# Lab Mice Join the Zoomers Club

In a recent post I summarized the Zoomer model, a population level version of Multi-scaled random walk (MRW). Interestingly (in fact, thrillingly!) I just discovered a recent paper on the statistical properties of movement of the B6 strain of laboratory mice (Shoji 2016), which indirectly supports an important statistical-mechanical assumption both under the zoomer concept and the MRW.

Before turning towards movement of mice, I recap a basic property of the Zoomer model, and its coherence with the complementary individual-level process.

The illustration above is copied from Gautestad and Mysterud (2005), and illustrates the zoomer principle from the individual level; i.e., from the perspective of Multi-scaled random walk (MRW). To the left: individual zoomers are in the process of relocating themselves at length L at respective spatial scales (indicated by virtual grid resolutions) during a given time period T=M*t, where t is the unit time scale and series length M is large. Consider that t is small, meaning that we study the spatial relocation process at relatively high temporal frequency [in an upcoming post I elaborate on the effect from high versus low frequency. In the meantime I refer to my book, and to Gautestad (2012)].

However, assume that the unit time increment t is still large enough to allow for a sufficiently deep hidden layer to ensure a statistical-mechanical interpretation of the set of displacements, since both MRW and the Zoomer model rely on this framework. In other words, the arrows in the left-hand “sandwich” of scale levels reflect the individuals’ position at two successive points in time, but these relocation vectors do not reveal finer-grained moves at even finer time increments. Hence, the respective displacements cannot be interpreted mechanistically at this temporal resolution. Only the set as a whole (or subsets, if M is large enough) provides important information about the underlying space use behaviour, from the statistical-mechanical perspective.

To the right, the coloured graph shows the expected distribution F(L²) of number of zooming events to respective spatial scales during T, as revealed by a sample M. The spatial scale k on the abscissa is in this case expressed by a two-dimensional area unit, u = k² ∝ L² of respective virtual grid cells, and the bin intervals are scaling log-linearly (by comparison, the grey inset shows the expected – less fat-tailed – step length distribution from a scale-specific process, and thus presented with arithmetic bin intervals).

From a statistical-mechanical perspective, F(L²) may also represent the multi-scaled redistribution in a population sample of M individuals – a statistical “ensemble” – during a given time increment t (the zoomer model’s population perspective), rather than a given individual’s redistribution pattern at frequency M = T/t during T (the MRW perspective).

Observe that the area of each of the coloured rectangles in F(L²) is constant, meaning that the individual/population on average has distributed its space use intensity “uniformly” over the actual scale range. In other words, the ordinate and abscissa values for respective bins are inversely proportional. Uniformly distributed space use intensity over a scale range – the “equal weight” condition – is satisfied by the scaling parameter’s default value β=2 in the MRW model and by the complementary constant inter-scale redistribution rate r’ = r/(m-1) in the Zoomer model.

Now, let the statistical-mechanical Zoomer/MRW framework meet the real world. Is it realistic? In other words, what empirical evidence points towards this kind of cognitively driven scale-free space use?

First, we have the steadily accumulating support for scale-free movement, which for example may be Lévy walk or MRW. I refer to Gautestad (2012) and Gautestad and Mysterud (2013) for a proposed method to differentiate between these two universality classes (the classes are explained in The scaling cube post).

Second, by Hiroto Shoji’s results on movement by lab mice in a homogeneous field we now have an additional and complementary empirical zoomer support.

Shoji (2016), studied scale-free movement from the perspective of time scale rather than spatial scale. He explored apparatus size on rodent locomotor activity (the image to the right is copied from his paper), specifically with respect to how resting and walking periods are interwoven. When the diameter of the test apparatus was greater than 75 cm (and thus allowing for less influence from area constraint), the duration of both the resting and moving periods both obeyed bounded power-law distribution functions (like truncated Lévy walk).

Shoji’s aim was not to explore specific behavioural traits of movement from the present context of the zoomer/MRW framework, but incidentally it seems like his results have broader implications.

