INTRODUCTION

Top Abstract Introduction Results & Discussion Concluding Remarks References

The weakly electric fish Gnathonemus petersii emits pulse-type electric organ discharges (EODs) with an electric organ in its caudal peduncle. During each discharge an electrical field is built up around the fish's body, which is perceived by the sender fish and by other electric fish nearby. The electroreceptor organs of G. petersii are of three distinct types: mormyromasts, knollenorgans, and ampullary organs (Bennett, 1965). Mormyromasts are the only type of electroreceptor organs used to sense the self-emitted EODs during active electrolocation.

Each mormyromast organ contains an outer and an inner sensory chamber (Szabo and Wersall, 1970; Bell et al., 1989). The outer sensory chamber is connected to the outside surface of the skin by a plug of loosely packed epithelial cells and is connected to the lower sensory chamber by a canal. There are two types of mormyromast sensory cells: A-cells and B-cells. Each sensory cell is contacted by an afferent fiber. Impulse generation in the afferent fiber is induced by transmitter release, which occurs when the basal membrane potential of the sensory cells is depolarized by an EOD. The spike latencies of the afferent fiber encode the amplitude and/or distortion of the EOD waveform (Szabo and Hagiwara, 1967; Bell, 1990; Hall et al., 1995; von der Emde and Bleckmann, 1997).

During active electrolocation, when an object with electric properties different from those of the surrounding water comes close, the fish's electric field will be distorted and the pattern of voltage gradients at the mormyromast electroreceptors is changed. Animate objects, like other fishes, plants, or insect larvae, have a complex impedance consisting of a resistive and a capacitive component (Schwan, 1963; Heiligenberg, 1973). Depending on their electric properties, electrolocation objects can alter the amplitude of the local EOD and/or its waveform. EOD amplitude depends on the impedance of the object under investigation: an impedance higher than that of the water decreases local EOD amplitudes, while objects with a lower impedance than the water increase local EOD amplitudes (Heiligenberg, 1973). Capacitive objects cause alterations of the waveform of the locally received EOD owing to frequency-dependent amplitude attenuation and phase shift of the EOD (Heiligenberg, 1973; Bastian, 1986; von der Emde, 1990).

A-receptors are pure amplitude coders (Bell, 1990; von der Emde 1990; von der Emde and Bleckmann, 1992a, b). EOD waveform distortions, which are caused only by capacitive objects, slightly alter the response of A-receptors in a similar way, as does a reduction in stimulus amplitude (von der Emde and Bleckmann, 1992a, b). B-receptors respond similarly to A-receptors with respect to stimulus amplitude changes (Bell, 1990; von der Emde and Bleckmann, 1992a, b; 1997). In contrast to A-receptors, however, B-receptors are extremely sensitive to EOD waveform distortions (von der Emde and Bleckmann, 1992a,b, 1997). Their afferent fibers respond to a distorted EOD with shorter latency and an increased number of spikes per burst, even if the EOD amplitude has been kept constant. Thus, they respond to an EOD phase-shift as if EOD amplitude had increased. As a result, mormyrids unequivocally discriminate between resistive and capacitive objects during electrolocation (von der Emde and Bell, 1994). The fish can also discriminate between objects of different capacitive values (von der Emde, 1993). Waveform tuning of electroreceptors of G. petersii is discussed in detail by von der Emde and Bleckmann (1997). Experiments with computer-generated stimuli reveal that the strong sensitivity of B-receptors to EOD waveform distortions cannot be attributed to any of seven waveform parameters tested (von der Emde and Bleckmann, 1997). It is still unclear to which parameter of the distorted EOD B-receptors are sensitive.

In this series of papers we are concerned with the nonlinear mechanism of information processing by A- and B-receptors, and try to answer the following questions. What are the microscopic neural dynamics of A- and B-receptors? What differences cause A-receptors to be amplitude coders while B-receptors are waveform coders? What parameters of the distorted EOD are B-receptors sensitive to? Why are B-receptors more sensitive to EODs than A-receptors?

