Simulation of Spiking Neural P Systems Using Pnet Lab Authors Padmavati Metta Kamala Krithivasan...
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Simulation of Spiking Neural P Systems Using Pnet Lab
Authors
Padmavati Metta
Kamala Krithivasan
Deepak Garg
Outline
CMC-12 2
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
7CMC-12
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
11CMC-12
< 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
14CMC-12
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
17CMC-12
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.
29CMC-12
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Thank Thank YouYou