In particular, the scale-free “resting” period with a distribution log[F(resting)] with slope -1 was compliant with the zoomer/MRW model under the “equal weight” condition for multi-scaled space use. Thus, the mice showed similar cognitively driven multi-scaled behaviour as Cole (1995) found for Drosophila; a concept Cole termed “fractal time”, which he linked both to Lévy walk and to optimal foraging.

I put Shoji’s (2016) term “resting” in quotes, since these intervals were defined similarly as “patch staying times” in the ecological literature. In other words, the mouse had to show locomotion above a given threshold to count as “active” during the defined time increment. The “resting” periods for the mice then showed a scaling exponent typically in the range from -1.1 to -1.2.

Above I have again copied from Shoji (2016), but with an added twist. The inset shows an example of a distribution of displacement pr. unit time, measured in “pixel units”. I sampled this graph, and present here the result with log-log transformed axes. The log-log linearity confirms a scale-free distribution of step lengths.

I present the entire distribution, rather than focusing on the distribution below and above the threshold separately. Further, I interpret the “threshold” as a choice of spatio-temporal unit scales, u and t, in the Zoomer model.

Acccording to the Zoomer/MRW model, displacements by a given individual from a given scale level u (the “threshold”, if you wish) to the next higher level u+1, should happen 1/u as often. In other words, the lab mouse example above satisfies this prediction. Since the environment in this experiment is homogeneous, the scaling behaviour is intrinsically driven. Also this system property satisfies the Zoomer/MRW model, since it complies with the parallel processing postulate for multi-scaled space use.

The slope from the mouse’s distribution is not exactly -1, showing a power exponent of circa -1.12. According to the Zoomer/MRW framework, a steeper slope (but still log-log linear) simply implies that the animal(s) has/have skewed its/their space use utilization slightly towards finer scales, on expense of coarser scales, but still in a scale-free manner. A slope of exactly -1 only represents the default Zoomer/MRW condition; i.e., the ideal model.

The classic Markov-based framework for animal movement and population redistribution does not predict log-log linearity in the graph above, but a steadily steeper slope with increasing abscissa value; in other words, a negative exponential function (see grey inset in the first illustration in this post, but observe its arithmetic axes).

Thanks to Shoji’s experiment, both MRW and the Zoomer model seems to have found additional support from behavioural studies on real animals.

REFERENCES

Cole, B. J. (1995) Fractal time in animal behaviour: the moment activity of Drosophila. Anim. Behav., 50, 1317-1324.

Gautestad, A. O. (2012) Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface, 9, 2332-2340.

Gautestad, A. O. & Mysterud, I. (2005) Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist, 165, 44-55.

Gautestad, A. O. & Mysterud, A. (2013) The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology, 1, 1-18.

Shoji, H. (2016) Scaling law in free walking of mice in circular open fields of various diameters. J. Biol. Phys., DOI: 10.1007/s10867-015-9406-z.

# Towards Complex Population Kinetics

Presented for the first time in  my book, Multi-scaled random walk (MRW) hereby also has a complementary formulation for population kinetics, “the Zoomer model” (see Chapter 6. Modelling parallel processing). While population dynamics primarily is occupied with the time scale of seasons, years and generation times, population kinetics is typically also including the shorter time range of intra-season redistribution of individuals. In other words, this concept also covers higher-frequency and finer-grained spatio-temporal variability of a population’s distribution – as a mixture of intrinsic and extrinsic factors – in a more explicit manner than traditional population dynamical modelling.

If movement is “simple”, i.e., obeying classic diffusion laws and thus the mean field principle at the population level, the effect from individual-level system details may conveniently be “averaged out”, allowing for quite simply-structured models.

In physics and probability theory, mean field theory (MFT, also known as self-consistent field theory) studies the behavior of large and complex stochastic models by studying a simpler model. Such models consider a large number of small individual components which interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem. (Wikipedia)

On the other hand, if individual-level movement is complex (e.g., scale-free and influenced by spatial memory in a non-Markovian manner), the mean field approximation should not a priori be expected to hold as a realistic representation of population dynamics.

