theory of the locomotion of nematodes: control of the somatic motor neurons by interneurons

Theory of the Locomotion of Nematodes: Control of the Somatic Motor Neurons by Interneurons

ERNST NIEBUR* AND PAUL ERD& Institute of Theoretical Physics, University of Lausanne, Lausanne, Switzerland

Received 7 July 1992; revised 8 December 1992

ABSTRACT

The only animal of which the complete neural circuitry is known at the sub- microscopical level is the nematode Caenorhabditis elegans. This anatomical knowledge is complemented by functional insight from electrophysiological experiments in the related nematode Ascaris lumbricoides, which show that Ascaris motor neurons transmit signals electrotonically and not with unattenuated spikes. We developed a mathematical model for electrotonic neural networks and applied it to the motor nervous system of nematodes. This enabled us to reproduce experimental results in Ascaris quantitatively. In particular, our computed result of the velocity v I 6 cm/s of neural excitations in the Ascarik interneurons supports the simple hypothesis that the so-called rapidly moving muscular wave is produced by a neural excitation traveling at the same speed in the interneuron as the muscular wave. In C. elegans, the computed velocity v I 8-30 cm/s of signals in the interneurons is much larger than the observed velocity v 2 0.2 cm/s of the body wave. Therefore, the hypothesis that the muscular wave is produced by a synchronous neural excitation wave cannot hold for C. elegans. We argue that stretch receptor control is the most likely mechanism for the generation of body waves used in the locomotion of C. elegans. Extending the simulation to larger groups of neurons, we found that the neural system of C. elegans can operate purely electrotonically. We demonstrate that the same conclusion cannot be drawn for the nervous system of Ascati, because in the long (1= 30 cm) interneurons the electrotonic signals would be too strongly attenuated. This conclusion is not in contradiction with the experimental findings of electrotonic signal propagation in the motor neurons of Ascaris because the latter are shorter (I ~5 cm) than the interneurons.

*Current address: Computation and Neural Systems Program, Division of Biology 216-76, California Institute of Technology, Pasadena, CA 91125.

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52 ERNST NIEBUR AND PAUL ERD6S

1. INTRODUCTION

Various theoretical approaches have been tried to discover and understand the principles that govern the interactions between neurons and how they lead to the complex phenomena of animal behavior. In this work, we have analyzed the simple nervous system of a primitive animal to learn how a biological neural network functions. The small number of neurons in the nervous system in question and their morphological simplicity allows us to use physiologically realistic models for neurons and synapses. These models are closely related to cable theory and allow us to describe a substantial part of the nervous system of the animal.

Nematodes (round worms> are animals with a very simple nervous system, which has been studied for over a century [ll, 12,211 in some of the larger species of this class, in particular Ascaris Zumbticoz&s (of length up to 300 mm). More recently, intracellular recordings were performed in identified Ascaris neurons (see, e.g., [24]).

A decade of experimental work has led to the electron microscopical reconstruction of the complete nervous system of another nematode, C. eleguns, a free-living nematode of about 1 mm length and 0.1 mm diameter. It was found that each adult, wild-type, hermaphrodite C. elegans has exactly 302 neurons among its 956 nongonadal cells. The neurons are connected by about 5000 chemical and 2000 electrical synapses (gap junctions). The shapes of all neurons and the positions of all synapses are known from this work, which thus yielded the complete “wiring diagram” of the nervous circuitry of this nematode [30,31]. This anatomical knowledge is supplemented by functional information from laser ablation experiments, in which neurons are killed selectively and conclusions are drawn on their function by comparing the behavior of treated and untreated animals [51.

The simplicity of the nematode nervous system and the availability of complementary data from two species with similar nervous systems made nematodes our choice for a detailed modeling approach. We did not treat the whole system of the 302 neurons of C. elegans, but concentrated our efforts on the neural circuitry that controls the undulatory locomotion of the worm.

Our aim was the simulation of the behavior of (1) individual neurons and (2) of the neural network used by C. eleguns for its forward locomotion. The individual neurons simulated were (1) Ascaris motor neurons; (2) Ascaris interneurons; (3) C. eZeguns motor neurons; and (4) C. elegans interneurons. The goal of the simulation was the determination of the velocity and attenuation of the nerve signals. The results were then used to decide which of the different hypotheses concerning

CONTROL OF THE LOCOMOTION OF NEMATODES 53

the functioning of the locomotive system of both species might be correct.

In Section 2, we give a justification for our choice of these four types of neurons and describe them in some detail. Section 3 describes a general model of electrotonic neural networks, of which the neural system under study is an example. We formulate a mathematical model of this neural system that enables us to simulate its behavior on a computer. In Section 4, we use the model to study single neurons as well as network properties. On the basis of our single-cell results we conjecture a simple mechanism that explains the propagation of fast muscular waves in Ascaris. This mechanism does not, however, explain C. eleguns behavior. By computer simulations of the network that generates forward locomotion, we demonstrate that the C. eleguns motor nervous system is capable of exerting global control over the somatic musculature using only electrotonic spread of potential (without spikes). We also show that in Ascaris, other forms of signal propagation must be present. Section 5 is devoted to discussion and conclusions.

To simulate the behavior of the motor nervous system, we have to make assumptions on the input and output of this system. Our basic hypothesis is that the input to this system is such that for movement with constant forward thrust (usually leading to a constant velocity), a specific pair of interneurons receives a constant synaptic excitation. The output is a relatively complicated time- and space-dependent excitation pattern of the motor neurons. Following other workers, we conjecture in Section 2 that this pattern can be generated by proprioceptive organelles in the motor neurons. It remains to be shown that the forces generated by this mechanism are suitable to propel the worm. This is a question that is not related to signal propagation in nervous systems but to the mechanics of self-propelling bodies and has been treated in two previous publications [9,20].

2. AN EXPERIMENT-BASED HYPOTHESIS ON THE FUNCTION OF THE NEMATODE NERVOUS SYSTEM

2.1. GROSS ANATOMY

Each of the 302 neurons of C. eleguns has been given a unique label [31]; groups of neurons that are believed to have a similar function and that differ from each other only in their position in the body have been assigned to the same class. There are 118 classes in C. eleguns, each class containing 1-13 neurons. Class names consist of either two or three uppercase letters. The class members are distinguished from each other by adding to the class name a number and, if appropriate, a side

54 ERNST NIEBUR AND PAUL ERDiiS

descriptor (L for the left and R for the right side of the body, respectively).

Most neurons, including many sensory cells and a group of neurons controlling the pharynx, are situated in the anterior part of the body, as is the “nerve ring,” the most complicated structure of the C. elegans nervous system. This is an extensive region of neuropile of toroi- da1 shape that encircles the pharynx and in which many synaptic connections are found.

Another group of neurons is situated in the caudal ganglia in the posterior part of the body. These are connected to the nerve ring by several process bundles, the most prominent of which are the ventral cord and the dorsal cord. Both of these bundles run parallel to the long axis of the body and are interconnected by commissures, which emanate from motor neurons whose somata are situated in the ventral cord.

