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# Coherence and Stochastic Resonances in Fitz-Hugh-Nagumo Model

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### Coherence and Stochastic Resonances in Fitz-Hugh-Nagumo Model

1. 1. Coherence and Stochastic Resonances in the FitzHugh Nagumo Model M.Sc. Dissertation Project Stage I Pratik TarafdarProject Guide : M.Sc. 2nd YearDr. Punit Parmananda Dept. of Physics IIT Bombay
2. 2. Outline of the Presentation• Introduction• Stochastic Resonances• Coherence Resonance• The FHN Model• Simulations and Results• Future Plans
3. 3. Introduction• Noise-induced regularity or coherence resonance and information transmission through stochastic resonances are well known phenomena in nonlinear systems with excitable dynamics.• Coherence, periodic stochastic and aperiodic stochastic resonances have been demonstrated in the FitzHugh Nagumo model through numerical simulation.
4. 4. Constructive Role of NoiseCoherence Resonance Stochastic Resonance
5. 5. STOCHASTIC RESONANCE
6. 6. Input Signal Nonlinear System Output Noise • Noise aids in Signal Transmission • QUESTION : When is the transmission OPTIMUM ??
7. 7. ANSWER :There is a FINITE OPTIMAL level of noise atwhich the response of the system ismaximum.STOCHASTIC RESONANCE (SR)NOISE is a FRIEND…!!
8. 8. MECHANISMInput Signal Nonlinear System Output Noise What is it that happens inside the BLUE BOX ??
9. 9. Let’s try to Understand……WEAK periodic signal Output Noise • Zero Noise ? : Particle oscillates within one well • Finite Noise ? : Particle can jump between the wells
10. 10. A Pinch of History…. THE ICE AGE !!!!Benzi et al (1981, 1982), C. Nicolis (1982) Why do ice ages recur periodically ?
11. 11. The SR Explanation• Global climate Double well potential• Small modulation of earths orbital eccentricity Weak periodic forcing• Short term climate fluctuations Noise
12. 12. First Experimental Verification of SRSchmitt Trigger Device - Fauve and Heslot (1983) Output A cos(ωt) + Dξ(t) Input Signal to Noise ratio maximum at an optimal level of noise
13. 13. Applications of SRHuge amount of applications throughout a large spectrum offields. About 1000 publications since 1981 till date –• Optics• Biology• Neurology• Psychophysics
14. 14. An interesting example in Nature Hungry Fish (Predator) Cray Fish having hydrodynamic sensors (Prey)Noise : Underwater turbulencePeriodic force : Water vibrations generated by thepredator’s tailThe Cray fish can detect its predator more easily in thebackground of underwater turbulence.
15. 15. COHERENCE RESONANCE
16. 16. Noise Nonlinear System Output“Stochastic Resonance without External Periodic Forcing” (Gang et al PRL 1993)
17. 17. • SR : Response of a bistable system to an external periodic forcing, with noise present.• CR : Coherent motion stimulated by the INTRINSIC dynamics of the system. “It has attracted considerable interest theoretically as well as experimentally, as quite counter-intuitively ORDER ARISES WITH THE AID OF TUNED RANDOMNESS” (D. Das, P. Parmananda, A. Sain, S. Biswas et al PRE 2009)
18. 18. MECHANISM OF CR Two time scales Activation Time Excursion Time• Time between end of one spike • Decay Time of unstable and beginning of another. state.• Strong dependence on Noise • Much weaker noise Intensity. dependence.• Follows Kramer’s-like formula – (Ta e(ΔV/D2) ) Pikovsky and Kurths et al PRL (1997)
19. 19. APPLICATIONS OF CR• Neuronal and biological systems.• Chemical models.• Electronic circuits.• Semiconductor lasers.
20. 20. HOW DO WE MEASURE COHERENCE AND STOCHASTIC RESONANCES ??
