A Model for Optic Flow Integration in Locust Central-Complex Neurons Tuned to
Head Direction
Kathrin Pabst¹, Frederick Zittrell²,Uwe Homberg²,Dominik Endres¹
¹Department of Psychology, Gutenbergstraße 18, 35032 Marburg, Germany
²Department of Biology, Karl-von-Frisch-Straße 8, 35043 Marburg, Germany
{kathrin.pabst, dominik.endres}@uni-marburg.de, {zittrell, homberg}@biologie.uni-marburg.de
This is a pre-print of the paper with the same title accepted at the CogSci conference 2022. It is published under the CC-BY license.
Abstract
Navigation is a fundamental cognitive function of virtually all
moving animals. Several navigation strategies require an es-
timate of the current traveling direction that is updated con-
tinuously. In the central complex of the insect brain, multi-
modal cues are fused into a compass-like head direction rep-
resentation. Based on the proposed connectivity of colum-
nar neurons in the central complex of the desert locust we
designed a computational model to examine how these neu-
rons could maintain a stable representation of heading direc-
tion and how shifts occur by optic flow signals when the an-
imal turns. Our model differs from previous architectures in
the excitation-inhibition interactions. Consequently, the activ-
ity of head direction-encoding CL1a neurons remains stable
if it is mirrored by CL2 neurons, which are another class of
columnar neurons. Shifts of the compass signal occur via mod-
ulation of the network connectivity. Our model can be used to
deduce testable hypotheses where data are lacking, inspiring
new avenues of experimental investigations.
Keywords: insect brain; central complex; sky compass orien-
tation; optic flow; desert locust
Introduction
Animals display an abundance of navigational strategies al-
lowing them to find their way in complex environments. Irre-
spective of the time frame and range of navigation behaviors,
most of these strategies require a sense of the current relative
heading direction (Heinze, 2017). Many insects, such as bees
(von Frisch, 1946), ants (Fent, 1986), butterflies (Perez, Tay-
lor, & Jander, 1997), and fruit flies (Weir & Dickinson, 2012),
infer their traveling direction from celestial cues. A robust es-
timate emerges when celestial cues are combined with possi-
bly redundant signals of other modalities (Honkanen, Adden,
Freitas, & Heinze, 2019). We are interested in the compu-
tations involved in the maintenance of the animal’s current
heading direction.
Multimodal sensory inputs are processed in the central
complex (CX), a prominent formation of neuropils that has
emerged as the navigation hub of the insect brain (Green &
Maimon, 2018; Honkanen et al., 2019; Pfeiffer & Homberg,
2014). It is characterized by a modular architecture with
the protocerebral bridge (PB), the upper (CBU) and lower
(CBL) divisions of the central body (CB, also termed ellip-
soid and fan-shaped body in some species) being divided into
columns, and the CB and paired noduli (NO) further divided
into layers (Figure 1A; (Homberg, 2008; Pfeiffer & Homberg,
2014)). Columnar neurons interconnect the PB, CB, and the
NO (Homberg, 2008). They arborize in PB and CB columns
in a distinct projection pattern (Heinze & Homberg, 2008;
M¨
uller, Homberg, & K¨
uhn, 1997; Wolff, Iyer, & Rubin,
2015) (see Figure 1B). Tangential neurons provide the CX
with inputs from various brain regions. They innervate most
or all columns of the PB or entire layers of the CB, some have
additional arborizations in the NO (von Hadeln et al., 2020).
Tangential neurons (SpsP in the fly, TB7 in the locust) from
the posterior superior slope innervate one hemisphere of the
PB and likely contact CL1a and CL2 neurons across the PB
depending on turn direction, like SpsP neurons in the fly (von
Hadeln et al., 2020; Green & Maimon, 2018; Lu et al., 2022).
CX neurons in various insect species respond to celestial
cues (Homberg, Heinze, Pfeiffer, Kinoshita, & Jundi, 2011).