This group of motor neurons has been subdivided into seven classes: AS,DA,DB,DD,VA, VB, VD. It is known from experiments that the somatic muscles that are used for the undulatory motion are controlled by these somatomuscular motor neurons.

The members of two of these classes, VA and VB, do not have commissures; that is, their neural processes do not leave the ventral cord. They are presynaptic in the ventral cord to ventral muscle cells and to the neurons of class DD.

The neurons in classes AS, DA, and DB, which are endowed with commissures, are presynaptic to dorsal muscle and to the neurons VD. Neurons of classes DD and VD are presynaptic to ventral and dorsal muscles, respectively. All somatic motor neurons with the exception of the DD and VD neurons are postsynaptic to inter-neurons in the ventral cord. Among these interneurons, four classes, each consisting of two members, deserve our special attention: AVA, AVB, AVD, and PVC. These are the only neurons that are presynaptic to the somatic motor neurons and extend along the entire ventral cord. For this reason we suppose that the control of the worm’s undulatory locomotion is mediated by these neurons.

Figure 1 shows the interneurons and motor neurons involved in forward locomotion, and Figure 2 shows the general structure of the nematode motor nervous system. For several reasons, this part of the nervous system is well suited for modeling. First of all, we know its purpose-the production of the undulatory motion of the worm. Furthermore, we are able to identify the neurons that are part of this subsystem and their global purpose: The interneurons AVA, AVB, AVD, and PVC receive synaptic input in the nerve ring and in the caudal ganglia, and they control the somatic motor neurons, which in turn control the somatic muscles. More detailed information is obtained from the experiments described below.


DB I

- VD VB 10

I, VB 11

FIG. 1. Schematic diagram of the neural circuitry used for forward locomotion by C. elegans. This is the circuitry simulated in the calculations described in the text. The circuitry is reconstructed from electron microscopic data obtained by J. White and collaborators (unpublished; used with permission). The diagram represents the interneurons AVBL and AVBR, the motor neurons of classes DB and VB, and the interconnections between all these neurons. The interneurons AVBL and AVBR are represented by the thick horizontal lines. They extend along the entire ventral cord, from the nerve ring (left, not shown) to the caudal ganglion (right, not shown). The ventral motor neurons VBl-VBll are represented by horizontal lines below those representing the interneurons. The seven dorsal motor neurons DBl-DB7 are shown above the interneurons. Each of these neurons has one process in the ventral cord, which is represented by a horizontal line between the interneurons and the dash-dotted line, and one dorsal process, represented by a horizontal line above the dash-dotted line. The two processes are connected by one commissure per neuron, represented by a curved vertical line. Straight vertical lines represent multiple gap junctions, dashed vertical lines indicate simple gap junctions.

2.2. ELECTROPHYSIOLOGICAL EXPERIMENTS IN ASCARIS

Ascaris lumbricoides is a large nematode whose nervous system is similar in structure to that of C. elegans. Despite the great difference in body length (C. elegans, 1 mm; Ascaris l., 300 mm), there are similar numbers of neurons in both nematode species (C. elegans, 302; Ascaris I., about 250). In particular, for each neuron class in the motor nervous system of C. elegans there exists a corresponding class in Ascaris. We surmise that the similarity of the structure reflects similarity in the neural function. In fact, identical neurotransmitters have been found in

56 ERNST NIEBUR AND PAUL ERD&

head /

dorsal motor neurons

\

nerve ring

ventral cord with interneurons

\ ventral motor neurons /

FIG. 2. Schematic diagram of a part of the nematode motor nervous system. The nerve ring, the interneurons emanating from it, and the motor neurons that are postsynaptic to these interneurons are shown. The synapses between intemeurons and motor neurons are represented by shaded boxes, and the neuromuscular junctions between motor neurons and muscles are white boxes. Two of the 11 ventral motor neurons and two of the seven dorsal motor neurons are shown.

the neurons of corresponding classes in these two species (S. L. McIn- tire, personal communication).

Penetration of C. eleguns neurons by microelectrodes and observation of the intracellular electrical potential is at present impossible because of the small diameter of the neural processes. In Ascaris l., however, the neurons are sufficiently large to allow intracellular recording. Furthermore, it was found that the somatic motor neuron commissures appear in the same sequence from head to tail in every Ascaris [24]. This observation allows the identification of the members of classes AS, DA, DB, DD, and VD by the position of their commissures, which permitted the cited authors to record the intracellular potential in identified neurons.

One of the results of the experiments (see Stretton et al. [24] for references) is the observation that none of the inspected neurons ever produced all-or-nothing action potentials, either spontaneously or after depolarization or hyperpolarization. The neurons are thus electrotonic, which means that signals propagate in them by means of graded, attenuated electrical potentials, which can take on a continuum of values. We will come back to this in more detail in Section 3.

2.3. LASER ABLATION EXPERIMENTS

It is possible to ablate chosen C. elegans neurons by a laser micro- beam.’ This allows one to probe for the function of a given neuron by

‘Usually it is not the neuron itself that is killed but its precursor before the mitose. This is possible because the complete cell lineage of C. eleguns, from the fertilized egg to the adult worm, is known [25,26].


comparing the behavior of a worm in which this neuron is destroyed with the behavior of an untreated worm. Systematic research using this technique has been undertaken for the neural circuitry that mediates the touch reflex in C. &galas L.51.

If an intact C. elegans is touched at its head, it moves backward; if it is touched at its tail, it moves forward. If the AVA and AVD neurons are killed, the worm is no longer able to move forward, although it moves backward normally. If only the AVA neurons are killed, the worm is sensitive to touch at the head and tail and moves forward normally, but it has serious problems moving backward. If only the AVD are killed, the worm moves forward and backward normally and is touch-sensitive at its tail but not at its head.’ The opposite effect is obtained when the AVB and PVC neurons are killed: The animal moves forward normally but is unable to move backward. Destroying only AVB neurons, it was found that these neurons are used for forward movement. Ablating only PVC neurons yields animals that are touch-insensitive at the tail.

For technical reasons, the experiments described were made with larvae, which do not yet have the complete set of somatomuscular motor neurons but have only classes DA, DB, and DD. Killing part of the DA neurons yields larvae that are unable to move forward but move backward normally. Animals lacking most of the DB neurons move forward normally but have problems moving backward. Finally, killing part of the DD neurons yields worms whose movement is badly coordi- nated when moving forward as well as when moving backward. The results are shown schematically in Table 1.

2.4. INPUT TO THE MOTOR NERVOUS SYSTEM: STEADY EXCITATION BY THE NERVE RING

We are now in a position to combine the results on the function and those on the anatomy of the nervous systems of C. elegans and Ascaris. The results cited in the previous subsections can be interpreted consis- tently as follows. C. efegans has two distinct neural circuits, one for moving forward and one for moving backward. Forward motion is invoked by an excitation of the interneurons AVB and PVC, which excite the neurons DB and VB. Backward motion is generated by an excitation of the interneurons AVA and AVD, which excite the neurons AS, DA, and VA. The only neurons that are used for both forward and backward movement are DD and VD, whose function is the coordina-

‘In larvae. Later in development, another type of interneuron (AVM) appears. AVM neurons are capable of assuming a part of the AVBs’ tasks. See [S] for details.


