Sequential Spiking Neural P Systems with Structural Plasticity Based on Max/Min Spike Number

Sequential Spiking Neural P Systems with Structural Plasticity Based on Max/Min Spike Number Francis George C. Cabarle1, Henry N. Adorna1, Mario J. P´erez-Jim´enez2 1Algorithms & Complexity Lab, Department of Computer Science University of the Philippines Diliman Diliman, 1101, Quezon City, Philippines; 2Department of Computer Science and AI University of Sevilla Avda. Reina Mercedes s/n, 41012, Sevilla, Spain email: [email protected], [email protected]d.edu.ph, marp[email protected] Abstract. Spiking neural P systems (in short, SNP systems) are parallel, distributed, and nondeterministic computing devices inspired by biological spiking neurons. Recently, a class of SNP systems known as SNP systems with structural plasticity (in short, SNPSP systems) were introduced. SNPSP systems represent a class of SNP systems that have dynamism applied to the synapses, i.e. neurons can use plasticity rules to create or remove synapses. In this work, we impose the restriction of sequentiality on SNPSP systems, using four modes: max, min, maxpseudo, and min-pseudo sequentiality. We also impose a normal form for SNPSP systems as number acceptors and generators. Conditions for (non)universality are then provided. Specifically, acceptors are universal in all modes, while generators need a nondeterminism source in two modes, which in this work is provided by the plasticity rules. Key words: Membrane computing, Spiking neural P systems, Structural plasticity, Sequential systems, Turing universality 1 Introduction Membrane computing, a branch of natural computing, aims to abstract ideas from living cells for computing use [25]. More recently, the actively investigated area of spiking neurons1was introduced in membrane computing as spiking neural P (in short, SNP) systems [8]. SNP systems consist of neurons (often drawn as ovals), and synapses between neurons (arcs between ovals). Neurons are placed in vertices of a directed graph: they process only one type of indistinct signal called a spike (denoted by the symbol a) by firing or applying spiking rules. SNP systems use time to encode information using spikes, e.g. the value n=t0−t 1See e.g. [13] and [14] and references therein. 2 F.G.C. Cabarle, H.N. Adorna, M.J. P´erez-Jim´enez is computed by neuron σ1if σ1produced consecutive spikes at steps tand t0, where t6=t0, t0> t. Time therefore plays a crucial role in information processing, and is not merely a background for computation. Many theoretical results on computability or computational complexity have been investigated in SNP systems2e.g., SNP systems are known to be Turing universal as: number generating or accepting devices in [8] and [28]; language generating devices in [5] and [21]; function computing devices in [21] and [24]. SNP systems have also been used to efficiently solve computationally hard problems in [12] and [22], trading space for time. SNP systems often use parallel, nondeterministic, or synchronous features in their computations. These features are powerful “ingredients” for achieving Turing universality, and many works (with biological inspiration) have investigated restrictions by removing at least one of these features, e.g. deterministic SNP systems have been considered since [8], asynchronous SNP systems as in [4], [6], and [30], and sequential SNP systems as in [6], [7], [10], and [31]. In SNP systems, neurons apply their rules in parallel but at most one rule is applied once in each neuron. Other ways of applying rules have also been considered, such as in [9][21] where rules were applied in a maximally parallel (i.e. exhaustive) way. A generalized rule application was introduced in [35], where neurons can nondeterministically choose the number of times a rule is applied. More neuroscience ideas and motivations have also been included, hence many variants of SNP systems have been introduced, e.g. the use of weighted synapses [20], neuron division and budding [22], the use of astrocytes [23], and anti-spikes [18]. Other biologically inspired restrictions imposed on SNP systems and its variants are the notions of simplicity and homogeneity: a neuron is simple if it precisely has one rule; an SNP system is simple if all neurons