In the locust they topographically map the solar azimuth
(Pegel, Pfeiffer, Zittrell, Scholtyssek, & Homberg, 2019) and
polarization pattern of the sky (Zittrell, Pfeiffer, & Homberg,
2020), giving rise to an internal sky compass (Figure 1B). In
many insects, the CX features a representation of the current
relative heading direction, in tune with external signals like
sky compass cues or based solely on self- generated signals in
their absence (Seelig & Jayaraman, 2015; Turner-Evans et al.,
2017). Central place foragers like ants (Ronacher & Wehner,
1995) and bees (Srinivasan, Zhang, & Bidwell, 1997) rely
on celestial cues for a sense of direction and on optic flow-
based speed cues to estimate the traveled distance. Rotation-
induced optic flow information could also be used for self-
motion dependent direction updates when idiothetic sky sig-
nals are unavailable or noisy.
It has been shown that compass cell activity is modulated
by visual motion in the locust (Rosner, Pegel, & Homberg,
2019). However, the neural circuitry performing optic flow
integration in the CX remains largely unknown. Recently,
computational models for angular path integration in the CX
based on data from Drosophila and the sweat bee Megalopta
have been proposed (Stone et al., 2017; Su, Lee, Huang,
Wang, & Lo, 2017; Turner-Evans et al., 2020). We chose a
similar approach to explore the connectivity of a subset of CX
neurons involved in head direction encoding in the desert lo-
cust (Schistocerca gregaria) and computational mechanisms
that produce sky compass-like characteristics. In the next
section, we describe a neural firing rate model that respects
known connectivity constraints from the locust and from the
fruit fly (Drosophila) where locust data are missing. We first
describe how the stereotypic projection pattern of columnar
Sun
TN
TB7
Figure 1: The abstracted connectivity diagram of the locust central complex, based on Heinze and Homberg (2008) and Zittrell
et al. (2020). (A) Schematic of the CX with a subset of the involved neuron types: CL1a and CL2 neurons are connected to
one another in the protocerebral bridge (PB) and lower division of the central body (CBL). CL2 neurons, in addition, have
postsynaptic arborizations in the noduli (NO) where they receive possibly rotation-dependent input from tangential neurons
(TN) from the lateral complexes. Rotation inputs may, likewise, be conveyed in the PB by tangential neurons (TB7) from the
superior posterior slope. CL1a neurons are topographically tuned to solar azimuth along the PB (black arrows). Consequently,
CL1a neurons with dendrites in adjacent columns of the CBL have 180° shifted tuning preference. (B) Full population of CL1a
and CL2 neurons and CL1a activity pattern: With the sun 90° left of the locust (bottom), the CL1a population activity (top) has
a distinct maximum according to their neural tuning (highlighted arrows in PB and CBL).
neurons can maintain stable head direction estimates when
the locust stands still and then explore possible computational
mechanisms that would update this internal representation
during turns using optic flow-based rotation signals, presum-
ably provided by TN and/or TB7 neurons. Finally, we discuss
the ramifications of our work, comparing the locust network
to that of the fly which has inspired previous models, and
indicating where further research is needed to support our as-
sumptions.
The Model
All computations were performed with the python program-
ming language (version 3.8.5) and the pytorch library (ver-
sion 1.10.0). Plots were created with the matplotlib library
(version 3.3.2). The code is available in this repository.
Model Design
Our model entails a subset of the columnar neurons: CL1a
neurons corresponding to E-PG neurons in the fly tracking
head direction in the CX (Seelig & Jayaraman, 2015; Zit-
trell et al., 2020), and CL2 neurons corresponding to P-EN
neurons responding to turning motion (Green et al., 2017;
Turner-Evans et al., 2017; Zittrell et al., 2020).