To systematically clarify the microscopic origins of most of the observed properties of A- and B-receptors of weakly electric fish G. petersii, we present a model that consists of a sensory cell perceiving the EOD and an afferent nerve fiber innervating the cell. The system considered is of active electroreception. Due to its resistance-capacitance structure, the sensory cell perceives not only the EOD amplitude, but also the variation of the EOD waveform. With the model, we show that the functional differences between A- and B-receptors are caused by differences in morphology of the sensory cells, as well as by differences of physical properties of the sensory cells, i.e., the different conductance of the leak ion channels of the apical membrane and the different capacitance of A- and B-cells. As a result, A-receptors mainly respond to the EOD amplitude, while B-receptors are sensitive to the variation of the EOD waveform. These differences also cause B-receptors to be more sensitive to stimuli than A-receptors. In the present paper we mainly concentrate on the investigation of the dynamics of the electroreceptor. In the second paper (Shuai, Kashimori, Kambara, and von der Emde, manuscript in preparation), we will discuss the responses of A- and B-receptor models to EODs distorted by various resistive and capacitive objects, to phase-shifted EODs, and to single periodic sinusoidal stimuli.

	MODELS

Kashimori et al. (1996) described a general model of P- and T-electroreceptor cells in the wavetype electric fish, Eigenmannia. In the model, the ionic currents across the apical and basal membrane of the receptor cell and the support cells are taken into account. Leak channels of Na⁺, K⁺, and Cl are included in the apical and basal membranes that contribute mainly to the determination of the resting potential of the sensory cell. The basal membrane of the cell has voltage-sensitive Ca²⁺ channels and Ca²⁺-activated K⁺ channels that are responsible for their regenerative voltage responses.

In the present sensory cell model, the contribution of supporting cells is ignored due to their high resistance. For simplicity, the assumption of a unified effective conductance has been used for the leak channels of Na⁺, K⁺ and Cl in the basal and apical membranes, as shown in Fig. 1. The cell membrane is separated by the supporting cells into apical and basal membranes. While the apical membrane contains only the effective leak channels, the basal membrane contains voltage-sensitive Ca²⁺ channels, voltage-sensitive and Ca²⁺-activated K⁺ channels, and effective leak channels.

View larger version (26K): [in this window] [in a new window]   FIGURE 1   Schematic representation of a mormyromast electroreceptor of weakly electric fish. Supporting cells divide the receptor cell membrane into apical and basal membranes. The apical membrane has only leak channels. The basal membrane has Ca2+ and K+-active channels besides the leak channels.

The afferent fiber is described by the modified Hodgkin-Huxley equations in which the voltage-sensitive gate of the K⁺ channels is a dynamic variable. Spike generation in the afferent fiber is induced by transmitter release at the synapse, which occurs when the basal membrane of the sensory cell is depolarized.

Sensory cell model

The equivalent circuit model of a sensory cell system is given in Fig. 2. The electric current density I_A in the apical membrane of the cell consists of the displacement current caused by the capacitance of the apical membrane and the ionic current flowing through the effective leak channel, i.e.,

(1)

where C₁ is the capacitance of the unit area of apical membrane, _A is the potential of the apical membrane, g₀ is the conductance of the effective leak ion channels of the unit area in the apical membrane, and ₀ is the equilibrium potential of leak channels.

View larger version (19K): [in this window] [in a new window]   FIGURE 2   The equivalent circuit of the receptor model. A and B are the potentials of the apical and basal membranes of the cell, respectively; C1 and C2 are the capacitances of the unit area of apical and basal membranes, respectively; g0 is the conductance of the leak ion channels of the unit area in the apical membrane; 0 is the equilibrium potential of leak channels; gCaB, gKB, and gLB are conductances of Ca2+, K+, and effective leak channels of the unit area in the basal membrane, respectively; Ca, K, and L are the equilibrium potentials of ion Ca2+, K+, and mixed leak channels across the basal membrane, respectively. The arrows attached to IA and IB indicate the positive directions of relevant current.

The electric current density I_B in the basal membrane consists of the capacitive displacement current and ionic currents flowing through the Ca²⁺, K⁺, and leak channels:

(2)

with

(3)

where C₂ is the capacitance of the unit area of basal membrane; _B is the potential of the basal membrane; g_Ca^B, g_K^B, and g_L^B are conductance of Ca²⁺, K⁺, and leak channels of the unit area in the basal membrane, respectively; and _Ca, _K, and _L are the equilibrium potentials of ion Ca²⁺, K⁺, and leak channels across the basal membrane, respectively.

For the equivalent circuit, there are two conservation laws as

(4)

where S₁ and S₂ are the areas of apical and basal membranes, respectively, and V_stim is the stimulation potential applied to the cell.

