Scale-invariant spatial or temporal patterns and L\'evy flight motion have
been observed in a large variety of biological systems. It has been argued that
animals in general might perform L\'evy flight motion with power law
distribution of times between two changes of the direction of motion. Here we
study the temporal behaviour of nesting gilts. The time spent by a gilt in a
given form of activity has power law probability distribution without finite
average. Further analysis reveals intermittent eruption of certain periodic
behavioural sequences which are responsible for the scaling behaviour and
indicates the existence of a critical state. We show that this behaviour is in
close analogy with temporal sequences of velocity found in turbulent flows,
where random and regular sequences alternate and form an intermittent sequence.Comment: 10 page

A Harnos

A.B Lawrence

Ben-Mizrachi

Bouchaud

Cole

G Horváth

G Vattay

Geisel

Ghashghaie

Klafter

Mandelbrot

Manneville

Mantegna

Schlesinger

Shlesinger

Viswanathan

West

English

arXiv

Harnos, A

Horvath, G

Lawrence, AB

Vattay, G

SRUC - Scotland's Rural College

Scaling and intermittency in animal behaviour

ELTE Digital Institutional Repository (EDIT)

arXiv:chao-dyn/9901009v1  12 Jan 1999Scaling and Intermittency in Animal BehaviorA. Harnos1, G. Horváth1, A. B. Lawrence2 and G. Vattay31 University of Veterinary Science, H-1078 Budapest, István u. 2, Hungary2 Genetics and Behavioural Sciences Department, The Scottish Agricultural College,Bush Estate Penicuik, EH26 0QE, UK3 Eötvös University, Department of Physics of Complex Systems,H-1117 Budapest, Pázmány Péter sétány 1/A, HungaryScale-invariant spatial or temporal patterns and Lévy flight motion have been observed in alarge variety of biological systems. It has been argued that animals in general might perform Lévyflight motion with power law distribution of times between two changes of the direction of motion.Here we study the temporal behaviour of nesting gilts. The time spent by a gilt in a given formof activity has power law probability distribution without finite average. Further analysis revealsintermittent eruption of certain periodic behavioural sequences which are responsible for the scalingbehaviour and indicates the existence of a critical state. We show that this behaviour is in closeanalogy with temporal sequences of velocity found in turbulent flows, where random and regularsequences alternate and form an intermittent sequence.Scale-invariant spatial and temporal patterns have been observed in a large variety of biological systems [1]. It hasbeen demonstrated that ants [2], Drosohyla [3] and the wandering albatross, Diomedea exulants [4] perform motionwith power law distribution of times between two changes of the direction of motion. The power law distributionof times then leads to an anomalous Lévy type diffusion in space. In the last few years an increasing interest hasbeen devoted to these superdiffusive processes in physics [5–7]. and in econophysics [8–11]. Inspite of the extensiveexperimental studies the detailed mechanism responsible for the creation of the underlying power law distributionsis not well understood. In this letter we demonstrate the first time that the power law and scaling observed in thebehaviour of certain animals is related to intermittency, a phenomenon familiar from the theory of dynamical systemsand turbulence.It is well known that non-hyperbolic dynamical systems show superdiffusive behaviour. In dissipative systemsit is caused by the trapping of trajectories in the neighborhood of marginally unstable periodic orbits [12]. Theparadigmatic system showing such behaviour is Manneville’s one dimensional map [13]xn+1 = xn + cxzn(mod1), (1)where z ≥ 1. Here the sequence xn spends long time trapped in the neighborhood of the marginally unstable periodicorbit (a fixed point) x = 0. In analytic maps typically z = 2. The invatiant density behaves as ̺(x) ∼ x1−z nearthe origin and it is not normalizable for z ≥ 2. Accordingly the distribution of times spent by the sequence near theunstable periodic orbit has a power law tail.Next we will show experimental evidence on the existence of unstable periodic patterns in