We adopted the schemes proposed by Heinze and Homberg
(2008) for the CL1a and CL2 connectivities. We hypothe-
sized that, as shown for E-PG and P-EN neurons in the fly
(Turner-Evans et al., 2017), CL1a neurons provide synap-
tic inputs to CL2 neurons in the PB, which in turn pro-
vide synaptic inputs to CL1a neurons in the CBL. Unlike
Turner-Evans et al. (2017) who proposed a similar model for
Drosophila, we do not assume excitatory connections in both
sites paired with global inhibition, but instead a sign reversal
Figure 2: The connectivity matrix Mderived from the wiring
scheme in Figure 1A (upper left and lower right quadrants)
with added excitatory self- recurrent connections (main diag-
onal).
within the CL1a-CL2 connectivity, presumably at the synapse
in the CBL where inhibitory inputs could reverse the oppos-
ing tuning preference of CL1a neurons in neighboring CBL
columns otherwise resulting from the connectivity (see Fig-
ure 1). Based on these assumptions, we constructed a connec-
tivity matrix M. For modeling purposes, we added excitatory
self-recurrent connections to all neurons to enable them to
maintain a baseline activity. Note that the self-recurrent con-
nections are not a faithful representation of the known neu-
roarchitecture. Other mechanisms of activity maintenance,
e.g., loops across several neurons or change of membrane
properties, are conceivable too, and have been implemented
in previous models. We chose self- recurrent loops for our
model because they comprise the simplest solution, and we
have no data from the locust that require otherwise. We as-
signed uniform weights to all excitatory and inhibitory con-
nections, 0.5 and -0.5, respectively. We made this choice to
obtain eigenvalues equal to one, see next section for details.
See Figure 2 for a visualization of M.
We employed a discrete time framework where the activity
deviation of all neurons from their baseline at time tis repre-
sented by a vector xtwith components xt,1:16 and xt,17:32 cov-
ering the CL1a and CL2 neurons, respectively. We assume
only small deviations, hence we can use a linear network as
an approximation to non-linear dynamics. The network is re-
current and iterated across time steps such that the activity at
the next time step can be computed from the current activity:
xt+1=Mxt.(1)
Maintenance of a Stable Head Direction Encoding
We first set out to test whether the proposed network could
maintain a stable head direction encoding in the CL1a activ-
ity.
Methods In the framework outlined above, maintenance
of the head direction representation or CL1a activity pat-
tern x1:16 translates to an equality of xt1:16 at time point tand
xt+11:16 at the following time point, t+1:
xt1:16 =xt+11:16 .(2)
According to equation 1, this is given if Mxt=xt. This equal-
ity holds for all xtthat are eigenvectors uof Mwhere δr=1,
with δrdenoting the real part of the corresponding eigen-
value, and zero imaginary part. We refer to these eigenvectors
as ‘stable’. Therefore, any linear combination of these stable
eigenvectors will be a stable state of the network, too. We
tested whether any of these stable states resembled activity
observed in the locust CX. Based on the 1 × 360° compass to-
pography reported in the locust PB (Pegel et al., 2019; Zittrell
et al., 2020) (see Figure 1), we defined n=1...16 different
CL1a activity targets ˆx, each with the maximum in a different
column:
ˆ
Xn
t+11:16 =cos(
~
φCL1a+n
~
φ)/kcos(
~
φCL1a+n
~
φ)kF.(3)
where ˆ
Xis a matrix containing all ˆxand kkFdenotes the
Frobenius norm. ~
φCL1ais the vector of preferred azimuths
of the CL1a neurons, approximated to be evenly distributed
across the PB from left to right in a range [0,2π],~
φ= (φ...φ)
with ~
φ=2π
16 , the angle covered by a PB column (see Figure
1). These activity targets are thus cosine-shaped across the
PB with the maximum and minimum 8 columns apart (see
Fig. 1B). Note that all activities should be understood as fir-
ing rate differences to a baseline firing rate. To find the linear
coefficient vectors awhich result in CL1a activities equal to
the target CL1a activities (see equation 3), we employed the
L-BFGS algorithm to optimize A, a matrix comprising all a,
with respect to a loss function
L=k(Uλr=1A)1:16 −ˆ
Xt+11:16 k2
F+0.1kAkF,(4)
where Uλr=1is the matrix of stable eigenvectors sorted by
kλkF, the Frobenius norm of the real and imaginary parts
of the corresponding λ. The first term of this loss function
measures the divergence of the CL1a activations from the tar-
gets, the second term regularizes the solution: of all Athat
represent a given target state, the optimizer picks the one that
minimizes the length of A, thus avoiding solutions where con-
tributions from different eigenvectors cancel each other. Op-
timization was iterated until convergence to four digits. We
tested whether the maintenance criterion was fulfilled;
[M(Uλr=1A)]1:16 = (Uλr=1A)1:16.(5)
Results A matrix Aof coefficient vectors for the linear com-
bination of stable eigenvectors Uλ=1was found that repro-
duced ˆ
Xup to machine precision. The optimization process
employed here always converged at Uλ=1A1:16 =Uλ=1A17:32
(see Figure 3A: Three targets ˆx(solid lines) along with the
matching activities xresulting from the optimization (dots)
are exemplarily shown in different colors). In other words,
with our proposed model and CL1a activity targets, when
the latter are stable head direction estimates, the CL2 ac-
tivation patterns are exactly the same; xt+1,1:16 =xt,1:16 if
xt,1:16 =xt,17:32.
