Abstract
Neural oscillations, which appear in several areas of the nervous system and cover a wide frequency range, are a prominent issue in current neuroscience. Extracellularly recorded oscillations are generally thought to be a manifestation of a neural population with synchronized electrical activity resulting from coupling mechanisms. The vertebrate olfactory neuroepithelium exhibits βband oscillations, termed peripheral waves (PWs), in their population response to odor stimulation. Here, we examine PWs in the channel catfish and propose that their properties could be explained as the superposition of asynchronous oscillators. Our model shows that the intriguing random pattern of amplitudemodulated PWs could be explained by Rayleigh fading, an interference phenomenon well known in physics and recognizable using statistical methods and signal analysis. We are proposing a mathematical fingerprint to characterize neural signals generated by the addition of random phase oscillators. Our interpretation of PWs as arising from asynchronous oscillators could be generalized to other neuronal populations, because it suggests that neural oscillations, detected in local field potential recordings within a narrow frequency band, do not necessarily originate from synchronization events.
 peripheral waves
 Rayleigh fading
 olfactory epithelium
 olfactory neuron
 Kuramoto model
 neuronal synchronization
Introduction
For over 15 years, neuronal synchronization has been considered a fundamental step in sensory, motor, and cognitive processing (Buzsaki and Draguhn, 2004; Lledo et al., 2005). In some systems, neuronal synchronization was detected by explicit crosscorrelation between spike trains, but in many studies, it was inferred by the existence of narrowband oscillations in recorded local field potentials (LFPs) (Engel et al., 2001; Varela et al., 2001). The rationale for this identification is the assumption that narrowband oscillations must necessarily reflect the constructive summation of synchronous oscillators. Thus, a great amount of theoretical effort was directed to elucidate the coupling mechanism(s) producing synchronous firing (White et al., 1998; Ermentrout and Chow, 2002; Jones, 2002; Averbeck and Lee, 2004); however, this search can be expanded by taking into account aspects such as a statistical analysis of the degree of synchrony of the underlying oscillator population or an estimation of the number of oscillators involved. For example, very different phenomena can arise in LFP recordings, which involve only a mesoscopic quantity of oscillators (few tens of oscillators) when compared with macroscopic techniques (EEG), where millions of oscillators are involved. Furthermore, powerful and stereotypical neural oscillations can be detected in systems, like the vertebrate olfactory epithelium (OE), where no coupling mechanisms (interneurons, gap junctions) likely exist between the elementary oscillators. It is thus worthwhile to consider whether neural oscillations can arise in some circumstances by a mechanism not involving coupling. To test the hypothesis that neural oscillations can be generated by the summation of uncoupled or uncorrelated oscillators, we studied the statistical properties of “peripheral waves” (PWs), a classic case of neural oscillations found in the olfactory epithelium of the vertebrate (Ottoson, 1956; Nikonov et al., 2002).
Since the mid 1950s, it was known that potent odors induce strong, mainly βband oscillations termed peripheral waves (Adrian, 1955; Ottoson, 1956) in local field potentials of the olfactory epithelium. Initially, PWs were described as “unstable signals” or “random shapes” (Takagi and Shibuya, 1961), because they exhibit amplitude modulation patterns that never repeat themselves (see Fig. 1A). The current view is that PWs originate from neural activity within the olfactory organ and not by centrifugal influence or electrotonic spread from the olfactory bulb, because transection of the olfactory nerve does not eliminate the waves (Sutterlin and Sutterlin, 1971; Dorries and Kauer, 2000; Nakazawa et al., 2000). Although examples of PW are abundant (Nikonov et al., 2002), their generating mechanisms, as well as their possible function(s), are still unclear. The dominant theoretical notion is that PWs represent intermittent synchronous activity of groups of olfactory receptor neurons (ORNs) (Ottoson, 1956); however, because there are no established neural connections between ORNs, the mechanism for this purported peripheral synchrony remains unclear, and PWs have been linked to gaseous second messengers, gap junctions, ephaptic interactions, and electric field effects (Tucker, 1975a,b; Dorries and Kauer, 2000; Suzuki et al., 2004). PWs provide an appropriate framework to test our model, which considers the possibility that neural oscillations may arise from the random addition of individual uncoupled oscillators.
