Modelling and analysis of spiking neural P systems with anti-spikes using Pnet lab
Introduction
Spiking neural P systems (shortly called SN P systems), introduced in [3] as a variant of P systems [10], are mathematical models inspired by the neurobiological behaviour of neurons sending electrical pulses of identical voltages called spikes to neighbouring neurons.
An SN P system consists of a set of neurons placed in the nodes of a directed graph with each neuron having spiking and forgetting rules. The rules involve the spikes present in the neuron in the form of occurrences of a symbol . The application of spiking rules transfer spikes to neighbouring neurons along the arcs of the graph (they are called synapses), whereas the forgetting rules simply forget some spikes present in the neuron.
In a standard SN P system there are only one type of objects called spikes which are moved, created and destroyed but never modified into another form. An SN P system with anti-spikes (shortly called SN PA system) introduced in [7], is a variant of an SN P system consisting of two types of objects, spikes (denoted as ) and anti-spikes (denoted as ). The inhibitory impulses/spikes are represented using anti-spikes. The anti-spikes behave in a similar way as spikes by participating in spiking and forgetting rules. They are produced from usual spikes by means of usual spiking rules; in turn, rules consuming anti-spikes can produce spikes or anti-spikes (here we avoid the rule anti-spike producing anti-spike). Each neuron in the system consists of an implicit annihilation rule of the form ; if an anti-spike and a spike meet in a given neuron, they annihilate each other. This rule has the highest priority and does not consume any time. So at any instant of time, a neuron in an SN P system with anti-spikes can have spikes or anti-spikes but not both.
The initial configuration of the system is described as initial number of spikes or anti-spikes present in each neuron. The SN PA system evolves in a synchronous fashion, meaning that a global clock is assumed and in each time unit all neurons work in parallel with each neuron which can use a rule should do it, but using only one rule at a time (sequential locally). Using the rules, we can define transitions among configurations. A sequence of transitions among configurations, starting from initial configuration is called a computation. A computation halts if it reaches a configuration where no rule can be used. With any computation whether halting or not together with output produced in such a case, yielding notions of functionality and computational power of SN PA systems including various aspects of computing.
It is extremely important to simulate these models to portray the system behaviour. Such models can shed insight into complex processes and suggest new directions for research. Scientists can study and analyse such models to make predictions about the behaviour of the system under different conditions and to discuss novel relationships among the different components of a system. The ability to predict system behaviour with a model helps to evaluate model completeness as well as improve our understanding of the system.
A modelling methodology that is especially tailored for representing and simulating concurrent dynamic systems is Petri Nets. An advantage of Petri nets is that they have a visual representation and simulation that facilitates user comprehension. Petri Net tools enable users to verify system properties, verify system soundness, and to simulate the dynamic behaviour.
Different variants of P systems are translated into Petri nets to complement the functional characterization of their behaviour. In [5], [4], Petri nets with localities were introduced to represent some variants of membrane systems. In [8], [9], SN P systems with delay and SN P systems with anti-spikes are translated into new variants of Petri net models. However, all these new variants of Petri nets typically lack the tools for building models, for executing and observing simulation experiments. In [1], a tool for simulating a simple and extended SN P system is introduced that yields only the transition diagram of a given system in a step-by-step mode and it lacks step-by-step graphical simulation of the system. Spiking neural P systems with anti-spikes are new variants of SN P systems. We studied the languages generated by the SN P systems with anti-spikes in [6] and proved that some of the languages that cannot be generated using standard SN P systems can be generated using SN P systems with anti-spikes but no tools are available to simulate these variants.
This paper introduces the direct translation of standard SN PA systems into Petri net models that can be simulated using existing Petri net tools. As the procedure is direct, it involves less complexity in translation and also using the notions and tools already developed for Petri nets, one can describe the internal process occurring during a computation in the SN PA system in a graphical way. Perhaps the greatest advantages of Petri nets are a solid mathematical foundation and the large number of techniques being developed for their analysis. These include reachability analysis, invariant analysis (a technique using linear algebra), transformations (including reductions) preserving desired properties, structure theory and formal language theory. We considered Pnet Lab—a Java based simulation tool for Petri nets to analyse the SN P systems. Pnet Lab allows the parallel firing of all enabled transitions after resolving the conflicts that can efficiently simulate the parallel use of rules in all neurons in each step. It also allows the user defined guard function that can encode the regular expression associated with each rule. Last but not least, Pnet Lab has the advantage that it is extremely light-weight and being implemented in Java it is platform independent.
This paper is organized as follows. We start with Section 2 by giving brief introduction about SN PA system. In Section 3, we discuss the Petri net model considered for translations. Section 4 gives a brief introduction about Pnet Lab. Using these definitions as basis in Section 5, we translate an SN PA system into an equivalent Petri net model that can be simulated using Pnet Lab. Section 6 gives analysis results for the SN PA system considered in Section 2 through Pnet Lab.
We recall here a few definitions and notations related to formal languages and automata theory.
is a finite set of symbols called alphabet. A string over is a sequence of symbols drawn from . denotes the empty string. is the set of all string over . is denoted by . The length of a string is denoted by . A language over is a set of strings over .
