Migrating eastern North American monarch butterflies use a time-compensated sun compass to adjust their flight to the southwest direction. Although the antennal genetic circadian clock and the azimuth of the sun are instrumental for proper function of the compass, it is unclear how these signals are represented on a neuronal level and how they are integrated to produce flight control. To address these questions, we constructed a receptive field model of the compound eye that encodes the solar azimuth. We then derived a neural circuit model that integrates azimuthal and circadian signals to correct flight direction. The model demonstrates an integration mechanism, which produces robust trajectories reaching the southwest regardless of the time of day and includes a configuration for remigration. Comparison of model simulations with flight trajectories of butterflies in a flight simulator shows analogous behaviors and affirms the prediction that midday is the optimal time for migratory flight.

A long-standing, fundamental question about monarch migration has been how the circadian clock interacts with the changing position of the sun to form a time-compensated sun compass that directs flight. Here, we propose a neuronal model for both encoding of the sun’s azimuthal position, and molecular timekeeping signals, and how they can be compared to form a time-compensated sun compass. Our results propose a simple neural mechanism capable of producing a robust time-compensated sun compass navigation system through which monarch butterflies could maintain a constant heading during their migratory flight.

While the monarch time-compensation clocks are housed in the antennae, the sun’s azimuthal position is detected by the eyes. Neuronal signals originating from photoreceptors in each ommatidial unit of the compound eye are processed by several optic neuropils before they are relayed to the central brain. A main target for visual neurons in the central brain is the anterior optic tubercle (AoTU), a structure that is connected to the central complex (CX), the presumed site of the sun compass (). Within the CX, the lateral accessory lobes (LALs) are of particular relevance because neurons within them connect to descending motor pathways that ultimately control behavior.

Monarch antennae contain an intracellular, light-sensitive clock mechanism, which has been shown to be responsible for calibrating the sun compass (). As in Drosophila and mice, the primary molecular mechanism of the monarch clock is an autoregulatory transcriptional feedback loop. In the monarch, the feedback loop utilizes two distinct cryptochromes (CRYs), a blue light circadian photoreceptor (CRY1) and a transcriptional repressor (CRY2). Transcription factors CLOCK (CLK) and CYCLE (CYC) drive the transcription of the period (per), timeless (tim), and cry2 genes, which are translated into PER, TIM, and CRY2 proteins, respectively ().

Each fall, eastern North American monarch butterflies (Danaus plexippus) fly up to 4,000 km to specific overwintering sites in central Mexico. Throughout this journey, monarchs constantly correct their flight direction to maintain a southerly orientation, using a time-compensated sun compass. Laboratory observations from a flight simulator, capable of tracking flight direction, show that migrant monarchs orient toward the southwest (SW) direction by visual cues, relying primarily on the horizontal (azimuthal) position of the sun (). Migrants use their circadian clocks to compensate for the changing sun position and thereby maintain a fixed flight bearing.

Results

Colwell, 2011 Colwell C.S. Linking neural activity and molecular oscillations in the SCN. Belle et al., 2009 Belle M.D.C.

Diekman C.O.

Forger D.B.

Piggins H.D. Daily electrical silencing in the mammalian circadian clock. A basic assumption of our model is that time and solar signals are encoded by neuronal firing rate. This allows us to propose a mechanism, which uses a small number of neurons, to compare the firing rate of azimuthal neurons, responding to the luminance detected by the eyes, with neurons whose firing rate shows a circadian rhythm, as seen in “clock” neurons in other species, (see) and as an outcome directs flight. We first define the neural input signals into such a mechanism.

Merlin et al., 2009 Merlin C.

Gegear R.J.

Reppert S.M. Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. ( N C L K 1 = B G − N C L K 1 _ C ) . As a result, the NCLK1 firing rate is anti-phase to the oscillation pattern of cry2. NCLK2 follows the oscillation pattern of per/tim transcription factors directly ( DeWoskin et al., 2015 DeWoskin D.

Myung J.

Belle M.D.

Piggins H.D.

Takumi T.

Forger D.B. Distinct roles for GABA across multiple timescales in mammalian circadian timekeeping. Myung et al., 2015 Myung J.

Sungho H.

DeWoskin D.

De Schutter E.

Forger D.B.

