Thinking about spiking neural P systems: some theories, tools, and research topics

Thinking aboutspiking neural P systems: some theories, tools, andresearch topics FrancisGeorgeC.Cabarle1,2 Received: 5 December 2023 / Accepted: 28 March 2024 © The Author(s) 2024 Abstract The present work focuses on three notions about spiking neural P systems (SN P systems), namely normal forms, homogeneous systems, and software tools for easy access and visual simulation of such systems. The three notions are presented in general and specific ways: their backgrounds and motivations, with detailed and up to date results. The aim of the work is to outline many results on these notions, mainly for research and pedagogy. SN P systems with normal or homogeneous forms, having many biological and computing inspirations, have much to contribute in the opinion of the author to membrane computing at least. The software we here mention aims to support both learning and research of such systems. We provide a brief survey of results in chronological order, using a unified notation to aid in more detailed comparisons of results. Lastly, we provide a list of open problems or research topics on the three notions and related areas, with the hope to further extend the theory and applications of SN P systems. Keywords Membrane computing· Spiking neural P systems· Normal forms· Visual simulators 1 Introduction Membrane computing is an active area of investigation under the more general field of natural computing, being part of a four-volume handbook in [89]. P systems, the models in membrane computing started in [78] (earlier as a technical report in [77]), are investigated both for their theory and practice. Some evidence for the enthusiasm in the investigations of P systems includes a recent survey in [90], a bibliometric analysis in [87], and the acceptance of the Journal of Membrane Computing (established in 2019) in the ESCI and SCOPUS indexing.1 A P system variant known as spiking neural P systems (in short, SN P systems) introduced in 2006 [55] is one of many P system variants with vigorous research activity. One of the main reasons for the great interest in spiking neuron models such as SN P systems, according to [82], is that, while nature proves to be a rich source of ideas for computing, the human brain is a “gold mine” of such idea source. Furthermore, the human brain consumes only about 20 watts of power, but it is capable of parallel and distributed processing of images, speech, consciousness, and so on. For these reasons and more, the research on SN P systems is quite active. For instance in the bibliometric analysis in [87], we see the following numbers: 10 of the top 20 papers are about SN P systems or variants; SN P systems, together with membrane algorithms, are the top in the 9 largest co-citation clusters. SN P systems and their variants, whether biologically or mathematically inspired, have been used to solve many computational problems. Further details can be found in [55, 82] and the SN P systems chapter in the membrane computing handbook [84]. Some works on the computing power of SN P systems and variants include [55, 69, 75] with recent works such as [5, 67, 74]. Such works mainly look at the effects or requirements of certain features with respect to classes of languages or problems that can or cannot be solved. Such features, with some given more details in the sections below, include delays in sending the spike signals, amount of spikes produced, the types of rules or *Francis George C. Cabarle [email protected]; [email protected] 1 Research Group onNatural Computing, Department ofComputer Science andArtificial Intelligence, SCORE lab, I3US, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012Sevilla, Spain 2 Department ofComputer Science, University ofthePhilippines Diliman, Diliman, 1101QuezonCity, Philippines 1 See https:// www. sprin ger. com/ journ al/ 41965/ updat es/ 19161 752 F.G.C.Cabarle neurons in the system. The classes of languages or problems are often with regards to classical models such as automata, grammars, Turing machines. Some works on computing efficiency include [27, 62, 73, 100] with recent works in [7, 19, 108]. These works on efficiency are mainly concerned with how much space or time is required to solve certain, usually computationally hard, (classes of) problems. In the case of SN P systems, space usually refers to the total number of neurons, rules, synapses, and so on, while time usually refers to (non)deterministic “clock steps” to halting after a solution (or lack thereof) is reached. Besides these theoretical results, much advancements in applications of SN P systems and variants appear in the last decade such as [95, 96, 109] with earlier surveys in [28, 34, 36]. Further details, especially when contributing to the three main notions of this present and brief survey, are provided in their respective sections in the following sections. Several PhD theses have also been done on SN P systems and their variants listed in [86] and the International Membrane Computing Society [54]. The present work is a largely revised and extended version of the abstract and talk given by the author in [12], with many references taken from a bibliography, as of March 2022, in [20]. The revisions and extensions include some up to date results as of March 2024 by the groups of the authors on the three main