My research interests are in the development and application of novel statistical physics and complex system methods. During my PhD I worked promarily on mathematical physics and statistical mechanics. Since I began working at Calgary I have been slowly migrating interestes towards biolgical problems. This I have been doing by reading the lastest research, interacting with various biologists at the University of Calgary, and discussing biological systems with colleagues withing the complexity science group. My latest research work has been on statistical physics problems motivated by biological phenomena as well as network analysis techniques applied to biological inference.
While I have learned the mathematical aspects of statistical mechanics and complex networks, through the Transdisciplinary Initiative, led by my supervisor Prof. Paczuski I have also had the oppurtunity to learn from various experts working in areas of biological and social sciences. These experts include microbiologists Michael Surette and Anthony Schryvers, computer scientist Przemyslaw Prusinkiewicz, sociologist Richard Hawkins, policy analyst Ron Bouchard, geomatician Danielle Marceau, systems biologist Sui Huang, and neuroscientist Naweed Syed. In these exchanges I have experienced the difficulties that arise when communicating across traditional disciplines. However, after many such discussions, I can confidently say that now I can consistently transcend the barriers of language and jargon, and grasp the heart of many diverse scientific problems. In collaboration with researchers at the University of Calgary and elsewhere, I am currently studying statistical physics models for several of these problems as well as developing new methods to analyse data from biological experiments. In the following I provide a synopsis of my current research and how this extends to my future interests.
Mathematical abstractions like an interaction newtork of biological data, rely heavily on the accuracy of inferences drawn from the experimental data. Extensive research on protein interaction networks (PINs) has pointed out the need for the theoretician to be extra careful while drawing general conclusions from experimental data. The state of art technique used to find protein interactions is the tandem-affinity purification followed by mass spectroscopy (TAP-MS). TAP-MS identifies stable protein complexes which are then used to infer pair-wise protein protein interactions. With graduate student Orion Penner and Prof. Maya Paczuski, we are tyring to develop a mathematical model for protein interactions, based on a protein's constituent domains. We will use this model to simulate the TAP-MS experiment on a computer. The simulated proxy data for stable protein complexes from our model will allow us to critically analyse the various commonly used inference protocols. Identifying the dominant features of these model PINs will let us recognze the properties of inferred PINs that are artifacts of the construction mehtod. Success in this endeavor could provide a powerful new tool for understanding the impact of inference methods and experimental techniques upon PIN structure, thereby by improving the experimental construction of PINs.
Certain experimental developments in the field have attracted my attention in particular. These include the observation of avalanches in neural media consisting of cortical slice cultures, and the functional connectivity network of the brain obtaine d using fMRI techniques. I have studied the mathematical description used to explain the observed properties, such as branching processes and self organized criticality, and would like to explore this connection further. We have been in contact with an experimental group at the University of Calgary to obtain a electrode chip data. This data is both for a larger system, as well as more detailed than the data used before.
My general interest in neuroscience has lead me to explore other neuronal phenomena. Plasticity of neural connections is governed by the on-going activity within the neural network. Old connections can become weak and new ones can form when the input to the network is changed. This phenomena underlies cross-modal plasticity, in which a set of neurons which respond to a specific input can become specialised to a different stimulus when its inputs are rewired. For example, the auditory neurons can start responding to visual inputs when the auditory input is rewired to the visual pathway in a mouse. Detailed mathematical models have been developed to explain this phenomena, but they have not kept pace with the vast leaps in our experimental knowledge about synaptic plasticity. Along with Dr. S. Sreenivasan at the University of Notre-Dame, I am developing a simple mathematical model for a small neuronal circuit capable of exhibiting this cross-modal plasticity. With a simple model, we hope to study different plasticity rules and their impact on cross-modal plasticity.
Plasticity also governs how a young neuron develops and reaches its mature shape. Availability of data from Prof. Syed's lab at the University of Calgary, will help us develop models for neurite growth at various levels of detail. A neuron's dendritic tree resembles the structure of a branching polymer studied in statistical physics. A starting point to model neurite growth will be to study the growth branching polymer. Modeling synaptic plasticity and growth of synaptic cones in response to neuronal signals will help us understand the departure from a simple branching polymer model.
