Stochastic synchronization of electroreceptors in the paddlefish

Figure 1: Photo of a small ( ~ 20 cm) paddlefish, Polyodon spathula, striking at a pair of wires presenting a weak oscillatory electrical signal, during "electrical feeding''.

Our experimental system is the paddlefish (Fig. 1), Polyodon spathula, named for its long flattened spatula-like appendage extending in front of the head, the "rostrum'' (Fig. 2). The function of the rostrum has been debated since the species was described in the late 1700's. Our present understanding began with the work of anatomists [Jø rgensen et al., 1972] who showed that the rostrum is covered with tens of thousands of sensory receptors, morphologically similar to the ampullae of Lorenzini of sharks and rays, well-known to be passive electroreceptors [Murray, 1974; Bullock, 1982]. Clusters of electroreceptors also cover the head and the gill covers. However, behavioral and neurophysiological evidence remained limited concerning the function of these receptors [Kalmijn, 1974; New & Bodznick, 1985] until our own work since 1993 finally established conclusively that these are indeed passive ampullary-type electroreceptors responding to the microvolt-scale electrical signals emitted by planktonic prey such as Daphnia, and that the electroreceptors are used by paddlefish to locate plankton during feeding behavior [Wilkens et al., 1997]. The location of the rostrum, out in front of the mouth, allows it to function as an ``early warning system'' for approaching prey, as the fish swims forward continuously. The electroreceptors may also mediate obstacle avoidance [Gurgens, 1998]. Hence the rostrum functions as an antenna, carrying arrays of electrosensors.

Figure 2: (A) Photo of the underside of the rostrum from a 21 cm paddlefish. (B) Enlarged photo of clusters of electroreceptor pores; the rostrum edge runs vertically near the left side. Width of corner bar = 0.5 mm.

Although present-day paddlefish are highly evolved, nevertheless they are often cited as a ``living fossil'' species, thought to have branched off long ago from the main evolutionary line leading to teleost fish. Fossil paddlefish with a recognizable rostrum have been found from geological strata of the Upper Cretaceous and Paleocene periods [Grande & Bemis, 1991], dating from ~ 65 million years ago. Sturgeons are the closest relatives to paddlefish; together they are classified in the Order Aciperseriformes, and are also termed Chondrosteans, referring to their largely cartilaginous skeletons.

The habitat of the two living species of paddlefish includes large rivers and floodplain lakes of the Mississippi River drainage in North America (Polyodon spathula), and the upper Yangtze River in China (Psephurus gladius). Such large rivers are laden with silt, and are murky and turbid, rendering vision ineffective. The eyes of paddlefish are small and directed laterally, and paddlefish hardly respond to changes in illumination or shadows overhead, unlike most fish. Electroreception appears to have largely replaced vision as a primary sensory modality, although olfaction and lateral-line mechanoreception are important modalities also. Testaments to the sensory prowess of paddlefish include the large body size of adult Polyodon, which commonly grow to total lengths > 1 m and 40 kg, with some trophy fish reaching 1.7 m and 55 kg, as the longest freshwater fish in North America. They are also long-lived, with many living 30 years and some 40 years. In Polyodon, analyses of stomach contents have revealed mainly zooplankton, especially Daphnia, although insect larvae are also taken. Adult paddlefish are ram suspension feeders: they open their cavernous mouth and swim forward to feed, straining masses of zooplankton (plus much silt and debris) from the water. Paddlefish are pelagic, swimming continuously throughout life. Their mouth is normally partly open, allowing them to breathe by ram ventilation [Burggren & Bemis, 1992]. This also presumably makes paddlefish electrically ``quiet'', by avoiding the ventilatory electrical interference demonstrated in other fish [New & Bodznick, 1990].

Movement sequence during capture of a brineshrimp.

Small juvenile paddlefish ( < 20 cm) feed in a different manner, capturing individual plankton one-by-one (``particulate feeding''). This requires the juvenile fish to detect and locate an individual Daphnia as it approaches, then turn and move to intercept and capture the selected prey. We study this feeding behavior in the laboratory in a recirculating stream (``swim mill'') [Vogel & LaBarbera, 1978; Burggren & Bemis, 1992; Wilkens et al., 1997], in which a propeller drives water around a closed circuit at the same velocity as a fish is swimming forward in a viewing chamber, such that the fish remains stationary with respect to two co-aligned video cameras viewing the fish from the side and below (via a mirror). Our experiments have shown that electrosense suffices, and other sensory modalities are not needed, for prey capture. We routinely use near-infrared illumination (_max = 880 nm) from arrays of light-emitting diodes, which is invisible to certain sturgeon and presumably also paddlefish, to exclude vision as a basis for prey capture. The water flow is laminar, and the plankton are swept along more-or-less straight paths, parallel to the long axis of the fish's rostrum. Hence the relative distance and direction from the rostrum axis to prey can be measured. Approximately 95% of captured plankton are < 40 mm from the rostrum's long axis, as they approach. However some are farther away, up to 93 mm.