Materials and Methods
Experimental procedures.
Channel catfish (Ictalurus punctatus) were anesthetized and manipulated according to the guidelines of the Animal Ethics Committee of Louisiana State University. Extracellular recordings and stimulation of the catfish olfactory epithelium were performed as described previously (Parker et al., 2000; Nikonov et al., 2002). The stimulus used in this study was a mixture of the lisomers of 0.1 mm methionine, arginine, and alanine applied for 5 s. A minimum of 3 min interstimulus intervals (ISIs) were used to avoid adaptation. Low impedance (∼50 KΩ) Wood's metal microelectrodes plated with platinum were used. LFPs were amplified, bandpass filtered between 3 and 300 Hz, and stored on magnetic tape. After acquisition (sampled at 10 kHz), the signal was digitally filtered between 3 and 80 Hz.
Eight sequential presentations of identical stimuli to the same animal produced eight instances of PWs, and their slowvarying envelopes, A(t), were calculated using the Hilbert transform. For each set of eight envelopes A(t), the average across repetitions [Ā(t)], and the SD (also across repetitions) σ_{A}(t) were calculated. The coefficient of variation (CV) over time CV(t) [defined as σ_{A}(t)/Ā(t)] was calculated, as well as the time average 〈CV(t)〉 (Fig. 1A, bottom trace).
Kuramoto and MatthewsMirolloStrogatz model systems.
Using Kuramoto dynamics (Strogatz, 2000; Izhikevich and Kuramoto, 2006), we simulated artificial LFPs generated by the superposition of signals from a population of 25 weakly coupled oscillators. To obtain variable degrees of coupling, we set the Kuramoto's coupling strength K to different values: 0 (corresponding to no coupling), 2, 4, 6, …, 42 (corresponding to a very tight coupling). For each simulated set of LFPs (eight elements), we calculated the same statistical values described for real signals. Simulated LFP signals using the MatthewsMirolloStrogatz model (MMS) regimes were also calculated for n = 25 oscillators with coupling parameter between K = 0 (random) and K = 1.7 (high degree of coupling) in steps of 0.005. The bandwidth (Δ) varied between 0.2 and 1.8 Hz in 0.025 Hz steps. Differential equations, described by Matthews et al. (1991), were numerically solved using fourthorder Runge–Kutta methods coded in Igor (WaveMetrics, Lake Oswego, OR).
Summation of oscillators with variable frequency and amplitude.
Artificial PWs were also obtained by the addition of uncoupled oscillators mimicking the amplitude and frequency signatures of empirical PWs (Fig. 1B). The amplitude profile of each oscillator changed as depicted in Figure 5A (top trace), whereas their individual frequency was represented by realizations of the random process defined by the following: with the empirical values y_{0} = 2.94 ± 0.30; A_{1} = 6.27 ± 3.00; A_{2} = 17.60 ± 0.30; τ_{1} = 0.52 ± 0.02; and τ_{2} = 4.760 ± 0.001 and a phase uniformly distributed in [π,π]. This random process simulated the widespread frequency change during real PWs (Fig. 1B). Each artificial PW was generated by the addition of 25 oscillators.
Summation of spike trains.
PWs were also simulated by the addition of uncorrelated spike trains. Each spike train (m) was simulated by three random variables (T_{m}, t_{0,m}, and a jitter_{m}). T_{m} corresponds to the average train ISI obtained from a Gaussian probability density function (PDF) (33.3 ± 0.6 ms). t_{0,m} [simulated by a uniform random variable in the range (0, T_{m})] corresponds to the time of the initial spike and thus serves to define an initial relative phase between trains. Neuronal noise was represented by jitter_{i}, a zero mean Gaussian random variable. The timing t_{i,m} for spike i was recursively computed as follows: Finally, at t_{i,m}, a smoothed, real recorded spike template was inserted. These simulated trains served to illustrate the basic idea that PW can be generated by regular spiking activity. Each artificial PW was generated by the addition of 100 of such spike trains.
