Mathematical modeling of navigational decisions based on intensity versus directionality in Drosophila larval phototaxis

Organisms use environmental cues for directed navigation. Depending on the sensory modality and complexity of the involved sensory organs, different types of information may be processed. Understanding the basic logic behind navigational decisions critically depends on the complexity of the nervous system. Due to the comparably simple organization of the nervous system of the fruit fly larva, it stands as a powerful model to study decision-making processes that underlie directed navigation. Here, we formulate a stochastic method based on biased Markov chains to model the behavioral basis of negative phototaxis. We have quantitatively measured phototaxis in response to defined sensory inputs. We find that larvae make navigational decisions by taking into account both light intensities and its spatial gradients, and our model allows us to quantify how larvae minimize their exposure to light intensity and at the same time maximize their distance to the source of light. The response to the light field is a non-linear response and saturates above an intensity threshold. Our mathematical model simulates and predicts larval behavioral dynamics only using light intensity and directionality as input parameters. Moreover, it allows us to evaluate the relative importance of these two factors governing visual navigation. The model has been validated with experimental biological data yielding insight into the strategy that larvae use to achieve their goal with respect to the navigational cue of light, paving the way for future work to study the role of the different neuronal components in this mechanism. Author Summary Navigational decision-making is a complex process during which the nervous system is able to decipher external input through molecular and cellular mechanisms to produce a spatially-coordinated behavioral output. Drosophila larvae provide an excellent model to understand these decision-making mechanisms as we can measure the behavioral output (larval navigation) in response to quantifiable external input (different light conditions). We have performed experiments to quantify larval light avoidance in order to subsequently design a mathematical model that quantitatively reproduces larval behavior. Our results allow us to characterize the relative importance of light intensity and directionality and yield insight into the neural algorithms used in the decision-making mechanism of larval phototaxis.

However, in a similar process to the equilibration pattern followed by a 2 8 2 thermodynamics system, we have found both in our simulations and in our state after some initial fluctuating steps. Therefore, these indicators converge to be taken as a free parameter that is chosen to obtain the best fit to experiments. In particular, we have found that ݊ ൌ 4 is an optimal value. Possible choices for Our mathematical model yields more targeted paths when the intensity and its  Our results show that larval navigation depends on light intensity and its 3 1 2 gradient and that larvae navigate more efficiently (larger ܰ ‫ܫ‬ ) when the light 3 1 3 intensity is higher and when the gradient of the intensity is steeper. However, 3 1 4 this non-linear behavior (Eqs (1) and (2)) saturates for intensities higher than 3 1 5 ‫ܫ‬ 2 0 W/m 2 ( Fig 1C) and for intensity gradients higher than ‫ܫ‬ ᇱ 0 . 2 W/m 2 /cm 3 1 6 ( Fig 1D). As commented above, we do not assign such a saturation behavior to 3 1 7 a lack of physical response from the larvae since the mean velocity is nearly independent of the illumination conditions, but rather to a limited capacity for 3 1 9 processing information in the underlying neural network in the larval brain. From only ones that they can process in their brains without requiring an expensive 3 2 4 memory process to record a string of magnitudes all along their paths.

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Saturation regarding the light intensity can be understood from a limited ability 3 2 6 to process the input signal of too many photons. On the one hand, the 3 2 7 experimental evidence that larvae navigate differently depending on the 3 2 8 gradient of the intensity implies that larvae must read the gradient of the field of light (‫ܫ‬Ԣ). Such an operation needs to measure the intensity in two close points and then proceed to compare them. Therefore, it involves a memory process if 3 3 1 it is to be done at two subsequent times along the larval path. Similarly, in the model, whether a transition in the Markov chain happens or not is based on variables that can be locally obtained using only the larval starting  probably linked to the general principles governing the generalized Metropolis- Hastings algorithm, which takes care of the statistical behavior in a way that is 3 5 9 known to work well for many different complex systems found in nature. The The related Metropolis-Hastings algorithm has been successfully used to such as the travelling salesman problem (17). In contrast, we remark that our into account local data (intensities and gradients in the immediate surroundings 3 6 8 of the organism) and proceed with a limited amount of neural circuitry.