Electrical signals from planktonic prey. Aquatic animals are known to produce electric fields. For example, Kalmijn [1974] measured DC fields from a variety of fish and marine invertebrates. We have characterized the electrical signals produced by Daphnia (Fig. 3A), the small (2-3 mm) freshwater crustacean used as prey in the feeding experiments. A high-impedance amplifier is used to measure the voltage difference between a stable Ag-AgCl recording electrode positioned near a Daphnia, and a distant reference electrode, in water of controlled conductivity.

Daphnia produce AC oscillations (Fig. 3B) which can be correlated with motor activities. Large-amplitude but irregular waves correlate with beating motions of the antennae, for locomotion. A low-amplitude continuous ~ 6 Hz sinusoid correlates with rhythmic beating of the legs, which create a current of water for filter-feeding. An amplitude spectrum (Fig. 3C) of the electrical signal shows peaks at these frequencies. The 5-7 Hz waves from the feeding legs are well-matched to the peak frequency sensitivity of paddlefish electroreceptors at ~ 5 Hz.

Figure 3: (A) Side-view photo of a Daphnia, the planktonic prey of paddlefish. Its head and antennae are to the right. (B) Oscillatory electrical activity recorded near a tethered Daphnia. The large waves are due to episodic beats of the antennae for locomotion. The smaller 6-7 Hz waves are due to continuous rhythmic beating of the feeding legs. (C) Power spectrum of Daphnia electrical signals.

Daphnia also produce a standing DC dipole field, which has been demonstrated by approaching a tethered Daphnia with a DC-coupled electrode. We model Daphnia as dipoles, and assume that their signals decline in amplitude with distance along approximately an inverse cube relation. Whether paddlefish pay attention to certain ones or all of these AC and DC signals from Daphnia is still being debated.

Structure and function of electroreceptors. Peripheral cells responding to weak electrical gradients of , defined as electroreceptors, have been identified in elasmobranch marine fish (the ampullae of Lorenzini of sharks, rays, skates, and ratfish), in several ``primitive'' freshwater fish (sturgeons, paddlefish, lampreys, lungfish, bichirs), and in several ``advanced'' teleost freshwater fish inhabiting murky waters (catfish, gymnotids, knifefish), as well as in certain amphibians [Bullock et al., 1983]. Electroreceptors are thought to have been ``reinvented'' by fish at least three times. Nevertheless, electrosense is relatively uncommon among the 22,000+ known species of fish, most of which instead use vision, chemoreception, and lateral-line mechanosense as their primary sensory modalities for the outside world.

Electroreceptors are of two main types. ``Ampullary'' electroreceptors are named for their resemblance to a vase: a gel-filled canal leads from the outside water to a sac below the skin, in whose wall are embedded the electrosensitive cells. Ampullary receptors have high sensitivity, and respond to low frequencies in the 0.1-20 Hz range. Those in paddlefish, sharks, and other primitive fish are excited by cathodal stimuli (i.e., their discharge rate accelerates when a nearby electrode becomes negative), and inhibited by anodal stimuli. The opposite is true in catfish and other teleost fish, whose ampullary receptors are excited by anodal stimuli (nearby electrode positive). This difference in polarity-sensitivity is due to different locations of the voltage-sensing ion channels in the receptor cells (see below), and reflects distinct evolutionary lineages. The other main class of electroreceptors are the ``tuberous'' type, of which several subvarieties have been distinguished. They are less sensitive, respond to high frequencies (up to 1 kHz), are excited by anodal stimuli, and are found only in ``weakly electric'' teleost fish which actively probe their environment with self-generated oscillatory or pulse-like electric fields.

Figure 4: (A) Cross-section photo of one side of the rostrum, showing the pores (p) and canals of eight ampullary electroreceptors. more...

The ampullary electroreceptors in paddlefish form a passive sensory system, meaning that paddlefish only receive signals from external sources. An external opening (pore) in the skin, 80-210 diameter, leads into a short canal ~ 200 long (Fig. 4A, B). The pores are organized into clusters of 5-8 on the rostrum (Fig. 2B), but there are much larger clusters on the head, gill covers, and near the mouth. The internal end of each canal is covered with a sensory epithelium (Fig. 4B, C). An epithelium is a layer of cells, one cell thick, typically lining a hollow organ. The cells of an epithelium are typically ``polarized'', meaning that the inner face towards the hollow space (the ``apical'' face) has different membrane ion channels and transporters than the outside (``basal'') face. The sensory epithelium in paddlefish electroreceptors is sometimes flat, but usually resembles a dome. The epithelium contains two types of cells. It is the ``hair cells'' which are considered electrosensensitive (h, Fig. 4C), named for their kinocilium projecting into the lumen of the ampulla. They are oval or pear-shaped, and in Fig. 4C are ~ 9 high and ~ 7 in largest diameter. Hair cells are not neurons: they have a different embryonic origin than the nervous system. Such ciliated receptor cells are common in a variety of sensory systems, e.g. for hearing, taste, or balance, and have been studied extensively. No counts have been made of the number of receptor cells per epithelium in paddlefish, but it is 400, assuming a half-spherical epithelium of ~ 100 diameter, and a ~ 7 hair cell diameter. The hair cells are interspersed among T-shaped ``support cells'', which secrete a gel-like substance into the lumen of the ampulla, and may have other functions. The support and hair cells form ``tight'' intercellular junctions, or high-resistance seals, which block extracellular paths from the canal (the ``apical'' or ``mucosal'' face of the epithelium) to the interior of the body (the ``basal'' or ``serosal'' face). Similar tight junctions exist between neighboring cells in the walls of the canal. Hence a voltage applied at the pore will, with little attenuation, be imposed across the sensory epithelium, i.e., between the apical and basal faces of the hair cells.