Takum T. GABA-mediated repulsive coupling between circadian clock neurons in the SCN encodes seasonal time. Ukai-Tadenuma et al., 2008 Ukai-Tadenuma M.

Kasukawa T.

Ueda H.R. Proof-by-synthesis of the transcriptional logic of mammalian circadian clocks. Figure 1 Circadian and Solar Azimuth Signals Show full caption (A) The antennal clock of the monarch. 1 ) Molecular time course curves involved in keeping the circadian rhythm. The per/tim/cry2 (blue/red/green) relative mRNA levels measured in the antennae ( Merlin et al., 2009 Merlin C.

Gegear R.J.

Reppert S.M. Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. (A) Molecular time course curves involved in keeping the circadian rhythm. The per/tim/cry2 (blue/red/green) relative mRNA levels measured in the antennae (). Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. Science 325, 1700–1704. (Reprinted with permission from AAAS.) (A 2 ) Signals recorded from the antenna using EAG over 2 days (D1-cyan and D2-magenta; normal L:D [12:12] regime; and freq. >65 Hz). The points in each light segment are fitted with a third order polynomial fit (dashed curves) and globally with a cosine fit (solid blue curve: m a x = 9 Z T , m i n = 18 Z T ). (A 3 ) Firing rate profiles of neurons NCLK1_C (light green; accords with cry2), NCLK1 (cyan), NCLK2_C (dark green), and NCLK2 (navy; accords with per/tim). (B) “The butterfly eyes”, Graphium cloanthus kuge butterfly eyes reflecting the direction of the light (Credit: Yung Samuel. The Butterfly eyes. http://www.samuelphotos.com ). (B 1 ) Luminance of rotating light source captured by two hemispheres that model the monarch eyes. (B 1 ) Luminance captured by left/right hemispheres (top). (B 1 ) Luminance captured by back/front hemispheres (bottom). (B 2 ) Firing rate curves of NS1 and NS2 neurons which receptive fields cover the right and the front hemisphere, respectively. For neural representation of the clock, we postulate two neurons, NCLK1 and NCLK2, whose firing rates oscillate in accordance with two primary molecular oscillations generated by the monarch circadian clock ( Figure 1 A1 ) (). NCLK1 is linked to the oscillations of cry2 transcription factor through a neuron called NCLK1_C. In the absence of input, NCLK1 fires with constant background firing rate (BG) and receives inhibitory input from NCLK1_C that follows oscillations of cry2. As a result, the NCLK1 firing rate is anti-phase to the oscillation pattern of cry2. NCLK2 follows the oscillation pattern of per/tim transcription factors directly ( Figure 1 A3). Analogously, we introduce a neuron labeled NCLK2_C with a firing rate that is anti-phase to the oscillation patterns found in NCLK2 and per/tim. These links between neural and molecular representations of the circadian rhythm are motivated by recent work describing the neuronal activity of clock neurons in various species (). The particular configuration of these links is phenomenological and aimed to show that a neural mechanism from molecular rhythms to firing rate with these phase relationships could exist.

T = 0 and a peak near mid afternoon a t T = 8 . This is surprisingly similar to the predicted activity of NCLK1. Non-monotonicity of the electric antennal signal has been also detected in Drosophila ( Tanoue et al., 2004 Tanoue S.

Krishnan P.

Krishnan B.

Dryer S.E.

Hardin P.E. Circadian clocks in antennal neurons are necessary and sufficient for olfaction rhythms in Drosophila. Each of the firing rate curves is non-monotonic, that is, they do not produce a unique firing rate during the daylight hours. This property precludes a single curve from being used to identify the time of day. However, we note that due to the 6 hr phase shift between the curves, a simple correction for this ambiguity could be to consider a sum of the NCLK1 and NCLK2 curves, which produces a monotonic curve. To gain further information on electrical signals possibly related to the circadian clock, we performed electro-antennogram (EAG) recordings over the course of several days under normal light/dark (L:D) conditions (see Supplemental Information ). Frequency analysis of the electrical activity suggests that intrinsic antennal electrical activity increases during the day and decreases during the night, illustrated in Figure 1 A2. During the day, when the sun compass is active, the signal power curve appears to be non-monotonic with lowest activity atand a peak near mid afternoon. This is surprisingly similar to the predicted activity of NCLK1. Non-monotonicity of the electric antennal signal has been also detected in Drosophila ().

