The minimum endpoint variance trajectory depends on the profile of the signal-dependent noise

It has been proposed that the central nervous system determines reaching movement trajectories so as to minimize the positional variance of the endpoint in the presence of signal-dependent noise. The hypothesis well reproduces the empirical movement trajectories for noise to the control signal whose variance is proportional to the second power of the amplitude of the control signal. However, empirical studies do not necessarily exhibit such a simple signal-noise relationship. The studies exhibit a wide distribution of estimates of the value of the exponent. This discrepancy raises the question of how the minimum endpoint variance trajectory depends on the value of the exponent. To address this question, we calculated minimum endpoint variance trajectories in simulations in which the value of the exponent was varied from 0 to 3. We found that the optimal trajectories differed according to the value of the exponent, and the profiles of optimal trajectories gradually diverged from the empirical ones as the value approached 0, though this change was not remarkable for larger values. Moreover, the optimal trajectories failed to replicate Fitts' law when the value was not equal to 2. These results suggest that the acceptability of the minimum endpoint variance theory depends on the value of the exponent in our motor system.     

Formation and control of optimal trajectory in human multijoint arm movement

In this paper, we study trajectory planning and control in voluntary, human arm movements. When a hand is moved to a target, the central nervous system must select one specific trajectory among an infinite number of possible trajectories that lead to the target position. First, we discuss what criterion is adopted for trajectory determination. Several researchers measured the hand trajectories of skilled movements and found common invariant features. For example, when moving the hand between a pair of targets, subjects tended to generate roughly straight hand paths with bell-shaped speed profiles. On the basis of these observations and dynamic optimization theory, we propose a mathematical model which accounts for formation of hand trajectories. This model is formulated by defining an objective function, a measure of performance for any possible movement: square of the rate of change of torque integrated over the entire movement. That is, the objective function CT is defined as follows: (formula; see text) We overcome this difficult by developing an iterative scheme, with which the optimal trajectory and the associated motor command are simultaneously computed. To evaluate our model, human hand trajectories were experimentally measured under various behavioral situations. These results supported the idea that the human hand trajectory is planned and controlled in accordance with the minimum torque-change criterion.     

Spatial learning and discrimination of food patches in the European badger (Meles meles L.)

Theories of foraging often assume that concentrations of prey (‘patches’) can be discriminated by a forager, but there have been few studies of how this is achieved when patches are not recognizable by means of an obvious proximal cue. We observed the search trajectories of two badgers (Meles meles L.) foraging for peanuts in artificial patches to see how efficiently they could map a new patch in the first place, and whether they would remember the location and extent of a previously visited patch. The results suggest that when a patch is encountered for the first time, a strategy of area-restricted searching keeps the animal's trajectory largely within the patch boundary. After a single exposure to a novel patch, however, badgers show evidence of being able to remember its location and extent, apparently with reference to distal landmarks.     

The functional response predicts the effect of resource distribution on the optimal movement rate of consumers

Understanding how often individuals should move when foraging over patchy habitats is a central question in ecology. By combining optimality and functional response theories, we show analytically how the optimal movement rate varies with the average resource level (enrichment) and resource distribution (patch heterogeneity). We find that the type of functional response predicts the effect of enrichment in homogeneous habitats: enrichment should decrease movement for decelerating functional responses, but increase movement for accelerating responses. An intermediate resource level thus maximises movement for type‐III responses. Counterintuitively, greater movement costs favour an increase in movement. In heterogeneous habitats predictions further depend on how enrichment alters the variance of resource distribution. Greater patch variance always increases the optimal rate of movement, except for type‐IV functional responses. While the functional response is well established as a fundamental determinant of consumer–resource dynamics, our results indicate its importance extends to the understanding of individual movement strategies.     

Activity of descending contralateral movement detector neurons and collision avoidance behaviour in response to head-on visual stimuli in locusts

We recorded the activity of the right and left descending contralateral movement detectors responding to 10-cm (small) or 20-cm (large) computer-generated spheres approaching along different trajectories in the locust's frontal field of view. In separate experiments we examined the steering responses of tethered flying locusts to identical stimuli. The descending contralateral movement detectors were more sensitive to variations in target trajectory in the horizontal plane than in the vertical plane. Descending contralateral movement detector activity was related to target trajectory and to target size and was most sensitive to small objects converging on a direct collision course from above and to one side. Small objects failed to induce collision avoidance manoeuvres whereas large objects produced reliable collision avoidance responses. Large targets approaching along a converging trajectory produced steering responses that were either away from or toward the side of approach of the object, whereas targets approaching along trajectories that were offset from the locust's mid-longitudinal body axis primarily evoked responses away from the target. We detected no differences in the discharge properties of the descending contralateral movement detector pair that could account for the different collision avoidance behaviours evoked by varying the target size and trajectories. We suggest that descending contralateral movement detector properties are better suited to predator evasion than collision avoidance.     

