Simulation of Spiking Neural P Systems Using Pnet Lab Authors Padmavati Metta Kamala Krithivasan Deepak Garg

Outline CMC-122 Spiking Neural P (or SN P) system without delay Petri net SN P system without delay to Petri net Simulation using PNetLab

Spiking Neural P system Ionescu, M., P ă un, Gh., Yokomori, T.: Spiking Neural P Systems, Fund. Infor. 71, (2006). Spiking Neural P system is a computational model that has been inspired by neurobiology Distributed and parallel computing model Variant of Membrane System (P System) Uses one type of object called spike (a) Computationally complete CMC-123

O = { a }, the alphabet. a is called spike and ā is called anti-spike. m neurons - σ 1, σ 2, σ 3,..., σ m Syn - Synapses among the neurons. Spike emitted by a neuron i will pass immediately to all neurons j connected to i through synapses. i 0 – Output neuron Spiking Neural P system without delay (Contd.) Π =(O, σ 1, σ 2, σ 3,..., σ m, syn, i 0 ) CMC-124

5 Each neuron σi contains – n i -- initial number of spikes – R i -- finite set of rules of the form 1.Spiking Rules E / a r → a – used when a neuron has n spikes/anti-spikes such that a n ∈ L(E) and n ≥ r where E is a regular expression over { a} r ≥ 0, number of spikes are consumed and a spike is sent to all neighbouring neurons. E is omitted if L(E)=a r SN P systems without delay (contd.)

CMC-126 SN P system without delay (contd.) 2.Forgetting Rules a s →λ - used when a neuron has s number of spikes s ≥ 0, number of spikes are forgotten by the neuron. a s should not be in L(E) for any spiking rule E/a r → a in R i. Configuration of SN P system The configuration of a system at any time is, where n i is the number of spikes present in neuron σ i

a 2 r11 : a 2 / a  a r12 : a 2  a r13: a  λ a 3 r31 : a 3  a r32 : a 2  λ r33: a  a An SN P System without delay п 2 a r21: a  a 3 7CMC-12 1 Initial Configuration

Working of an SN P System CMC-128 A global clock is there and all neurons work in parallel but each neuron can use one rule at a time. There can be more than one rule enabled at any time in a neuron, then a rule is chosen in a non-deterministic way. Using the rules, we pass from one configuration of the system to another configuration. Such a step is called transition. A computation of an SN P system is finite or infinite sequence of transitions starting from the initial configuration.

r12 : a 2  a r13: a  λ r32 : a 2  λ r33: a  a CMC r11:a 2 / a →a a3a3 r21 :a →a a r31 : a 3 →a Evolution 11, 21, 31 a a a a2a2 STEP - 1 a a An SN P System without delay п

Thus as long as neuron 1 uses the rule a 2 /a →a, it sends a spike to other two neurons. One spikes will remains in it and receives one spike from neurons 2 thus a total of 2 spikes in it and the system will be in the same configuration. Evolution 10CMC-12 11, 21, 31 An SN P System without delay п 11, 21, 32

At any moment, neuron 1 can choose the rule a 2 →a, This means all spikes of neuron 1 are consumed so in the next step, it will have one spike instead of two reaching a configuration Evolution 11CMC-12 11, 21, 31 12, 21, 31 12, 21, 32 An SN P System without delay п 11, 21, 32

CMC-1212 r11:a 2 / a →a r12 : a 2  a r31 : a 3 →a r33: a  a r13: a  λ a2a2 r21 :a →a a r32 : a 2  λ Evolution a NEXT STEP a 11, 21, 31 12, 21, 31 12, 21, 32 13, 21, 32 An SN P System without delay п 11, 21, 32

CMC-1213 r11:a 2 / a →a r12 : a 2  a r31 : a 3 →a r32 : a 2  λ r13: a  λ r21 :a →a a r33: a  a Evolution LAST STEP a 11, 21, 31 12, 21, 31 12, 21, 32 13, 21, 32 13, 20, 33 An SN P System without delay п 11, 21, 32

Petri net with guard Petri Nets are formal and graphical models to represent concurrent events Consists of set of places and transitions. Arcs connecting transitions and places, have weights Transitions are associated with enabling conditions called guard functions P1 P3 P2 T G(T)=true if #(P1)=3 14CMC-12

Petri Net Marking A transition t j  T is enabled when each input place has at least a number of tokens equal to the weight of the arc and guard function associated with t i returns true. When a transition fires it removes a number of tokens (equal to the weight of each input arc) from each input place and deposits a number of tokens (equal to the weight of each output arc) to each output place. A marking is an m (no. of places)-vector, containing number of tokens each place. 15CMC-12

Objective of the Paper To design algorithm for translating SN P systems into equivalent Petri net model. To simulate the obtained model using a Java based Petri net tool called PNetLab. 16CMC-12

Translation - SN P system and Petri net 17CMC-12 SN P SystemPetri Net Neurons and EnvironmentPlaces SpikesTokens SynapsesArcs Forgetting RulesSink Transitions (no output place) Spiking RulesTransitions Regular ExpressionGuard function ConfigurationMarking

Translation- Execution Semantics 18CMC-12 SN P SystemPetri Net Sequential at neuron level Synchronizing place is maintained for each place corresponding neuron, so that only one transition is enabled from each input place. Parallel at System LevelParallel execution of transitions

(SN P System to Petri net) a 2 r11: a 2 / a →a Petri Net 1 32 P1 P3 P2 P11-synchronizing place for P1 T11 - Transition corresponding to rule r11 Methodology G(T11)=true if #(P1)=2 19CMC-12 P11

About PNetLab CMC-1220 Java based Petri net tool Allows parallel execution of transitions after resolving conflicts. We can write user defined guard functions in C/C++ Provides step-by-step execution of net in a graphical way It can find Transition-invariants, Place-invariants, minimal siphons, traps, pre-incidence, post-incidence and incidence matrices and coverability tree.

CMC-1221 Petri net in PnetLab for SN P System п

CMC-1222 Specifying conflict management in PnetLab

CMC-1223 Simulation in PnetLab – Step 1

CMC-1224 Simulation in PnetLab – Step 2

CMC-1225 Simulation in PnetLab – Step 4

CMC-1226 Simulation in PnetLab – Step 5

CMC-1227 Markings during Simulation in PnetLab

CMC-1228 If we consider the sub marking-the marking of first three place we get Evolution of SN P System 11, 21, 31 12, 21, 31 12, 21, 32 13, 21, 32 13, 20, 33 11, 21, 32 Which is same as the evolution of the SN P systems

The Significance of the Study To verify and analyze the working of SN P systems without delay. Petri nets can aid in the analysis and verification of SN P systems. Other analytical and verification techniques developed for Petri nets can be deployed to deal with SN P systems. 29CMC-12

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