Stochastic Resonance addition to BEMS paper
Kendra Krueger
University of Colorado at Boulder,
Electrical Engineering Department
October 2011
Kendra.Krueger@gmail.com
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.
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 (Benzi, Sutera, & Vulpiani, 1981). 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. This behavior can be modeled using non-linear bistable
equations.
Figure 1: Double well potential model of dynamical SR
One intuitive way to imagine a bistable system is to think of a particle in a double well,
illustrated in figure 1. 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 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 easily determine 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. This result was observed in a study which sought to measure the
effects of noise on the generations of action potentials in sensory nerves within crayfish
tailfins (Douglass, Wilkends, Pantazelou, & Moss, 1993). The SNR obtained matched
the behavior seen in figure 2, a fast rise to maximum SNR, and then a steady decrease as
noise intensity is increased.
This represents the classic SR response in a system,
suggesting an optimal noise level. The maximum signal to noise ratio occurs when the
noise energy is half the barrier energy [Bulsara 1996].
Figure 2: SNR behavior in SR systems
To further evaluate this type of system one can determine a power spectrum analysis.
This sort of method examines the critical frequency required to enhance the input signal.
This is direct evidence of a resonance phenomenon, a fundamental frequency at which
the system will uniquely respond to an external signal. The noise enhancement of the
power spectral density by crayfish mechanoreceptors has been measured to be 4.5dB at
the optimal noise intensity [Douglass 1993].
Figure 3: Power Spectrum behavior in SR systems
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. 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 (Gingl, Kiss, & Moss, 1995). 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 data, information which
would remain undetectable without the addition of noise.
Figure 4: 'Non-dynamic' SR model
For experiments which rely on this type of 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. Mechanoreceptors in rat afferent nerves
have shown SR behavior with the non-dynamical model (Collins, Imhoff, & Grigg,
1996).
Figure 5: Information Correlation behavior of SR
Biochemical Mechanisms of Stochastic Resonance
After these initial studies, researchers have attempted to dig deeper to find the
biochemical source of stochastic resonance. 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 (Kandel, Schwartz, &
Jessell, 2000). 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 (Goychuk & Hanggi, 2000). This calls to attention the probabilistic
behavior of the ion channel, which will, in effect, cause noise within the membrane
voltage potential. 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’ (RS).
Figure 6: The effects of neuronal noise on relative spread of firing probabilities
This relative spread can be directly correlated with an increase in channel noise.
This suggests that with higher amounts of noise, the probability of a response occurring
at low amplitudes is higher than for conditions with low noise. 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.
Closing Remarks
It 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 hindrance or resistance, and may
actually provide the more energetically favorable route.
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 (West &
Grigolini, 2010). 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.
Works Cited
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