Journal of Theoretical Biology 349 (2014) 163–166
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Letter to Editor
Path integration and coordinate systems Path integration (PI, also referred to as dead-reckoning) is a site-independent updating process that enables an animal to keep track of its current position (location and orientation) with respect to its starting point (home) thanks to information collected en route about its movement. With movements modelled, for mathematical convenience, as series of discrete steps alternating with changes of direction (turns), movement information corresponds to sensory inputs about step length (translational information) and either step orientation (directional information) or turn (rotational information). Thus, two types of PI can be distinguished: allothetic PI, in which step orientations (i.e. headings) are estimated with respect to a compass, and idiothetic PI, in which turns are estimated using kinaesthetic (e.g. vestibular or other inertial) systems. In both types of PI, step lengths are assumed to be estimated by kinaesthetic systems and/or the visual flow from the ground. It has been shown that allothetic PI is much more efficient than idiothetic PI, because the use of a compass prevents PI from quickly accumulating errors (Benhamou et al., 1990). Recently, Cheung and Vickerstaff (2010) and Cheung (2014) claimed that the coordinate system used in noisy PI computation is of key importance. Basically, four types of coordinate system can be distinguished, depending on whether the frame of reference is bound to the environment (exocentric, allocentric or geocentric) and used to update the animal's position, or bounded to the animal's current position (egocentric or autocentric) and used to update the home location, and the coordinates used are Cartesian or polar (Séguinot, 2000; Merkle et al., 2006; Vickerstaff and Cheung, 2010). Note that, for etymological consistency, it is preferable to use ego vs. exo (Latin root) or auto vs. allo (Greek root) rather than mixing these terms as it is often done. When based on mathematical exact formulations without movement estimation errors, PI computation obviously provides the true result whatever the coordinate system used. However, Cheung and Vickerstaff (2010) showed that only Cartesian exocentric coding can prevent PI fed with noisy movement estimates from accumulating errors very fast, and concluded that the neural structures performing PI should have evolved to work in this way. Based on what he claimed to be “exact closed form mathematical solutions”, Cheung (2014) further asserted that “Using canonical equations of PI, it is shown that polar PI using a compass accumulates uncertainty in a manner similar to Cartesian PI without a compass”. This sounds like a definite answer which urges theoretical and experimental neurobiologists to look for neural networks able to perform PI computation based on Cartesian exocentric coding. However, the approach used by Cheung and Vickerstaff (2010) and Cheung (2014) appears to be flawed. Indeed, I show here, using simple recurrent formulae, that noisy PI generates exactly the same output errors whatever the coordinate
http://dx.doi.org/10.1016/j.jtbi.2014.02.012 0022-5193 & 2014 Elsevier Ltd. All rights reserved.
system used and the degree of movement estimation errors, in both allothetic and idiothetic forms. Let θti and lti be the true orientation and length of the ith step, and αti ¼ θti þ 1–θti , the ith true turn. The actual path can be modelled with recurrent formulae Xti ¼ Xti 1 þlti cos(θti ) and Yti ¼ Yti 1 þlti sin(θti ). For convenience without loss of generality, the X-axis is used as the compass-based reference direction in allothetic PI but is defined by θt1 in idiothetic PI, for which there is no particular reference direction. The true homing vector can be defined after each step in terms of distance between the current animal's location and home location, Dti ¼[(Xti –Xt0)2 þ(Yti –Yt0)2]0.5, and of home direction expressed either with respect to the animal's body axis, ωti ¼atan2(Yt0–Yti , Xt0–Xti ) – θti (which corresponds to the turn the animal has to do to get its home in front of it, and thus appears to be particularly useful in idiothetic PI) or with respect to the X-axis, γti ¼atan2(Yt0–Yti , Xt0–Xti ) (which corresponds to the home compass bearing, and thus appears to be particularly useful in allothetic PI), with atan2(y, x)¼arctan(y/x)7b 1801 with b¼ 0 for x4 0 and b¼1 for xo 0. A related variable is Φti ¼ atan2(Yti –Yt0, Xti –Xt0)¼ γti 71801, which corresponds to the compass bearing of the animal's current location from home (Fig. 1). Angular random noise δ applies to angular sensory inputs, which correspond to estimates of either step orientations θ in allothetic PI or turns α in idiothetic PI. However, as shown below, angular PI inputs correspond either to step orientations θ in the exocentric frame of reference or to turns α in the egocentric frame of reference (with both types of coordinates). Consequently, in the exocentric frame of reference, step orientations are directly estimated in allothetic PI (θi ¼θti þδi) but must be inferred from turns in idiothetic PI (θi ¼θi 1 þαi 1, with αi 1 ¼ αti 1 þδi 1), whereas in the egocentric frame of reference, turns are directly estimated in idiothetic PI (αi ¼αti þ δi) but must be inferred from step orientations in allothetic PI (αi ¼θi þ 1 θi, with θi ¼ θti þδi). In all cases, step lengths are directly estimated: li ¼ lti þλi where λ is a linear random noise. PI being an updating process, it can be modelled using the following mathematically exact recurrent formulae (with noisy estimates li and θi or αi 1). EXOCENTRIC FRAME OF REFERENCE, CARTESIAN COORDINATES (X, Y) Initialisation: X0 ¼ Xt0 and Y0 ¼Yt0, and for idiothetic PI, θ1 ¼ 0 PI computation (for iZ 1) X i ¼ X i 1 þ li cos ðθi Þ
ð1aÞ
Y i ¼ Y i 1 þ li sin ðθi Þ
ð1bÞ
Homing Vector Di ¼ ½ðX i –X 0 Þ2 þ ðY i –Y 0 Þ2 0:5
ð1cÞ
ωi ¼ atan2 ðY 0 –Y i ; X 0 –X i Þ–θi
ð1dÞ
164
Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
ωi ¼ ωi 1 –αi 1 þ atan2 ½ sin ðωi 1 –αi 1 Þ; Di 1 =li – cos ðωi 1 –αi 1 Þ ¼ atan2 ½ sin ðωi 1 –αi 1 Þ; cos ðωi 1 –αi 1 Þ–li =Di 1 ð4bÞ
Homing Vector: Di and ωi directly provided by Eqs. (4a) and (4b) γ i ¼ ωi þθi
ð4cÞ
Note: with very small step lengths l, discrete step models converge towards continuous ones, and Eqs. (2a) and (2b) can be approximated in a simpler form as Di ¼Di 1 þ cos(θi–Φi 1)li and Φi ¼Φi 1 þsin(θi–Φi 1)li/Di 1, and Eqs. (4a) and (4b) as Di ¼Di 1 cos(ωi 1–αi 1)li and ωi ¼ωi 1–αi 1 þsin(ωi 1–αi 1) li/Di 1 (Benhamou and Séguinot, 1995; Séguinot, 2000).