TABLE 1

Effect of Laser Ablation of Selected Neurons in C. elegun9

Head touch Tail touch Forward Backward Neurons killed sensitivity sensitivity movement movement

PVC Yes No Yes Yes AVD Partial Yes Yes Yes AVA Yes Yes Yes Partial AVB Yes Yes Partial Yes AVA and AVD Partial Yes Yes No AVB and PVC Yes No No Yes DA Yes Yes Yes No DB Yes Yes No Yes DD Yes Yes Partial Partial

aThe first column shows the neuron class whose members were destroyed; the other four columns show which part of the behavior is curtailed (“no”), or remains normal (“yes”). For details see [5].

tion of the movement (see Figure 3). The results of experiments [5] suggest that AVA and AVB are the most important interneurons for the control of locomotion and that AVD and PVC are used mainly for transmitting sensory information.

We thus formulate the following hypothesis on the input to the motor nervous system. When C. elegans is touched at its head, the anterior touch receptors are excited. They are presynaptic to AVA and AVD, which are, in turn, electrically coupled to AS, DA, and VA. This is the pathway that leads to backward movement after anterior touch. An analogous pathway exists for posterior touch. In the posterior ganglia, the posterior touch receptors are presynaptic to PVC; this interneuron runs along the ventral cord and is presynaptic to AVB in the nerve ring. Since AVB interneurons are electrically coupled to VB and DB (responsible for forward motion), posterior touch leads eventually to forward motion.

Generalizing, we assume that the anterior part of AVB is excited whenever the worm “decides” to move forward (e.g., because it was touched at its tail) and that the anterior part of AVA is excited whenever the worm “decides” to move backward. If neither of these neurons are excited, the worm does not move at all.

2.5. OUTPUT FROM THE MOTOR NERVOUS SYSTEM: MUSCLE CONTROL BY STRETCH RECEPTORS

The undulatory motion of a creeping nematode requires sequential excitation of the muscles in such a way that an excitation wave will propagate along its body. It is assumed that in higher animals, for


inter-

neurons neurons

FIG. 3. Interconnection scheme of the circuitry for forward movement of C. &guns. The boxes represent the classes of neurons identified in the text; the circles are muscles. Arrows indicate excitatory chemical synapses, black dots are inhibitory chemical synapses. Short bars represent electrical synapses. The hypothetical stretch receptors are to be included in boxes DB and VB. The circuitry for backward movement (not shown) is similar except that there are three excitatory motor neuron classes instead of two. The inhibitory motor neurons DD and VD are the only neuron classes common to both the fonvard and backward circuits.

example, snakes, the central nervous system produces this wave by use of the detailed innervation of the musculature, which makes it possible to excite the appropriate muscle at the appropriate time. C. eleguns does not have a detailed innervation of its body muscles by the nerve ring; there is little, if any, signal output from the nerve ring to the somatic muscles except for the control of somatic motor neurons along the entire body via the four pairs of interneurons AVA, AVB, AVD, and PVC. The simplest assumption is that this output consists of a continuous depolarization. If this is correct, the question arises as to how the excitation of individual muscles is controlled. When the worm advances by one wavelength, the excitation status of each somatic muscle cell undergoes a complete cycle-for example, from relaxed through contracted and back to relaxed. According to the model described, the signal coming from the nerve ring remains the same throughout the cycle. Thus, there must be another mechanism or mechanisms that locally control the muscle excitation. These mechanisms enable C. eleguns to move either forward or backward with

60 ERNST NIEBUR AND PAUL ERDhS

variable velocity and to halt its motion. This requires the generation of excitation patterns that depend on the momentary posture of the worm.

Posture-dependent control is also expected to be advantageous for C. eleguns because it lives in an inhomogeneous environment, the inter- stices of soil particles. According to the geometry of the surrounding soil particles, the body of a nematode moving in this environment has to change shape. In order to yield a sufficient propulsive force at any instant, the muscle excitation pattern has to adapt to the body shape.

The fact that posture information is necessary induced us to discard an otherwise attractive model for muscle control proposed by Stretton et al. [241. The model was developed for Ascaris and proposed that the somatic motor neurons form a chain of coupled oscillators that interact with each other to preserve the correct phase relation necessary for the generation of a traveling excitation wave. All oscillators are submitted to the same input from the interneurons. The input changes the parameters of the oscillators and yields excitation waves that travel with different velocities, either forward or backward.

Another argument against the validity of this model is cell economy: A chain of coupled oscillators can generate excitations that propagate along the chain, either forward or backward [lo]. If nematodes used this mechanism, there would be no reason to have two distinct circuits for forward or backward movement. It will be seen that the model we propose imperatively requires two separate circuits and that it predicts that the destruction of one of them will abolish locomotion in one direction without interfering with locomotion in the other direction. This is precisely what is observed experimentally IS].

Various methods are conceivable by which the worm, whether in motion or at rest, can keep track of its body shape. The simplest possibility is that the worm obtains the information about the actual shape of its body by monitoring it continuously by stretch receptors that send out a signal when they are stretched. The existence of such receptors in the motor neurons of C. elegans has indeed been proposed for morphological reasons. R. L. Russell (personal communication) pointed out that the distal parts of the somatic motor neurons DA, DB, VA, and VB do not have synapses or other specializations and might serve as stretch receptors. Moreover, they are situated immediately adjacent to the cuticle and are thus in an ideal position for this role. What remained unclear were questions such as

(1) Why are the prolongations of the somatic excitatory motor neurons approximately one-fifth to one-fourth of the body length?

(2) Why are there two distinct neural circuits, one for moving forward and the other for moving backward?


(3) Why are the putative stretch receptors of those neurons that are responsible for forward or backward movement situated respectively posterior or anterior to the muscles they are presumed to control?

Answers to these questions are provided by the analysis of the forces involved in undulatory locomotion.

To answer question 1, note that typically a neural stretch receptor has an excitatory reaction (i.e., its transmembrane potential is depolar- ized) when it is elongated.3 It is shown in [19] that a stretch receptor with this property must be located about one-fourth of the length of the body wave away from the muscle that it controls. This corresponds to the length of the prolongation of the somatic motor neurons. To answer questions 2 and 3, it has to be assumed that, in order to generate an excitation pattern that leads to forward locomotion, one needs a set of stretch receptors that are located posterior to the muscles for forward motion and another set located anterior to the muscles for backward motion. These two sets must be controlled by two separate neural circuits.

We concentrate in this paper on the global control of locomotion and leave the questions of local control and the mechanics of undulatory locomotion to another publication [19].

2.6. SUMMARY OF THE ASSUMPTIONS USED IN MODELING THE NEURAL NETWORK

In order to have a definite model suitable for mathematical simulation, we had to make a number of assumptions.