Distance in motion: response trajectories reveal the dynamics of number comparison

Cognitive and neuroscientific evidence has challenged the widespread view that perception, cognition and action constitute independent, discrete stages. For example, in continuous response trajectories toward a target response location, evidence suggests that a decision on which target to reach for (i.e., the cognition stage) is not reached before the movement starts (i.e., the action stage). As a result, instead of a straight trajectory to the correct target response, movement trajectories may curve toward competing responses or away from inhibited responses. In the present study, we examined response trajectories during a number comparison task. Participants had to decide whether a target number was smaller or larger than 5. They had to respond by moving to a left or a right response location. Replicating previous results, response trajectories were more curved toward the incorrect response location when distance to 5 was small (e.g., target number 4) than when distance to 5 was large (e.g., target number 1). Importantly, we manipulated the response mapping, which allowed us to demonstrate that this response trajectory effect results from the relative amount of evidence for the available responses across time. In this way, the present study stresses the tight coupling of number representations (i.e., cognition) and response related processes (i.e., action) and shows that these stages are not separable in time.     

Emerged or imposed: a theory on the role of physical templates and self‐organisation for vegetation patchiness

In this article, we develop a unifying framework for the understanding of spatial vegetation patterns in heterogeneous landscapes. While much recent research has focused on self‐organised vegetation the prevailing view is still that biological patchiness is mostly due to top‐down control by the physical landscape template, disturbances or predators. We suggest that vegetation patchiness in real landscapes is controlled both by the physical template and by self‐organisation simultaneously, and introduce a conceptual model for the relative roles of the two mechanisms. The model considers four factors that control whether vegetation patchiness is emerged or imposed: soil patch size, plant size, resource input and resource availability. The last three factors determine the plant‐patch size, and the plant‐to‐soil patch size ratio determines the impact of self‐organisation, which becomes important when this ratio is sufficiently small. A field study and numerical simulations of a mathematical model support the conceptual model and give further insight by providing examples of self‐organised and template‐controlled vegetation patterns co‐occurring in the same landscape. We conclude that real landscapes are generally mixtures of template‐induced and self‐organised patchiness. Patchiness variability increases due to source–sink resource relations, and decreases for species of larger patch sizes.     

Dispersion movements in ants: spatial structuring and density-dependent effects

This paper examines whether the characteristics of individual dispersion movements in ants are changed when workers are moving solitarily or in a group. We analyzed the trajectories of workers of the species Messor sancta moving solitarily or in groups of different size (5, 10, 15 individuals), tested for density-dependent effects on their trajectory characteristics and investigated through resampling techniques whether ants are able to spatially structure their movements through direct (e.g. contact) or indirect (pheromone deposited on the ground) interactions. In addition to group size, the effects of the nutritional state of the colony and of the state of the area on which ants were dispersing were also examined. Solitary ants moved faster and had more sinuous trajectories than ants moving in a group. We found however no significant differences in trajectory characteristics between groups of different size. Whatever the group size, ants from starved colonies moved more slowly and had more direct trajectories than their counterpart coming from fed colonies. On the other hand, the state of the area on which ants were moving had no direct significant effect on dispersion movement. Ants dispersing in a group moved independently and did not coordinate their movements through direct or indirect interactions. However, the geometry of their path was changed not only through the effect of random encounters with other workers but also through an active modification of their movement when they perceived directly or indirectly the presence of nearby workers.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995     

Change in Auditory Image Movement Trajectories under Conditions of Direct Nonsimultaneous Masking

The authors studied fused auditory image (FAI) movement trajectories under conditions of direct nonsimultaneous masking. This movement was created by a gradual change in a dichotically presented series of clicks with interaural differences in stimulation from 0 to ±700 s or from ±700 to 0 s. Binaurally presented transmissions of wide-band noise served as maskers. The location and length of the trajectories were evaluated without a masker and with five values of the time lag between the signal beginning and masker end. When the test signal duration was 200 ms, the length of the trajectories was 33–44° without a masker. In the first test group, this trajectory lay close to the median line of the head without a masker (irrespective of the movement direction) and moved away from it under masking conditions. When the FAI moved from the median line towards the right or left ear, the initial part of the trajectory was masked; when the movement direction was opposite, the final part was masked. In the second group, the trajectories were located near the ears when the FAI moved from either ear and shifted towards the median line as a result of masking. When the movement direction was opposite, they were close to the median line and shifted towards the ear under masking conditions. When the FAI moved along all trajectories, their initial parts were masked.     