Neri, 2006 Neri P. Spatial integration of optic flow signals in fly motion-sensitive neurons. Heinze et al., 2013 Heinze S.

Florman J.

Asokaraj S.

El Jundi B.

Reppert S.M. Anatomical basis of sun compass navigation II: the neuronal composition of the central complex of the monarch butterfly. For neural encoding of the solar azimuth, we consider neurons with visual receptive fields carrying information from the compound eye. A simple and robust way for detecting the solar azimuth is integration of luminance captured by all ommatidia in a wide receptive field that a neuron subtends (see Figure 1 B and Supplemental Information ). We note that when the eyes are modeled as two hemispheres separated by 180°, as approximately appears in the monarch, integration of the luminance from a single eye indeed provides azimuthal information, however, in just one Cartesian direction. This implies that azimuth detection is ambiguous. For example, azimuthal sun positions in front and behind the animal would yield the same integrated luminance in both left and right eye ( Figure 1 B1). A simple and sufficient way to measure the solar position in two Cartesian dimensions would be to consider receptive fields arranged 90° apart instead of 180°, implying that the solar azimuth is minimally encoded with two rhythms. We follow this reasoning and introduce two types of neurons, NS1 and NS2, with the simplest anatomical arrangement of receptive fields that allow unambiguous encoding of solar azimuth. The receptive field of NS1 comprises all ommatidia in a single eye (e.g., right eye), and its firing rate is proportional to the integrated luminance signal of the eye it subtends. The NS2 neuron has a receptive field composed of two quadrants of the two eyes (e.g., frontal hemisphere) and will produce a luminance signal and firing rate curve shifted by 90° relative to the NS1 response, see Figure 1 B2. The curves that we propose have been described in the insect visual system. Similar curves were obtained from recordings from Drosophila lobular plate tangential neurons (H and V neurons) (). Further, recent studies in monarchs and other insects have documented single, wide receptive field tangential neurons with dendrites that cover the whole lobula plate (). We expect NS1 and NS2 neurons to be of a similar type.

f l and f r . These neurons receive clock and azimuth signals and match them according to a particular wiring, denoted as I l and I r , to indicate whether a current flight angle is correct or whether to steer toward left ( f l ) or right ( f r ) directions. We then propose the following equations for modeling the firing rates of left and right control neurons, d f l d t = − α f l + βφ ( I l ) , d f r d t = − α f r + βφ ( I r ) , φ ( x ) = { 0 x < 0 , x x ≥ 0. (Equation 1)

The parameter α denotes the decay rate of the firing rate in the absence of input, and βφ ( x ) denotes that only positive inputs elicit response. Since neurons can encode motion, we assume that input wirings I l and I r correspond to the force used for steering and the firing rates will indicate the angular velocity used by the monarch for correction in each direction. Given the axisymmetric anatomy of the monarch, it is reasonable to assume that there are two parallel pathways, controlling leftward and rightward turns. Generally, in insect flight, turns come from opposite forces being applied to the left and right wings. Similarly, in our model, the total angular velocity is represented by the difference between left and right firing rate units, F = f l − f r = d A d t , (Equation 2)

with A representing the azimuthal angle between the sun’s position and the SW. We note that although the system in Bergou et al., 2010 Bergou A.J.

Ristroph L.

Guckenheimer J.

Cohen I.

Wang Z.J. Fruit flies modulate passive wing pitching to generate in-flight turns. m d 2 A d t 2 + v d A d t = I , (Equation 3)

