# Characterizations of Some Restricted Spiking Neural P Systems

## Abstract

A *k*-output spiking neural P system (SNP) with output neurons, *O* _{1}, ⋯, *O* _{ k }, generates a tuple (*n* _{1}, ⋯, *n* _{ k }) of positive integers if, starting from the initial configuration, there is a sequence of steps such that during the computation, each *O* _{ i } 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 *n* _{ i }. 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 (*n* _{1}, ⋯, *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 *n* _{ i }, 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*,2*m*) | *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 strictly monotonic counters.

## Keywords

Output Neuron Output Module Single Spike Counter Machine Membrane Computing## Preview

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