Rotation-induced Shifts of Compass Activity
An internal compass representation must adapt to a new head-
ing direction when the animal turns. In the fly, it has been ob-
served that the compass activity bump shifts counter- phasic
to the turn, i.e., such that the activity pattern signals a fixed
allocentric direction (Giraldo et al., 2018; Seelig & Jayara-
man, 2015). We therefore investigated how the stable CL1a
activation patterns of our model would be shifted in the oppo-
site direction of the animal’s turn direction by an input repre-
senting optic flow information. We hypothesize that such an
input could be mediated by TN and TB7 neurons as they sup-
ply the CX with various inputs. Furthermore, in Drosophila,
their counterparts (LNO and SpsP neurons) have been shown
to respond to turning motion (Lu et al., 2022).
Methods We conceptualized such shifts as switches be-
tween consecutive time points tand t+1 from one of the acti-
vation patterns to another, with the activity maximum moving
to the nearest neighboring PB column, i.e., the yaw angle is
always ±2π
16 . We thus searched for a computational mecha-
nism that would produce, starting at xn
t,xn
t+1=xn−1
tfor left
turns and xn
t+1=xn+1
tfor right turns. Note that we expect
xt,1:16 =xt,17:32 at all times teven though an offset between
E-PG and P-EN peak activity has been observed during turns
in flies (Turner-Evans et al., 2017). This is due to the discrete
time framework employed here where the dynamics of transi-
tions between time points are not made explicit. We explored
two different ways of changing the network’s activity x: An
additive feed forward input exciting or inhibiting the CL1a
and CL2 neurons, and a multiplicative input effect modulat-
ing the network’s synaptic weights M. To test whether an
additive effect could produce these shifts, we optimized two
vectors yLand yRto minimize
L=k[Mˆ
Xt+yL]−ˆ
XL
t+1k2
F+0.1kyLkF(6)
and
L=k[Mˆ
Xt+yR]−ˆ
XR
t+1k2
F+0.1kyRkF(7)
for left and right turns of the animal resulting in right and left
shifts of the activity pattern, respectively. Here, ˆ
XL,1
t+1=ˆ
X32
t+1
and ˆ
XL,2:32
t+1=ˆ
X1:31
t+1, and ˆ
XR,1:31
t+1=ˆ
X2:32
t+1and ˆ
XR,32
t+1=ˆ
X1
t+1; i.e.,
ˆ
XL
t+1and ˆ
XR
t+1are the activation pattern targets learned in the
maintenance experiment described above shifted one position
to the right and left, respectively. Note that the superscript
specifies the animal’s turn direction.
