Notes on wave optics

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The wave equation

  • Because light exhibits wave-particle duality, wave-based descriptions of light are often appropriate in optical physics, allowing the establishment of an electromagnetic theory of light.
  • As electric fields can be generated by time-varying magnetic fields and magnetic fields can be generated time-varying electric fields, electromagnetic waves are perpendicular oscillating waves of electric and magnetic fields that propagate through space. For lossless media, the E and B field waves are in phase.
  • By manipulating Maxwell’s equations of electromagnetism, two relatively concise vector expressions that describe the propagation of electric and magnetic fields in free space are found. Recall that the constants ε0 and μ0 are the permittivity and permeability of free space respectively.


  • Since an electromagnetic wave consists of perpendicular electric and magnetic waves that are in phase, light can be described using the wave equation (which is equivalent to the expressions above). Note that the speed of light c = (ε0μ0)-1/2. Electromagnetic waves represent solutions to the wave equation.


  • Either the electric or the magnetic field can used to represent the electromagnetic wave since they propagate with the same phase and direction. With the exception of the wave equation above, the electric field E will instead be used to represent both waves. Note that either the electric or magnetic field can be employed to compute amplitudes.

Solutions to the wave equation

  • Plane waves represent an important class of solutions to the wave equation. The parameter k is the wavevector (which points in the direction of the wave’s propagation) with a magnitude equal to the wavenumber 2π/λ. In a 1-dimensional system, the dot product k•r is replaced by kx. The parameter ω is the angular frequency 2πf and φ is a phase shift.


  • To simplify calculations, Euler’s formula can be used to convert the equation above into complex exponential form. Only the real part describes the wave as the real part corresponds to the cosine term.


  • Spherical waves are another useful solution to the wave equation (though they are an approximation and truly spherical waves cannot exist). Because of their geometry, the electric field of a spherical wave is only dependent on distance from the origin. As such, the equation for a spherical wave can be written as seen below with origin r0.


  • Gaussian beams are a solution to the wave equation that can be used to model light from lasers or light propagating through lenses. If a Gaussian beam propagates in the z direction, then from the perspective of the xy plane, it shows a Gaussian Fig. 1intensity distribution. For a Gaussian beam, the amplitude decays over the direction of propagation according to some function A(z), R(z) represents the radius of curvature of the wavefront, and w(z) is the radius of the wave on the xy plane at distance z from the emitter. Often these functions can be approximated as constants.


Intensity and energy of electromagnetic waves

  • The Poynting vector S is oriented in the direction of a wave’s propagation (assuming that the wave’s energy flows in the direction of its propagation).


  • The magnitude of the Poynting vector represents the power per unit area (W/m2) or intensity crossing a surface with a normal parallel to S. Note that this is an approximation since, according to a quantum mechanical description of electromagnetic waves, the energy should be quantized.


  • Power per unit area (intensity) of plane waves, spherical waves, and Gaussian beams can also be calculated using the equations below. The formula for the Gaussian beam’s power represents the power at a plane perpendicular to the direction of light propagation z.


  • For electromagnetic waves, instantaneous energy per unit area is difficult to measure, so the average energy per unit area over a period of time Δt is often worked with instead. Since waves are continuous functions, taking their time-average requires an integral.


  • After using the above integral on the function eiωt and then taking the real and imaginary parts of the result, the time-averages of the functions cos(ωt) and sin(ωt) are found.


Superposition of waves

  • Let two waves E1 and E2 of the same frequency traveling in the same direction undergo superposition. E1 and E2 may or may not possess the same amplitude or phase. The substitution α = –(kx+φ) will be carried out.


  • If the phases of the waves are different, some special equations are necessary to find the amplitude E0 and the phase α of the resulting wave.


  • For the superposition of any number of waves, the equations above can be extended.


Polarization of light

  • The waves comprising linearly polarized light are all oriented at the same angle which is defined by the direction of the electric field of the light waves. For linearly polarized plane waves with electric fields oriented along the x or y axes that propagate in the z direction, the following equations describe their electric fields.