Statistical analysis.
To obtain valid statistics to compare CV obtained from real signals from those generated by Kuramoto dynamics, we applied resampling methods. For each coupling strengths K (0, 2, 4, 6, … 42), 13 groups of eight PWs were calculated. For each group of eight, we calculated their CV. To obtain their empirical probability density distribution, we resampled each population of 13 CV. This PDF was used to calculate confidence intervals (95 and 99%) for the CV mean. We also resampled the set (n = 13) of CV experimentally obtained.
Results
PWs, which are triggered by intense odor stimuli, are βband amplitude modulated (AM) signals (Fig. 1A), where their carrier frequency varies with a temporal profile that depends on stimulus properties (analysis not shown in this work). For the particular biologically relevant olfactory stimuli (i.e., amino acids) for fish used in this work, the instantaneous frequency of PWs (determined using the Hilbert transform) declined from 30 to 10 Hz and was well described as a double exponential function (Fig. 1B). Although the pattern of amplitude modulation differs for each stimulus presentation (Fig. 1A), the frequency time course is strongly conserved (Fig. 1B).
Because no previous investigation was able to show electrical coupling between ORNs or demonstrate synchrony in responses from a population of ORNs, we developed an alternative hypothesis to explain the origin of PWs. The basis of our arguments is the fact, well known in statistical physics (Goodman, 1985), that the addition of N sinusoidal oscillators as follows:
with amplitude a, and where the frequency f_{k} of individual oscillators is obtained from a narrow frequency distribution centered in f_{0} and dispersion σ_{f} as follows:
and the initial phases φ_{k} are uniformly distributed in [π, π], results in an AMlike signal as follows:
The signal S_{N}(t) can be interpreted as a carrier, represented by sin[2πf_{0}t + β(t)], multiplied by an envelope (or modulating) function A(t), which may vary between 0 and a · N. In addition, A(t) follows a Rayleigh distribution with an RMS amplitude equal to a
Although Rayleigh fading is the fingerprint of totally asynchronous oscillators, a more general statistical framework to describe intermediate situations where oscillators could be partially synchronized (Izhikevich and Kuramoto, 2006) is required to analyze LFP signals.
Adopting the phasor notation, each elementary oscillator is represented by a rotating vector v⃗_{k}(t) having a magnitude a and a time varying phase φ_{k}(t) as follows: then A(t) [the envelope of wave S_{N}(t)] corresponds to the time varying magnitude of the vectorial summation of such phasors as follows: and the envelope of an LPF is the magnitude of the vectorial sum of v⃗_{k}(t). Although the relative phase difference among vectors v⃗_{k}(t) changes moment by moment as the vectors rotate with different frequencies, an instantaneous phase dispersion σ̂_{φ}(t) can be defined from a snapshot of the orientations of the vectors (Fig. 3, p, q). The standard deviation, σ_{φ}, of the probability density function (PDF) of the vectors' phases over time, defined as the time average σ_{φ} = 〈σ̂_{φ}(t)〉, measures the degree of synchrony of the population of oscillators. Thus, the problem of quantifying the properties of a signal envelope, a time varying function, can be reduced to consider vectorial sums with different σ_{φ}, thus subsuming time in a pure statistical framework.
Figure 3A shows the rms amplitude (A_{rms}) and its SD (σ_{A}) versus phase dispersion (σ_{φ}) for the result of numerically modeled sums of N = 25 oscillators. σ_{φ} changes from total synchrony (σ_{φ} = 0), with A_{rms} = N, to total asynchrony (σ_{φ} → π), a situation analogous to the twodimensional random walk, yielding an A_{rms} =
Figure 3B shows the coefficient of variation of the envelope (CV = σ_{A}/Ā; Ā is the average envelope amplitude) for groups of N = 15, 25, 50, and 100 oscillators. For intermediate values of σ_{φ}, the profile of the curve depends on N, but for extreme values, it is independent of N. Because the CV of the Rayleigh distribution is
To infer the degree of synchrony in the network of oscillators that produce an AMlike signal, the CV of its envelope must be computed, ideally, over an infinitely long period. In experimental situations where signals are finite and the temporal profile of an individual oscillator is unknown, the empirical CV can be calculated from the experimental data as the time average of the ratio between the instantaneous SD and the instantaneous mean of a family of timelocked envelopes, obtained as repetitions of the same experiment (i.e., CV = 〈CV(t)〉 = 〈σ_{Ai}(t)/Ā_{i}(t)〉) (Fig. 4).