To test whether a modulatory input could yield the desired
shifts, we optimized matrices YLand YRto minimize
L=k[MsYLˆ
Xt]−ˆ
XL
t+1k2
F+0.1kYLkF(8)
and
L=k[MsYRˆ
Xt]−ˆ
XR
t+1k2
F+0.1kYRkF,(9)
where Msdenotes Mconvolved with an RBF Kernel with ker-
nel width of 1 column, in other words a smoothed version of
the connectivity matrix, corresponding to broader arboriza-
tions (see Figure 3B). This way, the modulatory input can
manipulate any connection existing in Mbut also add con-
nections, with their possible strength decreasing with their
distance to existing connections. This approach was chosen
since a modulation of only the connections indicated by the
wiring scheme (see Fig. 1) and present in Mcould not yield
the desired shift results. The smoothed matrix was inspired
by the E-PN and P-EG connectivity reported in Drosophila
(Hulse et al., 2021). In the locust, arborizations of CL2 neu-
rons in the CBL are confined to single columns, while those
of CL1 neurons usually extend across at least three columns
(Heinze & Homberg, 2008). Optimization was iterated until
convergence to four digits.
Results No vectors yLand yRcould be found that would
produce the desired phase shift to the right or left for all xn.
Optimization converged at Xt+1=Xt(see Figure 3C and C’:
Three initial states xt(dashed lines) and the matching target
states ˆxt+1(solid lines) along with the activities xt+1resulting
from the optimization (dots) are exemplarily shown in differ-
ent colors). This demonstrates that purely feed forward input
to the CL1a/CL2 neurons cannot account for the observed
shift behavior. Instead, we found two matrices YLand YR
that would move any of the 16 predefined activation patterns
to the right or left (see Figure 3D and D’: As in 3C and C’,
three matching initial, target, and trained activity states are
shown). In both modulated matrices, activity is suppressed
in adjacent columns on one but increased in the ones on the
other side in addition to the existing connections, thereby ef-
fecting a shift of the activation pattern (see Figure 3E and E’).
Discussion
As the computational mechanisms underlying the locust sky
compass are largely unknown, we constructed a simplified
model of the columnar neurons of the locust central complex
to explore the circuit on the algorithmic level of analysis, in
the sense of Marr and Poggio (1979).
Constrained by available physiological and anatomical
data, we built a computational model of CX neurons in the
locust capable of maintaining and shifting a stable head di-
rection representation. Where no locust data was available,
our model was inspired by data from the fruit fly, leading
to a resemblance to models for angular velocity integration
in the Drosophila CX compass (Kakaria & De Bivort, 2017;
Turner-Evans et al., 2017; Pisokas, Heinze, & Webb, 2020;
Turner-Evans et al., 2020; Hulse et al., 2021). Key differences
however are the 1 × 360° angular representation in the locust
Text
A B
C'C
D D'
E'E
Figure 3: Modeling results. (A) When CL1a activities x1:16 (dots) converge with CL1a activity targets ˆx1:16 (solid lines), CL1a
and CL2 activity are equal. As shown for three different pairs of xand ˆx. (B) The smoothed matrix Ms, which we used to derive
the connectivity matrix for rotational movements. (C) No feed forward input yLcould be found that would shift the initial
activity xt(dashed lines) to the target activity ˆxt+1(solid lines). xt+1=xt+yL(dots) does not differ from xt. Three examples
are shown in different colors. (C”) Likewise, no yRcould be found that would shift the activity to the left. (D) A modulatory
input YLproduces the targeted activity shift to the right. (D’) Likewise, modulation by YRcauses a transition to the left. (E)
YL, the modulated connectivity matrix causing a shift to the right. (E’) YR, the modulated connectivity matrix causing a shift to
the left.
PB (Pegel et al., 2019; Zittrell et al., 2020) compared to the
2 × 360° representation of space in the Drosophila PB and,
contrasting the excitatory E-PG/P-EN loops in the fly, the as-
sumed inhibitory synapses from CL2 onto CL1a neurons in
the CBL which need to be supported by future physiological
analyses.
