INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Stochastic resonance (SR) is a phenomenon whereby the detection of a low-level signal is enhanced in a nonlinear system by the introduction of noise. SR originally was used to explain the periodicity of ice ages (Benzi et al. 1981) and since also has been detected in Schmitt triggers (Fauve and Heslot 1983), ring lasers (McNamara et al. 1988), sensory neurons (Bulsara et al. 1991; Collins et al. 1996; Douglass et al. 1993; Levin and Miller 1996; Pei et al. 1996), and other applications (Wiesenfeld and Moss 1995). In each system, the detection of a subthreshold signal was improved when a certain level of noise was added. The discovery of SR in sensory neurons has led to new insights about the role of noise and signal detection in the peripheral nervous system.

The physiological effect of SR in neurons is enhancement of signal detection, which is a primary function in many neurons. SR is characterized by a specific relationship between the signal-to-noise ratio (SNR) and the noise intensity. This relationship, describing how random fluctuations will affect the crossing of a barrier, originally was derived based on Kramers rate theory and applied to bistable phenomena (for review, see e.g., Dykman et al. 1995; Moss et al. 1994; Wiesenfeld and Moss 1995). The theory then was expanded to include monostable ("circular") states (Stocks et al. 1993; Wiesenfeld et al. 1994). Equation 1 is the monostable solution for SNR with varying amounts of noise, where  is the signal strength, U is the threshold barrier height, and D is the noise intensity.

(1)

The curve defined by Eq. 1 contains a resonant peak corresponding to the level of noise intensity that produces maximum SNR. This equation is valid for a periodic input into a monostable system, which is a good representation of a neuron receiving a periodic signal. Many sensory cells function in an inherently noisy background, which seemingly would make detection more difficult. For example, the crayfish mechanoreceptor is exposed to a persistent noisy input from turbulence in a stream. The noise created by the stream can improve signal detection due to the effects of SR (Douglass et al. 1993). Like sensory neurons, the primary function of many central neurons is signal detection in a noisy environment. SR has been observed in rat hippocampal brain slices when a sinusoidal signal and Gaussian noise are administered with parallel field electrodes to a slice in high-potassium solution (Gluckman et al. 1996). However, SR also could be involved in the detection of subthreshold synaptic inputs by central neurons with physiological noise sources.

In the CNS, determining what is "signal" and what is "noise" is a complex task. In terms of detection of any particular signal, any other input that is uncorrelated with that signal can be considered "noise" in SR analysis. This fact makes SR a powerful tool for analysis of signal detection in the CNS, especially when considering the number of connections some neurons have. Hippocampal CA1 cells have large apical dendritic trees that receive tens of thousands of synaptic inputs: a large potential noise source. Because multiple coactive synaptic connections are required to discharge the cell (Andersen 1990), there is a large potential source of subthreshold signals. In addition, the length of the apical dendrites (500 µm) produces a long electrotonic distance (Rall 1977). Synaptic inputs located at distal positions on the tree can attenuate to a point where they cannot have any significant summative effects on the soma (Spruston et al. 1993). Previous work has proposed that active channels in the dendrites could be responsible for amplifying the distal signals (Cook and Johnston 1997; Magee and Johnston 1995; Poolos and Kocsis 1990). SR describes an alternative mechanism capable of enhancing detection of distal synaptic inputs with the presence of noise. We have tested the hypothesis that SR can improve detection of subthreshold synaptic inputs in the presence of synaptic noise. Results were obtained using computer simulations of hippocampal CA1 cells and in vitro hippocampal slices. The noise levels capable of evoking SR were compared with published physiological noise data to evaluate the role of SR under normal physiological conditions.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

NEURON model

A CA1 cell was modeled with the neural modeling software NEURON (Hines 1993) with the parameters shown in Table 1. The neuron was constructed as shown in Fig. 1A using equivalent dendritic trees (Rall 1977). In the dendrites, the membrane specific resistance (R_m) was halved and capacitance (C_m) doubled to account for the increased surface area of dendritic spines (Spruston and Johnston 1992). The dendrites contained no active channels. The soma contained one sodium, one calcium, and four active potassium channels adapted from a previous model of CA1 cells (Warman et al. 1994) (see APPENDIX).

