Neuroethology of Insect Walking

## Contents

* 1 Introduction
* 2 Basic Leg movements Associated with Walking
* 3 Local Control of Basic Leg Movements: Reflexes and CPGs
* 4 Negotiating obstacles
* 5 A role for brain circuits in motor control
* 6 Summary of motor control in complex environments
* 7 Acknowledgements
* 8 References

## [edit] Introduction

Insects are by most criteria the most successful creatures in the animal kingdom. Certainly, their agility contributes greatly to this status. Few if any forms of terrain present an insurmountable barrier to all insects. Although many insects fly over complex terrain, all insects are capable of terrestrial locomotion. We find insects that walk slowly over floors or branches, scurry under rocks, and climb up walls and over ceilings or jump over barriers that if scaled to human dimensions would represent achievements beyond comprehension by the most accomplished athletes. These abilities have not been lost on engineers who have turned to observing insect locomotion as inspiration for legged robotic devices. Here, we will describe our current understanding of how insect nervous systems generate basic movements and then modify them as needed to move efficiently through complex natural terrain.

Although these creatures are often described as “simple systems” a close examination of their abilities reveals mechanisms that are elegant but not really simplistic. Insect locomotion represents a remarkable combination of mechanical principles, neural control, and sensory input leading to efficient muscular movement of leg joints. Although very talented scientists have studied these systems for many decades, they are only now beginning to be understood.

Brief historical overview: The history of investigations into insect walking parallels the overall study of neuroethology and includes many of the leaders of the field. A thorough historical discussion would be instructive but would easily take up an entire Scholarpedia review. As such it is beyond the scope of this review, but the following points to many of the main investigators and highlights to get the interested reader started.

Many early observers were fascinated by the seemingly rigid and repetitive patterns of leg movements and coordination in walking insects (Morgan 1887; reviewed by Hughes 1952). However, precise study showed that walking movements were not fixed but highly adaptable (Buddenbrock 1920; von Holst 1943; Wendler 1966). Individual legs can function largely as independent units and the walking pattern emerges from coordinating pathways within the nervous system (Bässler 1983; Delcomyn 1985; Cruse 1990; Cruse et al. 1995; Dürr et al. 2004; Dürr and Ebeling, 2005), the effects of sensory inputs (Pearson 1995) and mechanical coupling through the ground (Zill et 2009). The neuronal circuitry underlying walking has been extensively investigated using techniques of intracellular recordings and dye staining, which demonstrated that many neurons of insects (like other invertebrates) could be individually identified (Pearson and Fourtner 1975; Bässler 1983; Burrows 1996; Goldammer et al. 2012). These investigations have been complemented and extended by studies of walking in freely moving animals (Bleasing and Cruse 2004; Dürr et al. 2004; Wosnitza et al. 2013). This has generated a large body of knowledge on the structure of the nervous system, neuromuscular and biomechanical mechanisms underlying insect walking (reviews Zill et al. 2004; Ritzmann and Büschges, 2007; Büschges and Gruhn 2008; Koditschek, Full and Buehler, 2004). These mechanisms have also been successfully modeled using a number of approaches (Ekeberge et al. 2004; Daun-Gruhn and Büschges 2011; Schilling et al. 2013; Knops et al. 2013). As is the case in the overall field of Neuroethology, progress is made only when diverse strategies are employed. Here detailed neurobiological studies (Büschges and Gruhn 2008) are complemented by detailed behavioral investigations (Schilling et al. 2013) and brought together by diverse modeling efforts (Daun-Gruhn and Büschges 2011; Schilling et al. 2013).

The problem of insect walking, therefore, encompasses a wide range of issues from biomechanics to both central and peripheral neurobiological factors as well as force development in muscle. No review could possibly cover all of these topics in any depth and we will not attempt to do so in the space allotted to us. Rather we will concentrate on the local neurobiological circuits of the thoracic ganglia that generate basic leg movements while allowing for some modification in the timing and level of motor activity (Cruse et al. 2007). These circuits allow insects to adjust to many features such as inclines, holes or even walls. However, some barriers require the insect to actively re-direct leg movements to alter posture or turn its body. These actions require further processing in brain regions leading to interactions between brain and thoracic circuits. Absent this higher control, an insect could walk with a normal gait and deal with small barriers, but the exceptional behaviors that were mentioned above would not occur. Thus, we will also describe more recent findings regarding the interactions between brain and thoracic ganglia.

