{"title": "Neural Network Models of Chemotaxis in the Nematode Caenorhabditis Elegans", "book": "Advances in Neural Information Processing Systems", "page_first": 55, "page_last": 61, "abstract": null, "full_text": "Neural network models of chemotaxis \nthe nematode Caenorhabditis elegans \n\n\u2022 In \n\nThomas C. Ferree, Ben A. Marcotte, Shawn R. Lockery \n\nInstitute of Neuroscience, University of Oregon, Eugene, Oregon 97403 \n\nAbstract \n\nWe train recurrent networks to control chemotaxis in a computer \nmodel of the nematode C. elegans. The model presented is based \nclosely on the body mechanics, behavioral analyses, neuroanatomy \nand neurophysiology of C. elegans, each imposing constraints rel(cid:173)\nevant for information processing. Simulated worms moving au(cid:173)\ntonomously in simulated chemical environments display a variety \nof chemotaxis strategies similar to those of biological worms. \n\n1 \n\nINTRODUCTION \n\nThe nematode C. elegans provides a unique opportunity to study the neuronal ba(cid:173)\nsis of neural computation in an animal capable of complex goal-oriented behaviors. \nThe adult hermaphrodite is only 1 mm long, and has exactly 302 neurons and 95 \nmuscle cells. The morphology of every cell and the location of most electrical and \nchemical synapses are known precisely (White et al., 1986), making C. elegans espe(cid:173)\ncially attractive for study. Whole-cell recordings are now being made on identified \nneurons in the nerve ring of C. elegans to determine electrophysiological properties \nwhich underly information processing in this animal (Lockery and Goodman, un(cid:173)\npublished). However, the strengths and polarities of synaptic connections are not \nknown, so we use neural network optimization to find sets of synaptic strengths \nwhich reproduce actual nematode behavior in a simulated worm. \n\nWe focus on chemotaxis, the ability to move up (or down) a gradient of chemical \nattractants (or repellants). In the laboratory, flat Petri dishes (radius = 4.25 cm) \nare prepared with a Gaussian-shaped field of attractant at the center, and worms \nare allowed to move freely about. Worms propel themselves forward by generating \nan undulatory body wave, which produces sinusoidal movement. In chemotaxis, the \nnervous system generates motor commands which bias this movement and direct \n\n\f56 \n\nT. C. Ferree, B. A. Marcotte and S. R. Lockery \n\nthe animal toward higher attractant concentration. \n\nAnatomical constraints pose important problems for C. elegans during chemotaxis. \nIn particular, the animal detects the presence of chemicals with a pair of sensory \norgans (amphids) at the tip of the nose, each containing the processes of multiple \nchemosensory neurons. During normal locomotion, however, the animal moves on \nits side so that the two amphids are perpendicular to the Petri dish. C. elegans can(cid:173)\nnot, therefore, sense the gradient directly. One possible strategy for chemotaxis, \nwhich has been suggested previously (Ward, 1973), is that the animal computes a \ntemporal derivative of the local concentration during a single head sweep, and com(cid:173)\nbines this with some form of proprioceptive feedback indicating muscle contraction \nand the direction of head sweep, to compute the spatial gradient for chemotaxis. \nThe existence of this and other strategies is discussed later. \nIn Section 2, we derive a simple model of the nematode body which produces realis(cid:173)\ntic sinusoidal trajectories in response to motor commands from the nervous system. \nIn Section 3, we give a simple model of the C. elegans nervous system based on \npreliminary physiological data. In Section 4, we use a stochastic optimization algo(cid:173)\nrithm to determine sets of synaptic weights which control chemotaxis, and discuss \nsolutions. \n\n2 BIOMECHANICS OF NEMATODE ORIENTATION \n\nNematode locomotion has been studied in detail (Niebur and Erdos, 1991; Niebur \nand Erdos, 1993). These authors derived Newtonian force equations for each mus(cid:173)\ncular segment of the body, which can be solved numerically to generate forward \nsinusoidal movement. Unfortunately, such a thorough treatment is computation(cid:173)\nally intensive and not practical to use with network optimization. To simplify the \nproblem we first recognize that chemotaxis is a behavior more of orientation than of \nlocomotion. We therefore derive a set of biomechanical equations which direct the \nhead to generate sinusoidal movement, which can be biased by the network toward \nhigher chemical concentrations. \nWe focus our attention on the point (x,1/) at the tip of the nose, since that is where \nthe animal senses the chemical environment. As shown in Figure 1 ( a), we assign \na velocity vector fJ directed along the midline of the first body segment, i.e., the \nhead. Assuming that the worm moves forward at constant speed v, we can write \nthe velocity vector as \n\nfJ(t) = (~;, ~~) = (vcos(}(t),vsin(}(t)) \n\n(1) \n\nwhere