One of the biggest challenges in unravelling the complexity of living systems, is to fully understand the neural logic that translates sensory input into the highly nonlinear motor outputs that are observed when simple organisms crawl. Recent work has shown that organisms such as larvae that exhibit klinotaxis (i.e., orientation through lateral movements of portions of the body) can perform normal exploratory practices even in the absence of a brain. Abdominal and thoracic networks control the alternation between crawls and turns. This motivates the search for
Biological systems, as well as many economic and social systems, are characterized by highly nonlinear, complex dynamics which reflect the high degree of adaptability that they possess to the ever-changing environment that surrounds them. Such complex adaptive systems are often modeled as large collections of interacting agents that evolve in time to produce a complicated interplay between deterministic and stochastic outputs [
In bacteria, a goal-directed behavior is realized through biased random walks where paths are extended in the direction of the stimulus (klinokinesis) [
Carefully designed anatomical models of larvae following a combination of weathervaning and head casts have been proposed to explain several features of the organisms’ behavior [
Here, we present such a decentralized model that regulates the direction of motion of a system (e.g., larva) in search of a specific target, and as a result produces nonlinear motion that is consistent with that observed empirically. We stress that our model is not unique, and is purposely minimal in that it lacks a wealth of known biological details. Hence, it should be seen very much as a possible prototype. However, its value comes from the fact that, despite its minimal structure, it does produce nontrivial nonlinear behaviors, which are consistent with the observed empirical measurements. This in turn suggests that our minimal model may indeed be capturing some core principles (e.g., reward/penalty mechanism), albeit in a very crude way, which may eventually become generalizable to other organisms and systems in the future. The nonlinear output trajectories produced by our model share some interesting commonalities with those observed in klinotaxis dynamics. In addition to using temporal sampling to regulate turns, our analysis shows that the relationship between the turning rate and the optimal direction is akin to that found in the chemotaxis of organisms such as
Consider a system comprising
Initially, the system is located at an arbitrary position away from the target, and pointing in an arbitrary initial direction. Suppose that at the end of the timestep
(a) Parametrization of the trajectory of a moving system advancing towards a well-defined target. The velocity vector
Figure
Given the binary nature of the action, there are
To determine whether an individual action is good or not, we look at the change it produces to the direction of the system at time
It is important to understand the correlations among the different strategies, which in turn affect the system behavior in the model. The root of these correlations lies in the specifics of the strategy space for a given value of
Sample trajectories from our model are shown in Figure
(a) Sample trajectories from our navigation model moving from a specific point on the
For
Figure
A navigation strategy known as proportional navigation has the objective of maintaining the line of sight (LOS) angle fixed while the system moves towards the target [
(a) System reorientation as it moves toward the target. The angle
Figure
We find that the sinusoidal pattern is robust for a wide range of choices of the number of agents
A comparable turning rate pattern is found for the nematode
(a) Crawling chemotaxis trajectories from
Though we are certainly not arguing that the neural circuit responsible for the motion in these organisms follows the same mechanism as our model, it is curious and suggestive that there are nontrivial commonalities about the resulting nonlinear motion that could provide new insights into how these neural circuits connect the sensory and motor neurons. This becomes particularly interesting for light avoidance klinotaxis in
Our model can indeed be extended to account for more complex environments beyond a single target point, for example, considering a gradually (or temporally) changing signal often used to model chemotaxis. Indeed, considering a fixed-point target is a reasonable strong first step that accounts for basic but illustrative environments in phototaxis as well as chemotaxis. In addition, for a very large initial system-target separation, compared to a typical trajectory length, our current model could also be used to study one-dimensional thermotaxis.
Our work contributes to the undergoing transition of the biological sciences to a more quantitative subject and where nonlinear approximations can play a central role [
In summary, we have presented a decentralized multiagent model that, by using a decision-making mechanism based on strategies and scores, drives the trajectory of a system towards a specific target. Our model is scalable to any number of agents
The views and conclusions contained herein are solely those of the authors and do not represent official policies or endorsements by any of the entities named in this paper.
The authors declare that they have no conflicts of interest.
Neil Johnson gratefully acknowledges funding under National Science Foundation (NSF) Grant no. CNS 1522693 and Air Force Office of Scientific Research (AFOSR) Grant no. FA9550-16-1-0247. Pak Ming Hui acknowledges the support of a Direct Grant for Research in 2017–2018 from the Faculty of Science at the Chinese University of Hong Kong.