in which the force ( I ) equals the mass times the acceleration and a damping coefficient ( v ) times the angular velocity. In our model, F represents d A / d t such that the system in F positive designates correction to the left, and F negative designates correction to the right. When the flight angle relative to the sun is correct, the firing rates of the two units will be equal and F = 0 . We define this state as the balanced state (fixed point). To propose a mechanism for integration of clock and azimuth neural signals into a time-compensated sun compass, we introduce two control neurons, whose firing rate is denoted byand. These neurons receive clock and azimuth signals and match them according to a particular wiring, denoted asand, to indicate whether a current flight angle is correct or whether to steer toward leftor rightdirections. We then propose the following equations for modeling the firing rates of left and right control neurons,The parameterdenotes the decay rate of the firing rate in the absence of input, anddenotes that only positive inputs elicit response. Since neurons can encode motion, we assume that input wiringsandcorrespond to the force used for steering and the firing rates will indicate the angular velocity used by the monarch for correction in each direction. Given the axisymmetric anatomy of the monarch, it is reasonable to assume that there are two parallel pathways, controlling leftward and rightward turns. Generally, in insect flight, turns come from opposite forces being applied to the left and right wings. Similarly, in our model, the total angular velocity is represented by the difference between left and right firing rate units,withrepresenting the azimuthal angle between the sun’s position and the SW. We note that although the system in Equation 1 was derived based on neural firing rate principles, it is directly related to flight commands, as it is analogous to a formalism based on Newton’s first law (),in which the forceequals the mass times the acceleration and a damping coefficienttimes the angular velocity. In our model,representssuch that the system in Equation 1 is equivalent to the system in Equation 3 positive designates correction to the left, andnegative designates correction to the right. When the flight angle relative to the sun is correct, the firing rates of the two units will be equal and. We define this state as the balanced state (fixed point).

( A f p ) in a plane consisting of the time of day ( T ) and the azimuthal angle ( A ) . We require that the integration mechanism will return to the balanced state at any time ( T ) , i.e., requiring the system in A f p as a line attractor ( Amit, 1992 Amit D.J. Modeling Brain Function: The World of Attractor Neural Networks. I l and I r that implement such a mechanism? We address this question by deriving a potential function V ( A , T ) (see A f p is a line attractor, and we use it to detect candidate integration mechanisms. From this formalism, we first note that straightforward matching of single clock and sun signals (e.g., NCLK1 and NS1), the simplest candidate mechanism, will not satisfy the required conditions, as this results in multiple balanced states. Alternatively, we find that matching all four signals, in a particular way, such that the left unit subtracts the sun azimuth inputs from the clock inputs and the right unit implements the opposite, I l = D 1 + D 2 , D 1 = N C L K 1 − N S 1 , I r = − D 1 − D 2 = − l l , D 2 = N C L K 2 − N S 2 , (Equation 4)

will satisfy the derived conditions. We illustrate this input wiring and the associated potential function V ( A , T ) in Figure 2 Integration Mechanism for Flight Orientation Control Show full caption (A) Fall migration to the SW direction. (A 1 ) SW measurements (early fall, MA, USA from the United States Naval Observatory) with respect to the solar azimuth (gray). Line approximation ( A f p ) to these points shown in blue. (A 2 ) Control requirements for the line-attractor: all points in the plane will converge to the SW. (A 3 ) Potential (Lyapunov) function V ( A , T ) that determines the direction of the change of the angular position for each point in ( T , A ) plane. As required, V ( A , T ) > 0 everywhere except on the line attractor. Its derivative, ∂ V ( A , T ) / ∂ A , points toward the line attractor. (A 4 ) Input wiring diagram of L and R neurons indicative into which direction to turn. Such wiring produces the function V ( A , T ) , which guarantees a line attractor. (A 5 ) Angular position of fixed points (stable-black and unstable-cyan), directions of correction of flight (blue-right and red-left), and the angle A f p (gray) in the compass view at different times of day (T = 0, 3, 6, 9, and 12). (B) Schematic indicating spring remigration. (B 1 ) Reflected integration circuit (NCLK1_C and NCLK2_C replace NCLK1 and NCLK2; sign of connections is flipped) produces a circuit that keeps the NE flight direction (used in spring remigration). For the integration mechanism to be viable, it has to maintain the balanced state once achieved. Because the solar azimuth increases approximately linearly over the day ( Figure 2 A1 ), the balanced state is represented as a linein a plane consisting of the time of dayand the azimuthal angle. We require that the integration mechanism will return to the balanced state at any time, i.e., requiring the system in Equation 1 to implementas a line attractor ( Figure 2 A2) (). We then ask, for the given firing rate curves of clock neurons NCLK1 and NCLK2 and azimuth neurons NS1 and NS2, what are the input wiringsandthat implement such a mechanism? We address this question by deriving a potential function(see Supplemental Information for the derivation), which specifies the conditions for testing given clock and azimuth wirings, as to whether the lineis a line attractor, and we use it to detect candidate integration mechanisms. From this formalism, we first note that straightforward matching of single clock and sun signals (e.g., NCLK1 and NS1), the simplest candidate mechanism, will not satisfy the required conditions, as this results in multiple balanced states. Alternatively, we find that matching all four signals, in a particular way, such that the left unit subtracts the sun azimuth inputs from the clock inputs and the right unit implements the opposite,will satisfy the derived conditions. We illustrate this input wiring and the associated potential functionin Figures 2 A3 and 2A4. We also show that for the specified input signals and their addition or subtraction (256 variations), this is the only wiring possibility that produces a robust flight direction for the duration of the day (see Supplemental Information ).