are simple; an SNP system is homogeneous if all of its neurons precisely have the same set of rules; Simple SNP systems usually require more neurons to achieve universality due to the simplicity of their neurons [31][32]. Homogeneous SNP systems can become more “compact”, i.e. they can require less neurons, due to the fact that they can have more than one rule in a neuron as in [11] and [33]. Both notions can have interesting biological and computational interpretations, for theoretical and practical use: the connectivity of the neurons in the system is important for achieving a certain level of computing power; also, neurons do not need to be very complex (in the case of simple systems) or do not need to be very varied (in the case of homogeneous systems). Another restriction imposed on SNP systems and its variants is the type(s) of rule(s) present in a neuron: standard (spiking) rules (at each step, at most one spike is sent from a σ1to a σ2); extended (spiking) rules (at each step, more than one spike can be sent from a σ1to a σ2); spiking rules with delay (if σ1sends a spike at step t,σ2receives the spike at t+d, d ≥1); forgetting rules (rules that remove spikes from the system). Extended rules allow for more compact systems in terms of neuron count, due to the ability to produce more than one spike each step as in [5] and [24]. Both forgetting rules and rules with delays were used 2An overview in [29] and the SNP systems chapter in [28]. Sequential SNPSP Systems Based on Max/Min Spike Number 3 in [8], while in [19] for example, it was shown that universality is still achieved given the normal form of having no forgetting rules and rules with delays. The SNP system variants we consider in this work are SNP systems with structural plasticity, in short, SNPSP systems.3Structural plasticity refers to any changes in the connectivity between neurons, i.e. synapses can be created or removed. Synapses can also be rewired, i.e. if synapse (a, b) exists between neurons σaand σb,σacan remove (a, b) and create novel synapse (a, c) to neuron σcinstead. SNPSP systems also provide a response to an open problem in [29] where “dynamism” is only applied to synapses, in contrast to both neuron and (implicit) synapse dynamism for example in [22]. SNPSP systems represent the class of SNP systems that explicitly focus on synapse graph dynamism only, since the collection of neurons in the system remains static. Other than standard rules, the only other type of rule in SNPSP systems are plasticity rules: rules that allow neurons to create, remove, or rewire synapses. In this work, the restriction we impose on SNPSP systems is sequentiality. Specifically, and as in [7], we induce sequential operation based on the number of spikes stored in a neuron. For example, in the max sequentiality or maxs mode, if neurons σa,σb, and σccontain 1, 3, and 2 spikes at the same step respectively, only σbis allowed to apply its rule. If however σcalso stored 3 spikes, only one among σbor σcwill apply its rule (nondeterministically chosen). Our results also exhibit (biologically and computationally) two interesting features, also known as a normal form, as in [19] and [31]: a neuron has two rules (the maximum per neuron) if and only if it is purely plastic, i.e. only contains plasticity rules; if a neuron has a standard rule, then it is simple. These two features are interesting because while certain biological neurons seem to have more specific or complex functions (e.g. they create or remove synapses), other neurons seem more simple or generic (e.g. they simply function as spike repositories or relays). More related normal forms in this work include: almost simple SNP or ASSNP systems in [31], where the system has only one neuron that is not simple; simple and homogeneous SNP systems with astrocytes or SHSNPA systems in [23], where SHSNPA systems maintained universality despite being simple and homogeneous, due to the use of additional neuroscience structures (the astrocytes). Another distinction of SNPSP systems, again biologically and computationally motivated, is an alternative source for nondeterminism. A more common source of nondeterminism in SNP systems is rule-level (in short, ndrule): if more than one rule can be applied in a step, only one is chosen nondeterministically. In SNPSP systems, there is instead synapse-level nondeterminism (in short, ndsyn): selecting which synapses a neuron will create or remove. Our results show the conditions, with or without ndsyn (among other levels, except ndrule) by which our systems become (non)universal. Our systems, under the normal form mentioned above, are investigated under four modes: max, min, max-pseudo, and min-pseudo sequentiality. We organize this work as follows: in Section 2 we provide some prerequisites for our results; Section 3 introduces the syntax and semantics of SNPSP systems; 3Introduced in [1] and improved and extended in [2]. 