On the basal side of the epithelium, the hair cells make excitatory chemical synapses onto ``primary afferent'' axons, themselves considered to be insensitive to sensory stimuli: they require the hair cells as transducers. The synapse from each hair cell, together with the spiking properties of the primary afferent endings, converts the analog signal from the hair cells into spike trains (series of action potentials), coding the electrosensory information as a time series (the intervals between spikes). The spikes then propagate to the brain. The primary afferents typically discharge repetitively at ~ 30-50 Hz in the absence of any stimulus, and so are said to possess ``pacemaker'' properties, presumably in a segment of axon near the sensory epithelium. The primary afferent axons form the purely sensory anterior lateral line nerves (ALLn), which are ``extra'' cranial nerves present in fish. Electrosense is considered part of the octavolateralis system, related to hearing, balance, and mechanosense from the lateral line. The term ``electroreceptor'' is ambiguous, since the entire structure of pore + canal + epithelium + axon is needed for electrosensitivity (Fig. 4D). Although the hair cells are the actual sensors, the spike-train coded output is what is most often recorded, using a microelectrode placed in the sensory ganglion (collection of nerve cell bodies; Fig. 4D) of the ALLn, located near but outside the brain.

The accessory structures of the skin, canal, and gills play important roles in electroreceptor function [Murray, 1974; Kalmijn, 1974]. The great sensitivity of marine fish is in part attributed to their long canals, up to 20 cm long, such that hair cells of the epithelium measure large potential differences between the distant ends of the canals, when the animal is in a voltage gradient. Inversely, the short length of the canals in paddlefish and other freshwater fish reduces their electrosensitivity. Freshwater fish are considered to have high-impedance skin, to limit loss of salt. The body interior is thought to be isopotential, and connected electrically to the environmental water at the gills, which then serve as a ``reference'' electrode for electroreceptors [Kalmijn, 1974]. The same may apply to paddlefish. Canals also confer directional sensitivity: responses are largest when the electric field vector of a stimulus is parallel to the length of the canal [Murray, 1974]. Although the canals in paddlefish are short, we have data showing that they do confer significant directional sensitivity.

The frequency response of electroreceptors is measured by recording afferent spike trains while applying sinewave stimuli. Paddlefish electroreceptors respond maximally in the 0.5-20 Hz range, with reduced responsiveness over the 0.01-120 Hz range. However, in behavioral experiments, paddlefish show an even more restricted range of preferred frequencies, 5-10 Hz. Ampullary receptors are insensitive to DC voltage gradients. That is, an external voltage step will evoke a transient response from a receptor at the step, but within a few seconds the discharge rate of the primary afferent will return to the pre-step value, adapting completely. The restricted bandwidth of ampullary electroreceptors serves to reduce their noise level, increasing the signal-to-noise ratio for weak signals at frequencies within their bandwidth. However the cellular basis for the restricted bandwidth of all known ampullary electroreceptors is unexplained. It cannot be explained as low-pass RC filtering in the canal since the ``end bud'' electroreceptors in lampreys have no canal at all, yet respond best at 1 Hz [Zakon, 1986].

Two approaches have been used to estimate the sensitivity of ampullary electroreceptors, i.e., the minimal voltage gradient eliciting a response: behavior of whole animals, or analysis of spike trains from individual electroreceptors. In general, behavioral experiments have demonstrated ~ 10-fold greater sensitivity. In a shark, behavioral sensitivity to a 10 nV·cm^-1 external voltage gradient was assayed from changes in heartbeat rate [Kalmijn, 1974]. Behavioral tests on freshwater catfish have estimated the limit of sensitivity as ~ 1 Vcm^-1 [Peters & van Wijland, 1974; Bullock, 1982], i.e., 100-fold less sensitive than marine fish.