animal behaviour.Members of the species Sus scrofa invest considerable time and effort into building a nest before farrowing. Ouraim was to investigate the temporal pattern of this highly motivated activity. Using time-lapse video we recorded thebehaviour of 27 gilts and analyzed the last 24 hours preceding the farrowing. The experimental subjects were LargeWhite X Landrace gilts (Cotswold Pig Development Company, Lincoln, UK). On day 109 of pregnancy they weremoved to their individual farrowing accommodations. A behavioural collection program [15] was run to take datafrom the tapes. The behaviour of gilts was classified into eight mutually exclusive categories (see Table I). Furtherdetails of the experiment are published in Ref. [16].As a first step we assigned a symbol 0,1,...,6 to the 7 different types of behaviour listed on Table I. The records ofthe 27 gilts contain approximately 24.000 symbols. Then we computed the probabilities of the occurrences of symbolicsequences formed from the symbols. This has been done by evaluating the decimal value of the base seven numbercoded by the sequence. For example the code sequence 0156 is evaluated to be 0 ∗ 1 + 1 ∗ 7 + 5 ∗ 72 + 6 ∗ 73 = 2310.On the histograms of Fig. 1a, 1b and 1c we show the probabilities of the length 3, 4 and 5 sequences respectively.One can see that certain symbol sequences occur with high probability which does not decrease with increasing symbollength significantly, while the majority of symbols have relatively low probability and the occurrence of each particularsymbol decreases with increasing length.On the histogram of Fig. 2 we ordered the symbols of lengths L = 2, 3 and 4 according to decreasing probability.One can see that the tail of the histogram is exponential. An exponential histogram of probability ordered symbols isa property of random texts and reflects the fact that most of the behavioural patterns are generated by a Markovianprocess in a purely stochastic way. On the other hand, the most probable part cannot be considered as a result of1random uncorrelated processes. On the histogram we can see that the probability of the most probable sequencesdoes not decay significantly with increasing length and stays approximately constant for L = 2, 3 and 4.We have identified these most probable behavioural sequences. For gilts kept in pens the most likely sequence isthe cyclic repetition of ”nosing floor”-”alert”-”nosing floor”-... pattern. This sequence is the consequence of the nestbuilding instinct which governs the behaviour of the animal most of the time. In crates however there is no possibilityto try to build a nest due to the limited space and the absence of straw. Gilts become frustrated and the ideal ”nosingfloor”-”alert”-”nosing floor”-... sequence is occasionally interrupted by periods of rest. On Fig. 3 we show a typicalrecord from a gilt kept in pen. We can see that the deterministic sequences of 1-6-1-... are dominant interruptedeventually by short random-like sequences of other symbols. This is very reminiscent of temporal sequences of velocityfound in turbulent flows, where random and regular sequences alternate and form an intermittent sequence.We can quantify this qualitative analogy by studying the probability of periodic sequences. On Figure 4a we showthe probability to observe the periodic sequences ”4-6-4-6-...” and ”1-4-1-4-...” as a function of the length L of thesequence. These are typical periodic symbolic sequences. To add a new symbol to an existing sequence in this caseis approximately an uncorrelated, Markovian process. The probability decays exponentially with the length. On theother hand, the probability to observe the special periodic sequence ”1-6-1-6-...” decays very slowly, according to thepower law 1/L2 for a wide range of L values until an upper cutoff L ≈ 30 is reached (Fig. 4b). This is in closeanalogy with intermittent flows, where the probability of regular velocity patterns decays according to a power law.The occurrence of the correlated sequences has a drastic effect on the probability distribution of the time the animalspends engaged in a given type of behaviour as we show next. On Fig. 5 we show the probability distribution of thesetimes. The