4 F.G.C. Cabarle, H.N. Adorna, M.J. P´erez-Jim´enez our (non)universality results for the four modes are presented in Section 4; We provide final remarks and further directions in Section 5. 2 Preliminaries It is assumed that the readers are familiar with the basics of membrane computing4and formal language theory (available in many monographs). We only briefly mention notions and notations which will be useful throughout the paper. Let Vbe an alphabet, V∗is the set of all finite strings over Vwith respect to concatenation and the identity element λ(the empty string). The set of all non-empty strings over Vis denoted as V+,so V+=V∗− {λ}. If V={a}, we simply write a∗and a+instead of {a}∗and {a}+. If a∈V, we write a0=λ, and we write the language associated with a regular expression Eas L(E). In proving computational universality, we use the notion of register machines. A register machine is a construct M= (m, I, l0, lh, R), where mis the number of registers, Iis a finite set of instruction labels, l0is the start label, lhis the halt label, and Ris a finite set of instructions. Every label li∈Iuniquely labels only one instruction in R. Instructions in Rhave the following forms, given the value nstored in register r: –li: (ADD(r), lj, lk), increase nby 1, then nondeterministically go to either lj or lk; –li: (SUB(r), lj, lk), if n≥1,then subtract 1 from nand go to lj, otherwise perform no operation on rand go to lk; –lh:HALT, the halt instruction. We say Mcomputes or generates a number nas follows: Mstarts with all its registers empty. The register machine then applies its instructions starting with the instruction labeled l0. Without loss of generality, we assume that l0labels an ADD instruction, and that the content of the output register (e.g. register 1) is never decremented, only added to during computation, i.e. no SUB instruction is applied to it. If Mreaches the halt instruction lh, then the number n stored during this time in register 1 is said to be computed by M. We denote the set of all numbers computed by Mas N(M). It was proven that register machines compute all sets of numbers computed by a Turing machine, therefore characterizing the set NRE [17]. Register machines can also work in an accepting mode: a number nis stored in register 1 of M(all other registers are empty). If Mhalts, given this initial configuration, then nis said to be accepted or computed by M. It is also known that if Mis deterministic, where an ADD instruction is simply written as li: (ADD(r), lj), Mstill characterizes NRE. As a convention, when comparing the power of two number generating devices D1and D2, the number zero is ignored, i.e. N(D1) = N(D2) if and only if N(D1)−{0}=N(D2)−{0}. This convention 4A good introduction is [26] and the P systems webpage at http://ppage.psystems. eu/, with a handbook in [28]. Sequential SNPSP Systems Based on Max/Min Spike Number 5 corresponds to the common practice in language and automata theory to ignore the empty string. 3 Spiking neural P systems with structural plasticity In this section we define SNP systems with structural plasticity. Initial motivations and results are included in the seminal paper in [8]. A spiking neural P system with structural plasticity (SNPSP systems) of degree m≥1 is a construct of the form Π= (O, σ1, . . . , σm, syn, out), where: –O={a}is the singleton alphabet (ais called spike); –σ1, . . . , σmare neurons of the form (ni, Ri),1≤i≤m;ni≥0 indicates the initial number of spikes in σi;Riis a finite rule set of σiwith two forms: 1. Spiking rule: E/ac→a, where Eis a regular expression over O,c≥1; 2. Plasticity rule: E/ac→αk(i, N), where Eis a regular