Early efforts to measure sensitivity from afferent spike trains adopted the criterion of a 10% change in firing rate [Murray, 1974]. An interesting approach was used by Teeter, Szamier & Bennett [1980], who put the rostrum of a sturgeon in air (for electrical insulation, creating an open circuit condition), then delivered stimuli to individual electroreceptor pores via a micropipet. A 10% change in the firing rate of primary afferents could be evoked by transepithelial stimuli of a few hundred microvolts, for this freshwater fish. Equivalent tests on ampullae of Lorenzini from marine fish have shown changes in primary afferent firing rate in response to transepithelial stimuli of 2-5 V [Bromm et al., 1976; Lu & Fishman, 1995a]. Hence the greater electrosensitivity of marine fish is due in part to a greater intrinsic sensitivity of their epithelia, as well as to the longer canals, compared to freshwater fish. In recent measurements of the sensitivity of the ampullae of Lorenzini of a marine ray, Tricas & New [1998] applied uniform sinusoidal external fields (i.e., the canal length boosted the apparent sensitivity), and found responses to fields of 20-40 nV·cm^-1, comparable to results from behavioral experiments. They also showed that the gain (change in firing rate per unit of stimulus amplitude) is higher for weak stimuli, and pointed out that it is easy to overstimulate electroreceptors. We would conjecture that if a change in firing rate can be seen with the unassisted eye in raw spike trains, then the stimulus is probably too strong, and that the encoding of very weak stimuli into time series (spike trains) may have novel features. For example, power spectra of spike trains can readily reveal responses to sinusoidal stimuli not detectable by inspection. Even at the weak stimulus levels used in behavioral tests, the stimulus information must be encoded in the spike trains from individual receptors. An alternate possibility is that a subset of electroreceptors might be more sensitive than the others. Another possibility is that the apparently higher sensitivity in behavioral assays may be due to the brain receiving many parallel channels of incoming partly coherent spike trains, referred to as ``spatial summation'' [Bromm et al., 1976].

Cellular mechanisms of electroreceptors. The following is based on data from marine fish as well as sturgeon and catfish. The voltage sensors are thought to be voltage-gated calcium channels [Bennett & Obara, 1986; Lu & Fishman, 1995a, b], probably ``L'' type calcium channels [Fox et al., 1986] since electrosensitivity is reduced by dihydropyridine compounds, e.g. nitrendipine [Lu & Fishman, 1995a, b]. Such voltage-gated calcium channels endow the hair cells with negative resistance characteristics, tending to generate regenerative (positive feedback) shifts of membrane potential. Indeed, threshold-dependent action potentials can be recorded from the canals of ampullary receptors under open-circuit experimental conditions, i.e., when the two ends of a canal are electrically isolated [Bennett & Obara, 1986]. However, potassium and chloride channels in the hair cells [Lu & Fishman, 1995b], as well as electrical loading, probably damp their regenerativeness normally [Bennett & Obara, 1986]. Other hair cells are known to undergo damped membrane potential oscillations. An intrigueing suggestion by Murray [1974] is that external stimuli may serve to synchronize the regenerative potentials in different hair cells of an epithelium.

The voltage-sensing calcium channels are located in the apical membrane of hair cells excited by cathodal stimuli (e.g. in skates and probably also paddlefish), as shown by analysis of current flow [Bennett & Obara, 1986] or experimentally by local drug application [Lu & Fishman, 1995a]. The voltage-sensing channels are located in the basal membrane in anodally excited hair cells (e.g. in catfish) [Bennett & Obara, 1986].

A seeming problem is that all known voltage-gated channels change their open/closed state over ranges of transmembrane potential of tens of millivolts, some 2-4 orders of magnitude larger than the microvolt-scale voltages known to excite electroreceptors. One explanation is that the hair cell membrane potential may be biased to a level at which the voltage-sensitive calcium channels are at a metastable point on their negative resistance current-voltage relation [Murray, 1974; Bennett & Obara, 1986; Lu & Fishman, 1995b]. This would serve to amplify small perturbations of the membrane potential. The hair cells are reported to possess ion channels and pumps in their basal membrane for such biasing of the membrane potential [Lu & Fishman, 1995b]. A relevant characteristic of L-type calcium channels, necessary for the preceeding scheme, is that L-type channels inactivate slowly or not at all when held at a membrane potential slightly negative to their threshold for opening [Fox et al., 1986]. These cellular mechanisms have the net action of amplifying microvolt-scale stimuli into millivolt-scale changes in hair cell membrane potential, which evoke the release of neurotransmitter.

The excitatory chemical synapses from hair cells to primary afferents have the specialized morphology of ``ribbon synapses'', which in paddlefish protrude from the hair cells into trough-like depressions in the primary afferent membrane [Jø rgensen et al., 1972]. Ribbon synapses are typically found where neurotransmitter is released continuously, and slow changes in the membrane potential of the presynaptic cell modulate the ongoing transmitter release, sometimes called ``nonspiking synaptic transmission''. Several groups have carried out pharmacological experiments to identify the neurotransmitter released from ampullary hair cells, and to characterize the postsynaptic receptor proteins which bind the neurotransmitter, using ampullae isolated from an animal and kept alive in a saline bath [Bennett & Obara, 1986; Okano, 1988; Akoev et al., 1991; Andrianov et al., 1994, 1997]. Glutamate is the leading candidate as the transmitter. There may be two types of transmitter receptor proteins on the primary afferent membrane, since a brief excitatory stimulus evokes a biphasic (fast + slow) change in membrane potential in a primary afferent. These two phases may respectively be mediated by AMPA- and NMDA-type glutamate receptors.