F = 0 . Stability analysis of this fixed point shows that it is unstable and as such serves as a separatrix; trajectories initiated near it are repelled. The balanced state and the separatrix merge at the SW direction, at T = 0 and T = 12 , and become apart with maximal separation at T = 6 ( Tammero et al. (2004) Tammero L.F.

Frye M.A.

Dickinson M.H. Spatial organization of visuomotor reflexes in Drosophila. F , we observe some qualitative features that our model indicates, in particular, presence of fixed points and opposite signals in between the fixed points (see In addition to the line attractor, there is another solution (fixed flight angle) that satisfies the equation. Stability analysis of this fixed point shows that it is unstable and as such serves as a separatrix; trajectories initiated near it are repelled. The balanced state and the separatrix merge at the SW direction, atand, and become apart with maximal separation at Figure 2 A5). For further insight, we performed inceptive intracellular recordings from interneurons within the LALs responding to rotating light stimuli at different times of day. We find the recorded firing rates sensitive to time (firing rate increases with time), azimuthal position of the light, and rotation direction. By taking the difference between clockwise (left) and counter-clockwise (right) rotation responses, as in, and comparing the outcome with, we observe some qualitative features that our model indicates, in particular, presence of fixed points and opposite signals in between the fixed points (see Figure S1 ).

Guerra and Reppert, 2013 Guerra P.A.

Reppert S.M. Coldness triggers northward flight in remigrant monarch butterflies. Our formalism allows us to examine rewiring of input signals (e.g., inclusion of NCLK1_C and NCLK2_C) to test whether the modified circuit could support other stable directions. We find that such a configuration exists; when clock input signals NCLK1 and NCLK2 are replaced by NCLK1_C and NCLK2_C and the connections are flipped (excitatory connections become inhibitory and vice versa), the circuit will implement a stable northeast (NE) flight for the whole light phase of the day ( Figure 2 B1). This mechanism is unique as the mechanism for SW flight. We thus conclude that for all neural input signals that we defined in Figure 1 , the wirings, depicted in Figure 2 , are the only two stable plausible mechanisms. As such, the integration supports only two flight directions: SW or NE. Remarkably, the NE flight direction is actually used by remigrant monarchs in the spring ().

F (angular velocity) and compute the angular position A from it ( α , initial angular position A , and initial velocities f l and f r . We explore these dynamics by computational simulations and compare them with experimental flight paths recorded during tethered flight ( Guerra et al., 2012 Guerra P.A.

Merlin C.

Gegear R.J.

Reppert S.M. Discordant timing between antennae disrupts sun compass orientation in migratory monarch butterflies. Merlin et al., 2009 Merlin C.

Gegear R.J.

Reppert S.M. Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. Merlin et al., 2009 Merlin C.

Gegear R.J.

Reppert S.M. Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. Mouritsen and Frost, 2002 Mouritsen H.

