Spiking Neural P Systems With Enzymes

The neurotransmitter is a chemical substance that transmits information between neurons. Its metabolic process includes four links: synthesis, storage, release and inactivation. As one of the important chemical components of neurotransmitters, acetylcholine is synthesized under the catalysis of acetylcholine coenzyme A and choline acetylase. Inspired by the biological fact that enzymes exist in neurons and that enzymes are involved in neurotransmitter synthesis, we propose spiking neural P systems with enzymes (SNPE). Different from the previous spiking neural P systems and their variants, each neuron of SNPE contains two classes of objects, and each spiking rule has the participation of enzymes. In addition, the number of spikes and enzymes in a neuron can also serve as a consumption condition for controlling whether a reaction (rule execution) occurs. When the number of enzymes meets the requirements of a specific biochemical reaction, the number of occurrences of the reaction can also be controlled. As number generation and acceptance devices, the proposed SNPE systems are proved to be Turing universal. In addition, 61 neurons are used to construct an SNPE system that realizes function computation, which proves the Turing universality in this mode. Finally, we also explore using a uniform SNPE model to solve the subset sum problem and compare it with the standard SN P and its several variants.

theory of membrane computing was first proposed by Pȃun [1]. 36 Therefore, the membrane system is also referred to as the P 37 system [2] for short. The membrane computing models have 38 attracted a large number of scholars to conduct extensive and 39 in-depth research due to their outstanding characteristics such 40 as distribution, parallelism and non-determinism. According 41 to different topological structures and biological principles, 42 generally speaking, P systems mainly involve three types 43 in topology [3]: cell-like P systems (hierarchical topological 44 features), tissue-like P systems (network-like topological fea-45 tures), and neural-like P systems (topological features appear 46 as directed graphs). 47 In the nervous system of living organisms, nerve cells 48 (neurons) exchange information by transmitting spikes through 49 synapses. Inspired by this biological fact, spiking neural 50 P systems (SN P) were first proposed [4] with their con-51 cise expression and efficient structures. Since then, further 52 inspired by various biological facts such as the biochemical 53 reactions and functional structure of the biological nervous 54 system, various variants based on the original SN P sys-55 tems framework are constantly being developed. SN P has 56 become the most promising membrane computing model. 57 Considering the excitatory and inhibitory effects of astro-58 cytes on synapses, Pȃun [5] and Pan et al. [6] studied the 59 SN P systems with astrocytes. On this basis, Aman and 60 Ciobanu [7] further studied the SN P systems with astrocytes 61 that can produce calcium ions. Pan and Pȃun [8] abstracted 62 the concept of anti-pulse from the biological phenomenon of 63 inhibitory pulse, and proposed the SN P systems with anti-64 spikes. By introducing white hole rules, SN P systems with 65 white hole neurons [9] were constructively proposed. The 66 SN P systems with polarizations were fully discussed and 67 studied [10], [11]. Peng et al. [12], Huang et al. [13], and 68 Lv et al. [3] discussed the SN P systems with multiple chan-69 nels rules. Cavaliere et al. [14] proposed the SN P systems 70 with non-synchronous (i.e., asynchronous) rules for the first 71 time and proved its equivalence with Turing machines. Based 72 on this, Pan et al. [15] further proposed a new working mode 73 called limited asynchronous SN P systems. Peng et al. [16] 74 studied fuzzy reasoning SN P and applied it in the field 75 of fault diagnosis. In terms of the execution of rules, the 76 existing mainstream modes are divided into the following three 77 categories: sequential mode [4], exhaustive mode [17], [18], 78 and flat maximal parallel mode [19], [20]. Wu et al. [21] 79 first introduced the flat maximally parallel mode into the 80 SN P systems. 81 Zeng et al. [22] first introduced the concept and idea of 82 threshold to SN P systems. Inspired by the phenomenon 83 of synchronized oscillations of neurons in cat's visual cor-84 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ tex, Peng et al. [23] further conceived the dynamic threshold coenzyme A and choline acetylase, and then transferred to 141 vesicles for storage. When the nerve impulse reaches the 142 nerve terminal, the vesicle membrane and the presynaptic 143 membrane fuse to release acetylcholine into the synaptic cleft. 144 At the same time, acetylcholine is hydrolyzed into choline and 145 acetic acid by cholinesterase (ChE) in the nerve endings and 146 inactivated. Part of the choline is once again taken up by the 147 cholinergic nerve endings and participates in the synthesis of 148 new acetylcholine. 149 This work is inspired by the above biological principles, 150 and proposes new spiking neural P systems with enzymes 151 (SNPE). Compared with the standard SN P and its variants, 152 SNPE has made improvements in terms of objects, rules, 153 and system operation. As a function computing device, the 154 proposed SNPE is compared with 6 SN P variants published in 155 the past 3 years (2019∼2021). In addition, we also explore the 156 computational power of SNPE systems to uniformly solve the 157 subset sum problem attributed to NP-complete. The detailed 158 inspiration, motivation, innovations and contributions of this 159 work will be given in the next section. 160 The logical structure of this work is arranged below. 161 Section II describes the relevant basic knowledge. Motiva-162 tions, the formal definition of the SNPE system and an 163 example are presented in Section III. Section IV gives two 164 proofs of Turing universality of SNPE systems as number 165 generation and acceptance devices, respectively. The SNPE 166 system as a functional computing device and its computing 167 power are shown in Section V. Section VI demonstrates the 168 computational power of the SNPE system to uniformly solve 169 the subset sum problem attributed to NP-complete. A summary 170 and outlook are in Section VII.

