Elsevier

Biosystems

Volume 108, Issues 1–3, April–June 2012, Pages 52-62
Biosystems

Sequential spiking neural P systems with exhaustive use of rules

https://doi.org/10.1016/j.biosystems.2012.01.007Get rights and content

Abstract

Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes, where neurons work in parallel in the sense that each neuron that can fire should fire, but the work in each neuron is sequential in the sense that at most one rule can be applied at each computation step. In this work, we consider SN P systems with the restriction that at most one neuron can fire at each step, and each neuron works in an exhaustive manner (a kind of local parallelism – an applicable rule in a neuron is used as many times as possible). Such SN P systems are called sequential SN P systems with exhaustive use of rules. The computation power of sequential SN P systems with exhaustive use of rules is investigated. Specifically, characterizations of Turing computability and of semilinear sets of numbers are obtained, as well as a strict superclass of semilinear sets is generated. The results show that the computation power of sequential SN P systems with exhaustive use of rules is closely related with the types of spiking rules in neurons.

Introduction

Membrane computing is one of the recent branches of natural computing; it was initiated in Păun (2000) and has developed rapidly (already in 2003, ISI considered membrane computing as “fast emerging research area in computer science”, see http://esi-topics.com). The aim is to abstract computing ideas (data structures, operations with data, ways to control operations, computing models, etc.) from the structure and the functioning of a single cell and from complexes of cells, such as tissues and organs including the brain. The models obtained are distributed and parallel computing devices, usually called P systems. There are three main classes of P systems investigated: cell-like P systems (based on a cell-like (hence hierarchical) arrangement of membranes delimiting compartments where multisets of chemicals evolve according to given evolution rules) (Păun, 2000), tissue-like P systems (instead of hierarchical arrangement of membranes, consider arbitrary graphs as underlying structures, with membranes placed in the nodes while edges correspond to communication channels) (Martin-Vide et al., 2003), neural-like P systems. Many variants of all these systems have been considered; an overview of the field can be found in Păun (2002) and Păun et al. (2010), with up-to-date information available at the membrane computing website (http://ppage.psystems.eu). For an introduction to membrane computing, one may consult (Păun and Rozenberg, 2002, Păun et al., 2010). The present work deals with a class of neural-like P systems, called spiking neural P systems (SN P systems, for short), introduced in Ionescu et al. (2006).

SN P systems are a class of distributed and parallel computing models inspired by spiking neurons. As we know, neurons are one of the most interesting cell-types in the human body. A large number of neurons working in a cooperative manner are able to perform tasks that are not yet matched by the tools we can build with our current technology, e.g., thought, self-awareness, intuition. We believe the distributed manner in which the brain processing information is important in obtaining better performance, thus we are interested in SN P systems defined as a computation model. We would like to stress here that SN P systems in this work are a subject of investigation for theoretical computer science without intention to propose a platform for modeling biological processes.

Briefly, an SN P system consists of a set of neurons which are placed in the nodes of a directed graph. Each neuron contains a number of copies of a single object type called spike, which is denoted by the symbol a in what follows. The communications between neurons are achieved by sending signals (in the form of spikes) along synapses (arcs of the graph). The objects evolve by means of spiking rules, which are of the form E/ac  ap ; d, where E is a regular expression over {a} and c, p, d are natural numbers, c  1, p  1, d  0, with the restriction c  p. The objects can also be removed from the neurons by rules of the form E/ac  λ (i.e., the forgetting rules). A rule with E of the form ac is called a bounded rule and it is called an unbounded rule when the rule has E of the form ai(aj)* or ai(aj)+, i  0, j  1. A bounded neuron is the one in which each rule is bounded and an unbounded neuron is the one in which each rule is unbounded. If a neuron contains both bounded rules and unbounded rules, then it is called a general neuron. The application of the rules is controlled by the regular expression E in the sense that a neuron containing k spikes such that ak  L(E), k  c, can apply the rules with the same E. If the rule E/ac  ap ; d is applied in a neuron, then c spikes are consumed from the neuron, and p spikes are produced after a delay of d steps. These spikes are sent to all neurons connected by an outgoing synapse from the neuron where the rule was applied. The use of the rule E/ac  λ will remove c spikes from the neuron. One of the neurons is designated as the output neuron of the system, and its spikes are also sent to the environment. The moments of time when a spike is emitted by the output neuron are marked with 1, the other moments are marked with 0. This binary sequence is called the spike train of the system. Various numbers can be associated with a spike train, which can be considered as computed (we also say generated) by an SN P system. In this work, the result of a computation is defined as the total number of spikes sent to the environment by the output neuron.

