Handling editor: Phanikrishna Thota

Nonlinear Sciences Group at Kobe University

by Tamiki Komatsuzaki, Department of Earth and Planetary Sciences, Kobe University.

The nonlinear sciences group at Kobe University was founded in the late 1970s. This is the first and unique nonlinear sciences group in a Department of Earth and Planetary Sciences in Japan. From the 1970s, as one of the centers of nonlinear sciences in Japan, about eight professors and more than 30 PhD students have organized, amplified and developed new fields of nonlinear sciences involving emergent behavior, fractals, cognition, economy, chemical and biological physics. Most of the PhD students have continued their researches in academic research positions at universities and national institutes. Now there are two professors, three postdocs (from the University of Texas at Austin, the University of Chicago, and Nagoya University), and fifteen PhD students (nine of them from other universities and three from other departments (physics, biology and agriculture) at Kobe University), five MSc students (one from the US) and four BS students.

Students, postdocs, and staff in summer 2006 insets from left: Yuka Tauchi, Soichiro Tsuda, Moto Kamiura; standing from left: Akira Fukano, Junichiro Wakatsuki, Shinpei Tatsumi, Hiroyuki Ohta, Tomohiro Shirakawa, Kazuto Sei, Asaki Nishikawa, Satoshi Noda, Chun-Biu Li; half-rising posture from left: Sohei Wakisaka, Taichi Haruna, Fumihiro Matsuura; front from left: Yasuhiro Matsunaga, Mustafa Demirplak, Tamiki Komatsuzaki, Yukio-Pegio Gunji, Eugene Schneider Kitamura, Daisuke Uragami.

Main research interests of the current members are the establishment of a new mathematical framework for describing regularity buried in chaos in changes of the state, emergent behavior, and robustness of the occurrence (e.g., functions of biological systems) in complex systems. The mathematical tools on which our current research is based range from dynamical system theory (normal form, normally hyperbolic invariant manifolds, bifurcation theory, embedology), information theory (computational mechanics) to internal measurement/endo-physics (sheaf, lattice, category, group theory). The focus research projects are:

• Biological Organization in Internal Perspective
  (Yukio-Pegio Gunji)

  The organization of living things is characterized by dynamical hierarchical structures inheriting discrepancy among levels. It can be expressed as a system consisting of two layers; the microscopic perspective (Extent) defined by a collection of elements and the macroscopic perspective (Intent) defined by the property as a whole, and the interplay between them. First we show that if the microscopic and macroscopic perspectives are consistent with each other (an ideal case), then the operation between the two layers can be expressed as a sheaf between a lattice and a quotient lattice, where a sheaf is a mathematical operation representing the gluing process. Second, we introduce an observer who cannot look out over the whole world, and this reveals discrepancy between the two layers. Third, we introduce a new mathematical construction, called skeleton, that is derived by the sheaf operation. The skeleton reduces the discrepancy between the micro- and macroscopic perspectives, which reveals the perpetual transition between the perspectives. This process yields a basic framework of biological organizations. Finally, we argue that the skeleton mediating between the two levels is a particular expression for the material cause. These ideas are implemented by using sheaf theory (functor from a topological space to sets) and a lattice and quotient lattice described with respect to algebra [1, 9, 12].

• Internal measurement/endo-physics
  (Yukio-Pegio Gunji)

  The essential feature of the endo-perspective is examined, and a formal model of the endo-perspective is proposed, by introducing the mixture of intra- and inter-operations. Because such a mixture, in its naive realization, entails a paradox within a formal system, we weaken the inter-operation in order to allow the formal system to be endowed with that mixture without a contradiction. The weakened inter-operation is related to the infomorphism proposed by Barwise. The formal model of the endo-perspective is thereby expressed as the dynamic infomorphism driven by that mixture. The endo-perspective is described as a formal system that includes the outside of the occupied perspective. If such an inclusion is applied to the common definition of a set, it entails Russel's paradox. Retaining the outside can be expressed as the mixture of the intent and the extent of a set together with the mixture of intra-operations within the intent (or extent) and inter-operations between intent and extent. The endo-perspective, therefore, consists of two subsystems corresponding to the intent and the extent, respectively, and is defined as a system involving a particular mathematical tool (i.e., infomorphism) that allows for retaining the outside without a contradiction. Within that framework, the mixture of the intra- and the inter-operation drives the dynamical transition of the system, but it can be terminated by its collapse. This collapse can be predicted from the internal logic defined within the system. The model is constructed through the verification of `a weakened paradox'. Because the definition of the system involves a weakened paradox only, it does not always lead to a contradiction, although the collapse of the system corresponds to a contradiction. The double standards can be embedded into the system via a domain with truth-values (the inside) and a domain in which the collapse of the logic can occur (the outside) [4, 7, 10, 11].

