Mapping Between Neural and Physical 
Activities of the Lobster Gastric Mill 
Kenji Doya 
Mary E.T. Boyle 
Allen I. Selverston 
Department of Biology 
University of California, San Diego 
La Jolla, CA 92093-0322 
Abstract 
A computer model of the musculoskeletal system of the lobster 
gastric mill was constructed in order to provide a behavioral in- 
terpretation of the rhythmic patterns obtained from isolated stom- 
atogastric ganglion. The model was based on Hill's muscle model 
and quasi-static approximation of the skeletal dynamics and could 
simulate the change of chewing patterns by the effect of neuromod- 
ulators. 
I THE STOMATOGASTRIC NERVOUS SYSTEM 
The crustacean stomatogastric ganglion (STG) is a circuit of 30 neurons that con- 
trols rhythmic movement of the foregut. It is one of the best elucidated neural 
circuits. All the neurons and the synaptic connections between them are identi- 
fied and the effects of neuromodulators on the oscillation patterns and neuronal 
characteristics have been extensively studied (Selverston and Moulins 1987, Harris- 
Warrick et al. 1992). However, STG's function as a controller of ingestive behavior 
is not fully understood in part because of our poor understanding of the controlled 
object: the musculoskeletal dynamics of the foregut. We constructed a mathemat- 
ical model of the gastric mill, three teeth in the stomach, in order to predict motor 
patterns from the neural oscillation patterns which are recorded from the isolated 
ganglion. 
The animal we used was the Californian spiny lobster (Panulirus interruptus), which 
913 
914 Doya, Boyle, and Selverston 
(a) 
STG 
medial tooth 
cardiac sac 
pylorus 
lateral teeth 
 esophagus 
flexible endoscope 
(b) 
I Inhibitory  Functional Inhibitory 
 Excitatoy ], Funconal Excitatoy 
 Electronic 
Figure 1: The lobster stomatogastric system. (a) Cross section of the foregut 
(objects are not to scale). (b) The gastric circuit. 
is available locally. The stomatogastric nervous system controls four parts of the 
foregut: esophagus, cardiac sac (stomach), gastric mill, and pylorus (entrance to 
the intestine) (Figure 1.a). The gastric mill is composed of one medial tooth and 
two lateral teeth. These grind large chunks of foods (mollusks, algae, crabs, sea 
urchins, etc.) into smaller pieces and mix them with digestive fluids. The chewing 
period ranges from 5 to 10 seconds. Several different chewing patterns have been 
analyzed using an endoscope (Heinzel 1988a, Boyle et al. 1990). Figure 2 shows 
two of the typical chewing patterns: "cut and grind" and "cut and squeeze". 
The STG is located in the opthalmic artery which runs from the heart to brain over 
the dorsal surface of the stomach. When it is taken out with two other ganglia (the 
esophageal ganglion and the commissural ganglion), it can still generate rhythmic 
motor outputs. This isolated preparation is ideal for studying the mechanism of 
rhythmic pattern generation by a neural circuit. From pairwise stimulus and 
response of the neurons, the map of synaptic connections has been established. 
Figure 1 (b) shows a subset of the STG circuit which controls the motion of the 
gastric mill. It consists of 11 neurons of 7 types. GM and DG neurons control the 
medial tooth and LPG, MG, and LG neurons control the lateral teeth. A question of 
interest is how this simple neural network is utilized to control the various movement 
patterns of the gastric mill, which is a fairly complex musculoskeletal system. 
The oscillation pattern of the isolated ganglion can be modulated by perfusing it 
with of several neuromodulators, e.g. proctolin, octopamine (Heinzel and Selver- 
ston 1988), CCK (Turrigiano 1990), and pilocarpine (rison and Selverston 1992). 
However, the behavioral interpretation of these different activity patterns is not 
well understood. The gastric mill is composed of 7 ossicles (small bones) which is 
loosely suspended by more than 20 muscles and connective tissues. That makes it is 
very difficult to intuitively estimate the effect of the change of neural firing patterns 
in terms of the teeth movement. Therefore we, decided to construct a quantitative 
model of the musculoskeletal system of the gastric mill. 
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill 915 
(a) 
(b) 
Figure 2: Typical chewing patterns of the gastric mill. (a) cut and grind. (b) cut 
and squeeze. 
2 PHYSIOLOGICAL EXPERIMENTS 
In order to design a model and determine its parameters, we performed anatomical 
and physiological experiments described below. 
Anatomical experiments: The carapace and the skin above the stomach mill 
was removed to expose a dorsal view of the ossicles and the muscles which control 
the gastric mill. Usually, the gastric mill was quiescent without any stimuli. The 
positions of the ossicles and the lengths of the muscles at the resting state was 
measured. After the behavioral experiments mentioned below, the gastric mill was 
taken out and the size of the ossicles and the positions of the attachment points of 
the muscles were measured. 