View larger version (25K): [in this window] [in a new window]   Fig. 1. Computational CA1 model for evaluation of stochastic resonance (SR). A: model was configured to contain 5 cylinders with the dimensions in Table 1. "Noise" sources were placed at each of the positions indicated by the crosses in the dendrites. "Signal" was added to the cell at positions a, b, and c, corresponding to the electrotonic lengths indicated. B: effect of noise on a 5-Hz input signal (top). When noise is applied, the cell fires action potentials (middle) synchronized with the input. A pulse is generated whenever an action potential occurs (membrane voltage >0 mV) (bottom). Time series of detected output pulses is used for calculation of the power spectral density. C: plot of the power spectrum of the detected series in B. Signal-to-noise ratio (SNR) is computed by dividing the power at the signal frequency (indicated by the head of the arrow) by the average baseband noise level near the input frequency (indicated by the tail of the arrow).

Synaptic transmission was simulated using a model of AMPA synapses. Equations 2 and 3 describe the current from the AMPA synapse, where g_max is the peak conductance and [O] is the fraction of open channels (Destexhe 1998; Destexhe et al. 1994, 1998).

(2)

(3)

The peak current amplitudes of both the synaptic input signal and the synaptic noise were adjusted by changing the g_max parameter. For the remainder of this paper, g_max will be used as the indicator of input amplitude. Each synaptic channel was designed to represent a population of synapses on a segment of the dendritic tree. The subthreshold signals to be detected were introduced as synaptic events occurring at 5 Hz with g_max^s adjusted to a value that did not produce a somatic action potential. The noise signal was added to simulate random synaptic events on the CA1 dendrites. Although spontaneous synaptic events contain both AMPA and N-methyl-D-aspartate (NMDA) receptor components (McBain and Dingledine 1992), only AMPA receptor components were used in this simulation because the AMPA portion dominates in normal artificial cerebrospinal fluid (ACSF) (Manabe et al. 1992). Each noise synapse fired action potentials independently at random intervals, a Poisson process (Cox and Lewis 1996). Events were triggered at random intervals by generating a uniform-distributed random number (0-1) at every time step and comparing it with a threshold value. Triggering occurred whenever the random number was lower than the threshold. For a uniform-distributed set of random numbers from 0 to 1, the cumulative distribution function yields F_X(x) = x, so the threshold parameter value was equal to the event probability for one time step. The threshold was chosen to produce noise events with a mean interval of 0.05 (20 Hz) and 0.01 s (100 Hz). A histogram and a power spectrum were obtained to verify the uniformity of the random-number generator and white spectrum of the random event frequency (data not shown). The amplitude of each noise event was modulated with the g_maxⁿ parameter. To plot the data as a function of noise intensity D, the variance of the input noise for each level of g_maxⁿ was computed by measuring the current entering the synapse. The variance was directly proportional to the square of the input amplitude.

Experimental methods

PREPARATION OF SLICES. Mature Sprague-Dawley rats were anesthetized and decapitated, and their brains were removed quickly. Surgery followed protocol approved by the University Animal Resource Center. Hippocampal slices 400-µm thick were cut using a Starrett tissue chopper and placed in a perfusion chamber. Slices were bathed for over an hour in room temperature ACSF consisting of (in mM) 124 NaCl, 3.75 KCl, 1.25 KH₂PO₄, 2 CaCl₂, 2 MgSO₄, 26 NaHCO₃, and 10 dextrose and aerated with 95% O₂-5% CO₂. Each slice was then transferred to another perfusion chamber and incubated to 35°C for implementation.

