A First Model for Hebbian Learning with Spiking Neural P Systems

A First Model for Hebbian Learning with Spiking Neural P Systems Miguel A. Guti´errez-Naranjo, Mario J. P´erez-Jim´enez Research Group on Natural Computing Department of Computer Science and Artificial Intelligence University of Sevilla Avda. Reina Mercedes s/n, 41012, Sevilla, Spain E-mails: {magutier,marper}@us.es Summary. Spiking neural P systems and artificial neural networks are computational devices which share a biological inspiration based on the transmission of information among neurons. In this paper we present a first model for Hebbian learning in the framework of Spiking Neural P systems by using concepts borrowed from neuroscience and artificial neural network theory. 1 Introduction When an axon of cell Ais near enough to excite cell Bor repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased. D. O. Hebb (1949) [13] Neuroscience has been a fruitful research area since the pioneering work of Ram´on y Cajal in 1909 [22] and after a century full of results on the man and the mind, many interesting questions are today open problems. Two of such problems of current neuroscience are the understanding of neural plasticity and the neural coding. The first one, the understanding of neural plasticity, is related to the changes in the amplitude of the postsynaptic response to an incoming action potential. Electrophysiological experiments show that the response amplitude is not fixed over time. Since the 1970’s a large body of experimental results on synaptic plasticity has been accumulated. Many of these experiments are inspired by Hebb’s postulated (see above). In the integrate-and-fire formal spiking neuron model [9] and also in artificial neural networks [12] is usual to consider a factor was a measure of the efficacy of the synapse from neuron to another. 212 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez The second one, the neural coding, is related to the way in which one neuron sends information to other ones. It is interested on the information contained in the spatio-temporal pattern of pulses and on the code used by the neurons to transmit information. This research area wonders how other neurons decode the signal or if the code can be read by external observers and understand the message. At present, a definite answer to these questions is not known. The elementary processing units in the central nervous system are neurons which are connected to each other in an intricate pattern. Cortical neurons and their connections are packed into a dense network with more than 104cell bodies per cubic millimeter. A single neuron in a vertebrate cortex often connects to more than 104postsynaptic neurons. The neuronal signals consist of short electrical pulses (also called action potentials or spikes) and can be observed by placing a fine electrode close to the soma or axon of a neuron. The junction between two neurons is a synapse and it is common to refer to the sending neuron as a presynaptic cell and to the receiving neuron as the postsynaptic cell. Since all spikes of a given neuron look alike, the form of the action potential does not carry any information. Rather, it is the number and the timing of spikes which matter. Traditionally, it has been thought that most, if not all, of the relevant information was contained in the mean firing rate of the neuron. The concept of mean firing rates has been successfully applied during the last 80 years (see, e.g., [18] or [14]) from the pioneering work of Adrian [1, 2]. Nonetheless, more and more experimental evidence has been accumulated during recent years which suggests that a straightforward firing rate concept based on temporal averaging may be too simplistic to describe brain activity. One of the main arguments is that reaction times in behavioral experiment are often too short to allow long temporal averages. Humans can recognize and respond to visual scenes in less than 400ms [24]. Recognition and reaction involve several processing steps from the retinal input to the finger movement at the output. If at each processing steps, neurons had to wait and perform a temporal average in order to read the message of the presynaptic neurons, the reaction time would be much longer. Many other studies show the evidence of precise temporal correlations between pulses of different neurons and stimulus-dependent synchronization of the activity in populations of neurons (see, for example, [5, 11, 10, 6, 23]). Most of these data are inconsistent with a concept of coding by mean firing rates where the exact timing of spikes should play no role. Instead of considering mean firing rates, we consider the realistic situation in which a neuron abruptly receives an input and for each neuron the timing of the first spike after the