21. 21. COHERENCE RESONANCE• Co-efficient of Variation ( Normalized variance) T Interspike Interval• Power Spectral Density (PSD)• Auto Correlation Function (ACF)• Interspike Interval Histograms• Effective Diffusion Co-efficients (Deff)
22. 22. STOCHASTIC RESONANCEPeriodic Stochastic Resonance :• Co-efficient of Variation (VN)Aperiodic Stochastic Resonance :• Cross Correlation Coefficient (C0) C0 = <(x1-<x1>t)(x2-<x2>t)>tx1 Time Series of Aperiodic Input Signalx2 Time Series of Noise Induced Output Signal<>t Time Average
23. 23. The FitzHugh Nagumo Model
24. 24. The Fitz Hugh Nagumo model, named after Richard FitzHugh(1922–2007) and J. Nagumo et al approximately at the sametime, describes a prototype of an excitable system (e.g., aneuron).If the external stimulus exceeds a certain threshold value, thesystem will exhibit a characteristic excursion in phase space,before the variables relax back to their rest values.This behaviour is typical for spike generations ( shortelevation of membrane voltage ) in a neuron afterstimulation by an external input current.The Fitz Hugh Nagumo model is a simplified version of theHodgkin–Huxley model which models in a detailed manneractivation and deactivation dynamics of a spiking neuron. Theequivalent circuit was suggested by Jin-ichi Nagumo, SuguruArimoto, and Shuji Yoshizawa.
25. 25. • a, D, ξ are parameters• |a| > 1 Stable focus• |a| < 1 Limit cycle• |a| = 1 Centre• |a| > 2 Stable node• D Amplitude of Gaussian noise ξ(t)• <ξ(t)> = 0 (Random)• <ξ(t)ξ(t’)> = δ(t-t’) (Uncorrelated)
26. 26. SIMULATION AND RESULTS
27. 27. COHERENCE RESONANCE
28. 28. Time Series for LOW NOISE Figure 1
29. 29. Time Series for HIGH NOISE Figure 3
30. 30. Time Series for OPTIMAL NOISE Figure 2
31. 31. COEFFICIENT OF VARIATION versus NOISE INTENSITY
32. 32. STOCHASTIC RESONANCES
33. 33. PERIODIC STOCHASTIC RESONANCE
34. 34. Time Series for LOW NOISE
35. 35. Time Series for HIGH NOISE
36. 36. Time Series for OPTIMAL NOISE
37. 37. COEFFICIENT OF VARIATION versus NOISE INTENSITY
38. 38. APERIODIC STOCHASTIC RESONANCE
39. 39. Time Series for LOW NOISE
40. 40. Time Series for HIGH NOISE
41. 41. Time Series for OPTIMAL NOISE
42. 42. CROSS CORRELATION COEFFICIENT versus NOISE INTENSITY
43. 43. FUTURE PLANS• To study the response of Fitz Hugh Nagumo system after interaction with noise of fixed intensity, by varying the system parameter.• To study the interaction of Fitz Hugh Nagumo system with noise, by fixing the system parameter on oscillatory side instead of the conventional fixed point side.
44. 44. BIBLIOGRAPHY• Santidan Biswas, Dibyendu Das, P. Parmananda and Anirban Sain : Predicting the coherence resonance curve using a semianalytical treatment, PhysRevE 80, 046220 (2009)• G.J. Escalera Santos, M. Rivera, J. Escalona and P. Parmananda : Interaction of noise with excitable dynamics, Phil. Trans. R. Soc. A(2008) 366, 369-380• G.J. Escalera Santos, M. Rivera, M.Eiswirth and P. Parmananda : Effects of near a homoclinic bifurcation in an electrochemical system , PhysRevE 70, 021103 (2004)• G.J. Escalera Santos, M. Rivera and P. Parmananda : Experimental Evidence of Coexisting Periodic Stochastic Resonance and Coherence Resonance Phenomenon, PhysRevLett 92 230601 (2004)• P.Parmananda, G.J. Escalera Santos, M. Rivera, Kenneth Showalter : Stochastic resonance of electrochemical aperiodic spike trains, PhysRevE 71 031110 (2005)• Steven H. Strogatz : Nonlinear Dynamics and Chaos, Advanced Book Program, Perseus Books, Reading, Massachusetts, http://www.aw.com/gb/• http://www.arxiv.org• http://www.scholarpedia.org• http://www.wikipedia.org
45. 45. Acknowledgement• Dr. Punit Parmananda, Dept. of Physics, IIT Bombay• Dr. Dibyendu Das , Dept. of Physics, IIT Bombay• Dr. Sitabhra Sinha, IMSc Chennai• Santidan Biswas , Dept. of Physics, IIT Bombay• Supravat Dey , Dept. of Physics, IIT Bombay• All my friends and co-learners who have shared their views and have encouraged me to strive forward.
46. 46. THANK YOU FOR YOUR PATIENCE AND KIND ATTENTION….