ELECTRODE PLACEMENT AND SIGNAL GENERATION. Tungsten microelectrodes were used to introduce signal and noise in the form of synaptic events on CA1 cells (see Fig. 2). The signal electrode was placed in the Schaffer collateral layer of the hippocampal slice. The noise electrode was placed in the stratum oriens. A glass microelectrode filled with 150 mM NaCl was used for extracellular recording in the CA1 region. Extracellular recording was amplified using an Axoclamp 2A and recorded onto digital audio tapes.

View larger version (10K): [in this window] [in a new window]   Fig. 2. Evaluation of SR in hippocampal slice. Schematic of in vitro experiment. Recording electrode was placed extracellularly in the CA1 region. Noise electrode was placed in the stratum oriens, inputting randomly spaced, uniform-width (250 µs) current pulses. Signal electrode injected 250-µs, 5-Hz current pulses into the Schaffer collaterals. Thus the signal pulses synapsed onto the apical dendrites and the noise pulses onto the basal dendrites.

EVALUATION OF SR. In accordance with previous work on SR (Douglass et al. 1993; Gluckman et al. 1996; Wiesenfeld and Moss 1995), raw voltage data were converted to an output time series that contained binary pulses corresponding to the occurrence of action potentials (see Fig. 1B). The power spectral density (PSD) of the detected output was obtained in Matlab (50,000 point Hanning window; 45,000 point overlap). The SNR was calculated by the dividing the power at 5 Hz by the average of the baseband noise power on both sides of the 5-Hz peak (see Fig. 1C). Therefore any action potential in phase with the input signal is regarded as "signal" by contributing power to the input signal frequency. Spikes that are not in phase with the 5-Hz signal contribute to other frequencies, creating the baseband noise. Data were fitted to Eq. 1 by using the Microsoft Excel (Excel 97) solver to minimize the error by changing the U and  parameters.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Computer simulation

The goal of the simulations was to test whether detection of a subthreshold signal can be improved by SR with synaptic noise sources. The simulations investigated the noise characteristics necessary to produce SR in a CA1 cell model. The effect of both distributed and single noise sources on the SNR were tested.

DISTRIBUTED NOISE SOURCES. The model first was configured to generate noise at multiple synapses throughout the neuron simultaneously. An independent noise source was placed at each of the ten positions indicated by the crosses in Fig. 1A. This distribution is representative of the physiological synaptic noise in CA1 (Andersen 1990; Bekkers et al. 1990). Signal synapses were placed at three positions along the apical dendrite to model synaptic events located at different distances from the soma. The same input intensity (g_max^s) was used for each position. The intensity was chosen to be subthreshold for the distal positions (b and c) but above threshold at the proximal position (a). An independent simulation was performed for each signal position.

View larger version (16K): [in this window] [in a new window]   Fig. 3. SR in computational model with basal noise source. A: SNR obtained with 10 distributed noise sources. Response to signal at each of the 3 positions is shown with error bars for standard deviation. Data from positions b and c are shown fitted to the SR equation. At position a, SR is not present because the signal was not subthreshold. As the noise increases, SNR at position a decreases as expected. Detection at the subthreshold positions increases as predicted by SR. B: expanded plot from A, showing the SR effects even at very low noise levels. - - -, input noise that resulted in 12,000 µV2 variance at the soma, the physiological baseline. There are 1 data point in b and 2 in c that resulted in 0 SNR. C: equalizing effect of noise on detection of electrotonically distant signals for both 20- and 100-Hz noise. Data are arranged to show how detection for 4 different noise levels (shown in the legend in µV2) changes with electrotonic distance. Same data are in the "20 Hz noise" plot as in A. For small noise intensity (0.36 and 0.2 µV2), signal at positions b and c cannot be detected, whereas a has nearly perfect detection. As the noise increases, SNR at position a decreases. Detection at the subthreshold positions, however, increases as dictated by SR. For higher levels of noise, the difference in SNR between neighboring positions becomes progressively less. Of note are the nearly flat lines at higher noise levels, which was more pronounced for the 100-Hz noise. Although the SNR has been reduced greatly proximally, detection is much less dependent on the position of the inputs, equalizing signal detection from electrotonically distant inputs.