Accounting for this shortcoming of the standard range of population models may have broad implications for population dynamical modelling. As I describe in Chapter 1 of my book, classic tools like differential equations (temporally continuous dynamics, averaged over space), partial differential equations (dynamics in continuous time and space) and even coupled map lattice models (discrete time and space dynamics) may have to be substituted if space use is complex. Substituting with what? To my knowledge, there are no candidate models!

Thus, the Zoomer model represents the first (as far as I know) proposal for such a candidate model. It translates memory-influenced, scale-free space use at the individual level to the population level, in a statistical-mechanically coherent manner. The Zoomer model  includes all the four standard BIDE rates (Birth, Immigration, Death and Emigration), and it is also spatially explicit. However, contrary to standard coupled map lattice models, spatial scale (the “lattice”) is implemented in a multi-scaled manner. This “scale range” approach allows for formulation of various aspects of complex population kinetics.

For example, the Zoomer model allows for explicit modelling of intraspecific cohesion (conspecific attraction), which is a complex process that depends on both temporal and spatial memory utilization at the individual level. In a simplified scenario, consider that a tendency for conspecific attraction is the main driver of the population kinetics. Further, consider that we study the process at sufficiently fine temporal and spatial scales to allow us to disregard the “slower” Birth and Death rates in the BIDE terms (it will be trivial to account for these processes as well, and I will do it in later posts). In other words, in this example focus is kept on one particular aspect of the intra-population redistribution process, the relatively high-pace influence from conspecific attraction on a populations spatial dispersion.

Consider that this attraction process takes place in parallel over a range (1, 2, …, k, .., m) spatial scales, where m is the defined maximum extent for memory-influenced “strategic” moves by the M individuals during the defined time increment from t to t+1. In this (simplified) scenario the Zoomer model for a population of Mkj individuals within respective set of virtual grid cells in a lattice of cell size size u at spatial scale level k can be summarized as:

During a given time increment a given fraction of the population (expressed by the rate r, which by default is identical at respective levels k) redistributes itself in self-organizing manner, leading to a statistically self-similar (“fractal”) distribution pattern over a range of spatial scales 1, …, m in compliance with the conceptual visualization in this post. However, in the present population-level example scenario the respective “zoomers” Z to respective scale levels k and in respective cells j at level k are collectively attracted towards their neighbourhood with the currently highest abundance of conspecifics at this scale (this neighbourhood thus receives all zoomers from its neighbourhood at this scale level during the given time increment; E=1 here and E=0 for the surrounding cells). Often the zoomers tend to return to the same neighbourhood, though, since this is where the pre-zooming abundance typically is largest. However, discovering a better location in the neighbourhood during zooming will lead to re-distribution of these individuals.

A complete description of this equation and its modifications for improved realism (e.g., the response to local overcrowding) is outside the scope of a single blog post. However, the model’s output seems to share many characteristics of real population kinetics. I will return to some examples in future posts.

In compliance with the parallel processing postulate for individual space use behaviour, as expressed by the Multi-scaled random walk (MRW) model, the Zoomer model is primarily developed to explore the process of population re-distribution of individuals over a local range of spatial scales. “Local” may cover quite a wide area, due to the model’s unique characterization of individual movement as a multi-scaled process (in analogy, consider the broad spatial range of the long tail of the step length distributuion from a Lévy walk, with the clear distinction that Lévy walk – as opposed to MRW – does not include spatial memory). At a temporal scale that is substantially smaller than the average reproduction cycle, movement of individuals determines adaptation to local ecological conditions – including conspecific abundance – to a larger extent than death and birth rates.

Like I advocated for the individual level modelling (see this post), it is a good approach to start exploring a new modelling concept under homogeneous simulation conditions to study the process’ intrinsic properties (in this case, the scale-free zoomer re-distribution potential), before turning towards increased complicatedness and realism by introducing environmental heterogeneity. However, often the dynamic and statistical response to the latter can quite trivially be deducted/predicted from the homogeneous condition.

# A Statistical-Mechanical Perspective on Site Fidelity – Part II

One particular class of complex space use – Multi-scaled random walk (MRW) – implies that movement is influenced both by spatially explicit memory utilization and temporally multi-scaled goal execution; in short, it covers spatially explicit strategic locomotion, which may be processed over a range of temporal scales (the parallel processing conjecture; see the December post that summarizes the scaling cube).