U i ¼ U i 1 cos ðαi 1 Þ þV i 1 sin ðαi 1 Þ–li
ð3aÞ
Contrary to the complex closed-form formulae that Cheung (2014) attempted to develop to directly provide the output error distributions after a given step number, the recurrent formulae make it possible to obtain these distributions only through heavy computer simulations. Nevertheless, it is relatively easy to check on numerical examples that, after any step number, the noisy estimates of Dti and ωti obtained indirectly through Cartesian exocentric (Eqs. (1c) and (1d)) or egocentric (Eqs. (3c) and (3d)) coding, or in part indirectly through polar exocentric coding (Eqs. (2a) and (2c)), or directly through polar egocentric coding ((4a) and (4b)) are exactly the same, in both allothetic and idiothetic PI, and to check that the noisy estimates of the home bearing γti obtained with any type of coding (Eqs. (1e), (2d), (3e) or (4c)) are also equal. Table 1 provides an example obtained with a 10-step correlated random walk starting at Xt0 ¼ 0 and Yt0 ¼ 0. Thus, the animal's location after 10 steps is defined by Xt10 ¼ 5.599 and Yt10 ¼7.763, but its estimated location in the Cartesian exocentric coordinate system obtained through allothetic PI corresponds to X10 ¼ 5.480 and Y10 ¼6.648. According to Eqs. (1c), (1d) and (1e), the estimated homing vector is therefore provided by D10 ¼ (5.4802 þ6.6482)0.5 ¼8.615 step lengths and γ10 ¼atan2( 6.648, 5.480) ¼ 50.501 or ω10 ¼γ10 θ10 ¼ 151.931, that is exactly the same length and direction components obtained with allothetic PI performed in the polar egocentric coordinate system (Eqs. (4a)– (4c)). The reader can check that this holds true after any step number and in any coordinate system, for both allothetic and idiothetic PI. Computing PI in Cartesian or Polar coordinates in an egocentric or exocentric frame of reference has therefore strictly no impact on the accuracy of the homing vector: noisy PI accumulates uncertainty in exactly the same way in any coordinate system. This is not a surprising result: with mathematically exact formulae, there is no reason to obtain different PI outputs not only when there are no estimation errors but also when exactly the same estimation errors are introduced at each step in translations and directions (allothetic PI) or rotations (idiothetic PI). Accordingly, the resultant PI errors expressed in terms of position (location and orientation)
V i ¼ V i 1 cos ðαi 1 Þ–U i 1 sin ðαi 1 Þ
ð3bÞ
εPi ¼ jjðDi ; ωi Þ–ðDti ; ωti Þjj ¼ ½ðU i –U ti Þ2 þ ðV i –V ti Þ2 0:5
Fig. 1. Illustration of the different variables involved in PI computation (adapted from Benhamou and Séguinot, 1995). Although this figure illustrates the true values of the variables (no movement estimation noise), the superscript t has been omitted for clarity.
γ i ¼ atan2 ðY 0 –Y i ; X 0 –X i Þ
ð1eÞ
EXOCENTRIC FRAME OF REFERENCE, POLAR COORDINATES (D, Φ) Initialisation: D1 ¼l1, Φ1 ¼θ1 and, for idiothetic PI, θ1 ¼ 0 PI computation (for i Z2) 2
Di ¼ ½Di 1 2 þ li þ 2Di 1 li cos ðθi –Φi 1 Þ0:5
ð2aÞ
Φi ¼ Φi 1 þ atan2 ½ sin ðθi –Φi 1 Þ; Di 1 =li þ cos ðθi –Φi 1 Þ
ð2bÞ
Homing Vector: Di directly provided by Eq. (2a) ωi ¼ Φi –θi 7 1801
ð2cÞ
γ i ¼ Φi 7 1801
ð2dÞ
EGOCENTRIC FRAME OF REFERENCE, CARTESIAN COORDINATES (U, V) Initialisation: U1 ¼–l1 and V1 ¼ 0 PI computation (for iZ2)
¼ ½Di 2 þ Dti 2 –2Di Dti cos ðωi –ωti Þ0:5
Homing Vector: Di ¼ ðU i 2 þV i 2 Þ0:5
ð3cÞ
or in terms of location only
ωi ¼ atan2 ðV i; U i Þ
ð3dÞ
εLi ¼ jjðDi ; γ i Þ–ðDti ; γ ti Þjj ¼ ½ðX i –X ti Þ2 þ ðY i –Y ti Þ2 0:5
γ i ¼ atan2 ðV i; U i Þ þ θi
ð3eÞ
EGOCENTRIC FRAME OF REFERENCE, POLAR COORDINATES (D, ω) Initialisation: D1 ¼l1 and ω1 ¼ 71801 PI computation (for iZ2) 2
Di ¼ ½D2i 1 þ li –2Di 1 li cos ðωi 1 –αi 1 Þ0:5
ð4aÞ
¼ ½Di 2 þ Dti 2 –2Di Dti cos ðγ i –γ ti Þ0:5 dramatically depend on the type of PI (allothetic or idiothetic), but do not depend on the coordinate system used. Note that, because PI is used during both the outward and return paths (i.e. until one gets Di ¼ 0; Wehner et al., 1996), the final resultant PI error should be computed as the distance between the animal's current location and the home location at the end of the return path to the estimated home location (εPi = εLi = Dti ).