SINGLE NOISE SOURCE. The model then was configured to analyze the effect of the number and position of the noise synapses on SR. A periodic synaptic signal was applied to position b on the apical dendrite. The minimum signal amplitude that produced a somatic action potential was g_max^s = 8 nS; therefore the simulation was run using subthreshold values of 7 and 5 nS. A single noise synapse at the midpoint of the basal dendrite was activated (see Fig. 1A). Because only one source was active, the event probability was increased 10-fold from the distributed-source simulation to maintain mean noise frequencies of 20 and 100 Hz.

View larger version (11K): [in this window] [in a new window]   Fig. 4. SNR obtained by single noise sources. A: signal synapse was placed at position b (Fig. 1A) and gmaxs set to 2 subthreshold values. Data are fit to Eq. 1. Increased noise from 1 synapse produced SR just as in the previous simulation. Similar results were obtained when the single noise source was moved to different positions on the neuron. SNR improvement was greater for the signal closer to threshold. B: expanded plot from A, indicating physiological noise level. SNR for the 7-nS signal shows considerable improvement even at low noise.

Experimental results

To detect the presence of SR in CA1 cells, an in vitro experiment was designed to test the prediction of the second simulation protocol. Electrodes were placed in the stratum radiatum and s. oriens to introduce periodic inputs and random synaptic events, respectively (Fig. 2). Ten experiments were performed using slices from six different rats. Extracellular population spikes were recorded, and a threshold was chosen to represent a critical number of cells firing (a population spike of 200 µV). Noise improved detection of a subthreshold signal in all slices tested. An example of the effect of noise is shown in Fig. 5A. The addition of synaptic noise clearly improves detection of the signal within a certain range of noise intensity. All 10 experiments exhibited the characteristic SR curve (Fig. 5C). For low noise intensity, the SNR was increased, at times reaching values over 100, a result comparable with those obtained in the simulations. As the noise increased, it evoked action potentials independent of the periodic signal, adding false positive outputs and decreasing the SNR. There was no appreciable change in SR parameters for the three values of mean noise frequency generated (not shown). In addition to the subthreshold signal data, Fig. 5B shows the result of noise added to a suprathreshold signal. As demonstrated by the simulations (Fig. 3), high noise serves to equalize the detection of signals with different thresholds in the slice. The results show that SR is present in CA1 cells as predicted in the simulation and has a similar effect of equalizing detection of sub- and suprathreshold signals. Although it is not possible to compare directly the noise levels used in the experiment and the simulation, the results are qualitatively the same. The SNR profiles for both the simulation and the experiment exhibited similar SR behavior. The SNR reached similar values and in both cases the data fit the SR equation. This is, to our knowledge, the first demonstration of SR in hippocampal neurons using synaptic inputs.

View larger version (31K): [in this window] [in a new window]   Fig. 5. SR in hippocampal slices. A, top: reference input at 5 Hz. Bottom: raw extracellular voltage data. Noise pulses are added to the slice as shown by the bar above the data. Scale is indicated by the bar in the right. Large, thin spikes are truncated stimulation artifacts. - - -, threshold detected by the processing, ignoring artifacts. Output trace below the data shows the resultant detection, producing a time series detected output equivalent to those produced for the computer simulation in Fig. 1. Two boxed portions of the raw data and detected output are shown in expanded form as indicated. B: same slice as in A, except with a larger noise input. These data show how more noise causes "false" action potentials that decrease the SNR. C: averaged SNR data from 10 experiments are shown as a function of squared noise pulse amplitude. For all slices tested, amplitudes 400 nA2 remained subthreshold. More than 3,600 nA2, the slices began to saturate and the response degrade, so no values above this level were recorded. As seen in the data, the average response has the characteristic SR curve. Each of the 10 individual experiments showed similar results. Also included is the data from a typical suprathreshold signal. Comparison of the suprathreshold response with the average subthreshold response shows a result similar to that in Fig. 3C, where high noise variance equalizes the detection of supra- and subthreshold signals.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