In this Part II where I continue to focus on statistical-mechanical properties I elaborate on the entropy aspect of movement. Specifically, I describe how this key property may be coherently maintained – albeit in a surprising manner – even in the non-classic kind of space use, as represented by the MRW model.

According to the standard framework, one should not expect a change of entropy when “zooming” in and out of a given system of a given extent. This property will now be rattled as we move to the scale-free condition.

First, the classic system condition. Consider that a virtual grid is superimposed on the spatial scatter of serially non-auto-correlated GPS fixes (see Part I of this group of posts). Whether the grid has small or large cell sizes, it is a matter of trivial re-calculation to express the home range’s utilization distribution at respective pixel sizes. Half as large pixel size (√2 smaller spatial scale) implies half as large density of fixes at this finer scale, calculated in terms of the new scale unit.

In short, “zooming in” towards finer resolutions does not improve information about the animal’s pixel location at the next relocation in the series. A finer grid reveals more details of the utilization distribution, given that the sample size of fixes is sufficiently large to resolve these details. However, the embedded entropy is expected to be trivially maintained at different pixel sizes for the defined macroscopic system – the home range.

Now, be prepared for a surprise as we change from a classic to a complex kind of space use.

The illustration above is copied from a 11-year old paper (Gautestad A. O. & I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 65:44-55). It shows three replicates of two model conditions of complex space use, represented by the MRW model. The upper row shows a scenario where the model animal on average has put stronger weight on short-term relative to longer term goals (MRW, β=3; see the post for the scaling cube). The lower row illustrates an animal that has executed goals more equally distributed over a range of temporal scales (MRW, β=2). As a consequence, in the latter case the space use on one hand seems to become less “dense” over the range of pixel sizes due to the higher level of fragmentation, and on the other hand space use seems more dense when the fragments alone are considered (more empty grid cells in the virtual grid at a given resolution, leading to fix concentration in the others). This difference between upper and lower row can be quantified by the fractal dimension of the spatial pattern of presence/absence (non-empty grid cells, called incidence, at respective resolutions), changing from approximately 1.4<D<1.5 towards D≈1. In the interior (centre) part of the scatter, D=2 and D=1, respectively; see Eq. 15-17 and Figure 63 in the book.

The upper row above is closer to the classic home range concept (see Part I of this group of blogs), but still qualitatively different. For example, in Part I moveent was constrained inside a “box” (conceptually similar to a centre-pointing advection) to avoid free dispersal; here in Part II movement is constrained by site fidelity due to spatial memory.

Then, finally – what about entropy in this comparison of scenaria? Above I have reproduced Figure 5 from Gautestad and Mysterud (2005). It shows the local density of fixes in the “interior” part of the home range as a function of spatial scale (square root of cell area), using two of the fix dispersion patterns in the first illustration above; one from MRW with β=3 and one from the fully expressed complexity scenario, MRW with β=2. Both axes are log-transformed for the sake of stressing the respective scenaria’s power law (scale-free) properties. The local density of fixes equals the number of locations per grid cell, calculated over cells with nonzero abundance of locations (incidence). When the spatial resolution is increased from right to left, the density decreases because the given number of fixes is spread over an increasing number of smaller grid cells, and density is calculated as locations pr. non-empty cell at the scale of the new cell size.

The data from the top left panel in the first illustration above show a slope close to 2 (filled circles). This slope is trivially compliant with the classic entropy expectation, to be explained in more detail below. This result verifies that “disorder”; i.e., entropy, is maintained at the same level over a range of spatial resolutions. In other words, when the slope equals 2, entropy is both “space-filling” and resolution-independent within a given area extent, and thus complies with standard statistical mechanics. In the present scenaria the environment was defined to be homogeneous, to emphasize the fractal dimension’s relevance in the context of intrinsically driven space use behaviour (see this post for the relevance of simulating under a homogeneous condition).

The non-standard property is shown by the smaller slope of magnitude 1.06 for the D=1 condition. Here, density per non-empty cell is increasing non-trivially relative to the given area unit for density calculation.