Substituting Eqs. 1-3 into Eq. 4, the equation for _B can be obtained as

(5)

with C₀ = C₁S₁ + C₂S₂.

In the present model, the Ca²⁺ channel of the basal membrane that allows extracellular Ca²⁺ to enter the cell is voltage-sensitive, i.e., g_Ca^B = _Ca^Bd(_B). The K⁺ channel that allows intracellular K⁺ to leave the cell contains a voltage-sensitive activation gate f(_B) and a free intracellular [Ca²⁺]-activated gate g([Ca²⁺]), i.e., g_K^B = _K^Bfg. The opening of the d-gate and f-gate is assumed to take place very fast. Then they are treated reasonably as the nondynamic variables and can be approximated by the steady-state values d and f, which are sigmoid functions of basal membrane potential. These function can be expressed by the Boltzmann equation (Rorsman and Trube, 1986; Chay, 1990b) as

(6)

where S_d, S_f and V_d, V_f are the steepness factors and the half-maximal activation potentials, respectively. The [Ca²⁺]-activated gate g is written as

(7)

Here we consider that the [Ca²⁺]-activated gate g of K⁺ channels depends only on the compartmentalized calcium ion concentration [Ca²⁺]_sub in the submembrane space (Chay, 1990a, 1993).

The free intracellular calcium concentration [Ca²⁺]_sub in the submembrane space is treated as a dynamic variable in the model, and its changes with time arise from the following two terms: the influx of extracellular calcium ions into the cell through the voltage-sensitive calcium channel and the efflux of free intracellular Ca²⁺ ions from the submembrane space to the extracellular medium by the pump action and to the intracellular medium by absorption. We thus have

(8)

where  is the rate constant for the efflux pump and intracellular absorption of calcium ions;  = 3/(4r³F), where F is the Faraday constant and r the effective radius of the cell.  is a measure of how fast the free calcium concentration [Ca²⁺]_sub changes in the submembrane space. We assume that  is a function of _B,

(9)

Note that  = _min when _B =  and  = _min + ₀ when _B = . S and V are the steepness factor and the half-maximal activation potential, respectively. Equation 9 is derived based on the assumption that the Ca²⁺ ions in the submembrane space of the cell move faster when the depolarized potential of the cell is low. In pancreatic -cells or HIT cells, the experiments have shown that calcium ions can oscillate even though the calcium ion channels are closed (Penner and Neher, 1988; Lund et al., 1991; Prentki et al., 1988). This implies that the intracellular calcium ions are not always free and are distributed heterogeneously. There are some calcium ion compartments or stores in the cell (Velasco and Petersen, 1987; Chay, 1990a, 1993). The [Ca²⁺]-activated gate g of K⁺ channels depends on the compartmentalized calcium ion concentration in the submembrane space that corresponds to the observable concentration of free intracellular calcium ions. Large depolarization of the cell causes a large inward current carried by calcium ions; in this case, many calcium ions flow into the cell rapidly. Although the total calcium concentration in the cell increases greatly, most of the calcium ions are driven into compartments within the cell. Therefore, the number of free intracellular calcium ions in the submembrane space is changed slowly, i.e., the rate of change  of free intracellular calcium ions in the submembrane space is small. In contrast, for small depolarization, the calcium ions move into the cell slowly, so that more calcium ions remain in the submembrane space of the cell, which activates the gate g of K⁺ channels. That is,  becomes large.

Afferent fiber model

Spike generation in the afferent nerve fiber is induced by transmitter release, which occurs when the basal membrane potential of the cell is depolarized sufficiently (Sanchez and Zakon, 1990; Kashimori et al., 1996). To calculate the impulse trains in the afferent fiber, the modified Hodgkin-Huxley equations were used. As shown by Hodgkin and Huxley (1952), the membrane potential V at the axon terminal is determined by

(10)

where C is the membrane capacitance of the afferent fiber, g_Na and g_K are the maximum conductance of the active Na⁺ and K⁺ channels, respectively, when all channels are open; g_L is the conductance of leak ion channels; m, h, and n are the gating parameters; and V_Na, V_K, and V_L are the equilibrium potentials of Na⁺, K⁺, and leak ions, respectively.

