Characterizations of Some Restricted Spiking Neural P Systems

Abstract

Abstract. A k-output spiking neural P system (SNP) with output neurons, O1, ..., O k , generates a tuple (n1, ..., n k ) of positive integers if, starting from the initial configuration, there is a sequence of steps such that during the computation, each Oi generates exactly two spikes a a (the times the pair a a are generated may be different for different output neurons) and the time interval between the first a and the second a is ni. After the output neurons have generated their pairs of spikes, the system eventually halts. Another model, called k-train SNP, has only one output neuron. It generates a k-tuple (n1, ..., n k ) if, starting from the initial configuration, the output neuron O generates the spike train aa...a with exactly k + 1 a's such that the interval between the i th a and the i + 1 st a is ni, and the system eventually halts. We assume, without loss of generality, that each neuron in the SNP is either bounded or unbounded. (Bounded here means that there is a fixed constant c such that at any time during the computation, the number of spikes in the neuron is at most c. Otherwise, the neuron is unbounded.) It is known that 1-output SNPs (= 1-train SNPs) are universal, i.e., they generate exactly the recursively enumerable sets over N . Here, we show the following:1. For k ≥ 1, a set Q ⊆ N k is semilinear if and only if it can be generated by a k-output SNP, where every unbounded neuron satisfies the property that once it starts "spiking" it will no longer receive future spikes (but can continue spiking). This result also holds for k-train SNP. 2. The set Q = {(m, 2m) | m ≥ 1} (which is semilinear) cannot be generated by any 2-output bounded SNP (i.e., SNP all of whose neurons are bounded). Thus, for k ≥ 2, there are semilinear sets over N k that cannot be generated by k-output bounded SNPs. This contrasts a known result that 1-output bounded SNPs generate all semilinear sets over N . 3. For k ≥ 2, k-output bounded SNPs are computationally more powerful than k-train bounded SNPs. (They are identical when k = 1.) 4. For k ≥ 1, k-output bounded SNPs and k-train bounded SNPs can be characterized by certain classes of nondeterministic finite automata with monotonic counters.

Asynchronous spiking neural P systems

“…We find this problem worth investigating (a non-universality result -as we expect it will be the case -can show an interesting difference between synchronized and nonsynchronized devices, with the loss in power compensated by the additional "programming capacity" of extended rules). The non-synchronized case remains to be considered also for other issues specific to SN P systems, such as looking for small universal systems as in [17], for normal forms as in [10], for generating languages or processing finite or infinite sequences, [3], [4], [21], characterizations of multi-dimensional semilinear sets of numbers as in [8], using the rules in the in exhaustive mode, as in [12], etc.…”

A quick overview of membrane computing with some details about spiking neural P systems

“…[29,30], where several output neurons were considered, thus producing vectors of numbers, not only numbers. A detailed typology of systems (and generated sets of vectors) is investigated in the two papers mentioned above, with classes of vectors found in between the semilinear and the recursively enumerable ones.…”

Homogeneous Spiking Neural P Systems

“…Actually, some normal forms are already given, e.g. [5,6,12]. In this paper, we focus on the restriction that each neuron has the same set of rules.…”