A minireview of modelling in Drosophila and related systems

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Sunday 19 September 2021

Invertebrates including crayfish, lobsters and fruit fly larvae have been used to characterise the mechanisms of central pattern generators (CPGs), semi-autonomous oscillatory neural networks whose intrinsic activity outputs rhythmic motor patterns, including walking, swimming and crawling, in the absence of external time-coded stimulation [1][2]. In contrast to models of CPGs in limbed terrestrial vertebrates like cats and rats, or swimming models of lampreys and tadpoles, characterised by reciprocally inhibitory ‘half-centres’ that generate alternate bursts of left-right activity, D. melanogaster larvae exhibit a range of apparently antagonistic behaviours including synchronous peristaltic waves and asymmetric lateral head sweeps.

Larval behaviours Larvae can crawl at different velocities via synchronous waves of peristalsis that propagate predominantly in a posterior-anterior direction, resulting in coordinated waves of segmental muscle contraction and latent relaxation that enables the larva to move forwards; during goal-directed movement, crawling can become highly stereotyped and result in highly consistent waves of contraction [3]. Therefore, successful models of larval locomotion should consider different rates of movement, under different conditions. In contrast, larvae can also perform asynchronous lateral head sweeps when surveying the environment in response to chemotactic cues [4]. No single model has yet captured all the stereotyped behaviours of larvae, or fully captured how the CPG network located in the ventral nerve ganglia contribute to behaviours, despite evidence that segmentally controlled CPG networks can reproduce a spectrum of different motor patterns, without involvement of higher brain centres.

Various models on the mechanisms behind this alternation of motor programmes have been proposed, including whether there be a component of motor pattern selection involved in choosing between behaviours [5], simultaneous co-existence of both oscillatory behaviours that are facilitated or inhibited by sensory information [6], or as a biomechanical by-product of asymmetric forces between the larva and its environment [7], and statistical models of unbiased Brownian motion that determine probabilistic rates of head sweeps and peristaltic waves affected by sensory input, where larvae retain a basic memory of recent turns (Table 1) [4][8].

Biomechanical and rate-based models

The biomechanical approach consists of taking the structural properties and physical constraints of a living system to model the motion of the system. This approach is advantageous in allowing researchers to apply the same analytical techniques in engineering as to soft-bodied larvae. For instance, J. Loveless et. al (2019) designed a model to explain quantitatively axial propagation and transverse head sweeps, consisting of physical constraints and a neuromuscular model comprising segmental reflex circuits and far-acting reciprocally inhibitory connections across distant segments [7]. By exploring the parameter space of the model, forward peristaltic waves and chaotic lateral head sweeps could be reproduced. They concluded that, since their neuromuscular circuit lacked any intrinsic pattern generator, the alternation between motor programs must arise at the scale of larva-substrate interactions and asymmetries in the frictional force along the longitudinal axis of the larva.

An alternative method involved an analysis of traces of fictive locomotion in an isolated VNC and modelling the CPG using segmentally interconnected Wilson-Cowan (WC) compartments containing excitatory (E) and inhibitory (I) populations [9], capturing the mean-field rates of E and I firing, E_j in unit time τ. Despite concerns on how such a model could be validated with insufficient information about the connections and properties of interneurons in the CPG network [10], this insightful approach can still qualitatively reproduce many aspects of peristaltic locomotion using the same network with different stimulation amplitudes. A future approach might attempt to combine this model with motor neuron and muscle activity, or even combine the findings of the neuromechanical models.

Kuramoto and Winfree oscillator models

Cycles pervade at every scale in the universe, entering spontaneous states of synchrony without conscious effort [11], whether in swathes of Malaysian fireflies flashing in unison or the phase-locked rhythmic firing of neurons in the sinoatrial node. Particularly in the past two decades, the principle of ‘phase oscillators’ and spontaneous order have been applied to neural networks [12]–[19]. This description of neurons as oscillators lies in the Goldilocks zone of abstraction; it enables a simple but representative model of neurons via a single variable, phase, without the burden of simulating the properties of all the currents of the neurons involved in the oscillatory neural network (fig 2) [22][23].

This oscillator reduction can be applied to different single-neuron models, including Hodgkin-Huxley neurons, as used in modelling the leech heartbeat network [12], comprising a timing network of four pairs of heart interneurons [12]; leaky integrate-and-fire (LIF); and on larger population-wide scales [17][21]. By determining the limit-cycle of a given neuron, the low-level resistor-capacitance properties of individual neurons can be subsumed into the model by assuming that the neuron fires periodically and that synaptic connections among neurons only result in phase shifts (timing of spikes) of their neighbouring neurons [19][23].

Consequently, this type of analysis is well suited to multi-neuron rhythmic oscillatory networks like those involved in CPGs where synchronised firing underlies the regularity of different forms of locomotion and can be subjected to well established techniques of graph theory to analyse connectivity between nodes. In crayfish swimmerets where swimming is mediated by rhythmic motor neuron bursts [22], the CPG networks were modelled as a series of segmentally coupled half-centre oscillators with long-range connections where each segment could independently generate an oscillatory rhythm [12][22]. In Drosophila larvae, unlike the leech heartbeat model, the 250 bilaterally arranged pairs of excitatory (cholinergic) and inhibitory (GABAergic or glutamatergic) interneurons lie in the VNC [23], perhaps make single-neuron representation more difficult. The alternative approach by A. Wystrach et al. (2016) uses a similar approach to Cohen et al. (1992), who modelled lamprey swimming with a series of segmental population-wide oscillators, in studying Drosophila larval taxis and head-turn oscillation rate when placed on an odour gradient using a single left-right oscillator.

Table 1. Summary of different model approaches used within and outside the Drosophila community. Key models have been included in the references.

Overall, current Drosophila models are predominantly high-level, with varying levels of abstraction, ranging from the biomechanical, considering larvae as mechanistic catenaries consisting of repeat units of muscle motors and simple neural reflex circuits [8][11] to segmentally arranged oscillators representing the underlying CPG networks. Potentially, future models of Drosophila larval behaviour could take the bottom-top approach by using a medium-level representation of the segmental CPG network as nodes of key neuronal population oscillators, emulating the method of [12] in the Drosophila community. With increasing knowledge of the CPG circuitry through connectomics and in combination with powerful neuroevolutionary simulations [27]–[29], a holistic model of larval movement may be the next stop aboard the locomotive.

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