II. PRELIMINARY KNOWLEDGE 172
In this section, we briefly review some necessary knowledge 173 about formal languages and automata theory. At the same time, 174 the commonly used notations and their meanings throughout 175 this work are given.

176
Suppose that V is an alphabet, a set with non-emptiness 177 and finiteness. A finite sequence of elements (also called 178 characters) in alphabet V formed one after another in a certain 179 order is called a string. The empty string λ does not contain 180 any characters. The meaning of V * is similar to the Kleene 181 closure, and V + is similar to the positive closure. The union 182 of V + and λ is equivalent to V * . For a detailed introduction 183 to formal languages, readers can also refer to [2] and [46].

184
The formal definition of a register machine is denoted by 185 the tuple M = (m, H, l 0 , l h , I ), where, m denotes the number 186 of registers. H represents a set of instruction labels. l 0 and 187 l h mean the starting label and the halting label, respectively. 188 I is the set of all possible instructions. Every label in H 189 corresponds to an instruction in I one-to-one. The instructions 190 in I are divided into three types, that is, the addition instruction 191 l i : (ADD(r ), l j , l k ), the subtraction instruction l i : (SUB(r ), 192 l j , l k ) and the termination instruction l h : H ALT . For a further 193 detailed description of register machine, please refer to [47]. 194 The computing power of a new computing model is usually 195 verified by realizing the simulation of the register machine. 196 It is usually considered to verify separately from the per-197 spective of the two modes. One is the generation mode, 198 and the working principle of this mode is as follows.  tially, the stored value in each register is empty. The calcula-     3) syn ⊆ {1, 2, · · · , m} × {1, 2, · · · , m} represents the set 268 of synapses.

271
The SNPE systems consist of three important components, 272 namely, system objects, executing rules, and system structure. 273 It can be seen from the above definition that an enzyme as 274 a new object participates in all spiking rules. Since enzymes 275 only exist inside nerve cells, they do not follow the informa-276 tion (spikes) to transmit between neurons. Furthermore, the 277 enzymes involved in biochemical reactions (rules) can then be 278 automatically synthesized in nerve cells for supplementation. 279 In other words, in SNPE systems, a spiking rule can produce 280 spikes and enzymes but only spikes are transmitted to external 281 neurons while the enzymes still remain in the same neuron. Suppose u i (t) is used to represent the number of spikes 283 in σ i at step t, and v i (t) is used to represent the number of 284 enzymes in σ i at step (time) t. At the next moment (t + 1) 285 after applying the rule E/(a u , e v ) → (a p , e q ); d, the number 286 of spikes and the number of enzymes in σ i can be calculated 287 by (2) and (3), The n in equation (2) represents the number of spikes that 291 σ i receives from its predecessor. u and v represent the number 292 of spikes and enzymes needed to enforce the rule, respectively. 293 p is the number of spikes generated by this rule, which are 294 sent to the successor neurons along synaptic connections. q is 295 the number of enzymes generated by this rule, which are 296 only present in the current neuron to ensure the continued 297 availability of the rule. d means the time delay between the 298 use of the rule and the emission of spikes produced by that 299 rule. For example, assuming rule (a u , e v ) → (a p , e q ); d in σ i 300 is enabled at step t, then the p spikes generated will leave σ i 301 at step t + d. From step t until step t + d − 1, σ i is closed 302 and is in the refractory period. That is, during these d steps, 303 σ i can neither receive any spikes nor fire again. When d is 304 not specified, its value is 0 by default.

