick, R. M., Marder, E., Selverston, A. 1., and Moulins, M. 1992. Dy(cid:173)\nThe Stomatogastric Nervous System. MIT Press, \n\nnamic Biological Networks -\nCambridge, MA. \n\nHeinzel, H. G. 1988. Gastric mill activity in the lobster. I: Spontaneous modes of \n\nchewing. Journal of Neurophysiology 59, 528-550. \n\nHeinzel, H. G. 1988. Gastric mill activity in the lobster. II: Proctolin and oc(cid:173)\ntopamine initiate and modulate chewing. Journal of Neurophysiology 59, 551-\n565. \n\nHeinzel, H. G. and Selverston, A. 1. 1988. Gastric mill activity in the lobster. \nIII: Effects of proctolin on the isolated central pattern generator. Journal of \nNeurophysiology 59, 566-585. \n\nHill, A. V. 1938. The heat of shortening and the dynamic constants of muscle. \n\nProceedings of the Royal Sciety of London, Series B 126, 136-195 . \n\nHogan, N. 1984. Adaptive control of mechanical impedance by coactivation of \n\nantagonist muscles. IEEE Transactions on Automatic Control 29, 681-690. \n\nMaynard, D. M. and Dando, M. R. 1974. The structure ofthe stomatogastric neuro(cid:173)\n\nmuscular system in callinectes sapidus, homarus americanus and panulirus argus \n(decapoda crustacea). Philosophical Transactions of Royal Society of London, \nBiology 268, 161- 220. \n\nRowat, P. F. and Selverston, A. 1. Modeling the gastric mill central pattern gener(cid:173)\n\nator of the lobster with a relaxation-oscillator network. submitted. \n\nSelverston, A. 1. and Moulins, M. 1987. The Crustacean Stomatogastric System. \n\nSpringer-Verlag, New York, NY. \n\nTurrigiano, G . G. and Selverston, A. 1. 1990. A cholecystokinin-like hormone acti(cid:173)\n\nvates a feeding-related neural circuit in lobster . Nature 344, 866-868 . \n\nWinters, J. M. 1990. Hill-based muscle models: A systems engineering perspective. \n\nIn Winters, J. M. and Woo, S. 1.-Y., editors, Multiplie Muscle Systems: Biome(cid:173)\nchanics and Movement Organization, chapter 5, pages 69-93. Springer-Verlag, \nNew York, NY. \n\nZahalak, G. I. 1990. Modeling muscle mechanics (and energetics). In Winters, \nJ. M. and Woo, S. L.-Y., editors, Multiplie Muscle Systems: Biomechanics and \nMovement Organization, chapter 1, pages 1-23. Springer-Verlag, New York, NY. \n\n\f", "award": [], "sourceid": 662, "authors": [{"given_name": "Kenji", "family_name": "Doya", "institution": null}, {"given_name": "Mary", "family_name": "Boyle", "institution": null}, {"given_name": "Allen", "family_name": "Selverston", "institution": null}]}