Abstract
Perception of biological motion has long been studied in psychology – humans can identify the activity and identity of moving objects from a few reference points. Similarly, from a crowd of reference points, one can determine which points describe an individual. With the assistance of the Dance Department, utilizing motion capture technology, I will investigate the mathematical implications of this phenomenon, discover an analytic description of curves traced by human perambulation, and develop an algorithm (and computational implementation) to recognize patterns of motion in a group of points and classify the data into coherent subgroups. I hope to generalize the underlying mathematics to model dynamics in complex systems, such as patterns of dynamics in gene expression, and discover constraints on the architecture of gene networks.
Research Proposal
Perception of biological motion has been an intriguing subject of study in the psychological field. As Gunnar Johansson discovered, movement of the human body can be described using a limited number of strategically placed points. A single picture of the points is unrecognizable, but when put in motion, the mind easily interprets the correlation of points in their separate curves and can not only discern the approximation of a body but also the particular activity of the body, whether it be walking, running, dancing, et cetera; though every individual human being has a unique gait, which varying psychologists claim is affected by everything from gender to emotional state. This implies that for a standard movement, there is a set of curves along which these points of the body travel with a certain range of probabilistic variation.
In comparison, the character of the curves followed by a dancer's movements is completely different from those of a standard pedestrian walk by design of the movement. A ballerina will have a much smoother curve than a pedestrian, a tap dancer's path will be much sharper and more "erratic", and a ballroom dancer performing a waltz will have a smoother curve with a 2:1 ratio of elevation to depression (as opposed to a pedestrian's more even 'bounce'). But again, within these curves lies an inherent variation imposed by the body performing the action. Due to the more rigorous formal training in dance movements, this variation may in fact be smaller than that of the walking curves, a minor hypothesis I will also examine in the course of the first phase of research. After determining a general curve with probabilistic variation, however, my primary focus will be to determine a specific rule for any individual's particular variation through analysis of their curve.
It would appear relatively easy to generalize the curves described by the movements of a single dancer to examine a group of dancers moving in unison through choreography – the curves used have their slight variation, but every dancer (theoretically) moves through the same approximate portion of the curve at the same instant in time. A group of dancers interacting within choreography will not move along identical curves, but their curves will still have a similar character (defined by the curvature, continuity, or types of discontinuity, for example) that distinguishes them from another style of dance or pedestrian movement.
A group of people walking will follow the same basic idea, but with a greater degree of stochastic variation – the same type of pedestrian curve described at varying times with disparate velocities by several different body types in a myriad of emotional states. Since psychological research has shown that these various attributes are discernable on viewing by a human observer, I take this to imply that each attribute creates a distinctive variation in the described curves. The first major challenge we will face will be developing an algorithm to recognize patterns of motion in a composite data set of many individuals and classify the data into coherent subgroups corresponding to each individual.
Discussion with Professor Assadi has provided invaluable insight into the possible realms of generalization of the basic geometric structure described by this motion capture analysis and subsequent algorithm development, specifically in biochemistry research. The dynamics of a single gene can be described in a state-space model, as with T-cells in prior research, but can also be treated geometrically as with a single reference point on an object in motion.
The network of proteins and genes in a heart, for example, can be likened in a very real sense to the network of data points making up the image of an individual walking or dancing. Through analysis of the network of genes and proteins, I hope to discover specific rules for their behavior, as with the individual dancers and pedestrians. Furthermore, by likening the entire structure to that of the many pedestrians, I hope to generalize the algorithm and implementation for determining individuals within a composite scene to discovering subnetworks within the genes and proteins.
Prior research (e.g. Rangel C et al (2004), Stewart (2004)) utilized directional graphs and linear state-space models (SSMs) for analysis of genes and cellular behavior. I propose to approach the problem from a different angle, utilizing the new advanced tools from topology and differential geometry that have not yet been incorporated into statistics and other such traditional analysis. Dynamical systems theory together with the SSMs provide a host of extremely useful tools for analysis of the coherent motion of points, which can be extrapolated as explained above to the dynamics of gene expression and protein architecture.
This novel approach to analysis in bioinformatics is a field virtually untouched by prior study and an exciting venue for research. Through this unexplored angle, I propose to study common dynamic features in both the physical realization of coherent points in a dance and the patterns of dynamics in a genetic network.
Timeline for Research
· STEP Ia. Collect data for biological motion with the assistance of several students from the Dance and Inter-Arts and Technology (IATECH) Departments, under the support of Karen McShane-Hellenbrand of the Dance Department. Two undergraduates in the Computer Science Department will assist in the setup and collection of motion capture data, including development of useful software for the data collection process. Using Matlab© programs developed by Randolph Blake of Vanderbilt University, create a mathematical description of the curves traced by individuals walking and performing short combinations of dance material. (One month)
· STEP Ib. Develop and implement an algorithm to recognize these patterns of motion. Create composite scenes of the data, and refine the algorithm and its implementation to discern an individual from a group of moving points. I may also include an intermediate step – create a composite scene of several pedestrians and a dancer, and first develop an algorithm to discern the dancer's reference points. The curves described by a dancer will be chosen to differ significantly from those of a pedestrian, to test the effectiveness of the generic walk curve and its variance. (Through May 2006)
· STEP II. Collaborate with Professor Assadi and Professor G. Michelotti of Duke University to begin research generalization to the biology of the heart system, focusing specifically on the network of biomolecules relevant to the adaptor α_{1}AR. This will also include learning bioinformatics – the relevant computational biology and the use of biological databases – for ease and expediency of later computations and analysis, as well as images and dynamic gene expression data for familiarity with the types of data in systems biology and biological networks. (Summer 2006)
· STEP III. Re-examine the differential geometry and dynamical system theory from Step I, and generalize the algorithms and methods initially created for motion perception analysis to treat dynamic gene expression data. The networks of genes are indeed close to the networks of reference points, however the constraints between related points are expected to be considerably more elastic, and data is unfortunately much sparser than available while developing the initial algorithm. (September – November 2006)
· STEP IV. Apply the results of Step III to data on dynamic gene expression provided by Duke researchers. Much of this data is available now (February 2006) and more will be provided over the course of the year. In this final step I hope to determine the constraints on the architecture of gene/protein networks based on their dynamic features, and validate results against existing biological literature available to Professor Michelotti and the Duke researchers. (December – January 2006)
References.
Bach, Michael, Biological Motion. 08 Aug 2005. http://www.michaelbach.de/ot/mot_biomot/
Blake, Randolph, Perception of Biological Motion. http://www.psy.vanderbilt.edu/faculty/blake/BM/BioMot.html
Johansson, G (1973) Visual perception of biological motion and a model for its analysis. Perception and Psychophysics 14:201-211.
Rangel C, Angus J, Ghahramani Z, Lioumi M, Sotheran E, Gaiba A, Wild D & Falciani F (2004) Modeling T-cell activation using gene expression profiling and state-space models. Bioinformatics 20:1361-1372.
Stewart, Ian (2004) Networking Opportunity. Nature 427:601-604.