693 research outputs found
Excitable systems with noise and delay with applications to control: renewal theory approach
We present an approach for the analytical treatment of excitable systems with
noise-induced dynamics in the presence of time delay. An excitable system is
modeled as a bistable system with a time delay, while another delay enters as a
control term taken after [Pyragas 1992] as a difference between the current
system state and its state "tau" time units before. This approach combines the
elements of renewal theory to estimate the essential features of the resulting
stochastic process as functions of the parameters of the controlling term
Synchronization of a large number of continuous one-dimensional stochastic elements with time delayed mean field coupling
We study synchronization as a means of control of collective behavior of an ensemble
of coupled stochastic units in which oscillations are induced merely by external noise.
We determine the boundary of the synchronization domain of a large number of onedimensional
continuous stochastic elements with time delayed non-homogeneous
mean-field coupling. Exact location of the synchronization threshold is shown to
be a solution of the boundary value problem (BVP) which was derived from the
linearized Fokker-Planck equation. Here the synchronization threshold is found by
solving this BVP numerically. Approximate analytics is obtained by expanding the
solution of the linearized Fokker-Planck equation into a series of eigenfunctions of
the stationary Fokker-Planck operator. Bistable systems with a polynomial and
piece-wise linear potential are considered as examples. Multistability and hysteresis
is observed in the Langevin equations for finite noise intensity. In the limit of small
noise intensities the critical coupling strength was shown to remain finite
Dynamical system with plastic self-organized velocity field as an alternative conceptual model of a cognitive system
It is well known that architecturally the brain is a neural network, i.e. a collection of many relatively simple units coupled flexibly. However, it has been unclear how the possession of this architecture enables higher-level cognitive functions, which are unique to the brain. Here, we consider the brain from the viewpoint of dynamical systems theory and hypothesize that the unique feature of the brain, the self-organized plasticity of its architecture, could represent the means of enabling the self-organized plasticity of its velocity vector field. We propose that, conceptually, the principle of cognition could amount to the existence of appropriate rules governing self-organization of the velocity field of a dynamical system with an appropriate account of stimuli. To support this hypothesis, we propose a simple non-neuromorphic mathematical model with a plastic self-organized velocity field, which has no prototype in physical world. This system is shown to be capable of basic cognition, which is illustrated numerically and with musical data. Our conceptual model could provide an additional insight into the working principles of the brain. Moreover, hardware implementations of plastic velocity fields self-organizing according to various rules could pave the way to creating artificial intelligence of a novel type
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