Simulation of Spiking Neural P Systems Using Pnet Lab

Authors

Padmavati Metta

Kamala Krithivasan

Deepak Garg

Outline

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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., Păun, Gh., Yokomori, T.: Spiking Neural P Systems, Fund. Infor. 71, 279-308 (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

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• O = {a}, the alphabet. a is called spike and ā is called anti-spike.

• m neurons - σ1, σ2, σ3 ,. . . , σm

• Syn - Synapses among the neurons. Spike emitted by a neuron i

will pass immediately to all neurons j connected to i through

synapses.

• i0 – Output neuron

Spiking Neural P system without delay (Contd.)

Π =(O, σ1, σ2, σ3 ,. . . , σm , syn , i0)

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Each neuron σi contains

– ni -- initial number of spikes

– Ri -- finite set of rules of the form

1. Spiking Rules

• E / ar→ a – used when a neuron has n spikes/anti-spikes such that an 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)=ar

SN P systems without delay (contd.)

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SN P system without delay (contd.)

2. Forgetting Rules

2. as →λ - used when a neuron has s number of spikes3. s ≥ 0, number of spikes are forgotten by the

neuron.4. as should not be in L(E) for any spiking rule E/ar→

a in Ri. Configuration of SN P system

The configuration of a system at any time is <n1, n2, …, nm>, where

ni is the number of spikes present in neuron σi

a2

r11 : a2/ a a

r12 : a2 ar13: a λ

a3

r31 : a3a r32 : a2 λ

r33: a a

An SN P System without delay п

2

ar21: a a

3

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1

< 2 , 1 , 3>

Initial Configuration

Working of an SN P System

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• 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 : a2 a

r13: aλ

r32 : a2

λr33: a

a

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1

2

3

r11:a2/ a →a

a3

r21 :a →a

a

r31 : a3

→a

Evolution

< 2, 1 , 3 >

11, 21, 31

aa

a

a2

STEP - 1

< 2, 1 , 2 >

aa

An SN P System without delay п

Thus as long as neuron 1 uses the rule a2/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

< 2, 1, 3 >

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11, 21, 31

< 2, 1 , 2 >

An SN P System without delay п

11, 21, 32

At any moment, neuron 1 can choose the rule a2→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 < 1, 1, 2> Evolution

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< 2, 1, 3 >

11, 21, 31

< 2, 1 , 2 >

< 1, 1 , 2 >

12, 21, 31

12, 21, 32

An SN P System without delay п

11, 21, 32

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r11:a2/ a

→ar12 : a2

a

r31 : a3

→a

r33: a a

1

2

3

r13: aλ

a2

r21 :a →a

a

r32 : a2 λ

Evolution

a

NEXT STEP

a < 2, 1, 3 >

11, 21, 31

< 2, 1 , 2 >

< 1, 1 , 2 >

12, 21, 31

12, 21, 32

13, 21, 32

< 1, 0 , 1 >

An SN P System without delay п

11, 21, 32

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r11:a2/ a

→ar12 : a2

a

r31 : a3

→ar32 : a2

λ

1

2

3

r13: aλ

r21 :a →a

a

r33: a a

EvolutionLAST STEP

a< 2, 1, 3 >

11, 21, 31

< 2, 1 , 2 >

< 1, 1 , 2 >

12, 21, 31

12, 21, 32

13, 21, 32

< 1, 0 , 1 >13, 20, 33

< 0, 0 , 0 >

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.

2 1

2

P1

P3

P2

TG(T)=true if #(P1)=3

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Petri Net Marking

• A transition tj 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 ti 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.

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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 .

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Translation - SN P system and Petri net

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SN P System Petri Net

Neurons and Environment Places

Spikes Tokens

Synapses Arcs

Forgetting Rules Sink Transitions (no output place)

Spiking Rules Transitions

Regular Expression Guard function

Configuration Marking

Translation- Execution Semantics

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SN P System Petri Net

Sequential at neuron levelSynchronizing place is

maintained for each place

corresponding neuron, so

that only one transition is

enabled from each input

place.Parallel at System Level Parallel execution of

transitions

(SN P System to Petri net)

a2

r11: a2/ a →a

Petri NetPetri Net

1

32

P1

P3

P2

P11-synchronizing place for P1

T11 - Transition corresponding to rule r11

Methodology

G(T11)=true if #(P1)=2

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P11

About PNetLab

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•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.

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Petri net in PnetLab for SN P System п

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Specifying conflict management in PnetLab

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Simulation in PnetLab – Step 1

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Simulation in PnetLab – Step 2

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Simulation in PnetLab – Step 4

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Simulation in PnetLab – Step 5

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Markings during Simulation in PnetLab

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< 2, 1, 3 >

< 0, 0 , 0 >

< 2, 1 , 2 >

< 1, 1 , 2 >

< 1, 0 , 1 >

If we consider the sub marking-the marking of first three place we get

Evolution of SN P System

< 2, 1, 3 >

11, 21, 31

< 2, 1 , 2 >

< 1, 1 , 2 >

12, 21, 31

12, 21, 32

13, 21, 32

< 1, 0 , 1 >13, 20, 33

< 0, 0 , 0 >

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.

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Thank Thank YouYou