Evolution of a Heterogeneously Structured Population
Many different variants of the same bacterial species are found in nature. Using information theory, this variation can be related to the variability in a bacterium's environment (Kussell and Leibler 2006). With graduate student Jacob Foster, I have been studying the potential application of the theory developed by Bergstrom and Lachmann to experiments performed in the lab of Prof. Mike Surette. These experiments measure rpoS-mutS polymorphism in Salmonella. Given the constraints on a bacterial population's phenotypic response, we will characterize the amount of variability in the environment that leads to genetic mutations.
Random walks are a fascinating subject, both for their intrinsic mathematical beauty and for their wide range of applications. One of the most celebrated result is Polya's transience/recurrence property of a random walk on a "street-network". This result states that a random walk starting at the origin of an infinite lattice will eventually return to the origin, or that the random walk is recurrent, when the dimension of the lattice is 1 or 2, but in higher dimensions the random walk will wander off to infinity with positive probability, or that the random walk is transient. Following this early result, random walks have been studied on non-regular lattices, such as percolation structures, and graph structures such as Erdos-Renyi (ER) random graph and scale-free networks and Cayley graphs of groups.
In my research I am primarily interested in understanding the properties of random walks on general graphs. On regular lattices or percolation clusters on regular lattices, as well as Cayley graphs of groups, underlying symmetries and a metric structure can be used to understand the properties of a random walk. These properties include, besides the transience properties, the asymptotic speed of a random walk. On a general graph, however, the natural properties of a random walk to be studied are its first-passage properties. These first-passage properties can then be related to and understood in terms of the underlying topological structure of the network. This work formed the subject of my paper,
First-passage properties of the Erdos–Renyi random graph,
where we found that the percolation transition on a random graph induces non-monotonic behaviour of the mean first passage time of the random walk between a pair of nodes, as a function of link probability.
Whether the random walk will return or escape to infinity depends on how fast the graph grows with distance from the origin of the random walk. On an ER random graph the number of nodes grows exponentially with distance, causing random walks to escape. In order to arrive at a non-trivial problem in my paper with Peter Grassberger.
Localization Transition of Biased Random Walks on Random Networks,
we discuss walks biased towards a randomly chosen but fixed "target" node. We show that there is a phase transition from recurrence (or localization ) to delocalization at a critical bias strength.
Biological Regulatory Dynamics
Recent work has shown that real asynchronous protein-gene interactions can be modeled as simple deterministic boolean interactions. The results from these dynamics are startling in their prediction of complicated biological dynamics such as the segmentation patters oof the Drosophila embryo. We are trying to understand the range of applicability of of these deterministic models.
In a recent paper we related the critical behaviour of the dynamics of a Random Boolean Netwoork (RBN) to the topological structure of its state space network (SSN). We characterised both the local and the global structures of the SSN. RBNs have been used as caricatures of gene regulatory networks, and we aim to extend our state space analysis to the dynamics of gene regulation. We chose discrete time synchronous updates for the RBN. In order to extend our analysis to the real networks of gene regulation we will first characterize the complexity of stochastic asynchronous dynamics for the RBN.
The central dogma of modern economic theory is the assumption that an equilibrium exists and is realized. However, recent papers have shown that sensible adaptive dynamics do not guarantee convergence to Nash equilibrium (Hart 2003). Simple games like prisoner's dilemma have been studied using statistical mechanics formulations, with focus on the nature of the Nash equilibrium rather than on the dynamical approach to such an equilibrium.
We are trying to understand the approach to a Nash equilibrium of a population playing the bargaining and ultimatum games. Developing model dynamics for two players, which yield convergence to the Nash equilibrium, I will apply these interaction dynamics to a population of agents who trade with their neighbours on a complex network.
Social Dynamics and Opinion Formation
With Prof. S. Redner and Dr. T. Antal, I showed that the effective persuasiveness of an opinion can depend strongly on the structure of social contacts. A natural extension of this study is to analyse the case when the individuals themselves have heterogeneous persuasiveness and/or persuadability. I will use mathematical insights gained from my current work to study social dynamics in a real setting, eg. social structures such as Facebook, and massively multi-player online games. Using the insights from fundamental statistical physics, I would also like to understand the effect of social structure and individual abilities on the evolution of humanly unique cognitive abilities.
V. Sood and S. Redner,
Phys. Rev. Lett. 94, 178701 (2005).
T. Antal, S. Redner and V. Sood
Phys. Rev. Lett. 96, 188104 (2006)
V. Sood, T. Antal and S. Redner
Voter Models on Heterogeneous Graphs.