Frost B.J. Virtual migration in tethered flying monarch butterflies reveals their orientation mechanisms. Figure 3 Flight Tracks Simulator Show full caption (A) Simulated flight tracks in a compass view (top), with the sun at an angular position according to time of day (here Z T = 3 : 9 a m ). The butterfly rotates to find the SW. Rotation angle is sampled every 0.3 s and marked by blue dots. The simulated flight tracks projected on the potential function V ( A , T ) are shown (bottom). The black ball marks the current potential. As time progresses, the ball slides on the surface of the potential function toward the fixed point located on the stable line of fixed points (red) without crossing the separatrix line (blue). (B) Illustration of convergence to the SW from rest and initial angular position 110° to the left of the sun, sampled at t = 2, 8.5, 11.5, and 25 s. In the morning scenario (top), the initial position falls on the right of the separatrix (cyan bar) and is followed by rightward rotation to the SW (red bar). In the afternoon scenario (bottom), the initial position falls on the left of the separatrix, and followed by leftward rotation to the SW, produces a longer cycle, unlike the morning scenario. Figure 4 Model Simulated Flight Angle Tracks Compared with Tethered Flight Show full caption (A) Convergence to the proper direction in the model and outdoors experiments. (A 1 ) Mean timescales for convergence to the SW in the model starting at rest from uniformly distributed angular positions (5° bins) computed at different times of day. (A 2 ) Spatial distribution of convergence timescales at three times of day. (A 3 ) Spatial distributions in the afternoon ( Z T = 6 ) with shifted L:D cycle by −5 hr or +5 hr; analogous to morning/evening distributions rotated by 90° clockwise/counterclockwise. Guerra et al. (2012) Guerra P.A.

Merlin C.

Gegear R.J.

Reppert S.M. Discordant timing between antennae disrupts sun compass orientation in migratory monarch butterflies. (B) Changes of direction (turns/rotations) analyzed from raw data recorded in. See Figure S4 for particular datasets used here. (B 1 ) Direction changes in flight tracks. (B 2 ) Spatial distribution of turns. Each turn is marked by a magenta dot or, according to their direction, green: left and red: right. The red and blue radii denote the positions of predicted stable and separatrix fixed points. (B 3 ) Rotations count. (B 4 ) Probability for full rotation (within ± 2 s) from uniformly distributed angular positions (6° bins). (C) Typical flight signatures (10 s duration) within recorded (top) and model (for ZT = 8) (bottom) tracks. In the model, displacement is achieved by perturbing the input force until angular position reaches 300° (small), 15° (large), or 65° (ease-in). (D) Tracks for long period of flight (10 min) in a radial plot (time is the radius) and statistics of the angle/angular velocity. (D 1 ) Flight track in the afternoon for a monarch with regular L:D cycle (L is 6 am:6 pm). (D 2 ) Flight track in the afternoon for a monarch with a shifted L:D cycle by 6 hr backward (L is 12 pm:12 am). To simulate the time-compensated sun compass, we use the control units model ( Equation 1 ), which produces the signal(angular velocity) and compute the angular positionfrom it ( Equation 2 ). The outcome is a “self-correcting” model in which the simulated dynamics (flight tracks) converge to the balanced state from any initial angular position (except the degenerate case of zero velocity exactly at the separatrix). The transient dynamics, however, will depend on the choice of the parameter, initial angular position, and initial velocitiesand. We explore these dynamics by computational simulations and compare them with experimental flight paths recorded during tethered flight () in which dynamical features have not been analyzed before. As a particular example, we illustrate the convergence to the balanced state for two scenarios, morning and afternoon ( Figure 3 Movies S1 and S2 ). When the simulations are initiated with zero angular velocity, the dynamics follow the direction that does not require separatrix crossing. These trajectories may reach the balanced state at different times and with non-zero velocity, continue beyond it, change their direction, and return back to the fixed point with slower velocity in the opposite direction exhibiting “ease-in” dynamics to the balanced state. Indeed, such dynamics are characteristic to butterflies in a flight simulator (as we show in Figure 4 C and observed in).

10 s e c ( ± 8 s e c ) . However, for morning and evening times, we observe much slower convergence, with average times of about 30 sec ( ± 25 s e c ) . To further explore this variance, we depict the spatial distribution of convergence times. In the afternoon, most trajectories (except near the separatrix) quickly converge to the SW. However, in the morning and in the evening, slow convergence is extended up to a semi-circle, which includes the separatrix. Slow convergence in this region stems from angular acceleration being small, hindering the trajectory from gaining velocity. Our model, therefore, predicts that there is a difference in convergence timescales between control and clock-shifted tracks recorded in the afternoon. In particular, we expect clock-shifted tracks to be scattered in the slow convergence region. Such behaviors are typical for clock-shifted tracks recordings as reported in Guerra et al. (2012) Guerra P.A.

Merlin C.

Gegear R.J.