## [edit] Basic Leg movements Associated with Walking

All insects have six legs. At slow speeds, the legs follow a metachronal pattern moving from the hind legs to middle and then to front legs on either side (Hughes 1952; Wilson 1966). Each leg alternates between a stance phase when the tarsi are on the ground and the animal is pushed forward and a swing phase when the tarsus is moved forward through the air. At intermediate speeds, the pattern of leg movements shifts as the duration of the stance phase is shortened (this is termed gliding coordination, Wendler 1966; Cruse 1990) and more than one leg may be lifted at the same time (ex. tetrapod gait, Grabowska et al. 2012; Wosnitza et al. 2013). At higher speeds, the modification of the metachronal pattern leads to a tripod gait (Figure 1). Here the front and rear legs on one side of the animal move as a unit with the middle leg on the opposite side. This tripod alternates between swing and stance with the tripod made up of the remaining legs. The tripod gait is very stable, because at most speeds the animal’s center of mass remains within the base of support and it will not tip over. In cockroaches, the tripod gait can be further subdivided into a slower “amble”, a faster “trot” which has much less variability in leg coordination, and an even faster escape run which is used only about 1% of the time (Bender et al., 2011).

Figure 1: Tripod gait pattern for a cockroach on a tether. A. Video frames showing the extreme rear position during stance of the two tripods of legs. B. Stance and swing timing of each leg separated by color for the two tripods. Boxes indicate times when each leg is in swing. Note the right front and hind legs move in time with the left middle leg as do the remaining three legs.

Each leg is made up of segments that are similar from leg to leg, but differ in dimensions (Cruse 1976). From the most proximal to distal location, the leg segments are the coxa, trochanter, femur, tibia and a series of tarsal segments ending in a retractable claw (Figure 2). In cockroach, the most important joints for walking are the coxa-trochanter (CTr) joint and the femur-tibia (FTi) joint. The CTr joint actually moves the femur relative to the coxa because the trochanter-femur (TrF) joint makes only small movements (Frantsevich and Wang 2009). Although flexion of the TrF joint effectively rotates the tarsus and, in the middle and hind legs, this action is critical to initiating the swing phase of walking (Bender et al., 2010b), during many movements it acts mechanically as a fused joint. In stick insects and locusts, the coxa is much smaller and no movement occurs at the TrF joint. The relative proportions of the other leg segments are also very different. These factors may reflect the locus of connection on the side of the body rather than the ventral surface, as in cockroach. There are also many specializations in leg design found in various insects. Thus, for example, the hind leg of a locust has a very long muscular femur making a powerful jumping leg and the front legs of praying mantises are modified to form effective raptorial organs.

Figure 2: Diagrams describing the leg segments of the front and hind legs of the cockroach. A. The positions of the legs on the body are shown at various different parts of the leg cycle. Note that the movements through swing and stance are different for front and hind legs (see text for details.) B. Photographs of the front and hind legs. C. The various segments are color coded and names are designated. Note that the same segments are present in both front and hind legs, but their relative sizes differ.

Although the legs within a tripod move their feet as a unit, the joint movements and resulting forces are unique for each pair of legs (Full et al., 1991; Tryba and Ritzmann, 2000; Watson and Ritzmann, 1998a). In cockroaches, the hind legs make powerful movements that drive the animal forward. To accomplish this action, the CTr joint and the FTi joint move in near synchrony. This action allows these rotary joints to direct the movements of the tarsi (feet) in a line nearly parallel to the long axis of the animal’s body. The middle legs make similar movements, but with smaller CTr movements. The resulting actions of the middle legs generate forces that first brake the forward movement of the animal and then push it forward (Full et al., 1991). In contrast, the front legs make much greater use of the thoraco-coxa (ThC) joint, which attaches the leg to the thorax (Bender et al., 2010b; Ritzmann et al., 2004). This joint has three degrees of freedom similar to the ball and socket joint in a human shoulder. Movements of the ThC joint swings the front legs far forward much like a human arm. The CTr and FTi joints then pull the foot back towards the body. The resulting ground reaction forces slow the forward movement of the body and keep the animal from losing control (Full et al., 1991). Clearly the neural control of this leg is much different from the other two pairs of legs. Nevertheless, the sum of the ground reaction forces of all three pairs of legs is similar to that seen in the bipedal leg movements of a human (Full et al., 1991).