Table 1 Example of values obtained with a 10-step Correlated Random Walk with constant step length lt=1 (arbitrary unit) and directional correlation set to 0.9 starting at Xt0=0 and Yt0=0, with linear noise λ~N(0,0.2) in step length estimations and angular noise δ~N(0,17°) that applies either to step orientations θ (allothetic PI) or turns α (idiothetic PI). The value of θt1 was drawn at random between −180° and 180°. A) True values of various parameters i
αti
θti
a
1 2 3 4 5 6 7 8 9 10
-17.10 -4.61 19.28 -21.02 26.41 18.93 8.28 -20.52 -16.83 ————
126.62 109.51 104.90 124.18 103.16 129.57 148.50 156.78 136.26 119.43
Xti
a
-0.596 -0.931 -1.188 -1.749 -1.977 -2.614 -3.467 -4.386 -5.108 -5.599
Yti
a
0.803 1.745 2.712 3.539 4.513 5.283 5.806 6.200 6.892 7.763
γti
a
-53.38 -61.93 -66.35 -63.69 -66.34 -63.67 -59.16 -54.73 -53.45 -54.20
θti
b
0.00 -17.10 -21.71 -2.44 -23.46 2.95 21.88 30.16 9.64 -7.19
Xti
b
1.000 1.956 2.885 3.884 4.801 5.800 6.728 7.593 8.578 9.571
Yti
b
0.000 -0.294 -0.664 -0.707 -1.105 -1.053 -0.681 -0.178 -0.011 -0.136
γti
b
180.00 171.45 167.04 169.69 167.04 169.71 174.22 178.66 179.93 179.19
Uti
Vti
Dti
ωti
-1.000 -1.956 -2.926 -3.910 -4.844 -5.738 -5.990 -6.475 -8.456 -9.512
0.000 -0.294 -0.450 0.541 -0.898 1.350 3.139 3.969 1.447 -1.063
1.000 1.978 2.960 3.948 4.927 5.895 6.762 7.595 8.578 9.572
180.00 -171.45 -171.25 172.13 -169.50 166.76 152.34 148.50 170.29 -173.63
Input parameters
Exo. Cart.
Ego. Cart.
Exo. and ego. Polar d
i
li
θi
αi¼θi+1–θi
Xi
Yi
Ui
Vi
Di
1 2 3 4 5 6 7 8 9 10
0.920 1.151 0.630 0.878 1.105 1.016 1.042 0.865 1.348 1.059
158.66 106.51 110.02 110.33 96.69 164.47 167.32 -176.04 115.38 101.43
-52.15 3.51 0.31 -13.65 67.78 2.85 16.64 -68.58 -13.95 ————
-0.857 -1.184 -1.400 -1.705 -1.833 -2.812 -3.829 -4.692 -5.270 -5.480
0.335 1.438 2.030 2.853 3.950 4.223 4.451 4.392 5.610 6.648
-0.920 -1.715 -2.386 -3.267 -4.137 -3.840 -4.713 -4.377 -7.327 -7.602
0.000 -0.726 -0.620 -0.607 -1.361 3.315 3.502 4.705 -2.356 -4.053
0.920 1.862 2.466 3.323 4.355 5.073 5.872 6.426 7.697 8.615
c
Error d
Φi
ωi
γI
158.66 129.46 124.58 120.86 114.89 123.67 130.70 136.89 133.21 129.50
180.00 -157.05 -165.44 -169.48 -161.79 139.20 143.39 132.93 -162.18 -151.93
-21.34 -50.54 -55.42 -59.14 -65.11 -56.33 -49.30 -43.11 -46.79 -50.50
εPi
εLi
0.080 0.495 0.565 1.316 0.845 2.732 1.327 2.224 3.967 3.548
0.535 0.398 0.714 0.687 0.580 1.079 1.402 1.834 1.292 1.121
C) Idiothetic PI Input parameters
Exo. Cart.
Ego. Cart.
Exo. and Ego. Polar d
i
li
αi
θi¼θi-1+αi-1
Xi
Yi
Ui
Vi
Di
1 2 3 4 5 6 7 8 9 10
0.920 1.151 0.630 0.878 1.105 1.016 1.042 0.865 1.348 1.059
-20.10 0.51 5.43 -27.49 61.31 37.75 35.46 -41.39 -34.82 ———
0.00 -20.10 -19.59 -14.16 -41.66 19.65 57.40 92.87 51.48 16.65
0.920 2.000 2.594 3.445 4.271 5.228 5.789 5.746 6.586 7.601
0.000 -0.395 -0.607 -0.822 -1.556 -1.214 -0.336 0.527 1.582 1.885
-0.920 -2.014 -2.647 -3.541 -4.225 -4.515 -2.835 -0.239 -5.339 -7.822
0.000 -0.316 -0.298 -0.046 -1.676 2.902 5.058 5.765 4.167 0.372
0.920 2.039 2.664 3.541 4.545 5.367 5.799 5.770 6.773 7.831
c
Error d
Φi
ωi
γI
0.00 -11.18 -13.17 -13.41 -20.02 -13.08 -3.33 5.24 13.51 13.93
180.00 -171.08 -173.57 -179.25 -158.36 147.27 119.27 92.38 142.03 177.28
180.00 168.82 166.83 166.59 159.98 166.92 176.67 -174.76 -166.49 -166.07
εPi
εLi
0.080 0.063 0.317 0.694 0.994 1.976 3.693 6.490 4.136 2.217
0.080 0.111 0.296 0.454 0.697 0.595 1.000 1.977 2.551 2.823
Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
B) Allothetic PI
a
computed with respect to X axis in standard position for allothetic PI. computed with respect to X axis rotated by θt1 for idiothetic PI. computed directly using recurrent PI formulae within the exocentric (Eqs 2a and 2b) or egocentric (Eqs 4a and 4b) frame of reference, but one can check that the very same values of D, ω and γ can also be obtained indirectly from X, Y and θ values (by applying Eqs. 1c, 1d and 1e) or from U, V and θ values (by applying Eqs. 3c, 3d and 3e). d D and γ estimates take the very same values when computed within exocentric (Eqs. 2a and 2d), egocentric (Eqs. 4a and 4c) or mixed (Eqs. 5a and 5b) frame of reference. b c
165
166
Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
