Stochastic Resonance and its implications for the brain Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST

Deterministic vs stochastic appraoches

Ordinary differential equations Deterministic vs stochastic appraoches Ordinary differential equations Stochastic differential equations Discrete stochastic simulations

Sources of noise Intrinsic noise Noise resulting form the probabilistic character of the (bio)chemical reactions. It is particularly important when the number of reacting molecules is low. It is inherent to the dynamics of any genetic or biochemical systems. Extrinsic noise Noise due to the random fluctuations in environmental parameters (such as cell-to-cell variation in temperature, pH, kinetics parameters, number of ribosomes,...). Both Intrinsic and extrinsic noise lead to fluctuations in a single cell and results in cell-to-cell variability

Noise in biology

Noise in biology Conformation of the protein Post-translational changes of protein Protein complexes formation Proteins and mRNA degradation Transportation of proteins to nucleus ... Regulation and binding to DNA Transcription to mRNA Splicing of mRNA Transportation of mRNA to cytoplasm Translation to protein

Noise-producing steps in biology Noise in biology Noise-producing steps in biology Promoter state mRNA Protein Kaufmann & van Oudenaarden (2007) Curr. Opin. Gen. Dev. , in press

Effects of noise Imprecision in the timing of genetic events Destructive effect of noise Imprecision in the timing of genetic events Imprecision in biological clocks Phenotypic variations Constructive effect noise Noise-induced behaviors Stochastic resonance Stochastic focusing

What is Stochastic Resonance? A stochastic resonance is a phenomenon in which a nonlinear system is subjected to a periodic modulated signal so weak as to be normally undetectable, but it becomes detectable due to resonance between the weak deterministic signal and stochastic noise. The earliest definition of stochastic resonance was the maximum of the output signal strength as a function of noise (Bulsara and Gammaitoni 1996).

Ice-age cycle of Earth

Noise-aided hopping events Surmounting the barrier requires a certain amount of force. Suppose the ball is subjected to a force which varies in time sinusoidally, but is too weak to push the ball over the barrier. If we add a random “noise” component to the forcing, then the ball will occasionally be able to hop over the barrier. The presence of the sinusoid can then be seen as a peak in the power spectrum of the time series of noise-aided hopping events.

Signal and noise We can visualize the sinusoidal forcing as a tilting of the container. In the time series below, the gray background represents the time-varying depth of the wells with respect to the barrier. The red trace represents the position of the ball.

Physical picture of Stochastic Resonance If the particle is excited by a small sinusoidal force, it will oscillate within one of the two wells. But if the particle is also excited by a random force (i.e. noise plus sine) it will hop from one well to the other, more or less according to the frequency of the sine: the periodical force tends to be amplified. It can intuitively be sensed that if the particle is excited by the sine plus a very small noise it will hop a few times. In return, if the noise is too powerful, the system will become completely randomized. Between these two extreme situations, there exists an optimal power of input noise for which the cooperative effect between the sine and the noise is optimal.

Bistability in Stochastic Resonance

Power spectra of hopping events In this power spectra of hopping events, the gray bars mark integer multiples of the sinusoidal forcing frequency.

Kramers rate for Stochastic Resonance Physically, the sine posses a characteristic time that is its period. The dynamical system has also a characteristic time system that is the mean residence time in the absence of the sine, i.e. the mean (in statistical sense) time spent by the particle inside one well. This time is the inverse of the transition rate, known as Kramers rate, and is function of the noise level. (i.e. the inverse of the average switch rate induced by the sole noise: the stochastic time scale). For the optimal noise level, there is a synchronization between the Kramers rate and the frequency of the sine, justifying the term of resonance. Since this resonance is tuned by the noise level, it was called stochastic resonance (SR).

Peak SNRs correspond to maximum spatiotemporal synchronization. SNR of the middle oscillator of an array of 65 as a function of noise for two different coupling strengths, 0.1 and 10.

Stochastic Resonance in multi Array oscillators Time evolution (up) of an array of 65 oscillators, subject to different noise power and coupling strength. The temporal scale of the patterns decreases with increasing noise while the spatial scale of the patterns increases with increasing coupling strength. For this range of noise and coupling, spatiotemporal synchronization (and peak SNR) correspond to a coupling of about 10 and a noise of about 35 dB, as indicated by the striped pattern in the third column of the second row from the top.

Stochastic resonance in spatially extended systems  In spatially extended systems of coupled stochastic units, each exhibiting SR, driven by uncorrelated noises and a common periodic signal, the maximum SNR from a single element can be enhanced for proper coupling, due to noise-assisted spatiotemporal synchronization between the periodic signal and the system (array-enhanced SR) [e.g., J.F. Lindner et al., Phys. Rev. Lett. 75, 3 (1995)].  SR in spatially extended systems can be observed also for signals periodic only in space (without time dependence). In this case, periodic spatial structures are best visible in the system response for optimum noise intensity [e.g., Z. Néda et al., Phys. Rev. E 60, R3463 (1999); J.M.G. Vilar, and J.M. Rubí, Physica A 277, 327 (2000)].  The phenomenon of SR was also observed for spatiotemporal periodic signals, e.g., in the Ising model with thermal noise, driven by a plane wave [L. Schimansky-Geier, and U. Siewert, in Lecture Notes in Physics, ed. L. Schimansky-Geier, T. Pöschel, vol. 484, p. 245 (Springer, Germany, 1997)]. The enhancement of SR due to coupling is also observed, however, the strength of the effect is weaker than in the case of spatially uniform, periodic in time signal.

Stochastic resonance paddle fish Here, we show that stochastic resonance enhances the normal feeding behaviour of paddle fish (Polyodon spathula) which use passive electroreceptors to detect electrical signals from planktonic prey (Daphnia).

Bioengineering applications of stochastic resonance for noise-enhanced balance control

People always sway a small amount even when they are trying to stand still. The amount of sway increases with age. But under the influence of a small amount of vibration, which improves the mechanical senses in the feet, both old and young sway much less. Remarkably, noise made people in their 70s sway about as much as people in their 20s swayed without noise.

Balancing act using vibrating insoles Using a phenomenon called stochastic resonance, the human body can make use of random vibrations to help maintain its balance. In experiments on people in their 20s and people in their 70s, actuators embedded in gel insoles generated noisy vibrations with such a small amount of force that a person standing on the insoles could not feel them. A reflective marker was fixed to the research subject's shoulder, and a video camera recorded its position.

SR-based techniques SR-based techniques has been used to create a novel class of medical devices (such as vibrating insoles) for enhancing sensory and motor function in the elderly, patients with diabetic neuropathy, and patients with stroke.