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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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On structures and behaviors of spiking neural P systems and Petri nets<br />

If N is an AND-join net such that i, j ∈ •t and k ∈ t•, then Π N has neur<strong>on</strong>s<br />

σ t , σ i , σ j , σ k with synapses (i, t), (j, t), (t, k) and R t = {(a + ) v → a} for v = | • t|<br />

(in the example figure, v = 2). Given M 0 = (1, 1, 0) for N i.e. <strong>on</strong>ly k has no<br />

marking, the final marking after the firing of t will be (0, 0, 1) since N is an<br />

AND-split net, combining the tokens from i and j and producing <strong>on</strong>e token<br />

to k. Similarly for Π N there is C 0 = 〈1, 1, 0, 0〉 and the final c<strong>on</strong>figurati<strong>on</strong> is<br />

〈0, 0, 0, 1〉. The rule in σ t combines the spikes from σ i and σ j and produces <strong>on</strong>e<br />

spike to σ k . However if M 0 = 〈0, 1, 0〉 then t cannot fire. Similarly σ x will not<br />

spike given C 0 = 〈0, 1, 0, 0〉. Therefore Π N also performs an AND-join. ⊓⊔<br />

Fig. 4. SNP system (a) AND-join, and (b) AND-split.<br />

Observati<strong>on</strong> 1 If N is a n<strong>on</strong>safe Petri net that performs an AND-join, using<br />

the c<strong>on</strong>structi<strong>on</strong> for Lemma 2, SNP system Π N does not perform an AND-join.<br />

An example of Observati<strong>on</strong> 1 is shown in Fig. 5: the SNP system does not<br />

perform an AND-join since the joining neur<strong>on</strong> σ t still spikes <strong>on</strong>ce accumulating<br />

two spikes from its top input neur<strong>on</strong> σ j (and from the other spike from σ i ). In<br />

the Petri net however, transiti<strong>on</strong> t is never fired (a deadlock) since k is never<br />

marked, and j is a n<strong>on</strong>safe place.<br />

Lemma 3. Given a Petri net N that performs an OR-split (OR-join) of a token,<br />

there exists an SNP system Π N that performs an OR-split (OR-join) of a spike<br />

simulating N.<br />

Fig. 5. A n<strong>on</strong>safe AND-join Petri net and the “bad” AND-join SNP system<br />

simulating the net.<br />

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