Table 1.
The cross-insect properties of our model.
Fig 1.
Overview of the modelling pipeline.
Our simulation consists of four consecutive models (left to right). Given the position of the sun, a realistic skylight model [44] provides the predicted luminance, degree and angle of linear polarisation for every point in the sky-dome. This provides input to the eye model, based on biological data from insects [10, 38], which defines an array of polarisation sensitive photo-receptors facing different parts of the sky, and uses opponent processing to produce luminance-independent POL-neuron responses. The compass model provides a hypothesis for the unknown neural process that converts the POL-neuron response to a true compass signal in the TB1 neurons; this also utilises information about tilt of the sensor array, and allows for the movement of the sun with passing time. Finally, the compass neurons’ output is used along with speed as input to an anatomically grounded model of the central complex [35] which performs path integration and produces an output signal that can steer the insect back home. Blue boxes represent known systems in the pathway of the skylight; the red box represents a fuzzy/unknown system, which is the main focus of modelling in this paper.
Fig 2.
Sample output from the skylight dome model [44].
(A) The luminance pattern of the sky is proportional to its intensity and describes the amount of light per area unit existing in a specific direction. Along with the chromaticity coordinates, it can provide spectral information. (B) The degree of linear polarisation pattern in the sky based on the scattering of the light on atmospheric particles. It is defined by the fraction of the polarised portion over the total intensity. The red line on the colour-bar showing the d = 0.75 indicates the maximum DOP observed in the skylight simulation. (C) The angle of polarisation pattern in the sky is defined by the average e-vector (electric part of an electromagnetic wave) orientation of the photons. The black circle in the figures denotes the horizon. In all panels the sun position is 30° south.
Fig 3.
Processing stages of light in the biological and artificial DRA.
(A) Top view of the fan-like arrangement of the ommatidia on the Cataglyphis DRA for both the right (green) and left (red) eyes; adapted and modified after [36]. (B) A closer look at the DRA, which is composed of hexagonal ommatidia. (C) An ommatidium on the DRA of the compound eye of the Cataglyphis has 8 photo-receptor cells, with parallel microvilli direction in 2, 3, 4, 6, 7 and 8, and perpendicular in 1 and 5; the colour violet indicates sensitivity to ultraviolet light. (D) Top view of the fan-like arrangement of the units on our sensor. The dashed lines show the overlap with the areas of the left (red) and right (green) Cataglyphis DRAs. (E) 3D representation of the sensor array in the eye model with visual field ω = 56°: the 60 discs on the dome are different units (ommatidia) with acceptance angle ρ = 5.4°; the orientation of the lines on the circles denote the direction of the main (parallel) polarisation filter. (F) Model of a POL-unit: the photo-receptor neurons combine a UV-sensor (photo-receptor) and a polarisation filter (microvilli), and have a square-root activation function. The normalised difference of the photo-receptor neurons is calculated by the POL interneurons. The empty triangular and dashed synapses denote excitatory and inhibitory connections respectively. (G) Simulated response of the two photo-receptors in one unit in partially linearly polarised light of intensity I = 1 and degree of polarisation d = 0.9 against different e-vector orientations. (H) Simulated response of the POL-neuron to the input of Fig 3G; the dashed line shows the response of the POL-OP interneuron, and the solid line is the response of the POL-neuron (normalised difference). B and C figures adapted and modified from [46]. F, G and H are after [10].
Fig 4.
Overview of the compass model.
The light information stimulates the photo-receptor neurons. Using the eye model, this is transformed to a POL-neuron population code. Tilting information (orange) is propagated through the gating function and creates a mask for the POL-neurons’ response, altering the relative weighting of information from different parts of the sensor array (see Fig 5). By combining information across the array, the response of SOL-neurons encodes an estimate of the solar azimuth and elevation [black arrows indicate the direction]. The TCL-layer uses the elevation information along with passing time as an ephemeris function (green) which modulates its input weights to rotate the SOL-neuron output and provide a true north estimate [black arrows indicate the
direction]. The response across the TCL-neurons is in the form of a sine-wave which can be decoded to determine an estimate of the azimuth (phase) and of the confidence of that estimate (amplitude). Note however that CX path integration circuit can use the TCL-neuron activity directly, without this explicit decoding.