Our model maintains a stable CL1a head direction encod-
ing if the CL1a and CL2 activity are identical. This is in line
with the minimal offsets between the E-PG and P-EN activ-
ity bumps observed in fruit flies walking in darkness at an-
gular velocities below 30°/s (Turner-Evans et al., 2017). On
the other hand, equal CL1a and CL2 activities are not in line
with different responses of CL1a and CL2 neurons to polar-
ized light as reported by (Heinze & Homberg, n.d.). This
could be accounted for by adjusting the network weights in
future versions of our model. Future models could also ex-
plore how properties of the network depend on context and
available cues.
We investigated two different ways of shifting the CL1a
compass activity, presumably mediated by TN and/or TB7
neurons in the locust. In our model, an additive or feed for-
ward input to the CL1a and CL2 neurons could not reproduce
the shift of activity observed in the CX. The locust model
proposed by Pisokas et al. (2020), in contrast, shifts the com-
pass bump via excitation of P-EN (CL2) neurons in one hemi-
sphere, and inhibition of those in the other, depending on the
turn direction. This is in line with the P-EN responses of
fruit flies walking in darkness reported by Turner-Evans et
al. (2017). Our model, however, would require connections
between CL2 and CL1a neurons in the same column to shift
the compass activity via this mechanism. Comparable con-
nections exist between rotation- and head direction cells in
the model of the rat’s internal compass proposed by Skaggs,
Knierim, Kudrimoti, and McNaughton (1994). Skaggs et al.
further assume that the weights of synapses from visual fea-
ture detectors - cells that respond to relative positions of stim-
uli such as landmarks or the sun - on head direction cells are
modulated by the firing rate of the latter. In our model, too,
a multiplicative or modulatory effect of rotation input on the
CL1a/CL2 connectivity successfully shifts the activity pat-
tern to the left or right. We originally assumed a connectivity
based on single column innervations of each neuron, and this
was sufficient for the maintenance of a compass signal. The
compass shift, however, requires CL1a and CL2 neurons ex-
citing and inhibiting each other in three adjacent columns.
Likewise, it requires connections among neighboring CL1a
neurons and among neighboring CL2 neurons, respectively.
Substantial overlap of ramifications across columns has been
observed for CL1a neurons in the CBL of the locust (Heinze
& Homberg, 2008), but whether connections actually exist
among CL1a and CL2 neurons, respectively, and between
CL2- and CL1a neurons of neighboring columns requires fur-
ther research. Furthermore, it needs to be examined how the
proposed modulation could act in a turn direction-dependent
manner and how rotation cues of different modalities, such
as optic flow, proprioceptive feedback, or efferent copies are
processed and integrated in the CX.
The linear model and binary concept of movement em-
ployed here are abstractions of the neuronal and behavioral
characteristics of the locust. In particular, our model has
time-discrete dynamics; it switches between activity states
and does not make the dynamics driving the transitions ex-
plicit. This means that it cannot model a spatiotemporal rela-
tionship between the CL1a and CL2 neurons like the lead-lag
relationship between E-PG and P-EN observed by Turner-
Evans et al. (2017) in the fly. Furthermore, our model is a
single compartment model, meaning that each neuron is as-
signed one activity. Thus, a phenomenon like the reversal of
the lead-lag relationship of E-PG and P-EN between the PB
and EB reported by the same authors cannot be exhibited in
our model. We aim to increase the model’s biological plau-
sibility by implementing velocity dependence in future work
but expect the general principles of maintaining and updating
the compass bump to hold independently of the level of anal-
ysis. Nevertheless, it will be interesting to consider sources
of non-linearity in neural signal processing and transmission.
The locust model proposed by Pisokas et al. (2020) is noise
tolerant and the authors attribute this to recurrent connections.
We aim to test if the same applies to our model. We further in-
tend to explore if the integration of rotation information from
different sources can increase the robustness of the compass
signaling and how the compass responds to contradictory in-
puts from different modalities.
Acknowledgements
This work was supported by the DFG (452193090, HO
950/28-1, and EN 1152/3-1) and “The Adaptive Mind”,
funded by the Excellence Program of the Hessian Ministry
for Science and the Arts.