notions focused by this present work. The organization of this work is as follows: Sect.2 begins with some very brief preliminaries, followed by three general notions; Sect.2.1 defines the basic or standard model of SN P systems, with an illustrative example in Sect.2.2. Sect.2.3 and Sect.2.4 provide a chronological view of the results on normal forms and homogeneous systems, respectively, for SN P systems and variants; Besides the chronological order of results, Sect.2.3 and Sect.2.4 provide a more unified notation to compare many results on their respective ideas; For instance, a convention for providing results is to focus on a normal form of some variant with a certain feature, with less or no focus on comparable features of a normal form for another variant. Section2.5 introduces pedagogical software for the improved ease of access, visualization, and interaction of SN P systems and variants. Section3 provides a long list of open problems and research topics based on, but not limited to, the previous sections. 2 SN P stories: definitions andmain notions We consider two theoretical notions (Sects.2.3 and 2.4) and one practical notion (Sect.2.5) also in the hopes of furthering both theory and practice of SN P systems. Both theoretical notions have some biological inspiration: bio-neurons can be seen as rather “simple” or restricted compared to other types of cells; bio-neurons also seem rather “similar” or homogeneous at least for certain parts of the brain. The practical notion is an easy to access and visual tool for pedagogical purposes, not just for experts but also for students or new researchers. Definitions of SN P systems, their syntax and semantics, are only mentioned in brief below to be able to focus more on three main notions in the present section. Excellent introductions to and definitions of SN P systems include the seminal work in [55], the dedicated chapter in the handbook in [89], with open access tutorials in [61] and [80]. 2.1 SN P systems: thebasic model Very briefly, an SN P system includes spike processors (the neurons) as nodes in a directed graph, where edges are called synapses. The neurons communicate using signals known as spikes which are sent to other neurons using synapses. Specifically, an SN P system is a tuple or construct Π=({a},𝜎1,𝜎2,…,𝜎m,syn,in,out) which uses a single symbol a as the spike symbol, each 𝜎i is a neuron, syn ⊂{1, 2 …,m}×{1, 2, …,m} is the set of synapses between the m neurons (a reflexive synapse (i,i) is not allowed in this basic model), with in and out as the labels of the input and output neurons, respectively. Each neuron 𝜎i=(ni,Ri) includes the number ni of spikes contained in 𝜎i , and a rule set Ri . Rules in Ri include spiking rules of the form E∕ a c →a; d where E is a regular expression over {a}, with natural numbers c≥1, d≥0. A spiking rule is enabled if neuron 𝜎i contains ni spikes written as string an i , such that an i ∈L(E), that is, there is a regular set L(E) to “check” if all ani spikes are “covered” in the language L(E). The application or firing of a spiking rule consumes c spikes, so only ni −c spikes remain in 𝜎i , followed by sending of one spike to each 𝜎j such that (i,j)∈syn. Checking, consuming, and firing of spikes are performed in a single step t, except when the delay d≥1 in which the spike is received by “neighbor” neurons at step t+d . From step t until t+d−1 the neuron is closed, inspired by refractory periods of bio-neurons: a closed neuron is waiting to release or fire its spike at step t+d , it cannot apply other rules, and it cannot accept new spikes. Spikes sent to a closed neuron are considered lost or removed from the system. The neuron becomes open at step t+d when new spikes can be received. A rule can be applied next by the open neuron at step t+d+1. The other rule form in rule set Ri is a forgetting rule of the form as → 𝜆. The meaning is that if neuron 𝜎i contains exactly s spikes these spikes are erased or removed from 𝜎i . Note that in the basic model, the enabling of forgetting rules and spiking rules are mutually exclusive, that is, we can have all spikes in 𝜎i as ani enabling either forgetting rules only, or spiking rules only. More specifically, if 𝜎i has a forgetting Thinking aboutspiking neural P systems: some theories, tools, andresearch topics rule as→𝜆 then as∉L(E) for any E of any spiking rule of 𝜎i . It is possible to have more than one rule in a neuron 𝜎i that can be applied, that is, there exist at least two rules with expressions E1 and E2 where L(E1)∩L(E2) ≠ �. In this case, the rule to be applied is chosen nondeterministically. The system Π is globally parallel, that is, all neurons can fire at each step, but Π is locally sequential, that is, at most one rule in each neuron is applied. Π has a global clock to synchronize all neurons, that is, at each step if a neuron can apply a rule then it must do so. A system can have an out (in, respectively) neuron only if acting as a generator (acceptor, respectively), or both in and out neurons as a transducer or device for computing functions. Even with the basic or standard definition of an SN P system in this present section, there are various ways to interpret or obtain the output. Here we mention for now one common way to obtain the output, using a time interval between the first two spikes of the output neuron out to the environment. Visually, the out neuron has a synapse going out of the system given by an arrow that does not point to any neuron. Specifically, if neuron out fires its first and second spikes at steps t and t′ , respectively, the output is said to be the number n=t�−t. Thus, the basic model computes or generates numbers, or sets of numbers in the case of nondeterministic systems. This way of obtaining the output applies whether the system halts or not. A system halts if no more rules can be enabled. 