Fig 5.
The gating function that compensates for tilt.
The differing weightings of ommatidia input under three levels of tilt [(A) δ = 0° (θt = 90°), (B) δ = 30° (θt = 60°) and (C) δ = 60° (θt = 30°)] are shown, with darker shading indicating higher weighting. The inner dashed black circle delineates the actual receptive field of the simulated sensor (28° radius, equivalent to ω = 56° receptive field). The extended array (greyed out units) illustrates how this weighting adheres to a Gaussian function defined on the sky dome. The blue circle shows the dominant focus of the sensor (θg = 40°), and the green arrow shows the smoothing parameter (σg = 13°).
Fig 6.
Using confidence of the estimate to compensate for time.
(A) The confidence value of the compass response varies with the solar elevation (black dots). Within the range 23° − 72° this relationship can be used to estimate the elevation using Eq (8) (red dots). (B) The rate of change of the solar azimuth over time depends on the elevation [coloured dots represent different times of the day from morning (blue) to the evening (red)] and can be approximated using Eq (9) (black line). (C) Showing the solar elevation with respect to the solar azimuth for different times of the year. Each of the 12 imaginary curves in B and C correspond to the 21st of every month; the sampling rate in each day is every 10 minutes from sunrise to sunset.
Fig 7.
Step-by-step processing of the compass model.
The white, orange and green areas show the response of the POL-, SOL- and TCL-neurons respectively; red denotes excitation and blue suppression (see colour-bar for values). The set of synaptic weights connecting each layer is shown under the disc (values based on the same colour-bar). Orange background means that the activity is affected by the gating function, and green that it is affected by the time compensation mechanism. In the white disc, different points are the relative positions of the POL units on the sensor; the numbering starts from the centre and unwraps clockwise towards the outline like a spiral; the round green mark is the point of the sensor that is aligned with the zenith of the sky, and the yellow circle is the sun position. The black arrow with the dashed line is the decoded prediction of the solar azimuth from the TCL-neurons; the numbering refers to the identities of the neurons in weight matrices below. The weight matrices show the synaptic weight between consecutive layers [defined by Eqs (6) and (7) for the SOL and TCL respectively; values based on the same colour-bar]; the horizontal is the input and the vertical the output axis. (A) The sum-of-sinusoids mechanism detects the solar azimuth; the zenith (green) point is aligned with the sensor orientation; the solar azimuth is encoded in both the SOL- and TCL-layers and the activation code (i.e. phase) of the two layers look identical, as the time compensation mechanism has been deactivated. (B) The tilt compensation mechanism corrects the predicted solar azimuth using tilting information; the sensor has been tilted 30° NNW (so now the zenith point is 30° SSE); the gating function has changed the focus on the specific ommatidia, as shown in the WSOL ⋅ g matrix. (C) The time compensation mechanism corrects for the solar azimuth changes using the solar elevation; 8 hours have passed so the sun has moved 120° clockwise but the compass is still aligned to the same direction due to the updated WTCL weights (see also the difference in SOL and TCL responses compare to previous steps).
Fig 8.
The objective function and the accuracy of the compass.
(A) Schematic representation of the objective function; the yellow and green suns illustrate the real and estimated sun position; the disk around the green sun denotes the uncertainty of the estimation, . (B) Mean absolute angular error (coloured solid lines), MAE ± SE, and confidence (black dashed line), τs, against the solar elevation for different disturbance levels, when the sensor points towards the zenith. (C) The mean absolute angular error (black solid line), uncertainty (σs; black dashed line) and confidence (τs; grey dashed line) against different disturbance levels. On the bottom there are some examples of the responses of the POL units in (D) a non-disturbed condition; (E) with η = 33% disturbance; (F) with η = 66% disturbance; and (G) with η = 99% disturbance. The insets show a sample of the sky that caused the responses and the yellow mark on it shows the position of the sun.
Fig 9.
Dealing with time and light disturbance.