Based on indirect evidence, I have always considered that PI should involve a polar egocentric coding (Benhamou et al., 1990; Benhamou and Séguinot, 1995; Benhamou, 1997). Indeed, sensory inputs as well as motor outputs are necessarily expressed egocentrically, and PI (and other navigation systems) outputs seem to be suitably expressed in terms of distance and direction. To implement PI in an exocentric frame of reference, a neural structure would have to perform a two-fold conversion of frame of reference (ego–exo–ego). Such a conversion is theoretically necessary in eidetic navigation, where the goal location is coded with respect to landmarks. However, contrary to mammals which are indeed able to build up exocentric spatial representations thanks to neural structures such as the hippocampus and the post-subiculum, insects appear to by-pass the frame conversion to solve the task in a purely egocentric mode, but in a much more rigid way than mammals (Benhamou, 1997, 1998). As PI can be implemented in any frame of reference with the same final accuracy, it seems quite unlikely that the dedicated neural system would bother with frame conversions. When looking for mathematically exact formulae, the most suitable way to model idiothetic PI thus appears to be provided by Eqs. (4a) and (4b). In allothetic PI, angular sensory inputs correspond to compass-based heading estimates (θ) and the home direction should be estimated in terms of compass bearing (γ). It may be therefore more relevant for allothetic PI to use recurrent polar formulae expressed in a mixed ego/exo-centric frame of reference, which is centred on the animal's current location but oriented with respect to the compass reference direction, to directly provide the distance to home and its compass bearing: 2
Di ¼ ½D2i 1 þ li –2Di 1 li cos ðγ i 1 –θi Þ0:5
ð5aÞ
γ i ¼ γ i 1 þ atan2 ½ sin ðγ i 1 –θi Þ; Di 1 =li – cos ðγ i 1 –θi Þ
ð5bÞ
with D1 ¼l1 and γ1 ¼θ1 71801. If necessary, ωi can be obtained from Eq. (5b) as ωi ¼γi θi. It is however worth noting that actual PI rests on neural network computations that can only approximate mathematically exact formulae. This explains why an unexpected systematic bias occurs in the homing vector (in addition to random errors) when animals are forced to perform paths whose right and left turns are not balanced (e.g. L-shaped and U-shaped paths), both in species performing allothetic PI such as ants (Müller and Wehner, 1988; Merkle et al., 2006) and in species performing idiothetic PI such as hamsters and dogs (Séguinot et al., 1993, 1998; Séguinot 2000). We therefore need to develop neural network models able to perform PI updating in a biologically relevant way. Previous attempts by Hartmann and Wehner (1995) and Wittmann and Schwegler (1995) corresponded only to neural networks that implement approximate (based on Müller and Wehner, 1988) or mathematically exact, respectively, recurrent PI formulae. More
recently, Bernardet et al. (2008) developed a more biologically inspired neural network model. Contrary to what Cheung and Vickerstaff (2010) and Cheung (2014) claimed, there is no need to think about neural networks that specifically rely on some form of Cartesian exocentric coding. Acknowledgements I sincerely thank two anonymous referees for their constructive comments on a previous draft of this letter. References Benhamou, S., 1997. On systems of reference involved in spatial memory. Behav. Process. 40, 149–163. Benhamou, S., 1998. Place navigation in mammals: a configuration-based model. Anim. Cognit. 1, 55–63. Benhamou, S., Sauvé, J.-P., Bovet, P., 1990. Spatial memory in large scale movements: efficiency and limitation of the egocentric coding process. J. Theor. Biol. 145, 1–12. Benhamou, S., Séguinot, V., 1995. How to find one's way in the labyrinth of path integration models. J. Theor. Biol. 174, 463–466. Bernardet, U., Bermúdez i Badia, S., Verschure, P.F., 2008. A model for the neuronal substrate of dead reckoning and memory in arthropods: a comparative computational and behavioral study. Theory Biosci. 127, 163–175. Cheung, A., 2014. Animal path integration: a model of positional uncertainty along tortuous paths. J. Theor. Biol. 341, 17–33. Cheung, A., Vickerstaff, R.J., 2010. Finding the way with a noisy brain. PLoS Comp. Biol. 6, 1000992. Hartmann, G., Wehner, R., 1995. The ant's path integration system: a neural architecture. Biol. Cybern. 73, 483–497. Merkle, T., Rost, M., Alt, W., 2006. Egocentric path integration models and their application to desert arthropods. J. Theor. Biol. 240, 385–399. Müller, M., Wehner, R., 1988. Path integration in desert ants, Cataglyphis fortis. Proc. Natl. Acad. Sci. U.S.A 85, 5287–5290. Séguinot, V., 2000. l'intégration du trajet: approches théorique et expérimentale chez le hamster et le chien (Ph.D. Thèses). Université de Genève Séguinot, V., Cattet, J., Benhamou, S., 1998. Path integration in dogs. Anim. Behav. 55, 787–797. Séguinot, V., Maurer, R., Etienne, A.S., 1993. Dead-reckoning in a small mammal: the evaluation of distance. J. Comp. Physiol. A 173, 103–113. Vickerstaff, R.J., Cheung, A., 2010. Which coordinate system for modelling path integration? J. Theor. Biol. 263, 242–261. Wehner, R., Michel, B., Antonsen, P., 1996. Visual navigation in insects: coupling of egocentric and geocentric information. J. Exp. Biol. 199, 129–140. Wittmann, T., Schwegler, H., 1995. Path integration – a network model. Biol. Cybern. 73, 569–575.