2.2 An example: generating numbers Consider a small example as follows, labeled as SN P system Π1 . Let Π1= ({a},𝜎1,𝜎2,𝜎3,syn,3), where 1. 𝜎1=( 2, {a2∕a → a ;0, a →𝜆 }), that is, neuron 1 initially contains 2 spikes and its rule set R1 contains two rules. 2. 𝜎2=(1, a → a;0, a → a;1), here neuron 2 is the only nondeterministic neuron in Π1∶ both of its rules require exactly one spike a, but one rule has no delay while the other rule has a delay of 1 step when firing its spike. A writing convention is seen here: if E=ac then E∕ac → a;d is written simply as ac → a;d. 3. 𝜎3=( 3, a3 → a ;0, a → a ;1, a2 →𝜆 ), with neuron 3 as the output neuron. 4. syn = {(1, 2),(2, 1),(1, 3),(2, 3)} are synapses of Π1 . We briefly describe the computation of Π1 as follows: Step 1: all 3 neurons can fire, but only neuron 2 has a nondeterministic choice. Neuron 1 must apply its rule a2∕a → a;0 which consumes one spike (only one spike remains in 𝜎1 ) and sends one spike each to 𝜎2 and 𝜎3 . Assume for now that neuron 𝜎2 applies rule a → a;0, no spikes remain in 𝜎2 and one spike each is sent to 𝜎1 and 𝜎3 . Neuron 3 applies rule a3 → a;0 so at step t=1 the output neuron fires its first spike to the environment, consuming all of its spikes. Step 2: 𝜎1 has two spikes in total from step 1, so it must apply the same rule from step 1. At this step, as long as neuron 𝜎2 decides to apply its first rule a → a;0 then 𝜎1 always behaves as in step 1, while 𝜎3 always contains two spikes so it must always apply its forgetting rule a2→𝜆. The output is obtained soon after 𝜎2 decides to apply its second rule. Assume here at step 2 that 𝜎2 applies its second rule instead: it is closed from step 2 until step 2+1−1=2, that is, closed for only one step. Step 3: The spike from 𝜎1 to 𝜎2 in step 2 is lost since 𝜎2 was closed in step 2. Only the forgetting rule a→𝜆 of 𝜎1 can be used so it contains no more spikes. Note that the spike from the opening of 𝜎2 here at step 3 can only be used by 𝜎1 at the next step 4. Neuron 2 is now open and sends one spike each to 𝜎1 and 𝜎3 . Neuron 3 has only one spike from 𝜎1 in step 2 so a → a;1 is applied and 𝜎3 closes. Step 4: neuron 1 applies its forgetting rule, so here both 𝜎1 and 𝜎2 contain no spikes. Neuron 3 fires its second spike to the environment at step t�=4. Π1 halts since no more rules can be applied, with the output of this nondeterministic branch of computation as t�−t=4−1=3. The reader can easily verify that Π1 can generate smaller or larger numbers depending on the step when 𝜎2 decides to apply its second rule. In this way, we say Π1 computes or generates all natural numbers starting from 2, that is, L(Π1)={k∣k ≥ 2}. From the definition in Sect.2.1 and the seminal work in [55], we make a quick note: a minor “bug” in [61] where Π1 appears as Fig.1 to compute the set {k�∣k� ≥1 } , that is, all natural numbers starting from 1. A visual representation of Π1 appears later in Fig.5 of Sect.2.5. 2.3 Normal forms In short, a normal form is a simplifying set of restrictions for a given machine, model, or system. In language theory, a well-known normal form is the Chomsky normal form (in short, CNF) for context-free sets from [29] which says that instead of writing context-free rules in a large number of forms, it is enough to use two forms to describe all such sets. In other words, we maintain the computing power of context-free grammars to generate context-free sets while restricting the forms of rules of such grammars. The CNF is of theoretical and practical use, allowing less complicated proofs and further results. It is thus of interest to apply normal forms to other models, certainly to SN P systems. We set up some basic notations for use in the following sections, although a few more are introduced later when they are more relevant. We note that in the cited works there is not a standard convention, though there are F.G.C.Cabarle many intersections, for the notations that follow. Further details of the following notations, including the preliminaries of computations in SN P systems, can be found in tutorials, monographs, surveys as in [61, 80, 84]. We write as SpikiPj(rulek,consl,forgm,dleyn) the set of numbers computed by the basic model of SN P systems as defined in Sect.2.1. The parameter i∈{2, acc} depends on how the result is obtained: as a number generator the result is the interval between the first two spikes from the output neuron, hence the subscript 2; as a number acceptor, the result is halting the system, given an input as the interval between the first two spikes to the input neuron. The parameters j,k,l,m, and n give the upper bounds for the number neurons, rules, consumed spikes, forgetting rules, and rules with delay, respectively. If the bound is finite but arbitrary we replace the parameter with the ∗ symbol. A D is appended at the start of the set of numbers to emphasize deterministic computation since we assume by default that