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

Outline of the Presentation

• Introduction• Stochastic Resonances• Coherence Resonance• The FHN Model• Simulations and Results• Future Plans

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.

Constructive Role of Noise

Coherence Resonance Stochastic Resonance

STOCHASTIC RESONANCE

Nonlinear SystemInput Signal

NoiseOutput

• Noise aids in Signal Transmission

• QUESTION : When is the transmission OPTIMUM ??

ANSWER :

There is a FINITE OPTIMAL level of noise at which the response of the system ismaximum.

STOCHASTIC RESONANCE (SR)

NOISE is a FRIEND…!!

What is it that happens inside the BLUE BOX ??

MECHANISM

Nonlinear SystemInput Signal

NoiseOutput

WEAK periodic signal

NoiseOutput

Let’s try to Understand……

• Zero Noise ? : Particle oscillates within one well• Finite Noise ? : Particle can jump between the wells

A Pinch of History….THE ICE AGE !!!!

Benzi et al (1981, 1982), C. Nicolis (1982)Why do ice ages recur periodically ?

• Global climate Double well potential • Small modulation of earth's orbital eccentricity

Weak periodic forcing • Short term climate fluctuations Noise

The SR Explanation

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

Applications of SR

Huge amount of applications throughout a large spectrum of fields. About 1000 publications since 1981 till date –• Optics• Biology• Neurology• Psychophysics

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)

COHERENCE RESONANCE

Nonlinear SystemNoise Output

“Stochastic Resonance without External Periodic Forcing” (Gang et al PRL 1993)

• 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)

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.

APPLICATIONS OF CR

• Neuronal and biological systems.• Chemical models.• Electronic circuits.• Semiconductor lasers.

HOW DO WE MEASURE COHERENCE AND STOCHASTIC RESONANCES ??

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)

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

The FitzHugh Nagumo Model

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.

• 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

SIMULATION AND RESULTS

COHERENCE RESONANCE

Time Series for LOW NOISE

Figure 1

Time Series for HIGH NOISE

Figure 3

Time Series for OPTIMAL NOISE

Figure 2

COEFFICIENT OF VARIATION versus NOISE INTENSITY

STOCHASTIC RESONANCES

PERIODIC STOCHASTIC RESONANCE

Time Series for LOW NOISE

Time Series for HIGH NOISE

Time Series for OPTIMAL NOISE

COEFFICIENT OF VARIATION versus NOISE INTENSITY

APERIODIC STOCHASTIC RESONANCE

Time Series for LOW NOISE

Time Series for HIGH NOISE

Time Series for OPTIMAL NOISE

CROSS CORRELATION COEFFICIENT versus NOISE INTENSITY

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.

• 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

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.

THANK YOU FOR YOUR PATIENCE ANDKIND ATTENTION….