305
The SNPE system can realize the parallel execution mode of 306 a single rule in one neuron, which is similar to the exhaustive 307 mode [18]. That is to say, if a neuron contains exactly an 308 integer multiple (such as n) of the number of spikes and 309 enzymes required by a rule, then this rule will be executed For convenience, we say 325 that a rule satisfies its firing condition if it satisfies both its 326 trigger condition and its consumption condition.

327
When multiple rules satisfy their firing conditions, the 328 SNPE system follows the maximum spike-consumption strat-329 egy (mentioned in [30]). That is, when multiple rules are 330 available, the one that consumes the largest number of spikes 331 is selected for execution. For instance, if u 1 > u 2 , then 332 rule E 1 /(a u 1 , e v 1 ) → (a p 1 , e q 1 ) will be activated while rule 333 E 2 /(a u 2 , e v 2 ) → (a p 2 , e q 2 ) will not be executed. In addition,  After σ 2 receives two spikes, the situation is similar to σ 1 360 in the initial state. At t + 1, σ 2 fires 2 spikes to σ 3 and σ 4 361 respectively. After receiving the two spikes sent by σ 1 , because 362 the trigger condition is not met, no rule in σ 3 is activated. After 363 neuron 4 (written as σ 4 ) receives the two spikes sent by σ 2 , 364 it satisfies the firing condition and sends out one spike for the 365 first time at t + 2. Until another 2 spikes are obtained from 366 σ 2 , the four rules in σ 3 meet the trigger conditions. However, 367 considering the maximum spike-consumption strategy, rule 368 a 3 (a) * / (a 3 , e 2 ) → (a 2 , e 2 ) cannot be enabled. Also, because 369 σ 3 initially contains only two enzymes, rule a 2 (a 2 ) + / (a 4 , 370 e 3 ) → (a 2 , e 2 ) does not satisfy its consumption condition and 371 therefore cannot be activated. The remaining rules a 2 (a 2 ) + / 372 (a 4 , e 2 ) → (a 2 , e 2 ) and a 2 (a 2 ) + / (a 4 , e 2 ) → (a 3 , e 3 ) will be 373 executed non-deterministically, which will cause the system to 374 have different output results. Specifically, the following two 375 scenarios will lead to completely different computations. 376 1) Assuming that rule a 2 (a 2 ) + /(a 4 , e 2 ) → (a 2 , e 2 ) is 377 activated at t + 2, σ 3 will fire 2 spikes to σ 4 . σ 4 will run 378 rule (a 2 ) + /(a 2 , e) → (a, e) at t + 3 and send out one 379 spike outside again. At this time, the system terminates 380 the computation, and the final result of the output binary 381 sequence is "0011".
382 2) Assuming that rule a 2 (a 2 ) + /(a 4 , e 2 ) → (a 3 , e 3 ) is 383 activated at t + 2, σ 3 will fire 3 spikes to σ 4 . σ 4 will 384 execute the forgetting rule a 3 → λ at t + 3. The 385 3 spikes received directly disappear and no spikes are 386 launched outside. At this time, the system terminates the 387 computation, and the final result of the output binary 388 sequence is "0010".

