{"title": "Circuit Optimization Predicts Dynamic Networks for Chemosensory Orientation in Nematode C. elegans", "book": "Advances in Neural Information Processing Systems", "page_first": 1279, "page_last": 1286, "abstract": "", "full_text": "Circuit Optimization Predicts Dynamic\n\nNetworks for Chemosensory Orientation in the\n\nNematode Caenorhabditis elegans\n\nNathan A. Dunn\n\nJohn S. Conery\n\nDept. of Computer Science\n\nUniversity of Oregon\nEugene, OR 97403\n\n{ndunn,conery}@cs.uoregon.edu\n\nShawn R. Lockery\n\nInstitute of Neuroscience\n\nUniversity of Oregon\nEugene, OR 97403\n\nshawn@lox.uoregon.edu \u2217\n\nAbstract\n\nThe connectivity of the nervous system of the nematode Caenorhabdi-\ntis elegans has been described completely, but the analysis of the neu-\nronal basis of behavior in this system is just beginning. Here, we used\nan optimization algorithm to search for patterns of connectivity suf\ufb01-\ncient to compute the sensorimotor transformation underlying C. elegans\nchemotaxis, a simple form of spatial orientation behavior in which turn-\ning probability is modulated by the rate of change of chemical concen-\ntration. Optimization produced differentiator networks with inhibitory\nfeedback among all neurons. Further analysis showed that feedback reg-\nulates the latency between sensory input and behavior. Common patterns\nof connectivity between the model and biological networks suggest new\nfunctions for previously identi\ufb01ed connections in the C. elegans nervous\nsystem.\n\n1 Introduction\n\nThe complete description of the morphology and synaptic connectivity of all 302 neurons\nin the nematode Caenorhabditis elegans [15] raised the prospect of the \ufb01rst comprehensive\nunderstanding of the neuronal basis of an animal\u2019s entire behavioral repertoire. The advent\nof new electrophysiological and functional imaging techniques for C. elegans neurons [7, 8]\nhas made this project more realistic than before. Further progress would be accelerated,\nhowever, by prior knowledge of the sensorimotor transformations underlying the behaviors\nof C. elegans, together with knowledge of how these transformations could be implemented\nwith C. elegans-like neuronal elements.\n\nIn previous work, we and others have identi\ufb01ed the main features of the sensorimotor trans-\nformation underlying C. elegans chemotaxis [5, 11], one of two forms of spatial orientation\nidenti\ufb01ed in this species. Locomotion consists of periods of sinusoidal forward movement,\ncalled \u201cruns,\u201d which are punctuated by bouts of turning [12] that have been termed \u201cpirou-\nettes\u201d [11]. Pirouette probability is modulated by the rate of change of chemical concen-\ntration (dC(t)/dt). When dC(t)/dt < 0, pirouette probability is increased whereas when\n\n\u2217To whom correspondence should be addressed.\n\n\fdC(t)/dt > 0, pirouette probability is decreased. Thus, runs down the gradient are trun-\ncated and runs up the gradient are extended, resulting in net movement toward the gradient\npeak.\n\nThe process of identifying the neurons that compute this sensorimotor transformation is\njust beginning. The chemosensory neurons responsible for the input representation are\nknown[1], as are the premotor interneurons for turning behavior[2]. Much less is known\nabout the interneurons that link inputs to outputs. To gain insight into how this transfor-\nmation might be computed at the interneuronal level, we used an unbiased parameter opti-\nmization algorithm to construct model neural networks capable of computing the transfor-\nmation using C. elegans-like neurons. We found that networks with one or two interneurons\nwere suf\ufb01cient. A common but unexpected feature of all networks was inhibitory feedback\namong all neurons. We propose that the main function of this feedback is to regulate the\nlatency between sensory input and behavior.