reference signal contains all the information about the new stimulus. Spiking neural P systems (SN P systems, for short) were introduced in [15] with the aim of incorporating in membrane computing1ideas specific to spike-based 1The foundations of membrane computing can be found in [20] and updated bibliography at [25]. A First Model for Hebbian Learning with SN P Systems 213 neuron models. The intuitive goal was to have a directed graph were the nodes represent the neurons and the edges represent de synaptic connections among the neurons. The flow of information is carried on the action potentials, which are encoded by objects of the same type, the spikes, which is placed inside the neurons and can be sent from presynaptic to postsynaptic neurons according to specific rules and making use of the time as a support of information. This paper is a first answer to the question proposed by Gh. P˘aun in [21] related to link the study of SN P systems with neural computing and as he suggests, the starting point has been not only neural computing, but also recent discoveries in neurology. The paper is organized as follows: first we discuss about SN P systems with input and delay and a new computational device called Hebbian SN P system unit is presented. In section 3 we present our model of learning with SN P systems based on Hebb’s postulate. An illustrative experiment carried out with the corresponding software is shown in section 4. Finally, some conclusions and further discussion on some topics of the paper are given in the last section. 2 SN P Systems with Input and Decay An SN P system consists of a set of neurons placed in the nodes of a directed graph and sending signals (called spikes) along the arcs of the graph (called synapses). The objects evolve according to a set of rules (called spiking rules). The idea is that a neuron containing a certain amount of spikes can consume some of them and produce other ones. The produced spikes are sent (maybe with a delay of some steps) to all neurons to which a synapse exists outgoing from the neuron where the rule was applied. A global clock is assumed and in each time unit each neuron which can use a rule should do it, but only (at most) one rule is used in each neuron. One of the neurons is considered to be the output neuron, and its spikes are also sent to the environment (a detailed description of SN P systems can be found in [21] and the references therein). In this section we introduce the Hebbian SN P system unit which is an SN P system with m+ 1 neurons (mpresynaptic neurons linked to one postsynaptic neuron) endowed with input and decay. At the starting point all the neurons are inactive. At rest, the membrane of biological neurons has a negative polarization of about −65mV , but we will consider the inactivity by considering the the number of spikes inside the neuron is zero. The dynamics of a Hebbian SN P system unit is quite natural. At the starting point, all neurons are at rest and in a certain moment the presynaptic neurons receive spikes enough to activate some rules. The instant of the arrival of the spikes can be different for each presynaptic neuron. These spikes activate one rule inside the neurons and the presynaptic neurons send spikes to the postsynaptic neuron. In the postsynaptic neuron a new rule can be triggered or not, depending on the arrival of spikes and it may send a spike to the environment. 214 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez 2.1 The Input The basic idea in SN P systems taken from biological spiking neuron models is the codification of the information in time. The information in a Hebbian SN P system unit is also encoded in the time in which the spikes arrive to the neuron and the time in which the new spikes are emitted. The input will be also encoded in time. The idea behind this codification is that the presynaptic neurons may not be activated at the same moment. If we consider a Hebbian SN P system unit as part of a wide neural network, it is quite natural to think that the spikes will not arrive to the presynaptic neurons (and consequently, their rules are not activated) at the same time. In this way, if we consider a Hebbian SN P system unit with mpresynaptic neurons {u1, . . . , um}, an input will consist of a vector ~x ={x1, . . . , xm}of non-negative integers where xirepresents the time unit of the global clock in which the neuron uiis activated2. 