The gating variables m, h, and n are time-dependent such that the opening of the gates takes place by a simple two-state mechanism. If the opening and closing of channels are very fast, the probability in the open state can be replaced by its steady-state expression. In the present model, the usual time dependencies of m and h in the Hodgkin-Huxley equations are assumed to be replaced by their steady-state values m and h. The probability, n, is a dynamic variable whose value can be obtained by solving the following first-order equation:

(11)

Let y stand for m, h, and n; then the explicit expressions for m, h, and n can be written as

(12)

where _y and _y are the activation and deactivation rates of the two-state mechanism, respectively, and expressed as (Chay, 1985)

(13)

The relaxation time constant of the n-gate is expressed in terms of _n and _n as

(14)

The postsynaptic current I_ps in Eq. 10 is induced by the neural transmitter released from the presynaptic membrane of the sensory cell. The magnitude of I_ps is proportional to the intensity of released transmitter, which is determined by the Ca²⁺ current I_Ca across the basal membrane. Following Kashimori et al. (1996), the relation of I_ps to I_Ca is written as a sigmoid function

(15)

where I_Ca is given by Eq. 3, w is the strength of synaptic connection,  is the threshold value for the transmitter release, and  is the parameter determining the rising of the sigmoid curve. When I_Ca > , a large stimulus current I_ps is induced.

	RESULTS AND DISCUSSION

Top Abstract Introduction Results & Discussion Concluding Remarks References

In the present paper, first we discuss the dynamics of the sensory cell and the afferent fiber models in detail. Stimulated by an EOD, the sensory cell is excited to a large depolarization state with a duration of ~10 ms. Thus, a high calcium ion current is achieved with a duration of ~10 ms. The current causes the release of transmitter to the afferent fiber. The afferent fiber is then driven to fire after a certain time interval. One of the purposes of the present study is to clarify the origin of the functional differences between A- and B-receptors. Experimental results of receptors responding to a square wave stimulus (Bell, 1990) are considered based on the present model. We use the parameter values listed in Tables 1 and 2 for the computation. The parameter values of the equilibrium potentials and the conductance of ion channels are typical ones. For the other parameters, the reasons why we adopted the values listed in the tables are given at suitable points in the following three subsections. It is noted in Table 1 that, besides the different morphology of the A- and B-cells, the conductance of the leak channels of the apical membrane and the capacitance of the apical membrane are different in A- and B-cells.

Dynamics of the sensory cell

In the sensory cell model, we propose that the EOD stimulus drives the cell to a large depolarized state with a duration of ~10 ms. Here we show how the model can achieve this dynamical property.

First, we consider the dynamical properties of the current density in Eq. 5, i.e., S₁g₀(_B + ₀)  S₂(I_Ca + I_K + I_L), for a fixed value of [Ca²⁺]_sub. Because the values of g₀S₁/S₂ = 300 µS/cm² for both A- and B-cells, we consider the following current density I_c:

(16)

In Fig. 3 the current density I_c versus the basal membrane potential _B are plotted for different fixed values of Ca²⁺ concentration. The simulation shows that there are two stable fixed points (I_c = 0) when [Ca²⁺]_sub < 0.01867 mM. From I_c = 0 and d[Ca²⁺]_sub/dt = 0 (Eq. 8), the resting equilibrium state of the cell is obtained with _B^r = 52 mV and [Ca²⁺]_sub^r = 0.01 mM. When 0.01 mM < [Ca²⁺]_sub < 0.01867 mM, one stable fixed point (_B^h) corresponds to the hyperpolarized state, which is quite near the resting equilibrium state _B^r, and another point corresponds to the greatly depolarized state _B^d. Corresponding to the two fixed points there are two stable attracting regions that are separated by the unstable fixed point _B^u, i.e., the resting attraction region with the attractor point _B^h and the exciting attraction region with the attractor point _B^d. However, [Ca²⁺]_sub is a dynamical variable determined by Eq. 8. Once the sensory cell is depolarized, [Ca²⁺]_sub increases gradually. When [Ca²⁺]_sub > 0.01867 mM, the Ca²⁺-activated gate g becomes so large that the outward currents exceed the inward Ca²⁺-current. As shown in Fig. 3 with the dotted line, the greatly depolarized state becomes unstable and then only the hyperpolarized state _B^h is stable.

View larger version (20K): [in this window] [in a new window]   FIGURE 3   The curves of current Ic through the basal membrane versus the basal potential B for different fixed [Ca2+]sub (0.01, 0.012, 0.016, and 0.02 mM). Br is the resting equilibrium state when [Ca2+]sub = 0.01 mM; Bd is the greatly depolarized state. The unstable fixed points Bu separate the attraction regions to the resting and depolarized ones.