\n\n2 Assumptions\n\nWe used simulated annealing to search for patterns of connectivity suf\ufb01cient for computing\nthe chemotaxis sensorimotor transformation. The algorithm was constrained by three main\nassumptions:\n\n1. Primary chemosensory neurons in C. elegans report attractant concentration at a\n\nsingle point in space.\n\n2. Chemosensory interneurons converge on a network of locomotory command neu-\n\nrons to regulate turning probability.\n\n3. The sensorimotor transformation in C. elegans is computed mainly at the network\n\nlevel, not at the cellular level.\n\nAssumption (1) follows from the anatomy and distribution of chemosensory organs in C.\nelegans[1, 13, 14]. Assumption (2) follows from anatomical reconstructions of the C. ele-\ngans nervous system [15], together with the fact that laser ablation studies have identi\ufb01ed\nfour pairs of pre-motor interneurons that are necessary for turning in C. elegans[2]. As-\nsumption (3) is heuristic.\n\n3 Network\n\nNeurons were modeled by the equation:\n\ndAi(t)\n\ndt\n\n\u03c4i\n\n= \u2212Ai(t) + \u03c3(Ii), with\n\nIi =X\n\nj\n\n(wjiAj(t)) + bi\n\n(1)\n\nwhere Ai is activation level of neuron i in the network, \u03c3(Ii) is the logistic function\n1/(1 + e\u2212Ii), wji is the synaptic strength from neuron j to neuron i, and bi is static\nbias. The time constant \u03c4i determines how rapidly the activation approaches its steady-\nstate value for constant Ii. Equation 1 embodies the additional assumption that, on the\ntime scale of chemotaxis behavior, C. elegans neurons are effectively passive, isopoten-\ntial nodes that release neurotransmitter in graded fashion. This assumption follows from\npreliminary electrophysiological recordings from neurons and muscles in C. elegans and\nAscaris, another species of nematode[3, 4, 6].\n\nThe model of the chemosensory network had one input neuron, eight interneurons, and one\noutput neuron (Figure 1). The input neuron (i = 0) was a lumped representation of all\n\n\fFigure 1: Model chemosensory net-\nwork. Model neurons were passive,\nisopoential nodes. The network con-\ntained one sensory neuron, one out-\nput neuron, and eight interneurons.\nInput to the sensory neuron was the\ntime course of chemoattractant con-\ncentration C(t). The activation of\nthe output neuron was mapped to\nturning probability by the function\nF (t) given in Equation 2. The net-\nwork was fully connected with self-\nconnections (not shown).\n\nthe chemosensory neurons in the real animal. Sensory input to the network was C(t), the\ntime course of attractant concentration experienced by a real worm in an actual chemotaxis\nassay[11]. C(t) was added to the net input of the sensory neuron (i = 0). The interneurons\nin the model (1 \u2264 i \u2264 8) represented all the chemosensory interneurons in the real animal.\nThe activity level of the output neuron (i = 9) determined the behavioral state of the model,\ni.e. turning probability[11], according to the piecewise function:\n\n( Phigh\n\nPrest\nPlow\n\nF (t) =\n\nA9(t) \u2264 T1\nT1 < A9(t) < T2\nA9(t) \u2265 T2\n\n(2)\n\nwhere T1 and T2 are arbitrary thresholds and the three P values represent the indicated\nlevels of turning probability.\n\n4 Optimization\n\nThe chemosensory network model was optimized to compute an idealized version of the\ntrue sensorimotor transformation linking C(t) to turning probability[11]. To construct the\nidealized transformation, we mapped the instantaneous derivative of C(t) to desired turning\nprobability G(t) as follows:\n\n( Phigh\n\nG(t) =\n\ndC(t)/dt \u2264 \u2212U\ndC(t)/dt \u2265 +U\n\nPrest \u2212U < dC(t)/dt < +U\nPlow\n\n(3)\n\nwhere U is a threshold derived from previous behavioral observations (Figure 7 in [11]).\nThe goal of the optimization was to make the network\u2019s turning probability F (t) equal to\nthe desired turning probability G(t) at all t. Optimization was carried out by annealing\nthree parameter types: weights, time constants, and biases. Optimized networks were fully\nconnected and self-connections were allowed.