We are aiming at constructing a novel coherent viewpoint on biological system. This can be phrased as ``the dynamical behavior first, and the recursive rule second". Last year we have proposed isologous diversification theory for a system with inter-intra-dynamics, that is a system composed of units with internal dynamics and interaction, whose number is assumed to change through the internal dynamics. The theory is summarized as

**heterogeneity by intrinsic diversification <--> clustering due to orbital in each internal dynamics****macroscopic robustness <--> collective dynamics of coupled chaotic system, in particular homeochaos****recursivity <--> choice of initial conditions with digitalization of differentiated states**

The isologous diversification theory for cell differentiation is based on simulations of interacting cells with metabolic networks and cell division process following consumption of some chemicals. According to the simulations of the interaction-based dynamical systems model, the following scenario of the cell differentiation is proposed.

- Up to some threshold number, divisions bring about almost identical cells with synchronized metabolic oscillations.
- As the number is increased the oscillations lose the synchrony, leading to groups of cells with different phases of oscillations.
- Amplitudes of oscillation and averaged chemical compositions start to differ by groups of cells. The differentiated behavior of states is transmitted to daughter cells.
- Recursivity is formed so that the daughter cells keep the identical chemical character. This ``memory" is made possible through the transfer of initial conditions.
- Successive differentiation proceeds. Mechanism of tumor cell formation, origin of stem cells, anomalous differentiation by transplantations, apoptosis and other features of cell differentiation process are also discussed, with some novel predictions.

Although the theory is first intended to study the cell differentiation process, it is
expected to be applied to other systems with interacting agents. This year we have
extended the theory to include the following topics .

Successive differentiation process is found in a class of interacting cell models. For
example, differentiation from type-0 cell to two different types ("1" and
"2") and then the differentiation from 1 to three different types of cells are
observed. Hierarchical rule of differentiation is thus generated. It is found that the
choice of the path ( e.g., ("0" -> "0", "0" ->
"1", or "0" -> "2") follows essentially a stochastic
rule, as is observed in the stem cells in a hemopoietic system.

The rate of differentiation and reproduction vary as the function of the distribution of cell types. The global stability of the whole system is obtained, which is sustained by spontaneous regulation of the rates of the differentiations. As a couple dynamical system, the memory of cell types is given in a state stabilized by interactions. This state is not necessarily an attractor as a single cell dynamics, but is a "partial attractor" stabilized only in the presence of suitable interactions provided by the distribution of other cells. The recursivity of cell types is supported by such discrete stable states, while the global stability of cell society is assured through the interaction. Indeed, such partial attractors lose the stability and switch to other cell types when the interaction by the cell distribution is not suitable. It should be noted that two types of memory coexist, analogue and digital ones. The former gives information on the cell society, i.e., the distribution of cell types, hile the latter gives a distinct internal state on cell differentiation. We believe that such dual memory structure is a general feature in a biological system, that exists as an interface between external environment and the internal dynamics.

In an organism, there often appears a higher level of differentiations, leading to several distinct types of tissues. They consist of different types of cells and/or different distribution of cell types. We have confirmed that our cell society has several ( e.g., 4) distinct states of cell distribution, leading to a different colony. As a dynamical system, the present results provides the first example that the dynamics of a 2-step higher level is formed.

Several extensions of coupled maps are studied to clarify the collective dynamics of active elements.

- Coupled maps with heterogeneous parameters: Clear and regular collective behaviors are found in the case when the parameter is distributed. Its mechanism is clarified as a window-feedback mechanism.
- Coupled Map Gas: A model allowing for the motion of elements in real space is studied. The interference among internal dynamics, motion, and interaction lead to a novel type of order formation.
- Growing coupled maps: A model allowing for the growth of the number of elements is studied, inspired by the cell differentiation problem. The growth of elements is made possible with the clustering of phases, and then the differentiation of roles. The former leads to the time sharing of resources, while the latter leads to the separation of roles for the growth. The mechanism of the separation of characters of elements is studied.

Stability of attractor is studied by the return rate to itself after perturbations, for a multi-attractor state of a globally coupled dynamical system. Attractors are characterized according to the strength and basin volume, where the dominance of a fragile attractor is noted in the partially ordered phase. It is found that some attractors are selected rather sharply with the addition of noise, reflecting on the complex connection structure among attractors. Relevance of the weak attractors to chaotic itinerancy is also discussed.

Problems of the choice of attractors is very important in a biological system. In neural dynamics, Freeman has found a chaotic attractor corresponding to a searching state for a variety of attractors.

The weak attractors provide a candidate for such a searching state, because of connection to a variety of stronger attractors which play the role of memorized states.

Another relevance is the application to cell biology. Differentiation of cell types has recently been studied in the context of dynamical systems where the differentiation is

related with the selection of attractors. Cellular states at an earlier stage are transmitted to their offsprings while the same character is kept at determined cells that appear at later generations. This process is understood as a switch from a weak attractor at the initial state to a strong attractor later, due to the cell-to-cell interactions.

To study the origin of a unit of multi-cellular organism, it is essential how an ensemble of cells is formed which acts as an individual unit for growth. To study the problem, a system composed of interacting dynamic elements in a 2-dimenisional space is adopted. As an abstract model, we choose the coupled map gas model with division and death , which locally interacts in space and moves following the forces by neighboring cells. It is shown that the unit of strongly interacting elements is formed, which acts as a dividing unit for growth. The formation of such units is a balance between intra-unit interactions, and inter-unit perturbations. Such structure is made possible with the clustering of phases and the differentiation of roles.

A non-zero-sum 3-person coalition game is studied, for the evolution of complexity and diversity in communication and strategies, where the population dynamics of players with strategies is given according to their scores in the iterated game and mutations . Two types of differentiations emerge initially; a biased one to form

classes and a temporal one to change their roles for coalition. Communication rules are self-organized in a society through evolution. The co-evolution of diversity and complexity of strategies and communications are found at later stages of the simulation.

Tile Automaton is designed as a model for the origins of life, emerging from complex metabolic pathways of chemical reactions. Like computer game ``Tetris'', tiles with various shapes stand for molecules. They move on a plane (for a spatial version) or are stored in a tank (for a tank version). Their shapes are changed by the reactions induced by the collisions. The rules of reaction are deterministic and dependent on their mutual shape-to-shape relations .

We have obtained self-organization of ever-creating sets from a small number of simple-shaped tiles in the spatial version and temporally differentiated pathways of reaction in the tank version. These evolutions are realized through many-body reactions, spatial relationship and the interference of contexts which is believed to be essential for emergence of life-like phenomena.

Universal Turing machine is studied as a dynamical system. It is shown that the ``basin boundary" dimension for the halting state approaches 2, the phase space dimension, as the tape length is increased. The result characterizes the geometric nature of the undecidability, and suggests a new class of dynamics different from chaos . Indeed, the transient before falling on the halting state decays equal to or slower than the power law, in contrast with the exponential decay in the chaotic transient dynamics.

Consider a black-box with input-output relation, which is characterized by a function. Assume that this relation is changed according to the inputs, which are reflected by its own output. This situation leads to dynamics of the function itself. We have been investigating several types of such function maps, partly trying to see the mechanism how the world is categorized .