Fig. 4 shows the response of the model of an A-cell to a half-period positive sine stimulus (Fig. 4 a) given by A sin(t) with A = 5.0 mV and  = 1000 s¹. A similar result can be obtained for the model of a B-cell. With a large enough stimulus, the potential of the cell is depolarized (Fig. 4 b). When the depolarized potential exceeds the unstable fixed point, i.e., _B > _B^u, the cell is driven to the exciting attractor region from its resting attractor region. Once it enters this region, the cell automatically becomes greatly depolarized. Because of the short duration of the positive sinusoidal signal (0.5 ms), the calcium ion concentration change is small. Our simulation results show that when t = 0.5 ms, [Ca²⁺]_sub becomes 0.01034 mM. At this small calcium ion concentration, the stable depolarized potential of the basal membrane becomes 5.78 mV. The inward calcium ion current density I_Ca increases to 4982 µA/cm² from its resting equilibrium current density 175.0 µA/cm². Although many calcium ions move into the cell during the greatly depolarized state, the free intracellular calcium ion concentration in the submembrane space increases very slowly because most of the calcium ions move rapidly into the interior of the cell; then the greatly depolarized potential decreases very slowly. We adjusted the values of parameters _min, ₀, V, and S in Eq. 9 as shown in Table 2 so that the greatly depolarized state could be maintained for ~10 ms. However, the gradual increase of [Ca²⁺]_sub causes the depolarization to decrease which, in turn, causes [Ca²⁺]_sub to increase more. When the basal membrane potential decreases to 6.2 mV, the relaxation time  becomes small enough, and consequently [Ca²⁺]_sub and the membrane potential change rapidly. When [Ca²⁺]_sub becomes larger than 0.01867 mM, the Ca²⁺-activation gate g becomes so large that the outward current exceeds the inward current, and the greatly depolarized state becomes unstable. As a result, the potential of the cell returns to the hyperpolarized state _B^h in the resting attractor region. Then the inward Ca²⁺ current becomes small again. The free Ca²⁺ with high concentration in the submembrane space of the cell is pumped out. After a transient process, the cell gradually goes back to the resting equilibrium state.

View larger version (16K): [in this window] [in a new window]   FIGURE 4   The response properties of the A-cell model stimulated by a half period positive sinusoidal stimulus given by A sin(t) with A = 5 mV and  = 1000 s1. (a) Stimulus; (b) basal membrane potential B; (c) free calcium concentration in the submembrane space of the cell; (d) inward calcium current ICa; (e) outward K+ current IK.

Larger stimulus amplitude A induces the transition from the resting equilibrium state to the greatly depolarized state in a shorter time, which causes a smaller increase of [Ca²⁺]_sub. The smaller increase of [Ca²⁺]_sub causes a higher depolarized potential of the basal membrane with a longer duration. As a result, a larger calcium ion inward current with longer duration is achieved. In Fig. 5 the maximum difference between the largest depolarized potential and the resting equilibrium potential, i.e., _Bmax = _Bmax^d  _B^r, is shown as a function of the stimulus amplitude A, together with the curve of the duration of the greatly depolarized state. Here, the greatly depolarized state is defined as the state in which the potential of the basal membrane is larger than 7.0 mV.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   The difference Bmax between the maximum depolarized potential and the resting equilibrium potential of the basal membrane Bmax = Bmaxd  Br and the duration of greatly depolarized state as functions of the amplitude of the stimulus A sin(t) with  = 1000 s1. Here the greatly depolarized state is defined as its basal membrane potential is >7.0 mV.

Dynamics of the afferent fiber

Our model of an afferent fiber is based on the Chay model (Chay, 1985; Fan and Chay, 1994) which is described by three differential equations. As two bifurcation parameters are varied, a three-variable model can lead to an enormously complex bifurcation structure, including several types of chaos. In the Chay model, the calcium concentration is a slowly varying dynamic variable that generates bursting, while the gating kinetics variable of K⁺ channels is a voltage- and time-dependent fast varying dynamical variable that generates the spikes. The third dynamical variable is the membrane potential V, which is a variable depending on both the calcium concentration and the gating kinetics variable. To obtain the oscillation behavior in our model, we simplify the Chay model to two differential equations. In the present model, the fast gating kinetics variable of the K⁺ channels is maintained. The values of the parameters appearing in Eq. 13 are the same as those used in the Chay model. In the Chay model, the basic time scale is millisecond, while the microsecond scale is required for the dynamics in A- and B-receptors. Therefore, we modify the values of the conductance and the equilibrium potentials of the three kinds of channel gates, and the relaxation time _n0 of n-gate.