## [edit] Local Control of Basic Leg Movements: Reflexes and CPGs

Even though differences are found in the detailed morphology of various insect legs, common neural control systems are found even in insects as different as cockroaches and stick insects,(two insects that have been extensively studied and represent the focus of this review). The muscles that extend leg joints are typically innervated by very few motor neurons. Most of the muscles that extend the CTr joints of cockroaches are innervated by two excitatory motor neurons: one fast and one slow motor neuron. In slow movements, the slow neuron fires in bursts that lead to a series of facilitating muscle potentials that generate joint extension (Fig 3, Tryba and Ritzmann 2000b). At faster speeds, the fast motor neurons are added (Watson and Ritzmann, 1998b). These neurons generate few spikes, but each causes a rapid extension of the joint greatly increasing the joint velocity. Flexion of the joint is generated by antagonistic muscles that are typically controlled by a larger number of motor neurons. It should be noted that in addition to neural components, muscle properties also play an essential role in how motor neurons generate movements. A thorough review of muscle properties is beyond the scope of this review. The reader is directed to the following reviews for this important aspect of insect locomotion: http://www.scholarpedia.org/article/Muscle_Physiology_and_Modeling; see also recent investigations in insect muscle properties in (Blümel et al. 2012)).

Figure 3: Motor Neuron Activity in Walking - Activity of the slow Trochanteral Extensor motor neuron (Ds) in a cockroach that was tethered and walked on a glass plate. The lower trace shows electrical activity recorded in the extensor muscle, the upper trace is an intracellular recording from the motor neuron in the thoracic ganglion. The middle trace shows the CTr joint angle (measured from video recording). The extensor fires during the stance phase and produces extension of the CTr joint. Reproduced from Tryba and Ritzmann, 2000b

How are these motor neurons activated in patterns that will move the legs efficiently? The local control systems found in the thoracic ganglia must be able to generate basic swing-stance patterns but still be open to adjustments in both timing and force. First, the individual joints must be made to alternate between extension and flexion. In many neural systems, this basic timing function is often performed by central pattern generation (CPG) circuits. Then the joints of each leg must be coordinated to create an effective movement. It would be possible to have one overarching CPG that imposes a rhythm on all joints of each leg or even all of the legs. However, observations in stick insects strongly suggest that no single CPG exists for all legs or even for all joints of a single leg (Büschges et al., 1995). This turns out to be an important design feature, since intra-joint coordination can differ in front, middle and hind legs. Inter-leg coordination must also change from time to time (e.g. turning or walking backward). On top of this timing pattern, the forces generated by stance phase muscles must also adjust as the insect walks up and down inclines or walls, carries items, or even climbs over substantial barriers.

To accomplish these control functions, insects have evolved local control circuits that incorporate an elegant interplay between CPGs for each leg joint and sensory reflexes. (Büschges and Manira 1988). The existence of joint CPGs was demonstrated in stick insects by cutting the peripheral nerves and recording from the proximal leg stumps. In this de-afferented condition, application of the muscarinic agonist pilocarpine causes the motor neurons to burst. However, the timing of motor bursts serving various joints of deafferented preparations is not coordinated (Büschges et al., 1995).

What then is the role of numerous sensors that are found on each leg? In intact insects, muscle contractions and joint movements are tuned and coordinated by sensory inputs. For example, sensory receptors that monitor joint angles (called chordotonal organs) affect the timing of phase changes of motor neurons serving the leg joints and segments that they monitor as well as adjacent joints (Hess and Büschges, 1997; Akay et al., 2004). Campaniform sensilla are embedded in specific areas of the cuticle and respond to changes in strain that allow the insect’s nervous system to adjust to changing load (Zill et al., 2004; Zill and Seyfarth, 1996) (Figure 4). Signals of loads and muscle forces detected by campaniform sensilla can influence groups of muscles at different joints (e.g. leg extensors) to insure that they are tuned to work together as synergists (Akay et al., 2001; Zill et al., 2012).