\n\nThe result of a typical optimization run is illustrated in Figure 2(a), which shows good\nagreement between network and desired turning probabilities. Results similar to Figure\n2(a) were found for 369 networks out of 401 runs (92%). We noted that in most networks,\nmany interneurons had a constant offset but showed little or no response to changes in\nsensory input. We found that we could eliminate these interneurons by a pruning procedure\nin which the tonic effect of the offset was absorbed into the bias term of postsynaptic\nneurons. Pruning had little or no effect on network performance (Figure 2(b)), suggesting\n\nF(t)C(t)(0)sensoryneuron(1)inter-neuron(2)inter-neuron(8)inter-neuron(9)outputneuron\fthat the eliminated neurons were nonfunctional. By this procedure, 67% of the networks\ncould be reduced to one interneuron and 27% could be reduced to two interneurons. A key\nquestion is whether the network generalizes to a C(t) time course that it has not seen before.\nGeneralization was tested by challenging pruned networks with the C(t) time course from\na second real chemotaxis assay. There was good agreement between network and desired\nturning probability, indicating an acceptable level of generalization (Figure 2(c)).\n\nFigure 2: Network performance after optimization. In each panel, the upper trace repre-\nsents G(t), the desired turning probability in response to a particular C(t) time course (not\nshown), whereas the lower trace represents F (t), the resulting network turning probabil-\nity. Shading signi\ufb01es turning probability (black = Phigh, grey = Prest, white = Plow). (a)\nPerformance of a typical network after optimization. (b) Performance of the same network\nafter pruning. (c) Performance of the pruned network when stimulated by a different C(t)\ntime course. Network turning probability is delayed relative to desired turning probability\nbecause of the time required for sensory input to affect behavioral state.\n\n5 Results\n\nHere we focus on the largest class of networks, those with a single interneuron (Figure\n3(a)). All single-interneuron networks had three common features (Figure 3(b)). First,\nthe direct pathway from sensory neuron to output neuron was excitatory, whereas the indi-\nrect pathway via the interneuron was inhibitory. Such a circuit computes an approximate\nderivative of its input by subtracting a delayed version of the input from its present value[9].\nSecond, all neurons had signi\ufb01cant inhibitory self-connections. We noted that inhibitory\nself-connections were strongest on the input and output neurons, the two neurons compris-\ning the direct pathway representing current sensory input. We hypothesized that the func-\ntion of inhibitory self-connections was to decrease response latency in the direct pathway.\nSuch a decrease would be a means of compensating for the fact that G(t) was an instanta-\nneous function of C(t), whereas the neuronal time constant \u03c4i tends to introduce a delay\nbetween C(t) and the network\u2019s output. Third, the net effect of all disynaptic recurrent con-\nnections was also inhibitory. By analogy to inhibitory self-connections, we hypothesized\nthat the function of the recurrent pathways was also to regulate response latency.\n\nTo test the hypothetical functions of the self-connections and recurrent connections, we in-\ntroduced an explicit time delay (\u2206t) between dC(t)/dt and the desired turning probability\nG(t) such that:\n\n(4)\nG0(t) was then substituted for G(t) during optimization. We then repeated the optimization\nprocedure with a range of \u2206t values and looked for systematic effects on connectivity.