Asynchronous spiking neural P systems

Abstract

We consider here spiking neural P systems with a non-synchronized (i.e., asynchronous) use of rules: in any step, a neuron can apply or not apply its rules which are enabled by the number of spikes it contains (further spikes can come, thus changing the rules enabled in the next step). Because the time between two firings of the output neuron is now irrelevant, the result of a computation is the number of spikes sent out by the system, not the distance between certain spikes leaving the system. The additional non-determinism introduced in the functioning of the system by the nonsynchronization is proved not to decrease the computing power in the case of using extended rules (several spikes can be produced by a rule). That is, we obtain again the equivalence with Turing machines (interpreted as generators of sets of (vectors of) numbers). However, this problem remains open for the case of restricted spiking neural P systems, whose rules can only produce one spike. On the other hand we prove that asynchronous systems, with extended rules, and where each neuron is either bounded or unbounded, are not computationally complete. For these systems, the configuration reachability, membership (in terms of generated vectors), emptiness, infiniteness, and disjointness problems are shown to be decidable. However, containment and equivalence are undecidable. In short, an SN P system consists of a set of neurons placed in the nodes of a directed graph and sending signals (spikes, denoted in what follows by the symbol a) along the arcs of the graph (they are called synapses). Thus, the architecture is that of a tissue-like P system, with only one kind of object present in the cells (the reader is referred to [18] for an introduction to membrane computing and to [23] for the up-to-date information about this research area). The objects evolve by means of standard spiking rules, which are of the form E/a c → a; d, where E is a regular expression over {a} and c, d are natural numbers, c ≥ 1, d ≥ 0. The meaning is that a neuron containing k spikes such that a k ∈ L(E), k ≥ c, can consume c spikes and produce one spike, after a delay of d steps. This spike is sent to all neurons connected by an outgoing synapse from the neuron where the rule was applied. There also are forgetting rules, of the form a s → λ, with the meaning that s ≥ 1 spikes are removed, provided that the neuron contains exactly s spikes. Extended rules were considered in [4], [17]: these rules are of the form E/a c → a p ; d, with the meaning that when using the rule, c spikes are consumed and p spikes are produced. Because p can be 0 or greater than 0, we obtain a generalization of both standard spiking and forgetting rules. In this paper we consider extended spiking rules with restrictions on the type of the regular expressions used. In particular, we consider two types of rules. The first type are called bounded rules and are of the form a i /a c → a p ; d, where 1 ≤ c ≤ i, p ≥ 0, and d ≥ 0. We also consider unbounded rules of the form a i (a j) * /a c → a p ; d...

Asynchronous Spiking Neural P Systems with Structural Plasticity

“…In weighted SNP systems with standard rules, the weights can allow neurons to produce more than one spike each step, similar to having extended rules. In this way, our next result parallels the result that asynchronous SNP systems with extended rules are universal in [5]. However, our next result uses nd syn with asyn mode, while in [5] their systems use nd rule with asyn mode.…”

“…In this way, our next result parallels the result that asynchronous SNP systems with extended rules are universal in [5]. However, our next result uses nd syn with asyn mode, while in [5] their systems use nd rule with asyn mode. We also add the additional parameter l in our universality result, where the synapse weight in the system is at most l. Our universality result also makes use of the normal form given in Section 3.…”

“…The open problem in [3] whether asynchronous bounded SNP systems with standard rules are universal is conjectured to be false. Also, asynchronous SNP systems with extended rules are known to be universal [5]. Our results provide some support to the conjecture, since neurons in SNPSP systems produce at most one spike each step (similar to standard rules) while synapses with weights function similar to extended rules (more than one spike can be produced each step).…”

Asynchronous Spiking Neural P Systems: Decidability and Undecidability

“…The output neuron is identified by both its label, out, and pictorially by a short arrow exiting the membrane and pointing to the environment. Examples of SN P systems working in an asynchronous way can be found in the technical report [1].…”

“…In a similar way we construct the ADD and FIN modules of an SN P system Π (the reader can find the detailed construction in the technical report [1]). Hence N nsyn gen (Π) = N (M ).…”