Biased Brownian dynamics for rate constant calculation

An enhanced sampling method-biased Brownian dynamics-is developed for the calculation of diffusion-limited biomolecular association reaction rates with high energy or entropy barriers. Biased Brownian dynamics introduces a biasing force in addition to the electrostatic force between the reactants, and it associates a probability weight with each trajectory. A simulation loses weight when movement is along the biasing force and gains weight when movement is against the biasing force. The sampling of trajectories is then biased, but the sampling is unbiased when the trajectory outcomes are multiplied by their weights. With a suitable choice of the biasing force, more reacted trajectories are sampled. As a consequence, the variance of the estimate is reduced. In our test case, biased Brownian dynamics gives a sevenfold improvement in central processing unit (CPU) time with the choice of a simple centripetal biasing force.     

Language comprehension in the balance: the robustness of the action-compatibility effect (ACE)

How does language comprehension interact with motor activity? We investigated the conditions under which comprehending an action sentence affects people''s balance. We performed two experiments to assess whether sentences describing forward or backward movement modulate the lateral movements made by subjects who made sensibility judgments about the sentences. In one experiment subjects were standing on a balance board and in the other they were seated on a balance board that was mounted on a chair. This allowed us to investigate whether the action compatibility effect (ACE) is robust and persists in the face of salient incompatibilities between sentence content and subject movement. Growth-curve analysis of the movement trajectories produced by the subjects in response to the sentences suggests that the ACE is indeed robust. Sentence content influenced movement trajectory despite salient inconsistencies between implied and actual movement. These results are interpreted in the context of the current discussion of embodied, or grounded, language comprehension and meaning representation.     

Integrating individual search and navigation behaviors in mechanistic movement models

Understanding complex movement behaviors via mechanistic models is one key challenge in movement ecology. We built a theoretical simulation model using evolutionarily trained artificial neural networks (ANNs) wherein individuals evolve movement behaviors in response to resource landscapes on which they search and navigate. We distinguished among non-oriented movements in response to proximate stimuli, oriented movements utilizing perceptual cues from distant targets, and memory mechanisms that assume prior knowledge of a target??s location and then tested the relevance of these three movement behaviors in relation to size of resource patches, predictability of resource landscapes, and the occurrence of movement barriers. Individuals were more efficient in locating resources under larger patch sizes and predictable landscapes when memory was advantageous. However, memory was also frequently used in unpredictable landscapes with intermediate patch sizes to systematically search the entire spatial domain, and because of this, we suggest that memory may be important in explaining super-diffusion observed in many empirical studies. The sudden imposition of movement barriers had the greatest effect under predictable landscapes and temporarily eliminated the benefits of memory. Overall, we demonstrate how movement behaviors that are linked to certain cognitive abilities can be represented by state variables in ANNs and how, by altering these state variables, the relevance of different behaviors under different spatiotemporal resource dynamics can be tested. If adapted to fit empirical movement paths, methods described here could help reveal behavioral mechanisms of real animals and predict effects of anthropogenic landscape changes on animal movement.     

A model of handwriting

The research reported here is concerned with hand trajectory planning for the class of movements involved in handwriting. Previous studies show that the kinematics of human two-joint arm movements in the horizontal plane can be described by a model which is based on dynamic minimization of the square of the third derivative of hand position (jerk), integrated over the entire movement. We extend this approach to both the analysis and the synthesis of the trajectories occurring in the generation of handwritten characters. Several basic strokes are identified and possible stroke concatenation rules are suggested. Given a concise symbolic representation of a stroke shape, a simple algorithm computes the complete kinematic specification of the corresponding trajectory. A handwriting generation model based on a kinematics from shape principle and on dynamic optimization is formulated and tested. Good qualitative and quantitative agreement was found between subject recordings and trajectories generated by the model. The simple symbolic representation of hand motion suggested here may permit the central nervous system to learn, store and modify motor action plans for writing in an efficient manner.     

Computational model of motor learning and perceptual change

Motor learning in the context of arm reaching movements has been frequently investigated using the paradigm of force-field learning. It has been recently shown that changes to somatosensory perception are likewise associated with motor learning. Changes in perceptual function may be the reason that when the perturbation is removed following motor learning, the hand trajectory does not return to a straight line path even after several dozen trials. To explain the computational mechanisms that produce these characteristics, we propose a motor control and learning scheme using a simplified two-link system in the horizontal plane: We represent learning as the adjustment of desired joint-angular trajectories so as to achieve the reference trajectory of the hand. The convergence of the actual hand movement to the reference trajectory is proved by using a Lyapunov-like lemma, and the result is confirmed using computer simulations. The model assumes that changes in the desired hand trajectory influence the perception of hand position and this in turn affects movement control. Our computer simulations support the idea that perceptual change may come as a result of adjustments to movement planning with motor learning.     