389
IV. TURING UNIVERSALITY OF SNPE SYSTEMS 390 As mentioned earlier, the simulation of the register machine 391 is usually divided into two modes, that is, the generation mode 392 and the acceptance mode. Each register r in M has a one-393 to-one correspondence with a neuron σ r in SNPE systems. 394 The number n stored in the register r corresponds to 2n 395 spikes in the neuron σ r . That is, there are twice as many 396 spikes in the neuron σ r as the value stored in the register r . 397 Similarly, each instruction l ∈ H corresponds to a neuron 398 σ l in SNPE systems one-to-one. In addition, there are some 399 auxiliary neurons, denoted as σ b i . In the following, the Turing 400 universality of SNPE systems in these two modes is explored 401 and proved. It should be noted that in the completeness proof 402 of the following two modes, two spikes are introduced as the 403 starting mechanism for all involved modules. istically. As a result, the following two scenarios will appear.   the instruction l k . At t + 2, 4 spikes will be accumulated 457 in σ b 2 , but these 4 spikes will be immediately forgotten 458 by the internal forgetting rule a 4 → λ. In this scenario, 459 σ l j will not receive any spikes, which means that the l j 460 instruction will not be activated and executed. Table I 461 shows the computational process in the above scenario. 462 The numbers in square brackets represent the number 463 of spikes and enzymes at the corresponding step in the 464 neuron, respectively. 465 b. Suppose that at t + 1, the rule (a 2 ) + /(a 2 , e) → (a, e); 466 0 is activated. σ b 1 will fire a spike to σ b 2 and σ l k simulta-467 neously. Similarly, the spike received by neuron σ l k will 468 be directly consumed by its internal rule a → λ. The 469 instruction l k cannot be executed. At t + 2, the neuron 470 σ b 2 has accumulated three spikes, which satisfies the 471 firing condition of rule a(a 2 ) + /(a 3 , e 2 ) → (a 2 , e 2 ); 0. 472 At t +3, σ l j obtains 2 spikes, which marks the instruction 473 l j will be executed. The specific computational process 474 under this scenario is presented in Table II. In summary, the addition instruction has been activated since 476 σ l i received 2 spikes. Next, σ r receives 2 spikes, which is  a. If the number existed in r is not empty (i.e., n > 0), that 502 is, the number of spikes in σ r is not less than 2. Then, 503 at t + 1, the spikes contained in σ r is at least 3. At this 504 time, only the rule a(a 2 ) + /(a 3 , e 2 ) → (a, e 2 ); 0 will be 505 activated and fire one spike to σ l k and σ l j , respectively.

506
At t + 2, σ l j will contain exactly 2 spikes, which 507 means that instruction l j will be activated and executed.

508
In addition, because the rule in σ b 2 uses delay, σ l k will 509 receive one spike from σ r and σ b 2 at t + 2 and t + 3 510 respectively. That is, the two spikes that do not simul-511 taneously arrive at σ l k will be forgotten separately. The 512 specific evolution process of the SUB module in this 513 scenario is presented in Table III.   514   TABLE III  THE COMPUTATIONAL PROCESS OF THE SUB UNDER SCENARIO A   TABLE IV  THE COMPUTATIONAL PROCESS OF THE SUB UNDER SCENARIO B b. If the number existed in the register r is empty 515 (i.e., n = 0), that is, the number of spikes contained 516 in σ r is 0. At t + 1, σ r has only one spike sent from σ l i . 517 At this time, only rule a/(a, e) → (a, e); 1 will be 518 activated and executed. Since the available rules in σ r 519 and σ b 2 both have the same time delay, at t + 3, σ l k will 520 receive one spike from σ r and σ b 2 , respectively. This 521 causes σ l k to be activated, meaning that the l k instruction 522 begins to be executed. Similar to the previous scenario, 523 the single spike received by σ l j successively from σ b 1 524 and σ r will be respectively forgotten. Table IV shows 525 the computational process of the configuration in each 526 neuron under this scenario.

527
In summary, since the neuron σ l i received 2 spikes, the 528 subtraction instruction became active. When the number of 529 spikes in σ r is not empty, 2 spikes of them are consumed, 530 and σ l j is further activated. Conversely, when the number of 531 spikes in σ r is zero, σ l k will be directly activated. The process 532 successfully simulates the subtraction instruction.