Transformation of the compass response to solar elevation, and to the derivative of the solar azimuth function. (A–C) Function of the elevation with respect to the response with disturbance η = 6%, η = 26% and η = 43%; the red dots are the predicted solar elevation given the compass response, and the black dot denote the real values; the dashed lines give the range of the function. (D) Function of the real derivative of the solar azimuth against its prediction using Eq (9); black line shows perfect match of the two derivatives; colour denotes the time: blue is for morning and red is for evening.
Fig 10.
Dealing with tilt for a variety of gating parameters.
Angular error of the expected direction of the sensor for different tilted angles and gating parameters. Black arrows show the axes; red shading shows the value of the objective function (J)—darker shading is for higher error values; red shaded points show the values of J for different sun positions; green discs show the zenith angle, θt; black dashed lines visualise this angle for any tilt direction, ϕt (red arrows). (A-F) Visualisation of the azimuthal angular error with respect to the sun position, i.e. ϵ ∈ [0, 90] and α ∈ [0, 360), for three tilting angles of the sensor—(A, D) δ ≈ 0° (θt = 90°), (B, E) δ ≈ 30° (θt = 60°), (C, F) δ ≈ 60° (θt = 30°); without [top row (A-C)] and with gating [bottom row (D-F)]. (G) Average angular error for different gating parameters. The lowest cost (green star; J = 10.47° ± 0.12°, N = 8, 500) is for ring radius θg = 40° and width (variance) σg = 13°.
Table 2.
Mean absolute error before and after using the gating function.
Fig 11.
Optimal compass structural parameters.
The performance of the compass for different topological parameters. (A) Values of the objective function on the ω × n plane; red shades illustrate the degree of error; black arrows show the axes; the green line shows the receptive field value associated with the minimum error for different number of units. (B) The error as a function of the number of units, n; the receptive field is fixed at ω = 56°; inset demonstrates the 56° wide sensor with n = 360 units; with green are marked the 60 units closest to the ones we chose. (C) The error as a function of the receptive field, ω; the resolution (ratio between ω and n) is fixed so that the number of units is n = 60 for ω = 56°; inset demonstrates the different visual fields including the optimal one with n = 60 units.
Fig 12.
Behavioural simulation for the path integration task.
Testing the celestial compass on path integration tasks. We set up the experiments to take place at 10am in Seville, Spain (37°23′33.03′′N, 5°53′01.95′′W). The altitude variance is 0.8 m and the maximum tilting angle noticed in all the experiments is δ = 47°. (A) Five representative routes of ants in different sky disturbance levels for an even [(B) uneven] terrain and their respective inward paths; different colours are for different disturbance levels (see legend); the faded lines are the outward paths and the bold ones are the inward. (C) The uneven terrain map; green colour denote hills and purple valleys; the marked region is the one cropped for the A and B plots. (D) Deviation from the best possible route during homing for different disturbance levels for even [(E) uneven] terrain. We scale up our experimental arena (by a factor of 120) to enable longer runs that demonstrate the performance of the time compensation mechanism. (F) Comparison of the path integration performance in terms of tortuosity with (solid black line) and without using the time compensation mechanism (dashed line). (G) The actual paths generated by the above experiment; green arrows show the direction of the sun at the beginning (10:00 am, 103.65° clockwise from north) and end of the route (11:16 am, 127.47° clockwise from north).
Fig 13.
Real and simulated response of compass neurons for artificial and natural polarised light.
E-vector orientation resulting in maximum excitation, ϕmax, for TB1-neurons in different columns of the CX (A,B) and TCL neurons in our simulation, determined by circular statistics (Rayleigh test [55]): (A) Real data of TB1-neurons (N = 15) in the PB of the locust brain, reconstructed from original data supplied by Stanley Heinze after [33] (inset shows the naming of the columns in the PB), (C) simulated TCL-neurons under a rotating linear polariser, and (E) under a rotating natural sky (N = 100, η = 50%, θs = 30°); black dots show individual samples, red squares show the mean ϕmax and red stars show the ϕmax in a condition without external disturbance (η = 0%); red lines are best linear fit to the data points. (B,D,F) show corresponding examples for specific neurons from four of the CX columns; bin width 10°; black solid lines indicate background activity; red lines indicate the ϕmax direction.
Table 3.
Properties of dorsal rim ommatidia in different species.