expression over O,c≥1, α∈ {+,−,±,∓},k≥1, and N⊆ {1, . . . , m}−{i}; –syn ⊆ {1, . . . , m} × {1, . . . , m}, with (i, i)/∈syn for 1 ≤i≤m(synapses between neurons); –out ∈ {1, . . . , m}indicates the output neuron. Given neuron σi(we also say neuron ior simply σi) we denote the set of neuron labels with σias their presynaptic (postsynaptic, resp.) neuron as pres(i), i.e. pres(i) = {j|(i, j)∈syn}(as pos(i) = {j|(j, i)∈syn}, resp.). Spiking rule semantics in SNPSP systems are similar with SNP systems in [8]. We do not use forgetting rules or rules with delays. Spiking rules are applied as follows: If neuron σicontains bspikes and ab∈L(E), with b≥c, then a rule E/ac→a∈Ri can be applied. Applying such a rule means consuming cspikes from σi, thus only b−cspikes remain in σi. Neuron isends one spike to every neuron with a label in pres(i) at the same step as rule application. If a rule E/ac→ahas L(E) = {ac}, we simply write this as ac→a. Plasticity rules are applied as follows. If at step twe have that σihas b≥c spikes and ab∈L(E), a rule E/ac→αk(i, N)∈Rican be applied. The set Nis a collection of neurons to which σican create a synapse to, or remove a synapse from, using the applied plasticity rule. The rule consumes cspikes and performs one of the following, depending on α: –If α:= + and N−pres(i) = ∅, or if α:= −and pres(i) = ∅, then there is nothing more to do, i.e. cspikes are consumed but no synapses are created or removed. Notice that with these semantics, a plasticity rule can replace a forgetting rule, i.e. the former can be used to consume spikes without producing any spike. –for α:= +,if |N−pres(i)| ≤ k, deterministically create a synapse to every σl,l∈Nj−pres(i). If however |N−pres(i)|> k, nondeterministically select kneurons in N−pres(i), and create one synapse to each selected neuron. –for α:= −,if |pres(i)| ≤ k, deterministically delete all synapses in pres(i). If however |pres(i)|> k, nondeterministically select kneurons in pres(i), and delete each synapse to the selected neurons. 6 F.G.C. Cabarle, H.N. Adorna, M.J. P´erez-Jim´enez If α:= ±(α:= ∓, respectively), create (delete, resp.) synapses at step tand then delete (create, resp.) synapses at step t+ 1. Only the application priority of synapse creation or deletion is changed, but the semantics of synapse creation and deletion remain the same as when α∈ {+,−}. Neuron ican receive spikes from tuntil t+ 1, but σican only apply another rule at time t+ 2. An important note is that for σiapplying a rule with α∈ {+,±,∓}, creating a synapse always involves a sending of one spike when σiconnects to a neuron. This single spike is sent at the time the synapse creation is applied, i.e. whenever synapse (i, j) is created between σiand σjduring synapse creation, we have σi immediately transferring one spike to σj. Let tbe some step during a computation: we say σiis activated at step tif there is at least one r∈Rithat can be applied (i.e. the stored spikes in σisatisfy the regular expression of r); we say σiis active at tif for some step t0< t,σiwas activated by applying r∈Ri, and is currently fulfilling the requirements for the application of r. Also, we say σiis simple if |Ri|= 1. Given an SNPSP system Π, we can have the following levels of nondeterminism: system-level, if given at least two activated neurons, Πchooses exactly one neuron to fire; rule-level, if at least one neuron has at least two rules with regular expressions E1and E2 such that E16=E2and L(E1)∩L(E2)6=∅;synapse-level, if at least one neuron can nondeterministically create or delete a synapse; By default SNP and SNPSP systems are locally sequential (at most one rule is applied per neuron) but globally parallel (all activated neurons must apply a rule). Note that the application of rules in neurons are synchronized, i.e. a global clock is assumed and if a neuron can apply a rule then it must do so. A configuration of an SNPSP system is based on (a) distribution of spikes in neurons, and (b) neuron connections based on syn. For some step t, we can represent: (a) as hs1, . . . , smiwhere si,1≤i≤m, is the number of spikes contained in σi; for (b) we can derive pres(i) and pos(i) from syn, for a given σi. The initial configuration therefore is represented as hn1, . . . , nmi, with the possibility of a disconnected