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Therefore, the weights governing the simulation in Eq (3) should be considered as a local solution to the problem rather than a global one. The value of the effective ܶ in the simulations is adjusted so that the currents of ratios that match the actual production at a given temperature. On the other 3 7 7 hand, such a parameter lends itself to a biological interpretation; it controls the 3 7 8 larval probability of taking risks by either going to higher intensity regions or by getting closer to the source of light. Such a behavior is known to be a useful in order to explore their environment more efficiently. Regarding the different models that we have tried for ݂ ሺ ߙ ሻ , the functions that candidates have been highly non-linear functions. Therefore, the larval behavior decisions, which works on a non-linear function, which is a common feature to 3 9 5 neural circuits organized in layers (25). So far, the model does not take into account the larval dimensions. However, it case, we could take into account the fact that they cannot go to places already  An additional feature of larval taxis, studied for chemotaxis, is weathervaning, which is defined as miniature head-sweeping during runs resulting in curved weathervaning for larval chemotaxis to conclude that it is the least crucial 4 1 4 navigational parameter according to their model. factors are additive towards larval navigation as suggested in (12). parental flies, ensuring that they would correspond to the 3 rd larval stage (L3).

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Larvae were kept for at least 10 minutes in the dark with food before the The experimental setup consists of a 23x23 cm agarose plate where the larvae 4 5 0 can move freely ( Fig 1A). Larval movements were recorded with a Basler illuminated with red LEDs that do not influence larval behavior but enable the 4 5 5 image recollection with the camera (Fig 1A).  Fig 1A). The custom-made LabView software (32, 3) was used to record the larval intensity variation was merely due to variation of the photon flux with the 4 7 0 distance to the projector ( Fig 1B and S1 Fig 1). In the patterns "Pos", "Neg", and 4 7 1 "Tilted" used to explore directionality, an artificial modulation in the light gradient

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axis with a gradient that was about 5 times steeper (S1 Table 1 4 7 6 and S2 Table 1). "Neg" was a 1 8 0 ° rotation of "Pos"; therefore, the intensity field this case, the light gradient artificially varied along the y direction, therefore the 4 8 0 intensity field decreased along the െ ‫ݕ‬ axis (Fig 2A, S2 Fig 1C). spectra, but to exclude the red one (S1 Fig 1 and S2 Fig 1). Integrals have been The expected variation of light intensity on the plate by a uniform source of light 4 9 7 is described by assuming a steady rate of generation of photons. For the actual 4 9 8 parameters of the geometrical setup, this has the implication of an approximate 4 9 9 linear variation, which has been corroborated by measuring intensities on the 5 0 0 plate (S1 Fig 1 and S2 Fig 1). Therefore, the spatial variation of intensities has 5 0 1 been represented by a linear fit with directions. Values for these coefficients are given in S1 Table 1 and S2 Table 1. The air conditioning was turned on at 25°C during the experiments to ensure a The acquired images of the larval tracks were analyzed with the MAGAT from the videos and these data were analyzed using a custom-made software 5 1 4 written in MATLAB (34). Statistical analysis of the data was calculated using the Welch's unpaired t-test 5 1 7 to compare results with different genotypes and a regular unpaired t-test was In our simulations, we assign transition probabilities between states in the The standard deviation sets up a length scale that we adjust to the 5 3 9 observation that the larvae approximately advance a distance equivalent to the length of its body in about ten moves. Therefore, the standard space Markov chain of transition Kernel: thermodynamics system it would be the equilibrium temperature. High The experimental recording stops tracking larvae that hit the border of the 5 5 9 agarose plate. Accordingly, we introduce a similar boundary condition in our Depending on the illumination conditions, this usually happens after a few  The angular part in equation (3) has been modelled to take into account the from it ( Fig 3A).