Behavioral experiments: With the carapace removed and the gastric mill ex- 
posed, one video camera was used to record the movement of the ossicles and the 
muscles. Another video camera attached to a flexible endoscope was used to record 
the motion of the teeth from inside the stomach. In the resting state, muscles were 
stimulated by a wire electrode to determine the behavioral effects. In order to in- 
duce chewing, neuromodulators such as proctolin and pilocarpine were injected into 
the artery in which STG is located. 
Single muscle experiments: The gml, the largest of the gastric mill muscles, 
was used to estimate the parameters of the muscle model mentioned below. It was 
removed without disrupting the carapace or ossicle attachment points and fixed to a 
tension measurement apparatus. The nerve fiber aln that innervates gml was stim- 
ulated using a suction electrode. The time course of isometric tension was recorded 
at different muscle lengths and stimulus frequencies. The parameters obtained from 
the gml muscle experiment were applied to other muscles by considering their rel- 
ative length and thickness. 
916 Doya, Boyle, and Selverston 
(a) 
contraction element (CE) 
parallel elasticity (PE) 
(c) 
fo 
0 v 0 v c 
 o 
serial elasticity (SE) 
(d) 
0 lco lc lsO ls 
Figure 3: The Hill-based muscle model. 
3 MODELING THE MUSCULOSKELETAL SYSTEM 
3.1 MUSCULAR DYNAMICS 
There are many ways to model muscles. In the simplest models, the tension or the 
length of a muscle is regarded as an instantaneous function of the spike frequency 
of the motor nerve. In some engineering approaches, a muscle is considered as 
a spring whose resting length and stiffness are modulated by the nervous input 
(Hogan 1984). Since these models are a linear static approximation of the nonlinear 
dynamical characteristics of muscles, their parameters must be changed to simulate 
different motor tasks (Winters90). Molecular models (Zahalak 1990), which are 
based on the binding mechanisms of actin and myosin fibers, can explain the widest 
range of muscular characteristics found in physiological experiments. However, 
these complex models have many parameters which are difficult to estimate. 
The model we employed was a nonlinear macroscopic model based on A. V. Hill's 
formulation (Hill 1938, Winters 1990). The model is composed of a contractile 
element (CE), a serial elasticity (SE), and a parallel elasticity (PE) (Figure 3.a). 
This model is based on empirical data about nonlinear characteristics of muscles 
and its parameters can be determined by physiological experiments. 
The output force fc of the CE is a function of its length lc and its contraction speed 
vc : -dl/dt (Figure 3.b) 
where f0 is the isometric output force (at vc = 0) and v0 is the maximal contraction 
velocity. The parameters of the f-v curve were a - 0.25 and/ = 0.3. The isometric 
force f0 was given as the function of CE length lc and the activation level a(t) of 
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill 917 
the muscle (Figure 3.c) 
fo(lc,a(t)) = 0 
o < tc < (2) 
otherwise, 
where It0 is the resting length of the CE and 7 = 1.5. 
The SE was modeled as an exponential spring (Figure 3.d) 
{ k(exp[k.]-l) l klo, 
f,(l,) = 0 1 < 1o, (3) 
where f, is the output force, 1,0 is the resting length, and k and k2 are stiffness 
parameters. The PE was supposed to have the same exponential elasticity (3). 
In the simulations, the CE length lc was taken as the state variable. The total 
muscle length 1, = 1 + 1, is given by the skeletal model and the muscle activation 
a(t) is given by the the activation dynamics described below. The SE length is 
given from 1, = 1, - lc and then the output force f, (1,) = f + fp = f, is given 
ate is derived from the inverse of (1) at 
by (3). The contraction velocity v = -- 
f = f,(l,)- fp(/) and then integrated to update the CE length lc. 
The activation level a(t) of a muscle is determined by the free calcium concentration 
in muscle fibers. Since we don't have enough data about the calcium dynamics in 
muscle cells, the activation dynamics was crudely approximated by the following 
equations. 
de(t) -e(t) + n(t)  (4) 
aa(t) _ -a(t) + e(t) and we at 
r, d---- ' = ' 
where n(t) is the normalized firing frequency of the nerve input and e(t) is the elec- 
tric activity of the muscle fibers. The nonlinearity in the nervous input represents 
strong facilitation of the postsynaptic potential (Govind and Lingle 1987). 
We incorporated seven of the gastric mill muscles: gml, gm2, gm3a, gm3c, gm4, 
gm6b, and gm9a (Maynard and Dando 1974). The muscles gml, gm2, gm3a, and 
gm3c are extrinsic muscles that have one end attached to the carapace and gm4, 
gm6b, and gm9a are intrinsic muscles both ends of which are attached of the ossicles. 
