Spiking neural P systems with multiple channels and anti-spikes
Introduction
Spiking neural P systems (SN P systems, in short) were inspired from the way that the neurons process and communicate data to each other by sending spikes along synapses (Ionescu et al., 2006). SN P systems are a class of distributed parallel computing models in membrane computing (Păun et al., 2010, Peng et al., 2016, Peng et al., 2017c). SN P systems have been investigated for more than 10 years. It has been proved that SN P systems are computationally complete as generating/accepting number devices (Păun et al., 2010, Peng et al., 2017c), function computing devices (Neary, 2010, Păun and Păun, 2007), and language generators (Zhang et al., 2007, Chen et al., 2007). In addition, inspired by some biological phenomena and mathematical methods, a lot of SN P systems with different features were proposed: SN P systems with rules on synapses (Păun et al., 2006, Song et al., 2014a, Song et al., 2015a, Song and Pan, 2015, Song and Pan, 2014), SN P systems with astrocyte-like control (Zhang et al., 2014a, Păun, 2007, Pan et al., 2012a), SN P systems with white hole neurons (Macías-Ramos and Pérez-Jiménez, 2012, Song et al., 2016a), axon P systems (Artiom et al., 2015), asynchronous SN P systems (Zhang et al., 2015, Song et al., 2012a), homogeneous SN P systems (Song et al., 2015b, Song et al., 2014b, Zeng et al., 2009, Jiang et al., 2015), fuzzy reasoning SN P systems (Song et al., 2016b, Peng et al., 2017a, Peng et al., 2017b, Wang and Peng, 2013, Wang et al., 2015a), weighted fuzzy SN P systems (Wang et al., 2015b, Wang et al., 2015c, Wang et al., 2013, Wang et al., 2010, Pan et al., 2012b), sequential SN P systems (Zeng et al., 2014a, Ibarra et al., 2009, Song et al., 2013a, Zhang et al., 2014b, Zhang et al., 2012) and SN P systems with thresholds (Cabarle et al., 2016). Most of these variants can generate/accept natural numbers, compute Turing computable functions or generate string languages.
For the SN P systems mentioned above, only “positive” spikes are used, which are usually denoted by single symbol a. Inspired from the inhibitory spikes in communications among neurons, anti-spikes (denoted by symbol ) were introduced into SN P systems, which is called SN P system with anti-spikes (ASN P systems) (Zeng et al., 2014b). In ASN P systems, spiking rules are of the form E/bc → b′ . E is the regular expression over {a} or . The forgetting rules, bs → λ (), can moved out s spikes or anti-spikes from the neuron. Annihilating rule works in every neuron, which means a and cannot stay together. There are four kinds of rules, including ac → a, , and . Here only the first three kinds of rules are used.
Till now, more than a dozen of papers on SN P systems have focused on anti-spikes. SN P systems with anti-spikes can work as function computing devices (Pan and Păun, 2009, Song et al., 2013b, Metta and Kelemenová, 2014a) and language generators (Metta and Kelemenová, 2014b, Krithivasan et al., 2011a). Asynchronous, homogenous and sequential SN P systems with anti-spikes were discussed in Metta and Krithivasan (2014), Song et al. (2015c) and Song et al. (2013c), respectively. Also in Jiang and Pan (2016), the normal forms of SN P systems with anti-spikes were studied. In most of SN P systems with anti-spikes, when anti-spikes meet spikes, they annihilate instantaneously without consuming any time, which means annihilating rules have priority over any other rules. However, in (Song et al., 2012b), regardless of the annihilating priority, number acceptors in SN P systems with anti-spikes were proposed. Moreover, SN P systems with anti-spikes were also used to simulate Boolean circuits and logic gates (Tan et al., 2014, Krithivasan et al., 2011b).
Recently, SN P systems with multiple channels was proposed in Metta et al. (2012). In these systems, the spiking rules with the form of E/ac → ap(l) are used, which means each of the synapse has its own channel label. When rule E/ac → ap(l) is used, only the neurons connected by channel (l) will receive p spikes from the neuron. It was proved that SN P systems with multiple channels are universal as number computing devices (Metta et al., 2012). As a good further research topic pointed out in (Metta et al., 2012), we can integrate other strategies in SN P systems with multiple channels to find other variants.