Figure 4: Campaniform sensilla of the stick insect. A. Drawing of a stick insect indicating the position of the middle leg trochanter. B. Scanning electron micrograph of the trochanter cuticle. Many groups of sensory receptors are found in the trochanter including a hair plate and groups of campaniform sensilla. C. Confocal micrograph of cuticular caps of two groups of trochanteral campaniform sensilla. Strains in the exoskeleton are detected by processes of sensory neurons that attach to the caps. (Micrographs: B, Josef Schmitz and Annelie Exter; C. Sumaiya Chaudhry and David Neff)

Sense organs can, therefore, have effects both on the amount (magnitude) and timing of muscle contractions. These effects are thought to occur through two different types of pathways (Figure 5). Sense organs can influence the size of muscle contractions as reflexes through direct pathways to motor neurons or by pathways with intervening interneurons. The effects on the timing of motor outputs occur because some sensory neurons provide inputs to the pattern (rhythm) generators. These inputs can speed up, prolong or delay the muscle contractions.

Figure 5: Effects of sense organs on motor outputs - Sensory signals can modify motor activity by reflex action (based on direct or polysynaptic pathways through interneurons). Sense organs can also have affects upon activities of rhythm generators (CPGs). These inputs can change the timing of motor patterns, thereby in turn affect motor activity.

The coupling of joint CPGs can be produced by interneurons that activate muscles at different joints simultaneously (Brunn 1998; von Uckermann and Büschges 2009) and by interjoint sensory reflexes to generate coordinated leg movements that are appropriate to the actions required at any time (Akay et al., 2004; Hess and Büschges, 1997). An advantage in determining joint coordination by sensory feedback is that it can be easily modulated (Stein and Schmitz, 1999; Schmitz and Stein, 2000). Experimentally this can be demonstrated as 'reflex reversals' (Bässler 1983). When reflexes ‘reverse’, different groups of muscles are activated by sensory signals according to the specific behavior being performed. Thus, when a stick insect walks backwards the nervous system changes the inter-joint reflexes to generate new leg coordination that is appropriate to that particular movement (Akay et al., 2007). Similarly, turning movements are associated with selective reflex modulation in appropriate legs (Hellekes et al., 2012).

Leg sensors also impact coordination between legs. Experiments in stick insects demonstrated that when the front legs start walking the muscles of the middle legs will show bursting activities (Borgmann et al., 2009). Stimulation of sensors located on the trochanter leg segment was necessary to establish the appropriate phase of bursting in which the middle legs alternated with the front legs. In freely walking animals, coordination of leg movements may also be aided by mechanical forces transmitted through the substrate (ground), without relying on connections within the nervous system. In cockroaches, discharges of campaniform sensilla that detect unloading occur after an adjacent leg is placed in stance and pushes down on the substrate (Zill et al., 2009). These signals insure that the leg is not lifted until another leg is providing adequate support (Cruse 1990; Cruse et al. 2007).

Many of these effects have been reproduced in simulations (Daun-Gruhn and Büschges, 2011; Ekeberg et al., 2004; Schilling et al. 2013). In the simulations, sensory reflexes are used to adjust the strength of motor activity, thereby creating effective movements for any situation in which the insect finds itself (e.g. walking on a flat horizontal surface, going up or down an incline or climbing over an object). A controller that was inspired by this simulation has also been developed that can control a robotic stick insect or cockroach leg (Rutter et al., 2008).

More detail on the organization of the local control system found in thoracic ganglia of insects can be found in several excellent reviews (Büschges, 2012; Büschges et al., 2008; Büschges and Gruhn, 2008; Ritzmann and Büschges, 2007; Zill et al., 2004).

## [edit] Negotiating obstacles

The local control that we described above will allow insects to walk on horizontal surfaces, up and down inclines or even on walls. However, some barriers require more profound adjustments, and the agile locomotion associated with insects clearly demonstrates that they are up to the task. Stick insects are known to turn as a result of visual cues (Gruhn et al. 2009) and to use tactile information from the antennae to direct foreleg movements in climbing (Schütz and Dürr 2011) and crossing gaps in the substrate (Blaesing and Cruse, 2004). Drosophila use visual cues to traverse similar gaps (Pick and Strauss, 2005). Cockroaches actively investigate near field objects with their antenna prior to moving toward or around them (Okada and Toh, 2006; Staudacher et al., 2005). Faced with a large block, a cockroach must rear up before it can place its front legs on the top of the barrier. It can do that without actually contacting the front of the object with its legs Figure 6), by using information gained from sensors on its head (Watson et al., 2002b). In particular, the cockroach appears to use its antennae to judge the height of the barrier (Harley et al., 2009) then climbs accordingly.

