The nematode C. elegans provides a unique opportunity to study the neural computations which underly animal orientation. It has a relatively simple nervous system, but exhibits complex, goal-oriented behaviors.
Thus a physiologically and anatomically detailed model of the entire nervous system is an attainable goal.
As a first step, we study idealized models of the nervous system, which are optimized to control chemotaxis in a worm-like physical body. We first derive a mathematical model of the nematode body, which is controlled by a neural network. We then motivate a simple network model, based on isopotential neurons with graded signalling. The network senses the local chemical environment, and sends output to the motor neurons, to direct movement up the gradient. The strength and polarity of synaptic connections in C. elegans are not known. We therefore use a computer search algorithm (simulated annealing) to find sets of synaptic weights that produce chemotaxis. Many such sets have been found.
By studying the behavior of the simulated worm in various chemical environments, one can infer the computations being made by the network to control chemotaxis. The model worm shown here appears to use a combination of two strategies: one for orientation, and one for dwelling.
Figure 1: During chemotaxis, nematodes orient up gradients of chemical attractants. Three examples of worms, which were placed in a Petri dish containing a Gaussian-shaped gradient of chemical attractant (Cl-). After initial movements, all three worms oriented almost directly up the gradient and remained there for the duration of the experiment, here about 15 min. (J. T. Pierce and S. R. Lockery, unpublished.)
Figure 2: Rate of turning is controlled by neck angle. We focus on the movement of the tip of the nose, located at the point (x,y). We assume that worm moves forward at constant speed v=0.22 mm/s, and at heading angle theta(t). The rate of turning, dtheta/dt, is given by the neck angle alpha(t)=theta1(t)-theta2(t), (see Fig. 3), and the time delay between segments dt=L/Nv.
Figure 3: Neck angle is controlled by motor neurons. A single body segment is shown, with dorsal (D) and ventral (V) muscles indicated. The main result is that the neck angle alpha is controlled by the voltage of dorsal and ventral motor neurons.
Figure 4: Nematodes sense the chemical environment at a single point. a) Side view showing chemosensory neurons (green) sending processes to the laterally-arranged amphids. b) End view showing that, since the worm creeps on its side, the amphids (green) are oriented perpendicular to the gradient during chemotaxis in a Petri dish.
Figure 5: Neurons involved in chemotaxis have been identified by laser ablation. The chemotaxis circuit includes 11 pairs of chemosensory neurons (one pair is shown), 4 pairs of interneurons, and 5 pairs of motor neurons. Chemical synapses are indicated by solid lines and electrical synapses by dotted lines. Line thickness indicates the number of contacts. (C. Bargmann, unpublished.)
Figure 6: Simplified neural network model. The model network has a single sensory neuron, interneurons, and two motor neurons. (In the result shown here, we used 9 interneurons.) Model neurons are isopotential and have graded potentials, and each neuron can be connected to any other. In addition to synaptic input, the motor neurons also receive input from a central pattern generator (CPG), which generates ordinary sinusoidal movement.
Figure 7: Neural network formulae. Since C. elegans neurons do not fire action potentials, (continuous) voltage is the state variable. Pairwise recordings from Ascaris suum indicate that nematode synapses release neurotransmitter tonically, and that postsynaptic voltage is a sigmoidal function of presynaptic voltage. This simple model captures these features. Chemosensory input is assumed to be proportional to the stimulus C, and the motor neurons are driven sinusoidally at frequency w, corresponding to a period T = 4.2 sec.
Figure 8: Chemotaxis control by neural network. A) The model worm was started from three different initial conditions (position and heading angle) in a Gaussian gradient. B) Behavior of the same model worm with ~1% noise added to the voltage of each neuron. This shows that the network is somewhat robust to neuronal noise.
Figure 9: Model worm behavior in a planar gradient. The model worm was started from a single initial condition, in a planar gradient, in which the concentration varies linearly from C to C 0. The shape of this track suggests that a strategy in which the worm makes tighter turns at higher concentrations (stimulus-strength strategy) may be sufficient to produce chemotaxis.
Figure 10: Stimulus-strength strategy was not sufficient to produce chemotaxis. A) To test whether this strategy was sufficient to produce chemotaxis, we determined the radius of curvature R as a function of constant chemical concentration C, for the network in Figs. 8 and 9. B) We then used this rule to control the model worm directly. The model worm did not chemotax. This shows that the model worm controlled by the neural network is using higher-order properties of the stimulus, e.g., dC/dt.
Figure 11: Gradient decomposition analysis. To control orientation, this model worm must use dC/dt to infer the gradient. Future work will focus on what aspect of the gradient is used, i.e., components of the gradient which are parallel and/or perpendicular to the worm's mean path.
Thanks to M. B. Goodman and J. T. Pierce for valuable discussions. This work is supported by NIMH MH11373, NIMH MH51383, NSF IBN 9458102, ONR N00014-94-1-0642, the Sloan Foundation, and The Searle Scholars Program.