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}]}