graph. A computation is defined as a sequence of configuration transitions, from an initial configuration, and following rule application semantics. A computation halts if the system reaches a halting configuration, i.e. a configuration where no rules can be applied. A result of a computation can be defined in several ways in SNP systems literature. For sequential SNPSP systems, we use the following as in [11]: we only consider the first two consecutive steps t1and t2that σout spikes. Their difference minus one, i.e. the number n= (t2−t1)−1, is said to be computed by Π. We refer to Πas generator, if Πcomputes in this manner. Πcan also work as an acceptor, as follows: nspikes are stored in a defined neuron in an initial configuration. Πthen accepts nif the computation halts. The following two features are used in our systems as the normal form: (i) only purely plastic neurons (i.e. neurons with plasticity rules only) have at most 2 rules (the maximum in any neuron of the system), and (ii) neurons with standard rules are simple. We denote the family of sets computed by nondeterministic SNPSP systems (in the mentioned normal form) as gen- Sequential SNPSP Systems Based on Max/Min Spike Number 7 erators as N2,genSNPSPγ: subscript 2 indicates the first 2 spikes of σout as the result; we replace 2 and gen with acc for acceptors; we have the mode γ∈ {maxs, maxps, mins, minps}(details given shortly); Given an SNPSP system Π, we denote the computed set by Πas Nm(Π), m∈ {gen, acc}. To further parametrize our results in this work we have the following: +synk(−synj, resp.) where at most k(j, resp.) synapses are created (deleted, resp.) each step; ndβ, β ∈ {syn, rule, sys}indicate additional levels of nondeterminism source; rulemindicates at most mrules (either standard or plasticity) per neuron; Since our results for kand jfor +synkand −synjare equal, we write them instead in the compressed form ±synk, where ±in this sense is not the same as when α:= ±. Induced sequentiality is as follows: In one step of an SNPSP system Π, only the neuron with the maximum (minimum, resp.) number of stored spikes becomes activated in maxs (mins, resp) mode. If there is more than one such neuron, then exactly one is chosen among them (i.e. ndsys is used). In max pseudosequential or maxps (min pseudosequential or minps, resp.) mode, Π allows all neurons with the maximum (minimum, resp.) stored spikes to become activated (i.e. no ndsys). In all four modes, the system is sequential, induced by the global maximum (minimum, resp.) spike number. The ndsys was used in [7] (denoted as nondeterminism at the level of the system) and [10]. In [7], they denoted ndrule as nondeterminism at the level of neurons. Note that if γ for example takes the value maxs,ndsys is implied so that ndsys is omitted from writing.              j a→a k a→a la a→a ia a→ ±1(i, {j, k}) Fig. 1. An SNPSP system Πej . To illustrate the notions and semantics in SNPSP systems, we take as an example the SNPSP system Πej in Figure 1, and describe its computations. The initial configuration is as follows: spike distribution is h1,0,0,1ifor the neuron order σi,σj,σk,σl, respectively; syn ={(j, k),(k, l)}; output neuron is σl, indicated by the outgoing synapse to the environment. Let tbe the step when σiand σlbecome activated, i.e. they can apply their rules because the regular expressions are satisfied. At tthen we have the following: σlapplies its spiking rule and sends the first (out of two) spike to the environment; since k= 1 <|{j, k}|,σinondeterministically selects whether to create synapse (i, j) or (i, k); if (i, j) ((i, k), resp.) is created, a spike is sent from σito σj(σk, resp.) at tdue to the immediate sending of a spike during synapse creation, and σlfires for the second and final time at t+ 3 (t+ 2, resp.); also, if (i, j) is created then syn0:= syn ∪ {(i, j)}, otherwise syn00 := syn ∪ {(i, k)}. At t+ 1, σideletes the synapse created at t(since α:= ±), and we have syn again. By definition, σiand σlbecome activated at t, while only σiis active at 8 F.G.C. Cabarle, H.N. Adorna, M.J. P´erez-Jim´enez t+ 1 (while it deletes a synapse). The output of Πej is either (t+ 3 −t)−1,or (t+ 2 −t)−1, so that N2(Πej) = {1,2}. Computations of Πej generating {1} and {2}are given in Table 1 and Table 2, respectively, where (!) indicates the spiking of the output neuron. time σiσjσkσlsyn t01 0 0 1 syn t0 0 1 0(!) syn00 =syn ∪ {(i, k)} t+ 1 0 0 0 1 syn t+ 2 0 0 0 0(!) syn Table 1. Computation of Πej generating {1}, with syn ={(j, k),(k, l)}. time σiσjσkσlsyn t01 0 0 1 syn t0 1 0 0(!) syn0=syn ∪ {(i, j)} t+ 1 0 0 1 0 syn t+ 2 0 0 0 1 syn t+ 3 0 0 0 0(!) syn Table 2. Computation of Πej generating {2}, with syn ={(j, k),(k, l)}. 