Stochastic Resonance and Receptor Function

Introduction

This simulation is another in the series of simulations dealing with the function of sensory receptors in organisms. Other simulations in this series allow you to investigate the basic aspects of receptor function such as dynamic range, gain, and frequency modulation. This simulation is intended to extend your understanding to more subtle aspects of receptor function.

This simulation is based on the function of a mechanoreceptor found in the tail of the crayfish. Functionally similar to the receptor cells in the cochlea of the vertebrate ear, this receptor is sensitive to the vibrations set up in the water by swimming fish. Output from the receptor connects via a reflex arc to the flexor muscles of the crayfish’s tail, and can cause strong and rapid contractions of the muscles. No input from the crayfish’s brain is required for this response, so this system functions similarly to the withdrawal reflex you have probably learned about in your physiology course. The effect of the flexor muscles’ contraction is to cause the crayfish to swim rapidly (backwards) away from the direction of the vibration’s source. Since fish are one of the principal predators of crayfish, the function of this receptor is clearly crucial to the crayfish’s survival.

Given the importance of the receptor’s function, we would expect that it would be to the crayfish’s benefit to have the receptor be able to detect weak vibrations set up by a fish while it’s still far enough away from the crayfish for the crayfish to make good its escape. It seems reasonable to expect the receptor to be very sensitive to weak vibrations, which physiologists would term a “weak, or low-amplitude, signal”.

Consider the problem of detecting a weak signal. We might expect the best design for a receptor would be to have it (i) very precisely tuned to the signal it’s supposed to detect, and (ii) able to reject all other forms of information. This would work well for a detector of a signal that was invariant, i.e., a signal whose frequency (for example) didn’t change so that the receptor would only need to detect a single frequency.

However, as always seems to be the case for living organisms, there are constraints to the design of the crayfish’s vibration receptor. It can’t be precisely tuned to a single frequency of vibration, if for no other reason than the vibrations it must detect will vary in frequency, depending on how rapidly the fish is swimming. In fact, the frequency of the sound probably conveys important information to the crayfish about what the fish is up to: is it swimming rapidly or slowly, is it accelerating rapidly (preparatory to an attack, for example), is it decelerating, etc.? That aside, it would probably be difficult or impossible for evolution to produce such a finely-tuned receptor anyway.

So, the crayfish’s vibration receptor necessarily ends up being sensitive to a range of frequencies. This in turn means the receptor is sensitive to vibrations generated by sources other than swimming fish. In other words, it becomes sensitive to noise, which we may define in this context as signals that convey no useful information, perhaps because they’re random or “stochastic” in their frequencies and amplitudes. Needless to say, the aquatic environment of the crayfish is full of noise in the form of vibrations set up by wind, waves, splashes, bubbles, ripples, swimming animals other than fish, animals walking along the shore, and so forth.

Common sense would tell most of us that all this noise would pose real problems for the crayfish. A priori, most would expect that noise would interfere with a receptor system’s ability to separate the wheat from the chaff, to detect the (perhaps weak) signal in the face of competing inputs from noise sources. However, common sense fails us in this circumstance. As you will learn when you perform the exercises, the crayfish’s vibration takes advantage of a phenomenon termed stochastic resonance, in which the presence of the ‘right kind’ and right amount of noise actually enhances the sensitivity of receptors to the signal they’re supposed to detect.

As you probably suspect from the term, stochastic resonance is envisioned as involving some sort of random (stochastic) process resonating with a regular, predictable periodic signal to produce a unpredictably large effect. The theory of stochastic resonance was originally developed in the early 1980’s to account for the abrupt shifts in earth’s climate (from ice age to warm interglacial and back again) that have occurred with an approximately 100,000 year cycle over the past one million years or so. Since then it has found applications in fields as disparate as biology, laser technology, and quantum theory.

This simulation is based on what is referred to in the literature as “non-dynamical threshold” stochastic resonance. In this model, a signal and a noise component are algebraically summed to yield a total stimulus impinging on a system. The system responds whenever the input exceeds a threshold. I have modified the basic model slightly by the inclusion of a threshold that you can vary. This feature is more realistic in that the threshold of neurons and receptors is often not constant, neither over short time scales (seconds to minutes) nor over evolutionary time scales.

There are three requirements for non-dynamical threshold stochastic resonance to occur. In terms of the features of the crayfish vibration receptor system, they are:

1. The output of the system exhibits a response threshold, giving no response to inputs that aren’t strong enough to drive the system to threshold, but giving an intense response once the threshold is reached. In the crayfish, the receptor’s membrane has a threshold V_m which, if exceeded, triggers action potential generation on its sensory neuron. This is what elicits the reflex contraction of the flexor muscles

2. The system responds to input that is periodic in nature but which may not be ‘strong’ enough by itself to drive the system to its threshold. The vibrations (sound waves) that the crayfish’s receptor is monitoring produces sine-wave fluctuations in water pressure, which is what the receptor is responding to. Low-amplitude vibrations may not contain enough energy to drive the receptor’s membrane to threshold.