\n\nG0(t) = G(t \u2212 \u2206t)\n\n8006004002000G(t)F(t)(a)8006004002000time (seconds)G(t)F(t)(c)8006004002000G(t)F(t)(b)\f(a)\n\nC(t)\n\nInput\nNeuron\n\nOutput\nNeuron\n\nF(t)\n\nExcitatory\nInhibitory\n\n(b)\n\nFeature\n\nInter-\nneuron\n\ndirect excitatory\n\ndelayed \ninhibitory\n\nself-connection\n\ninhibitory \nrecurrent\nconnection\n\nFigure\n+ slow\n\nFunction\n\ndifferentiation\n\n+\n\nhypothesis: \nregulation of \n\nresponse latency\n\nFigure 3: Connectivity and common features of single-interneuron networks. (a) Average\nsign and strength of connections. Line thickness is proportional to connection strength. In\nother single-interneuron networks, the sign of the connections to and from the interneuron\nwere reversed (not shown). (b) The three common features of single-interneuron networks.\n\nEffects on self-connections. We found that the magnitude of self-connections on the\ninput and output neurons was inversely related to \u2206\nt (Figure 4(a)). This result suggests\nthat the function of these self-connections is to regulate response latency, as hypothesized.\nWe noted that the interneuron self-connection remains comparatively small regardless of\nt. This result is consistent with the function of the disynaptic pathway, which is to present\n\na delayed version of the input to the output neuron.\n\n(cid:31) t, Target Delay for G(t) (seconds)\n\n(a)\n\n0\n\n(cid:31) t (seconds)\n3\n\n2\n\n1\n\n4\n\n5\n\ninterneuron\n\n-5\n\ni\n\nt\nh\ng\ne\nW\n \nn\no\ni\nt\nc\ne\nn\nn\no\nc\n-\nf\nl\ne\nS\n\n-10\n\n-15\n\n-20\n\n output neuron\n\n input neuron\n\ns\nt\nh\ng\ne\nW\n\ni\n\n \nt\nn\ne\nr\nr\nu\nc\ne\nR\n \nf\no\n \nt\nc\nu\nd\no\nr\nP\n\n-100\n\n-150\n\n-200\n\n-250\n\n(b)\n\n0\n\n1\n\n(cid:31) t (seconds)\n3\n\n2\n\n4\n\n5\n\n-50\n\n input - output\n\n interneuron - output\n\n input - interneuron\n\n interneuron - output\n\n input neuron\n interneuron\n output neuron\n\n input - interneuron\n\n input - interneuron\n input - output\n interneuron - output\n\n input - interneuron\n input - output\n interneuron - output\n\nFigure 4: The effect on connectivity of introducing time delays between input and output\nduring optimization. (a) The effect on self-connections. (b) The effect on recurrent connec-\ntions. Recurrent connection strength was quanti\ufb01ed by taking the product of the weights\nalong each of the three recurrent loops in Figure 3(a).\n\n(cid:31) t, Target Delay for G(t) (seconds)\n0\n\n3\n\n2\n\n1\n\n5\n\n4\n\n-50\n\n-150\n\n input - output\n\nEffects on recurrent connections. We quanti\ufb01ed the strength of the recurrent connec-\ntions by taking the product of the two weights along each of the three recurrent loops in the\nnetwork. We found that the strengths of the two recurrent loops that included the interneu-\nron was inversely related to \u2206\nt (Figure 4(b)). This result suggests that the function of these\n interneuron - output\nloops is to regulate response latency and supports the hypothetical function of the recur-\n\n-200\n\n input - interneuron\n\ns\nt\nh\ng\ne\nW\n\ni\n\n \nt\nn\ne\nr\nr\nu\nc\ne\nR\n \nf\no\n \nt\nc\nu\nd\no\nr\nP\n\n-100\n\n-250\n\n\u2206\n\frent connections. Interestingly, however, the strength of the recurrent loop between input\nand output neurons was not affected by changes in \u2206t. Comparing the overall patterns of\nchanges in weights produced by changes in \u2206t showed that the optimization algorithm uti-\nlized self-connections to adjust delays along the direct pathway and recurrent connections\nto adjust delays along the indirect pathway. The reason for this pattern is presently unclear.