Detection of a substantial object in its path, leads to a rotation of the ThC joint of the middle legs (Figure 6) (Watson et al., 2002b). With this change in leg orientation, extension of the more distal leg joints now raises the front of the body upward. In this new posture, the cockroach can now easily swing the front legs to the top of the block then, by extending the hind legs, push its body up to the top surface. Although the rearing and climbing movements use the same distal leg joints moving through the same joint angles, the motor activity is typically enhanced as the cockroach moves upward against gravity (Watson et al., 2002a). This observation suggests that the same reflex circuits that were described previously come into play here. Having altered its posture, the cockroach moves upward against gravity which may increase the strain on its cuticle as well as on the actual muscles. These changes should be detected by campaniform sensilla along the leg which would then activate reflexes that increase motor activity. Thus, the entire behavior to surmount the block would involve detection of the large object with sensors such as antennae or eyes, leading to descending commands that alter posture through changes in orientation of the middle leg’s ThC joints. After the cockroach rears up, control is returned to the local circuits of the thoracic ganglia that generate normal leg joint extensions, but now with enhanced force to push the insect upward.

Figure 6: Sequence of climbing movements made by cockroach negotiating a block. 50 msec elapses between each frame. Note that the front legs never contact the front of the block. Also note, at 150 msec, the middle leg has rotated at the ThC joint (red arrow) to bring that legs tibia nearly perpendicular to the substrate (blue arrow). Thus, when the leg extends in the next frame the body is pushed upward rather than forward.

If the block is replaced with a shelf, there are two possible outcomes. The cockroach can either climb over the shelf or tunnel under it. Again the antennae are critical to this decision. If they tap the shelf from above, the cockroach typically climbs, but if they tap the under surface, they tunnel under the shelf (Harley et al., 2009). Interestingly, this decision is also influenced by ambient light levels. In the light, there is a much greater tendency for the cockroach to tunnel. However, in the dark, there is no significant difference between climbing and tunneling. This observation suggests that in the light, these nocturnal insects are predisposed to seek out shelter, whereas in the dark they perform more natural foraging behaviors. The detection of light that is important to this process is performed by the ocelli (two simple eyes on the head). If they are covered, there is no longer a preference for tunneling even in the light, whereas if the compound eyes are covered, there is no change in the behavior (Harley et al., 2009).

## [edit] A role for brain circuits in motor control

The observation that multiple sensory inputs detected by sensors on the head affect climbing decisions suggests that association areas of the brain may be very much involved in these decisions. This conclusion is supported by observations of cockroach behavior after ablation of connectives that link higher centers to the thoracic ganglia (Ridgel and Ritzmann, 2005). With neck connectives cut, cockroaches make very small movements if any. However, if the circumoesophageal connectives are cut removing the brain, but retaining connections between the suboesophageal and thoracic ganglia, normal tripod gait movements are observed but with little ability to alter walking patterns in response to objects in the insect’s path. In effect, removal of the brain appears to have reduced movement to a robotic pattern that loses many of the complex behaviors that attract robotic engineers to insects as model systems.

Within the brain, the central complex (CC, Figure 7) has been specifically suggested to be involved in supervising such motor activity (Strausfeld, 1999; Strausfeld, 2012). The CC is made up of several prominent neuropils on the midline of the brain of all insects. These neuropils are highly columnar with fibers projecting between neuropils in a very regular array that suggests that it may be comparing left and right inputs. In locusts and monarch butterflies, the CC contains numerous neurons that are sensitive to polarized light and that could be used to guide them as they migrate (Heinze and Homberg, 2009; Heinze and Reppert, 2011). In cockroaches, neurons sensitive to both mechanical stimulation of antennae and changes in ambient light levels have been recorded using extracellular methods (Ritzmann et al., 2008).