Simon Benhamou CEFE, CNRS Montpellier, France E-mail address: [email protected] Received 25 October 2013 Available online 22 February 2014
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Letter to Editor
Path integration and coordinate systems Path integration (PI, also referred to as dead-reckoning) is a site-independent updating process that enables an animal to keep track of its current position (location and orientation) with respect to its starting point (home) thanks to information collected en route about its movement. With movements modelled, for mathematical convenience, as series of discrete steps alternating with changes of direction (turns), movement information corresponds to sensory inputs about step length (translational information) and either step orientation (directional information) or turn (rotational information). Thus, two types of PI can be distinguished: allothetic PI, in which step orientations (i.e. headings) are estimated with respect to a compass, and idiothetic PI, in which turns are estimated using kinaesthetic (e.g. vestibular or other inertial) systems. In both types of PI, step lengths are assumed to be estimated by kinaesthetic systems and/or the visual flow from the ground. It has been shown that allothetic PI is much more efficient than idiothetic PI, because the use of a compass prevents PI from quickly accumulating errors (Benhamou et al., 1990). Recently, Cheung and Vickerstaff (2010) and Cheung (2014) claimed that the coordinate system used in noisy PI computation is of key importance. Basically, four types of coordinate system can be distinguished, depending on whether the frame of reference is bound to the environment (exocentric, allocentric or geocentric) and used to update the animal's position, or bounded to the animal's current position (egocentric or autocentric) and used to update the home location, and the coordinates used are Cartesian or polar (Séguinot, 2000; Merkle et al., 2006; Vickerstaff and Cheung, 2010). Note that, for etymological consistency, it is preferable to use ego vs. exo (Latin root) or auto vs. allo (Greek root) rather than mixing these terms as it is often done. When based on mathematical exact formulations without movement estimation errors, PI computation obviously provides the true result whatever the coordinate system used. However, Cheung and Vickerstaff (2010) showed that only Cartesian exocentric coding can prevent PI fed with noisy movement estimates from accumulating errors very fast, and concluded that the neural structures performing PI should have evolved to work in this way. Based on what he claimed to be “exact closed form mathematical solutions”, Cheung (2014) further asserted that “Using canonical equations of PI, it is shown that polar PI using a compass accumulates uncertainty in a manner similar to Cartesian PI without a compass”. This sounds like a definite answer which urges theoretical and experimental neurobiologists to look for neural networks able to perform PI computation based on Cartesian exocentric coding. However, the approach used by Cheung and Vickerstaff (2010) and Cheung (2014) appears to be flawed. Indeed, I show here, using simple recurrent formulae, that noisy PI generates exactly the same output errors whatever the coordinate
http://dx.doi.org/10.1016/j.jtbi.2014.02.012 0022-5193 & 2014 Elsevier Ltd. All rights reserved.
system used and the degree of movement estimation errors, in both allothetic and idiothetic forms. Let θti and lti be the true orientation and length of the ith step, and αti ¼ θti þ 1–θti , the ith true turn. The actual path can be modelled with recurrent formulae Xti ¼ Xti 1 þlti cos(θti ) and Yti ¼ Yti 1 þlti sin(θti ). For convenience without loss of generality, the X-axis is used as the compass-based reference direction in allothetic PI but is defined by θt1 in idiothetic PI, for which there is no particular reference direction. The true homing vector can be defined after each step in terms of distance between the current animal's location and home location, Dti ¼[(Xti –Xt0)2 þ(Yti –Yt0)2]0.5, and of home direction expressed either with respect to the animal's body axis, ωti ¼atan2(Yt0–Yti , Xt0–Xti ) – θti (which corresponds to the turn the animal has to do to get its home in front of it, and thus appears to be particularly useful in idiothetic PI) or with respect to the X-axis, γti ¼atan2(Yt0–Yti , Xt0–Xti ) (which corresponds to the home compass bearing, and thus appears to be particularly useful in allothetic PI), with atan2(y, x)¼arctan(y/x)7b 1801 with b¼ 0 for x4 0 and b¼1 for xo 0. A related variable is Φti ¼ atan2(Yti –Yt0, Xti –Xt0)¼ γti 71801, which corresponds to the compass bearing of the animal's current location from home (Fig. 1). Angular random noise δ applies to angular sensory inputs, which correspond to estimates of either step orientations θ in allothetic PI or turns α in idiothetic PI. However, as shown below, angular PI inputs correspond either to step orientations θ in the exocentric frame of reference or to turns α in the egocentric frame of reference (with both types of coordinates). Consequently, in the exocentric frame of reference, step orientations are directly estimated in allothetic PI (θi ¼θti þδi) but must be inferred from turns in idiothetic PI (θi ¼θi 1 þαi 1, with αi 1 ¼ αti 1 þδi 1), whereas in the egocentric frame of reference, turns are directly estimated in idiothetic PI (αi ¼αti þ δi) but must be inferred from step orientations in allothetic PI (αi ¼θi þ 1 θi, with θi ¼ θti þδi). In all cases, step lengths are directly estimated: li ¼ lti þλi where λ is a linear random noise. PI being an updating process, it can be modelled using the following mathematically exact recurrent formulae (with noisy estimates li and θi or αi 1). EXOCENTRIC FRAME OF REFERENCE, CARTESIAN COORDINATES (X, Y) Initialisation: X0 ¼ Xt0 and Y0 ¼Yt0, and for idiothetic PI, θ1 ¼ 0 PI computation (for iZ 1) X i ¼ X i 1 þ li cos ðθi Þ
ð1aÞ
Y i ¼ Y i 1 þ li sin ðθi Þ
ð1bÞ
Homing Vector Di ¼ ½ðX i –X 0 Þ2 þ ðY i –Y 0 Þ2 0:5
ð1cÞ
ωi ¼ atan2 ðY 0 –Y i ; X 0 –X i Þ–θi
ð1dÞ
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Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
ωi ¼ ωi 1 –αi 1 þ atan2 ½ sin ðωi 1 –αi 1 Þ; Di 1 =li – cos ðωi 1 –αi 1 Þ ¼ atan2 ½ sin ðωi 1 –αi 1 Þ; cos ðωi 1 –αi 1 Þ–li =Di 1 ð4bÞ
Homing Vector: Di and ωi directly provided by Eqs. (4a) and (4b) γ i ¼ ωi þθi
ð4cÞ
Note: with very small step lengths l, discrete step models converge towards continuous ones, and Eqs. (2a) and (2b) can be approximated in a simpler form as Di ¼Di 1 þ cos(θi–Φi 1)li and Φi ¼Φi 1 þsin(θi–Φi 1)li/Di 1, and Eqs. (4a) and (4b) as Di ¼Di 1 cos(ωi 1–αi 1)li and ωi ¼ωi 1–αi 1 þsin(ωi 1–αi 1) li/Di 1 (Benhamou and Séguinot, 1995; Séguinot, 2000).