Many computational properties of SN P systems have been studied. SN P systems were proved to be computationally complete (equivalent with Turing machines or other equivalent computing devices; we also say that SN P systems are universal) as number computing devices (Ionescu et al., 2006), language generators (Chen et al., 2008, Chen et al., 2007), and function computing devices (Păun and Păun, 2007). SN P systems were also used to (theoretically) solve computationally hard problems in a feasible time (see, e.g., Ishdorj et al., 2010, Pan et al., 2011). All systems in Chen et al., 2008, Chen et al., 2007, Ionescu et al., 2006, Ishdorj et al., 2010, Pan et al., 2011 and Păun and Păun (2007) function in the following way: all neurons work in parallel in the sense that at each step each neuron that can apply a rule should do it, while the rule in each neuron is applied in a sequential manner with the meaning that at most one rule is applied in each neuron. Generally, the work of an SN P system is determined both by the way how the neurons operate and by the way how the rules are applied in each neuron.

Although biological processes in living organisms happen in parallel, they are not synchronized by a universal clock as assumed in SN P systems. Several authors have noticed that the maximal parallelism way of rule application is rather non-realistic and considered various “strategies” of rule application, e.g., Ciobanu et al. (2007) and Freund (2005). In Ibarra et al. (2006), SN P systems were considered to function in a sequential manner, i.e., at each step at most one neuron can fire (if several neurons can fire at a step, then one neuron is non-deterministically chosen and fires). Such systems are called sequential SN P systems. It was shown that certain classes of sequential SN P systems are not universal, while others are universal (Ibarra et al., 2006).

At the level of neurons, a kind of local parallelism, called exhaustive use of rules, was proposed in Ionescu et al. (2007), where in each neuron an applicable rule is used as many times as possible. The biological motivation of exhaustive use of rules is that an enabled chemical reaction consumes related substances as many as possible. It was proved that SN P systems with exhaustive use of rules are universal if neurons work in parallel (Ionescu et al., 2007, Zhang et al., 2008).

In this work, we investigate the computation power of sequential SN P systems with exhaustive use of rules. It is proved that sequential SN P systems with exhaustive use of rules are universal if the systems consist of bounded neurons and general neurons. A characterization of semilinear sets of numbers is obtained if all neurons are restricted to be bounded. Sequential SN P systems with exhaustive use of rules that consist of bounded neurons and unbounded neurons can generate more than semilinear sets of numbers. The results show that the computation power of sequential SN P systems with exhaustive use of rules is closely related with the types of spiking rules in neurons, which can be interpreted as that neurons from different blocks of the brain have different functioning.

The rest of this paper is organized as follows. In Section 2, we recall some necessary preliminaries. The sequential SN P systems with exhaustive use of rules are introduced in Section 3. The computation power of sequential SN P systems with exhaustive use of rules is investigated in Section 4. Finally, conclusions and remarks are presented in Section 5.

Section snippets

Preliminaries

It is useful for readers to have some familiarity with basic elements of language theory, e.g., from Rozenberg and Salomaa (1997). We here only introduce the necessary prerequisites.

For an alphabet V, V* denotes the set of all finite strings over V, with the empty string denoted by λ. The set of all non-empty strings over V is denoted by V+. When V = {a} is a singleton, then we simply write a* and a+ instead of {a}*, {a}+. The families of right-linear, regular and recursively enumerable languages

Sequential SN P Systems with Exhaustive Use of Rules

In this section, we introduce sequential SN P systems with exhaustive use of rules.

A sequential SN P system with exhaustive use of rules, of degree m  1, is a construct of the formΠ=(O,σ1,,σm,syn,i0),where

  • O = {a} is the singleton alphabet (a is called spike);

  • σ1, …, σm are neurons, of the form

    σi=(ni,Ri),1im,where

    • (a)

      ni  0 is the initial number of spikes contained in σi;

    • (b)

      Ri is a finite set of rules of the form E/ac  ap ; d, where E is a regular expression over {a}, and c  1, p  0 and d  0, with the

Computation Power of Sequential SN P Systems with Exhaustive Use of Rules

In this section, we investigate the computation power of sequential SN P systems with exhaustive use of rules. In Section 4.1, we consider sequential SN P systems with exhaustive use of rules that consist of bounded neurons; in Section 4.2, we consider sequential SN P systems with exhaustive use of rules that consist of bounded neurons and unbounded neurons; in Section 4.3, we consider sequential SN P systems with exhaustive use of rules that consist of bounded neurons and general neurons.

Conclusions and Remarks

In this work, we investigate the computation power of sequential SN P systems with exhaustive use of rules. Results show that the computation power of sequential SN P systems with exhaustive use of rules depends on the types of neurons used in the systems. Specifically, it is proved that sequential SN P systems with exhaustive use of rules are universal if the systems consist of bounded neurons and general neurons. A characterization of semilinear sets of numbers is obtained if all neurons are

Acknowledgements

We gratefully acknowledge the anonymous referees that helped improve significantly the presentation of the paper.

This work was supported by National Natural Science Foundation of China (61033003, 91130034 and 61003038), Ph.D. Programs Foundation of Ministry of Education of China (20100142110072), Fundamental Research Funds for the Central Universities (2010ZD001), Natural Science Foundation of Hubei Province (2011CDA027), and Scientific Research Foundation for Doctor of Anhui University (

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