(1) Corresponding classes of neurons in Ascaris 1. and C. efegans have corresponding functions. Reason: Corresponding classes have

anatomical-that is, positional and structural-similarities in the two species. Identical neurotransmitters have been found in corresponding classes (S. L. McIntire, personal communication).

(2) The control of the simulated worm’s undulatory locomotion is mediated by the classes of interneurons AVA, AVB, AVD, and PVC. Reason: These are the only neurons that are presynaptic to the somatic motor neurons and extend along the entire ventral cord.

(3) C. elegans has two distinct neural circuits, one for moving forward and one for moving backward. Forward motion is invoked by the excitation of the interneurons AVB and PVC, which in turn excite the motor neurons DB and VB. Backward motion is generated by the excitation of the inter-neurons AVA and AVD, which in turn

‘We would like to thank Prof. E. Florey for pointing this out to us.

62 ERNST NIEBLJR AND PAUL ERD6S

excite the motor neurons AS, DA, and VA. Reason: The ultrastructural evidence shows the existence of the two separate circuits; laser ablation experiments specify their function.

(4) Forward movement is maintained by a constant excitation of the circuit for forward movement by the nerve ring. Reason: There is no compelling reason to assume a time-varying excitation. It will be shown that a constant excitation explains the locomotion.

(5) Any motor neuron is activated by the synergic action of the constant excitation by the nerve ring (see 4) and excitation by a stretch receptor in the cuticle. Reason: See extensive discussion in Section 2.5.

(6) The nerve cell membranes are passive; that is, the transmembrane conductivity is essentially independent of voltage and is not capable of producing all-or-nothing action potentials. Reason: Experi- mental evidence obtained for Ascaris motor neurons (e.g., [7]).

(7) Neurons are coupled by (a) electrical synapses, which are modeled by ohmic resistors, and (b) chemical synapses, which are modeled by conductivity changes in the postsynaptic membrane. The conductivity of the postsynaptic membrane is a function of the presynaptic voltage. This function is discussed in Section 3.4. Reason: (a) and (b) are coupling mechanisms known from the microstructural data 1311. No data are available for other possible types of coupling (e.g., hormonal).

(8) The ranges of values for the circuit parameters were chosen in accordance with measurements when such were available. For the resting conductivity of the cell membrane of Ascaris interneurons and all C. eleguns neurons, no data were available. We explored the ranges of these parameters that seemed reasonable to us (see Tables 2 and 3).

(9) Lack of more detailed knowledge and the wish to limit the number of parameters leads to the following additional assumptions:

(a) The transmembrane resting conductivity g, is constant along each fiber.

(b) g, has the same values for all members of a given neuron class.

(c) The specific conductivities for all electrical synapses between neurons AVB and DB are the same.

3. THEORY OF ELECTROTONIC NEURAL NETWORKS

3.1. NEURAL PROCESSES

The motor neurons of A. lumbricoides that have been studied by electrophysiological experiments behaved essentially electrotonically [24]. [Small deviations from the behavior of a passive cell membrane have been observed. These have been attributed to the presence of Ca*+ channels ([241 and Stretton and Davis, personal communication).]


The properties of such nonspiking neurons were first discussed by Burrows and Siegler [3,4]. We recently developed a mathematical model for electrotonic neural networks [17,18]. In this model, which is based on Rail’s theory of dendritic trees [22,23], the intracellular voltage l$x,t> in the neural process i is calculated in the absence of synaptic interactions by solving the partial differential equation (PDE)

The solution of this and related equations is described by Jack et al. [13]. The voltage c depends on the spatial coordinate x along the process i (assumed one-dimensional) and the time t. The parameters in Equation (1) are h,, the characteristic electrical length; ri, the characteristic time of the neural process i; and finally V,, the intracellular resting potential. We calculate A and r from the following equations, where the subscript i is omitted from all parameters:

A = [ D/4R,g,]“2 (2a)

and

7 =c,/g,* (2b)

Here R, is the volume resistivity of the cytoplasm, D is the diameter of the neural process (assumed to be cylindrical), g, is the transmembrane resting conductivity per unit area, and C, is the transmembrane capacity per unit area.

A branching neuron is represented by as many PEDs as there are branches, with appropriate boundary conditions ensuring current con- servation at the branching points. In this paper we deal only with monopolar and dipolar neurons, which are represented by one PDE each. Let x, and x, be the coordinates of the two ends of neuron i, respectively. The boundary conditions assumed for all neurons are

aJqx,t) dC(x,t) dx x=x” = dx X=X”= 0 for all t. (3)

Physiologically, this corresponds to a vanishing leakage current out of the distal ends of the neural fibers. The neural processes are coupled by synapses, and the modeling of these couplings is described in the Appendix.


3.2. PARAMETERS OF MODEL NEURONS

The parameters of the network have to be chosen appropriately for simulating a specific neural network. In the case of the motor nervous systems of C. eleguns and Ascaris 1. the only parameter sufficiently well known for all neurons is the diameter of the neural process. For Ascaris motor neurons, experimental data are also available for C,,,,, R,, V,, and g, ([7],[27], and personal communication). The values for CM and R, are close to those found in most cells, while the resting potential V, was found to be unusually high (approximately -35 mV). We used the mean of the measured values of CM, R,, and V, for all neurons. Values considered typical were chosen for the electrochemical reversal potential V, and for the conductivities of postsynaptic membranes, g(ijk, x, t) and g ‘c&n, X, t) defined in Section A.2 of the Appendix [15]. Since no experimental data for individual synapses were available, we used the same values for all synapses, namely g(ijk, x, t> = g, for all i, i, and k, and g’(ijm, X, t) = g, for all i, j, and m. Table 2 shows the parameters used for all neurons. See [16] for more details.

The diameters of the chemical synapses 4 in C. elegans vary between about 0.3 pm and 3 pm [31]. Accordingly, the postsynaptic part of our model synapses consists of a membrane patch with a surface area of 0.5 pm2 whose transmembrane conductivity varies as a function of the presynaptic voltage as described by Equation (A.2).

Electrical synapses in C. eleguns consist of plaques of adjacent cells in close apposition. The diameter of these plaques is about 350 nm [311. Usually, in the regions where an interneuron of the ventral cord makes

TABLE 2

Membrane Parameters of Neurons

Volume resistivity of cytoplasm, R, (n.rn) 1 Transmembrane capacity, C, (F/m2) 0.75 x lo- * Intracellular resting potential, V, (mV) -35 Reversal potential for excitatory synapses, V, (mV) 15 Transmembrane conductivity for excitatory

synapses [ll, g, (CI-‘m-2) 104 Transmembrane conductivity for electrical

synapses [2], g, (O-‘m-2) 104

4More precisely, the diameter of the presynaptic terminal. No postsynaptic specializations can be distinguished in the electron micrographs used. We assume in the following that the postsynaptic receptor sites are concentrated in a region with the same area as the presynaptic terminal.


electrical synapses with the somatic motor neurons, plaques are found in a whole series of adjacent electron microscopic slides. Apparently, plaques are closely packed in these regions, whose length ranges from 1 electron micrographic slide to about 10 slides (from 0.2 to 2 pm). We make the assumption that in these regions the gap junctions between an interneuron and a motor neuron cover the entire half-cylinder of the latter, which is the smaller of the two neurons. Since the circumference of the motor neurons is = 1 pm, we assume that the conductivity between the two synaptic partners is equal to g, in an area that is equal to the length of the synaptic region times a width of 0.5 Frn.

