Flávio C. Coelho
and Claudia T. Codeço
Department of Biology.
University of Texas at Arlington
phone: (817) 272-2871
FAX:: (817) 272-2855
Correspondence should be addressed to: Flávio C. Coelho.
Submitted for publication: November 11, 1998
Keywords: stochastic resonance, red noise, chemoreceptors, mechanoreceptors
The phenomenon of Stochastic Resonance (SR) has been described as the improved detection of weak signals by nonlinear systems, on the presence of optimal amounts of background random noise. Most studies of SR in biology have used white (infinite band) noise, although some authors have investigated the effect of colored noise on physical systems showing SR. Here we investigate by numerical experiments the importance of the input signal's frequency for the performance of the Fitzhugh-Nagumo neuronal model when the background noise is colored (narrow band), and discuss the implications of the properties of SR under red noise to the evolution of sensory structures and function.
In order to survive, an organism has to successfully perceive and extract useful information from the environment. This seems to be a difficult task since natural signals are often mixed with noise. However, Stochastic Resonance (SR) Theory suggests that nonlinearity and noise are actually the key to the extreme efficiency of sensory systems in retrieving weak (information rich) signals from the noisy surroundings. This phenomenon is easily understood if we consider the response of a sensory neuron to a weak (sub-threshold) periodic signal. Without noise, such signal is undetectable.
The addition of a small amount of noise to the signal increases the probability of a threshold-crossing event to occur (1; 2)when the signal reaches a peak. As more noise is added, this probability also increases as the signal's peak is boosted beyond the threshold. However, above certain level of noise, the probability of threshold-crossing events to occur at points other than the peaks increases significantly making the system respond to the noise dynamics instead of the signal's. In summary, there is an optimal amount of noise that allows the neuron to efficiently respond to a weak signal that, otherwise, would not be detected. Figure 1 illustrates a case in which sub-optimal amounts of noise are added to a sub-threshold signal. The system responds to some but not all of the signal upswings.
SR has been verified in experimental studies. In these, increasing levels of noise were applied in vitro to sensory cells of crayfish (2) and crickets (3) to measure neuronal response. The results are the same expected from simulation models. The correlation between signal and neural response increases as noise is added. After a maximum is reached, however, correlation drops. That is the signature of SR. As a rule, SR applies to any situation where there is a threshold value beyond which the system either switch to a different equilibrium, as in bi-stable systems, or make a large excursion through the phase space before returning to steady state, as in integrate-and-fire neuronal models.
Most studies on SR consider the effect of white noise. White noise is very efficient in boosting weak periodic signal no matter what the signal frequency is. Even aperiodic signals may be boosted (4). However, the infinite frequency band of white noise makes it poorly suited to represent the background noise in most biological systems. Hänggi et al. (5)refer to colored noise as a more "realistic noise". Sensory neurons are often encapsulated within some kind of tissue structure that acts as a band pass filter. As the high frequencies are filtered, the spectra of the noise that actually reaches the sensory unit is reddened (only low frequency components) (6).
In this study, we use the classical Fitzhugh-Nagumo neural model (FN) to illustrate how the coherence between signal and neural firing is affected when signal frequency is varied in the presence of red noise. Although the effects of colored noise on SR have already been discussed in the literature (7;8), a discussion on the interactions between the noise color and the input signal's frequency in the context of sensory biology is still lacking. FN is a two-dimensional limit-cycle oscillator that models the firing dynamics of neurons. Several investigations have addressed SR in FN models (9;10;11) under white noise.
MATERIALS AND METHODS
To study SR under red noise, we consider the following FN model:
where v (voltage) is the excitability of the system, w represents the combined forces that drives the system back to the resting state after an action potential (recovery variable), and the constants A (tonic activation signal), a, , and b were set for the same values utilized by Collins et al.(9) (a=0.5; b=0.15; =0.005). These parameter values lead to a voltage threshold of 0.211. S(t) is the sub-threshold periodic input signal (a simple sinusoid) and (t) represents the noise. The red noise utilized was obtained by passing a square window filter (low pass filter with a cutoff frequency of 0.09) over a gaussian white noise (as generated by MATLAB's gaussian random number generator) with zero mean and variance one. S(t) frequencies were chosen so that they would correspond to different regions of the noise spectrum (figure 2a).
The FN model was numerically integrated, for 5000 time steps, using a variable step-size Runge-Kutta algorithm. For the quantification of stochastic resonance we utilized the normalized power norm (9), i.e., the coefficient of correlation between the mean firing rate (MFR) and the input signal. The MFR was calculated by passing a square window filter over the spike train built from raw output of the model for the variable v.