  • The superposition of two linearly polarized plane waves that are orthogonal to each other (and out of phase) is the vector sum of each electric field.


  • The superposition of two linearly polarized plane waves that are orthogonal to each other (and in phase) is computed via the following equation and has a tilt angle θ determined by the ratio of amplitudes of the original waves. This process can also be performed in reverse with a superposed polarized wave undergoing decomposition into two orthogonal waves.

eq18Fig. 2

  • When two constituent waves possess equal amplitudes and a phase shift of nπ/2, the superposed wave is circularly polarized (as it can be expressed using a sine and a cosine term). Equations for the constituent waves and the superposed wave are given below.


  • When circularly polarized light propagates, it takes a helical path and so rotates. As such, a full rotation occurs after one wavelength. If a circularly polarized wave rotates clockwise, it is called right-circularly polarized and has a positive sine term. If a circularly polarized wave rotates counterclockwise, it is called left-circularly polarized and has a negative sine term.


  • If a right-circularly polarized light wave and a left-circularly polarized light wave of equal amplitude are superposed, then they create a linearly polarized light wave with twice the amplitude of the individual waves.


  • Linearly polarized and circularly polarized light are special cases of elliptically polarized light. For elliptically polarized light, the amplitudes of the superposed waves may differ and the relative phase shift does not need to be nπ/2. As such, the electric field traces an elliptical helix as it propagates along the z direction.


  • For elliptically polarized light with a positive phase shift φ, it is called right-elliptically polarized if E0x > E0y and left-elliptically polarized if E0x < E0y.
  • Most light is unpolarized (or more appropriately, a mixture of randomly polarized waves). To obtain polarized light, polarizing filters are often used.



Boudoux, C. (2017). Fundamentals of Biomedical Optics. Blurb, Incorporated.

Degiorgio, V., & Cristiani, I. (2015). Photonics: A Short Course. Springer International Publishing.

Hecht, E. (2017). Optics. Pearson Education, Incorporated.


Notes on fiber biomechanics

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PDF version: Notes on fiber biomechanics – Logan Thrasher Collins

Elastic fiber models

  • For an elastic fiber in which a linear relationship between force and change in length is assumed, the force is given by F = k(L – L0).
  • To normalize for other elastic fibers with different starting lengths, this equation is divided by L0 to give F = k(L/L0 – 1). It is common practice to represent L/L0 as a parameter λ (called the stretch ratio).
  • As such, F is found using the formula below. Note that the quantity λ – 1 is referred to as the strain.


  • While linear models are often useful, many real fibers exhibit finite extensibility (a nonlinear phenomenon) after exceeding a certain critical strain value λc. That is, the force necessary to extend the fiber farther after exceeding λc increases rapidly. Finite extensibility can be modeled using the following equation which divides k by a term dependent on λ and λc.


  • To model a muscle, let L0 represent the muscle’s length in its inactive state and Lcontracted represent the muscle’s length in its contracted state. Unlike the spring, the contracted state is used as the reference length. The contraction stretch is described by the ratio λcontracted = Lcontracted/L0 while the stretch ratio remains as λ = L/L0.


  • If this muscle is contracted without carrying a load such that F = 0, then λ = λcontracted. If the muscle acquires a load and so must maintain a constant length equal to its original length L0 (to “hold the load steady”), then the force in the muscle is F = k(1/λcontracted – 1).
  • To generalize this model for 3-dimensional space, the locations of the fiber’s endpoints A and B are used. The fiber’s length and orientation are given below.


  • The following force vectors can act on point B and on point A. The stretch ratio is still λ = L/L0.


Viscous fiber models

  • Purely viscous behavior (as with liquids) can be described 1-dimensionally using the equation below where cη is a damping coefficient.


  • The normalized rate of deformation is equivalent to the above formula without the damping coefficient. In addition, the rate of deformation can be written in terms of the stretch ratio λ = L/L0.