computations are nondeterministic. Even in the seminal paper of SN P systems in [55], we already have some results on normal forms, though such results were not labeled as “normal form” in the paper. For instance, the set of all finite sets of numbers NFIN can be generated by an SN P system with at most 2 neurons, each neuron with an unbounded number of rules, each rule consuming at most one spike, and without forgetting rules. We can summarize this result as follows: Theorem1 (Ionescu etal 2006 [82] ) We use parameters indi and outdj to mean a neuron has an indegree at most i and an outdegree at most j, respectively. For the indegree of SN P systems, Theorem2 is one result from [81] which limits the indegree of each neuron to at most two incoming synapses. In the sections that follow, we omit from writing some parameters such as ind and outd if such parameters are not the main focus of a work or result. Theorem2 (Păun etal 2006 [81]) An early and explicit work on normal forms for SN P systems is [48]. Results from [48] are about Turing completeness or the characterization of the set of all Turing computable sets of numbers NRE, as follows: Theorem3 (Ibarra etal 2007 [48]) The following sets of numbers are equivalent to NRE: 1. Spik2P∗(rule3,cons4,forg5,dley0,outd∗). 2. DSpik{2,acc}P∗(rule2,cons3,forg2,dley0,outd∗). 3. Spik2P∗(rule3,cons4,forg4,dley0,outd2) . Spik2P1(rule∗,cons1,forg0)=Spik2P2(rule∗,cons∗,forg∗)=NFIN. Spik2P∗(rule2,cons3,forg1,dley1,ind2,outd2)=NRE Fig. 1 A screen of Snapse, showing the software interface, four neurons (boxes labeled with 8, 9, 10, 12), synapses (arrows), a spike (yellow circle between neurons 9 and 10) Thinking aboutspiking neural P systems: some theories, tools, andresearch topics 4. Spik2P∗(rule2,cons3,forg0,dley1,outd2). 5. Spik2P∗(rule2 , cons2 , forg1 , dley2 , outd2) , ai for i≥1 or a+ . Some interesting results in Theorem3 from [48] include: removal of delay, for both the nondeterministic (item 1) and deterministic case (item 2); restricting the outdegree of each neuron (item 3); removal of forgetting rules (item 4); lastly, restricting the type of regular expression (item 5). Notice that results in Theorem3 allow only the removal in a mutually exclusive way for features such as delays, forgetting rules, or simplified regular expressions. In [40], some of these features are combined while maintaining the computing power of SN P systems. For instance, Theorem4 combines simplified regular expressions and removal of delays. Theorem4 (García-Arnau etal 2009 [40]) using rules of the form a∗∕a → a or ar → a, and as→𝜆 for r,s≤3. An open problem from [40] asks if more than two features of the system can be removed simultaneously while maintain Turing completeness. A positive answer to this open problem was provided in [71] as Theorem5. Theorem5 (Pan, Păun 2010 [71]) using only the following types of regular expressions: a(aa)∗ ,a(aaa)∗ , or (a2∪a). What is surprising about Theorem5 is how restricted an SN P system is where rulee k means that each neuron uses at most k rules and at most e types of regular expressions. Theorem5 not only answers an open problem from [40] but also shows how “simple” the neurons can be to be Turing complete: a neuron only needs at most two rules using exactly one of three types of regular expressions, and without delays and forgetting rules. An open problem from [71] is if the system can maintain its computing power with fewer types of regular expressions. Before we proceed to further normal forms for SN P systems, we make a short digression for some variants of SN P systems. SN P systems with anti-spikes or ASN P systems are SN P systems that allow for a second type of spike known as an anti-spike [72]. The idea of an anti-spike in [72] is inspired from the inhibitory features of biological neurons. Using a as the anti-spike symbol, a normal form for ASN P systems include Theorem6 (Theorem1 in [94]). Spik2P∗(rule3,cons3,forg3,dley0,outd2)=NRE Spik2 P ∗ (rule 1 2 ,cons 3 ,forg 0 ,dley 0 ,outd 4 )= NRE Theorem6 (Song etal 2013 [94]) using categories (a,a),(a,a). The feature prulek means each neuron uses at most k “pure” spiking rules of the form E∕bc → b� , and the regular expression E has L(E)=bc for b , b�∈{a , a}. That is, we simply write pure spiking rules as bc→b′ . The feature catl means there are at most l categories in each neuron, from a total of four possible categories: (a,a),(a,a),(a,a),(a,a). For instance, category (a,a) means rules can only consume anti-spikes and produce spikes. Another normal form was introduced in [92] for sequential SN P systems. Sequential SN P systems, as introduced in [49], are systems where neurons cannot operate in parallel. For instance, in the case of max-sequential mode, only the neuron with the most number of spikes can apply a rule: if more than one such neuron exists then one neuron is nondeterministically chosen to apply its rule. A normal form (Theorem3.1) from [92] improves a result in [49] by proving systems in max-sequential mode are Turing complete without forgetting rules, each neuron has exactly one rule, but with the use of delays. The last variant we consider for now are SN P systems with (structural) plasticity or SNPSP systems from [17]. SNPSP systems are inspired by the ability of biological neurons to create new or remove existing synapses. A parameter introduced in [17] was 𝛼∈ {+,−,±,∓} where 𝛼 