An intrigueing aspect of ampullae of Lorenzini is that the hair cells undergo oscillatory changes in membrane potential at ~ 35 Hz [Clusin & Bennett, 1979; Lu & Fishman, 1995a]. The oscillations are discernible in the total current across an ampullary epithelium, implying that the individual hair cells of the epithelium are coupled to produce partly synchronized oscillations. Pharmacological agents that abolish the oscillations also abolish synaptic transmission from the hair cells to the primary afferents. This oscillation in the hair cell may function as an internal source of noise, mediating stochastic resonance for weak stimuli [Wiesenfeld & Moss, 1995], as suggested using different terminology by Lu & Fishman [1995a]. Attempts to cross-correlate the hair cell oscillation with the periodic firing of primary afferents were inconclusive. Hence there may be two distinct pacemakers in ampullary electroreceptors, a spiking pacemaker in the endings of the primary afferents [Braun et al., 1974], and a nonspiking oscillator in the hair cells.

In certain other sensory systems that have hair cells, e.g. for balance or lateral line mechanosense, the brain sends out impulses along special axons which synapse onto and regulate the sensitivity of the peripheral receptors. Such ``efferent control'' has never been observed for any type of electroreceptor in any species [Bullock, 1986], including paddlefish [Jø rgensen et al., 1972].

Electroreceptors are usually assumed to detect voltage gradients in the water, but some investigators espouse the possibility that they may instead sense current or charge, like a particle detector. A line of evidence that is difficult to reconcile with voltage detection is the insensitivity of ampullary electroreceptors to the conductivity of the water. For example, Gurgens [1998] found no statistically significant effect of water conductivity, over the range of 160 to 1160 S·cm^-1 , on the distance at which paddlefish detected and avoided metal obstacles. The expected result was that fish would approach closer to metal obstacles in high-salt water, due to shunting of voltage gradients, but the experimental results turned out differently. Paddlefish in nature do encounter a wide range of water conductivity, from the low-salt northern rivers of Montana, to brackish Lake Pontchartrain near the Gulf of Mexico. Finally, ampullary electroreceptors are quite sensitive to temperature, pH, and touch, and some investigators consider them multimodal receptors [Murray, 1974; Braun et al., 1994].

Synchronization of periodic oscillators by external periodic fields is understood as adjustment of the oscillator rhythm to that of a periodic signal, or the appearance of phase locking. If (t) is the phase of the oscillator and (t) is the phase of the driving force, then the phase locking condition is:

where n and m are integer numbers. The phases of the oscillator and the signal as well as the phase difference (t) are defined on a whole real line. In the regime of synchronization, the phase difference therefore remains constant forever. In the simplest case of 1:1 synchronization, the response of the oscillator is represented by one complete cycle for each period of driving force. In the more general case of m:n synchronization, during m complete cycles of the driving signal will occur n complete cycles of the oscillator. For periodic oscillators the synchronization condition of Eq.(1) is equivalent to the notion of frequency locking n , where is the natural frequency of the oscillator and ₀ is the driving frequency. The synchronization conditions are fulfilled in synchronization regions, called Arnold tongues, in the parameter space of the system. Outside the synchronization regions the motion of the system is quasiperiodic and represented by ergodic two-dimensional tori in the phase space. Synchronization corresponds to the existence of a correspondent stable limit cycle lying on the torus.

The concept of synchronization for stochastic systems is not trivial. As is well known [Stratonovich, 1967], the influence of noise on a self-sustained oscillator results in the diffusion of its phase. That is why the properly defined phase difference also diffuses, so that the condition of Eq.(1) is never fulfilled in the presence of Gaussian noise. The phase locking may occur only for random periods of time, and be interrupted by so-called phase slips. Thus, synchronization in the presence of noise will appear to be ``blurred''. That is why the conditions of synchronization should be defined in a statistical manner.

Let us consider a simple example of 1:1 synchronization of a noisy Van der Pol type oscillator [Stratonovich, 1967]. The stochastic differential equation for the phase difference has the generic form of an Adler equation:

Figure 5: Time series of the phase difference generated by Eq.(2) with = 0.2, = -0.1, for the indicated values of noise intensity D.

The Fokker-Planck equation for the probability density of the phase p(,t) corresponding to the stochastic differential equation (2) is:

The phase difference (t) is an unbounded variable, and the correspondent stochastic process defined by Eq.(2) or Eq.(3) is nonstationary. It is convenient therefore to introduce a phase defined on the circle [-, ] (or [0,2]). Since coefficients of the Fokker-Planck equation are periodic with respect to , we can introduce the probability distribution P(,t) of the circle phase, which is bounded in [-, ]:

D 0

Let us now go back to Fig. 5, where the noise-induced diffusion of the unbounded phase difference is shown. Let in an initial state the distribution of the phase difference be concentrated at some initial value ₀, p(,t = 0) = (-₀), so that <²(t = 0)> = 0. In other words, in the initial state we have an ensemble of oscillators with exactly the same phase. Due to noise, the phase difference will diffuse according to the law <²(t)> D_eff ·t, where D_eff is the effective diffusion constant which measures the rate of diffusion:

The definition of synchronization of stochastic systems can be carried out by imposing some restrictions on statistical measures of the corresponding process. In particular, for our purposes, an effective synchronization can be defined based on:
(i) The stationary probability density of the cycle phase difference. In this case a peak in P_st() should be well-expressed in comparison to the uniform distribution [Tass et al, 1998].
(iii) The effective diffusion constant. This measure should be small enough that the phase locking segments are much longer than the period of external force. In other words, this restriction requires that the phase of the oscillator is locked during a considerable number of periods of the external signal, and can be expressed as:

The general case of n:m synchronization can be studied using the circle map, which represents a Poincaré return map for a periodically driven self-sustained oscillator:

Gaussian noise can be introduced additively to the r.h.s. of the circle map, giving the following stochastic difference equation:

Figure 6: Iteration sequences generated by the stochastic map of Eq.(10) at the 1:5 phase locking mode ( = 0.20488, K = 0.3) for different values of noise variance (from top to bottom): = 0, = 10-3, = 10-2.

Figure 7: Stationary probability densities corresponding to the sequences of Fig. 6.

Electrophysiological experiments has been performed with paddlefish 35-40 cm in length. A detailed description of the experimental setup can be found in Wilkens et al.[1997]. The primary afferents (see Fig. 4) receive input directly from the receptor cells, leading to the production of action potentials (spikes). Recordings from nerve fibers in vivo have shown that in the absence of any stimulation, the afferent neurons generate action potentials at fairly regular intervals. However, the average firing rates are different for different neurons, over a rather wide range of 20-85 spikes/sec [Pei et al., 1998].

Figure 8: Interspike interval histograms from four different primary afferents of paddlefish. The cells fired nearly periodically with fundamental frequencies of 25 Hz (a), 62 Hz (b), 50 Hz (c), or 58 Hz (d).

Recordings of spontaneous activity of primary afferents are presented in Fig. 8 as the interspike interval histograms for four different neurons. The histograms show a peak centered at a fundamental frequency f. The finite width of this peak indicates that periodic firings are contaminated by noise. Different cells possesses different natural frequencies and are characterized by different degree of stochasticity, which can be estimated through the width of the interspike interval histogram. Detailed experimental study have shown no dependency between the natural frequency of electrosensitive cells and the width of the corresponding interspike interval histograms.

From the last figure we realize a remarkable behavior of electroreceptors, which is closely akin to that of noisy self-sustained periodic oscillators [Stratonovich, 1967]. In the case of a self-sustained oscillator an analog of the interspike histogram can be obtained by counting intervals between crossing of a Poincaré secant plane in the phase space of the oscillator. Therefore, it is natural to expect that electroreceptor cells can be synchronized by a weak external periodic field.

Figure 9: Examples of recordings of spike train from an electroreceptor cell stimulated by a dipole electric field at the three different frequences listed.

To check this hypothesis, we stimulated a cell by a weak electric or magnetic field generated by a dipole consisting of a small coil located near the rostrum of the fish. The electric field strengths were comparable in magnitude to those generated by the zooplankton (a few tens of V/cm). We recorded the spike train generated by a primary afferent and the periodic electric signal from the dipole simultaneously. Three examples of these recording from the same cell are shown in Fig. 9 for different values of stimulus frequency.

The instantaneous phase of the spike train can be calculated from the times t_k when the electroreceptor neuron fires [Pikovsky et al, 1996]:

Figure 10: The cyclic phase difference of spike trains calculated using Eq.(12), for the indicated values of dipole electric field frequency.

Figure 11: The probability density of the cyclic phase difference obtained from the same data as in Fig.10.

The statistical evidence of synchronization behavior is presented in Fig. 11 as the probability density of the cyclic phase difference. In the case of strong 1:5 mode synchronization, the probability density consists of well expressed peaks corresponding to the phase-locking patterns in Fig. 10.

An alternative approach is based on the phase difference defined on a whole real line:

Qualitatively the same results (although for different mode locking regimes) were obtained for other electroreceptor cells. We also made several recordings with a large-amplitude electric field (hundreds of V/cm). Although such strong stimuli are not natural for paddlefish, it is interesting from the point of view of nonlinear dynamics to study the responses of primary afferents in order to learn more about underlying nonlinear processes in the system. As can be expected, large stimuli drove the system to highly non-linear regimes of oscillation, including chaotic.

Figure 12: Instantaneous phase difference, from Eq.(13), for a stimulus frequency of 17 Hz.

We have shown that stochastic synchronization can be used to study the encoding of weak electric and magnetic fields in the paddlefish electroreceptor system. We find that for signals whose strengths are in the range that the animal customarily encounters in the wild, synchronization coding offers a plausible alternative to the more usual rate coding. Significant synchronization was achieved by the noisy electroreceptors only at the lower range of frequencies studied. These correspond to the range of frequency sensitivity previously determined for the paddlefish receptor system, and the signature frequencies of the electric fields from its usual planktonic prey [Wilkens, et al, 1997]. The high frequency tested was well beyond this range and showed no statistically significant synchronization, serving as a control. As mentioned, the small paddlefish use electrosensitivity to feed on individual zooplankton, which generate nearly periodic low-frequency (5 to 12 Hz) electric fields. Since the feeding occurs during movements of both the fish and the zooplankton, a synchronization code could be a possible mechanism that the animal uses to track the target prey.