Three connective tissues were also incorporated and regarded as muscles without 
contraction elements. See Figure 4 for the attachment of these muscles and tissues 
to the ossicles. 
3.2 SKELETAL DYNAMICS 
The medial tooth was modeled as three rigid pieces P, P2and P3. P1 is the base of 
the roedial tooth. P2is the main body of the medial tooth. P3 forms the cusp and 
the V-shaped lever on the dorsal side. The lateral tooth was modeled as two rigid 
pieces P4 and P5. P4 is a L-shaped plate with a cusp at the angle and is connected 
to P3 at the dorsal end. P5 is a rod that is connected to P4 near the root of the 
cusp (Figure 4). 
We assumed that the motion is symmetric with respect to the midline. Therefore 
the motion of the roedial tooth was two-dimensional and only the left one of the 
918 Doya, Boyle, and Selverston 
x 
lO 
20 
~20 
lO 
o 
P1 
gm4 
gm3c 
 gm9a 
z 
gm3a 
-100 10 20 30 
Figure 4: The design of the gastric mill model. Ossicle P1 stands for the ossicles 
I and II, P2 for VII, Ps for VI, P4 for III, IV, and V, Ps for XIV in the standard 
description by Maynard and Dando (1974). 
two lateral teeth was considered. The coordinate system was taken so that x-axis 
points to the left, y-axis backward, and z-axis upward. The rotation angles of the 
ossicles around z, y, and z axes ware represented as 0, b, and p respectively. The 
configuration of the ossicles was determined by a l0 dimensional vector 
0 -- (yO,ZO, 01,02,03,04, (4, P4, 05, (5), (5) 
where (yo,zo) represents the position of the joint between P and P. and 0., 0.) 
represents the rotation angle of P1, P2 and Ps in the y-z plane. The rotation angles 
of P4 and Ps were represented as (04, 44, P4) and (0s, 4s) respectively. P has only 
two degrees of rotation freedom since it is regarded as a rod. 
We employed a quasi-static approximation. The configuration of the ossicles 0 
was determined by the static balance of force. Now let Lm and Fm be the vectors 
of the muscle lengths and forces. Then the balance of the generalized forces in the 
0 space (force for translation and torque for rotation) is given by 
T..,(O,F.)+Te =0, (6) 
where Tm and Te represent the generalized forces from muscles and external loads. 
The muscle force in the O space is given by 
T.(O,F..,) = J(O):T F, (7) 
where J(O) = c9L,/00 is the Jacobian matrix of the mapping 0  L, determined 
by the ossicle kinematics and the muscle attachment. Since it is very difficult to 
obtain a closed form solution of (6), we used a gradient descent equation 
dO _ -z(T.(O, F.) + T) = -s(J(O):TF. + T) (8) 
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill 919 
(a) t=O. t=2. t=4. t=6. 
(b) t=O. 
t=l.5 t=3 . t=4.5 
Figure 5: Chewing patterns predicted from oscillation patterns of isolated STG. (a) 
spontaneous pattern. (b) proctolin induced pattern. 
to find the approximate solution of O(t). This is equivalent to assuming a viscosity 
term e-dO/dt in the motion equation. 
4 SIMULATION RESULTS 
The musculoskeletal model is a 17-th order differential equation system and was 
integrated by Runge-Kutta method with a time step 1ms. Figure 5 shows examples 
of motion patterns predicted by the model. The motoneuron output of spontaneous 
oscillation of the isolated ganglion was used in (a) and the output under the effect 
of proctolin was used in (b). It has been reported in previous behavioral studies 
(Heinzel 1988b) that the dose of proctolin typically evokes "cut and grind" chewing 
pattern. The trajectory (b) predicted from the proctolin induced rhythm has a 
larger forward movement of the medial tooth while the lateral teeth are closed, 
which qualitatively agrees with the behavioral data. 
5 DISCUSSION 
The motor pattern generated by the model is considerably different from the chew- 
ing patterns observed in the intact animal using an endoscope. This is partly 
because of crude assumptions in model construction and errors in parameter esti- 
mation. However, this difference may also be due to the lack of sensory feedback in 
the isolated preparation. The future subject of this project is to refine the model so 
that we can reliably predict the motion from the neural outputs and to combine it 
with models of the gastric network (Rowat and Selverston, submitted) and sensory 
receptors. This will enable us to study how a biological control system integrates 
central pattern generation and sensory feedback. 
920 Doya, Boyle, and Selverston 
Acknowledgements 
We thank Mike Beauchamp for the gml muscle data. This work was supported by 
the grant from Office of Naval Research N00014-91-J-1720. 
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