Biological Illustrations of Stochastic ResonanceKendra Krueger

University of Colorado at Boulder, Electrical Engineering DepartmentKendra.Krueger@gmail.com

Abstract: Stochastic resonance is a phenomenon which allows for signals to be amplified with the aid of background noise. The mathematical construct has just come of age in the past decade, thus only recently have biological manifestations been observed. This paper

will identify the theoretical concept of stochastic resonance, involving aspects of noise sources and information theory. Various mechanisms that have been discovered within

biological systems will then be identified and discussed with comparisons to the theoretical models. Furthermore, the broader implications of change within the current scientific

paradigm due to these new perspectives will be addressed.

A New Outlook on Noise

Noise has always been seen as a nuisance, the intrinsic harbinger of chaos; unavoidable

and inherent of the universe. Every system that lingers away from the ideal model will

expectantly be plagued by some sort of random fluctuations which will cloud or degrade the

output result. However, this perspective of noise and its properties may be shifting. In the past

ten years researchers have begun to observe strange benefits within noisy systems. This

phenomenon has come to be known as stochastic resonance.

Noise itself is defined to be random background fluctuations, which over time should

average to zero. The term stochastic describes a system which behaves randomly with no

predictable nature, or is dominated by noisy processes. In most systems when an input signal is

combined with noise, the output will also have an increase in noise. However, in stochastic

resonance (SR) systems the addition of noise to an input signal will actually improve the

resultant output (1).

Dynamical Stochastic Resonance

Two models exists to explain this phenomenon, the first is known as dynamical SR which

requires a weak periodic input signal and noise that is greater in magnitude than the input. Now,

intuitively we could imagine that adding these two signals together will create a very noisy

periodic function at the output, but this, in fact, is not the case. Figure 1 illustrates how a stable

periodic signal can be produced with only small fluctuations at the output. This behavior can be

modeled using non-linear bistable equations.

One intuitive way to imagine a bistable system is to think of a particle in a double well,

illustrated in figure 2. Here the particle will tend to rest at points of low potential energy. As it

does so, the potential energy function is also fluctuating periodically, and is susceptible to

random fluctuations. With just the energy provided by the periodic oscillation, the particle will

Figure 2: Double well potential model of dynamical SR

Figure 1: Black box representation of input/output SR

never leave its comfortable position. However, with the additional push from random

fluctuations the particle is capable of overcoming the barrier and repositioning itself within the

adjacent state, or potential. The behavior of this particle can be modeled using the following

equation:

Where x describes the position of the particle, U is the potential of the well, the sin function acts

as the periodic oscillation, and ) is the noise. We can envision that the periodic function

modulates the depth of the well, making one or the other deeper at any given time. The noise

provides the extra push to get the particle over the hill. This combination of functions allows for

the particle to be in either state depending on conditions. If we think of the periodic function as

being caused by an input signal or voltage that is changing the system or potential we can further

evaluate the ideal amount of noise needed to reach optimal behavior.

Most input/output systems can be evaluated using a signal-to-noise ratio(SNR). In a

common intuitive system, as noise is added the SNR will decrease, and the signal will become

degraded or destroyed. However, in a SR system the noise will actually improve the SNR. The

data in figure 2b was obtained from a system exhibiting SR. Here the SNR is measured as a

function of input noise. We can see that the ratio increases until an optimal noise value is met

whereupon the SNR then begins to decrease again. The output power spectrum of the system

can also be analyzed for SR behavior. Figure 3a illustrates the output power of the same system

as a function of noise intensity. One of the first observations of SR in a biological system used

SNR and power spectrum analysis to identify the SR behavior.

Researchers at the University of Missouri at St. Louis were investigating the effects of

noise on the generations of action potentials in sensory nerves within crayfish tailfins (4). Each

tailfin is composed of mechanoreceptors that have cuticular hairs capable of detecting motion in

the surrounding aqueous environment, illustrated in figure 4.

Figure 4: Hair cells located in crayfish tailfin

These cells are believed to aid in the localization of predators or prey. When a large number of

hair cells are activated, an action potential is generated in the sensory nerves. In their

experiment, they sought to measure the firing rate or ‘interspike interval’ of action potentials for

cells in vitro, with an artificial stimulus. The weak periodic stimulus was a 55.2Hz generated

sinusoidal function and the noise applied was Gaussian distributed at three different intensities.

Figure 5 shows the power spectra of the output nerve signal for noise intensities of 0, 0.14 and

0.44 V rms.

Figure 5: Power Spectrum in nerve cell output for three different noise intensities

The peaks in the power spectrum represent the fundamental frequency of the periodic function

(55.2 Hz) and the associated harmonics. When the noise intensity is increased, the amplitude of

these peaks also increases up to a certain critical noise level. After this point information on the

input signal will degrade. This is also illustrated in the SNR plots in figure 6.

