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One-Dimensional Dynamics and Winnerless Competition of Patterns

Building dynamical models to study the neural basis of behavior has long history. Recently a dynamical principle, called winnerless competition (WLC), was suggested. In such
models, given by multidimensional dynamical systems, spatio-temporal coding is realized in the form of deterministic trajectories of a moving along heteroclinic orbits that connect certain saddle fixed points and saddle limit cycles in the state space. The separatrices connecting these saddle states correspond to sequential switching from one active state – specific neurons or groups of neurons – to another.
For modelling information of this kind, we propose to use one-dimensional maps with positive topological entropy. Such maps have a countable set of saddle cycles and an uncountable set of homoclinic and heteroclinic orbits connecting these cycles.
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