• Biologically motivated computing (slime-mold computing)
  (Yukio-Pegio Gunji)

  Although there are many attempts for emergent computing based on biological molecular devices (local non-linear dynamics, DNA-based, protein-based and amorphous computing), the notion of emergent computing is still ambiguous. Emergence and 'error' are both sides of the same coin. A biologically based computer could be an unstable machine, while it would have the potential for emergent computing. Clearly, not the error but the emergent computing is based on the relationship of trust between a user and a machine.
  		To manifest that aspect we implement Boolean gate as a biological device made of slime mold. The emergent computation is theoretically defined by the global coherence generated from local interactions, while that coherence cannot be predicted locally. It needs discrepancies between local and global mechanisms. To introduce those discrepancies we use a real, living slime mold, Physarum polycephalum as a computing device. Compared with DNA and/or proteins used in computing that are non-living things, living slime mold has living wholeness that cannot be predicted locally. That is why discrepancies between local interactions and living unity as a whole can be employed to emergent computing [3, 5, 8].
  Physarum polycephalum grown on the agar gel plate (left); a device of plasmodium on a dry negative mask on a moist agar surface connected to a robot through optical interfaces (right).

• Cognitive Systems / Consciousness
  (Yukio-Pegio Gunji)

  Psychologists previously proposed introspection: The perception of a red apple is based on a label of the red apple. Such an idea falls into infinite regression and/or a self-reference because it leads to the mixture of a label and referring to a label. In spite of the collapse of the introspectionism, the notion of qualia was recently proposed, and it brings us another concept different from types. On the one hand, introspection was prescribed as a kind of type, that is, why it is a particular abstract expression and/or ideal form. Then it is destined to be a self reference. On the other hand, qualia appear in the science of consciousness as tokens, individual things that exist in a universe in their own right. Brain scientists and philosophers of consciousness claim that brain computes or constructs not types but tokens that are qualia. If one separates real universe from formal universe, or description from what is to be described, then one is destined to take after the difficulty resulting from the discrepancies between real things and introspection. One has to claim that real things (tokens) cannot be described or cannot be expressed as types. Hence, the proposal of qualia has a new significance against introspection. Therefore, the qualia are destined to be hard-problems. We have to focus on the co-existence of types-computation and tokens-computation. It is easy to see that type-computation in a machine must be accompanied with materialistic phenomena (i.e., tokens-computation) as electric flow, adequate temperature and so on. Especially in a brain, type-computation (a particular computation in a particular area of a brain) is carried out with token-computation (surrounding activities in and out of a brain). There is no static duality between types- and tokens-computations. Even if one attempts to describe a consistent model using both types- and tokens-computations, tokens-computation must be cut off with a finite domain. As a result, computation co-existing with its own environment by which a computation is carried out must be expressed as a dynamical duality enhanced in phenomenal computing. The notion of dynamical duality is a key concept breaking through the notion of hard problems in the science of consciousness. In our lab, cognitive experiments and models in which consciousness are grasped as dynamical duality between types- and tokens-computations are conducted and proposed [2, 6].

• Dynamical Origin of Chemical Reactions
  (Tamiki Komatsuzaki)

  Chemical reactions are regarded as the most elementary prototype of `change' of the states or matter. How a rearrangement of atoms or molecules constituting a system occurs is one of the most intriguing fundamental questions from the day of alchemy. However why can the system react from reactants to the products? In other words, in the figure below, what origin differentiates the initial conditions which bring the system to the product (gray colored trajectory) or not (red colored trajectory)? Recently we revealed [13-15] that normally hyperbolic invariant manifolds (NHIMs) and the stable and unstable invariant manifolds survive even in a sea of chaos, and this can provide a regularized reaction pathway to mediate chemical reaction in the `stochastic' process of an n-particle system.
  Most chemical reaction theories in condensed phase divide the wholeness into the reacting system (part) and the residuals, which has been regarded as a heat bath. The chemical reactions are then regarded as being governed by `casting dice'. However, our findings indicate that nature may have, at least locally, known a priori the destination of chaotic reaction dynamics [17]. The stability and bifurcation of the NHIMs [20] are thus the most essential subjects in understanding "who knows the goal of the reaction" and "what the whole and parts are in the change of the state."

