The study of the emergent behaviors arising from the interaction of individual components in biological systems (cellular dynamics, pathways functions, protein functions, enzymatic and metabolic pathways), their modeling, prediction of cellular states driven by various internal and external stimuli etc are the main focus. Main works are development of computational tools and mathematical modeling of these systems using deterministic and stochastic approaches in the following problems: systems medicine, genetic oscillator, switch and regulation, biological rhythms, cancer dynamics, brain disorder and dynamics, synchronization of the dynamical systems, molecular basis communication, signal processing, DNA sequence analysis (Fourier transform, spectral analysis), stochastic resonance in cellular processes, and study of dynamic instability in microtubules using Monte Carlo techniques.

The causal behavior and dynamics of phenomena which have probabilistic representations of natural systems can be studied and analyzed through stochastic approaches. The random and non-linear nature of the fluctuating interactions among the components of the stochastic models (systems) give rise to completely new attributes which may or may not be similar to those of the components. We try to address and explore answers to the questions about how these fluctuation driven dynamical systems work at the fundamental level and their properties using the stochastic framework. Be it a simple phenomena like a two species chemical reaction or a complicated problem like protein folding or the formation of turbulent structures in galaxies and stars in faraway nebulae, we strive to find answers to some fundamental questions regarding these phenomena by using the powerful framework of stochastic models. We are interested in finding solutions to the Chapman-Kolmogorov equation (Master and Fokker-Planck-Kolmogorov equations) with related extensions both for memory and memoryless processes, fractional brownian motion etc using various techniques and their applications to complex systems.

We focus on the modeling complex dynamical systems: phase flow in low and high dimensional systems, linear stability analysis, bifurcation theory, lyapunov exponent and function, synchronization of coupled systems, coupling mechanisms, order parameters for detection of synchronization (permutation entropy, phase locking dynamics, correlation function, etc), generalized synchronization, reservoir computing (use of machine learning techniques in coupled systems) and applications to various dynamical systems.

The applications of fractals range from economics to geography to medical imaging to art. Fractal analysis measures the degree of complexity in the morphology of patterns involved in the system’s structure using the fractal parameters, namely fractal dimension, Hurst exponent, etc. In our Complex Dynamics Lab, we study multifractality in various data such as genomic (tuberculosis, SARS-Cov-2), financial, astrophysical, biomedical (EEG), etc. In addition to data analysis, we also study fractal/multifractal theory to explain the behavior and properties of the system. 

Though the subject of evolution and development of organisms is two-three centuries old, the progress in understanding even its most singular principle i.e. heredity is still largely incomplete. There are a multitude of questions, each of which has a myriad of dimensions, still standing. Molecular biological tools, to probe atomistic level structures (gene) and the functions these structures play, have helped shape our understanding about the hereditary process. Mathematical abstraction of the hereditary processes is carried out through the help of population genetics models. We use the knowledge of molecular level physio-chemical rules (mutation, dominance, epistasis, recombination, etc.) to build these mathematical models. The main aim of the endeavor is to find some causal understanding of how an organic being sustains itself in ever changing surroundings.

1. Identification of key regulators: Novel key regulators are generally an important focus of the study of complex networks. Because these key regulators could be potential target genes in drug discovery in disease networks, novel regulators of social network dynamics, etc. We focus on the development of techniques to identify these key regulators in complex networks.
2. Controllability theory: Whether a spacecraft has to be launched in space or a vaccine has to be developed, without control theory it can’t manifest into reality. Control theory provides a mathematical framework to control and analyze such dynamical systems. These dynamical systems can be represented through networks. We use controllability theory (linear and nonlinear) to identify the key driver nodes in the network and control dynamics via appropriate perturbation through key nodes.
3. Network medicine: We focus on the development of computational techniques and study of identification of key genes in disease networks to the drug discovery which is an important field in network science of biological systems.
4. Cognitive neuroscience: The structure and functions of the brain can be studied through representation of brain regions as a network. We apply the notion of controllability to study the cognitive functions and states of brain and neurodevelopmental disease through network controllability theory approach.