Effects of coupling

In the integrated model, communication between DB and VB neurons is mediated by a combination of muscle inhibition, neural inhibition and physical forces of the body and environment. This coupling is responsible for maintaining the correct antiphase relation-ship between dorsal and ventral neurons. In order to investigate the effects of these three forms of coupling, I developed a stripped down version of the model that is presented in Section 8.4.4. The model has three coupling parameters ¯w_NMJ, w^V₋ and α_physical which determine the strength of the muscle inhibition, neural inhibition and physical interaction respectively. When all of these parameters are set to zero, the dorsal and ventral sides oscillate totally independently, as shown in Figure 9.8 A. I will begin by demonstrating the effect of each form of coupling independently. The effect of neural inhibition is in-tuitive and very robust. Over a wide range of values for w^V₋ (from 0.1 to 10 was tested), neural inhibition imposes an antiphase relationship (see Figure 9.8 B). For values of 1 or more, this even works in the face of an inherent frequency difference. One way neural inhibition imposes synchrony very effectively – the main difference when compared to two way inhibition is that the amplitude and offset of the dorsal and ventral oscillations are different, but this is compensated for in the full model.

The effect of physical coupling is also quite intuitive. In the real worm, the pressure of the internal fluid will tend to maintain a constant volume in the body. In my physi-cal model, this effect is approximated by the diagonal springs. The physiphysi-cal coupling in this simplified model is implemented differently (see Section 8.4.4), but has an equiva-lent effect. Whenever the sum of the dorsal and ventral lengths is greater than (less than) two, the coupling will tend to shorten (lengthen) both sides. As expected, physical cou-pling effectively imposes and antiphase relationship over a wide range of α_physical (see Figure 9.8 C). For α_physical≥ 0.2, synchrony occurs within two or three cycles, except that oscillations break down for α_physical> 1.45. However, it is not clear how strong this synchronizing effect would be in the context of the full physical model. As it turns out, there are some cases where physical coupling is sufficient to synchronise the dorsal and

ventral oscillations correctly in the full model. This will be shown in Section 10.2.

Figure 9.8: Effect of coupling in the minimal oscillator model with one DB (red) and one VB (black) neuron. A) No coupling is included, allowing the dorsal and ventral neuron to maintain their starting phase lag. B) The addition of neural inhibition (w^V₋ = 0.4) ef-ficiently imposes an anti-phase relationship. C) Physical coupling (α_physical= 0.3) also imposes an anti-phase relationship, though it does so more slowly. D) Contrary to con-ventional wisdom, muscle inhibition ( ¯w_NMJ= 0.4) tends to impose in-phase synchrony.

The last effect to investigate is that of muscle inhibition. Contralateral inhibition of muscles by D class neurons is generally believed to contribute to locomotion by imposing the required antiphase relationship [84]. But in contrast to this hypothesis, I have found that muscle inhibition actually pushes the neurons towards in phase synchrony, as shown in Figure 9.8 D. The speed at which this happens depends on the strength of the coupling.

However, for ¯w_NMJ> 0.53 the oscillations are quickly quenched. For slightly lower val-ues, oscillation continues but at significantly reduced frequency. Both of these effects are due to the fact that once the neurons are in phase, they inhibit each other while they are on, thereby reducing the total output to the “muscles”.

To explain this counter intuitive result, one must carefully consider what is happening.

Assume that the neurons are oscillating with a phase lag close to 180^o. At the instant that the dorsal neuron switches on, the ventral neuron is slightly behind schedule and has not yet switched off. Now, if the dorsal neuron was inhibiting the ventral neuron directly, the onset of the inhibition would push the ventral neuron closer to its “off” threshold, thereby causing it to switch off sooner and hence reducing the phase error. If instead the dorsal neuron inhibits the ventral muscle, the effect of the inhibition is to relax the muscle.

Superficially this looks like the correct effect – the dorsal muscle is relaxing too late, and this makes it relax sooner. But what effect does this have on the underlying neural state?

Causing the muscle to relax and therefore lengthen will increase the stretch input into the dorsal neuron, pushing it further from its “off” threshold, thereby causing it to switch off later and hence making the phase error larger. The only time this would not happen is when the neurons are already in exact antiphase. Muscle inhibition is likely to have this effect in the context of any model that relies on stretch receptors that cause depolarization in response to elongation.

Next one can ask which effect would dominate when multiple forms of coupling are present. Physical coupling and neural inhibition have the same effect, so combining them is not particularly interesting. When muscle inhibition and neural inhibition are both included, neural inhibition is dominant. Even for the maximum value of ¯w_NMJ = 0.53, weak neural inhibition (w^V₋ ≥ 0.1) is sufficient to impose a nearly antiphase relationship.

For w^V₋ = 0.5, the muscle inhibition has no detectable effect on the phase lag (see Figure 9.9 A). The interaction of muscle inhibition with physical coupling is more complex.

There are ranges of combinations for which oscillations die out, for which the muscle inhibition dominates and for which the physical coupling dominates. Examples of these are illustrated in Figure 9.9 B-D.

Figure 9.9: Interaction between multiple forms of coupling. A) With neural and muscle inhibition, the effect of neural inhibition wins. When combining physical coupling with muscle inhibition, different effects are possible. B) If physical coupling is sufficiently weak, muscle inhibition dominates, whereas if muscle inhibition is sufficiently weak (C), physical coupling dominates. D) If neither is sufficiently weak, oscillations are quenched.