Figure 6: Signal to noise ratio plots of mechanoreceptor spike trains

These data points were generated by first developing an interspike interval histogram. The graph

in figure 7 visualizes the firing of action potentials over time; the more narrow the spikes, the

more coherent the output information will be. By summing the amplitude of these peaks over

specific intervals the SNR can be calculated. The plot of SNR in figure 6 displays the

characteristic behavior of a SR process, suggesting that these cells are in fact capable of at least

imitating SR in a forced setting. The implications of this effect have been theorized to be

evolutionary beneficial to the creature, giving them the ability to detect predators in murky

waters.

Figure 7: Interspike Histogram

Non-Dynamical Stochastic Resonance

A model more applicable to biological systems is known as ‘threshold SR’ or ‘non-

dynamical SR’. In these systems the noise aids in enhancing a non-detectable signal in order to

reach a given detection threshold. This sort of system is visualized in figure 4. For non-

dynamical SR, the input signal does not have to be temporally periodic, instead the frequency of

the oscillations may vary in time around an average value (2). The rightmost image in figure 8

represents the phase interpretation of this frequency variant signal known as a Rossler attractor.

As the signal + noise combination crosses threshold, an encoded discretized signal is produced.

It is believed that encoded within this signal is information about the original Rossler attractor,

but that information would be undetectable without the addition of noise.

Figure 8: 'Non-dynamic' SR model

The bottom portion of figure 8 is another method of identifying the optimal noise

intensity of a SR system. For this experiment, and others which rely on discreet data or

threshold detection, information correlation is used as a type of SNR. This information

correlation examines the similarity between input and output bits. For example in a similar study

to crayfish mechanoreceptors, rat afferent nerves were examined for SR behavior, but the data

was analyzed using a correlation function.

In the previous study, the driving inputs relied on periodic signals which could be

appropriated to dynamical SR. However, further studies have been undertaken with the use of

aperiodic signals. J. Collins et al at Boston University determined that mechanoreceptors in rat

afferent nerves also showed SR behavior with the non-dynamical model (5).

The experiment made use of rat skin samples which connected to afferent nerves. The

stimuli consisted of a mechanical indentation mechanism which was actuated based on a

computer generated signal. The amplitude and frequency of the generated signal was aperiodic,

and was applied in 60 s intervals. These signals were designed to be below the threshold for

initiating an action potential within the afferent nerve. Gaussian white noise was also generated

and superimposed on the aperiodic signal. In each trial, the variance of the noise was altered,

thus changing the intensity.

The main difference in this sort of analysis method is the importance of the firing rate.

The assumption is made that information in the neuron is transferred via changes in the firing

rate. Therefore in this experiment they observed and analyzed R(t) the time-varying mean firing

rate, also known as the number of spikes per second produced by the neuron. Figure 9 portrays

first the input signal (a), the spike trains generated in the afferent nerve corresponding to the

mean firing rate over time, and the average firing rate as a function of input noise (b).

Figure 9: Mean firing rate data in rat afferent nerve with aperiodic noisy signal

Instead of computing the signal-to-noise ratio, an information correlation was derived

from the input signal S(t) and the output firing rate R(t). This was accomplished by determining

a normalized power norm C1, a cross-correlation function. In the below plots, figure 11 show C1

for three different neurons. As is familiar in the classic SNR plots, there is a steep rise to a

maximum for a critical noise level, at which point the normalized power norm then decreases.

The authors suggest that this model is proof that SR exists in mechanoreceptors for aperiodic

functions as well as periodic.

Figure 11: Normalized power norm for three separate neurons as a function of noise

Biochemical Mechanisms of Stochastic Resonance

After these initial studies, researchers attempted to dig deeper to find the biochemical

source of stochastic resonance. Looking into the general framework of the neuron we can break

it down into three components; the input (dendrites), the processor (soma) and the output (axon).

The output has two possible states, either low (resting), or high (action potential firing). This

output is generated depending on the temporal and spatial summation of the hyperpolarizing or

depolarizing inputs of the dendrites. If depolarizing inputs dominate and are strong enough in

amplitude, the soma generates an action potential which propagates through the cell to the axon.

In this model the major component determining the probability of an action potential firing is the

level of depolarization. Noise can interfere with the depolarization from two significant sources;

the on-going random fluctuations in the membrane potential, and the triggers which cause gated

ion channels to open and further depolarize a cell.