Figure 1 illustrates a typical output of a simulation with a sub-threshold signal and sub-optimum amount of noise. Neuron firing occurs only at some peaks of the input signal. Figure 2b shows the effect of the addition of red noise on the correlation between signal and neuronal firing. The points are ensemble averages (50 runs of 5000 points each for each noise level) of the correlation between the mean firing rate and the input signal. Standard errors are also shown. The results are characteristic of stochastic resonance, with a quick rise of the correlation to a peak on a specific value of noise intensity followed by a gradual decrease for higher amounts of noise.
With respect to the effect of the red noise in contrast to white noise, some of the results predicted by Hänggi et al. (5 )for bistable systems can also be observed in the Fitzhugh-Nagumo model. (i) For the same signal-to-threshold distance, SR shows a reduced effect when driven by red noise as compared to white noise. (ii) The peak of SR is shifted to higher noise intensities when red noise is used. (Figure 3).
With the low frequency background noise (red noise), we observed that SR decreases drastically as the frequency of the input signal departs from the region of the noise spectrum that contains most of its energy (figure 2b). Figure 2a shows the spectrum of the red noise utilized superimposed with the spectra of the input signals utilized. The magnitude of the coefficient of correlation between mean firing rate and input signal is highest for the lowest frequency input signal, which is nearest to the peak of the red noise power spectrum. The highest frequency signal, whose frequency lies in a region of the noise's power spectrum of very low energy, is almost not boosted. The noise intensity of the SR peak is not affected by the frequency of the input signals, depending only on the amplitudes of both signals and noise.
DISCUSSION AND CONCLUSIONS
Red noise is less efficient than white noise to promote boosting of sub-threshold signals. However, the fact that red noise-driven stochastic resonance requires higher noise intensities to reach its peak may be translated into an important adaptation to biological sensors: the ability to benefit from SR at noise intensities that would be degrading for white noise driven SR. Moreover, living organisms are frequently operating "on the edge", i.e., every extra bit of information that can be obtained from the environment can mean the difference between success and failure. Thus organisms are frequently trying to detect signals that barely scratch the surface of the surrounding "sea of noise".
In many organisms, mechanoreceptors are composed of a sensory neuron which is frequently encapsulated in some sort of surrounding structure. This surrounding tissue may act as a band-pass filter causing the noise spectrum to converge on the particular frequency band to which the mechanoreceptor is tuned, and thus helping to improve the resonance effect. The use of internal noise sources is another possibility at an organism's disposal to boost its sensory capabilities. Although still speculative, these possibilities have strong implications on the evolution and function of sensory systems.
Experimental evidence supporting the role of SR in sensory neurons is still scarce but the few studies done so far are promissing. For a long time, mechanoreceptors were classified in terms of the stimulus frequencies they best responded to. Recently, Ivey et al. (12) found that the addition of white noise to periodic stimuli to rapidly adapting mechanoreceptors dramatically expands the range of frequencies to which the receptor responds. These findings are in accordance to the expected SR's dependence on a match between the frequency bands of both background noise and the stimulus frequencies. By widening the band of the available background noise, Ivey et al. (12) have been able to reduce the frequency specificity of the mechanoreceptor. Many examples are known where the discharge patterns of chemoreceptors may indicate the presence of SR. The resting activity of single fibers on the carotid sinus nerve (CSN) has been describe as random, possibly due to the random release of packets of excitatory neurotransmitters (13). This resting activity may serve as the background noise to enhance the detection of signals of interest. It is known that carotid bodies' chemoreceptors whose main function is the monitoring of PaO2 (arterial partial pressure of O2) levels are entrained by weak oscillatory signals of PaCO2 (arterial partial pressure of CO2). There is also evidence for variation of sensitivity depending on the frequency of the oscillatory signal. In the cat, the carotid sinus nerve discharge oscillations driven by the PaO2 signal are found to decrease and even disappear as the signal frequency increases from 15 to 30 /min even though the amplitude of the PaO2 oscillations remains constant (13). These CSN discharge oscillations disappear at breathing frequencies greater than 60/min. However, the system was show to follow PaCO2 oscillations of up to 72/min. This variation of the response of the system to varying input frequencies rather than being a physical limitation of the system, can be the result of the kind of frequency tuned stochastic resonance described in this paper.
To test experimentally if this behavior of chemoreceptors is derived from the property of stochastic resonance, it would be necessary to characterize the spectrum of the background neural "noise" and manipulate the oscillations of blood gases according to it. If the frequency at which the CSN discharge is best correlated with the oscillation of the input signals coincide with the band of maximal power of the background "noise" we would have evidence of the role of stochastic resonance in chemoreception.
There are many advantages to the wise use of background noise. Evidence is beginning to accumulate that noise is in fact used as a sensory boosting factor throughout the animal kingdom. This suggests that not only there is a clear co-evolution of environmental conditions and sensory systems, but also proves that organisms can adapt very rapidly to changes in their "informational" surroundings through quick "tune-ups" to their sensory apparatuses.
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