Basins of attraction in cellular automata

Self-organization of self-clearing beating patterns in an array of locally interacting ciliated cells formulated as an adaptive Boolean network

“…A more general definition for dynamical systems says that an attractor is a set of states to which the system evolves after a long enough time [37]. The transient time τ is the number of states a network undergoes (starting from an initial state), before it reaches an attractor [38][39][40]. Fig.15 depicts the temporal evolution of two ensembles consisting of 100 ensemble members differing only by their initial condition.…”

Self-organization of self-clearing beating patterns in an array of locally interacting ciliated cells formulated as an adaptive Boolean network

“…A more general definition for dynamical systems says that an attractor is a set of states to which the system evolves after a long enough time [37]. The transient time τ is the number of states a network undergoes (starting from an initial state), before it reaches an attractor [38][39][40]. Fig.15 depicts the temporal evolution of two ensembles consisting of 100 ensemble members differing only by their initial condition.…”

The Cellular Automaton Pulsing Model, Experiments with DDLab

“…Taking single-bit or single-value perturbations to attractor states as the simplest case, the jump-graph [14][17,#20.7] represents the probabilities of jumping between basins. Apart from the philosophical/mathematical implications [6], insights into the stability and adaptability of dynamics is relevant in attractor models of memory and learning [8,10,15] and of cell differentiation in genetic networks [2,12].…”

“…We consider finite, deterministic (noise free), discrete dynamical systems, including Cellular Automata (CA) [8,7], Random Boolean Networks (RBN) [2,3,9], the general case of Discrete Dynamical Networks (DDN) [17], and perhaps the most general of all -random maps [11][17,#29.8] 1 -random directed graphs with outdegree=1. A system's state at any given moment is a string of n elements, each with a value x ∈ (0, 1, .…”

The Placement of the Head that Minimizes Online Memory

“…A basin of attraction consists of an attractor and all the trajectories leading to this attractor (Wuensche, 2000;Riley & Holden, 2012). Figure 2 defines the trajectories and their impact on online memory cost when moving the head one position forward or backward, and thus defines the basin of attraction of word order evolution on the single dimension of online memory cost.…”

Self-organization of self-clearing beating patterns in an array of locally interacting ciliated cells formulated as an adaptive Boolean network

“…A more general definition for dynamical systems says that an attractor is a set of states to which the system evolves after a long enough time [37]. The transient time τ is the number of states a network undergoes (starting from an initial state), before it reaches an attractor [38][39][40]. Fig.15 depicts the temporal evolution of two ensembles consisting of 100 ensemble members differing only by their initial condition.…”

The Cellular Automaton Pulsing Model, Experiments with DDLab

“…Taking single-bit or single-value perturbations to attractor states as the simplest case, the jump-graph [14][17,#20.7] represents the probabilities of jumping between basins. Apart from the philosophical/mathematical implications [6], insights into the stability and adaptability of dynamics is relevant in attractor models of memory and learning [8,10,15] and of cell differentiation in genetic networks [2,12].…”

“…We consider finite, deterministic (noise free), discrete dynamical systems, including Cellular Automata (CA) [8,7], Random Boolean Networks (RBN) [2,3,9], the general case of Discrete Dynamical Networks (DDN) [17], and perhaps the most general of all -random maps [11][17,#29.8] 1 -random directed graphs with outdegree=1. A system's state at any given moment is a string of n elements, each with a value x ∈ (0, 1, .…”

The Placement of the Head that Minimizes Online Memory

“…A basin of attraction consists of an attractor and all the trajectories leading to this attractor (Wuensche, 2000;Riley & Holden, 2012). Figure 2 defines the trajectories and their impact on online memory cost when moving the head one position forward or backward, and thus defines the basin of attraction of word order evolution on the single dimension of online memory cost.…”

Self-organization of self-clearing beating patterns in an array of locally interacting ciliated cells formulated as an adaptive Boolean network

“…A more general definition for dynamical systems says that an attractor is a set of states to which the system evolves after a long enough time [37]. The transient time τ is the number of states a network undergoes (starting from an initial state), before it reaches an attractor [38][39][40]. Fig.15 depicts the temporal evolution of two ensembles consisting of 100 ensemble members differing only by their initial condition.…”

The Cellular Automaton Pulsing Model, Experiments with DDLab

“…Taking single-bit or single-value perturbations to attractor states as the simplest case, the jump-graph [14][17,#20.7] represents the probabilities of jumping between basins. Apart from the philosophical/mathematical implications [6], insights into the stability and adaptability of dynamics is relevant in attractor models of memory and learning [8,10,15] and of cell differentiation in genetic networks [2,12].…”

“…We consider finite, deterministic (noise free), discrete dynamical systems, including Cellular Automata (CA) [8,7], Random Boolean Networks (RBN) [2,3,9], the general case of Discrete Dynamical Networks (DDN) [17], and perhaps the most general of all -random maps [11][17,#29.8] 1 -random directed graphs with outdegree=1. A system's state at any given moment is a string of n elements, each with a value x ∈ (0, 1, .…”

The Placement of the Head that Minimizes Online Memory

“…A basin of attraction consists of an attractor and all the trajectories leading to this attractor (Wuensche, 2000;Riley & Holden, 2012). Figure 2 defines the trajectories and their impact on online memory cost when moving the head one position forward or backward, and thus defines the basin of attraction of word order evolution on the single dimension of online memory cost.…”