4 Main results In this section we consider SNPSP systems as acceptors or generators of sets of numbers in four modes: maxs,maxps,mins, and minps. It is known from [2] that SNPSP systems as generators or acceptors are universal, operating in the “usual way” in membrane computing, i.e. neurons operate in parallel. Using the four modes, we show in this section that we can still achieve universality despite the restriction of induced sequentiality and the normal form in Section 3. 4.1 Sequential SNPSP systems based on max spike number We first consider acceptors and generators in maxs and maxps modes: in maxs mode, at most one neuron with the most stored spikes is nondeterministically chosen to fire; in maxps mode, all neurons with the most number of spikes are allowed to fire. Our (non)universality results under a normal form are as follows (with details of the parameters provided shortly): for maxs, acceptors and generators are universal; for maxps, acceptors are universal while generators can become universal with nondeterminism provided in this work by ndsyn. Sequential SNPSP Systems Based on Max/Min Spike Number 9 Max sequentiality (maxs) mode Theorem 1 NRE =N2,genSNPSPmaxs(rule2,±synk, ndsyn), k ≥1. Proof. It is enough to simulate a register machine Mby means of an SNPSP system Π, with restrictions given in the Theorem statement. Before we construct Π, we provide a general description of the computation as follows: each register rin Mis associated with a σrin Π. If rstores the value n, then σrstores 2n+ 2 spikes. We will construct modules for Πthat will simulate addition, subtraction, and halting operations in M. If a rule with label liis applied in M, then the associated neuron σliin Πis activated in order to simulate the operation performed by li. In the initial configuration, all neurons are empty, except neuron σl0which contains one spike and is associated with the initial instruction l0of M. Module ADD: The module simulating li: (ADD(r), lj, lk) is shown in Figure 2. Once σliis activated it sends one spike each to σl1 iand σl2 i. At this step σl1 i stores two spikes, while σl2 istores one spike. Let step tbe the step where σl1 i is activated: σl1 iapplies its only rule so it consumes one spike and deletes the synapse (l1 i, r) if it exists. At step t+ 1 we have the following: σl1 ibecomes an active neuron to continue the application of its plasticity rule applied at t, while σl2 iis now the only activated neuron. At t+ 1, σl2 isends one spike each to σr and σp, while σl1 icreates synapse (l1 i, r), thus sending one spike to σr. The spikes stored in σrat t+ 1 increase by two, and σpis activated at the next step. We omit further details of σrat this point, but we note that no rule of σris activated while σrstores an even number of spikes. Since σpis activated, it will nondeterministically select whether to create synapse (p, lj) or (p, lk). Once σpselects one synapse to create, either σljor σlkis activated. The ADD module correctly simulates the addition operation: the spikes stored in σrare increased by two, simulating the increase of the value stored in rby one; then, either σljor σlkis activated nondeterministically, simulating the nondeterministic application of either ljor lkin M.                     @@ @R           ) ? li a→a l1 i a a2/a → ∓1(l1 i,{r})l2 i a→a rpa→ ±1(p, {lj, lk}) ljlk Fig. 2. Module ADD simulating li: (ADD(r), lj, lk) in the proof of Theorem 1. 