In this work, a new variant of SN P systems is considered, which is called ASN P systems with multiple channels. We investigate the computational power of such systems. It is proved that the systems are Turing universal as natural number generators/acceptors and function computing devices. By exploring some relationships between some consecutive and non-consecutive instructions, a small universal system with 65 neurons for computing functions is constructed.
In the next section, some necessary prerequisites related to register machines and universality are introduced. Section 3 gives the definition of ASN P systems with multiple channels, and an example is presented. The computational power of ASN P systems with multiple channels as number generator and acceptor is investigated in Section 4, and a small universal ASN P system with multiple channels for computing functions is given in Section 5. Finally, the conclusions are drawn.
Section snippets
Prerequisites
It is useful for readers to have some elementary knowledge of automata theory, formal language and P systems. In this section, some basic notations about automata theory and formal languages are introduced, and some definitions of register machines adopted in this paper are given.
For an alphabet Σ, Σ+ and Σ* denote nonempty strings and all finite strings respectively, and λ denotes empty string. {a}* and {a}+ are represented by a* and a+ respectively, when Σ = {a} is a singleton.
Over an
Definition
We define an ASN P system with multiple channels as follows:
- (1)
is the alphabet, where a and are called spike and anti-spike respectively;
- (2)
L = {1, 2, …, N} is channel labels;
- (3)
σ1, σ2, …, σm are neurons, with the form of σi = (ni, Li, Ri), 1 ≤ i ≤ m, where
- (a)
ni ≥ 0 is the number of spikes initially stored in σi;
- (b)
Li ⊆ L is the set of channel labels used in σi;
- (c)
Ri is the rules used in neuron σi, which have two forms:
- •
E/bc → b′(l), E is a regular expression over {a
- •
- (a)
Computational completeness
In this section, an ASN P system with multiple channels is constructed to investigate its computational power. We prove this system can generate and accept any recursively enumerable set of numbers.
A small universal ASN P system with multiple channels for computing functions
In this section, we proceed to construct a small universal ASN P system with multiple channels for computing functions. Theorem 3 There is a universal ASN P system with multiple channels for computing functions having 65 neurons. Proof Fig. 8 shows the main framework of Π′ to simulate , which is a universal ASN P system with multiple channels. Π′ consists of ADD, SUB, ADD-ADD, SUB-ADD-1, SUB-ADD-2, INPUT, OUTPUT and some composite modules. The ADD module, SUB module and OUTPUT module in system Π′ are same as
Conclusions
In this paper, ASN P systems with multiple channels, as a new kind of SN P systems, are investigated. We first prove their computational completeness as number generating and accepting device, and then investigate small universality problem for computing Turing computable functions. We construct a small universal system. With 65 neurons, this system can achieve the universality as a function computing device. Comparing with the small universal SN P system using anti-spikes shown in Song et al.
Acknowledgments
This work was partially supported by the Chunhui Project Foundation of the Education Department of China (Nos. Z2017082, Z2016143 and Z2016148), Foundation of Sichuan Province Key Laboratory of Power Electronics Energy-saving Technologies & Equipment (No. szjj2015-065), National Natural Science Foundation of China (No. 61472328), Sichuan Province Key Laboratory of Power Electronics Energy-saving Technologies & Equipment (No. szjj2016-048), Research Foundation of the Education Department of
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2021, Information and ComputationCitation Excerpt :By introducing anti-spike, Pan et al. [12] discussed SNP system with anti-spikes. Abstracted by the biological fact that the synapse has one or more chemical channels, SNP systems with multiple channels were discussed in Peng et al. [13] and Song et al. [14], respectively. Considering that the rules on synapses instead of neurons, SNP systems with rules on synapses were investigated in Song et al. [15], and another spike consumption strategy was adopted in Peng et al. [16].
Computational completeness of sequential spiking neural P systems with inhibitory rules
2021, Information and ComputationCitation Excerpt :SN P systems with astrocytes [27,28] have been proposed based on the fact that astrocytes have inhibitory and exciting effects on synapses, and the inhibitory spike of neurons also inspired the proposal of SN P systems with anti-spike [29]. SN P systems with multiple channels [30] were inspired by multiple chemical channels in the synapses of neurons, and their other working modes have been investigated in [31–33]. It is well-known that each neuron has a positive or negative charge to convey different biological information.