The discrepancy, marked by the Δ character, emerges in local density of locations, and shows how local intensity of habitat use – even in these scenaria of site fidelity in a homogeneous environment – increases in a heterogeneous manner towards finer resolutions: a given total number of locations inside the grid arena is concentrated on a smaller and smaller number of cells when cell size is reduced. Consequently, entropy – lack of information of the animal’s whereabouts – is apparently reduced towards finer resolutions.

By the way, the transition towards a common slope of 2 at the coarsest resolutions in this illustration is an artifact from choosing a grid extent that does not cover the total set of relocations in the given set (i.e., some parts of the distribution at the fringes of the home range were left out, for the sake of studying the home ranges’  “interior”). This effect could, to a large extent, have been avoided by choosing an area extent that covered all fixes.

However, here comes an even more interesting – and biophysically novel – observation: the slope ≈1 (in log-log transformtion) for complex space use in the illustration above means that density N/(incidence area) – where incidence area equals the sum grid cells with at least one relocation – is increasing proportionally with the square root of increased resolution (decreased cell size). I am now referring to the aggregated area of non-empty grid cells at respective cell sizes where density is transformed to a specific reference scale (like N/m²), rather than being “re-calibrated” to actual cell size at respective resolutions. This implies that the entropy reduction that is surprisingly revealed when observing the system towards finer spatial resolutions is exactly compensated for by entropy expansion from the Home range ghost concept: observed home range area (e.g., non-empty grid cells at a given resolution) expanding proportionally with I(N) ∝ √N! Simultaeously, the entropy reduction from increased observation intensity (larger N) happens in parallel and “non-observed” within each non-empty cell at the given resolution for the observed I(N).

Why this harmonic log-linear relationship between “inward” versus “outward” space use expansion? Entropy is a so-called extensive statistical-mechanical property, which means that it is expected to change proportionally with the log-transformed magnitude of system extent (in this case, home range size). In the present context of complex space use the sum of log(inward entropy reduction at smaller pixel scales) and log(outward expansion of system that follows from increased sample size of fixes that is necessary to reveal the increased order at the spatially finer resolution) equals:

“inward” change of entropy + “outward” change of entropy = -0.5 + 0.5 = 0

The sum of the power law exponents cancels out. Entropy theory is consequently maintained, otherwise it would have been a dubious extension of statistical-mechanical theory!

In this extended frame of reference for statistical mechanics, entropy (or information content) is distributed non-trivially over a scale range due to the parallel processing postulate for the MRW model. Within this extended system the zooming paradox I referred to above when such a system is interpreted from the perspective of the classic scenario is resolved by considering the totality of change of entropy towards finer resolutions and towards coarser resolutions in tandem.

Contrast this with the classic system, where there is no paradox in the first place, since the power exponents for change of entropy (represented here by change of standardized density of space use, like N/m²) are both zero; i.e.,

“inward” change of entropy + “outward” change of entropy = 0 + 0 = 0

In this case there is no fine-pixel surprise factor in the form of reduced entropy if the observer is zooming in (within a given area extent), and no non-trivial area expansion with increasing sample size, N. In the latter case the only area expansion comes from a small-sample artifact – the trivial N-dependency – as the observed area is approaching the true home range size asymptotically with increasing N.

Under conditions of complex space use, simulations are -at least for the time being – a prerequisite for exploring the system’s dynamic and statistical properties. Theory from classic statistical mechanics simply does not suffice to predict the model’s behaviour. Consequently, simulations contribute to extending the theoretical framework for complex space use through an interplay between inductive and deductive reasoning.

In statistical-mechanical terms our empirical tests have generally shown closer compliance with the lower row of the first illustration above. Consequently, a range of alternative methods for ecological inference related to home range data are presented in my book.

In a planned follow-up Part III-post on the present theme I show for the first time a fascinating “point of balance” scale between what I referred to above as inward contraction versus outward expansion of entropy. Hence, an additional bridge between GPS-based ecological studies on animal space use and the theory for complexity-extended statistical-mechanics is emerging.