  • If one endpoint of a filament of fluid is moved with constant velocity, its position is given by xB = L0 + vt. This means that the rate of deformation is v/xB.


  • Solving the above equation gives the following result. For a constant rate of elongation, the point xB must be displaced exponentially over time.


  • Given endpoint displacements uA and uB, the total displacement is ΔL = uB – uA. Using this quantity, the stretch λ and the strain ε can be written using the equations below.


  • For the small strains (i.e. |ε| is much less than 1) that result from small stretches, some approximations can be made which reduce the force equation to the following form.


Viscoelastic fiber models

  • Many biological materials exhibit viscoelastic behavior rather than elastic behavior. In viscoelastic systems, the force on a fiber with a constant length decreases over time and applying constant force causes the length to increase.
  • The strain response of a fiber to force is given as ε(t). The creep function J(t) describes the fiber’s tendency to permanently deform as strain is applied. Many possible creep functions can be devised depending on the system. The creep function is related to the strain by a factor of force increase F0.


  • Strain responses follow the principle of superposition. That is, if a force is applied at τ1 and then another force is applied at τ2, a total strain response can be expressed as a sum of the two individual strain responses.


  • For an arbitrary history of applied forces, an integral formulation of strain response is used. In this equation, the change in force at each infinitesimal time interval is multiplied by the creep function.


  • Similarly, the force resulting from an imposed strain history can be expressed as an integral where G is a function that describes the relaxation of the fiber with time (analogous to the creep function, but the opposite concept).


Reference: Oomens, C., Brekelmans, M., & Baaijens, F. (2009). Biomechanics: Concepts and Computation. Cambridge University Press.


Notes on nanoparticle self-assembly

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PDF version: Notes on nanoparticle self-assembly – Logan Thrasher Collins

Preparation of nanoparticle superlattices

  • Nanoparticle superlattices can be prepared using solvent evaporation, solvent destabilization, or gravitational sedimentation methods.
  • Solvent evaporation involves evaporating a nanoparticle-containing solvent to induce ordered aggregation of the particles. Note that many inorganic nanoparticles are insoluble in polar solvents and soluble in nonpolar solvents, though the presence of polar surface ligands can alter this behavior. Fig. 1
  • Solvent evaporation techniques include (i) placing a small droplet of nanoparticle-containing solvent on a solid substrate and allowing for evaporation to occur, (ii) evaporating a nanoparticle-containing solvent from a tilted vial so as to control the orientation of the meniscus, (iii) placing a small droplet of polar solvent on a solid substrate and then adding a nonpolar nanoparticle-containing solvent over the top to facilitate aggregation in the thin layer of nonpolar solvent, and (iv) filling a tray with a polar solvent and adding nonpolar nanoparticle-containing solvent over the top to facilitate aggregation in the thin layer of nonpolar solvent.
  • Solvent destabilization promotes gradual clustering of nanoparticle in solution via slowly changing the solvent conditions. Solvent destabilization techniques include (i) allowing for a polar and a nonpolar solvent to gradually intermingle and so facilitate a steady increase in the favorability of nanoparticle-nanoparticle interactions and (ii) heating a premixed solvent mixture that includes both polar and nonpolar components and so facilitating a controlled enrichment of the solvent with a higher boiling point. This increases the favorability of nanoparticle-nanoparticle interactions in a controlled fashion. Because many nonpolar liquids possess lower boiling points, the nanoparticle lattices prepared in this way may require nanoparticles equipped with polar surface ligands.
  • Gravitational sedimentation is less common than the other techniques since many nanoparticles are small enough to remain dissolved in spite of gravitational forces. But very large nanoparticles (100-1,000 nm) often do sediment, facilitating close-packing and the assembly of superlattices.