To test the hypothesis that PWs are an example of Rayleigh fading produced by uncoupled oscillators, we calculated the empirical CV(t) for families of PW envelopes. Figure 4A shows how the empirical CV(t) varied after stimulus onset for a family of eight PW traces. For these cases, CV(t) hovers around the theoretical value 0.523, and its experimental time average is CV = 0.503. Next, we calculated the CV for theoretical situations where oscillators were coupled to each other. To generate such surrogate data, we used the most common model in the literature, the Kuramoto model (Strogatz, 2000), for the synchronization of coupled oscillators. In this model, the frequency of each oscillator (φ_{i}) in the network depends on its natural frequency (ω_{i}) plus a perturbation produced by the collective influence of the rest of the network, controlled by a coupling strength parameter K (Strogatz, 2000) as follows: where r and ψ correspond, respectively, to the magnitude and phase of the normalized vectorial summation of the phasors representing individual oscillators. The polar coordinates of this mean field vector (also known as order parameter or centroid) can be obtained by the following: Thus, we simulated the resulting PWs by adding 25 oscillators under different Kuramoto coupling strengths and calculated their CV (Fig. 4B). The Kuramoto model features a bifurcation behavior, because K above a critical value (Kc) produces synchronization, but K < Kc produces only a very weak tendency to synchronization. Thus, in our numerical simulations, a null coupling strength (K = 0) produced Rayleigh fading, and the corresponding CV varied at ∼0.523, as expected. When the oscillators were forced to synchronize (K > Kc), the CV of the resultant waves was of much lower value, even for K = 16, which corresponds to a medium coupling strength that resulted in a partial synchronization (K≈ Kc). Initially, just after the stimulus onset, CV(t) is close to 0.523, because the oscillators start with random phases; however, as synchronization takes control, the CV(t) drops to lower values, because most oscillators adopt similar phases (Fig. 3). After systematically exploring Kuramoto coupling strengths (K) and the expected CVs, we concluded K decreases as CV increases (Fig. 4C). Our set of 13 CV measurements, obtained from empirical PWs, was compared against surrogate data obtained from Kuramoto systems using resampling techniques (Davidson and Hinkley, 1999). The PDF from the experimental data differs (p < 0.01) from the PDF of K = 10 (a coupling strength that barely departs from an asynchronous state) and cannot be separated from the PDF of K = 0 (Fig. 4D). Thus, considering our model, the value of the experimentally derived CV of PWs indicates lack of coupling between ONRs.
To further challenge our claim that CV could be considered a good indicator for random phase oscillators, we also analyzed the MatthewsMirolloStrogatz model (Matthews et al., 1991). The MMS model has important theoretical interest, because it considers not only phase relationships between oscillators but also allows oscillators to undergo amplitude variations. As a consequence, the MMS model exhibits a complex variety of behaviors (incoherence, locking, amplitude death, and several types of unsteady states). Thus, the MMS model has been used to check analytical methods in neuroscience (Rudrauf et al., 2006). As in the Kuramoto model, synchronization for the MMS model depends on a coupling parameter K and bandwidth (Δ). Figure 5A compares the variations of CV against K for Kuramoto and MMS models at a given bandwidth (Δ = 0.8). The CV for unsynchronized states lies close to 0.523, whereas large CV values (0.65 < CV < 0.9) are produced by unsteady states characterized by periodic pulsating envelopes (Fig. 5C, trace c) very unlike PWs. As soon as the system synchronizes, however, the envelope amplitude becomes constant (Fig. 5C, trace b), and the CV drops dramatically. To globally check the behavior of CV with respect to synchronization, we systematically explored the (K, Δ) phase space of MMS parameters for 25 oscillators. This computation revealed that CV by itself reproduces the qualitative phase diagram introduced by MMS [Matthews et al. (1991), their Fig. 17] as every behavior of MMS oscillators could be identified by particular CV values (Fig. 5B,C). In the asynchronous state, CV stays near 0.523 even in the presence of large changes in the amplitude of oscillators.