SR can improve detection of distal signals

SR in central neurons such as CA1 may help explain how distal synapses can be detected. Even with completely passive dendrites, we have shown that attenuated signals are detectable when synaptic noise is added to the neuron. SR thus explains a possible method that CA1 cells can use to improve detection of distal signals without the need for active channels in the dendrites.

Because the apical dendrites are so long, distal synapses in CA1 often are attenuated greatly and cannot generate action potentials. How do these distal synapses contribute to somatic integration? SR provides two answers to that question. First, because noise can improve detection, a source of noise is needed for there to be any effect. The high number of synapses in CA1 potentially provide a large, highly variable noise source. Second, our simulation has shown that SR can make detection less dependent on signal position. This finding agrees with experimental evidence showing that equivalent signals applied to various positions on the dendritic tree are detected equally at the soma (Andersen 1990), and is verified by the data in Fig. 5B, showing equalized detection of sub- and suprathreshold signals from a single position. Although this effect clearly relies on decreasing the fidelity of proximal signal detection, it provides a possible explanation of the experimental data using only passive dendrites. The effect could be enhanced if the model included active dendritic channels, which amplify distal signals (Magee et al. 1998). Because of these results, we suggest that SR using synaptic noise may serve to effectively decrease the electrotonic length in a cell, preferentially aiding detection of more distal synapses and decreasing dependence on input position under high-noise circumstances.

Choice of synaptic inputs

Although SR classically has been studied using sinusoidal input signals, the results reported here used a periodic pulse train input because pulses are required for synaptic inputs in most neurons. Our results agree with recent work showing that a pulse train can evoke SR (Chapeau-Blondeau et al. 1996). The periodicity of the input signal allows the use of power spectral density as a measure of the response to the signal input; however, an aperiodic input analyzed with other correlative methods would be expected to give similar SR results (Collins et al.1996; Heneghan et al. 1996; Levin and Miller 1996; Pei et al. 1996). SR is therefore a method of improving detection of any signal in a neuron regardless of whether the signal is a sinusoid or even aperiodic.

Physiological relevance of SR in CA1 neurons

For the presence of SR in neurons to be relevant, the effect must occur within physiological noise levels. There are more than 10,000 synapses on each CA1 pyramidal cell (Andersen 1990), creating the potential for a significant amount of synaptic noise. Brain-slice experiments have shown an intracellular noise variance of ~12,000 µV² (Turner 1988; Wahl et al. 1997). To evaluate SR in CA1 cells, the noise levels used in the data need to be compared with this physiological level.

The somatic noise variance was calculated by recording the voltage at the soma for each level of noise intensity and computing the variance of each data set in squared microvolts. The synaptic input that produced 12,000 µV² at the soma was 1.7 pA². This baseline level is indicated by a dashed line in Figs. 3B and 4B. The baseline falls within a region where the SNR clearly is enhanced by the noise, especially for signals close to threshold. This result strongly suggests that endogenous noise can improve detection of subthreshold signals. Because even the noise present in a brain slice is capable of generating SR, it is probable that CA1 cells and many other CNS neurons use SR to detect small signals. Furthermore, because the baseline lies on the steep upward slope of the SR curve, any small increase in noise intensity will produce a large improvement in signal detection.