Characterization of nanoparticle superlattices

  • Transmission electron microscopy (TEM) is used to visualize nanoparticle superlattices directly. As TEM requires very thin slices, it makes 2-dimensional images of nanoparticle superlattices.
  • TEM operates best when there is a high contrast between the atomic number of the nanoparticles and the atomic number of the background support structure. For instance, PbS is easily imaged on a carbon support.
  • To circumvent issues that arise with atomic number contrast, ultrathin (i.e. graphene) supports or supports that possess numerous holes can be used. Ultrathin supports absorb less electrons while “holey” supports allow some nanoparticles to be positioned over the holes during imaging, preventing background absorption.
  • TEM often requires a vacuum chamber and so necessitates dry samples, meaning  that superlattice structure can be visualized after removal of the solvent, but snapshots of the self-assembly process cannot be taken. However, recent investigations into designing liquid-cell TEM may circumvent this problem.
  • Scanning electron microscopy (SEM) generates 3-dimensional images via a scanning electron beam and so is useful for imaging nanoparticles and Fig. 2nanoparticle superlattices that exhibit some kinds of notable 3-dimensional geometric characteristics.
  • Atomic force microscopy, a technique in which a nanoscale probe is moved across a sample to reconstruct its shape via a “sense of touch,” has also been used for nanoparticle superlattice characterization.
  • Images of repetitive superlattices are amenable to processing with two-dimensional fast Fourier transforms (FFTs) that can reveal insights about the lattice’s characteristics. Performing this form of FFT upon an image of a repetitive crystal structure creates a plot of spatial frequencies. This plot is said to display reciprocal space (or Fourier space).
  • Distinct points on the reciprocal space plot correspond to certain properties of the lattice that are sometimes not apparent from the image prior to the FFT. In this way, very similar lattices can be clearly distinguished.

Kinetics of nanoparticle superlattice formation

  • Homogenous nucleation occurs in solution and requires overcoming a nucleation barrier while heterogenous nucleation occurs as nanoparticles are added to a preexisting seed crystal. Homogenous nucleation typically leads to disordered solids and is typically much slower than heterogenous nucleation. Fig. 3
  • In heterogenous nucleation, crystal growth occurs at differing rates depending on how many new contacts are formed (assuming attractive interparticle interactions). If more new contacts occur upon the addition of a nanoparticle, the process will exhibit grater energetic favorability and occur at a faster rate.
  • This means that adding nanoparticles to kinks and vacancies happens more rapidly than the adsorption of nanoparticles to steps, terraces, and “adatoms” (see figure at right). As such, large scale structures that minimize surface energy tend to form.

Thermodynamics of nanoparticle superlattices

  • As mentioned, if superlattice assembly occurs rapidly, disordered aggregates can form. Allowing the process to occur more slowly facilitates sampling of many states as assembly proceeds. As such, the most thermodynamically stable structures can form when gradual assembly is performed.
  • Van der Waals interactions between nanoparticles are often approximated using the following pair potential equation. U is the potential energy for the interparticle interaction, C represents a proportionality constant for the interparticle interaction, ρ1 and ρ2 are the number of atoms per unit volume in two interacting bodies, and r is the distance between the bodies.


  • For nanoparticles with volumes V1 and V2, the total van der Waals energy of attraction is obtained by the following integral that performs a pairwise summation of all the atomic van der Waals interactions.


  • If two nanoparticles are spherical with radii R1 and R2, the integral can be solved analytically to give their interparticle potential energy.


  • When the distance d between two nanoparticles is much less than the radius of either nanoparticle, the above equation can be approximated using the following formula.