is used in a new type of rule known as a plasticity rule. For 𝛼=+ , this means that a rule can only create new synapse, up to some number k, while 𝛼=± means a rule can create new synapses in the present step and delete synapses in the next step. A normal form from [91] is Theorem7. Theorem7 (Song, Pan 2015 [91]) Theorem7 has rule4 4 which means each neuron uses at most 4 types of regular expressions and at most 4 rules. Another normal form for SNPSP systems is Theorem8 from [66]. Normal forms for SNPSP systems under max- and min-sequential modes, as in [49], are given in [16]. Theorem8 (Macababayao etal 2019 [66]) where R is the maximum number of subtraction instructions associated with any register of the simulated register machine. Spik2ASNP∗(cat2,prule2,forg0,dley0)=NRE Spik2 SNPSP ∗ (𝛼∈ {+,−},rule 4 4 ,cons 4 ,forg 1 ,dley 0 ,outd 3 )=NRE . Spik2 SNPSP ∗ (𝛼∈ {±},rule 3 3 ,cons 4 ,forg 0 ,dley 0 ,outd R )= NRE F.G.C.Cabarle Explicit in the results of Theorem8 and later results is the idea of a register machine: actually all the results on Turing completeness in this work since the seminal paper in [55] are shown by simulating register machines. In short, register machines are Turing complete models which have finite and rudimentary instructions or programs. Further details on register machines and how SN P systems and variants simulate them are from [61, 84]. Theorem8 uses a single yet “compound” type for 𝛼 , unlike the “simple” types in Theorem7, with the trade-off that Theorem8 has smaller values for other ingredients, including the lack of delays and forgetting rules. Now we return to an open problem from [71]: can fewer than three types of regular expressions allow SN P systems to be Turing complete? That is, can we improve on Theorem5? The belief in [71] was in the negative. An early and positive answer to this open problem comes from improving Theorem8 for SNPSP systems and attempting to apply it to SN P systems, resulting in Theorem9 from [21]. Theorem9 (Macababayao etal 2019 [21]) Using only regular expressions a(aa)∗ or (a2∪a), the following sets of numbers are equivalent to NRE: and Even more interesting are improvements from [67] which lowered the values from [21]. Theorem10 (Macababayao etal 2022 [67]) Using only the regular expression a(aa)∗ , the following sets of numbers are equivalent to NRE: and where R is the maximum number of subtraction instructions associated with any register of the simulated register machine. 2.4 Homogeneous systems Next we look at a notion not too dissimilar to a normal form: the idea of a homogeneous system. At least in the context of SN P systems, such a system is homogeneous Spik2 P ∗ (rule 1 2 ,cons 3 ,forg 0 ,dley 0 ,outd 4 ) , Spik2 SNPSP ∗ (𝛼∈ {±},rule 1 1 ,cons 4 ,forg 0 ,dley 0 ,outd 4 ) . Spik 2P∗(rule 1 2,cons3,forg0,dley0,outd3), DSpikacc P ∗ (rule1 1 ,cons 3 ,forg 0 ,dley 0 ,outd 3 ) , Spik2 SNPSP ∗ (𝛼∈ {±},rule 1 1 ,cons 3 ,forg 0 ,dley 0 ,outd 2R) if every neuron has the same set of rules. The idea of homogeneous SN P system was first introduced in [114]. In some sense a homogeneous system is a “restricted” system, hence the similarity of the notions of normal forms and homogeneous systems. From the perspective of computing, a homogeneous system is where processors, or neurons in the case of SN P systems, are not distinct from each other. In terms of engineering for instance, this can mean that replacing a malfunctioning or damaged processor can be easily done: we simply swap the old processor with a new processor, since both processors are identical. Not only in engineering, such homogeneous systems may also reduce other necessary resources, such as during the design, manufacture, and maintenance of a single kind of processor. From the perspective of the human brain, the neocortex or “new brain” is visually uniform. A natural question for the study of the brain is how various functions, such as perception, language, vision, and so on, arise from such uniformity. A common answer to this question, though much details still elude us, is how neurons are connected: the “size” of each connection, number of connections, "distance” between connections, and so on. Thus, both from the computing and biological perspectives, the idea of homogeneous SN P systems is well-motivated. An early result is that SN P systems as number generators can be Turing machines with the same set of seven rules in every neuron, given by Theorem11 (Theorem4.1 from [114]). Since every neuron or processor in the SN P system has the same set of rules, the “programming” of the system is focused on the topology or connection of the system using synapses, rather than the “details” of each neuron. Theorem11 (Zeng etal 2009 [114]) The following sets of numbers are equivalent to NRE: and The weight5 feature in Theorem11 refers to synapses having integer weights of at most 5, to multiply the number of spikes sent to specific neurons. A way to remove the weight feature, resulting in “standard” synapses of weight of 1, is also included in [114]. As expected, removing the synapse weight increases the number of synapses and outdegree of the system. Further results on homogeneous SN P systems combine ideas from results on normal forms, such as removing delays. In [58] the max-sequential mode is applied to HSN P systems, resulting in Theorem12. Spik2 HSNP ∗ (rule 6 7 ,cons 7 ,forg 1 ,dley 1 ,outd 3 ,weight 5 ) , DSpikacc HSNP ∗ (rule 6 7 ,cons 7 ,forg 1 ,dley 1 ,outd 3 ,weight 6 ) . Thinking aboutspiking neural P systems: some theories, tools, andresearch topics Theorem12 (Jiang etal 2013 [58]) The following sets of numbers are equivalent to NRE: and An interesting result from Theorem12 using max-sequen- tial mode, hence the superscript ms, the delay is removed but values of other ingredients are increased, such as number of rules. An improvement on another computing parameter is given in [76] where they improve on [114] by giving “small” universal systems. We return later to small universal systems, where the main goal of such works is to reduce the number of neurons while maintaining Turing completeness. An interesting extension of Theorem11 is in [64] where the entire system is heterogeneous, but parts of the system are homogeneous. That is, the system is locally homogeneous. Specifically, neurons in modules are homogeneous, where such modules simulate operations of a register machine. Since there are three such operations, there are three sets of rules in [64] summarized in Theorem13. We note in Theorem13 that some of their modules now use negative weights, unlike previous results where the minimum weight is one: weight−2 5 means the minimum weight is -2 while maximum is 5. Theorem13 (Liu, Qi 2016 [64]) Before we end this section, we mention more recent works on homogeneous variants of SN P systems. In [31], homogeneous SNPSP systems were investigated. Each of the neurons in the homogeneous SNPSP systems in [31] uses the same set of 9 rules. In [117], the asynchronous mode is applied to homogeneous SN P systems. In asynchronous mode, a second level of nondeterminism is applied: if a neuron can apply a rule the neuron nondeterministically decides to apply the rule or not. Specifically, local synchronization is used where the entire system is asynchronous but sets of neurons operate in a synchronous manner, similar to the idea of locally homogeneous systems in [64]. Local synchronization is used since the lack of synchronization makes programming the system more difficult, see for instance details in [24]. Using local synchronization, at most 5 rules are required to characterize NRE, with trade-offs including an increase in the outdegree of neurons. An interesting and recent work on homogeneous SN P systems is in [33] which introduces H, a homogenization algorithm for an arbitrary SN P system Π , such that H(Π) = Π� where Π� is a homogeneous SN P system Spikms 2 HSNP ∗ (rule 8 8 ,cons 11 ,forg 1 ,dley 0 ,outd 3 ,weight 5 ) , DSpikms acc HSNP ∗ (rule 8 8 ,cons 7 ,forg 1 ,dley 0 ,outd 3 ,weight 6 ) . Spik2 LocHSNP ∗ (rule 3 4 ,cons 5 ,forg 0 ,dley 0 ,outd 3 ,weight −2 5 )=NRE . computing the same set as Π . Algorithm H requires some reasonable constraints, such as the lack of delay which is known to not affect their computational completeness, see for instance [67]. Using operations such as neuron translation and subsystem scaling, H introduces modifications in the number of spikes, rule sets, synapses, and neurons of Π to produce Π� . 2.5 Visual‑ andweb‑based simulators The practical notion is that of a visual simulator, for now at least, of SN P systems and some variants. At least one way to better engage or increase enthusiasm of new and old researchers is to have tools to allow easy and visual creation and experimentation of theoretical models. One such pedagogical tool for use in automata and grammar models is a well-known tool Java Formal Languages and Automata Package (in short, JFLAP) see for instance [42]. With JFLAP, novice and experts can easily and visually interact with notions from automata and formal languages, such as a step-by-step transformation of a nondeterministic finite automaton to a deterministic one or a regular grammar. For SN P systems, in a similar way, it is useful to have some examples of solutions to problems: applying notions such as normal forms from Sect.2.3 or homogeneous systems in Sect.2.4 can improve appreciation and increase engagement. For instance, an easy access and visual tool to create or interact with SN P systems generating certain languages, accepting numbers, or solving small instances of hard problems. An early attempt for such an easy access and visual tool is Snapse, see Fig.1, in [37] which includes a download link to the tool. Snapse allows a visual and (quite) easy to access tool to experiment with SN P systems, with features such as: animations for the sending of spikes between neurons, pseudorandom mode to simulate nondeterministic application of rules or guided mode to allow users to select which rules to apply, some examples such as a natural number generator, a bit comparator, and a bit adder. While Snapse was tested in a few platforms, such as Windows and Linux computers, it’s ease of access (among other features) can still be improved. Thus, WebSnapse was introduced in [35], see Fig.2, with links in [110]. Web- Snapse is mainly a web browser version of Snapse. WebSnapse allows an improved ease of access since its access is not limited to computers running Windows or Linux. Most modern web browsers, including those found in laptops, tablets, or mobile phones can be used to access WebSnapse. Besides the improved ease of access, WebSnapse includes new features such as the viewing of a computation history of an SN P system: for each time step, WebSnapse