• Single-molecule time series analysis: Dynamical Hierarchy, Molecular Memory, Non-Ergodicity, Abnormal Diffusion
  (Tamiki Komatsuzaki)

  The question "How can one learn from single-molecule time series about the underlying multi-dimensional free-energy landscape or state space?" is one of the most important and central subjects in biological research. In order to explore the mechanism of fast protein folding in competition with thermal fluctuations in a single molecule level, we focus on the following four subjects:
  Construction of the underlying state space structure and/or an effective free-energy landscape from single-molecule time series; Noise reduction to extract symbolic sequences of state-to-state transitions buried in experimental noise of single-molecule time series; Extraction of the underlying hierarchical dynamical structures and their mutual information flow from singlemolecule time series; Exploration of the geometrical structure of the high-dimensional state space from single-molecule time series.
  		We have been developing a new dynamical platform of nonlinear time series analysis by mixing the modern techniques of embedology (Wavelet-based embedology), information theory (computational mechanics), and dynamical system theory (finite size Liapunov exponent, epsilon-tau entropy). This project is supported by Japan Science and Technology Cooperation as a five-year project (2004-2009) with experimentalists in other institutes.
  How can one learn from single-molecule time series about the underlying multi-dimensional free-energy landscape or state space?

• Complexity of Protein Landscape and Dynamics: toward Molecular Sociology
  (Tamiki Komatsuzaki)

  Why can natural proteins find their unique native state among the candidates of states, whose number is estimated to be beyond the age of the universe? The key concept of protein folding problem has been interpreted as the `design' of the protein landscapes that the protein might have acquired via mutations. Natural (small) proteins are believed to be designed as having a funnel-like energy landscape introducing a strong energy bias toward the global minimum. However, how can one quantify the degree of funnel-like or ruggedness in terms of the underlying multi-dimensional energy landscape? What is the environment for protein in solutions, cells, or organisms? Can we put all the effect of the environment, e.g., water molecules in the vicinity of protein molecules, into the thermal bath and the friction source? We have been developing a new platform to resolve the important contemporary issues in complexity of protein landscape [19] and dynamics [16, 18] in solution in terms of several advanced techniques in data mining (principal component analysis), information theory, network theory and fluid mechanics (water vector field analysis, Lagrangian coherent structure). The goal of this project is to establish the undistinguishable, itinerant behavior among the dynamically moving `system' and the `bath' and its role for function occurring robustly in the thermal fluctuations.

We have also organized (or we have been involved in the organization of) many multidisciplinary conferences:

• Geometrical Structures of Phase Space in Multi-Dimensional Chaos: Applications to Chemical Reaction Dynamics in Complex Systems, Kyoto, October 26-November 1, 2003.
• Cross-Talk between Chemistry and Planetary Sciences, Kobe, November 17, 2004
• Dynamical Foundation of Functions and Energy Transfer in Protein Systems, Chicago , November 1-6, 2004, and Sapporo, November 26-28, 2005.

Recently, one of the staff edited a book entitled Geometric Structures of Phase Space in Multi-Dimensional Chaos: Applications to Chemical Reaction Dynamics in Complex Systems as a special volume of Advances in Chemical Physics, 1224 pages, John-Wiley & Sons, Inc. New York (2005). This book includes several reviews, for example, of recent developments concerning regularity buried in chaos in chemical reactions and celestial mechanics in terms of normally hyperbolic invariant manifolds in high-dimensional phase space.

Kobe is located between the sea and the Rokko mountain range. Kobe has been an important port city for many centuries as one of the first Japanese ports to be opened to foreign trade in the 1800s and it is one of Japan's most cosmopolitan cities. Every year we have foreign visitors working on several multidisciplinary researches (robotics, chemical and biological physics, single-molecule measurement, celestial mechanics, molecular informatics) and conduct long-term collaborative research with foreign research groups in the USA, Germany, UK, France and other countries. The current staff have been appointed as adjunct professors in other institutes and presented a set of tutorial lectures of their research in departments of mathematics, chemistry, physics, philosophy, sociology, and so forth, in more than nine universities (University of Tokyo, 1994, 1999, 2003; Tohoku University, 1995; Hokkaido University, 1996; Osaka University, 1997; Niigata University, 1998; Nagoya University, 2002; Waseda University, 2002; Ibaraki University, 2004; Hiroshima University, 2006).

Applicants to PhD programs in nonlinear sciences group at Department of Earth and Planetary Sciences, Kobe University, will be nominated for RA fellowships by the Graduate School of Science and Technology or COE RA fellowships by a 21st Century Center of Excellence (COE) Program Origin and Evolution of Planetary Systems. They are also encouraged to apply for funding via JSPS Postdoctoral Fellowship that are available every year.

