Design of logic gates using spiking neural P systems with homogeneous neurons and astrocytes-like control
Introduction
In biological nervous systems, the processing and transmission of information among the neurons had inspired and influenced computer scientists in designing powerful computational devices and intelligent machines for the past decades. In particular, the study of neural computing models based on spiking function is an emerging branch of natural computing which has been explored in great depth [19], [23], [24]. In 2006, a new class of neural computing models, namely spiking neural P systems (shortly named SN P systems) were proposed [16] through modelling the way neurons communicate via electrical impulses (spikes). Such systems are also known as a specific group of ingredients in membrane computing [29], and corresponding to a paradigm shift from cell-like to neural-like architectures.
An SN P system is essentially a networked model constructed with a number of neurons placed in the nodes of a directed graph. The neurons are able to transmit signals, i.e. spikes, to one other along the synapses, i.e. the arcs of the directed graph. The neurons then process the information received in the form of spikes by spiking (firing) and forgetting rules. The execution of a spiking rule is triggered by matching the number of spikes in a neuron with the regular expression associated with the rule. Using a spiking rule, a neuron can send a number of spikes to its neighbouring neurons, and the spikes are accumulated at the target neuron. With the execution of a forgetting rule, a predefined number of spikes will be removed from a neuron. In this system, the input neurons, act like the biological sensory neurons, receive spikes (information) in the nervous system, and the output neuron sends spikes to the external environment.
Taking ideas from various neurobiological phenomena, there have been several variants of SN P proposed, such as asynchronous SN P systems [5], asynchronous SN P systems with local synchronization [42], SN P systems with anti-spikes [20], [27], sequential SN P systems [15], [37], [53], SN P systems with rules on synapses in equal consumption mode [38], maximum spiking mode [39] and maximum spike consumption mode [40], SN P systems with plastic structure and self-organization [3], [49], SN P systems with request rules [41], cell-like SN P systems [50]. Most of these variants are Turing universal as number generating/accepting devices, language generating devices or function computing devices. There are some significant applications of SN P systems in solving issues in practical engineering and science fields, such as fuzzy reasoning SN P systems for fault diagnosis [33], [47], [48], fuzzy SN P systems for knowledge representation [45], [46], SN P systems for approximately solving combinatorial optimization problems [52], SN P systems based nuclear signal export identifying method [7] and a theoretical CPU model made of SN P systems [12].
In biological nervous systems, apart from the neurons, astrocytes are described as star-shaped glial cells spanning around neurons. One can refer to [1], [34] for the biological details of the interaction between neurons and astrocytes.
Through studying the “excitatory and inhibitory” behaviours of the astrocytes during the interaction among neurons, SN P systems with astrocyte-like control were proposed in [32]. In the systems, each astrocyte is associated with a pre-defined threshold, and is able to sense the number of spikes passing through its neighbouring synapses. At any moment, if the number of spikes passing through the neighbouring synapses of an astrocyte is greater than or equal to its threshold, the astrocyte will then have an inhibitory influence to remove these spikes passing along the synapses. On the other hand, if the number of spikes passing through the neighbouring synapses of an astrocyte is strictly less than its threshold, the astrocyte will then have an excitatory influence on the neighbouring synapses, i.e. the spikes can “safely” pass to the destination neurons. SN P systems with astrocyte-like control have been proven to be Turing universal as number computing devices [25]. It was suggested in [32] that finding some applications of SN P systems with astrocytes-like control is of great interest to the research community.
Bio-inspired computing models based Boolean gates and Boolean circuits have been heavily investigated, see e.g. membrane computing systems based logic gates in [6], [11], [17] and DNA computing models based logic gates in [21], [43], [44]. With DNA strand displacement strategy, the so-called “DNA neuron” was developed to perform logic computation in nano-scale, but these ”DNA neurons” sometimes cannot achieve the correct logic computation result since there are several spiking rules in a neuron, owing to the additional challenge that the proposed system used only the units of volume of DNA molecules to represent the stack of spikes [43]. In order to construct neural-like logic gates and circuits, SN P systems are used to design logic gates in [18], where it needs several types of neurons to simulate a logic gate. In [26], anti-spikes were introduced into SN P system to design NAND gate where by three types of neurons were needed to simulate a NAND gate, and the neuron had at least two spiking rules. In 2010, Metta et. al. [26] suggested a possible research direction to design logic gates and circuits using some other variants of SN P systems.