records the rules applied for each neuron, and the output of the system. WebSnapse also supports an XML F.G.C.Cabarle Fig. 2 A screen of WebSnapse version 1, showing the software interface, eight neurons, synapses, firing of spikes (arrows with broken lines) Fig. 3 A screen of WebSnapse V2, with some new features: improved access to edit parts (e.g., nodes or neurons, synapses) of the system; a slider at the top right, to change the simulation and animation speed; support for weights on synapse, such as weight of 1 for both synapses in the screen Thinking aboutspiking neural P systems: some theories, tools, andresearch topics file format, inspired by one of the formats of the well-known tool P-Lingua [68]. An improvement is in WebSnapse V2 (that is, version 2) from [30], also shown in Fig.3. In WebSnapse V2, all features of version 1 are included, together with further variants or ingredients, such as multiple input and output neurons, weights on synapses. WebSnapse V2 also includes new simulator features, such as increasing or decreasing the speed of the simulation and animation, improving the ways to create, remove, or edit parts of the system. More recently, a significant improvement is WebSnapse CS (that is, client–server) from [43] and shown in Fig.4. Together with another contemporary variant known as Web- Snapse V3 from [39], WebSnapse CS includes all features of WebSnapse V2 but with improved efficiency from the perspective of the user and from the software developer. For the user: the screen interface can be more intuitive or aligned toward recent software applications especially for mobile devices such as phones and tablets; improved ways to add, edit, or delete neurons, rules, spikes, and synapses using context-aware gestures or clicks of the computer mouse; support for LaTeX-like input of rules and regular expressions; support for improved representation of input or output trains of spikes, such as by writing the string 1 08 10 in the output neuron out in Fig.4 instead of the explicit string 10000000010 from previous versions; support for light and dark themes or modes to improve visibility depending on the screen or device used. For the developer, WebSnapse CS allows easier extension for future variants by improving the separation of the code for the user interface and the “logic” and simulations of the computation. Also, the CS or client–server part of WebSnapse CS allows the logic to be computed in a separate machine from the machine for the user interface (i.e., the computer for viewing the screen): in previous versions both the simulation and user interface can only be done in the same machine or computer; with WebSnapse CS it is possible to use a more powerful computer (e.g., server or workstation computer) to perform the computations of the simulation, while a less powerful computer (e.g., laptop or tablet computer) can be used to view the animations, results. In this way, larger systems can be simulated compared to previous versions of WebSnapse, while maintaining ease of access and visual simulations. WebSnapse is currently used not only for research but also for pedagogy in the research group of the author. For instance, at least in the undergraduate classes CS198 and CS199 for two semesters at the University of the Philippines Diliman, WebSnapse has been used since 2021 for some students to learn about SN P systems or variants. In both classes, undergraduate students are expected to prepare a technical or research project for the degree. The author and some colleagues use WebSnapse, as well as require the software for students focusing on SN P systems or related systems. Our experience is similar with the JFLAP tool for automata and formal languages, where WebSnapse allows improved understanding of SN P systems and their computations. Recalling the example Π1 from Sect.2.2, we see the system depicted in WebSnapse in Fig.5. Fig. 4 A screen of WebSnapse CS using dark theme or mode; at the bottom right is a minimap, a small directed graph to represent the simulated system and its placement in the software canvas; support for LaTeX-like inputs and outputs; support for client–server architecture or style of simulation F.G.C.Cabarle systems and variants is [28]. In recent years, advances in computer vision and natural language processing are in part due to parallel processors such as GPUs, and recent variants of artificial neural networks known as transformers. However, it seems that only exceptionally large companies or organizations can afford the training and maintenance of “large” models of natural languages using transformers and GPUs: the cost of training and maintaining such models, even with a large number of expensive GPUs, can be very prohibitive. Such costs are usually in terms of processor hours, in joules or watts of energy. While many approaches are considered to reduce the costs of training and maintenance of such models, perhaps ideas based on normal or homogeneous forms, certainly the human brain, can further reduce the costs. Besides such issues on software and hardware, much of the modern computers today including parallel processors are based on the von Neumann architecture. Creating hardware and software inspired by the brain is known as neuromorphic computing: unlike the von Neumann architectures which is inspired by the Turing machine, a neuromorphic processor can be analog