2.2 The Decay The effect of a spike on the postsynaptic neuron can be recorded with an intracellular electrode which measures the potential difference between the interior of the cell and its surroundings. Without any spike input, the neuron is at rest corresponding to a constant membrane potential. After the arrival of the spike, the potential changes and finally decays back to the resting potential. The spikes, have an amplitude of about 100mV and typically a duration of 1-2 ms. This means that if the total change of the potential due to the arrival of spikes is not enough to activate the postsynaptic neuron, it decays after some milliseconds and the neuron comes back to its resting potential (see Fig. 1). This biological fact is not implemented in current SN P systems, where the spikes can be inside the neuron for a long time if they are not consumed by any rule. In the Hebbian SN P system unit, we introduce the decay in the action potential of the neurons. When the impulse sent by a presynaptic neuron arrives to the postsynaptic neuron, if it is not consumed for triggering any rule in the postsynaptic neuron it decays and its contribution to the total change of potential in the postsynaptic neuron decreases with time. This decayed potential is still able to contribute to the activation of the postsynaptic rule if other spikes arrive to the neuron and the addition of all the spikes trigger any rule. If this one does not occur, the potential decays and after a short time the neuron reaches the potential at rest. Figure 2 shows a scheme in which two presynaptic neurons send two spikes each of them at different moments to a postsynaptic neuron. Figure 3 shows the changes of potential in the postsynaptic neuron till reaching the threshold for firing a response. In order to formalize the idea of decay in the framework of SN P systems we introduce a new type of extended rules: the rules with decay. They are rules of the form 2In Section 5 we discuss about other codings for the input. A First Model for Hebbian Learning with SN P Systems 215 Fig. 1. Dynamics of one spike Fig. 2. Two presynaptic and one postsynaptic neuron E/ak→(ap, S); d where, Eis a regular expression over {a},kand pare natural numbers with k≥p≥0, d≥0 and S= (s1, s2, . . . , sr) is a finite non-increasing sequence of natural numbers called the decaying sequence where s1=kand sr= 0 . If E=ak, we will write ak→(ap, S); dinstead of ak/ak→(ap, S); d. The intuition behind the decaying sequence is the following. When the rule E/ak→(ap, S); dis triggered at t0we look in S= (s1, . . . , sr) for the greatest lsuch that p≥sl. Such slspikes are sent to the postsynaptic neurons according with the delay din the usual way. Notice that slcan be equal to p, so at this point this new type of rule is a generalization of the usual extended rules. At t0+d+1, the slspikes arrive to the postsynaptic neurons. The decay of such spikes is determined by the decaying sequence. If the spikes are not consumed by the triggering of a rule in the postsynaptic neuron, they decay and at time t0+d+2 we will consider that sl−sl+1 spikes have disappeared and we only have sl+1 spikes in the postsynaptic neuron. If the spikes are not consumed in the following steps 216 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez by the triggering of a postsynaptic rule, at t0+d+ 1 + r−lthe number of spikes will be decreased to sr= 0 and the spikes are lost. This definition of decay3can be seen as a generalization of the decaying spikes presented in [7]. In that paper a decaying spike ais written in the form (a, e), where e≥1 is the period. From the moment a spike (a, e) arrives in a neuron, eis decremented by one in each step of computation. As soon as e= 0, the corresponding spike is lost and cannot be used anymore. In this way, a rule E/ak→ap;d(k > p) where apare pdecaying spikes (a, e) can be seen with our notation as E/ak→(ap, S); dwith S= (s1, . . . , se+2), s1=k, s2=· · · =se+1 =pand se+2 = 0. 2.3 Hebbian SN P System Units Hebbian SN P system units are SN P systems with a fixed topology endowed with input and decay. They have the following common features: •The initial number of the spikes inside the neurons is always zero in all Hebbian SN P system units, so we do not refer to them in the description of the unit. •All the presynaptic neurons are linked to the postsynaptic neuron and these are all the synapses in the SN P system, so they are not provided in the description. •The output neuron is the postsynaptic one. Bearing in mind these features, we describe a Hebbian SN P system unit in the following way. 3Further discussion about the decay can be found in Section 5. Fig. 3. The potential at the postsynaptic neuron A First Model for Hebbian Learning with SN P Systems 217 Definition 1. A Hebbian SN P system unit of degree mis a construct HΠ = (O, u1, . . . , um, v), where: •O={a}is the alphabet (the object ais called spike); •u1, . . . , umare the presynaptic neurons. Each presynaptic neuron uihas associated a set of rules Ri={Ri1, . . . , Rili}where for each i∈ {1, . . . , m}and j∈ {1, . . . , li},Rij is a decaying rule of the form: ak→(anij , S); dij We will call nij the presynaptic potential of