Selected Publications (2001-2006)

1.	Gunji, Y.-P. & Higashi, H. (2001) The origin of universality: making and invalidating free category. Physica D 156: 283-313.
2.	Gunji, Y.-P., Aono, M., Higashi, H. & Takachi, Y. (2001) The third wholeness as an endo-observer. In: Science of Interface (Diebner, H.H., Druckrey, T. & Weibel, P., eds.), Genista verlag, Tubingen, pp. 111-130.
3.	Takachi, Y. & Gunji, P.Y., (2003). Punctuated Equilibrium in Thoroughbred Evolution and its Model based on Asynchronous Clocks. Chaos, Solitons & Fractals 19: 555-562.
4.	Gunji, Y.-P., Takahashi, T. & Aono, M. (2004) Dynamical infomorphism: form of endo-perspective. Chaos, Solitons & Fractals 22: 1077-1101.
5.	Tsuda, S. Aono, M. & Gunji, Y.-P. (2004) Robust and emergent Physarum-computing. BioSystems 73: 45-55.
6.	Nabeshima, T. & Gunji, Y.-P. (2004) Zipf's law in phonograms and Wreibull distribution in ideograms: comparison of English with Japanese. BioSystems 73: 131-139.
7.	Gunji, Y.-P. & Kamiura, M. (2004) Observational heterarchy enhancing active coupling. Physica D 198: 74-105.
8.	Tsuda, S., Zauner, K. P. & Gunji, Y. P. (2005) Robot Control with Biological Cells. In Proceedings of Sixth International Workshop on Information Processing in Cells and Tissues, St. William's College, York, pp.202-216.
9.	Haruna, T. & Y.-P. Gunji (2005) Autonomous indefiniteness versus external indefiniteness: theory of weak topped C-structure and its application to elementary local cellular automaton. Physica D 202: 71-94.
10.	Gunji, Y.-P. Miyoshi, H. Takahashi, T. & Kamiura, M. (2005) Dynamical duality of type- and token-computation as an abstract brain. Chaos, Solitons & Fractals 27: 1187-1204.
11.	M.Kamiura & Y.-P. Gunji (2006) Robust and Ubiquitous on-off intermittency in active coupling. Physica D 218: 122-130.
12.	Gunji, Y.-P. Haruna, T. & Sawa, K. (2006) Principles of Biological organization: Local-global negotiation based on "Material cause". Physica D (in press).
13.	Komatsuzaki, T. & Berry, R. S. (2001) Dynamical Hierarchy in Transition States: Why and How Does a System Climb over the Mountain? Proceedings of National Academy of Sciences USA 98: 7666-7671.
14.	Komatsuzaki, T. & Berry, R. S. (2002) Chemical Reaction Dynamics: Many-Body Chaos and Regularity. Advances in Chemical Physics 123 (Prigogine, I., Rice, S.A. eds.) John Wiley & Sons, Inc. New York, pp. 79-152.
15.	Komatsuzaki, T. & Berry, R. S. (2002) A Dynamical Propensity Rule of Transitions in Chemical Reactions. Journal of Physical Chemistry A 106: 10945-10950.
16.	Matsunaga, Y., Kostov, K. S. & Komatsuzaki, T. (2002) Multi-Basin Dynamics in Off-Lattice Minimalist Protein Landscapes. Journal of Physical Chemistry A 106: 10898-10907.
17.	Komatsuzaki, T. & Berry, R. S. (2005) Regularity in Chaotic Transitions on Two-Basin Landscapes. Advances in Chemical Physics 130A (Toda, M., Komatsuzaki, T., Konishi, T., Berry, R.S. & Rice, S.A. eds.) John Wiley & Sons, Inc. New York, pp. 143-170.
18.	Komatsuzaki, T., Hoshino, K. & Matsunaga, Y. (2005) Regularity in Chaotic Transitions on Multi-Basin Landscapes Advances in Chemical Physics 130B (Toda, M., Komatsuzaki, T., Konishi, T., Berry, R.S. & Rice, S.A. eds.) John Wiley & Sons, Inc. New York, pp. 257-313.
19.	Komatsuzaki, T., Hoshino, K. & Matsunaga, Y., Rylance, G. J., Johnston, R. L. & Wales, D. J. (2005) How Many Dimension is Required to Approximate Potential Energy Landscape of A Model Protein? Journal of Chemical Physics 122: 084714.
20.	Li, C.-B., Shojiguchi, A., Toda, M. & Komatsuzaki, T. (2006) Definability of No-return Transition States in High Energy Regime Above Threshold Physical Review Letters 97: 028302-028305.