We propose here to develop a computer simulation environment for C. elegans, which will support basic research and education in C. elegans and systems computational neuroscience. The proposed simulator will be called NemaSys, referring to the Nematode being viewed as a complete biological System. It will implement essentially everything that is known about C. elegans body mechanics, neuron morphology and physiology, and synaptic connectivity. It will include sensory mechanisms and stimulus environments for studying chemotaxis, thermotaxis, and mechanical reflexes. It will also include a variety of fitness functions and optimization algorithms for evolving model worms toward particular behaviors. All computed quantities will be exported to text data files for flexible, subsequent analysis. Some analysis capability will also be provided in NemaSys, e.g., simulated whole-cell patch-clamp experiments, and simulated laser ablation experiments in optimized networks.
A mechanical worm body model will be implemented to generate forward and backward movement on a flat two-dimensional surface, simulating the Petri dish of most biological worm experiments.
Figure 2. A segmented body model illustrating (a) the instantaneous location, orientation, and velocity of the tip of the nose, and (b) how the relative length of opposing muscles in the neck segment determine the angle of bend in the neck, and thereby the rate of turning of the worm head.
The body model will be implemented in three steps of approximation:
• In the first step, only the location of the nose is represented, and forward movement is imposed ad hoc by assuming that
and that the speed v is constant. The turning rate dq/dt is controlled by the angular bend in the neck (Figure 2), which in turn is determined by the relative lengths of the neck muscles. This model achieves an efficient implementation, which is well-suited for network optimization on small computers. It is very limited, however, in that it does not generate forward movement
biologically via muscle contractions, nor does it interface the nervous system to the body properly via identified motor neurons. Backward movement is required for mechanical reflexes, and appears also to play a significant role in chemotaxis (Pierce et al., 1999).
• In the second step, the body is implemented as a continuous hydrostatic structure, sheathed by a stiff cuticle and controlled by longitudinal muscle contraction. But in this intermediate approximation, sinusoidal movement is also generated ad hoc, by imposing lateral oscillations on the nose or tail (Neibur and Erdos, 1991). Output from a neural circuit can still modulate the body movement, by increasing or decreasing the bend in the neck relative to the head sweeps, but this model also does not allow a detailed understanding of the interplay between sensory-driven control circuits and sinusoidal movement.
• In the third step, the body model and its interface to the nervous system are implemented as realistically as possible, according to theory and anatomical data (Neibur and Erdos, 1993). Opposing motor neurons with mutual inhibition will generate sinusoidal movement, and other neurons carrying sensory information will modulate this movement. This exact model generates forward and backward movement naturally, and includes proprioceptor cells, which may relay posture information back to the interneurons and play some role in a control algorithm.
Each of the 302 neurons will be implemented according to available anatomical and physiological data.
Figure 3 . Complete anatomical representation of neuron type RMD.
The neurons will also be implemented in three steps of approximation:
• In the first step, each cell is assumed to be isopotential, and represented as a single electrical compartment. Because C. elegans neurons are electrically compact, this approximation has been used in simulations by Wicks et al. (1996), Ferree et al. (1997) and Ferree and Lockery (1999). Cell parameters will be defined based upon whole-cell, patch-clamp experiments performed at the soma (Goodman et al., 1998).
• In the second step, cable attenuation in the one or two cylindrical processes typical in C. elegans neurons will be included phenomenologically, by computing analytically the voltage attenuation between the soma and any synapse using the cable equations. This approximation was made in Wicks et al. (1996). It accounts for cable attenuation reasonably, yet remains computationally efficient since each neuron is described by only a single voltage.
• In the third step, arbitrarily complex models of individual neurons will be enabled by interfacing NemaSys with the neural simulators NEURON and GENESIS. In this way, the stiff nature of multi-compartmental models will be solved efficiently, the cell membranes will be equippable with a wide array of active channel types, known to exist in C. elegans (Goodman et al. (1998), and researchers will be able to study the physiology of single neurons and small networks from the viewpoint of electrophysiology experiments. Because NEURON and GENESIS each have many users, we will seek to interface the NemaSys simulator with both. The Co-PI Malony will provide Computer Science expertise at this step, as well as general guidance on the overall development of NemaSys.
Every electrical and chemical synapse and neuromuscular junction in the nervous system will be implemented, according to the anatomical data from Wood et al. (1988) and White et al. (1986).
• The existence and number of connections between all pairs of neurons has been provided in electronically readable form (Achacoso and Yamamoto, 1992). We will make extensive use of their database and observations. These data were provided on a 5.5” floppy drive which accompanies the book, however, and is no longer practical for dissemination. We will enable NemaSys to look up the number and type of synaptic connections between any two neurons as needed. When users select particular neurons to include in simulations, the correct number of anatomical connections will automatically be included. The user will be able to modify these for research and save these changes, but the defaults will be set by data.