For the numerical solution of the partial differential equations, all neurons were modeled as consisting of a sequence of compartments of length 20 pm. This size was chosen to enable us to perform the numerical calculations within the allotted computing time. Within one compartment the intracellular voltage was assumed constant, but this voltage varied from compartment to compartment. The interneurons of the ventral cord (length 1 mm) were thus represented by 50 compartments, and the number of compartments used for the motor neurons depended on their respective length as obtained from electron microscopial data. For the numerically correct implementation of the boundary conditions [Eq. (311, see [20].

4. RESULTS

4.1. ASCARIS MOTOR NEURONS

In Ascaris l., motor neurons can be excited by a suction electrode in the ventral cord. By cutting all connections between dorsal and ventral parts of the body with the exception of the commissure of one motor neuron, the effect of the ventral stimulation of this neuron can be observed either directly in the commissure ([S] and personal communication) or in the dorsal musculature, where it is postsynaptic to this neuron [281. In both cases, the behavior typical of electronic neurons is observed: stimulation by a square pulse yields a steep voltage rise to a maximum and a somewhat flatter decay. (In the case of intracellular recordings in the muscle cells, one has to assume a linear relationship between voltage in the neuron and the muscle cell for this conclusion.) The maximum of the voltage is attained after a delay that depends on the distance between the stimulating and recording electrodes, and we define the velocity of the traveling excitation as the velocity of this maximum.

It is possible to simulate this experiment on a computer. The purpose of this simulation is twofold. First, it allows us to check the mathematical model expressed by Equations (A.lHA.3); second, it permits, with


certain limitations, the determination of parameters that are not known experimentally. Since only one neuron is involved, Equation (A.l) is used for the special case i = M = 1. Furthermore, since synaptic interactions are not important, the right-hand side of this equation is set identically to zero. The parameters h and r are calculated from Equa- tions (2a) and (2b), using R, and CM from Table 2. We use parameter values corresponding approximately to the mean of the data deduced from the observation of excitatory motor neurons (see Table 3).

Suction electrode stimulation by a square pulse of duration t, is simulated by setting

V( x,t) = v, for 0<x<5X10P3 m;O<t<t,.

The only variable not known from experiment is yY:,, the intracellular voltage in the stimulated region during the excitation. We found that Vs = 1.0 V leads to a rise to approximately the same calculated maximal voltage in the recording electrode as that found in the experiment. The conduction velocity of neural excitations (see below) is independent of Vs. The experimental observation of the intracellular voltage by a microelectrode is simulated by recording V(x(‘), t) as a function of t, where x(‘) is chosen as the distance between the suction and recording electrodes in the experiment.

It is found experimentally that the time span between simulation and maximum voltage observation in the recording electrode grows linearly with the distance between stimulating and recording electrodes [27,28]. A linear relationship is also found in the simulation (see Figure 4). Thus, the velocity of the passive spread of depolarization is the same for the entire neural process. Experimentally, in Ascaris motor neurons, this velocity varies between 21 and 38 cm/s (taking into account the spread of experimental results in [14], [28], and [29]). The simulation yields 22 cm/s.

4.2. ASCARIS INTERNEURONS

It has not been possible to measure the velocity of neural excitations in interneurons of Ascaris 1. However, we can perform computer simulations of such experiments using reasonable values of the parameters

TABLE 3

Parameters of the Ascaris 1. Excitatory Motor Neurons

Process diameter D=16X10-6m Transmembrane resting conductivity g, = 0.15K’m-* Process length I=5XlO-*m


0.3

0.2

0.1

0

0 I 2 3 4 5 6

x(cm)

FIG. 4. Simulation of an excitation propagating in Arcark motor neurons. At time t = 0, the neuron is excited by a square wave of amplitude Vs = 1.0 V and duration I,. The time when the excitation (defined as the voltage maximum) is observed at a given point of the neural process is shown on the vertical axis as a function of the distance x between this point and the stimulated region. Filled circles, tS = 100 ms; open circles, t,s = 10 ms.

in Equation (A.l). We chose for simulation the largest interneurons of the ventral cord (which are analogous to classes AVA and AVB in C. .&guns). These neurons are probably responsible for the control of undulatory locomotion.

If we use the values from Table 2 for R,, C,, and V,, the only unknown parameter is the transmembrane resting conductivity g,. We investigated two cases with two different transmembrane conductivities. The reasons for the choice of these two values will be explained shortly.

Case 1. g, = 1.25 x 10-3fiP’m-2. Using this value and Equation (2a), we obtain A = 1. The upper curve of Figure 5 shows the result of the simulation of a neural excitation traveling along the interneuron for this choice of g,. Here, x represents the distance traveled by the peak of the excitation during time T. It is evident from the changing slope of the curve that the velocity of the traveling excitation is not the same for the entire interneuron. Its mean value is 2.8 cm/s.

Note the acceleration of the peak close to the end of the neuron, which is due to the sealed-end boundary condition. Close to the sealed end, the current in the forward direction is smaller than it would be in a cable of infinite length. This forces the propagating peak to have a higher value close to x = 1 and to occur earlier than it would if the pulse

68 ERNSTNIEBURANDPAULERDbS

0 2 4 6 8 10 12

x(cm)

FIG. 5. Simulation of an excitation propagating in an Ascuris interneuron. At time t = 0, the neuron is excited at the point x = 0 by a square wave of amplitude Vs = 100 mV and duration ts = 10 ms. Filled circles, g, = 1.25 X 10e3 R-’ mA2; open circles, g, = 1.25X 10e2 R-’ mm2.

could propagate further. Although the peak travels with approximately constant velocity in a cable of infinite length, it accelerates slightly close to a sealed end, as is seen in Figure 5. The same behavior is found in the analytical solution of the cable equation (E. Niebur, unpublished).

Case 2. g, =1.25X 10-*fl-‘rn- 2. Using Equation (2a), we obtain A = 0.3161, which yields approximately the same ratio 1 /A as that found in Ascaris motor neurons. The lower curve of Figure 5 shows the result of the simulation of a traveling neural excitation. As in the case of Ascaris motor neurons, the relationship between x and 1 is approximately linear. The velocity of the neural excitation is 5.5 cm/s.

The curves shown in Figure 5 can be used to predict the propagation velocity and the shape of the function x(t) for a given value of g,, or vice versa. The results are summarized in Table 4.

