Spiking neural P systems with multiple channels and anti-spikes

Highlights

• Investigate the computational power of ASN P systems with multiple channels.

• Prove ASN P systems with multiple channels are Turing universal as natural number generators.

• Construct an ASN P system with multiple channels with 65 neurons for computing functions.

Abstract

Spiking neural P systems (SN P systems) with multiple channels are a variant of SN P systems presented recently. By introducing anti-spikes in neurons, SN P systems with multiple channels and anti-spikes are constructed in this work, where both spikes and anti-spikes are used in rules with channel labels. The Turing universality as number generating and accepting devices is proved at first, and then a universal SN P systems with multiple channels and anti-spikes for computing functions is investigated. At last, a small universal system using 65 neurons for computing any Turing computable function is given.

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 $\overline{a}$) 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/b^c → b′ $(b,b^{\prime }\in \{a,\overline{a}\})$. E is the regular expression over {a} or $\{\overline{a}\}$. The forgetting rules, b^s → λ ($b\in \{a,\overline{a}\}$), can moved out s spikes or anti-spikes from the neuron. Annihilating rule $a\overline{a}\to \lambda$ works in every neuron, which means a and $\overline{a}$ cannot stay together. There are four kinds of rules, including a^c → a, $a^c\to \overline{a}$, ${\overline{a}}^c\to a$ and ${\overline{a}}^c\to \overline{a}$. 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/a^c → a^p(l) are used, which means each of the synapse has its own channel label. When rule E/a^c → a^p(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.