Abstract: Classes of Mathematical Modeling for Brain Dynamics
We report 10 classes of mathematical modeling from neurons to a society of brains. Various mathematical models at hierarchical levels of function from a single neuron to a society of brains, namely, a level of communication have been proposed so far in the field of cognitive neuroscience. However, no clear classification of such mathematical models has been made. To clarify the functional significance of the mathematical models at each level, we present typical models for each class. Although our present classification may not be sufficient, and the examples provided may be limited, as they do not cover all important contributions, the importance of mathematical modeling in complex brain dynamics and neural correlates of functions will be shown clearly. In particular, the following type of mathematical modeling is challenging, as it may elucidate the mechanism of functional differentiation in the mammalian cortex. From the viewpoint of systems development, it is important to investigate the manner in which components emerge in a network system consisting of interacting units. In this respect, we report two mathematical models: one treats the emergence of neuron-like components from interacting maps, and the other addresses the emergence of hierarchical module-like components from interacting neuron-like units. In both models, maximum transmission of information was used as a “variational” principle. All mathematical models described in the present talk will probably be useful for future studies in the field of developmental robotics.
Ichiro Tsuda holds the position of Professor in the Research Institute for Electronic Science (RIES) and is the director of Research Center for Integrative Mathematics of Hokkaido University, as well as an invited professor at Department of Mechanical Engineering of Osaka University. He is also a member of the Advisory Board of Complex Systems Institute at Kalamazoo College, USA. He was a visiting professor at the Japan Advanced Institute of Science and Technology (JAIST), and was the group leader of the Basic Design Group, National Bioholonics Project of Exploratory Research for Advanced Technology (ERATO), Research Development Corporation of Japan (currently named JST). He has published widely in the field of chaotic dynamical systems and brain sciences. His research interest is mathematical modeling of higher brain function, including memory dynamics, thoughts, and inference processes, and also numerical studies of chaotic dynamical systems. He constructed a one-dimensional map to explain chaos and bifurcation structure in the BZ reaction sufficiently and found noise-induced order in this model. He also constructed neural network models for dynamic associative memory and for episodic memory based on physiological data concerning class I neurons and different types of synapses. In the former, he found a new dynamical state, termed chaotic itinerancy (the recent contribution is available in ) and proposed its dynamical interpretation in terms of a Milnor attractor, and for the latter, he proposed a new coding scheme, termed Cantor coding, for episodic memory formation.