534
This module is designed to simulate the termination instruc-535 tion l h , and yields the system's computation results. Suppose 536 σ l h gets 2 spikes at step t, marking the start of the termination 537 instruction to be simulated. Rule (a 2 , e) → (a, e); 0 is 538 executed, and 1 spike is fired to σ 1 .

539
Assuming that the value existed in register 1 is n, 540 correspondingly, the number of spikes in σ 1 is 2n. Then, 541 after neuron σ 1 obtains one spike at t + 1, the number of 542 spikes it contains will become odd. Rule a 3 (a 2 ) + /(a 2 , e) → 543 (a, e); 0 in σ 1 is activated and executed. It should be noted that 544 the regular expression for this rule requires that the number of 545 spikes contained in σ 1 cannot be less than 5. This is to avoid 546 a conflict with the applicability of another rule.

557
Through the above analysis, it can be easily found that 558 the generation mode can be successfully simulated by 1 .

559
Therefore, Theorem 1 is proved to be true. convenience. The specific reasoning is proved as follows.

573
The purpose of this module is to import binary sequence activated. This rule is executed twice in parallel simultane-579 ously, and 2 spikes are fired to σ b 1 , σ b 2 , and σ b 3 , respectively. 580 At t + 1, although σ b 1 obtains 2 spikes, it does not meet the 581 firing conditions of the rules contained in it. Therefore, its 582 internal rules cannot be enabled. Simultaneously, after the neu-583 rons σ b 2 and σ b 3 receive two spikes, the rule (a 2 ) + /(a, e) → 584 (a, e) within them is activated, and from this moment on, they 585 send two spikes to each other at the same time. At t + 2, σ 1 586 gets and stores 2 spikes for the first time. At t + n, σ in obtains 587 2 pulses from the outside again. At t + n + 1, the neurons 588 σ b 2 and σ b 3 will hold four spikes at the same time, which 589 meets the forgetting rule a 4 → λ. The four spikes accumulated 590 in neurons σ b 2 and σ b 3 are all forgotten at this moment. 591 Therefore, at t +n+1, it is the last time that neuron σ 1 gets two 592 spikes from σ b 2 . From t + 2 to t + n + 1, σ 1 has accumulated 593 2n pulses. This simulates the value existed in register 1 is n, 594 and n also happens to be the time difference between the two 595 pulses. On the other side, at t + n + 1, σ b 1 obtained two spikes 596 from σ in for the second time. Therefore, the number of spikes 597 accumulated inside is four, rule a 4 /(a 4 , e 3 ) → (a 2 , e 3 ) is 598 activated, and 2 spikes are fired to σ l 0 . After σ l 0 gets 2 spikes, 599 it indicates that the initial instruction l 0 will be simulated.

601
This module realizes the simulation of deterministic addi-602 tion instructions. Suppose that the neuron σ l i obtains 2 spikes 603 at step t. At this time, rule (a 2 ) + /(a, e) → (a, e) fires 2 spikes 604 to σ r and σ l j respectively. After σ r gets 2 spikes, it indicates 605 the existing storage value in the register r is increased by 1. 606 Simultaneously, σ l j received 2 spikes, which corresponds to 607 the deterministic execution of the instruction l j .

608
Through the above deduction, it can be clearly found that 609 the computation in this mode can be successfully simulated 610 by 2 . Hence, Theorem 2 is proved to be true.   and acceptance devices, for the sake of formal unification, 644 we introduce two spikes as the starting mechanism for the 645 proof of all involved modules. However, the function com-646 puting devices considered in this section need to redesign 647 the INPUT module according to its characteristics. Therefore, 648 slightly different from the input mechanism of the INPUT 649 module in acceptance mode, the INPUT module here uses 650 a single spike to correspond to 1 in the binary sequence. 651 In addition, the delay mechanism is not involved in this 652 module, so it is ignored.