U i ¼ U i 1 cos ðαi 1 Þ þV i 1 sin ðαi 1 Þ–li
ð3aÞ
Contrary to the complex closed-form formulae that Cheung (2014) attempted to develop to directly provide the output error distributions after a given step number, the recurrent formulae make it possible to obtain these distributions only through heavy computer simulations. Nevertheless, it is relatively easy to check on numerical examples that, after any step number, the noisy estimates of Dti and ωti obtained indirectly through Cartesian exocentric (Eqs. (1c) and (1d)) or egocentric (Eqs. (3c) and (3d)) coding, or in part indirectly through polar exocentric coding (Eqs. (2a) and (2c)), or directly through polar egocentric coding ((4a) and (4b)) are exactly the same, in both allothetic and idiothetic PI, and to check that the noisy estimates of the home bearing γti obtained with any type of coding (Eqs. (1e), (2d), (3e) or (4c)) are also equal. Table 1 provides an example obtained with a 10-step correlated random walk starting at Xt0 ¼ 0 and Yt0 ¼ 0. Thus, the animal's location after 10 steps is defined by Xt10 ¼ 5.599 and Yt10 ¼7.763, but its estimated location in the Cartesian exocentric coordinate system obtained through allothetic PI corresponds to X10 ¼ 5.480 and Y10 ¼6.648. According to Eqs. (1c), (1d) and (1e), the estimated homing vector is therefore provided by D10 ¼ (5.4802 þ6.6482)0.5 ¼8.615 step lengths and γ10 ¼atan2( 6.648, 5.480) ¼ 50.501 or ω10 ¼γ10 θ10 ¼ 151.931, that is exactly the same length and direction components obtained with allothetic PI performed in the polar egocentric coordinate system (Eqs. (4a)– (4c)). The reader can check that this holds true after any step number and in any coordinate system, for both allothetic and idiothetic PI. Computing PI in Cartesian or Polar coordinates in an egocentric or exocentric frame of reference has therefore strictly no impact on the accuracy of the homing vector: noisy PI accumulates uncertainty in exactly the same way in any coordinate system. This is not a surprising result: with mathematically exact formulae, there is no reason to obtain different PI outputs not only when there are no estimation errors but also when exactly the same estimation errors are introduced at each step in translations and directions (allothetic PI) or rotations (idiothetic PI). Accordingly, the resultant PI errors expressed in terms of position (location and orientation)
V i ¼ V i 1 cos ðαi 1 Þ–U i 1 sin ðαi 1 Þ
ð3bÞ
εPi ¼ jjðDi ; ωi Þ–ðDti ; ωti Þjj ¼ ½ðU i –U ti Þ2 þ ðV i –V ti Þ2 0:5
Fig. 1. Illustration of the different variables involved in PI computation (adapted from Benhamou and Séguinot, 1995). Although this figure illustrates the true values of the variables (no movement estimation noise), the superscript t has been omitted for clarity.
γ i ¼ atan2 ðY 0 –Y i ; X 0 –X i Þ
ð1eÞ
EXOCENTRIC FRAME OF REFERENCE, POLAR COORDINATES (D, Φ) Initialisation: D1 ¼l1, Φ1 ¼θ1 and, for idiothetic PI, θ1 ¼ 0 PI computation (for i Z2) 2
Di ¼ ½Di 1 2 þ li þ 2Di 1 li cos ðθi –Φi 1 Þ0:5
ð2aÞ
Φi ¼ Φi 1 þ atan2 ½ sin ðθi –Φi 1 Þ; Di 1 =li þ cos ðθi –Φi 1 Þ
ð2bÞ
Homing Vector: Di directly provided by Eq. (2a) ωi ¼ Φi –θi 7 1801
ð2cÞ
γ i ¼ Φi 7 1801
ð2dÞ
EGOCENTRIC FRAME OF REFERENCE, CARTESIAN COORDINATES (U, V) Initialisation: U1 ¼–l1 and V1 ¼ 0 PI computation (for iZ2)
¼ ½Di 2 þ Dti 2 –2Di Dti cos ðωi –ωti Þ0:5
Homing Vector: Di ¼ ðU i 2 þV i 2 Þ0:5
ð3cÞ
or in terms of location only
ωi ¼ atan2 ðV i; U i Þ
ð3dÞ
εLi ¼ jjðDi ; γ i Þ–ðDti ; γ ti Þjj ¼ ½ðX i –X ti Þ2 þ ðY i –Y ti Þ2 0:5
γ i ¼ atan2 ðV i; U i Þ þ θi
ð3eÞ
EGOCENTRIC FRAME OF REFERENCE, POLAR COORDINATES (D, ω) Initialisation: D1 ¼l1 and ω1 ¼ 71801 PI computation (for iZ2) 2
Di ¼ ½D2i 1 þ li –2Di 1 li cos ðωi 1 –αi 1 Þ0:5
ð4aÞ
¼ ½Di 2 þ Dti 2 –2Di Dti cos ðγ i –γ ti Þ0:5 dramatically depend on the type of PI (allothetic or idiothetic), but do not depend on the coordinate system used. Note that, because PI is used during both the outward and return paths (i.e. until one gets Di ¼ 0; Wehner et al., 1996), the final resultant PI error should be computed as the distance between the animal's current location and the home location at the end of the return path to the estimated home location (εPi = εLi = Dti ).
