Mammalian Brain As a Network of Networks

Acknowledgements AZ, SG and AL acknowledge support from the Russian Science Foundation (16-12-00077). Authors thank T. Kuznetsova for Fig. 6.Peer reviewedPublisher PD

Relevance of Dynamic Clustering to Biological Networks

Network of nonlinear dynamical elements often show clustering of synchronization by chaotic instability. Relevance of the clustering to ecological, immune, neural, and cellular networks is discussed, with the emphasis of partially ordered states with chaotic itinerancy. First, clustering with bit structures in a hypercubic lattice is studied. Spontaneous formation and destruction of relevant bits are found, which give self-organizing, and chaotic genetic algorithms. When spontaneous changes of effective couplings are introduced, chaotic itinerancy of clusterings is widely seen through a feedback mechanism, which supports dynamic stability allowing for complexity and diversity, known as homeochaos. Second, synaptic dynamics of couplings is studied in relation with neural dynamics. The clustering structure is formed with a balance between external inputs and internal dynamics. Last, an extension allowing for the growth of the number of elements is given, in connection with cell differentiation. Effective time sharing system of resources is formed in partially ordered states.Comment: submitted to Physica D, no figures include

Boolean Dynamics with Random Couplings

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical Sciences Serie

State Differentiation by Transient Truncation in Coupled Threshold Dynamics

Dynamics with a threshold input--output relation commonly exist in gene, signal-transduction, and neural networks. Coupled dynamical systems of such threshold elements are investigated, in an effort to find differentiation of elements induced by the interaction. Through global diffusive coupling, novel states are found to be generated that are not the original attractor of single-element threshold dynamics, but are sustained through the interaction with the elements located at the original attractor. This stabilization of the novel state(s) is not related to symmetry breaking, but is explained as the truncation of transient trajectories to the original attractor due to the coupling. Single-element dynamics with winding transient trajectories located at a low-dimensional manifold and having turning points are shown to be essential to the generation of such novel state(s) in a coupled system. Universality of this mechanism for the novel state generation and its relevance to biological cell differentiation are briefly discussed.Comment: 8 pages. Phys. Rev. E. in pres