  • For many nanoparticles without chemical ligands, these van der Waals interactions would cause rapid aggregation in solution. However, the presence of certain surface ligands gives repulsion that maintains colloidal solutions of nanoparticles.
  • In order to convey repulsive effects between nanoparticles, the surface ligands require a proper solvent. Such solvents exhibit negative free energy upon ligand-solvent mixing. That is, interactions between the surface ligand and the solvent are energetically favorable.
  • Surface ligand repulsion includes an osmotic component and an elastic component. As the solvent molecules are sterically blocked by surface ligands, when the surface ligands of two nanoparticles start to interact, the volume that the solvent cannot enter increases. This situation is osmotically unfavorable, so osmotic repulsion occurs. When surface ligands are compressed because two nanoparticles are close to each other, elastic repulsion occurs.
  • When a solution of nanoparticles with surface ligands that are attracted to each other (i.e. hydrophobic chains) is dried, the ligands begin to freeze together rather than experiencing repulsion.
  • Equilibrium superlattice structures minimize the free energy G in terms of enthalpy U and entropy S according to Gibb’s equation ΔG = ΔU – TΔS.
  • The contributions of cores and ligands can be decomposed into the terms ΔUcores, ΔUligands, ΔScores, and ΔSligands. The energy terms can be further broken down into the components of van der Waals interactions (London dispersion forces, dipole-induced dipole interactions, and dipole-dipole interactions). The entropy terms can be further broken down into configurational, rotational, and translational components.


Reference: Boles, M. A., Engel, M., & Talapin, D. V. (2016). Self-Assembly of Colloidal Nanocrystals: From Intricate Structures to Functional Materials. Chemical Reviews, 116(18), 11220–11289.




Notes on Honeybee Sensory Neurobiology

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Antennal lobes

  • Honeybee antennal lobes (ALs) are composed of about 160 regions called glomeruli in which olfactory receptor neurons from the antennae make synapses on projection neuron cell bodies as well as inhibitory local neurons.
  • The projection neurons send cholinergic axons to the mushroom bodies and to the lateral horn (LH) while the GABAergic local neurons facilitate olfactory computations within the antennal lobes.

Mushroom bodies

  • The mushroom bodies are paired structures located on either side of the central brain (CB). They are known to facilitate higher sensory integration as well as associative learning processes.Fig. 1
  • In honeybees, the mushroom bodies use cup-shaped medial calyces (MCAs) and lateral calyces (LCAs) as their major sensory input regions while using the pedunculi (PEDs) as their major sensory output regions.
  • The calyces contain Kenyon cells which receive cholinergic axons from the projection neurons of the antennal lobes and the pedunculi contain the efferent axons of the Kenyon cells.

Associative olfactory learning

  • Honeybee associative olfactory learning can occur where the olfactory pathway converges with other pathways.Fig. 2
  • Specific odors can serve as conditioned stimuli when they are associated with unconditioned stimuli of appetitive or aversive character.
  • Experimental evidence shows that the VUMmx1 neuron is sufficient for olfactory reward learning in bees. Its cell body is located within a region called the subesophageal ganglion and it synapses upon cells in the calyces, the lateral horn, and the antennal lobe.


Honeybee eyes

  • Honeybees possess two frontal compound eyes and three ocelli (simple eyes) located on the top of the head.
  • The retinas of honeybee compound eyes are composed of ommatidia, each with nine photoreceptor cells. The types of bee photoreceptor cells include S, M, and L photoreceptors corresponding to UV, blue, and green wavelengths respectively.
  • Ocellar retinas are composed of rod cells (note that they do not have ommatidia) and are covered by a lens. However, the focal plane of this lens is behind the actual retina, leading to much lower resolving power than that of the compound eyes. Although the function of ocelli is not entirely understood, they may operate as widefield detectors of illumination changes. In addition, ocellar retinas can be subdivided into dorsal and ventral regions which view the horizon and the sky respectively. Distinct neuronal pathways are associated with these subdivisions.