the rule and dij is the delay of the rule. Note that all rules are triggered by kspikes. The decaying sequence S will be discussed below. •vis the postsynaptic neuron which contains only one postsynaptic rule E∗ p/ap→ a; 0 where E∗ pis the set4of regular expressions {n∈N|n≥p}. We will call p the threshold of the postsynaptic potential of the Hebbian SN P system unit. By considering the decaying sequences we can distinguish among three types of Hebbian SN P system units: •Hebbian SN P system units with uniform decay. In this case the decaying sequence Sis the same for all the rules in the mpresynaptic neurons. •Hebbian SN P system units with locally uniform decay. In this case the decaying sequence Sis the same for all the rules in each presynaptic neuron. •Hebbian SN P system units with non-uniform decay. In this case each rule has associated a decaying sequence. A Hebbian SN P system unit is an abstract machine where a global clock is assumed (the system is synchronized). It takes an input and can provide an output or not, depending if the potential in the postsynaptic neuron reaches or not its threshold. The concept of input of a Hebbian SN P system unit is defined as follows: Definition 2. An input for a Hebbian SN P system unit of degree mis a vector ~x = (x1, . . . , xm)of mnon-negative integers xi. AHebbian SN P system unit with input is a pair (HΠ, ~x)where HΠ is Hebbian SN P system unit and ~x is an input for it. The intuitive idea behind the input is encoding the information in time. Each xirepresent the moment, according to the global clock, in which one spike is provided to each presynaptic neuron. 4This rule is an adaptation of the concept of a rule from an extended spiking neural P system with thresholds taken from [7]. 218 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez 2.4 How it works In this subsection we provide a description of the semantics of a Hebbian SN P system unit. As we saw before, each xiin the input ~x = (x1, . . . , xm) represents the time in which kspikes are provided to the neuron ui. At the moment xiin which the spike arrives to the neuron uione rule (ak→(anij , S); dij) is chosen in a non-deterministic way among all the rules of the neuron. Applying it means that kspikes are consumed and we look in S= (s1, . . . , sr) for the greatest lsuch that nij ≥sl. Such slspikes are sent to the postsynaptic neurons according to the delay dij in the usual way, i.e., slspike arrive to the postsynaptic neuron at the moment xi+dij + 1. The decay of such spikes is determined by the decaying sequence. As we saw above, if the spikes are not consumed by the triggering of a rule in the postsynaptic neuron, they decay and at time xi+dij + 2 we will consider that sl−sl+1 spikes have disappeared and we only have sl+1 spikes in the postsynaptic neuron. If the spikes are not consumed in the following steps by the triggering of a postsynaptic rule, at x0+dij +1+r−l the number of spikes will be decreased to sr= 0 and the spikes are lost. The potential on the postsynaptic neuron depends on the contributions of the chosen rules in the presynaptic neurons. Such rules send spikes that arrive to the postsynaptic neuron at different moments which depend on the input (the moment in which the presynaptic neuron is activated) and the delay of the chosen rule. The contribution of each rule to the postsynaptic neuron also changes along the time due to the decay. Formally, the potential of the postsynaptic neuron is a natural number calculated as a function R∗which depends on the time t, on the input ~x and on the rules chosen in each neuron R∗(R1i1, . . . , Rmim, ~x, t)∈N. Such a natural number represents the number of the spikes at the moment tin the postsynaptic neurons and it is the result of adding the contributions of the rules R1i1, . . . , Rmim. The Hebbian SN P system unit produces an output if the rule of the postsynaptic neuron v,E∗ p/ap→ais triggered, i.e., if at any moment tthe amount of spikes in the postsynaptic neuron is greater than or equal to the threshold p, then the rule is activated and triggered. If there does not exist such t, then the Hebbian SN P system unit does not send any spike to the environment. Bearing in mind the decay of the spikes in the postsynaptic neuron, if any spike has been sent out by the postsynaptic neuron after an appropriate number of steps, any spike will be sent to the environment. From a practical point of view we have a bound for the number of steps in which the spike can be expelled, so we have a decision method to determine if the input ~x provided to the Hebbian SN P system unit produces or not an output. Example 1. Let us consider the following Hebbian SN P system unit HΠ = (O, u1, u2, v) with non-uniform decay, where: A First Model for Hebbian Learning with SN P Systems 219 •O={a}is