653
The working mechanism of this module is stated as follows. 654 Initially, only σ in gets the first one pulse from the outside. Its 655 internal rule (a, e) → (a, e) is triggered, and fires 1 spike to 656 each of the five subsequent auxiliary neurons simultaneously. 657 But only the rule a/(a, e) → (a, e) in σ b 1 and σ b 2 meets the 658 firing condition and is activated. At each step from this time 659 on, σ b 1 and σ b 2 will complement each other with one spike, 660 and both send a spike to σ 1 simultaneously. Until σ b 1 and σ b 2 661 receive the second spike from σ in , during this period, σ 1 will 662 accumulatively obtain 2g(x) spikes.

663
When σ in gets the second spike from the outside, (a, e) → 664 (a, e) is triggered again and simultaneously fires 1 spike to 665 each of the five auxiliary neurons. At this time, none of the 666 rules in the neurons σ b 1 , σ b 2 , and σ b 5 satisfy the firing condi-667 tion, so none of them can be activated. However, two spikes 668 have been accumulated in neurons σ b 3 and σ b 4 , satisfying the 669 firing condition of (a 2 ) + /(a, e) → (a, e). At each step from 670 this moment on, the neurons σ b 3 and σ b 4 will consume and 671 complement each other with one spike, and both send one 672 spike to σ 2 simultaneously. It should be noted that during this 673 process, 2 spikes will always remain in σ b 3 and σ b 4 due to the 674 complementary mechanism. Until σ b 3 and σ b 4 receive the third 675 spike from σ in , during this period, neuron σ 2 will accumulate 676 2y spikes.   We take Fig. 10 as an example to give the evolution process 717 of the SU B − ADD 1 module. It should be emphasized that 718 this type of module has a common feature, that is, the first 719 output instruction of SU B is exactly the input instruction of 720 ADD. The initial scenario of this module is similar to the 721 SU B module shown in Fig. 3. The difference is that the rules 722 with 0 delay in σ r x and σ b 1 will send one spike to σ r y and σ l g 723 simultaneously when register r x of instruction l i is not empty. 724 In other words, σ r y and σ l g will receive two spikes simultane-725 ously. For instruction l j , this is equivalent to simultaneously 726 completing the two operations of adding 1 to the stored value 727 of register r y and deterministically going to instruction l g . The 728 single spike sent from σ r x and σ b 2 to σ l k will be forgotten 729 successively in σ l k due to the time interval. In addition, the 730 rules with 1 delay in σ r x and σ b 2 will be fired simultaneously 731 when the value stored in register r x is empty. In this scenario, 732 σ l k will receive 2 spikes simultaneously, which means that the 733 instruction l k starts to execute. Each of the 6 combinations 734 that can be classified as SU B − ADD 1 saves one neuron. 735 The evolution process of the SUB-ADD 1 module when the 736 register r x is not empty is shown in Table VI. Table VII shows 737  TABLE VI  THE EVOLUTION PROCESS OF THE SUB-ADD 1 MODULE WHEN THE  REGISTER r x IS NOT EMPTY   TABLE VII  THE EVOLUTION PROCESS OF THE    The subset sum problem is NP-complete and can be 759 described as follows. Given a set V of n positive integers and a 760 positive integer S, is there a subset B ⊆ V such that the sum of 761 all elements in B is exactly equal to S? Leporati et al. [52] and 762 Leporati et al. [53] have used standard SN P systems to solve 763 subset sum problem in a non-uniform and uniform manner, 764 respectively. Non-uniform way relies on specific instances to 765 design models, whereas a uniform way is problem-oriented 766 rather than concrete instances. In other words, solving the 767 subset sum problem in a uniform manner means that the design 768 of the system depends only on the size n of the problem, while 769 the specific elements of V and S need to be introduced into 770 the system. Obviously, the uniform way is more transparent 771 than the non-uniform way [53]. Therefore, this work explores 772 the use of SNPE systems to solve subset sum problem in a 773 uniform way. The construction of the SNPE model to solve the subset sum 776 problem is shown in Fig. 13. We follow the design philosophy 777 of Leporati et al. [53] that if the problem is solved then the 778 computation of the system stops automatically, otherwise the 779 computation of the system continues forever. The specific 780 reasoning process is as follows.  This causes rule (a 2 , e) → (a, e) in σ c i to be activated. Next, 797 the input neuron σ in i will receive one spike at step 3, which 798 makes the number of spikes in it become odd (i.e., 2v i + 1).