Table 1 Example of values obtained with a 10-step Correlated Random Walk with constant step length lt=1 (arbitrary unit) and directional correlation set to 0.9 starting at Xt0=0 and Yt0=0, with linear noise λ~N(0,0.2) in step length estimations and angular noise δ~N(0,17°) that applies either to step orientations θ (allothetic PI) or turns α (idiothetic PI). The value of θt1 was drawn at random between −180° and 180°. A) True values of various parameters i
αti
θti
a
1 2 3 4 5 6 7 8 9 10
-17.10 -4.61 19.28 -21.02 26.41 18.93 8.28 -20.52 -16.83 ————
126.62 109.51 104.90 124.18 103.16 129.57 148.50 156.78 136.26 119.43
Xti
a
-0.596 -0.931 -1.188 -1.749 -1.977 -2.614 -3.467 -4.386 -5.108 -5.599
Yti
a
0.803 1.745 2.712 3.539 4.513 5.283 5.806 6.200 6.892 7.763
γti
a
-53.38 -61.93 -66.35 -63.69 -66.34 -63.67 -59.16 -54.73 -53.45 -54.20
θti
b
0.00 -17.10 -21.71 -2.44 -23.46 2.95 21.88 30.16 9.64 -7.19
Xti
b
1.000 1.956 2.885 3.884 4.801 5.800 6.728 7.593 8.578 9.571
Yti
b
0.000 -0.294 -0.664 -0.707 -1.105 -1.053 -0.681 -0.178 -0.011 -0.136
γti
b
180.00 171.45 167.04 169.69 167.04 169.71 174.22 178.66 179.93 179.19
Uti
Vti
Dti
ωti
-1.000 -1.956 -2.926 -3.910 -4.844 -5.738 -5.990 -6.475 -8.456 -9.512
0.000 -0.294 -0.450 0.541 -0.898 1.350 3.139 3.969 1.447 -1.063
1.000 1.978 2.960 3.948 4.927 5.895 6.762 7.595 8.578 9.572
180.00 -171.45 -171.25 172.13 -169.50 166.76 152.34 148.50 170.29 -173.63
Input parameters
Exo. Cart.
Ego. Cart.
Exo. and ego. Polar d
i
li
θi
αi¼θi+1–θi
Xi
Yi
Ui
Vi
Di
1 2 3 4 5 6 7 8 9 10
0.920 1.151 0.630 0.878 1.105 1.016 1.042 0.865 1.348 1.059
158.66 106.51 110.02 110.33 96.69 164.47 167.32 -176.04 115.38 101.43
-52.15 3.51 0.31 -13.65 67.78 2.85 16.64 -68.58 -13.95 ————
-0.857 -1.184 -1.400 -1.705 -1.833 -2.812 -3.829 -4.692 -5.270 -5.480
0.335 1.438 2.030 2.853 3.950 4.223 4.451 4.392 5.610 6.648
-0.920 -1.715 -2.386 -3.267 -4.137 -3.840 -4.713 -4.377 -7.327 -7.602
0.000 -0.726 -0.620 -0.607 -1.361 3.315 3.502 4.705 -2.356 -4.053
0.920 1.862 2.466 3.323 4.355 5.073 5.872 6.426 7.697 8.615
c
Error d
Φi
ωi
γI
158.66 129.46 124.58 120.86 114.89 123.67 130.70 136.89 133.21 129.50
180.00 -157.05 -165.44 -169.48 -161.79 139.20 143.39 132.93 -162.18 -151.93
-21.34 -50.54 -55.42 -59.14 -65.11 -56.33 -49.30 -43.11 -46.79 -50.50
εPi
εLi
0.080 0.495 0.565 1.316 0.845 2.732 1.327 2.224 3.967 3.548
0.535 0.398 0.714 0.687 0.580 1.079 1.402 1.834 1.292 1.121
C) Idiothetic PI Input parameters
Exo. Cart.
Ego. Cart.
Exo. and Ego. Polar d
i
li
αi
θi¼θi-1+αi-1
Xi
Yi
Ui
Vi
Di
1 2 3 4 5 6 7 8 9 10
0.920 1.151 0.630 0.878 1.105 1.016 1.042 0.865 1.348 1.059
-20.10 0.51 5.43 -27.49 61.31 37.75 35.46 -41.39 -34.82 ———
0.00 -20.10 -19.59 -14.16 -41.66 19.65 57.40 92.87 51.48 16.65
0.920 2.000 2.594 3.445 4.271 5.228 5.789 5.746 6.586 7.601
0.000 -0.395 -0.607 -0.822 -1.556 -1.214 -0.336 0.527 1.582 1.885
-0.920 -2.014 -2.647 -3.541 -4.225 -4.515 -2.835 -0.239 -5.339 -7.822
0.000 -0.316 -0.298 -0.046 -1.676 2.902 5.058 5.765 4.167 0.372
0.920 2.039 2.664 3.541 4.545 5.367 5.799 5.770 6.773 7.831
c
Error d
Φi
ωi
γI
0.00 -11.18 -13.17 -13.41 -20.02 -13.08 -3.33 5.24 13.51 13.93
180.00 -171.08 -173.57 -179.25 -158.36 147.27 119.27 92.38 142.03 177.28
180.00 168.82 166.83 166.59 159.98 166.92 176.67 -174.76 -166.49 -166.07
εPi
εLi
0.080 0.063 0.317 0.694 0.994 1.976 3.693 6.490 4.136 2.217
0.080 0.111 0.296 0.454 0.697 0.595 1.000 1.977 2.551 2.823
Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
B) Allothetic PI
a
computed with respect to X axis in standard position for allothetic PI. computed with respect to X axis rotated by θt1 for idiothetic PI. computed directly using recurrent PI formulae within the exocentric (Eqs 2a and 2b) or egocentric (Eqs 4a and 4b) frame of reference, but one can check that the very same values of D, ω and γ can also be obtained indirectly from X, Y and θ values (by applying Eqs. 1c, 1d and 1e) or from U, V and θ values (by applying Eqs. 3c, 3d and 3e). d D and γ estimates take the very same values when computed within exocentric (Eqs. 2a and 2d), egocentric (Eqs. 4a and 4c) or mixed (Eqs. 5a and 5b) frame of reference. b c
165
166
Letter to Editor / Journal of Theoretical Biology 349 (2014) 163–166