Reppert S.M. Discordant timing between antennae disrupts sun compass orientation in migratory monarch butterflies. Merlin et al. (2009) Merlin C.

Gegear R.J.

Reppert S.M. Antennal circadian clocks coordinate sun compass orientation in migratory monarch butterflies. To determine the time it takes for the angular position to settle to the SW, we set the angular velocity to zero and initialize simulations from various angular positions ( Figure 4 A). During afternoon, (zeitgeber time = 4:8) average convergence time is on the order of. However, for morning and evening times, we observe much slower convergence, with average times of about. To further explore this variance, we depict the spatial distribution of convergence times. In the afternoon, most trajectories (except near the separatrix) quickly converge to the SW. However, in the morning and in the evening, slow convergence is extended up to a semi-circle, which includes the separatrix. Slow convergence in this region stems from angular acceleration being small, hindering the trajectory from gaining velocity. Our model, therefore, predicts that there is a difference in convergence timescales between control and clock-shifted tracks recorded in the afternoon. In particular, we expect clock-shifted tracks to be scattered in the slow convergence region. Such behaviors are typical for clock-shifted tracks recordings as reported inandand depicted in Figures 4 D and S4

Z T = 8 ), and the density in the right correction interval is higher than in the left interval, another feature of the asymmetric position of the separatrix. The directionality of turns is consistent with the predicted left and right correction regions in our model, indicating that turns in flight tracks are in accord with the correction response of the model. From analysis of rotations, defined as trajectories that cross the separatrix region, we observe that they typically occur in clusters (with average of three rotations per cluster) and occurrence of turns during rotations significantly decreases. Indeed, our model indicates that once the trajectory crosses the separatrix, the correction direction is alternated, making a full rotation to be favorable than a turn against the correction direction. To infer the location of the separatrix, we compute the positions from which full rotations are more probable. We find that positions in the region of 80–140° have the highest probability to perform a rotation, with a peak of P = 0.7 at 80–115° bin. Remarkably, this position is in close agreement with the model prediction. A central feature in our model is the separatrix, which determines the time and the directionality of the flight correction to the SW. We therefore validate its existence by analyzing the changes in directions (turns and rotations) in experimental flight track data (four tracks spanning 32 min of flight), see Figures 4 B and S4 . We find that turns (extreme points of slope >0.01, farther than 20° from the SW) are frequent and appear more regularly than rotations. Their distribution is dense near the SW and becomes sparse as distance from the SW increases. Particularly, we do not identify any turns in the interval 80–110°, the predicted location of the separatrix (positioned at 105° at), and the density in the right correction interval is higher than in the left interval, another feature of the asymmetric position of the separatrix. The directionality of turns is consistent with the predicted left and right correction regions in our model, indicating that turns in flight tracks are in accord with the correction response of the model. From analysis of rotations, defined as trajectories that cross the separatrix region, we observe that they typically occur in clusters (with average of three rotations per cluster) and occurrence of turns during rotations significantly decreases. Indeed, our model indicates that once the trajectory crosses the separatrix, the correction direction is alternated, making a full rotation to be favorable than a turn against the correction direction. To infer the location of the separatrix, we compute the positions from which full rotations are more probable. We find that positions in the region of 80–140° have the highest probability to perform a rotation, with a peak ofat 80–115° bin. Remarkably, this position is in close agreement with the model prediction.

Since the compass is being activated by changes in the course of flight, we examine the model’s behavior in the presence of input noise. When perturbations originate from white noise (e.g., due to cloudiness), the trajectory once converged, will hover around the SW, a radius proportional to the magnitude of the perturbations. Incorporating occasional kicks in the model, that simulate strong perturbations, for example, caused by wind bursts, occlusions, or effects of tethering, the angular position may rapidly diverge from the fixed point, as we indeed observe in recordings ( Figure 4 D). In such a case, the trajectory could cross the separatrix and exhibit a full rotation or several rotations before settling to the fixed point. Our model indicates that compass sensitivity to such behaviors depends on the distance between the SW and the separatrix, which is minimal in the morning and evening and maximal in the middle of the day. Therefore, we expect that monarchs are more prone to frequent rotations due to strong perturbations in the morning and evening than in the middle of the day (see Movies S4 S5 , and S6 ).