Acknowledgments The authors would like to acknowledge useful discussions with M. Rosenblum and J. Kurths. This work has been supported by the U.S. Office of Naval Research, Physical Science Division. A.N. is a recipient of the Fetzer Institute post-doctoral fellowship. X.P. is supported by the D.O.E.

References

Akoev, G., Andrianov, G.N., Szabo, T. & Bromm, B. [1991] "Neuropharmacological analysis of synaptic transmission in the Lorenzinian ampulla of the skate Raja clavata", J. Comp. Physiol. A 168, 639-646.

Andrianov, G.N., Bretschneider, F. & Peters, R.C. [1997] Ëlectrophysiological demonstration of N-methyl-D-aspartate receptors at the afferent synapse of catfish electroreceptor organs", Neuroscience 79, 1231-1237.

Andrianov, G.N., Peters, R.C. & Bretschneider, F. [1994] Ïdentification of AMPA receptors in catfish electroreceptor organs", Neuroreport 5, 1056-1058.

Andronov, A., Witt, A. & Khaykin, S. [1966] Theory of Oscillations (Pergamon Press, Oxford).

Arnold, V.I. [1978] Mathematical Methods of Classical Mechanics (Springer-Verlag, New York).

Bennett, M.V.L. & Obara, S. [1986] Ïonic mechanisms and pharmacology of electroreceptors", in Electroreception, eds. Bullock, T.H. & Heiligenberg, W. (Wiley, New York) pp. 157-181.

Blekhman, I. [1988] Synchronization in Science and Technology (ASME Press, New York).

Braun, H.A., Wissing, H., Schäfer & Hirsch, M.C. [1994] Öscillation and noise determine signal transduction in shark multimodal sensory cells", Nature 367, 270-273.

Bromm, B., Hensel, H. & Tagmat, A.T. [1976] "The electrosensitivity of the isolated Ampulla of Lorenzini", J. Comp. Physiol. 111, 127-136.

Bullock, T.H. [1982] Ëlectroreception", Ann. Rev. Neurosci. 5, 121-170.

Bullock, T.H. [1986] "Significance of findings on electroreception for general neurobiology", in Electroreception, eds. Bullock, T.H. & Heiligenberg, W. (Wiley, New York) pp. 651-674.

Bullock, T.H., Bodznick, D.A. & Northcutt, R.G. [1983] "The phylogenetic distribution of electroreception: evidence for convergent evolution of a primitive vertebrate sensory modality", Brain Res. Reviews 6, 25-46.

Burggren, W.W. & Bemis, W.E. [1992] "Metabolism and ram gill ventilation in juvenile paddlefish, Polyodon spathula (Chrondrostei: Polyodontidae)", Physiol. Zool. 65, 515-539.

Clusin, W.T. & Bennett, M.V.L. [1979] "The oscillatory responses of skate electroreceptors to small voltage stimuli", J. Gen. Physiol. 73, 685-702.

Elson, R.C., Selverston, A.I., Huerta, R., Rulkov, N.F., Rabinovich, M.I. & Abarbanel, H.D.I. [1998] ``Synchronous behavior of two coupled biological neurons'', Phys. Rev. Lett. 81, 5692-5695.

Fox, A.P., Nowycky, M.C. & Tsien, R.W. [1987] "Kinetic and pharmacological properties distinguishing three types of calcium currents in chick sensory neurones", J. Physiol. (Lond.) 394, 149-172.

Glass, L. & Mackey, M.C. [1988] From Clocks to Chaos. The Rhythms if Life (Princeton University Press, Princeton, NJ).

Grande, L. & Bemis, W.E. [1991] Östeology and phylogenetic relationships of fossil and recent paddlefishes (Polyodontidae) with comments on the interrelationships of Acipenseriformes", J. Vert. Paleontol. 11 (Mem. 1), i-121.

Gurgens, C.C. [1998] Electrosensory Avoidance Behavior of the Paddlefish (Polyodon Spathula) and the Effect of Coded-Wire Tags, Masters thesis, Biology Dept., Univ. of Missouri-St. Louis, 1-63.

Haken, H. [1983] Synergetics: An Introduction (Springer, Berlin, Heidelberg, New York).

Hamm, A. & Graham, R. [1992] `` Scaling for small random perturbations of golden critical circle maps'', Phys. Rev. A 46, 6323-6333.

Huygens, C. [1673] Horoloqium Oscilatorium (Parisiis, Paris, France).

Jø rgensen, J.M., Flock, Å. & Wersäll, J. [1972] "The Lorenzinian ampullae of Polyodon spathula", Z. Zellforsch. 130, 362-377.

Kalmijn, A.J. [1974] "The detection of electric fields from inanimate and animate sources other than electric organs", in Electroreceptors and Other Specialized Receptors in Lower Vertebrates (Handbook of Sensory Physiology, vol. III/3), ed. Fessard, A. (Springer-Verlag, Berlin) pp. 147-200.