x, y and (} are measured relative to fixed coordinates in the Petri dish. \nAssuming that the worm moves without lateral slipping and that the undulatory \nwave of muscular contraction initiated in the neck travels posteriorally without \nmodification, then each body segment simply follows the one previous (anterior) to \nit. In this way, the head directs the movement and the rest of the body simply \nfollows. \n\nFigure l(b) shows an expanded view of the neck segment. As the worm moves \nforward, the posterior boundary of that segment assumes the position held by its \nanterior neighbor at a slightly earlier time. If L is the total body length and N is \n\n\f57 \nNeural Network Models of Chemotaxis \nthe number of body segments, then this time delay is 6t ~ L/Nv. (For L = 1 mm, \nv = 0.22 mm/s and N = 10 we have 6t ~ 0.45 s, roughly an order of magnitude \nsmaller than the relevant behavioral time scale: the head-sweep period T ~ 4.2 s.) \nIf we define the neck angle aCt) == (h(t) - 92 (t), then the above arguments imply \n\naCt) = 91 (t) - 91 (t - 6t) ~ dt 6t \n\ndOl \n\n(2) \n\nwhere the second relation is essentially a backward-Euler algorithm for dOl/dt. \nSince 9 == 91 , we have reached the intuitive result that the neck angle a determines \nthe rate of turning dO/dt. Note that while 91 and 92 are defined relative to the \nfixed laboratory coordinates, their difference a is invariant under rotations of these \ncoordinates, and can therefore be viewed as intrinsic to the body. This allows us \nto derive an expression for a in terms of muscle cell contraction, or motor neuron \ndepolarization, as follows. \n\n(a) \n\nv \n\n(b) \n\nNeck \n\nT \nIv \n1 \n\nFigure 1: Nematode body mechanics. (a) Segmented model of the nematode body, \nshowing the direction of motion v. (b) Expanded view of the neck segment, showing \ndorsal (D) and ventral (V) neck muscles. \n\nNematodes maintain nearly constant volume during movement. To incorporate this \nconstraint, albeit approximately, we assume that at all times the geometry of each \nsegment is such that (ID -10) = -(Iv -10), where 10 == L/N is the equilibrium length \nof a relaxed segment. For small angles a, we have a ~ (Iv -ID)/d, where d is the \nbody diameter. The dashed lines in Figure 1(b) indicate dorsal and ventral muscles, \nwhich are believed to develop tension nearly independent of length (Toida et al., \n1975). When contracting, these muscles must work against the elasticity of the \ncuticle, internal fluid pressure, and elasticity and developed tension of the opposing \nmuscles. If these elastic forces act linearly, then TD-TV ~ k (Iv-ID), where TD and \nTv are dorsal and ventral muscle tensions, and k is an effective force constant. For \nsimplicity, we further assume that each muscle develops tension linearly in response \nto the voltage of its corresponding motor neuron, i.e., TD,v = E VD,V, where E is a \npositive constant, and VD and Vv are dorsal and ventral motor neuron voltages. \n\nCombining these results, we have finally \n\n~~ = \"1 (VD(t) - Vv(t\u00bb) \n\n(3) \n\n\f58 \n\nT. C. Ferree, B. A. Marcotte and S. R. Lockery \n\nwhere\"Y = (Nv/L)\u00b7 (E/kd). With appropriate motor commands, equations (I) and \n(3) can be integrated numerically to generate sinusoidal worm trajectories like those \nof biological worms. This model embodies the main anatomical features that are \nlikely to be important in C. elegans chemotaxis, yet is sufficiently compact to be \nembedded in a network optimization procedure. \n\n3 CHEMOTAXIS CONTROL CIRCUIT \n\nC. elegans neurons are tiny and have very simple morpologies: a typical neuron \nin the head has a spherical soma 1-2 pm in diameter, and a single cylindrical \nprocess 60-80 pm in length and 0.1-0.2 pm in diameter. Compartmental mod(cid:173)\nels, based on this morphology and preliminary physiological recordings, indicate \nthat C. elegans neurons are effectively isopotential (Lockery, 1995). Furthermore, \nC. elegans neurons do not fire classical all-or-none action potentials, but appear to \nrely primarily on graded signal propagation (Lockery and Goodman, unpublished). \nThus, a reasonable starting point for a network model is to represent each neuron \nby a single isopotential compartment, in which voltage is the state variable, and the \nmembrane conductance in purely ohmic. \n\nAnatomical data indicate that the C. elegans nervous system has both electrical and \nchemical synapses, but the synaptic transfer functions are not known. However, \nsteady-state synaptic transfer functions for chemical synapses have been measured \nin Ascaris s'U'Um, a related species of nematode, where it was found that postsynaptic \nvoltage is a graded function of presynaptic voltage, due to tonic neurotransmitter \nrelease (Davis and Stretton, 1989). This voltage dependence is sigmoidal, i.e., \nVpOlt \"-J tanh(Vpre ). A simple network model which captures all of these features is \n\nT~ = -Vi + Vmax tanh(f3 t Wij (Vj - Vj\u00bb) + ~Iilm{t) \n\n3=1 \n\n(4) \n\nwhere Vi is the voltage of the ith neuron. Here all voltages are measured relative \nto a common resting potential, Vmax is an arbitrary voltage scale which sets the \noperational range of the neurons, and f3 sets the voltage sensitivity of the synaptic \ntransfer function. The weight Wij represents the net strength and polarity of all \nsynaptic connections from neuron j to neuron i, and the constants Vj determine the \ncenter of each transfer function. The membane time constant T is assumed to be \nthe same for all cells, and will be discussed further later. Note that in (4), synap(cid:173)\ntic transmission occurs instantaneously: the time constant T arises from capacitive \ncurrent through the cell membrane, and is unrelated to synaptic transmission. Note \nalso that the way in which (4) sums multiple inputs is not unique, i.e., other sig(cid:173)\nmoidal models which sum inputs differently are equally plausible, since no data on \nsynaptic summation exists for either C. elegans or Ascaris 8'U'Uffl. \n\n. \n\nThe stimulus term ~8t1m(t) is used to introduce chemosensation and sinusoidal \nlocomotion to the network in (4). We use i = 1 to label a single chemosensory \nneuron at the tip of the nose, and i = n - 1 == D and i = n == V to label dorsal and \nventral motor neurons. For simplicity we assume that the chemosensory neuron \nvoltage responds linearly to the local chemical concentration: \n\n(5) \n\n\fNeural Network Models of Chemotaxis \n\n59 \n\nwhere Vchem is a positive constant, and the local concentration C(x, y) is always \nevaluated at the instantaneous nose position. \n\nIn the previous section, we emphasized that locomotion is effectively independent of \norientation. We therefore assume the existence of a central pattern generator (CPG) \nwhich is outside the chemotaxis control circuit (4). Thus, in addition to synaptic \ninput from other neurons, each motor neuron receives a sinusoidal stimulus \n\n(6) \n\nwhere VCPG and w = 211\" /T are positive constants. \n4 RESULTS AND DISCUSSION \nEquations (1), (3) and (4), together with (5) and (6), comprise a set of n + 3 \nfirst-order nonlinear differential equations, which can be solved numerically given \ninitial conditions and a specification of the chemical environment. We use a fourth(cid:173)\norder Runge-Kutta algorithm and find favorable stability and convergence. The \nnecessary body parameters have been measured by observing actual worms (Pierce \nand Lockery, unpublished): v = 0.022 cm/s, T = 4.2 s and '\"Y = 0.8/(2VcPG). The \nchemical environment is also chosen to agree roughly with experimental values: \nC(x,y) = Coexp(-(x2 + y2)/-Xb), with Co = 0.052 p.mol/cm3 and -Xc = 2.3 cm. \nTo optimize networks to control chemotaxis, we use a simple simulated annealing \nalgorithm which searches over the (n2 + 3)-dimensional space of parameters Wij, \n{3, Vchem and VCPG. In the results shown here, we used n = 12, and set V; = O. \nEach set of the resulting parameters represents a different nervous system for the \nmodel worm. At the beginning of each run, the worm is initialized by choosing an \ninitial position (xo, Yo), an initial angle 00 , and by setting V. = O. Upon numerically \nintegrating, simulated worms move autonomously in their environment for a prede(cid:173)\ntermined amount of time, typically the real-time equivalent of 10-15 minutes. We \nquantify the performance, or fitness, of each worm during chemotaxis by computing \nthe average chemical concentation at the tip of its nose over the duration of each \nrun. To avoid lucky scores, the actual score for each worm is obtained by averaging \nover several initial conditions. \n\nIn Figure 2, we show a comparison of tracks produced by (a) biological and (b) \nsimulated worms during chemotaxis. In each case, three worms were placed in a \ndish with a radial gradient and allowed to move freely for the real-time equivalent \nof 15 minutes. In (b), the three worms have the same neural parameters (Wij, (3, \nVchem, VCPG), but different initial angles 00 \u2022 In both (a) and (b), all three worms \nmake initial movements, then move toward the center of the dish and remain there. \nIn other optimizations, rather than orbit the center, the simulated worms may \napproach the center asymptotically from one side, make simple geometric patterns \nwhich pass through the center, or exhibit a variety of other distinct strategies for \nchemotaxis. This is similar to the situation with biological worms, which also have \nconsiderable variation in the details of their tracks. \nThe behavior shown in Figure 2 was produced using T = 500 ms. However, prelimi(cid:173)\nnary electrophysiological recordings from C. elegans neurons suggest that the actual \nvalue may be as much as an order of magnitude smaller, but not bigger (Lockery and \nGoodman, unpublished). This presents a potential problem for chemotaxis com-\n\n\f60 \n\nT. C. Ferree, B. A. Marcotte and S. R. Lockery \n\nputation, since shorter time constants require greater sensitivity to small changes \nin O(x, y) in order to compute a temporal derivative, which is believed to be re(cid:173)\nquired. During optimization, we have seen that for a fixed number of neurons n, \nfinding optimal solutions becomes more difficult as T is decreased. This observation \nis very difficult to quantify, however, due to the existence of local maxima in the \nfitness function. Nevertheless, this suggests that additional mechanisms may need \nto be included to understand neural computation in C. elegana. First, time- and \nvoltage-dependent conductances will modify the effective membrane time constant, \nand may increase the effective time scale for computation by individual neurons. \nSecond, more neurons and synaptic delays will also move the effective neuronal time \nscale closer to that of the behavior. Either of these will allow comparisons of O(z, II) \nacross greater distances, thereby requiring less sensitivity to compute the gradient, \nand potentially improving the ability of these networks to control chemotaxis. \n\n2em \n\nFigure 2: Nematodes performing chemotaxis: (a) biological (Pierce and Lockery, \nunpublished), and (b) simulated. \n\nWe also note, based on a variety of other results, not shown here, that the head(cid:173)\nsweep strategy, described in the introduction, is by no means the only strategy for \nchemotaxis in this system. In particular, we have optimized networks without a \nCPG, i.e., with VCPG = 0 in (6), and found parameter sets that successfully con(cid:173)\ntrol chemotaxis. This presents the possibility that even worms with a CPG do not \nnecessarily compute the gradient based on lateral movement of the head, but may \ninstead respond only to changes in concentration along their mean trajectory. Sim(cid:173)\nilar results have been reported previously, although based on a somewhat different \nbiomechanical model (Beer and Gallagher, 1992). \nFinally, we have also optimized discrete-time networks, obtained by setting T = 0 \nin (4) and updating all units synchronously. As is well-known, on relatively short \ntime scales (- T) such a system tends to \"overshoot\" at each successive time step, \nleading to sporadic behavior of the network and the body. Knowing this, it is \ninteresting that simulated worms with such a nervous system are capable of reliable \nbehavior over longer time scales, i.e., they successfully perform chemotaxis. \n\n\fNeural Network Models of Chemotaxis \n\n61 \n\n5 CONCLUSIONS AND FUTURE WORK \n\nThe main result of this paper is that a small nervous system, based on graded(cid:173)\npotential neurons, is capable of controlling chemotaxis in a worm-like physical body \nwith the dimensions of C. elegans. The model presented is based closely on the body \nmechanics, behavioral analyses, neuroanatomy and neurophysiology of C. elegans, \nand is a reliable starting point for more realistic models to follow. Furthermore, we \nhave established the existence of chemotaxis strategies that had not been anticipated \nbased on behavioral experiments with real worms. \n\nFuture work will involve both improvement of the model and analysis of the result(cid:173)\ning solutions. Improvements will include introducing voltage- and time-dependent \nmembrane conductances, as this data becomes available, and more realistic models \nof synaptic transmission. Also, laser ablation experiments have been performed that \nsuggest which interneurons and motor neurons in C. elegans may be important for \nchemotaxis (Bargmann, unpublished), and these data can be used to constrain the \nsynaptic connections during optimization. Analyses will be aimed at determining \nthe role of individual physiological and anatomical features, and how they func(cid:173)\ntion together to govern the collective properties of the network as a whole during \nchemotaxis. \n\nAcknowledgements \n\nThe authors would like to thank Miriam Goodman and Jon Pierce for helpful dis(cid:173)\ncussions. This work has been supported by NIMH MH11373, NIMH MH51383, \nNSF IBN 9458102, ONR N00014-94-1-0642, the Sloan Foundation, and The Searle \nScholars Program. \nReferences \nBeer, R. D. and J. C. Gallagher (1992). Evolving dynamical neural networks for \nadaptive behavior, Adaptive Behavior 1(1):91-122. \n\nDavis, R. E. and A. O. W. Stretton {1989}. Signaling properties of Ascaris ma(cid:173)\ntorneurons: Graded active responses, graded synaptic transmission, and tonic trans(cid:173)\nmitter release, J. Neurosci. 9:415-425. \nLockery, S. R. (1995). Signal propagation in the nerve ring of C. elegans, Soc. Neu(cid:173)\nrosd. 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London 314:1-340. \n\n\f", "award": [], "sourceid": 1199, "authors": [{"given_name": "Thomas", "family_name": "Ferr\u00e9e", "institution": null}, {"given_name": "Ben", "family_name": "Marcotte", "institution": null}, {"given_name": "Shawn", "family_name": "Lockery", "institution": null}]}