16 F.G.C. Cabarle, H.N. Adorna, M.J. P´erez-Jim´enez Min pseudo-sequentiality (minps) mode Without a source of nondeterminism, deterministic SNPSP system as generators in minps mode are not universal, i.e. NRE 6=DgenSNPSPminps(rule∗,±syn∗). This nonuniversality parallels the nonuniversality of deterministic generators in maxps mode. As in Theorem 4 for generators in maxps mode, generators in minps mode can become universal using ndsyn, as the next result shows. Theorem 8 NRE =N2,genSNPSPminps(rule2,±synk, ndsyn), k ≥2. Proof. We modify the ADD module in Figure 6, to remove ndsys and maintain only ndsyn, as follows: Replace σp1and σp2with a single neuron σp, and add synapse (l1 i, p); the rule set of σpis Rp={a→ ±1(p, {lj, lk})}. The SUB and FIN modules in Figure 7 and 8, respectively, can be used in minps mode. In particular, the SUB module is still correct because if σl2 i(σl3 i, resp.) contains two spikes, then σl3 iand every neuron with a label in Nj− {l2 i} (σl2 iand every neuron with a label in Nk− {l3 i}, resp.) contain one spike each. The neurons that contain one spike each must remove their spikes in parallel, before the next instruction is correctly simulated. ut Theorem 9 NRE =NaccSNP SPminps(rule2,±synk), k ≥2. Proof. All modules in the proof of Theorem 7 can be used in minps mode. ut 5 Final remarks We have shown that SNPSP systems as acceptors are more powerful than as generators, since the former do not need any source of nondeterminism. SNPSP systems maintain Turing universality despite the following restrictions: induced sequentiality, and under a normal form. The normal form is that purely plastic neurons have at most two rules (the maximum in the system), and neurons with standard rules are simple. Acceptors are universal in all four modes, while generators are not universal under maxps and minps modes. Replacing the commonly used ndrule (e.g. in [8] and [7]) with ndsyn as a nondeterminism source, we are able to: allow generators in maxps or minps mode to become universal; reduce the rules in each neuron to at most two, while making the modules more compact, i.e. require fewer neurons. Our results provide a family of uniform modules, i.e. at the expense of using ndsyn, our modules can be used in both maxs and maxps modes, or mins and minps modes. In [10], the problem of constructing a family of uniform ADD, SUB, and FIN modules that can be used in both mins and minps modes was open. Our work thus provides a positive hint to this open problem. The systems in [10] only contain spiking and forgetting rules, hence they use ndsys or ndrule instead of ndsyn as nondeterminism sources. Meanwhile, the sequential systems in [34] are investigated in the same four modes as in this work but with a different rule application semantic: instead of applying at most one rule once each step Sequential SNPSP Systems Based on Max/Min Spike Number 17 in a neuron (the common way of applying rules in SNP systems), the rules are applied in an exhaustive or maximally parallel way. It is also interesting to consider the notion of homogeneous systems. In [11], two types of neurons are enough for universality: one for maxs, and another for maxps. It is then proved that homogeneous SNP (HSNP) systems as acceptors and generators are universal in both modes. Since neurons are homogeneous, the connectivity of the neurons is important. For SNPSP systems, perhaps a pseudohomogeneous construction is reasonable, where only the set Nis different among the rules of neurons. Another restriction we investigate next, and which is related to sequentiality, is the removal of synchronization, i.e. asynchronous SNPSP systems [3]. However, much is left to be investigated for SNPSP systems. We here only list a few established results in membrane computing and SNP systems which can be applied to SNPSP systems: investigating the type of regular expression in standard or plasticity rules, e.g. bounded, unbounded, or general rules as in [4] and [30]; constructing small universal systems as in [24] and [32]; language generation as in [5]; processing (in)finite sequences as in [27]; exhaustive use of rules as in [21]. Further investigations on the structure or connectivity on SNP and SNPSP systems are also interesting. In [36] for example, a neural-like P system known as Axon P systems are proven to be universal despite having only a linear arrangement of the nodes. Such investigations can lead to further interesting results for building theoretical or practical neural systems with dynamic synapse graphs. Lastly, it is interesting to include SNPSP systems in the P-Lingua tool as in [15][16], which includes simulators for many SNP system variants. 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