Illustration of research question, experimental setup, and stimuli. A) Bumblebees experience rapid changes in the angle of polarization as they fly under natural skies. Left: polarization pattern of the sky. Angles of polarization are oriented along concentric circles around the sun. While direct sunlight is unpolarized, the degree of polarization, indicated by the thickness of the bars, increases with increasing angular distance from the sun. Right: naturalistic flight trace with saccadic yaw turns alternating with translational flight. B) Schematic illustration of intracellular recordings from compass neurons in the central complex of Bombus terrestris during stimulation with a rotatable polarizer backlit by a UV-LED (365 nm). C) Counter-clockwise rotation (180°–0°) of the polarizer at discrete angular velocities between 30°/s and 1920 °/s. Each rotation velocity was presented for 12 s. D) Upper panel: naturalistic stimulus sequence reproducing the dynamics of an actual flight (original flight data from Boeddeker et al., 2015). Lower panel: rotation velocities of the naturalistic stimulus used in this study.

CX neurons can reliably encode naturalistic stimuli. A) Direct volume rendering of dye-injected TL3 neuron. B) Neuronal responses of TL3 neuron to selected rotation velocities (30°/s, 120°/s, 480°/s, 1440°/s) of a counter-clockwise rotation stimulus. Grey areas show responses to 480°/s and 1440°/s at a stretched timescale. C) Circular response histograms for clockwise (cw; purple) and counter-clockwise (ccw; turquoise) rotation for all presented velocities showing the preferred firing directions of the tested TL3 neuron. The mean preferred firing directions and the mean vector length are indicated by the golden/black lines. The 95% confidence intervals are given by the black arcs. D) Proportion of significantly directed responses for all recordings at the different rotation velocities (cw and ccw plotted together; n=24, N=48). E) Normalized mean spike rate for all recordings at the different rotation velocities (cw and ccw plotted together; n=24, N=48). Boxplots show median, interquartile range (IQR), whiskers with 1.5 x IQR and outliers greater than 1.5 x IQR. F) Upper panel: Naturalistic stimulus sequence according to head orientations obtained from freely flying bumblebees (Boedekker et al., 2015). Red line shows the preferred angle of polarization (AoP), the blue line the anti-preferred AoP of this neuron. Middle panel: Intracellular recording trace of the same neuron as above responding to the naturalistic stimulus sequence. Lower panel: Raster plots of five consecutive trials of stimulation with the naturalistic stimulus sequence.

Measured and modeled TL3 neuron responses to continuously rotating and naturalistic stimuli. A) Preferred angle of polarization (AoP) for every velocity and direction (cw: lilac, ccw: turquoise) plotted against rotation velocity. The individual 12 s rotations were divided into two segments of 6 s (first segment, light circles; second segment, dark circles) to illustrate reproducibility of the responses. Circular linear regression analysis (Kempter et al., 2012) was carried out on the five highest rotation velocities. The absolute value of the slope of the regression indicates the delay of the neuron. Dashed line indicates extrapolation of regression towards slow rotation velocities. The stars symbolize the preferred AoP values obtained from the response of the mathematical model using the same stimuli and model parameters as obtained from fit to naturalistic stimulation of the same neuron. The close-up of the low-velocity range on the right shows deviation of the preferred AoPs from the extrapolated regression line (dashed line). Note that ccw values (lilac) in this velocity range are smaller than cw values (turquois), whereas they are bigger at higher velocities. B) Bumblebee CX-neurons show spiking history dependent responses like post-excitatory inhibition (left panel, after presenting preferred AoP for 10 s) and post-inhibitory rebound excitation (right panel, after presenting anti-preferred AoP for 10 s). Recorded data were digitally highpass-filtered using a 0.1 Hz first order Bessel filter. C) Upper panel: naturalistic stimulus sequence, lower panel: recorded neuronal responses (grey, peristimulus time histogram of all 8 stimulus repetitions) and fitted model with spiking history (orange) and without spiking history (blue). D) The difference of Akaike’s information criterion (AIC) for model with history and model without history calculated for each of the recordings (n=18). A value less than 0 indicates that the model with history fits better. E) Comparison of preferred AoP values that were recorded (dots in A), with those calculated from the model parameters that were fitted to the naturalistic stimulus response (stars in A) of the same neurons (n=5, N=10). The absolute difference between these values was calculated and plotted. Boxplots show median, interquartile range (IQR), whiskers with 1.5 x IQR and outliers greater than 1.5 x IQR.