data can be fitted very accurately with the power lawP (t) = C1(t+ t0)2, (2)where t0 = 21.3 ± 0.6 sec. The exponent 2 of this power law is in accordance with the similar power law found forthe probability of the ”1-6-1-6-...” sequences. This function is valid between some lower tl and upper tu cutoff times.Based on the available data we have not reached these and we can say that tl is less than 30 seconds and tu is morethan 2000 seconds. The distribution (2) is normalizable, however it has no first and higher moments. For examplethe average time spent in an activityt̄ =∫dttP (t) (3)does not exist. Taking into account that the validity of (2) is limited between lower and upper cutoffs the averagetime spent in an activity becomes t̄ ∼ t0 ln(tu/tl), which is a very slowly growing function of tu. However the varianceof the time spent in an activity ¯(∆t)2 ∼ t0tu and higher moments are very large making the behaviour of the animalvery unpredictable.It is important to note, that the observed power law distributions above are the same as those observable in theManneville system [13,14,12] for z = 2. This indicates that the sequence ”1-6-1-6-...” marks a marginally unstableperiodic orbit of the dynamics.As a summary we demonstrated here, that the scaling behaviour in the animal behaviour found here is not relatedto environmental factors like in case of foraging animals, where the distribution of food might be distributed in acomplex way forcing animals to follow Lévy flight patterns. The environment of gilts in pens and crates is almostabsolutely unmotivating. The source of this complex behaviour can come only from the neural system forced byhormonal stimulus due to nesting instincts. This is the first carefully examined case, where complex scaling behaviourof animals is related to the self-organization and possibly to some unstable critical state of the nervous system. Wehope, that further investigations of the behaviour of other types of animals subject to some internal hormonal pressurecan be investigated and the existence of such a critical state can be established.This work has been supported by the Hungarian Science Foundation OTKA (F17166/T17493/T25866) and theHungarian Ministry of Education.[1] West, B. J., Fractal Physiology and Chaos in Medicine (World Scientific, 1990).[2] Schlesinger, M. F. in On Growth and Form (ed. Stanley, H. E. and Ostrowsky, N.) 283 (Nijhoff, Dordrecht, 1986).2[3] Cole, B. J. Anim. Behav. 50, 1317-1324 (1995).[4] Viswanathan, G. M. et al. Nature 381, 413-415 (1996).[5] M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Nature (London) 363, 31 (1993)[6] M. F. Shlesinger, G. M. Zaslavsky and U. Frisch (Editors), Lévy Flights and Related Topics in Physics, Springer VerlagBerlin (1996)[7] J. Klafter, M. F. Shlesinger and G. Zumofen, Physics Today 49, 33 (1996)[8] B. B. Mandelbrot, Fractals and scaling in finance: discontinuity, concentration, risk Springer, New York (1997)[9] R. N. Mantegna, H. E. Stanley, Nature 376, 46-49 (1995); Nature 383, 587-588 (1996)[10] J.-P. Bouchaud, D. Sornette, M. Potters, Option pricing in the presence of extreme fluctuations in Mathematics of DerivativeSecurities, ed. MA.H. Dempster, S. R. Pliska, Cambridge University Press (1997)[11] S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, Y. Dodge, Nature 381, 767-770 (1996)[12] T. Geisel and S. Thomae, Phys. Rev. Lett 52, 1936 (1984); T. Geisel, J. Nierwetberg, A. Zacherl, Phys. Rev. Lett. 54,616 (1985)[13] P. Manneville J. Phys. (Paris) 41, 1235 (1980)[14] Ben-Mizrachi, A. Procaccia, I. Rosenberg, N. Schmidt, H. G. Schuster, Phys. Rev. A 31, 1830 (1985)[15] Deag, M. J. Keytime: a Program System for Recording and Analizing Behavioural Data. Copyright John M. Deag 1987-1993.