Coherence and Stochastic Resonances in Fitz-Hugh-Nagumo Model
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Transcript of Coherence and Stochastic Resonances in Fitz-Hugh-Nagumo Model
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Coherence and Stochastic Resonances in the FitzHugh Nagumo
Model
M.Sc. Dissertation ProjectStage I
Pratik TarafdarM.Sc. 2nd Year
Dept. of PhysicsIIT Bombay
Project Guide :Dr. Punit Parmananda
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Outline of the Presentation
• Introduction• Stochastic Resonances• Coherence Resonance• The FHN Model• Simulations and Results• Future Plans
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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.
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Constructive Role of Noise
Coherence Resonance Stochastic Resonance
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STOCHASTIC RESONANCE
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Nonlinear SystemInput Signal
NoiseOutput
• Noise aids in Signal Transmission
• QUESTION : When is the transmission OPTIMUM ??
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ANSWER :
There is a FINITE OPTIMAL level of noise at which the response of the system ismaximum.
STOCHASTIC RESONANCE (SR)
NOISE is a FRIEND…!!
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What is it that happens inside the BLUE BOX ??
MECHANISM
Nonlinear SystemInput Signal
NoiseOutput
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WEAK periodic signal
NoiseOutput
Let’s try to Understand……
• Zero Noise ? : Particle oscillates within one well• Finite Noise ? : Particle can jump between the wells
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A Pinch of History….THE ICE AGE !!!!
Benzi et al (1981, 1982), C. Nicolis (1982)Why do ice ages recur periodically ?
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• Global climate Double well potential • Small modulation of earth's orbital eccentricity
Weak periodic forcing • Short term climate fluctuations Noise
The SR Explanation
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First Experimental Verification of SR
Schmitt Trigger Device - Fauve and Heslot (1983)
A cos(ωt) + Dξ(t)Output
Input
Signal to Noise ratio maximum at an optimal level of noise
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Applications of SR
Huge amount of applications throughout a large spectrum of fields. About 1000 publications since 1981 till date –• Optics• Biology• Neurology• Psychophysics
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An interesting example in Nature
The Cray fish can detect its predator more easily in the background of underwater turbulence.
Noise : Underwater turbulencePeriodic force : Water vibrations generated by the predator’s tail
Hungry Fish (Predator) Cray Fish having hydrodynamic sensors (Prey)
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COHERENCE RESONANCE
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Nonlinear SystemNoise Output
“Stochastic Resonance without External Periodic Forcing” (Gang et al PRL 1993)
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• 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)
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MECHANISM OF CR
Two time scales
Activation Time Excursion Time
• Time between end of one spike and beginning of another.• Strong dependence on Noise Intensity.• Follows Kramer’s-like formula – (Ta e(ΔV/D2) )
Pikovsky and Kurths et al PRL (1997)
• Decay Time of unstable state.
• Much weaker noise dependence.
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APPLICATIONS OF CR
• Neuronal and biological systems.• Chemical models.• Electronic circuits.• Semiconductor lasers.
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HOW DO WE MEASURE COHERENCE AND STOCHASTIC RESONANCES ??
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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)
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STOCHASTIC RESONANCE
Periodic Stochastic Resonance :• Co-efficient of Variation (VN)
Aperiodic Stochastic Resonance :• Cross Correlation Coefficient (C0) C0 = <(x1-<x1>t)(x2-<x2>t)>t
x1 Time Series of Aperiodic Input Signal
x2 Time Series of Noise Induced Output Signal <>t Time Average
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The FitzHugh Nagumo Model
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The Fitz Hugh Nagumo model, named after Richard FitzHugh (1922–2007) and J. Nagumo et al approximately at the same time, describes a prototype of an excitable system (e.g., a neuron).If the external stimulus exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables relax back to their rest values.This behaviour is typical for spike generations ( short elevation of membrane voltage ) in a neuron after stimulation by an external input current.The Fitz Hugh Nagumo model is a simplified version of the Hodgkin–Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.
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• a, D, ξ are parameters• |a| > 1 • |a| < 1• |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)
Stable focusLimit cycle
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SIMULATION AND RESULTS
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COHERENCE RESONANCE
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Time Series for LOW NOISE
Figure 1
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Time Series for HIGH NOISE
Figure 3
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Time Series for OPTIMAL NOISE
Figure 2
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COEFFICIENT OF VARIATION versus NOISE INTENSITY
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STOCHASTIC RESONANCES
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PERIODIC STOCHASTIC RESONANCE
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Time Series for LOW NOISE
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Time Series for HIGH NOISE
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Time Series for OPTIMAL NOISE
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COEFFICIENT OF VARIATION versus NOISE INTENSITY
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APERIODIC STOCHASTIC RESONANCE
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Time Series for LOW NOISE
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Time Series for HIGH NOISE
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Time Series for OPTIMAL NOISE
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CROSS CORRELATION COEFFICIENT versus NOISE INTENSITY
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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.
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• 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
BIBLIOGRAPHY
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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.
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THANK YOU FOR YOUR PATIENCE ANDKIND ATTENTION….