Rayleigh fading is an extremely robust phenomenon that does not require equal or invariant amplitudes or special temporal profiles for the individual oscillators (as the sine functions of Fig. 2 would seem to imply). Figure 6 shows examples of Rayleigh fading obtained with profiles produced by common periodic neuronal activity. In Figure 6A, it was assumed that the individual oscillators were subthreshold membrane oscillations of ORNs with a temporal profile similar to an AM chirp. Alternatively, Figure 6B explores the scheme that the individual oscillators responsible for PW generation are the spike trains produced by the ORNs (Dorries and Kauer, 2000). In both cases, waves resembling PWs were obtained. The spike trains of excited ORNs are highly regular (Fig. 6B, bottom trace) and have frequency properties that match those of PWs (Baylin, 1979; Trotier and MacLeod, 1983; Dorries and Kauer, 2000). Thus, the elementary oscillators involved in PW genesis could be neuronal subthreshold oscillations, spike trains, or a combination of both (Desmaisons et al., 1999). CV is a noiseresistant indicator, as addition of a substantive zeromean noise processes (SD = 0.05 rad/s) still shows that locked states have small CVs (data not shown). Thus, although LFP data always contain noise components because the CV of the envelope reflects an intrinsic statistical property of the signal, the collection and averaging of enough instances of envelopes should allow the detection of the underlying CV. In contrast, our determination of the empirical CV assumes that no systematic drift exists between trials. Indirect evidence shows that this is the case, and thus the data are stationary (see below, Discussion).
Discussion
Because of the signal properties of the obtained PWs (i.e., random shape, uncorrelation with stimulus characteristics, CV close to 0.523) along with consideration of a simple generative model based on uncoupled oscillators, we suggest that PWs are the result of a superposition of asynchronous neuronal oscillators. Our interpretation might reconcile known anatomical data with theory. In effect, a puzzling fact in olfactory research was the conflict created by the existence of PWs and a lack of evidence for a coupling mechanism in the OE (Dorries and Kauer, 2000). Our analysis suggests an alternative path to resolve the contradiction, which arises from the implicit, but incorrect, expectation that neural oscillations must necessarily imply synchrony.
Because the observed CV is calculated by evaluating the SD across trials collected at different moments, it could be argued that this result is attributable, not to the interference produced by random oscillators, but to systematic drifts in the neural response (nonstationary effects). Because the existence of adaptation effects in the OE are well known, we followed established methods (Nikonov et al., 2002) to avoid them. In fact, the electroolfactograms (EOGs) collected following this procedure were identical, which suggests that the PWs that we collected do not have significant nonstationary effects.
Using simulated small populations of oscillators synchronized by Kuramoto or MMS dynamics, we indicated a relationship between a signal CV and the average degree of coupling among oscillators contributing to that signal. In these two models, the CV differentiates behaviors among oscillators (from incoherent to synchronization) and thus could be used as an indirect measurement of the underlying coupling parameter K. Although most modern synchronization models (like MMS) are derived from the Kuramoto model, it is still an open question whether a CV close to 0.523 could be produced by some type of synchronization regime. However, independently of the synchronization model considered, because synchronization implies necessarily a locking in the allowed values for phases and thus a restriction of the overall system variability, a low value of CV must necessarily ensue. In addition, because the amplitude variations of the oscillator are also a possible source of system variability, exploring the MMS model (where amplitudes change dynamically) provided us more general support for our claim that CV = 0.523 together with the characteristic patterns of Rayleigh fading are diagnostic features of random phase oscillators. Finally, the theoretical problem of whether a synchronization regime with a CV = 0.523 could exist or not is an interesting problem that, in conjunction with the possible influence of nonstationary effects, deserves additional study.