References
Fent, K. (1986). Polarized skylight orientation in the desert
ant cataglyphis. Journal of Comparative Physiology A,
158, 145-150.
Giraldo, Y. M., Leitch, K. J., Ros, I. G., Warren, T. L., Weir,
P. T., & Dickinson, M. H. (2018). Sun navigation requires
compass neurons in drosophila. Current Biology,28, 2845-
2852. doi: 10.1016, j.cub.2018.07.002
Green, J., Adachi, A., Shah, K. K., Hirokawa, J. D., Magani,
P. S., & Maimon, G. (2017). A neural circuit architecture
for angular integration in drosophila. Nature,546, 101-106.
doi: 10.1038, nature22343
Green, J., & Maimon, G. (2018). Building a heading signal
from anatomically defined neuron types in the drosophila
central complex. Current Opinion in Neurobiology,52,
156–164. doi: 10.1016, j.conb.2018.06.010
Heinze, S. (2017). Unraveling the neural basis of insect nav-
igation. Current Opinion in Insect Science,24, 58-67. doi:
10.1016, j.cois.2017.09.001
Heinze, S., & Homberg, U. (n.d.). Linking the input to the
output: new sets of neurons complement the polarization
vision network in the locust central complex.
doi: 10.1523, JNEUROSCI.0332-09.2009
Heinze, S., & Homberg, U. (2008). Neuroarchitecture of the
central complex of the desert locust: Intrinsic and columnar
neurons. Journal of Comparative Neurology,511, 454-478.
doi: 10.1002, cne.21842
Homberg, U. (2008). Evolution of the central complex
in the arthropod brain with respect to the visual system.
Arthropod Structure and Development,37, 680-687. doi:
10.1016, j.asd.2008.01.008
Homberg, U., Heinze, S., Pfeiffer, K., Kinoshita, M., & Jundi,
B. E. (2011). Central neural coding of sky polarization
in insects. Philosophical Transactions of the Royal Soci-
ety B: Biological Sciences,366, 680-687. doi: 10.1098,
rstb.2010.0199
Honkanen, A., Adden, A., Freitas, J. D. S., & Heinze, S.
(2019). The insect central complex and the neural basis of
navigational strategies. Journal of Experimental Biology,
222. doi: 10.1242, jeb.188854
Hulse, B. K., Haberkern, H., Franconville, R., Turner-Evans,
D. B., Takemura, S. Y., Wolff, T., . . . Jayaraman, V. (2021).
A connectome of the drosophila central complex reveals
network motifs suitable for flexible navigation and context-
dependent action selection. eLife,10. doi: 10.7554,
eLife.66039
Kakaria, K. S., & De Bivort, B. L. (2017). Ring attractor
dynamics emerge from a spiking model of the entire proto-
cerebral bridge. Frontiers in Behavioral Neuroscience,11,
8.
Lu, J., Behbahani, A. H., Hamburg, L., Westeinde, E. A.,
Dawson, P. M., Lyu, C., . . . Wilson, R. I. (2022). Trans-
forming representations of movement from body- to world-
centric space. Nature,601, 98-104. doi: 10.1038, s41586-
021-04191-x
Marr, D., & Poggio, T. (1979). A computational theory of hu-
man stereo vision. Proceedings of the Royal Society of Lon-
don. Series B. Biological Sciences,204(1156), 301–328.
M¨
uller, M., Homberg, U., & K¨
uhn, A. (1997). Neuroarchi-
tecture of the lower division of the central body in the brain
of the locust (Schistocerca gregaria). Cell and Tissue Re-
search,288, 159-176. doi: 10.1007, s004410050803
Pegel, U., Pfeiffer, K., Zittrell, F., Scholtyssek, C., &
Homberg, U. (2019). Two compasses in the central com-
plex of the locust brain. Journal of Neuroscience,39, 3070-
3080. doi: 10.1523, JNEUROSCI.0940-18.2019
Perez, S. M., Taylor, O. R., & Jander, R. (1997). A sun com-
pass in monarch butterflies. Nature,387, 29. doi: 10.1038,
387029a0
Pfeiffer, K., & Homberg, U. (2014). Organization and func-
tional roles of the central complex in the insect brain. An-
nual Review of Entomology,59, 165-184. doi: 10.1146,
annurev-ento-011613-162031
Pisokas, I., Heinze, S., & Webb, B. (2020). The head direc-
tion circuit of two insect species. Elife,9, e53985.