• Chemical synaptic currents will be included as membrane conductances multiplied by driving potentials. This is a key point. Many neural network simulations model synaptic current with a sigmoidal function of the sum of presynaptic voltages, or else a sum of sigmoidal functions of individual presynaptic voltages. Any physiologically correct model must include a driving potential, which depends on the postsynaptic rather than presynaptic potential. This form implies that even if the postsynaptic conductance is a linear function of presynaptic voltage, then the system is still nonlinear by virtue of the quadratic terms arising between pre- and post-synaptic potentials. The postsynaptic conductance is assumed here to be either a linear or sigmoidal function of presynaptic voltage.
• Electrical gap junction currents will be implemented as constant conductances multiplying the potential difference between connected neurons. It is generally assumed that all electrical synapses have approximately the same single-channel conductance. NemaSys will allow the user to optimize these parameters, but the most physiologically realistic results will be those for which the conductance of single gap junctions fall within a tight distribution.
Complete set of sensory modalities will be implemented supporting chemotaxis, thermotaxis, and mechanical reflex responses. Anatomical evidence suggests that there are a total of 39 sensory neurons enabling three sensory modalities: chemosensation, thermosensation, and mechanosensation (White et al., 1986). Figure 4 shows several chemosensory neurons, will cell bodies located in the anterior nerve ring, and processes extending to the tip of the nose, where they exit through the amphids.
Figure 4. Side view of worm head showing anatomical arrangement of chemosensory receptors and amphids. The animal moves via dorsal-ventral flexions, such that the amphids are oriented vertically during chemotaxis on a horizontal surface. Similar receptors are located in the tail, but appear unnecessary for chemotaxis.
Three sensory modalities will be implemented:
• Chemosensation: When nematodes are placed in a Petri dish with a gradient of chemical attractant (repellant) they orient up (down) the gradient. This behavior is initiated by twelve chemosensory neuron classes protruding anteriorly from the amphids (ASE, ASG, ASH, ASI, ASJ, ASK, ADF, ADL, AFD, AWA, AWB, AWC), and two posteriorily from the phasmids (PHA, PHB). The environment is usually a Gaussian-shaped gradient of attractant which diffuses through agar slowly compared to the duration of behavioral experiment. Other common assays include two Gaussian gradients superimposed to achieve a more peaked center (Ward, 1978), and multiple agar surfaces placed adjacently with different concentration levels.
• Thermosensation: When nematodes are placed in a Petri dish with a temperature gradient, they migrate to the temperature at which they were hatched (Mori and Ohshima, 1995). This behavior is initiated by the thermosensory neuron AFD, and by balancing input to circuits which control upward and downward movement (Mori and Ohshima, 1995). It is interesting that the circuits believed to control chemotaxis and thermotaxis share several interneurons, suggesting that these behaviors may interact. Such hypotheses may be pursued easily in NemaSys.
• Mechanosensation: When nematodes are touched, they move in the opposite direction, in a behavior called touch withdrawl. When their Petri dish is tapped, nematodes tend to either speed up or execute a three-point turn, in a behavior called tap reflex. Both behaviors are initiated by touch receptors (ALM, PLM, AVM, PVM) underneath the cuticle. To mimic mechanical stimulation, membrane conductances will be opened transiently in mechanosensory cells.
Optimization algorithms will be implemented to evolve simulated worms toward actual worm behaviors. Optimization methods will include simulated annealing and genetic algorithms.
• The parameter list for C. elegans, as represented in NemaSys, consists of a large number of real variables, and on the order of 3022 discrete variables indexing the polarity of the synaptic connection between each pair of cells, as is required under Dale’s principle. We will therefore develop optimization algorithms based on mixed real and integer parameters.
• A necessary requirement of a physically and biologically plausible network is that it have stable dynamics. Experiments in Ascaris suum have shown that at equilibrium each neuron releases neurotransmitter at half its maximum rate (Davis and Stretton, 1989a; 1989b). Wicks et al. (1996) have argued that this implies the existence of a stable fixed point in the absence of sensory input. This constraint will be included in computing the fitness of neural parameter sets: networks with unstable dynamics will be assigned zero fitness.
• Body and network parameters will be optimized using simulated annealing and genetic algorithms. The PI has already implemented several simulated annealing algorithms which generated the papers and results in this proposal. Several genetic algorithms will also be implemented, including those described in Masters (1995), for example.
© 2001 Thomas Charles Ferree
Computational Science Institute, University of Oregon