Rayleigh fading, in theory, can be observed with the population of an oscillator of any size. But, because electrodes are equipotential surfaces recording the spatial average of the electric fields surrounding their conductive surface (Nunez, 1981), the rms amplitude of the signal recorded by an electrode immersed in a population of N oscillators with equal amplitude (V_{0}) can vary between V_{0} (all oscillators in phase) and V_{0}/
In addition to demanding that the number of oscillators be mesoscopic, Rayleigh fading imposes another requirement concerning the dispersion of frequencies of the oscillators (i.e., the oscillators must have, in every instant, similar frequency). Although it was assumed that the inherent frequency of neural oscillators is an emergent property of the neural network (Manor et al., 2000), experiments in connexin36 knockout mice show that uncoupled neurons from the inferior olive retain their wildtype frequencies (Long et al., 2002). Therefore, the firing frequency of neurons is not likely a property derived from the network but is rather an intrinsic property of their biophysics (Hutcheon and Yarom, 2000). The present model describing the generation of PWs as the addition of uncoupled oscillators may have wide implications for our understanding of neuronal population dynamics in other systems where narrowband LFP signals have not been thoroughly analyzed. In fact, it is not that the narrow band per se that is the important condition but rather the population dispersion of frequencies should be small. Thus, because it is manifested in PWs, the frequency can change in a 1:3 ratio (Fig. 1B) (a rather extended relative bandwidth), but in each moment, the population dispersion is small.
In the model presented, the envelope of a PW carries no biologically relevant information, because it reflects the timevarying phase relationship between asynchronous oscillators and does not reflect an intrinsic property of the stimulus. Although profiles of PWs appear to be stimulus independent, the time course of the frequency change varies with the nature of the odorant and its concentration (data not shown). Thus, neural coding of odorant quality in the olfactory system could not depend on the relative phase between ORNs (because they appear to be random with respect to each other) but must exploit either the stereotyped change in frequency in ORNs or use distributed processing achieved by comparing the activity of idiotypic ORNs scattered across the OE that project to the same glomerulus (Mombaerts, 2006). Our results suggest that mitral cells act not as coincidence detectors but as resonant units (Hutcheon and Yarom, 2000; Izhikevich, 2001), requiring regular but not synchronous presynaptic input from ORNs.
Asynchronous neural oscillators with similar frequencies is possibly a widespread situation in the nervous system such that Rayleigh fading could be a pervasive phenomenon in many neural structures and, perhaps like the case described here for PWs, instances of Rayleigh fading could be confused with synchronization events. The analytical tool introduced here, which focuses on the envelopes of the signal and is analogous to standard procedures used in the wireless communications industry for the characterization of radio channels, could serve to evaluate the degree of synchrony in a population of neural oscillations. Finally, our model suggests that in the case of the OE, there may be a simpler interpretation of the observed data represented by PWs than the requirement of neural synchronization. Although we cannot rule out the possibility that ORNs are instead coupled and follow a synchronization model that because a particular covariation of phases and amplitudes produces PWs with a CV ≈ 0.523, our model based on uncoupled ORNs explains important experimental facts and provides an intriguing alternative hypothesis.
Footnotes

This work was supported by Comisión Nacional de Investigación Cientifica y Tecnológica Programa Bicentenario en Ciencia y Tecnología ACT 45, Ministerio de Planificación Iniciativa Científica Milenio P05001F (J.B.), Fondo Nacional de Desarrollo Científico y Tecnológico 1061108 (J.C.L.), and Grant National Science Foundation IBN0314970 (J.C.). J.B. holds a John S. Guggenheim Memorial Foundation Fellowship. We are indebted to Ranulfo Romo and Francisco Bezanilla for critical reading of this manuscript, Magdalena Sanhueza for invaluable discussions, Oliver Schmachtenberg for advice, and the important input of the anonymous reviewers.
 Correspondence should be addressed to Juan Bacigalupo, Department of Biology, University of Chile, Las Palmeras 3425, Nunoa, Santiago, Chile 7800023. bacigalu{at}uchile.cl