Interference competition,payoff asymmetries,and the social relationships of central place foragers

We present a central place foraging model that shows how payoff asymmetries originate in contests for access to resources. The essence of the model is that interference competition at resource points lowers the rate at which foragers can load prey, thereby depressing the rate of food delivery to the central place. We show that interference leads to asymmetric payoffs when contests involve foragers with (i) unequal travel distances between the central place and the contested resource points; (ii) inequalities in the rate of food delivery available from alternative foraging sites; (iii) differences in loading efficiency; or (iv) different abilities to interfere. We use the asymmetries to predict dominance rankings, and the patch exploitation tactics of individual foragers. We also consider the implications of the model for changes in the travel distance (= area) over which foragers can exclude competitors (= territoriality) as food density changes. Finally, incorporation of interference permits our model to predict the transition between scramble and contest competition.     

A field experiment on the influence of load transportation and patch distance on the locomotion velocity of <Emphasis Type="Italic">Dorymyrmex goetschi</Emphasis> (Hymenoptera,Formicidae)

Summary Locomotion velocity during foraging activities is determined by factors such as travel distance, habitat structure and load mass among others. However, few studies on foraging behavior have analyzed the influence of spatial heterogeneity and food transportation on the locomotion velocity of ants under natural conditions. In order to study the mentioned factors, we selected 20 nests of the ant Dorymyrmex goetschi (subfamily Dolichoderinae), in a lower Andes locality of central Chile. Half of the nests were offered a food patch located at 10 cm from the nest entrance, and at 20 cm for the other half. We measured the duration of trips between nest and food patch and vice versa, and the distances traveled. We also recorded spatial heterogeneity of the substratum and soil temperature. Temperature was used as a covariate in the statistical analysis. Travel speed was significantly slower when worker ants returned to the nest with a food load, compared to the velocity of foragers without load that traveled from the nest to the patch. When the food patch was located at greater distance, locomotion velocity was significantly faster. Spatial heterogeneity did not affect movement speed. The reduction in locomotion velocity in ants carrying a load of 5.6 mg represents an energetic cost of transportation equivalent to 79% of the costs involved in moving a body mass of 1.6 mg. Faster velocities at larger patch distances can be interpreted as a strategy to maintain an efficient resource exploitation, by way of decreasing the time exposed to higher predation risk.Received 28 April 2003; revised 11 November 2003; accepted 22 January 2004.     

A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility

In recent single-particle tracking (SPT) measurements on Listeria monocytogenes motility in cells [Kuo and McGrath (2000)], the actin-based stochastic dynamics of the bacterium movement has been analyzed statistically in terms of the mean-square displacement (MSD) of the trajectory. We present a stochastic analysis of a simplified polymerization Brownian ratchet (BR) model in which motions are limited by the bacterium movement. Analytical results are obtained and statistical data analyses are investigated. It is shown that the MSD of the stochastic bacterium movement is a monotonic quadratic function while the MSD for detrended trajectories is linear. Both the short-time relaxation and the long-time kinetics in terms the mean velocity and effective diffusion constant of the propelled bacterium are obtained from the MSD analysis. The MSD of the gap between actin tip and the bacterium exhibits an oscillatory behavior when there is a large resistant force from the bacterium. For comparison, a continuous diffusion formalism of the BR model with great analytical simplicity is also studied.     

Estimation of dynamic joint torques and trajectory formation from surface electromyography signals using a neural network model

In this study, human arm movement was re-constructed from electromyography (EMG) signals using a forward dynamics model acquired by an artificial neural network within a modular architecture. Dynamic joint torques at the elbow and shoulder were estimated for movements in the horizontal plane from the surface EMG signals of 10 flexor and extensor muscles. Using only the initial conditions of the arm and the EMG time course as input, the network reliably reconstructed a variety of movement trajectories. The results demonstrate that posture maintenance and multijoint movements, entailing complex via-point specification and co-contraction of muscles, can be accurately computed from multiple surface EMG signals. In addition to the model's empirical uses, such as calculation of arm stiffness during motion, it allows evaluation of hypothesized computational mechanisms of the central nervous system such as virtual trajectory control and optimal trajectory planning.     

Value-complexity tradeoff explains mouse navigational learning

We introduce a novel methodology for describing animal behavior as a tradeoff between value and complexity, using the Morris Water Maze navigation task as a concrete example. We develop a dynamical system model of the Water Maze navigation task, solve its optimal control under varying complexity constraints, and analyze the learning process in terms of the value and complexity of swimming trajectories. The value of a trajectory is related to its energetic cost and is correlated with swimming time. Complexity is a novel learning metric which measures how unlikely is a trajectory to be generated by a naive animal. Our model is analytically tractable, provides good fit to observed behavior and reveals that the learning process is characterized by early value optimization followed by complexity reduction. Furthermore, complexity sensitively characterizes behavioral differences between mouse strains.