[16] Horvath, G. and Lawrence, A. B. The effect of nest-building phase and the environment on the temporal pattern of gilts’neast-like activity to be published (1998)3Symbol Behaviour Definition0 Comfort scratching1 Alert all other behavioursnot defined here2 Nosing environment exploratory or manipulativebehaviour directed at fixedfeatures of the environmentabove floor level3 Excretion voiding of faeces or urine4 Rest lateral lying with the headlowered and not ’nosing’5 Feeding or drinking head placed inside thefeeding or drinking trough6 Nosing floor exploratory or manipulativebehaviour directed at thehorizontal surface of thefloor or at the substrate thereonTABLE I. The ethogram used to record the gilts’ preparturient activity400.020.040.060.080.10.120.140 50 100 150 200 250 300 350probability of the symbol sequencedecimal code of the symbol sequenceL=300.010.020.030.040.050.060.070.080.090 500 1000 1500 2000 2500probability of the symbol sequencedecimal code of the symbol sequenceL=4500.010.020.030.040.050.060.070 2000 4000 6000 8000 10000 12000 14000 16000 18000probability of the symbol sequencedecimal code of the symbol sequenceL=5FIG. 1. Histogram of the probabilities of the symbolic sequences coding behavioural patterns. The video recorded animalactivity has been translated into a code sequence of codes 0,1,...,6 using the classification scheme of Table I. Then the sequentialoccurrence of length 1,2,3,4 and 5 code combinations of symbols had been analyzed. The symbol sequences are coded accordingto the value of the symbol sequence as a base 7 number. For example the length 4 code sequence 6510 is represented by thenumber 0 ∗ 1 + 1 ∗ 7 + 5 ∗ 72 + 6 ∗ 73 = 2310 In this representation similar sequences get close to each other. This number isgiven on the horizontal axis and the corresponding probability is on the vertical axis and the length L = 3, 4 and 5 are shownon a,b and c respectively. We can see that the probability of high probability sequences changes very slowly with L, while therest of the probability is scattered among the increasing number of possible symbols.6-10-9-8-7-6-5-4-3-220 40 60 80 100 120 140 160 180 200log(p)symbols in descending probability orderL=2L=3L=4FIG. 2. The same histogram as on Fig. 1, however the symbols are now ordered according descending probability on thehorizontal axis and on the vertical axis the logarithm of the symbol sequence probability is shown. The tail of the histogramcan be fitted reasonably with an exponential. This is the typical behaviour of random symbol sequences generated by Markovprocesses. The head of the histogram remains almost in the same position indicating a highly correlated behaviour. These highprobability sequences are codings of the periodic sequence ”alert - nosing floor - alert -...” and its variants with ”rest”. Thesesequences are forced by the nest building instinct of the animals.701234567180 200 220 240 260symbolsdiscrete timeA typical recordFIG. 3. A typical sequence from the time series. The horizontal axis is the (discrete) time while the symbols 0 ... 6 areon the vertical axis. One can see clearly, that the ”1-6-1-...” = ”alert - nosing floor - alert -...” sequence dominates and it isinterrupted eventually by other sequences. The random-like changes between the regular ”1-6-1-6-...” parts are very similar toan intermittent time series typical in turbulence.80.0010.010.10 2 4 6 8 10 12 14 16 18 20ProbabilityLProbability of ..4646..Probability of ..1414..0.00010.0010.010.11101 10ProbabilityLProbability of ..1616..1/L^21/LFIG. 4. Probability of the periodic sequences ”1-4-1-4-...” and ”4-6-4-6-...” a,, and the probability of the periodic sequence”1-6-1-...” b, as a function of the length L of the sequence. One can see that the probability of periodic sequences decaysexponentially with the length except for the ”alert - nosing floor - alert -...” sequence. The 1/L2 power law decay is in closeanalogy with turbulent flows, where the probability of regular velocity patterns scales in a similar way.91e-0071e-0061e-0050.00010.0010.0110 100 1000 10000 100000probability densitytime in sec.measured distributionc/(t+t_0)^2FIG. 5. Probability distribution of times spent in a given activity. The whole distribution can be fitted with the formula (1),where t0 = 21.3 ± 0.6 and C = 16.6 ± 0.3. The limits of validity of this formula are beyond the dataset available. The scalingbehaviour of this distribution is a direct consequence of the neuro-hormonal behaviour of the animal. It is not a consequenceof some scaling factors in the local environment of the animal.10

Crossref

Physica A Statistical Mechanics and its Applications

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