Simulation of three-dimensional Turing pattern (Gyroid structure)
The primary areas of inquiry of our nonlinear-dynamics group are structure formation, dynamics, and statistics of non-equilibrium systems far from thermal equilibrium. The theoretical framework of the equilibrium statistical-thermal physics had been established in the early 20th century. Theories for linear non-equilibrium systems, which began with Einstein's analysis of Brownian motion and resulted in the fluctuation-dissipation relations and the linear-response formalism, had also been established in the middle of the last century. Those theoretical developments stimulated the development of the theories of solid-state physics, which gave rise to the great electronic industries in the late 20th century. In Japan, studies of nonlinear non-equilibrium systems started in the 1970's, which produced significant results such as projection operator formalisms, mode-coupling theories, and dynamical reduction theories. However, universal laws of non-equilibrium systems corresponding to the maximum-entropy principle in equilibrium statistical mechanics is still unknown, and the first-principle theories to calculate non-equilibrium observables still haven't reached maturity.
Phase diagram of FitzHugh-Nagumo equation
Recent experimental developments in material and biological sciences seem to foreshadow a revolution in current nonlinear non-equilibrium theories. Questions about how much information is contained in experimental data obtained by single-molecule measurements or by optical tweezers, or how to effectively control nano-scale non-equilibrium structures; these appear to be applications, but are directly related to the fundamental problems of non-equilibrium physics. To solve these problems, detailed analysis of minimal mathematical models that incorporate essential features of non-equilibrium phenomena, and development of phenomenological theories that capture the laws underlying experimental data, are indispensable.
Spike sequence of neuron
Another major research interest of ours is the dynamics of neuronal systems. Information from the outside world exists in the brain as a signal composed of neuronal spikes. We would like to understand the mechanism of spiking in a quantitative manner. Until recently, it was thought that spike trains in vivo could be thought of as a Poisson (random) process. With our most recent research, it is becoming clear that the spike train is not a uniform Poisson process, but differs from cell to cell, and is dependent on location and cortical layer in which it is located. Also, it is becoming clear that the existing neuron models do not adequately illuminate the spiking mechanism, so we are in the process of creating a new mathematical framework that better models neurons. We are enjoying the new results that arise and the variety of activity in this field, while we strive to understand the essential and fundamental laws that govern the information processing and the brain.