799
Note that the intrinsic number of enzymes in σ in i is 2,

815
Given this concern, the trigger neuron σ T is therefore designed 816 to determine whether all selected numbers have been sent to 817 the σ acc . The rule a 2t +1 → λ in σ T is available as long as the  When σ acc and σ in 0 receive one spike from σ T , the number 831 of spikes contained will become odd. Then the only rule within 832 σ acc and σ in 0 will be activated, sending one spike to σ e 0 833 while consuming two of its own spikes. After σ e 0 receives two 834 spikes, it immediately forgets them according to rule a 2 → λ. 835 The above process continues until one of the following three  that σ acc and σ in 0 contain exactly the same number of spikes. 838 In this case, after σ e 0 executes the forgetting rule for the last 839 time, the entire system will stop computing. This means that 840 the sum of all numbers of non-deterministic choices is exactly 841 equal to S, and the subset sum problem is solved. (ii) The 842 number of spikes stored in σ acc is more than that stored in σ in 0 . 843 (iii) Conversely, more spikes are stored in σ in 0 than in σ acc . 844 The latter two scenarios can be grouped into one category, that 845 is, σ e 0 will start to execute rule (a, e) → (a, λ) after executing 846 rule a 2 → λ for the last time. As mentioned earlier, rule 847 (a, e) → (a, λ) consumes one enzyme without reproducing it, 848 which causes this rule will no longer be available from now 849 on. Subsequently, σ g 1 and σ g 2 will work forever.

850
As can be seen from Fig. 13, the uniform SNPE model 851 requires only 3n + 10 neurons. And at most 2 n i=1 v i + 3 852 steps are required when the computation of this model is 853 stopped automatically, including the initial 2 steps, from σ in i 854 to σ acc at most max 1≤i≤n {v i } steps, from σ acc (or σ in 0 ) to σ e 0 at 855 most n i=1 v i steps, and one step from σ T to σ acc (and σ in 0 ). 856 Considering that max 1≤i≤n {v i } ≤ n i=1 v i must be true, we take 857 n i=1 v i as the upper bound of max 1≤i≤n {v i }. Therefore, when the 858 calculation of the system is automatically terminated, at most 859 2 n i=1 v i + 3 steps are required.

860
Finally, we compare with the standard SN P [53] and its 861 several state-of-the-art variants in terms of complexity (see 862  Table IX). It is evident from Table IX that the original SN P 863 has the largest number of neurons used and the number of steps 864 required to compute termination. Our SNPE is second only to 865 RSSN P [54] in terms of using the total number of neurons, but 866 outperforms the other three compared models. Whereas, the 867 proposed SNPE model outperforms all four compared models 868 in terms of the number of steps required. This work proposes new spiking neural P systems with 871 enzymes (SNPE). Compared with the standard SN P and its 872 variants, SNPE has made improvements in terms of objects, 873 rules, and system operation. While giving the formal definition 874 of the SNPE systems, it also gives the expression of the 875 change in the number of two kinds of objects in the neuron. 876 Turing computing ability of the proposed SNPE systems in 877 generation mode and acceptance mode is proved respectively. 878 The computing power of this system as a small universal 879 function computing device is demonstrated and compared 880 with 7 SN P variants. Finally, the performance of the system in 881 solving NP-complete problem is explored and compared with 882 the standard SN P and its several state-of-the-art variants.

883
It should be emphasized that the excellent performance 884 (see Table VIII) of the proposed SNPE as a small universal 885 function computing device is not independently and directly 886 contributed by the "enzyme", but the collaboration of multiple 887 mechanisms including the "delay". There are indeed some 888 SN P variants that use a smaller total number of neurons 889 than SNPE when simulating a function computing device. For 890 example, SN P with request rules [57] uses only 47 neurons, 891 yet the number of rules in its neurons is as high as 11, which 892 is much higher than all the comparison models in Table VIII. 893 In addition, it is precisely because of the reasonable control 894 of the "enzyme" over the sustainability of rule execution that 895 SNPE excels in solving the Subset Sum problem.