Spiking Neural P Systems and Petri Nets

Transcription

1 Spiking Neural P Systems and Petri Nets By M.V.Padmavati Bhilai Institute of Technology, Durg Co-Authors Dr. Deepak Garg Thapar University Dr. (Mrs.) Kamala Krithivasan IIT Madras, Chennai

2 Outline Spiking Neural P Systems Petri nets Objective and Methodology The Significance of the Study 2

3 Introduction to P Systems Computing with membranes (P systems) is a branch of Molecular Computing initiated by Gh. Paun. is a computing model which abstracts from the way the live cells process chemical compounds in their compartmental structure. Many of these variants lead to computationally universal systems. Spiking Neural P system is a variant 3

4 Spiking Neural P System Spiking Neural P systems is a computational model that has been inspired by neurobiology Soma Synaptic Terminals Dendrites Axon Synapse Components of a Neuron 4

5 Spiking Neural P Systems Some ideas incorporated are synapses, with the replication of impulses in the case of multiple synapses linking a neuron to several neighbouring neurons state of a neuron aspect captured is the fact that most of the neural impulses are almost identical, electrical signals of a given voltage, with a crucial role played by the time when these signals are issued, hence by the intervals between signals. 5

6 Spiking Neural P System SN P system is mathematically represented as {a} is a singleton alphabet called spike Synapses among the neurons. Spike emitted by a neuron i will pass immediately to all neurons j connected to i through synapses and both are open. Output neuron m number of neurons where each neuron contains initial value (of spikes) in each neuron finite set of rules of the form E/a r a ; t (Spiking Rules) a s λ (Forgetting Rules) 6

7 Outputs with SN P system From the output neuron, spikes are sent to the environment. The moments of time when a spike is emitted by the output neuron are marked with 1, the other moments are marked with 0. The sequence is called the spike train of the system With a spike train we can associate various numbers, which can be considered as computed (we also say generated) by an SN P system. The output can be taken as the number of spikes present in the output neurons at the end of a halting computation (reaching a configuration where no rule can be used) distance between the first two spikes of a spike train 7

8 SN P System Example 1 a 2 a 2 /a a ; 0 a λ 2 a a a ;0 a a ;1 a 3 a 3 a ;0 a a ;1 a 2 λ 3 8

9 Output of SN P System The output produced by the SN P system is N 2 (Π)= N - {1} Є Spik 2 P 3 (rule 3, cons 3, forg 2 ) The two spikes of the neuron 3 cannot be consecutive because of the waiting time imposed by the rule a a;1 of the neuron 3. Therefore at least two steps must exist in between. 9

10 Petri Net It is a diagrammatic tool to model concurrency and synchronization in distributed systems. It consists of P is a finite set of places, P={p1,,pn} T is a finite set of transitions, T={t1,,tm} A is the set of arcs from places to transitions and from transitions to places (pi, tj) or (tj, pi) represent the arcs w is the weight function on arcs The state of a Petri net is determined by the marking vector x =[x1,, xn] represents the number of tokens in each place. 10

11 Petri Net Example Arc Transition 0 Arc with w=1 Place Place 0 Place 1 Token Transition 1 Transition 11

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 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. 12

13 Petri Net Variations Colored Petri Nets In this case, tokens have various properties associated with them. This can be an attribute or an entire data structure. Timed Petri nets are similar to Petri nets with the addition of a clock structure associated with each timed transition A timed transition t j (denoted by a rectangle) once it becomes enabled fires after a delay v jk. 13

14 Timed Petri Net Example Transitions t1 and t3 fire after a delay given by the model clock structure t 1 p 1 p 3 Transition t2 fires immediately after it becomes enabled t 2 p 2 t 3 14

15 Similarities - SN P system and Petri nets The place in spiking Petri net corresponds to a neuron (cell body soma) in SN P system. Output place in spiking Petri net corresponds to environment in SN P system. The arc between the place and transition represents an axon. Tokens in Petri net place correspond to spikes in the neuron. The initial marking of the Petri net corresponds to initial configuration of SN P system. Both SN P systems and Petri nets are concurrent 15

16 Objective of the Paper Introducing a new variant of Petri net - Spiking Petri net - Input place is closed and does not accept any tokens during the transition delay. Design algorithms that translate one system into another. - SN P system to Spiking Petri net - Spiking Petri net to SN P system 16

17 Methodology 1 2 a 2 a 2 /a a ; 1 a λ SN P System 3 2 Petri Net a a Guard Function: if #(P2)=2 then enable Delay : 1 Time Unit a a a 0 0 a

18 The Significance of the Study To complement the functional characterisation of the behaviour of SN P systems. Using the notations and tools developed for Petri nets, one can describe what is actually going on during a computation of a SN P system. SN P systems are concurrent in nature and is a core feature of Petri nets. So Petri nets can support and analyse concurrency in its most fundamental fashion. 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. 18

19 Spiking Neural P Systems and Petri Nets THANK YOU! 19