Weighted spiking neural P systems with anti-spikes (AWSN P systems) are proposed by adding anti-spikes to spiking neural P systems with weighted synapses. Anti-spikes behave like spikes of inhibition of communication between neurons. Both spikes and anti-spikes are used in the rule expressions. An illustrative example is given to show the working process of the proposed AWSN P systems. The Turing universality of the proposed P systems as number generating and accepting devices is proved. Finally, a universal AWSN P system having 34 neurons is proved to work as a function computing device by using standard rules, and one having 30 neurons is proved to work as a number generator.
Membrane computing, introduced by Păun [
With different biological features and mathematical motivations, many variants of SN P systems have emerged. Some of them made changes on synapses between neurons, such as SN P systems with rules on synapses [
Since the SNP system was proposed, many scholars have explored its applications. At present, there are many applications of SN P systems, such as skeletonizing image processing [
Inspired by the spikes of inhibition of communication between neurons, a new type of SN P systems is proposed by adding anti-spikes to SN P systems, which is called spiking neural P systems with anti-spikes (ASN P systems) [
In [
The rest of this article is organized as follows. In Section
The universality of systems is proved by simulating a register machine
A register machine has two modes: a generating mode and an accepting mode. A register machine
Generally, a universal register machine is used to compute Turing computable functions for the purpose of analyzing the computing power of system. A universal register machine
The universal register machine
The proposed AWSN P system is described as follows: Spiking rules, Forgetting rules,
In the AWSN P system, each neuron has one or more spiking rules and some of them also have forgetting rules, and either spikes or anti-spikes exist in each neuron. If there are
If the forgetting rules
Through these rules, transitions between configurations can occur. Any sequence of transitions starting from the initial configuration is called a computation. A computation will stop when it reaches a configuration where all neurons are open and no rules can be used. To compute the function
Let
An example as graphically shown in Figure
An example of the AWSN P system.
The results of the example.
Step | ||||
---|---|---|---|---|
2 | 2 | 0 | 0 | |
1 | −1 | −3 | 0 | |
2 | 0 | −3 | 2 | |
1 | −3 | −6 | 2 | |
1 | −3 | 0 | 3 (fires) |
The system has four neurons as shown in Figure
A register machine
In the simulation process, a register
Module ADD (Shown in Figure
Module ADD: stimulating the ADD instruction
Module SUB (Shown in Figure
Module SUB: simulating the SUB instruction
Therefore, the SUB instruction can be simulated correctly by module SUB.
Module OUTPUT (Shown in Figure
Module OUTPUT.
The proof of this theorem is similar to that of Theorem
Module ADD (Shown in Figure
Module ADD: simulating the ADD instruction
Module INPUT (Shown in Figure
Module INPUT.
From the descriptions above about the three modules, it is clear that the register machine
There is a universal AWSN P system having 34 neurons which can be used to perform function computing.
A general framework of a system
Module INPUT works as follows: when neuron
When
As with the proof of Theorems 9 neurons for 9 registers 25 neurons for 25 labels 5 neurons for the module INPUT 1 neuron in each SUB instructions and 14 in total 2 neurons for the module OUTPUT
Therefore, totally 55 neurons are used.
The numbers of neurons can be decreased by exploring some relationships between some instructions of register machine
The SUB-ADD instructions can be divided into two cases, depending on the number of spikes placed in register
By using this module, 6 neurons can be saved. In the same way, the module shown in Figure
The module ADD-ADD shown in Figure
The SUB instructions share a common neuron when the labels of their registers are different, as shown in Figure
Two SUB modules dealing with the same register, as shown in Figure
From the above description about the numbers of neurons saved, the system uses the following: 9 neurons for 9 registers 17 neurons for 17 labels 5 neurons for the module INPUT 1 neuron for all the 14 SUB instructions 2 neurons for the module OUTPUT
A total of 21 neurons can be saved and the number of neurons in this system can be decreased from 55 to 34. The proof is complete.
General framework of the universal AWSN P system.
Module INPUT.
Module SUB-ADD-1: the sequence of the ADD and SUB instructions
Module SUB-ADD-2: the sequence of the ADD and SUB instructions
Module ADD-ADD: the sequence of ADD and ADD instructions
Module SUB-SUB with
Module SUB-SUB with
A small universal AWSN P system as a number generator is considered. The process of simulating universal number generators is similar to that of simulating general function computing devices, but the difference between them lies in the module INPUT. The system starts with the spike train
Furthermore, module INPUT and module OUTPUT can be combined. The module INPUT-OUTPUT is shown in Figure
Module INPUT-OUTPUT.
The computation process of the module INPUT-OUTPUT.
Step | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
0 | 1 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
1 | 1 | 1 | 1 | 2 | 6 | 1 | 0 | 2 (fire) | 0 | |
0 | 2 | 2 | 1 | 1 | 8 | 2 | 1 | 3 | 0 | |
0 | 2 | 2 | 0 | 0 | 4 | 2 | 2 | 3 | 0 | |
0 | 2 | 2 | 0 | −1 | 2 | 2 | 3 | 3 | 0 | |
0 | 2 | 2 | 0 | 0 | 1 | 2 | 4 | 4 (fire) | 1 |
Assume that
The string is read through neuron
Therefore, this system contains the following: 8 neurons for the 8 registers 14 neurons for the 14 labels ( 1 neuron for 13 SUB instructions 7 neurons in the module INPUT-OUTPUT
There is a universal AWSN P system having 30 neurons that can be used to perform number generating.
In this work, a variant of the SN P systems, called the AWSN P systems,is proposed. Because of the use of anti-spikes, the proposed systems are more biologically significant thanSN P systems, with inhibitory spikes in the communication between neurons. An example is used to illustrate the working process of this system. The computational universality is then proved in the case of generating mode and accepting mode, respectively. Finally, the Turing universality of AWSN P systems is proved. The function computing device can be realized by using 34 neurons. Compared with the small universal SN P system using anti-spikes introduced by Song [
The computational universality is proved for AWSN P systems with standard rules. There are three types of spiking rules,
No datasets were used in this article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (nos. 61876101, 61802234, and 61806114), the Social Science Fund Project of Shandong (16BGLJ06 and 11CGLJ22), China Postdoctoral Science Foundation Funded Project (2017M612339 and 2018M642695), Natural Science Foundation of Shandong Province (ZR2019QF007), China Postdoctoral Special Funding Project (2019T120607), and Youth Fund for Humanities and Social Sciences, Ministry of Education (19YJCZH244).