**by Juan Luis Cabrera ^{1,2} & Esther Desiree Gutiérrez^{2,3}**

^{1}Dep. de Física Aplicada, ETSIAE, UPM, Madrid, Spain;^{2}Lab. de Dinámica Estocástica, Centro de Física, IVIC, Caracas, Venezuela;^{3}Fac. Ciencias Nat y Mat, ESPL, Guayaquil, EcuadorEmail us!

The movement of many animals follow Lévy patterns.

The possible endogenous neuronal dynamics that generates this behaviour is unknown.

In this work we show the novel discovery of multifractality in winnerless competition (WLC) systems and how it reveals an encoding mechanism able to generate two dimensional superdiffusive Lévy movements from models and real data.

A superdiffusive Lévy movement is characterised by a Hurst exponent H>1/2 and a power law distribution of steps. It has been shown that a superdiffusive Lévy flight is an optimal search strategy.

Superdifusion of organisms was predicted in 1986 and is also observed in apparently unrelated problems as the one of a finger chasing the top of a stick balanced on a fingertip.

Examples of proposed mechanisms about the internal dynamics include anomalous diffusion as a result of stochastic time delayed dynamics in the nervous system (1), adaptive memory losses of previous behaviour (2), or suppression of the scale-free power law behaviour of fruit flies by blocking of synapses in the motor cortex (3). Here we explore the possibility that WLC may provide a framework to explain superdiffusive Lévy movements in animals. WLC has been proposed as a paradigm for information flow in the neural context remarking the important role played by transient dynamics in complex systems. It is know that May-Leonard systems are archetypical dynamical systems exhibiting WLC. In the present work we used results obtained from the study of the discrete Lotka-Volterra which allows for the analytical deduction of the behaviour of the times of residence on each fixed point participating in the heteroclinic circulation. We showed that residence times may exhibit different regimes, i.e., they can display situations of constant increments producing constant differences between residence times and the situation in which that constant equals to zero. This behaviour seems a to us perfect setup to establish a coding schema. We exemplify this idea building two schemes based on a simple recipe: 1) determine the residence time by calculating the difference between the time entering a fixed point and exiting its orbit. Note this is simply the width of a peak in the temporal series. 2) with this set of residence times a new series is built and now 3) from this we can calculate the difference between two residence times separated a distance D. A first scheme (scheme I) is obtained building a new temporal series with the difference obtained with a fixed value of D and a second scheme (scheme II) is obtained building the new temporal series with increasing values of D in the set (D, 2D, …, nD) . While the first scheme is useful to explain the basic idea the second scheme produces larger temporal series and is the one we use in the following. The figure at the right showing three panels corresponds to the temporal series of the residence times (top), the temporal series for the difference between residence times obtained with scheme II and the power spectra for that temporal series which was obtained for a particular value of the control parameter r in the Lotka-Volterra map above. When we calculate this same temporal series over different values of the control parameter r satisfying the necessary and sufficient conditions for the appearance of WLC the power spectra display the 3d behavior depicted in the blue-green surfaces. The one at the right is a zoom of the one at the left. Exhaustive inspection of these figures at different scales (not shown here) reveals some sort of imperfect self-similarity, i.e., not fully achieved.

_{(1), Cressoni, J.C.,daSilva, M.A.A. & Viswanathan,G.M. Amnestically induced persistence in random walks. Phys. Rev. Lett. 98, 070603 (2007).(2) Bartumeus, F. & Levin, S. A. Fractal reorientation clocks: Linking animal behavior to statistical patterns of search. Proc. Natl. Acad. Sci. USA 105, 19072–19077 (2008).(3) Martin, J. R., Faure, P. & Ernst, R. The power law distribution for walking-time intervals correlates with the ellipsoid-body in Drosophila. J. Neurogenet. 15, 205–219 (2001).}

This self-similarity aspect is better analysed representing on a plane (as a black dot) those values of the frequency (f) and the control parameter (r) where the power spectra vanishes For both schemes this representation shows an intricate pattern which when submitted to a multifractal analysis reveals a singularity spectrum (4) characteristic of multifractality for the pattern generated by the zero power values in the power spectra. Under this ground it is just straight forward to fix a value of the control parameter r to extract a fractal dust as the one we show at the left of this page. We have artificially coloured the points on it filling them with a cyan pattern surrounded by a red line on a yellow background. But please remember this are a set of black dots on the (r,f) plane for r fixed. It is easy to see that this fractal dust induces a superdiffusive 2-dimensional random walk. It can be directly done calculation the distance between the set of points forming the fractal and with this distance the (x,y) coordinates on the plane taking the orientation angle from a uniform distribution. The resulting walk describe the usual characteristics of successive small spatial excursions alternated with large jumps, as expected for a Lévy walk. Its mean square displacement is is characterised by a Hurst exponent H > 1/2 indicating superdiffusion as evaluated by two methods: direct determination and the generalised H one. We also analysed the case of bursting (chaotic) saddles in L-V model and a conductance model showing WLC showing similar results (not shown here).

Next we show that a superdiffusive Lévy random walk can be decoded from a real searching task studying the temporal series from multichannel simultaneous recordings made from layer CA1 of the right dorsal hippocampus of three Long-Evans rats during open field tasks in which the animals chased randomly placed drops of water or pieces of Froot Loops while on a elevated square platform. These are data obtained from the Collaborative Research in Computational Neuroscience Data sharing repository (crcns.org). For our study we used the directory ec013.527. The application of our recipe to the temporal series of this set reveals a superdiffusive Lévy walk given by a Hurst exponent H > 1/2 and power law distribution f steps with exponent in (1,3) . The analysed temporal series occupied 180 GB uncompressed and the power spectra was calculated with 2^{17} points.

_{(4) Chlabra, A. B., Meneveau, C., Jensen, R. V. & Sreenivasan, K. R. Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence. Phys. Rev. A 40, 5284–5294 (1989).}

Our study on decoding Lévy walks from experimental data was extended by analysing data from extracellular recordings from the anterior lateral motor cortex neurons related to voluntary movement in mice .(Total size of the raw voltage traces is about 715 GB and the power spectra was calculated with 2^10 points). Also from CRCNS.org. In this case the fractal dust induces a superdiffusive walk but the amount of data is not enought to allow us to determine whether the walk is Lévy. Also, from the CRCNS.org database we analysed experimental neuronal data Grasshopper auditory receptor cell finding that the obtained fractal dust induces a superdiffusive walk but, as in the mice data, the number of points is not enough to allow us to determine the Lévy character of the walk (the power spectra was calculated with 2^10 points).

The proposed mechanism provides the first plausible explanation for the neuro-dynamical fundamentals of spatial searching patterns observed in animals (including humans) and illustrates an until now unknown way to encode information in neuronal temporal series while revealing multifractal behaviour in WLC.

In 2019 Sims et al. provided the first conclusive evidence pointing out that supperdiffusive optimal searching is generated intrinsically.

*This work is published in Esther D. Gutiérrez & Juan Luis Cabrera, Scientific Reports 5:18009, DOI: 10.1038/srep18009*