Let the alphabet be the set . The letter distribution, , of a -word is the -tuple with the number of occurrences of in . The Parikh set, , of a -language is .
A language is said to be regular if there is a regular expression over such that . The regular expressions are defined using the following rules. (i) and each are regular expressions. (ii) if are regular expressions over , then and are regular expressions over , and (iii) nothing else is a regular expression over . With each regular expression , we associate a language .
When is a singleton, then the regular expression denotes the set of all strings formed using . i.e. the set . The positive closure . If is a singleton then the Parikh set of the language denoted by regular expression over is .
A multiset over a set is a function . Multiset is empty if there are no such that by which we mean that and . The cardinality of is . For two multisets and over , the sum is the multiset given by the formula for all , and if then is the multiset given by for all . We denote whenever for all , and if , then the difference is for all .
A multiset over may be represented as a string of elements from ; for example, (three occurrences of s and two occurrences of ) denotes the multiset over such that , and . The empty string (multiset) will be denoted by .
Section snippets
Spiking neural P system with anti-spikes
First we recall the definition of the SN P system with anti-spikes (or SN PA system). Definition 2.1 SN P System with Anti-Spikes Mathematically, we represent a spiking neural P system with anti-spikes of degree , in the form is the alphabet. is called spike and is called anti-spike. are neurons, of the form is the multiset of spikes or anti-spikes contained by the neuron. is a finite set of rules of the following two forms: where is a
Petri net
A Petri [13] net is a bipartite graph with two kinds of nodes, place nodes are represented with circles having tokens and transition nodes are represented with bars or boxes. The directed arcs connecting places to transitions and transitions to places may be labelled with an integer weight, but if unlabelled are assumed to have a weight equal to 1. Now we introduce the class of Petri nets with transitions having guards, to be used in the translation. Definition 3.1 Petri Net A Petri net with guard is represented by
Pnet Lab
Pnet Lab tool provides interactive simulation, analysis and supervision for Petri nets. It allows modelling and analysis of coloured Petri Nets, place-transition nets, timed/untimed. We can build an arc (guard) function by combining the built in functions or using several mathematical functions in accordance with the C/C++ syntax. This paper makes use of built in function that returns the number of tokens present in the place . We can also write user defined guard functions. The DFA
SN PA system to Petri net
In this section, we propose a formal method to translate SN PA systems into Petri nets suitable for simulation using Pnet Lab.
Three places are used to represent each neuron. The marking of the places and gives the number of spikes and anti-spikes present in the neuron respectively. The place is added to allow at most one transition to fire from each input place corresponding to . and are the places corresponding to the environment and shows the number of spikes
Simulation with Pnet Lab
We explain the simulation with an example. Fig. 2 shows the Petri net model for the SN P system in Example 2.1 modelled using Pnet Lab. Each transition is named as , where is the transition name given by the tool and is the transition name given as per methodology discussed in the previous section. We can also find the invariants. In [2], it is proved that finding invariants enables us to establish the soundness and completeness of the system. The tool also outputs the coverability
Conclusion
SN PA systems are biologically inspired computing models that involve the use of two types of objects called spikes and anti-spikes and thus model the systems working with binary data in a very natural way. A formalism to study these models and validating them is needed. The Petri net tool called Pnet lab allows the parallel execution of transitions. It enables us to model the globally parallel firing semantics of all SN PA systems. They also allow the definition of functions on arcs and
Acknowledgements
We are grateful to Marian Gheorghe for many inspiring discussions. We would also like to thank the referees for their helpful comments.
Venkata Padmavati Metta received her MCA in Computer Applications from MANIT, Bhopal and currently pursuing her Ph.D. in Computer Science and Engineering at Thapar University. She has 10 years teaching experience and is an Associate Professor at Bhilai Institute of Technology, India. Her main research fields are formal language theory, algorithms and membrane computing.
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Venkata Padmavati Metta received her MCA in Computer Applications from MANIT, Bhopal and currently pursuing her Ph.D. in Computer Science and Engineering at Thapar University. She has 10 years teaching experience and is an Associate Professor at Bhilai Institute of Technology, India. Her main research fields are formal language theory, algorithms and membrane computing.
Kamala Krithivasan received her Ph.D. from the University of Madras, and she joined the Indian Institute of Technology Madras (IITM) in 1975. With more than 30 years of teaching and research experience at IITM, she is currently Professor at the Department of Computer Science and Engineering, in which she served as Chairperson during 1992–1995. Her research interests include formal language theory and unconventional models of computing like DNA computing, membrane computing and discrete tomography. She has published a large number of research papers, lectured at many universities and gave numerous invited talks at recognized international conferences. A recipient of the Fulbright fellowship in 1986, Professor Kamala is also a fellow of the Indian National Academy of Engineering.
Deepak Garg has done his Ph.D. in the area of efficient algorithm design. He has 11 years teaching experience and is currently Professor at Thapar University, India. His active research interests are designing efficient algorithms, bioinformatics and knowledge management. He started his career as a Software Engineer in IBM Corporation Southbury, USA.