Characterizations of Some Restricted Spiking Neural P Systems

  • Oscar H. Ibarra
  • Sara Woodworth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4361)


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 aaa 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,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 strictly monotonic counters.


Output Neuron Output Module Single Spike Counter Machine Membrane Computing 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Oscar H. Ibarra
    • 1
  • Sara Woodworth
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

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