It is interesting to compare the results obtained by simulation with the behavior of Ascaris lumbricoides. Crofton [6] described three different types of muscular waves (i.e., alternating regions of the body where the muscles are alternately under tension and relaxed) that propagate along this nematode’s body. Two of these have velocities that are variable between 0 and 0.5 cm/s, while the waves of the third kind (“fast muscular waves”) have a constant velocity of about 6 cm/s. These fast muscular waves always propagate from the head to the tail, whereas

CONTROL OF THE LOCOMOTION OF NEMATODES

TABLE 4

Properties of Neural Excitations in Ascaris 1. Interneurons of Length I = 0.1 ma

g, (W’m-*I u (cm/s) f 7(s)

69

1.25x10-3 2.8 2.7 6 1.25 x 10-2 5.5 25 0.6

agR, transmembrane resting conductivity; V, propagation velocity; f, attenuation factor of the voltage pulse amplitude through propagation along the full length of the interneuron; r, approximate pulse transit time through the interneuron.

the other two types of waves may change direction. Crofton suspected that the fast muscular waves could be due to a neural excitation that travels along the nematode’s body with the same velocity as the muscular wave. Measurements indicate that the propagation velocity of excitations in the motor neurons is much greater than the velocity of the fast muscular waves. However, our simulations indicate that in the interneurons of the ventral cord of Ascaris the neural excitations travel with the same velocity as the fast muscular waves, assuming that the ratio A/f is the same in the interneurons as it is in the motor neurons.

Using the larger value of g, (Case 21, the excitation propagates with a speed of 5.5 cm/s. In this case, a depolarization of 50 mV in the nerve ring decays to a mere 2 mV near the posterior end of the process. In addition, the distance between the interneuron-motor neuron synapses and the neuromuscular junctions is also a multiple of A, so the 50 mV depolarization in the nerve ring leads to a depolarization of a fraction of a millivolt at the location of the neuromuscular junctions. This is certainly too small for reliable control of the muscle cells, and it is thus necessary to assume a “boosting” mechanism, which works against a too rapid loss of the amplitude as the excitation travels posteriorly along the body. At present, it seems impossible to decide whether this is realized by an action potential traveling at roughly the same speed as the electrotonic excitation or by a gradual enhancement of the transmembrane voltage due to voltage-dependent channels in the cell membrane. It can be excluded, however, that the interneurons work purely electrotonically and that the decay of the excitation is made acceptable by a small resting conductivity g,, which would lead to a less pro- nounced voltage decay along the interneuron. The reason is seen clearly in the data presented in Table 4. This choice of g, implies that neural excitations travel with a velocity of about 3 cm/s and are thus unable to control the fast muscular waves. Furthermore, this value of g, leads to


the nonrealistic reaction time T = 6 s [see Eqs. (211. Using even larger values of g, than in Case 2 leads to stronger signal decay and is therefore unrealistic.

4.3. C. ELEGANS NEURONS

To perform simulations in C. elegans neurons similar to those described above, we used the parameters shown in Table 5.

Membrane parameters were used from Table 2. Table 6 shows the propagation velocities of neural excitations as a function of g,. For both interneurons and motor neurons, g, was chosen in case (b) such that A and I are of the same order of magnitude. Since the lengths and diameters of interneurons and motor neurons are different, this implies different values for g, for these two types of neurons (see Table 5). In cases (al and (cl of Table 6, g, was chosen considerably smaller or greater, respectively, to cover the range of physiologically reasonable values.

The observed velocity of muscular waves in C. elegans is less than 0.2 cm/s [171. Since the velocities of the neural excitations as found by our simulation (Table 6) all exceed 8 cm/s, it follows that one cannot explain the propagation of muscular waves in C. elegans as simply as the fast muscular waves in Ascaris 1. Indeed, to obtain a velocity as small as 0.2 cm/s in the calculation, a transmembrane conductivity as small as 2.25 X lO^“R-’ mm2 would be required. This would lead to a characteristic time of T = 3000 s. In another publication [19] we argue that the propagation of muscular waves in C. elegans can be explained by a mechanism involving stretch receptors.

4.4. SIMULATION OF INTERNEURONAL CONTROL OF MOTOR NEURONS

In Sections 2.2, we developed a model of the C. elegans motor nervous system. According to this model, the system is composed of two

TABLE 5

Sizes of C. elegans Neuron?

Interneuron diameter D = lo-’ m Interneuron length I= 10e3 m Motor neuron diameter D = 0.33 X 10-O m Motor neuron length I=OSXlO~” m

‘Average values from [31].


TABLE 6

Signal Propagation Velocity in C. eleguns Neurons for Different Resting Conductivities gRa

g, (R-‘m+) v (cm/s)

lnterneuron (a) (b) (c) Motor neuron (a) (b) (c)

0.05 8 0.5 16 5 31

0.25 8 1 13

10 42

“See Section 4.3 for details.

circuits:

Circuit A consists of the neurons of classes AVA, AS, DA, and VA, which controls backward movement,

Circuit B consists of the neurons of classes AVB, DB, and VB, which controls forward movement.

The inhibitory neurons of classes DD and VD are used by both circuits. The morphology of these neurons and the results of electrophysiological experiments in Ascaris suggest that they function as cross-inhibitors, inhibiting ventral muscle action when dorsal muscle is excited and vice versa. Possibly, the function of DD and VD neurons is to restrict muscle excitation to those regions of the functional syncytium in which the neuromuscular junctions of the excitatory somatic motor neurons are excited (see [24] for details and experimental evidence).

In the following we want to answer the question of whether the neurons of the nerve ring are capable of controlling the somatic motor neurons by means of the interneurons of the ventral cord. We assume that the function of classes DD and VD is as described above. This allows us to perform a computer simulation of circuit B without including the inhibitory neurons DD and VD. The data used are listed in Table 7.

Membrane parameters from Table 2 were used. Neurons of classes AVB, DB, and VB do not show any branching, with the exception of minor ramifications in the anterior parts of VBl and VB2. These ramifications were neglected in the simulations, which made it possible to represent each neuron by one process. Since only the part of the reconstruction data covering the anterior half of the animal is at our


TABLE 7

Parameters for the Various Classes of C. eleguns Neuronsa

Diameter of interneurons of class AVB D = 10m6 m Resting conductivity of interneurons of

class AVB g, = 0.5 R-‘mm2 Length of interneurons of class AVB 1=10m3 m Resting conductivity of motor neurons

of classes DB and VB g, = 1 R-‘mm2 Diameter of motor neurons of classes

DB and VB D = 0.33 x 10m6 m Length of motor neurons of classes

DB and VB Different for each neuron; taken from electron microscopic datab

aSizes are averages of measured values. The values of g, are calculated from A = I (see Section 4.3).b J. White, personal communication.

disposal (the series N2U; see [31]), data for the posterior half were extrapolated from the structure of the anterior half.

Only electrical synapses have been found between AVB neurons and those of classes DB and VB. For the simulation, the position and length of each of these synapses were obtained from the reconstruction data.

Each AVB neuron is postsynaptic to about 100 chemical synapses in the nerve ring. According to the model developed earlier, the worm’s decision to go forward leads to the excitation of the anterior part of the AVB neurons. This excitation must be generated via some or all of these synapses. These synapses are probably also used for other functions, so it is unlikely that all of them are excited at the same time or that all are excitatory. On the other hand, we presume that the control of the motor neurons is the most important task of these interneurons. It can then be expected that a large fraction of the available synapses will be used for this task. We assumed, somewhat arbitrarily, that this fraction is one-half of the synapses, which means that the decision to move forward leads to the activation of 50 excitatory synapses. The consequences of other assumptions concerning the number of activated synapses are discussed in Section 5.