3. The process is sensitive to input that is stochastic in nature. As discussed above, the crayfish’s receptor is sensitive to at least some of all the noise present in the crayfish’s habitat. Much of this noise is indeed stochastic in nature.

There are other situations in which stochastic resonance can occur, but we will not deal with those. They are in general more applicable to physical and chemical systems, although recent theoretical work suggests a number of applications to biological systems may be forthcoming.

What Does The Stochastic Resonance Simulation Allow You To Do?

Stochastic resonance depends on the ability of a receptor to respond to noise as well as to the stimulus it ordinarily transduces. Basically, the receptor uses the noise to depolarize its membrane towards threshold. Under the right conditions, this can actually facilitate the receptor's ability to detect faint signals. This simulation provides you with controls that let you vary the value of four parameters that contribute to the stochastic resonance phenomenon: the receptor's threshold, the amplitude of the signal, the amplitude of the noise, and the type of noise.

What Does The Stochastic Resonance Simulation Display?

When you initiate a simulation run, you will see a display similar to:

The dominant feature of the display consists of three blank panels. The top (yellow) panel will display two bits of information: first, whenever the receptor's V_m reaches threshold, a vertical red line representing an action potential will be drawn; second, whenever a signal is present, a short horizontal blue line will be drawn during the time that the amplitude of the signal is within 80% of its maximum amplitude. This feature is included to help you detect and interpret any patterns in the action potentials being generated by the receptor. The top panel’s display is only cleared when you click the Clear button.

The middle panel displays the membrane potential (V_m) of the receptor. In the absence of any signal (sound), this graph will display the receptor’s resting V_m. When a signal is present, this graph will display the receptor’s electrical response to the signal, i.e., the receptor potential. For this simulation, the resting membrane potential (i.e., V_m in the absence of any stimulus) of the receptor is set to -70 mV. Three lines will be drawn on this set of axes:

1. A thin horizontal red line representing the threshold V_m of the receptor will be drawn as soon as you activate the simulation. During a simulation run, any time the receptor's V_m is less negative than threshold V_m, an action potential will be generated and a 'spike' will be drawn in the upper (yellow) panel. Note that for sake of clarity the action potential itself is not displayed on the V_m axes.

2. A thick blue line will be drawn to represent the signal. In this simulation, the receptor transduces sound waves generated in the water by a swimming predator, so the signal is modeled as a sine wave function of time that induces corresponding changes in V_m.

3. A second thin red line will depict the combined effect of signal and noise on V_m.

Unlike the top panel’s display, the middle panel’s display is cleared every time you click the Go button to do a new run.

The bottom panel displays the intensity (loudness) of the output of the simulation’s noise-generating function. You have the choice of two ‘types’ of noise that you may use in your simulation runs. The type of noise is controlled by the two “Noise Function” checkboxes located immediately below the bottom panel of the display and labeled “Uniform” and “Gaussian”. The uniform distribution function randomly selects numbers from the range of 0 - 1, and each number has the same probability of being selected as every other number. The Uniform noise function thus approximates what we would term “white noise”. The mean of this distribution is 0.5, and the standard deviation is 0.3. The Gaussian distribution function randomly selects numbers from a Gaussian, or “normal”, distribution with a mean of 0.8 and a standard deviation of 0.5, and returns the absolute value of the selected number. The values returned by the noise-generating function are unitless.

At the right side of the display, you will see three slider/text field combinations that let you adjust the Amplitude of the Signal, the Amplitude of the Noise, and the receptor's Threshold V_m. The two Amplitude-control sliders serve as multipliers of the Signal or of the noise-generator’s output. Thus, the numerical value of the Signal slider is multiplied by the numerical value of the signal’s sine function, allowing you to control the amplitude (loudness) of the signal reaching the receptor. Similarly, the Noise slider’s numerical value is multiplied by the output of the simulation’s noise-generating function, giving you control over the amplitude (loudness) of the Noise reaching the receptor.

The Threshold slider actually controls the difference between threshold and resting V_m, which is why it's presented as a dV_m, rather than an V_m, value When you change the Threshold slider’s setting, you are in effect varying the sensitivity of the receptor to the incoming Signal + Noise combination.

The Signal and the Noise (after being multiplied by their respective slider’s value) are summed to yield the total signal, which is calculated relative to the resting value of –70 mV. The resulting V_m is represented by a thin red line graphed in the middle panel.

Near the top right of the display are two "Responses" text fields, labelled “Total” and “False”. Together, these two text fields give you (i) the total number of action potentials generated by the receptor during the simulation run (“Total”) and (ii) the number of those action potentials that were due to noise alone (“False”). The data displayed in these two text fields may be used to construct an index of the receptor's ability to discriminate true signal from noise, which will aid you in designing an 'optimal' receptor, which is one of the exercises accompanying the Stochastic Resonance simulation (for further discussion of the significance of the numbers displayed in the Total and False text fields, read the essay entitled What Does The False Response Text Field Signify?