Intersegmental coordination of the central pattern generator via interleaved electrical and chemical synapses in zebrafish spinal cord

Abstract

A significant component of the repetitive dynamics during locomotion in vertebrates is generated within the spinal cord. The legged locomotion of mammals is most likely controled by a hierarchical, multi-layer spinal network structure, while the axial circuitry generating the undulatory swimming motion of animals like lamprey is thought to have only a single layer in each segment. Recent experiments have suggested a hybrid network structure in zebrafish larvae in which two types of excitatory interneurons (V2a-I and V2a-II) both make first-order connections to the brain and last-order connections to the motor pool. These neurons are connected by electrical and chemical synapses across segments. Through computational modeling and an asymptotic perturbation approach we show that this interleaved interaction between the two neuron populations allows the spinal network to quickly establish the correct activation sequence of the segments when starting from random initial conditions, as needed for a swimming spurt, and to reduce the dependence of the intersegmental phase difference (ISPD) of the oscillations on the swimming frequency. The latter reduces the frequency dependence of the waveform of the swimming motion. In the model the reduced frequency dependence is largely due to the different impact of chemical and electrical synapses on the ISPD and to the significant spike-frequency adaptation that has been observed experimentally in V2a-II neurons, but not in V2a-I neurons. Our model makes experimentally testable predictions and points to a benefit of the hybrid structure for undulatory locomotion that may not be relevant for legged locomotion.

Appendix 1

1.1 Adaptation of V2a-II neuron and $\phi$ control

Fig. 12

Adaptation of V2a-II neuron and $\phi$ control. (a-c) Numerical solutions for V2a-II neurons for different input frequencies f. (a) Bistabiliy between no-spiking for low-frequency periodic input ($f=77.5$) and spiking for high-frequency input ($f=112$). Arrow marks the peak of the overshoot. Dashed grey line: spiking threshold. (b) When switching the input from high to low frequency the V2a-II stops firing (red line). (c) Switching from low to high frequency does not trigger spiking in the V2a-II (blue line). Parameters: $g_{c}^{(II)}=0.055$, $\Delta I=-0.04$, $I_{\mathrm {II}}=1.4$, $N=8$

A key feature arising from the adaptation current is the overshoot in the voltage that occurs during the recovery phase after a spike (black arrow near $t=30.5$ in Fig. 12a, b). Synaptic input from the V2a-I neuron arriving during that overshoot can drive an action potential in the V2a-II neuron (blue triangles in (Fig. 12a), even if it is too weak to elicit a full action potential at a later time (red triangle near $t=38$). Thus, inputs arriving at a higher frequency will drive V2a-II spikes and reduce the ISDP $\phi$, while low-frequency inputs of the same amplitude will not elicit any spikes (Fig. 12a). The overshoot does not arise in the relevant time window if the previous input to the neuron only depolarized it without triggering an action potential (green arrow in Fig. 12b, c). In that case, even input that arises early does not evoke an action potential (blue triangle in Fig. 12c). Thus, in this regime the V2a-II neuron exhibits significant hysteresis: once a spike is triggered, inputs with sufficiently high frequency maintain the spiking (Fig. 12b), but they do not initiate a transition to spiking. However, spiking will cease, when the input frequency becomes too low (Fig. 12a).

For the synaptic strength used in most of the paper ($g_{c}^{(II)}=0.13$) the V2a-II neurons spike over the whole range of frequencies investigated. In Fig. 11 we consider therefore weaker synaptic strengths ($g_{c}^{(II)}=0.055$) for which the spiking depends on the timing of the inputs and therefore on the frequency of the wave.

For large f the V2a-II spike and reduce the ISPD $\phi$. When the frequency is lowered below the threshold $f_{th}\approx 91.2$, the V2a-II neurons stop spiking and $\phi$ jumps up to the upper branch (blue arrow). Due to the hysteresis in the spiking transition (Fig. 12c) it is not sufficient to merely increase the frequency now above $f_{th}$ to induce spiking and have $\phi$ jump down to the lower branch. However, a brief boost in the base input $I_{\mathrm {II}}$ whenever the frequency is changed would trigger an initial spike of the V2a-II neuron. The resulting overshoot would be sufficient to induce repetitive spiking for high frequencies, but not for low frequencies (Fig. 12a). It would therefore induce a jump to the lower branch (red arrow).

1.2 Wave propagation with rectifying and persistent gap junctions

A characteristic feature of most gap junctions is that they can be depolarizing and hyperpolarizing, depending on the voltage difference across the junction. In a wave propagating along the segments this difference will be negative during a brief time when the presynaptic cell has spiked already, but the postsynaptic cell is just about to spike (cf. Fig. 3b). To assess whether this hyperpolarization can suppress the propagation of the wave we have simulated also a version of the model in which the gap-junction current is rectified, i.e. it is set to 0 when it is hyperpolarizing. Indeed, in that case waves can propagate for significantly more negative values of $\Delta I$ (Fig. 13, open symbols vs filled symbols). Moreover, while for non-rectified gap junctions increasing their strength does not substantially enhance the propagation range, that range grows linearly for rectified gap junctions.

For the fast electrical coupling no hyperpolarizing current was observed in (Menelaou & McLean, 2019). Note, however that the currents shown in their Fig. 3 were measured in response to a single action potential in the presynaptic cell and no spike in the postsynaptic cell. In such a configuration a purely depolarizing gap-junction current is obtained also in our model without employing a rectifier, if $\Delta V$ is suitably reduced.

Fig. 13

Wave propagation without V2a-II. If the gap junctions are only depolarizing, the parameter range in which waves propagate extends to more negative values of the current difference (open symbols). Whether the gap junctions are active only during action potentials (blue symbols) or all the time (red symbols) affects the parameter range for propagation only modestly

Motivated by the apparent unidirectionality of the gap junctions in (Menelaou & McLean, 2019), we assumed that they are only effective during the strong voltage deflections associated with action potentials. This feature does, however, not have substantial impact on the wave propagation and merely shifts the propagation limits upward by some amount (Fig. 13, red vs blue symbols). Note that in (Menelaou & McLean, 2019) gap-junction currents (of both signs) were measured also for small voltage deflections. However, this coupling was attributed to a slow, indirect, electrical coupling between cells (Fig. 7 in (Menelaou & McLean, 2019)) and not the fast direct coupling between V2a-I neurons discussed here.

Fig. 14

Wave propagation with V2a-II. If the gap junctions are only depolarizing, the parameter range in which waves propagate extends to more negative values of the current difference (open symbols). Whether the gap junctions are active only during action potentials (blue symbols) or all the time (red symbols) has a moderate impact on the parameter range for propagation

In the presence of chemical synapses the effect of the hyperpolarization is less pronounced (Fig. 14).