Each synapse is simulated as described in Section 3, where the function G(V) in Equation (A.21 is chosen as

if the synapse is activated,

otherwise. (4)


We assume that all synapses between the nerve ring and the interneurons are located in the first compartment of the latter.

The nerve ring can control the somatomuscular motor neurons only if it produces a sufficiently great depolarization in them. It has been shown experimentally in Ascaris 1. that a depolarization of a few millivolts is needed for the release of neurotransmitter [241. Therefore we have to find out whether the excitation of the AVB neurons in the nerve ring leads to a voltage rise of some millivolts in the neuromuscular junction region of all somatic motor neurons. If this is so, control by the nerve ring is possible.

The answer obtained by solving the partial differential equations is shown in Figure 6. It is evident from this diagram that the excitation in the nerve ring leads first to an excitation of the interneuron fibers and then to the excitation of the somatic motor neurons. It is also clear from the figure that after 50 ms (which is a short time at the scale of the motion of C. eleguns) all somatic motor neurons are excited to at least 3 mV above the resting potential.

If the voltage distribution in the neurons is analyzed in more detail (see [16]), it turns out to be very heterogeneous. The neuromuscular junction region in VBl is excited by 33 mV above resting potential, while the corresponding excitation of DB7 is only 3 mV. The reason for this large difference is most likely attenuation along the electrotonic neural processes. Although different amounts of depolarization may be compensated for by the neuromuscular junctions if they are more sensitive to depolarization in the posterior parts of the body than in the anterior parts, excitation differences of a whole order of magnitude (3 mV vs. 33 mV) seem improbable. However, it can be shown by our calculations [16] that a much more homogeneous excitation pattern, in which the minimal and maximal depolarizations differ only by a factor of two, would result if the transmembrane conductivities were higher in the VB neurons (gR = lK’m_‘) and lower in the DB neurons (gR = 0.2552-l m-‘1 and in the AVB neurons (gR = 0.05K’m-2) than assumed previously.

5. DISCUSSION

5.1. C. ELEGANS

We have shown that the motor nervous system of C. elegans may function entirely electrotonically. Excitation of the interneurons AVB in the nerve ring yields a reasonable excitation pattern under the simplifying assumptions listed in Section 2.6. One of these assumptions is that the input to the interneuron is provided by 50 equal synapses, each modeled by Equation (4) (see Section 4.3). We chose 50 synapses


because the number of anatomically defined synapses between neurons in the nerve ring and the interneurons is about 100 [31] and we assumed that an appreciable part of this pool of available synapses would be used for locomotion, which is the only known function of the ventral cord interneurons. Other, as yet unknown, functions of the interneurons might exist; furthermore, some of the synapses might be inhibitory. Therefore, it seemed prudent to base the model on the assumption that about half of the available neurons are necessary for the simulated function.

We believe that our results do not substantially depend on this assumption for the following reasons. It is clear that no sufficient excitation will be achieved in the motor neuron if the number of

I .: : : : . . . . . . . . . . . ::::::::::“““”

.*.:_*:: ::: :;*:: ...... !I rT*:::::*-** *

. *

1 I

23 45 6 7

8 9 10 11

1 . . . . . . . :r:. . . . . . . . . . .p ,llll\ljjiiiilllt:~: . . ..r.!!!ii!!il!!Ii!!!iii”’

DB

VB

AVB

t = 0.1 ms

(a)

FIG. 6. Calculated electric potential distribution after excitation at time t = 0 by a signal from the nerve ring in the head of C. eZeguns in the motor neurons VBl-11 and DBl-7 and in the interneurons AVBL and AVBR. The number of each neuron is printed above its head-side (left) end. Full lines indicate neural regions where the potential exceeds -32 mV (“excited”); dotted lines represents regions of lesser depolarization (“nonexcited”). The curved arcs are the commissures. (a) t = 0.1 ms. Only some parts of the interneurons AVBL and AVBR and of the somatomuscular motor neurons VBl, VB2, DBl, and DB2 are excited. (b) t = 5 ms. The excitation has completely invaded the somatic motor neurons VBl-VB6 and partially invaded VB7-VBll and DBl-DB7 as well as interneurons AVBL and AVBR. (cl t = 50 ms. The intracellular potential exceeds - 32 mV everywhere.

CONTROL OF THE LOCOMOTION OF NEMATODES 7.5

456 7

23 fj g 10 11 VB 1 . . . .

~iiiiiiii;i:::

R

I AVB

L

t = 5.0 ms

(b)

DB

23 456 7 * g 10 11 VB

1

IR 1 AVB L II

t = 50.0 ms

cc)

FIG. 6. (Continued)

76 ERNST NIEBUR AND PAUL ERDdS

activated synapses (or the conductivity change per activated synapse) is too small. The point at which the excitation will no longer be sufficient depends on the depolarization level necessary for the reliable activation of the muscle cells.

The relationship between the number of active synapses and the depolarization is nonlinear [owing to voltage saturation; see Eq. (A.111 with a decreasing slope. This adds robustness to the system, insofar as the depolarization varies less than proportionally with the number of synapses.

Since our results show that the use of action potentials is not necessary even in the longest neurons of C. elegans, which are the interneurons of the ventral cord, we suggest that the nervous system of C. elegans uses only electrotonic potentials. If true, this situation renders the operation of the nervous system simple, since in this case it is not necessary to code and decode the intracellular voltage with the frequency of action potentials.

Our study implies that the motor nervous system of other nematodes of a size comparable to that of C. elegans may also function entirely electrotonically.

Another result of this study is that one has to reject the hypothesis that excitation waves traveling in the interneurons may explain the generation of muscular waves in C. elegans, because the propagation along the nerve fiber of the excitation in C. elegans is much faster than the fastest observed postural wave. (This is in contrast to the results obtained for Ascaris l., where the propagation velocity of a traveling excitation in an Ascaris interneuron coincides with that of the observed postural wave.) It is still possible that a traveling neural wave leads to a traveling postural wave (e.g., during swimming), which is for some (neural, mechanical, hydrodynamic, or other) reason slowed down considerably. We can conclude, however, that the simple and plausible explanation we found for Ascaris does not work in C. elegans.

We then discussed a mechanism that may produce traveling muscle excitation waves in C. elegans. This mechanism is based on an idea of R. Russell (personal communication) that there are stretch receptors under the cuticle of the animal. Experimental evidence that would show that the ends of the excitatory motor neurons function as stretch receptors is still lacking. However, assuming that control is exercised by stretch receptors, we explain anatomical features, in particular the length of certain processes and the associated position of the putative stretch receptors in the body, that have previously been unclear.

