Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

Let $[K_n,f,\pi]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_n$ using the update order $\pi\in S_n$ in which all vertex functions are equal to the same function $f\colon\mathbb F_2^n\to\mathbb F_2^n$. Let $\eta_n$ denote the maximum number of periodic orbits of period $2$ that an SDS map of the form $[K_n,f,\pi]$ can have. We show that $\eta_n$ is equal to the maximum number of codewords in a binary code of length $n-1$ with minimum distance at least $3$. This result is significant because it represents the first interpretation of this fascinating coding-theoretic sequence other than its original definition.Comment: 12 pages, 2 figure

Cycle Equivalence of Graph Dynamical Systems

Graph dynamical systems (GDSs) can be used to describe a wide range of distributed, nonlinear phenomena. In this paper we characterize cycle equivalence of a class of finite GDSs called sequential dynamical systems SDSs. In general, two finite GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs. Sequential dynamical systems may be thought of as generalized cellular automata, and use an update order to construct the dynamical system map. The main result of this paper is a characterization of cycle equivalence in terms of shifts and reflections of the SDS update order. We construct two graphs C(Y) and D(Y) whose components describe update orders that give rise to cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper bound for the number of cycle equivalence classes one can obtain, and we enumerate these quantities through a recursion relation for several graph classes. The components of these graphs encode dynamical neutrality, the component sizes represent periodic orbit structural stability, and the number of components can be viewed as a system complexity measure

Update schedules of sequential dynamical systems

AbstractSequential dynamical systems have the property, that the updates of states of individual cells occur sequentially, so that the global update of the system depends on the order of the individual updates. This order is given by an order on the set of vertices of the dependency graph. It turns out that only a partial suborder is necessary to describe the global update. This paper defines and studies this partial order and its influence on the global update function

On asynchronous dynamic neural field computation

The hallmark of most artificial neural networks is their supposed intrinsic parallelism where each unit is evaluated concurrently to other units in a distributed way. However, if one gives a closer look under the hood, one can soon realize that such a parallelism is an illusion since most implementations use what is referred to as synchronous evaluation. The present article propose to consider different evaluation methods (namely asynchronous evaluation methods) and to study their properties in some restricted but illustrative cases

Synchronous and Asynchronous Evaluation of Dynamic Neural Fields

International audienceIn \cite{rougier:2006}, we've introduced a dynamic model of visual attention based on the Continuum Neural Field Theory \cite{Taylor:1999} that explained attention as being an emergent property of a dynamic neural field. The fundamental property of the model is its facility to select a single stimulus out of several perfectly identical input stimuli by applying asynchronous computation. In the absence of external noise and with a zero initial state, the theoretical mathematical solution of the field equation predicts the final equilibrium state to equally represent all of the input stimuli. This finding is valid for synchronous numerical computation of the system dynamics where elements of the spatial field are computed all together at each time point. However, asynchronous computation, where elements of the spatial field are iterated in time one after the other yields different results leading the field to move towards a single stable input pattern. This behavior is in fact quite similar to the effect of noise on dynamic fields. The present work aims at studying this phenomenom in some details and characterizes the relation between noise, synchronous evaluation (the ``regular'' mathematical integration) and asynchronous evaluation in the case of a simple dual particle system. More generally, we aim at explaining the behavior of a general differential equation system when it is considered as a set of particles that may or may not iterated by synchronous computations