Based on indirect evidence, I have always considered that PI should involve a polar egocentric coding (Benhamou et al., 1990; Benhamou and Séguinot, 1995; Benhamou, 1997). Indeed, sensory inputs as well as motor outputs are necessarily expressed egocentrically, and PI (and other navigation systems) outputs seem to be suitably expressed in terms of distance and direction. To implement PI in an exocentric frame of reference, a neural structure would have to perform a two-fold conversion of frame of reference (ego–exo–ego). Such a conversion is theoretically necessary in eidetic navigation, where the goal location is coded with respect to landmarks. However, contrary to mammals which are indeed able to build up exocentric spatial representations thanks to neural structures such as the hippocampus and the post-subiculum, insects appear to by-pass the frame conversion to solve the task in a purely egocentric mode, but in a much more rigid way than mammals (Benhamou, 1997, 1998). As PI can be implemented in any frame of reference with the same final accuracy, it seems quite unlikely that the dedicated neural system would bother with frame conversions. When looking for mathematically exact formulae, the most suitable way to model idiothetic PI thus appears to be provided by Eqs. (4a) and (4b). In allothetic PI, angular sensory inputs correspond to compass-based heading estimates (θ) and the home direction should be estimated in terms of compass bearing (γ). It may be therefore more relevant for allothetic PI to use recurrent polar formulae expressed in a mixed ego/exo-centric frame of reference, which is centred on the animal's current location but oriented with respect to the compass reference direction, to directly provide the distance to home and its compass bearing: 2
Di ¼ ½D2i 1 þ li –2Di 1 li cos ðγ i 1 –θi Þ0:5
ð5aÞ
γ i ¼ γ i 1 þ atan2 ½ sin ðγ i 1 –θi Þ; Di 1 =li – cos ðγ i 1 –θi Þ
ð5bÞ
with D1 ¼l1 and γ1 ¼θ1 71801. If necessary, ωi can be obtained from Eq. (5b) as ωi ¼γi θi. It is however worth noting that actual PI rests on neural network computations that can only approximate mathematically exact formulae. This explains why an unexpected systematic bias occurs in the homing vector (in addition to random errors) when animals are forced to perform paths whose right and left turns are not balanced (e.g. L-shaped and U-shaped paths), both in species performing allothetic PI such as ants (Müller and Wehner, 1988; Merkle et al., 2006) and in species performing idiothetic PI such as hamsters and dogs (Séguinot et al., 1993, 1998; Séguinot 2000). We therefore need to develop neural network models able to perform PI updating in a biologically relevant way. Previous attempts by Hartmann and Wehner (1995) and Wittmann and Schwegler (1995) corresponded only to neural networks that implement approximate (based on Müller and Wehner, 1988) or mathematically exact, respectively, recurrent PI formulae. More
recently, Bernardet et al. (2008) developed a more biologically inspired neural network model. Contrary to what Cheung and Vickerstaff (2010) and Cheung (2014) claimed, there is no need to think about neural networks that specifically rely on some form of Cartesian exocentric coding. Acknowledgements I sincerely thank two anonymous referees for their constructive comments on a previous draft of this letter. References Benhamou, S., 1997. On systems of reference involved in spatial memory. Behav. Process. 40, 149–163. Benhamou, S., 1998. Place navigation in mammals: a configuration-based model. Anim. Cognit. 1, 55–63. Benhamou, S., Sauvé, J.-P., Bovet, P., 1990. Spatial memory in large scale movements: efficiency and limitation of the egocentric coding process. J. Theor. Biol. 145, 1–12. Benhamou, S., Séguinot, V., 1995. How to find one's way in the labyrinth of path integration models. J. Theor. Biol. 174, 463–466. Bernardet, U., Bermúdez i Badia, S., Verschure, P.F., 2008. A model for the neuronal substrate of dead reckoning and memory in arthropods: a comparative computational and behavioral study. Theory Biosci. 127, 163–175. Cheung, A., 2014. Animal path integration: a model of positional uncertainty along tortuous paths. J. Theor. Biol. 341, 17–33. Cheung, A., Vickerstaff, R.J., 2010. Finding the way with a noisy brain. PLoS Comp. Biol. 6, 1000992. Hartmann, G., Wehner, R., 1995. The ant's path integration system: a neural architecture. Biol. Cybern. 73, 483–497. Merkle, T., Rost, M., Alt, W., 2006. Egocentric path integration models and their application to desert arthropods. J. Theor. Biol. 240, 385–399. Müller, M., Wehner, R., 1988. Path integration in desert ants, Cataglyphis fortis. Proc. Natl. Acad. Sci. U.S.A 85, 5287–5290. Séguinot, V., 2000. l'intégration du trajet: approches théorique et expérimentale chez le hamster et le chien (Ph.D. Thèses). Université de Genève Séguinot, V., Cattet, J., Benhamou, S., 1998. Path integration in dogs. Anim. Behav. 55, 787–797. Séguinot, V., Maurer, R., Etienne, A.S., 1993. Dead-reckoning in a small mammal: the evaluation of distance. J. Comp. Physiol. A 173, 103–113. Vickerstaff, R.J., Cheung, A., 2010. Which coordinate system for modelling path integration? J. Theor. Biol. 263, 242–261. Wehner, R., Michel, B., Antonsen, P., 1996. Visual navigation in insects: coupling of egocentric and geocentric information. J. Exp. Biol. 199, 129–140. Wittmann, T., Schwegler, H., 1995. Path integration – a network model. Biol. Cybern. 73, 569–575.
Simon Benhamou CEFE, CNRS Montpellier, France E-mail address: [email protected] Received 25 October 2013 Available online 22 February 2014
