Figure 1: Nematode chemotaxis. The nematode C. elegans provides a unique opportunity to study biological computation, mainly because it has exactly 302 neurons, and the morphology of every neuron and the location of nearly every synaptic connection are known precisely. Among its many behaviors is chemotaxis, shown on the right. Four worms were placed in a Petri dish containing a Gaussian-shaped gradient of chemical attractant. After initial transient movements, all three worms oriented almost directly up the gradient and dwelled at the peak. Each run lasted about 5 minutes. (J. T. Pierce and S. R. Lockery, unpublished.)
Figure 2: Sensory apparatus. An essential aspect of the C. elegans chemotaxis control problem is the following. The nematode has a bilaterally arranged pair of sensory organs, each containing the processes of 11 chemosensory neurons. During movement on a flat surface, however, the nematode moves on its side, making dorsal (D) and ventral (V) contractions. This places the amphids perpendicular to the gradient, implying that the nematode performs chemotaxis while sensing the chemical concentration at effectively a single point in space. It must therefore make temporal comparisons of the absolute concentration to derive information about the spatial gradient.
Figure 3: Movement on a planar surface. Behavioral experiments indicate that during forward movement, nematodes maintain approximately constant speed v=0.22 mm/s (Pierce and Lockery, unpublished). On the left is an idealized model of the nematode body, which describes the movement of the location (x,y) of the tip of the nose, in direction theta. On the right is a model of the neck segment, which controls the direction of movement. Geometrical considerations show that the rate of turning is determined by the relative length of dorsal (D) and ventral (V) neck muscles, and subsequently the relative voltage of dorsal and ventral motorneurons.
Figure 4: Biological chemotaxis network. Laser ablations have determined that the chemotaxis circuit includes 11 pairs of chemosensory neurons (one pair is shown), 4 pairs of interneurons, and 5 pairs of motor neurons (C. Bargmann, unpublished). Chemical synapses are indicated by solid lines and electrical synapses by dotted lines. Line thickness indicates the number of contacts.
Figure 5: Idealized chemotaxis network. Work is underway to implement the biological network in Figure 4. In this work, however, we used an idealized model network, which has the same basic architecture. The model network has a single sensory neuron, three interneurons, and two motor neurons. Each neuron was connected to each other neuron. To test the hypothesis that sinusoidal movement of the head is not required for nematode-like chemotaxis, no central pattern generator was included.
Figure 6: Linearization of bioligical network. Whole-cell recordings indicate that C. elegans neurons have nonlinear membrane currents (Goodman and Lockery, unpublished). Anatomical data indicate that the biological network includes both electrical and chemical synapses, the latter introducing additional nonlinear voltage dependence. To test the hypothesis that these nonlinearities are not essential for nematode-like chemotaxis, we linearized the voltage dependence of the network.
Figure 7: Chemotaxis control by linear network. The linear network parameters were optimized to control chemotaxis on spatial and temporal scales like that of C. elegans. We used a simulated annealing algorithm to maximize a fitness function, equal to the average stimulus (chemical concentration) at the tip of the nose. On the left are real worm tracks, and on the right are model worm tracks generated by a linear network. These results suggest that small linear networks of graded-potential neurons are capable of controlling nematode-like chemotaxis.
Figure 8: Robot inspired by C. elegans. We constructed a simple, four-wheel robot, which mimics the biomechanical aspects of the nematode chemotaxis problem. It has two wheels which provide forward movement at constant speed, and a single sensor at the front. Because chemical gradients are difficult to set up on large spatial scales, we used a single photocell to sense a light environment.
Figure 9: Real robot. On the left, drive motors and steering servos are visible. On the right, the Handy Board and a single photocell are visible.
Figure 10: Demonstration of robot phototaxis. On the left are shown tracks made by the simulated robot after optimization of the network. The circular movements at the center reflect the finite turning radius of the real robot. On the right, real and simulated photocell values are compared for the same initial conditions, corresponding to the lower-right track shown at left. This result is typical, and demonstrates that the real robot performed nearly identically to the simulated robot.
Figure 11: Robustness to mechanical perturbations. To demonstrate that control of the robot by the optimized network is robust to mechanical perturbations, we plotted the fitness of the real and simulated robots as a function of speed and additive turning bias. The dots represent the fitness of the real robot averaged over four initial conditions. Error bars are plus-minus one standard deviation. The solid lines represent the fitness of the simulated robot averaged over the same initial conditions. Dotted lines are plus-minus one standard deviation. These results suggest that the real robot performs nearly identically to the simulated robot, and that both are robust to significant mechanical perturbations.
Figure 12: Goal of rule extraction. To demonstrate that the network in the robot controls phototaxis by a simple form of active perception, we seek to extract a computational rule from the network which reveals the computations being performed.
Figure 13: Mathematical extraction of computational rule. To extract a computational rule, we make use of the fact that the linear network equations are analytically soluble. The response of the robot to sensory input S(t') is described by the usual linear response formula. The linear kernel, or impulse response k(t-t'), oscillates and decays with an exponential envelope. Most important is the fact that it decays rapidly on the time scale of the behavior. Thus the input S(t') can be expanded about the time t, to yield a much simpler computational rule.
Figure 14: Phototaxis control by extracted rule. Calculation of the rule coefficients for this optimized network leads to the result shown above. We define the "order" of the expansion to be equal to the number of time-derivatives retained. The figures show that the first derivative is required for phototaxis control, and that higher derivatives are unimportant. Further investigation shows that the S-term is significant only near the top of the gradient, and there plays only a small role. Phototaxis control by this network can be understood, therefore, by noting that the sign of the dS/dt-term is opposite that of the constant term. Thus when dS/dt<0 the rate of turning is large, and when dS/dt>0 the rate of turning is small.
Figure 15: Result of rule extraction. We have replaced the optimized linear neural network with a simple computational rule, which elucidates computations being performed implicitly by the network. This shows that the robot is implementing a very simple form of active perception, in which it moves its single sensor continuously through the environment, and computes a time derivative in order to orient its movement up the light gradient.
Thanks to 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.