instead of digital, or use circuit elements other than silicon transistors. Again we ask how ideas on normal or homogeneous forms can aid in the design of such neuromorphic computers, since as late as around the first half of the twentieth century we know how similar and different human brains and digital computers can be [106]. Other models bearing many similarities to SN P systems, certainly in the structural or visual sense, include Boolean circuits and Petri nets. Ideas for Petri nets and SN P systems are mentioned in problem N in [82], with latter works such as [13]. Some recent works involving similar analyses and properties between Petri nets and SN P systems include [3, 4] including formal verification in [41]. Petri nets and Boolean circuits seem to represent many interesting ideas related to normal or homogeneous forms, also similar to SN P systems in [111]. Thus, work between such nets, circuits, SN P systems and variants, are likely to lead to interesting or useful ideas. For instance, the feature of probabilistic or stochastic computation is well-known in Petri nets, but can still be further investigated for SN P systems, with some results in [8, 60, 112]. It is interesting to include a feature in WebSnapse to convert certain classes of SN P systems to Petri nets and vice-versa, including the support of colored spikes inspired by colored tokens [99]. In relating SN P systems to Boolean circuits or similar models, we are likely to have or even require some normal or homogeneous form, to obtain further results on topology of SN P systems. What kinds of languages are computed by such systems with (non)planar and other types (for instance, kite or hammock, simple, acyclic) of graphs? How much time and space are required to solve hard problems using such systems? Further topics for investigations on SN P systems are in [85], with a recent book on applications, software, and hardware of P systems in [116]. 4 Final remarks A survey of results with the length of the present work, even confined on notions such as those in Sect.2 is expected to be incomplete: at least a few previous works on such notions are likely to be missing in this work. For instance, at the time of writing the present work, it is likely that at least some of those mentioned in the list in Sect.3 are already being investigated even if in preliminary form. Besides the results in Sect.2, the problems and topics in Sect.3 are mainly informed with the experience of the author working with SN P systems since around 2011, together with excellent and earlier lists such as [61, 82, 85, 88]. The present work is a brief survey on specific aspects of theory and pedagogy of SN P systems, with some recent surveys including more comprehensive details on theory (see [61]) and applications (see [36]). It is thus recommended to complement ideas from such recent and comprehensive surveys with problems and topics mentioned in Sect.3 of the present work. While the research on SN P systems and variants is quite active, the author and many others, suspect much is still to come in both theory and applications of such systems. We end the present work with the provocative and hopeful last line from [101] by Alan Turing: “We can only see a short distance ahead, but we can see plenty there that needs to be done.” Acknowledgements The author thanks the organizers for the chance to give an invited talk at 11th Asian Conference on Membrane Computing 2022 (ACMC2022) from 6 to 9 September 2022, held virtually (online) and organized at the Department of Computer Science, University of the Philippines Diliman (UPD), Quezon city, Philippines. This talk was presented simultaneously at a joint event, the 23rd International Conference on Membrane Computing (ICMC) at the University of Trieste, Italy. The author likewise thanks the ICMC2022 organizers. Author support provided by the Dean Ruben A. Garcia PCA, and Project No. 222211 ORG from the Office of the Vice Chancellor for Research and Development, both from UPD, Philippines. The author is also supported by QUAL21 008 USE project (PAIDI 2020 and FEDER 2014- 2020 funds). Author contributions Only one author prepared the submitted article. Thinking aboutspiking neural P systems: some theories, tools, andresearch topics Funding Funding for open access publishing: Universidad de Sevilla/ CBUA. Data Availability No datasets were generated or analysed during the current study. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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Springer. 117. Zhang, L., & Xu, F. (2022). Asynchronous homogenous spiking neural P systems with local rule synchronization. Theoretical Computer Science, 926, 51–61. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Francis George C. Cabarle received his PhD in Computer Science in 2015 from the University of the Philippines Diliman, where he is currently an associate professor. He was a visiting researcher with the Research Group of Natural Computing (RGNC) from 2014 to 2015, then headed by Prof. Mario de J. Pérez Jiménez, at the Universityof Seville, Spain. He performed postdoctoral work in 2016 to 2019 with Prof. Xiangxiang Zeng etal at Xiamen University, China, and from 2023 to 2024 with the RGNC now headed by Prof. Agustín Riscos Núñez. His research interests include the intersections of natural computing, computability, and parallel computing.