\n\n6 Analysis\n\nTo provide a theoretical explanation for the effects of time delays on the magnitude of self-\nconnections, we analyzed the step response of Equation 1 for a reduced system containing\na single linear neuron with a self-connection:\n\ndAi(t)\n\ndt\n\n\u03c4i\n\n= wiiAi(t) \u2212 Ai(t) + h(t)\n\n(5)\n\nwhere h(t) represents a generic external input (sensory or synaptic). Solving Equation 5\nfor h(t) equal to a step of amplitude M at t = 0 with A(0) = 0 gives:\n\n(cid:19)(cid:20)\n\n(cid:18) M\n\n1 \u2212 wii\n\nAi(t) =\n\n(cid:20)\n\n(cid:18)1 \u2212 wii\n\n(cid:19)(cid:21)(cid:21)\n\nt\n\n\u03c4i\n\n1 \u2212 exp\n\n\u2212\n\n(6)\n\nFrom Equation 6, when wii = 0 (no self-connection) the neuron relaxes at the rate 1/\u03c4i,\nwhereas when wii < 0 (inhibitory self-connection) the neuron relaxes at the higher rate\nof (1 + |wii|)/\u03c4i. Thus, response latency drops as the strength of the inhibitory self con-\nnection increases and, conversely, response latency rises as connection strength decreases.\nThis result explains the effect on self-connection strength of introducing a delay between\nbetween dC(t)/dt and turning probability (Figure 4(a)).\nWe made a similar analysis of the effects of time delays on the recurrent connections. Here,\nhowever, we studied a reduced system of two linear neurons with reciprocal synapses and\nan external input to one of the neurons.\n\ndAi(t)\n\ndt\n\n\u03c4i\n\n= wjiAj(t) \u2212 Ai(t) + h(t)\n\nand\n\n\u03c4j\n\ndAj(t)\n\ndt\n\n= wijAi(t) \u2212 Aj(t) (7)\n\nWe solved this system for the case where the external input h(t) = M sin(\u2126t). The solu-\ntion has the form:\n\nAi(t) = Di sin(\u2126t \u2212 \u03c6i)\n\nand\n\nwith\n\n\u03c6i = \u03c6j = arctan\n\n(cid:20)\n\nAj(t) = Dj sin(\u2126t \u2212 \u03c6j)\n\n(cid:21)\n\n2\u2126\u03c4\n\n1 \u2212 wijwji \u2212 \u21262\u03c4 2\n\n(8)\n\n(9)\n\nEquation (9) gives the phase delay between the sinusoidal external input and the sinusoidal\nresponse of the two neuron system. In Figure 5, the relationship between phase delay and\nthe strength of the recurrent connections is plotted with the connection strength on the\nordinate as in Figure 4(b). The graph shows an inverse relationship between connection\nstrength and phase delay that approximates the inverse relationship between connection\nstrength and time delay shown in Figure 4(b). The correspondence between the trends in\nFigure 4(b) and 5 explain the effects on recurrent connection strength of introducing a delay\nbetween between dC(t)/dt and turning probability.\n\n\fFigure 5:\nThe relationship between\nphase delay and recurrent connection\nstrength.\nEquation 9 is plotted for\nthree different driving frequencies, (Hz\n\u00d710\u22123): \u21261 = 50, \u21262 = 18.75, and\n\u21263 = 3.75. These frequencies span the\nfrequencies observed in a Fourier analy-\nsis of the C(t) time course used during\noptimization. There is an inverse rela-\ntionship between connection strength and\nphase delay. Axis have been reversed for\ncomparison with Figure 4(b).\n\nFigure 6: The network of chemosensory\ninterneurons in the real animal. Shown\nare the interneurons interposed between\nthe chemosensory neuron ASE and the\ntwo locomotory command neurons AVA\nand AVB. The diagram is restricted to in-\nterneuron pathways with less than three\nsynapses. Arrows are chemical synapses.\nDashed lines are gap junctions. Connec-\ntivity is inferred from the anatomical re-\nconstructions of reference [15].\n\n7 Discussion\n\nWe used simulated annealing to search for networks capable of computing an idealized\nversion of the chemotaxis sensorimotor transformation in C. elegans. We found that one\nclass of such networks is the three neuron differentiator with inhibitory feedback. The\nappearance of differentiator networks was not surprising [9] because the networks were\noptimized to report, in essence, the sign of dC(t)/dt (Equation 3). The prevalence of in-\nhibitory feedback, however, was unexpected. Inhibitory feedback was found at two levels:\nself-connections and recurrent connections. Combining an empirical and theoretical ap-\nproach, we have argued that inhibitory feedback at both levels functions to regulate the\nresponse latency of the system\u2019s output relative to its input. Such regulation could be func-\ntionally signi\ufb01cant in the C. elegans nervous system, where neurons may have an unusually\nhigh input resistance due to their small size. High input resistance could lead to long re-\nlaxation times because the membrane time constant is proportional to input resistance. The\ntypes of inhibitory feedback identi\ufb01ed here could also be used to mitigate this effect.