Model of neuronal population responses. The simulation was run using the averaged model parameters from all fitted TL3 neurons n=8 modelling a population of 180 neurons with preferred AoPs between 0° and 179°. A) and A’) Heat maps represent model responses including the spiking history (A) or without spiking history (A’) of the population model to the first 20 s of the stimulus (white) from Boeddeker et al. (2015). The response of each individual neuron in the population corresponds to the response profile along a horizontal line at each preferred AoP. B) and B’) Mean spiking activity of the model population including the spiking history (orange; B) excluding spiking history (blue; B’). Grey line shows average over time. C) Population Vector Average (PVA) with spiking history (orange) or without spiking history (blue). Grey shows the stimulus. Red circles: PVA overshoots the stimulus. Pink circles: PVA from model with history has decreased delay. D) Histogram of heading errors calculated as: heading error = AoPstimulus – PVA. Orange: with spiking history, blue: without spiking history.

Dynamic properties of neuronal population model. A) Angle of Polarization (AoP; black line), population vector average with history (orange) and without history (blue) for the first 20 s of the naturalistic stimulus (black) created from all flights from Boeddeker et al. (2015). B) First derivative of A), i.e. angular velocity. C) and C’) First and second boxplot: PVA change rate/AoP change rate either after a change of rotation direction (n = 138) or after no change of rotation direction (n = 315). Medians in (C) are significantly different (Mann-Whitney U test, p = 1.3*10-11). Third and fourth boxplot: PVA change rate/AoP change rate either after at least 200 ms of translation (n = 99) or after less than 200 ms of translation (n = 354). Medians in (C) are significantly different (Mann-Whitney U test, p = 2.3*10-11). Ratio in (C’) is 1 for all data points. Boxplots show median, interquartile range (IQR), whiskers with 1.5 x IQR and outliers greater than 1.5 x IQR. D) and D’) Influence of the duration of translational flight before a saccade on the PVA change rate. This is expressed as the ratio between the peak values of the PVA change rate and the AoP velocity. E) and E’) cross-correlation between mean activity of the population and absolute angular velocity of the AoP (C: with spiking history, C’: without spiking history).

Graphical illustration of model calculations. Neuronal firing rate (R) over time (t), elicited by the stimulus was modeled as a cos2 function of the angle of polarization (AoP). Where A is the response amplitude, del is the system’s delay and Φmax is the preferred angle of polarization of the neuron. To accommodate for spiking-history-effects, a spiking history (Rhis) was implemented into the model. Spiking history was defined by two parameters: the length of a rectangular window, over which past responses are averaged and Ahis, a scaling factor for the spiking-history-effect. Rhis was then subtracted from the response to the current stimulus (R), to give the total response Rtot. Since the model calculates spike rate, negative values were replaced by zeros.

Starting parameter and Bootstrapping for model fit. The parameters preferred AoP, delay, response amplitude, history duration (length of memory window), and amplitude of spiking history were used as free parameters. To avoid the detection of local minima in the fitting procedure, we repeated the fitting procedure 50 times for each recording and initialized the fitting parameters each time with random start values. The fitted parameter values were then obtained from the best of the 50 fits. To assess the quality of the fit, we used a bootstrapping method. We repeated the fitting procedure 1000 times, but instead of fitting the model to the original PSTHs, we randomly permuted the bins of the PSTHs before the fit. We then compared the sum of squared errors of the fit to the actual data to the sum of squared errors of the permuted data.

Results for concatenated flight tracks of all recorded bumblebee flights from Boeddeker et al. 2015. A) grey: Angle of Polarization (AoP), orange: Population vector average (PVA) for model with spiking history, blue: Population vector average (PVA) for model without spiking history. B) First derivative from the track from A showing angular velocities. The PVA values and their derivatives are shifted by the delay of the system for better comparison to the stimulus. C) The heatmap represents the response of the modelled population of neurons (with spike history) to the stimulus (white line). D) Grey shows the stimulus, orange the population vector average (PVA) shifted to the left by the neuronal delay and calculated from the responses in C. E) Heading Error calculated as AoP – PVA. F) Orange: Population mean activity from the responses including models spiking history, grey: averaged activity over time. G) The heatmap represents the responses for population of neurons to the stimulus (white line) without the models spiking history. H) Grey shows the stimulus, blue the population vector average (PVA) shifted to the left by the neuronal delay and calculated from the responses in G. I) Heading Error calculated as AoP – PVA. J) Blue: Population mean activity from the responses including models spiking history, grey: averaged activity over time.

Population model response to stimulus with sparse rotations. A) and A’) The heatmaps represent the responses for a modelled population of neurons to an artificial stimulus with sparse rotations (white line) including the models spiking history (A) or without the spiking history (A’). Note that population activity (with history) declines over time and is higher, the larger the rotation angle. B) Grey shows the stimulus, orange the population vector average (PVA, light orange) shifted to the left by the neuronal delay (dark orange) and calculated from the responses in A. B’) Grey shows the stimulus, blue the population vector average (PVA, light blue) shifted to the left by the neuronal delay (dark blue) and calculated from the responses in A. C) and C’) Heading Error calculated as AoP – PVA, in orange for model with (orange) and without (orange) spiking history. D) and D’) Population mean activity from the responses including models spiking history (orange) or without spiking history (blue), grey: averaged activity over time.