5.2. ASCARIS LUMBh’ICOIDES

The fastest observed muscular wave in Ascaris moves with a velocity of about 6 cm/s. Strikingly, this is about the value we obtain by


simulation for the velocity of a neural excitation traveling in the larger interneurons of the ventral cord (see Table 4). It is therefore possible that the fast muscular wave is due to a pulse of excitation traveling at the same speed in the interneurons.

Since our calculations predict an attenuation by a factor of 25 over a distance of 10 cm for a wave with this velocity, one may conclude that this type of electrotonic excitation alone cannot ensure the undulatory locomotion of Ascaris. This leads to the tentative conclusion that in Ascaris some signaling mechanism other than electrotonic propagation must be present to switch the motion on and off over the length of the body.

All the above conclusions are based on results obtained using specific model assumptions and should be tested experimentally. For instance, our prediction of the velocity of the traveling excitation in the interneuron in Ascaris could be tested by using an Ascarii preparation containing the ventral cord and at least two commissural motor neurons, which are separated by some distance along the body. Cutting the body completely between the motor neurons except for the ventral cord will make any signal propagation impossible except in the ventral cord. Stimulating the ventral cord by a suction electrode in one part of the body and recording intracellularly in the commissures of at least two motor neurons belonging to the same class should allow a measurement of the propagation velocity of excitations in the interneurons of the ventral cord. This propagation velocity is an interesting parameter in itself; moreover, knowledge of it would approximately determine other interneuron membrane parameters, in particular, g,, as well.

5.3. SUMMARY OF THE RESULTS

In this work we undertook the first detailed simulation of a substantial part of the complete nervous system of an organism-the motor nervous system of the nematode Caenorhabditis elegans. The complete neural circuit diagram of this nematode is known. This knowledge is complemented by data from intracellular recordings in identified neurons in the related nematode Ascaris lumbricoides.

We believe that the model of the motor nervous system of nematodes that we developed will be useful for further work and that the establish- ment of this model is an interesting result in itself. Other results presented in this paper include the following:

Theoretical study of a mechanism of the rapidly traveling muscular waves in Ascaris. This mechanism involves excitation waves traveling in the interneurons. We propose an experiment to test the underlying hypothesis.

78 ERNST NIEBUR AND PAUL ERDijS

Exclusion of the possibility that excitation waves traveling in the interneurons generate the muscular waves in C. elegans.

Demonstration that the complete nervous system of C. elegans might work electrotonically. As far as we are aware, such a possibility had not been suggested previously for any organism.

Demonstration that the nervous system of Ascaris cannot work purely electrotonically.

Provision of theoretical evidence that a special kind of electrical excitation in an Ascaris preparation travels in the motor neurons rather than in the syncytium. This is an experimentally unresolved question.

Discussion of the mechanism that produces traveling muscle excitation waves in nematodes. Taking up the idea proposed by R. Russell that control is exercised by stretch receptors, we can explain anatomical features, in particular, the position of the putative stretch receptors in the body. In a previous paper we showed by computer simulations that this mechanism does lead to active propulsion [20].

Under suitable experimental conditions, a second type of muscular wave is observed to travel in the body of Ascaris at a velocity of about 12 cm/s [14,28]. The authors of [14] and [28] concluded that this slower excitation travels in the functional syncytium formed by the nematode muscle cells, which are closely coupled by gap junctions. Earlier, Weisblat and Russell [29] also found two kinds of excitations traveling at different velocities (16 and 21 cm/s) and assumed, in contrast to the previously cited authors, that the faster excitation travels in the syncytium and the slower one in the neural fibers. Walrond and Stretton [28] provided experimental evidence for the opposite point of view (see Figure 1 and Table 1 of [27]). The result of our simulation, which showed that excitations in Ascaris motor neurons travel with a velocity of more than 20 cm/s, supports the conclusion of Stretlon and co- workers, which is that fast excitation travels in the neural processes.

We are grateful to Drs. R. E. Davis, A. 0. W Stretton, and J. G. White for the communication of unpublished material. Our thanks are due to the Swiss National Science Foundation for financial support through grants 2000-5.295 and 20-28846.90.

APPENDIX

A.l. SYNAPSES

The model neurons are presumed to interact by means of electrical and chemical synapses; all other (such as hormonal or ephaptic) interactions are neglected. Synapses introduce coupling terms between those partial differential equations that represent the synaptic partners.


Synapses are small (length = 1O-6 m) compared to neural processes

(length = lop3 m in C. elegans and = 10-l m in Ascaris 1.). We may

thus assume that the presynaptic part of the kth excitatory synapse whose presynaptic partner is neuron i and whose postsynaptic partner is neuron j, is at a point we denote by x(ijk). We cannot, however, make the assumption that electrical synapses or the postsynaptic parts of chemical synapses are pointlike without introducing infinite transmembrane conductivities. Therefore we denote by e(ijk) the interval that represents the length of the postsynaptic area of the kth synapse leading from neuron i to neuron j. We assume that the synaptic reversal potentials are the same for all excitatory synapses in the nervous system under study and that VR < V,. The quantity of the neurotransmitter substance released and the ensuing change of the postsynaptic transmembrane conductivity depend on the presynaptic voltage.

If e(ijm) is that interval of neuron i to which the mth electrical synapse of the connection between neuron i and neuron j leads, then e(jim) is the corresponding part of neuron j. Since the two latter intervals stand for immediately adjacent physical regions, every point in e(ijm> is very close to a point in e(jim). In the equations of Section A.2, these adjacent points may be considered to have the same x coordinate, as measured along the neurons from an arbitrarily chosen origin.

Our model of an electrical synapse consists of an ohmic conductivity g connecting the cytoplasms of the synaptic partners. This model is in accordance with experimentally determined properties of electrical synapses [ 11.

A.2. NEURAL NETWORK EQUATIONS

Combining our models for neural processes and synapses, we can formulate the equations that represent the nervous system under study. The mathematical model of this system consists of the following set of M coupled partial differential equations:

iE{l,..., M}. (A.1)


The left-hand side of this equation describes the behavior of a neural process in the absence of synaptic interactions [compare with Eq. Cl)], while the right-hand side contains all synaptic terms. There, the first term inside the curly brackets stands for the excitatory chemical synapses, in which neuron i is postsynaptic. According to the model described above, these chemical synapses are represented by the transmembrane conductivities g(ijk, X, t), which are defined by

g(ijk,x,t) = i

G(WJ)> if x’e(ijk),

0 otherwise. (A-2)

The function G describes the dependency of the posysynaptic transmembrane conductivity on the presynaptic voltage. In general, G is expected to be a sigmoidal function of its argument (see [17]).

The second term in the curly brackets of Equation (A.l) stands for the electrical synapses between neuron i and all other neurons, where g’(ijm, X, t) is defined by

if xEe(ijk),

otherwise. (A-3)

To solve this set of equations subject to the boundary conditions (3), we choose the initial conditions

y( x, t = 0) = V, for all points of all neurons i = 1,. . . , M. (A.4)

Physiologically, this means that there is no activity in the neural network at t = 0.

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82 ERNST NIEBUR AND PAUL ERDGS

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theory of the locomotion of nematodes: control of the somatic motor neurons by interneurons

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