Ion channels are responsible for regulating the electrochemical equilibrium of the cell

membrane by allowing specific types of ions to flow inwards or outwards, creating voltage

modifying currents. Each channel is made of a long string of molecules which forms a protein

that transcends the lipid bilayers of the membrane, creating a pathway between the extracellular

environment and the internal cytoplasm of the cell. These proteins are binary or bistable in

nature exhibiting a conformation associated with either open or closed states, no median state

exists. Furthermore there are many different activation triggers for each ion channel. Some can

be chemically activated, tension activated (such as those in mechanoreceptor cells) or voltage

gated. Voltage gated channels have been studied predominantly in the case of stochastic

resonance and as such will be discussed further here.

It has been observed that the neuronal response to a repeated stimulus is not identical

every time. This calls to attention the probabilistic behavior of the ion channel, which in turn

will create noise within the membrane voltage potential (5). The current generated by ion

channels along with the variance can be derived using the following equation

In these equations represents the conductance of a single ion channel, N is the number of

channels, p(V) is the probability of the channel being open and V is the membrane potential.

From these equations the noise can be calculated and is known as the coefficient of variation

(CV)

Looking at this equation, it can be inferred that the noise should decrease proportional to

the square root of the number of channels. This calculation has manifested itself in the broad

assumption that at large densities of ion channels, the effective noise can be ignored. This,

however, is not the case. When examining the firing threshold in a Hodgkin-Huxley signal

neuron model, the probability of firing as a function of input amplitude is not discreet. Instead

there is a gradual rise indicating a factor of ‘relative spread’, later to be correlated to neuronal

nose sources. Figure 12 displays some experiments which investigated the effects of noise on

Relative spread (RS) .

Figure 12: Data from experiments on the effects of neuronal noise on relative spread of firing

probabilities

Figure 12a shows the firing of action potentials in a frog Ranvier with repeated stimuli of

equal amplitude. 12b illustrates the gradual increase in probability as a function of input

amplitude; here the RS is also identified. 12c shows similar simulations with three different

channel densities. In these plots the RS can be calculated using

also known as the error function.

These results suggest that neuronal noise is more involved in threshold activity than

previously thought. Soon new models may need to be modified in order to include the effects of

noise. The concept of RS can be applied to other biological systems which incorporate threshold

detection.

Closing RemarksIt is apparent that stochastic resonance has established itself as a legitimate process in

which systems are able to amplify a subthreshold signal with the addition of noise. In all of these

models, both dynamical and non-dynamical, the noise is visualized as being entrained on the

incoming signal, riding atop it and aiding in the transferring of information to the detection

mechanism. However, another point of view may be to see the noise as altering the threshold

itself. This concept was approached in the channel noise study, but not fully recognized.

Examining this phenomenon with this perspective may lead to new insights on how systems

develop these sorts of capabilities.

This further leads to questions about how systems directly adapt to use this process, or if

it is just a byproduct of evolution. One might ask, however, if nature strived to detect

subthreshold signals, than why not make the threshold lower? We can’t forget that noise isn’t

something new; nature has evolved in the presence of noise and thus sought to benefit from the

start. By utilizing the energy that is already present within the background, the system does not

need to build a more robust and perhaps less efficient detection system. Nature will always

follow the path of least resistance. Counter-intuitively noise, usually thought of as a hinderance

or resistance, may actually provide the more energetically favorable route.

As science begins to uncover the strange phenomenon of stochastic resonance, abound in

so many biological processes, the connotation and stigma of noise may begin to shift. Besides

from re-evaluating the place of noise in our current paradigms, it also becomes relevant to

observe the interactions between the structured nature of biology and the questionably un-

deterministic nature of a constantly fluctuating universe.

A new hypothesis has emerged known as the Living Matter Way (LMW) which theorizes

that the complexity of information is key to biological communication, and that noise is the

vehicle through which this complex information is transferred (8). This theory revolves around

the production and transmission of 1/f noise, a spectrum of noise discovered by Schottky in

electrical devices. This form of noise is inherent in all sorts of process from biological to

economic. What arises through this analysis is the idea of high densities of information being

encoded in a low energy system. It is interesting to suddenly hear a mention of energy in this

sort of context. In all of the preceding experiments the quest was always to determine the

relationships of input and output information, but there was no mention as to the energy that may

also be transferred in this process. Investigations into this aspect of the process will inevitable

conjure up discussion on the entropy rates associated with stochastic resonance amplification and

rectification.

Stochastic resonance not only sheds light on the complex functioning of the nervous

system, but also begins to pave the way for greater understanding of the fundamentals of

information transfer and the values of noise in our natural paradigm. Technological

developments have historically fought against the probabilistic fatalism of noise, but with

inspiration from these newly discovered phenomena, we may in time learn to embrace an

inexorably stochastic universe and thrive within the chaos.

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