Lu, J. & Fishman, H.M. [1995] "Localiztion and function of the electrical oscillation in electroreceptive ampullary epithelium from skates", Biophysical J. 69, 2458-2466.

Lu, J. & Fishman, H.M. [1995] Ïon channels and transporters in the electroreceptive ampullary epithelium from skates", Biophysical J. 69, 2467-2475.

Murray, R.W. [1974] "The ampullae of Lorenzini", in Electroreceptors and Other Specialized Receptors in Lower Vertebrates (Handbook of Sensory Physiology, vol. III/3), ed. Fessard, A. (Springer-Verlag, Berlin) pp. 125-146.

Neiman, A., Feudel, U. & Kurths, J. [1995] ``The cumulant approach for investigating the noise influence on mode-locking bifurcations'', J. Phys. A: Math. Gen. 28, 2471-2480.

Neiman, A., Silchenko, A., Anishchenko, V. & Schimansky-Geier, L. [1998] ``Stochastic resonance: noise-enhanced phase coherence'', Phys. Rev. E 58, 7118-7125.

Neiman, A., Pei, X., Russell, D., Wojtenek, W., Wilkens, L., Moss, F., Braun, H.A., Huber, M.T. & Voigt, K. [1999] ``Synchronization of the electrosensitive cells in the paddlefish'', Phys. Rev. Lett. 82, 660-663.

New, J.G. & Bodznick, D. [1985] "Segregation of electroreceptive and mechanoreceptive lateral line afferents in the hindbrain of chondrostean fishes", Brain Res. 336, 89-98.

New, J.G. & Bodznick, D. [1990] "Medullary electrosensory processing in the little skate. II. Suppression of self-generated electrosensory interference during respiration", J. Comp. Physiol. 167, 295-307.

Okano, K. [1988] "Two pharmacologically distinct receptors mediate two postsynaptic potential (PSP) components in the afferent synapse of the Plotosus electroreceptor", Brain Res. 457, 89-97.

Pecora, L.M. & Carroll, T.L. [1990] ``Synchronization in chaotic systems'', Phys. Rev. Lett. 64, 821-824.

Peters, R.C. & van Wijland, F. [1974] Ëlectro-orientation in the passive electric catfish, Ictalurus nebulosus LeS.", J. Comp. Physiol. 92, 273-280.

Pei, X., Russell, D.F., Wilkens, L.A. & Moss, F. [1998] ``Dynamics of the electroreceptors in the paddlefish, Polyodon spathula'', Computational Neuroscience, ed. by Bower, J. (New York).

Pikovsky, A.S., Rosenblum, M.G., Osipov, G.V. & Kurths, J. [1996] ``Phase synchronization of chaotic oscillators by external driving'', Physica D 104, 219-238.

Poincaré, H. [1954] Ouevres I (Paris: Gauthier-Villar).

Schäfer, C., Rosenblum, M.G., Abel, H. & Kurths, J. [1998] ``Synchronization in the human cardiorespiratory system'', Phys. Rev. E 60 857-870.

Rosenblum, M.G., Pikovsky, A.S. & Kurths, J. [1996] ``Phase synchronization of chaotic oscillators'', Phys. Rev. Lett. 76, 1804-1807.

Schäfer, C., Rosenblum, M., Kurths J., & Abel, H. [1998] ``Heartbeat synchronized with ventilation'', Nature 392,239-240.

Soen, Y., Cohen, N., Lipson D., & Braun, E. [1999] ``Emergence of spontaneous rhythm disorders in self-assembled networks of heart cells'' Phys. Rev. Lett 82, 3556-3559.

Stratonovich, R.L. [1967] Topics in the Theory of Random Noise (Gordon and Breach, New York) Vol. 2.

Tass, P., Rosenblum, M., Weule, J., Kurths J.,. Pikovsky, A., Volkmann, J., Schnitzler A. & Freund, H. [1998] ``Detection of n:m phase locking from noisy data: Application to magnetoencephalography'', Phys. Rev. Lett. 81, 3291-3294.

Teeter, J.H., Szamier, R.B. & Bennett, M.V.L. [1980] Ämpullary electroreceptors in the sturgeon Scaphirhynchus platorynchus (Rafinesque)", J. Comp. Physiol. 138, 213-223.

Tricas, T.C. & New, J.G. [1998] "Sensitivity and response dynamics of elasmobranch electrosensory primary afferent neurons to near threshold fields", J. comp. Physiol. A 182, 89-101.

Wiesenfeld, K. & Moss, F. [1995] "Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS", Nature 373, 33-36.

Wilkens, L.A., Russell, D.F., Pei, X. & Gurgens, C. [1997] "The paddlefish rostrum functions as an electrosensory antenna in plankton feeding", Proc. Roy. Soc. (London) B 264, 1723-1729.

Winfree, A.T. [1980] The Geometry of Biological Time (Springer, New York).

Vogel, S. & LaBarbera, M. [1978] "Simple flow tanks for research and teaching", Bioscience 28, 638-643.