Figure 7: Diagram of the Central Complex and Thoracic Ganglia- The central complex is highly structured region of the brain. Recent experiments have shown that central complex can affect motor outputs by modulating reflex circuits in the thoracic ganglia.(Redrawn in part from Mizunami et al. 1998)

Thus, the CC appears to process various types of sensory cues. Does it affect movement? Genetic mutations in Drosophila that generate breaks in one of the CC neuropils (the protocerebral bridge) have difficulty walking (Strauss, 2002; Strauss et al., 1992). Moreover, electrolytic lesions within various regions of the CC in cockroach result in deficits in specific behaviors. For example, lesion in lateral regions of one of the CC neuropils called the fan-shaped body increases the number of wrong turns when the cockroach is asked to walk in a U-shaped track (Harley and Ritzmann, 2010). Lesions on the midline of this neuropil do not affect turning but do alter climbing over blocks or shelves. Finally, extracellular recordings in the CC’s of tethered cockroaches reveal neurons that alter their firing rate in tandem with and, for some neurons, in advance of changes in step frequency. Moreover, stimulation through the same electrodes evokes changes in step frequency (Bender et al., 2010a). Similarly, neurons in the lateral regions of the fan-shaped body are tuned to turning movements toward the side on which they are recorded and stimulation in that region reliably evokes turning in the appropriate direction (Guo and Ritzmann, 2013). This and other bits of evidence are consistent with the notion that the CC takes in various different forms of sensory information and uses them to generate or at least influence descending commands that alter activity in the thoracic ganglia leading to turning, climbing or tunneling movements (Ritzmann et al., 2012).

How might these descending commands re-direct leg movements? Again we must return to the local reflex circuits of the thoracic ganglia. Remember that the CPGs for each joint are coordinated by local reflexes. However, there appear to be multiple reflex circuits present. When a stick insect walks backward, stimulation of the trCS causes a switch in the ThC motor neurons from retraction to protraction (Akay et al., 2007). This effect is the opposite of what occurs when the stick insect is walking forward. The local reflex effects are, therefore, not constant but change according to the direction of walking.

A similar effect could account for changes in turning movements in the cockroach. When the cockroach turns on a tether, the middle leg on the inside of the turn changes it movements so that joint extension now occurs during swing rather than stance (Mu and Ritzmann, 2005). This change causes the leg to reach out laterally then after setting down pull medially in an attempt to move the body toward the tarsus. If opposing reflexes exist in the cockroach as in the stick insect, this switch could be caused by descending commands enhancing one reflex while reducing the strength of the opposite one.

Such a reflex reversal can be demonstrated by removing descending activity through bilateral ablation of neck connectives (Mu and Ritzmann, 2008). Stretching the femoral chordotonal organ (FCo) that monitors femur-tibia joint angle normally enhances activity in the trochanteral slow depressor (Ds) motor neuron. Relaxation of the FCo inhibits ongoing Ds activity. However, after cutting both neck connectives, the activation associated with FCo stretch is greatly reduced while the inhibition associated with relaxation actually reverses to now produce excitation. Similar reflex reversals are seen in appropriate legs during turning movements of stick insects (Hellekes et al., 2012). These studies suggest that the modulation or switching of reflexes may be an important function of higher centers.

## [edit] Summary of motor control in complex environments

The image that arises from the observations that are described above is one that includes complex interactions of neural circuits that reside in different regions of the insect CNS. Local reflexes and CPGs interact in thoracic ganglia to generate basic leg movements. Reflexes also account for changes in strength of motor activity at appropriate times, such as when walking up or down inclines or after postural adjustments to rear up in preparation for climbing over blocks. When the insect approaches an object that is too large to be negotiated with simple reflex adjustments, sensors on the head evaluate it and the resulting information is integrated within brain circuits such as those that reside in the CC. Here appropriate commands are formulated that descend to the thoracic ganglia where they can redirect leg movement by altering the strength of competing inter-joint reflexes.

With this type of hierarchical motor control, the insect has the best of both worlds. It has local reflex circuits that can quickly adjust movement as needed, but it also has a sophisticated brain that can take in large amounts of data from the wealth of sensors located on its head then use that information to temporarily re-direct leg movements as it initiates transient turning, climbing or tunneling movements while maintaining the stability that is inherent in the local feedback circuits. These systems allow insects to move through very diverse environments at will. As such they serve as excellent models for human engineered vehicles that are designed to move through a range of dangerous terrains, but only when all parts of the hierarchical controls that we describe are intact.

## [edit] Acknowledgements

This material is based in part upon work supported by the National Science Foundation under Grant No. IOS-1120305 and the AFOSR under grant FA9550-10-1-0054 to RER and NSF grants IBN-0235997 to SNZ and MRI 0959012 to Marshall University.

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