Ronacher, B., & Wehner, R. (1995). Desert ants cataglyphis
fortis use self-induced optic flow to measure distances trav-
elled. Journal of Comparative Physiology A,177, 21-27.
doi: 10.1007, BF00243395
Rosner, R., Pegel, U., & Homberg, U. (2019). Responses of
compass neurons in the locust brain to visual motion and
leg motor activity. Journal of Experimental Biology,222.
doi: 10.1242, jeb.196261
Seelig, J. D., & Jayaraman, V. (2015). Neural dynamics for
landmark orientation and angular path integration. Nature,
521, 186-191. doi: 10.1038, nature14446
Skaggs, W., Knierim, J., Kudrimoti, H., & McNaughton, B.
(1994). A model of the neural basis of the rat’s sense of
direction. Advances in Neural Information Processing Sys-
tems,7, 173-180.
Srinivasan, M. V., Zhang, S. W., & Bidwell, N. J. (1997).
Visually mediated odometry in honeybees. Journal of
Experimental Biology,200, 2513-2522. doi: 10.1242,
jeb.200.19.2513
Stone, T., Webb, B., Adden, A., Weddig, N. B., Honkanen,
A., Templin, R., . . . Heinze, S. (2017). An anatom-
ically constrained model for path integration in the bee
brain. Current Biology,27, 3069-3085. doi: 10.1016,
j.cub.2017.08.052
Su, T. S., Lee, W. J., Huang, Y. C., Wang, C. T., & Lo, C. C.
(2017). Coupled symmetric and asymmetric circuits under-
lying spatial orientation in fruit flies. Nature Communica-
tions,8, 1-15. doi: 10.1038, s41467-017-00191-6
Turner-Evans, D. B., Jensen, K. T., Ali, S., Paterson, T.,
Sheridan, A., Ray, R. P., . . . Jayaraman, V. (2020). The
neuroanatomical ultrastructure and function of a biologi-
cal ring attractor. Neuron,108, 145-163. doi: 10.1016,
j.neuron.2020.08.006
Turner-Evans, D. B., Wegener, S., Rouault, H., Franconville,
R., Wolff, T., Seelig, J. D., . . . Jayaraman, V. (2017). An-
gular velocity integration in a fly heading circuit. eLife,6.
doi: 10.7554, eLife.23496, e23496
von Frisch, K. (1946). Die T¨
anze der Bienen. ¨
Osterreichische
Zoologische Zeitschrift,1, 1-48.
von Hadeln, J., Hensgen, R., Bockhorst, T., Rosner, R., Hei-
dasch, R., Pegel, U., .. . Homberg, U. (2020). Neuroar-
chitecture of the central complex of the desert locust: Tan-
gential neurons. Journal of Comparative Neurology,528,
906-934. doi: 10.1002, cne.24796
Weir, P. T., & Dickinson, M. H. (2012). Flying Drosophila
orient to sky polarization. Current Biology,22(1), 21-27.
Wolff, T., Iyer, N. A., & Rubin, G. M. (2015). Neuroarchitec-
ture and neuroanatomy of the drosophila central complex:
A GAL4-based dissection of protocerebral bridge neurons
and circuits. Journal of Comparative Neurology,523, 887-
1037. doi: 10.1002, cne.23705
Zittrell, F., Pfeiffer, K., & Homberg, U. (2020). Matched-
filter coding of sky polarization results in an internal sun
compass in the brain of the desert locust. Proceedings of the
National Academy of Sciences of the United States of Amer-
ica,117, 25810-25817. doi: 10.1073, pnas.2005192117