the alphabet; •u1, u2are the presynaptic neurons. The presynaptic neurons u1, u2have associated the sets of rules R1={R11, R12, R13}and R2={R21, R22}, respectively, with R11 ≡a3→(a2,(3,2,0)); 0 R21 ≡a3→(a2,(3,2,0)); 1 R12 ≡a3→(a, (3,1,0)); 1 R22 ≡a3→(a, (3,1,0)); 0 R13 ≡a3→(a3,(3,0)); 0 •vis the postsynaptic neuron which contains only one postsynaptic rule E∗ 2/a2→a; 0. Notice that in this example, the rules send all the presynaptic potential to the postsynaptic neuron but it only lasts one time unit before being lost. If they are not consumed immediately, they disappear. Case 1: Let us consider the input ~x = (0,0), i.e., at t= 0 three spikes are placed in each presynaptic neuron. We represent the contribution of each rule for ~x = (0,0) in the following table. Notice that for t≥3 the contribution is zero for all the rules. R11 R12 R13 R21 R22 t= 1 2 0 3 0 1 t= 2 0 1 0 2 0 Considering the different contributions of the rules and bearing in mind that in each neuron only one rule is non-deterministically chosen, the changes in the postsynaptic potential for ~x = (0,0) are described in the following table. R11 R21 R12 R21 R13 R21 R11 R22 R12 R22 R13 R22 t= 1 203 3 14 t= 2 2 32010 Notice that with the input ~x = (0,0), the postsynaptic neuron activates the rule at t= 1 if the chosen rules are R11 R21,R13 R21,R11 R22 or R13 R22. If the chosen rules are R12 R21, then the rule is activated at t= 2 and if the chosen rules are R12 R22 then the postsynaptic rule is not activated. Case 2: Let us consider now the input ~x = (1,0), i.e., at t= 0 three spikes are placed in the presynaptic neuron u2and in t= 1 other three spikes are placed in u1. As above, we represent the contribution for ~x = (1,0) in the following table. R11 R12 R13 R21 R22 t= 1 0 0 0 0 1 t= 2 2 0 3 2 0 t= 3 0 1 0 0 0 The changes of the potential R∗in the postsynaptic potential for ~x = (1,0) are described in the following table. 226 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez •The rate of learning ²= 0.1 Step 1: Let us consider the input ~x = (0,2). The contribution can be summarised in the following table: R11 R12 R13 R21 R22 P t= 1 30 0 30 0 0 60 t= 2 15 70 15 0 0 100 t= 3 0 30 0 0 30 60 t= 4 0 15 0 80 15 110 t= 5 0 0 0 70 0 70 t= 6 0 0 0 30 0 30 t= 7 0 0 0 15 0 15 Therefore, at time t= 2 the potential of the postsynaptic neuron reaches a value greater than the threshold 70, then t(0,2) = 2. We can compute now the values t(0,2) ij =xi+dij + 1, s(0,2) ij =t(0,2) −t(0,2) ij and L(s(0,2) ij ) for every rule Rij. After computing the values L(s(0,2) ij ) for every rule Rij, the new weights are calculated as w0 ij =wij +² L(s(0,2) ij ) These values are summarised in the following table t(0,2) ij s(0,2) ij L(s(0,2) ij )wij w0 ij R11 1 1 2 1 1.2 R12 2 0 4 1 1.4 R13 1 1 2 1 1.2 R21 4−2−1 1 0.9 R22 3−1−1 1 0.9 Therefore, after this fist step the new weights are w0 11 = 1.2, w0 12 = 1.4, w0 13 = 1.2, w0 21 = 0.9 and w0 22 = 0.9. Step 2: Let us consider the new extended Hebbian SN P system unit built by replacing the initial weights by the new w0 ij and let us consider the second input ~x2= (0,0). The contribution can be summarized in the following table. R11 R12 R13 R21 R22 P t= 1 30 0 30 0 30 90 t= 2 15 80 15 70 15 195 t= 3 0 70 0 30 0 100 t= 4 0 30 0 15 0 45 t= 5 0 15 0 0 0 15 Therefore, at time t= 1 the potential of the postsynaptic neuron reaches a value greater than the threshold 70, then t(0,0) = 1. We can compute now the values t(0,0) ij =xi+dij + 1, s(0,0) ij =t(0,0) −t(0,0) ij and L(s(0,0) ij ) for every rule Rij. A First Model for Hebbian Learning with SN P Systems 227 After computing the values L(s(0,0) ij ) for every rule Rij, the new weights are calculated as w00 ij =w0 ij +² L(s(0,0) ij ) These values are summarized in the following table t(0,2) ij s(0,2) ij L(s(0,2) ij )w0 ij w00 ij R11 1 0 4 1.2 1.6 R12 2−1−1 1.4 1.3 R13 1 0 4 1.2 1.6 R21 2−1−1 0.9 0.8 R22 1 0 4 0.9 1.3 Therefore, after this fist step the new weights are w0 11 = 1.6, w0 12 = 1.3, w0 13 = 1.6, w0 21 = 0.8 and w0 22 = 1.3. The use of weights needs more discussion. The weights are defined as real numbers and membrane computing devices are discrete. If we want to deal with discrete computation in all the steps of the learning process we have to choose the parameters carefully. The following result gives a sufficient constraint for having an integer number of spikes at any moment. Theorem 1. Let abe the greatest non-negative integer such that for all presynaptic potential nij there exists an integer zij such that nij =xij ×10a. Let bbe the smallest non-negative integer such that for all initial weight wij and for the rate of learning ²there exist the integers kij and ksuch that wij =kij ×10b and ²=k×10b. If a−b≥0, then for all presynaptic potential nij and all the weights wobtained along the learning