Optic lobe

  • Honeybee vision (associated with the compound eyes) starts with the optic lobe’s three regions; the lamina (La), medulla (Me), and lobula (Lo).
  • The lamina is positioned directly under the compound eye’s photoreceptors. It receives inputs mainly from the L photoreceptors, which are involved in the achromatic pathway and exhibit fast response times. However, some very rough color processing may still occur in the lamina. Fig. 3
  • In the medulla, neurons are organized in a columnar retinotopic fashion with eight layers. The columns also possess horizontal connections (unlike the lamina) which likely facilitate color opponency. The medulla’s outer layers contain neurons that respond to specific wavelengths and neurons that respond to a broad range of wavelengths while the medulla’s inner layers contain color-opponent neurons that compare colors at center and surround regions of receptive fields.
  • The lobula consists of six layers. Its outer layers (1-4) are part of the achromatic pathway and exhibit motion sensitivity. Its inner layers (5-6) continue the color processing pathway. Some projections from the inner layers go to the mushroom bodies, possibly facilitating sensory crosstalk and learning.
  • Beyond the optic lobe, further visual processing of the achromatic and color pathways occurs in the protocerebrum and central brain.

Audition and antennal somatosensation

Johnston’s organ

  • Honeybees use Johnston’s organ as their sensory organ for audition. In bees, audition also acts as a form of somatosensation. Johnston’s organ is located on the antennae. It detects vibrations during the waggle dance and air currents during flight. Fig. 4
  • Johnston’s organ contains about 240 scolopidia, mechanosensory complexes which include bristles that deform and trigger action potentials along efferent axons.
  • The soma of neurons within Johnston’s organ are divided into dorsal (dJO), ventral (vJO), and anterior groups (aJO).

Projections from Johnston’s organ

  • The main axons from the soma within Johnston’s organ trifurcate into the fascicles called T6I, T6II, and T6III. The T6I axons terminate at the ventro-medial superior posterior slope (vmSPS), the T6II axons terminate at the antennal mechanosensory and motor center (AMMC), and the T6III axons terminate at the ventro-central superior posterior slope (vcSPS). Fig. 5
  • In the vmSPS, the axons show some degree of somatotopy arising from the dorsal, ventral, and anterior Johnston’s organ regions. Somatotopy is not observed in the AMMC or vcSPS.
  • All the sensory axons from Johnston’s organ also send small collateral branches to the bee’s dorsal lobe (DL).


  • The AMMC contains two classes of interneuron, AMMC-Int-1 and AMMC-Int-2. AMMC-Int-1 neurons have somas located in the honeybee’s primary auditory center, which is near the central brain. They densely arborize at the AMMC and thinly arborize in the ventral protocerebrum (the protocerebrum is a region of the insect brain that includes the mushroom bodies and central brain as well as several other structures). Their dense arborization in the AMMC runs close to the T6 collaterals at the dorsal lobe.
  • AMMC-Int-1 neurons demonstrate spontaneous spiking without sensory input.Fig. 6 During exposure to a vibratory stimulus, the spike rate slows slightly. After the stimulus is removed, the spike rate increases to a higher rate than that of the spontaneous spiking, but eventually returns to the basal rate. However, it should be noted that olfactory stimuli and other modulating factors can drastically alter the response properties of AMMC-Int-1 neurons.
  • AMMC-Int-2 neurons have somas located in the dorsal lobe. Their dendrites split into three main branches called x, y, and z. Branch y is the axon while branches x and z are dendritic. It sends a long process to the lateral protocerebrum (LP) and makes synapses. The x arborization represents the densest of the three branches and is located in the AMMC. Branch z passes through the dorsal lobe and into the lateral superior posterior slope (lateral SPS). Fig. 7
  • AMMC-Int-2 neurons respond to relatively high vibratory amplitudes, especially those which cause 30 μm (or greater) shifts in antennal position. Their sensitivity reaches a maximum at 265 Hz (a frequency that occurs during the waggle dance), though they also respond to other frequencies.


  • The SPS contains an interneuron known as SPS-D-1 which projects to the ipsilateral and contralateral SPS.
  • SPS-D-1 does not respond to 265 Hz alone. However, it responds to long-lasting 265 Hz vibratory stimulation with simultaneous olfactory stimulation at the contralateral antenna.