Pioneering works of designing logic gates by simple neurons was proposed in [2], [54], which constructed NAND gates by SN P systems with simple neurons and excitatory/inhibitory astrocytes. In this work, this result is further improved by creating Boolean logic gates with only inhibitory astrocytes-like control, simple and homogenous neurons. In particular, we look into the construction logic AND, OR, NOT, NOR, XOR and NAND gates by SN P systems with a unique type of neuron. The resulting systems are simple and homogeneous, which means each neuron in the systems has the same and unique spiking rule that makes each neuron to function like a “transmitter of information”, yet no forgetting rule is formulated. This arrangement leads to a new class of simple and homogeneous SN P systems. After that, an example of Boolean circuit is achieved by the logic gates of SN P systems and a synchronization module for making n steps delay. As such, we demonstrate that SN P systems with astrocyte-like control are capable of emulating finite computing devices (based on logic gates), such as the finite transducers. These results can be extended to a novel way for constructing logic circuits working in a neural-like manner, as well as offer some potential directions to design neural circuits theoretically.
The merits of our results are: (1) the individual neuron is simple in structure but universal, i.e. only one type of neuron is used for constructing any layout of Boolean circuits; and (2) with the modulation of astrocytes and cooperating with each other, a network of neurons and astrocytes can perform complex functions.
Section snippets
Spiking neural P systems with astrocyte-like control
Some necessary prerequisites of basic concepts and notions of formal language theory from [35] are firstly recalled. For an alphabet set V, V* denotes the set of all finite strings of symbols from V, the empty string is denoted by λ, and the set of all nonempty strings over V is denoted by . If is a singleton, we write simply a* and instead of {a}* and .
A regular expression over an alphabet V is defined as follows: (i) λ and each a ∈ V are regular expressions, (ii) if E1, E2 are
Simulating logic gates
In this section, SN P systems with astrocyte-like control are constructed to emulate operations of logic gates AND, OR, NOT, NOR, XOR and NAND, respectively. In the systems, only one type of neuron is used, which has a unique spiking rule a*/a → a, so we omit the spiking rule in the neuron from the graphical forms of the systems for a better and clear illustration.
In SN P systems with astrocyte-like control, multiple input neurons are used to receive spikes from the environment, but only one
An example of neural-like Boolean circuit
With the neural-like logic gates designed in Section 3, an example of neural-like Boolean circuit is developed by connecting neural-like logic gates. Since different logic gates may spend different steps to complete their computations, it needs synchronization modules to regulate the outputs of different logics gates to enter the other logic gate as inputs synchronously. The basic function of the synchronization module is to make certain steps of delay for spikes entering into certain neurons.
Conclusion
In this research, we proposed and established SN P systems with astrocyte-like control to emulate logic AND, OR, NOT, NOR, XOR and NAND gates. The obtained SN P systems are simple and homogeneous, which means that each neuron in the systems has the same and unique spiking rule. A Boolean circuit is built by the SN P systems for logic gates, where synchronization modules are designed to regulate the outputs of different logics gates to enter the other logic gate as inputs synchronously. These
Acknowledgements
The research is under the auspices of National Natural Science Foundation of China (No. 61402187, 61572522, 61572523 and 61502535), China Postdoctoral Science Foundation funded project (2016M592267), and Fundamental Research Funds for the Central Universities (R1607005A). One of the researchers of this work, Tao Song, is under the sponsorship of the visiting fellowship scheme of Swinburne University of Technology Sarawak Campus.
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