process, nij ×wis an integer number. In other words, if there exists aand bsuch that all the presynaptic potentials nij can be expressed as nij =xij ×10afor an appropriate integer xij and the initial weights wij and rate of learning ²can be expressed as wij =kij ×10b and ²=k×10bfor appropriate integer numbers kij, k and a−b≥0 then for all presynaptic potential nij and all the weights wobtained along the learning process, nij ×wis an integer number. Proof. It suffices to consider the recursive generation of new weights wn+1 =wn+ ²L(sn) and therefore wn+1 =w0+²(L(s0) + · · · +L(sn)). If we develop nij ×wn+1 according to the statement of the theorem, we have nij ×wn+1 =xij ×10a×[k0×10−b+ (k×10−b(L(s0) + ···+L(sn)))] = 10a−b×xij ×[k0+k(L(s0) + · · · +L(sn))] Since xij ×[k0+k(L(s0) + · · · +L(sn))] is an integer number, if a−b≥0 then nij ×wn+1 is an integer number. 228 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez 4 An Experiment Let us consider the Hebbian SN P system HΠ = (O, u1, u2, v) with uniform decay, where: •O={a}is the alphabet; •u1, u2are the presynaptic neurons. The presynaptic neurons u1, u2have associated the sets of rules R1={R11, R12, R13}and R2={R21, R22}, respectively, with R11 ≡a3000 →(a3000, S); 0 R21 ≡a3000 →a1000; 0 R12 ≡a3000 →(a2000, S); 1 R22 ≡a3000 →a3000; 3 R13 ≡a3000 →(a2000, S); 7 •The decaying sequence is S= (3000,2800,1000,500,0). •vis the postsynaptic neuron which contains only one postsynaptic rule E∗ 1200/a1200 →a; 0. Let EHΠ be the Hebbian SN P system unit HΠ extended with the initial weights w11 = 0.5, w12 = 0.5, w13 = 0.5, w21 = 0.5 and w22 = 0.5. Let us consider the learning problem (EHΠ, X, L, ²) where •EHΠ is the extended Hebbian SN P system unit described above, •Xis a set of 200 random inputs (x1 i, x2 i) with 1 ≤1≤200 and x1 i, x2 i∈ {0,1,...,5} •Lis the learning function L:Z→Z L(s) =    3 if s= 0 1 if s= 1 −1 otherwise •The rate of learning is ²= 0.001 We have programmed an appropriate software for dealing with this learning problems. After applying the learning algorithm, we obtain a new extended Hebbian SN P system unit similar to EHΠ but with the weights w11 = 0.754, w12 = 0.992, w13 = 0.3, w21 = 0.454, w22 = 0.460 Fig 4 shows the evolution of the weights of the synapses. The learning process shows clearly the differences among the rules. •The worst rule is R13. In a debugging process of the design of an SN P System network that rule should be removed. The value of the weight has decreased along all the learning process. This fact means that the rule has never contributed to the success of the unit and then it can be removed. The reason is A First Model for Hebbian Learning with SN P Systems 229 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 Weights Inputs Sin1 Sin2 Sin3 Sin4 Sin5 Fig. 4. The evolution of the weights clear. The rule emits four spikes and the postsynaptic rule is activated with two spikes. Even with the decay, the potential provided by the rule is too much for triggering the rule. •On the other extreme, the best rules are R11 and R21. In most of the cases, (not all) these rules have been involved in the success of the unit. •The other two rules R21 and R22 have eventually contributed to the success of the unit but not so clearly as R11 and R21. We can also guess the reasons. For R11, the presynaptic potential, 1000, has little influence in the postsynaptic potential and for R22, the presynaptic potential is larger than the threshold, but it has a large delay, so the arrival of its potential to the postsynaptic neuron is often later than the activation of the postsynaptic rule. 230 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez 5 Conclusions and Future Work The integration in an unique model of concepts from neuroscience, artificial neural networks and spiking neural P systems is not an easy task. Each of the three fields have its own concepts, languages and features. The work of integration consists on choosing ingredients from each field and trying to compose a computational device with the different parts. This means that some of the ingredients used in the devices presented in this paper are not usual in the SN P systems framework. Although the authors have tried to be as close to the SN P system spirit as possible some remarks should be considered. In the paper, the input of the device is provided as a vector (t1, . . . , tm) of non-negative integers, where tirepresents the moment in which one rule (nondeterministically chosen) of the neuron uiis activated. Obviously, this is not the usual way to provide the input to an SN P system. Nonetheless, the information encoded in the vector (t1, . . . , tm) can be provided to the input neurons by mspike trains were all the elements are 0’s and there is only one 1 in the position ti. In this way, the input is encoded