Gustatory sensilla

  • Gustatory receptor cells are found in sensilla, structures which resemble hairs or pegs. Sensilla are located on the glossa, antennae, labial palps, and several other parts of the bee’s body. Fig. 8
  • Each sensillum contains 3-5 gustatory receptor neurons that send dendrites up the shaft towards a pore at the sensillum’s tip. The somas of the receptor cells (along with a mechanoreceptor cell) are encapsulated by auxiliary cells and bathed in a receptor hemolymph. The gustatory receptor neurons likely use GPCRs to detect various food molecules while the mechanoreceptor facilitates evaluation of the food’s position and density.
  • Antennal sensilla respond in a dose-dependent and highly sensitive manner to sucrose solutions. In addition, antennal sensilla respond to aqueous NaCl. As the antennal sensilla do not respond to very low concentrations of KCl, they probably do not contain a receptor that responds to water alone (unlike in many other insects). Sensilla on the mouthparts respond to aqueous sucrose, glucose, fructose, LiCl, KCl, and NaCl. They do not respond to CaCl2 or MgCl2. Foreleg sensilla respond to sucrose as well as very low concentrations of KCl, suggesting that these sensilla may contain a receptor that responds to water alone (unlike the bee’s other sensilla).

Honeybee central gustatory processing

  • Honeybee central gustatory processing takes place primarily in their subesophageal ganglion (SEG). Axons of gustatory neurons and the mechanosensory neurons found in the sensilla project to the SEG’s mandibular, maxillary, and labial neuromeres via the mandibular, maxillary, and labial nerves respectively.
  • As mentioned, the SEG contains the VUMmx1 neuron, which facilitates pairing of olfactory and gustatory stimuli for reward learning. Other VUM neurons have been identified in the SEG, but their function remains unclear.
  • Beyond the SEG, other neurons might be involved in the honeybee’s gustatory processing. In the mushroom bodies, the PE1 neuron exhibits increased spiking in response to sucrose gustation. However, PE1 also responds to mechanical and olfactory inputs. Also located in the mushroom bodies are cells dubbed as “feedback neurons” which respond to odors and sucrose as well. In these cases, multisensory integration likely occurs.

With the exception of image created for the section “projections from Johnston’s organ,” images were modified from: (Steijven, Spaethe, Steffan-Dewenter, & Härtel, 2017), (R. Menzel, 2012), (Kiya & Kubo, 2011),  and (Galizia, Eisenhardt, & Giurfa, 2011).


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Galizia, C. G., Eisenhardt, D., & Giurfa, M. (2011). Honeybee Neurobiology and Behavior: A Tribute to Randolf Menzel. Springer Netherlands.

Heisenberg, M. (2003). Mushroom body memoir: from maps to models. Nature Reviews Neuroscience, 4, 266. Retrieved from

Hung, Y.-S., & Ibbotson, M. (2014). Ocellar structure and neural innervation in the honeybee. Frontiers in Neuroanatomy. Retrieved from

Kiya, T., & Kubo, T. (2011). Dance Type and Flight Parameters Are Associated with Different Mushroom Body Neural Activities in Worker Honeybee Brains. PLOS ONE, 6(4), e19301. Retrieved from

Menzel, R. (2012). The honeybee as a model for understanding the basis of cognition. Nature Reviews Neuroscience, 13, 758. Retrieved from

Mota, T., Yamagata, N., Giurfa, M., Gronenberg, W., & Sandoz, J.-C. (2011). Neural Organization and Visual Processing in the Anterior Optic Tubercle of the Honeybee Brain. The Journal of Neuroscience, 31(32), 11443 LP-11456. Retrieved from

Sandoz, J.-C. (2013). Chapter 30 – Neural Correlates of Olfactory Learning in the Primary Olfactory Center of the Honeybee Brain: The Antennal Lobe. In R. Menzel & P. R. B. T.-H. of B. N. Benjamin (Eds.), Invertebrate Learning and Memory (Vol. 22, pp. 416–432). Elsevier.

Steijven, K., Spaethe, J., Steffan-Dewenter, I., & Härtel, S. (2017). Learning performance and brain structure of artificially-reared honey bees fed with different quantities of food. PeerJ, 5, e3858.