Although these results show that endogenous noise could play an important role in signal integration in CA1 cells, the peak SNR levels occur at noise intensities much higher than the baseline level. The peak for synapse b in Fig. 1C occurred at 181,900 µV². Can physiological noise produce such high variance, thereby maximizing the detection predicted by SR? The answer to this question requires consideration of several noise sources that exist in the neural tissue. Our model includes noise similar to spontaneous miniature excitatory postsynaptic potentials (mEPSPs) because they are random synaptic events on AMPA receptors. The frequency of mEPSPs has been shown to be only ~1 Hz (Manabe et al. 1992; Wyllie et al. 1994), but to produce any noticeable effect, the frequency of the noise events had to be increased well 1 Hz. However, mEPSP frequency is quite dependent on experimental conditions such as stimulation frequency (Manabe et al. 1992), bath calcium (Raastad et al. 1992), and temperature (Finch et al. 1990) and is increased by paired pulse modulation (Mennerick and Zorumski 1995) and long-term potentiation (Malgaroli and Tsien 1992). In fact, noise representing only mEPSPs1-Hz pulses injecting a charge of 100 fC (Bekkers and Stevens 1990; Goda and Stevens 1994; Larkman et al. 1997; Manabe et al. 1992)produced a variance <6,000 µV². This result agrees with the finding that minis produce only a fraction of the total endogenous noise in slices (Wahl et al. 1997). Other forms of random synaptic activity also will contribute to synaptic noise. For example, random axonal firing contributes to synaptic noise, evoking larger amplitudes than minis (Turner 1988; Wahl et al. 1997). Uncorrelated action potentials from CA3 cells can greatly increase noise variance because any signal uncorrelated with the desired input can be considered "noise" in terms of signal detection.

The most important sources of noise, however, are those not active in a slice preparation but present in vivo. In vivo recordings demonstrate that the normal input to the hippocampus is behavior dependent, containing theta rhythms (White et al. 1998), random pulse trains (Leung 1982), and the powerfully depolarizing sharp waves (Kamondi et al. 1998). Studies of other central neurons reveal that there exist active neuronal states in vivo that are not present in slices (Wilson and Kawaguchi 1996) and can increase baseline variance 100-fold (Destexhe and Paré 1999). Taking into account these additional sources, we estimate that even normal in vivo activity could easily produce background noise 100 times greater than the 12,000 µV² baseline observed in slices, placing the SNR of the cell in the peak region of signal detection predicted by SR. This agrees with previous work suggesting that arrays of neurons have greatly increased detection (Moss and Pei 1995), an effect that also has been studied in SR systems (Inchiosa and Bulsara 1995; Lindner et al. 1995). Therefore, the amplitude of the noise can span a broad spectrum of variance, with a minimum value 12,000 µV² and an arbitrarily large maximum level. According to our simulations, this range encompasses the entire region of interest of the SR equation for near-threshold signals. Because of our finding that even minimum, endogenous noise levels are capable of producing SR, we hypothesize that SR is present in any threshold-bearing central neuron for signals near threshold. The neural response can be modulated by factors affecting the background noise. Quiescent states, with very low noise variance, will preferentially detect signals close to threshold and be very sensitive to small changes in the noise. Active states will move the SNR beyond the SR peak, to the region where detection is less dependent on signal strength or synaptic position. SR thus predicts a novel method of modulating signal detection in the CNS.

Conclusion

Many systems have been shown to use SR as a method of improving signal detection. Our results demonstrate that SR can function in rat hippocampal CA1 cells using only synaptic inputs for both signal and noise. The levels of noise necessary for SR were within practical physiological limits. SR provides a possible explanation for the finding that distal and proximal synapses are detected equally. Other theories to explain this phenomenon, such as the role of active voltage-gated channels in the dendrites (Magee et al. 1998), could be synergistic with SR to improve detection of the distal signals. SR can improve detection of many different subthreshold signals and so may help explain detection of low-amplitude inputs such as very weak electric fields (Weaver and Astumian 1990) and extremely low-frequency signals (Kavet and Banks 1986). Perhaps most intriguing is the finding that even minimum noise levels improve detection of some subthreshold signals. Noise thus could play a major role in signal processing by CNS neurons, both in slices and in vivo.