by mspike trains, which is closer to the standard inputs in SN P systems. The idea of providing the input with a spike train of 0’s and only one 1 in the position ticarries out new problems. In the literature of SN P systems, in the instant tionly one spike is supplied to the neuron ui. In our device we want that a rule of type ar→ap;dis activated with r > 1. At this point we can consider several choices. The first one is to consider that at time tithe spike train provides rspikes, but this choice lead us far from the SN P system theory. A second option is to consider that the spike trains have rconsecutive 1’s and each of them provide one spike. The remaining elements in the train are zeros. In this way the moment tiwill be the instant in which the rspikes have been provided to the neuron. A drawback for this proposal can be that rcan be a big number and this increase the number of steps of the device. A third choice is to consider amplifier modules as in Figure 5. The leftmost neuron receives a spike train where all the elements are 0’s but the ti−th which is 1. At the moment tionly one spike is supplied to the neuron. At ti+ 1, one spike arrives to the rpostsynaptic neurons, and each of them sends one spike to the rightmost neuron, so at ti+ 2 exactly rspikes arrive simultaneously to the last neuron. These three solutions can be an alternative to the use of the vector (t1, . . . , tm) and deserve to be considered for further research in this topic. Another main concept in this paper is the delay. It has strong biological intuition, but it is difficult to insert into the SN P systems theory. The main reason is that if we consider the spike as the information unit it does not make sense to talk about a half of a spike or a third of a spike. In that sense, the approach to decay from [7] is full of sense since one spike exists or it is lost, but its potential it is not decreasing in time. The key point for the decay in this paper is taken from the definition of extended SN P systems. In such devices, a neuron can send a different amount of spikes depending on the chosen rule. So, in such devices the information is not only A First Model for Hebbian Learning with SN P Systems 231 a a;0 a a;0 a a;0 a a;0 a a;0 aran;d ... ... Fig. 5. Amplifier module encoded in the time between two consecutive spikes, but on the number of spikes. This lead us to define the decay as a decrement in the number of spikes. In this way, we can consider that a pulse between two neurons is composed by a certain number of spikes which can be partially lost depending on the time. In this paper, such a decay has been implemented by extending the rules with a finite decreasing sequence which can be uniform, locally-uniform or non uniform for the set of rules. Other implementations are also possible. Probably, the decay can also be implemented with an extra neuron as in Figure 6 which sends to the final neuron a decaying sequence of spikes. a a;0 p a a ;d1 n1 k ak a ;dv nv ... wi Input neuron Decay neuron Postsynaptic neuron Fig. 6. Including a decay neuron The use of weights also deserves to be discussed. In Theorem 1 we provide sufficient conditions for handling at every moment an integer number of spikes. In this way, the presented devices keep the principle of discrete computation of SN P systems. Nonetheless, further questions should be considered. For example, the use of negative weights or weights greater than one. Should we consider negative weights and/or a negative contribution to the postsynaptic potential? On the other hand, the use weights greater than one leads us to consider that the contribution of 232 M.A. Guti´errez-Naranjo, M.J. P´erez-Jim´enez one rule to the postsynaptic potential is greater than its own presynaptic potential. Can the efficiency of the synapses amplify the potential beyond the number of emitted spikes? More technical questions are related to the rate of learning and to the algorithm of learning. Both concepts have been directly borrowed from artificial neural networks and need deeper study in order to adapt them to the specific features of SN P systems. As a final remark, we consider that this paper opens a promising line research bridging SN P systems and artificial neural networks without forgetting the biological inspiration and also opens a door to applications of SN P systems. Acknowledgements The authors acknowledge the support of the project TIN2006-13425 of the Ministerio de Educaci´on y Ciencia of Spain, cofinanced by FEDER funds, and the support of the project of excellence TIC-581 of the Junta de Andaluc´ıa. References 1. E.D. Adrian. The impulses produced by sensory nerve endings. J. Physiol. (Lond.), 61, 49-72, 1926. 2. E.D. Adrian. The basis of Sensation. W.W. Norton. 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