\n\nThere are intriguing parallels between our three-neuron network models and the biological\nnetwork. Figure 6 shows the network of interneurons interposed between the chemosensory\nneuron class ASE, the main chemosensory neurons for salt chemotaxis, and the locomotory\ncommand neurons classes AVB and AVA. The interneurons in Figure 6 are candidates for\ncomputing the sensorimotor transformation for chemotaxis C. elegans. Three parallels are\nprominent. First, there are two candidate differentiator circuits, as noted previously[16].\nThese circuits are formed by the neuronal triplets ASE-AIA-AIB and ASE-AWC-AIB.\nSecond, there are self-connections on three neuron classes in the circuit, including AWC,\none of the differentiator neurons. These self-connections represent anatomically identi\ufb01ed\nconnections between left and right members of the respective classes.\nIt remains to be\nseen, however, whether these connections are inhibitory in the biological network. Self-\nconnections could also be implemented at the cellular level by voltage dependent currents.\nA voltage-dependent potassium current, for example, can be functionally equivalent to an\n\n80x10-3604020Phase Delay (radians)-250-200-150-100-50Recurrent Product f = 0.00375 Hz f = 0.01875 Hz f = 0.05 Hz W1W2W3(cid:10)-250-200-150-100-50Recurrent Product80x10-3604020Recurrent ProductPhase Delay (radians)W1W2W3AIBAVAAFDASEAIARIBRIMSAADDVCFLPRIAAWCAVBRIFinter-neuronschemosensoryneuronscommandneuronsAIY\finhibitory self-connection. Electrophysiological recordings from ASE and other neurons in\nC. elegans con\ufb01rm the presence of such currents[6, 10]. Thus, it is conceivable that many\nneurons in the biological network have the cellular equivalent of self-connections. Third,\nthere are reciprocal connections between ASE and three of its four postsynaptic targets.\nThese connections could provide recurrent inhibition if they have the appropriate signs.\n\nCommon patterns of connectivity between the model and biological networks suggest new\nfunctionality for identi\ufb01ed connections in the C. elegans nervous system.\nIt should be\npossible to test these functions through physiological recordings and neuronal ablations.\n\nAcknowledgements\n\nWe are grateful Don Pate for his technical assistance. Supported by NSF IBN-0080068.\n\nReferences\n\n[1] C. I. Bargmann and H. R. Horvitz. Chemosensory neurons with overlapping functions direct\n\nchemotaxis to multiple chemicals in C. elegans. Neuron, 7:729\u2013742, 1991.\n\n[2] M. Chal\ufb01e, J.E. Sulston, J.G. White, E. Southgate, J.N. Thomson, and S. Brenner. The neural\n\ncircuit for touch sensitivity in C. elegans. J. of Neurosci., 5:956\u2013964, 1985.\n\n[3] R. E. Davis and A. O. Stretton. 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Proc of the Natl Acad Sci USA, 70:817\u2013821, 1973.\n\n[14] S. Ward, N. Thomson, J. G. White, and S. Brenner. Electron microscopical reconstruction\nof the anterior sensory anatomy of the nematode C. elegans. J. of Comparative Neurology,\n160:313\u2013338, 1975.\n\n[15] J. G White, E. Southgate, J. N. Thomson, and S. Brenner. The structure of the nervous system\n\nof the nematode C. elegans. Phil Trans of the R Soc Lond [Biol], 314:1\u2013340, 1986.\n\n[16] J.G. White. Neuronal connectivity in C. elegans. Trends in Neuroscience, 8:277\u2013283, 1985.\n\n\f", "award": [], "sourceid": 2369, "authors": [{"given_name": "Nathan", "family_name": "Dunn", "institution": null}, {"given_name": "John", "family_name": "Conery", "institution": null}, {"given_name": "Shawn", "family_name": "Lockery", "institution": null}]}