To discuss the dynamics of the afferent fiber model in detail, we let I_ps = 20 µA. The simulation results are shown in Fig. 6. When the afferent fiber is stimulated by a constant current I_ps, the membrane potential V of the afferent fiber is depolarized gradually. Then the voltage-sensitive Na⁺ channels are activated by the depolarization, and the inward Na⁺ current increases. This, in turn, accelerates the depolarization of potential V. The increasing potential activates the time-delayed voltage-dependent K⁺ channels. The rapidly increasing outward K⁺ current suppresses the increase of the potential. Then the potential begins to decrease, which causes the deactivation of Na⁺ channels. Because the Na⁺ current decreases, the repolarization of the membrane potential is accelerated. The time-delayed voltage-dependent K⁺ channels are also deactivated. As a result, the afferent potential approaches the resting state. However, the constant current I_ps continuously stimulates the afferent potential, and a new oscillation cycle occurs.

View larger version (24K): [in this window] [in a new window]   FIGURE 6   The response properties of the afferent model under application of a constant current Ips = 20 µA. (a) Stimulus; (b) membrane potential of the afferent V; (c) time-delayed voltage-sensitive gate of K+ channels; (d) inward Na+ current INa; (e) outward K+ current IK.

Larger constant current I_ps leads to a faster and a larger depolarization process. Larger depolarization, in turn, causes faster repolarization. Thus, the oscillation period and the latency duration between stimulus onset and the occurrence of the first spike in the afferent fiber is shortened by an increase in I_ps. The dependence of the oscillation period on the stimulus current is given in Fig. 7. Simulation results show that the oscillation period is 9.2 ms at the threshold current I_ps = 17.86 µA. The calcium ion current intensity of the excited sensory cell is ~5000-4900 µA/cm². We chose the values of parameters w, , and  in Eq. 15 as shown in Table 2 so that the corresponding current I_ps that can initiate afferent fiber oscillation is ~22.2-12.0 µA.

View larger version (9K): [in this window] [in a new window]   FIGURE 7   The dependence of the oscillation period in the afferent fiber on the stimulus current Ips from 17.8 to 52.0 µA. The oscillation periods decrease from 9.2 to 0.69 ms.

Now one can see how the A- and B-receptor models process the information contained in the EOD stimulus. Although the real EOD pulse is very short (duration < 0.5 ms), it can drive the cell to the greatly depolarized state. Due to the large relaxation time constant of free intracellular calcium ions in the submembrane space of the cell, the large calcium ion current is maintained for ~10 ms. As a result, the afferent fiber can oscillate with a duration of ~10 ms. Differently distorted EODs cause differently depolarized potentials and different durations in the A- and B-cells. The release of transmitter at the afferent fiber synapse induced by the various depolarizations then differs in intensity and duration. A higher intensity of transmitter release leads to a shorter spike latency in the afferent fiber. Coincidentally, the longer the duration of transmitter release, the larger the number of spikes.

Origin of functional differences between A- and B-receptors

A-receptors are known to respond to the intensity of the EOD. What kind of information can B-receptors process? It is seen from Eq. 5 that the sensory cell not only responds to the EOD intensity V_stim, but also senses the variation dV_stim/dt of the EOD waveform due to its resistance-capacitance structure. To answer the question, it is a reasonable consideration that A- and B-receptors respond mainly to one of the two terms, respectively.

An object with a considerable capacitive component has two effects on EOD signal (von der Emde, 1990, 1992b). 1) There is a change in the intensity if the impedance of the object differs from that of the surrounding water. The impedance of a capacitive object is frequency-dependent in contrast to that of a resistive object. Low-frequency components of an EOD are attenuated more strongly than high-frequency components. 2) There is a frequency-dependent phase shift of the signal: frequency components within a certain range, depending on the value of the capacitance, are phase-shifted more strongly than others. Objects with no capacitive components do not cause any phase shift of the signal. Both frequency-dependent attenuation and frequency-dependent phase shift can lead to a strong distortion in the waveform of the EOD, i.e., cause a large change in the variation of EOD waveform.

In this subsection we show that because of the difference in morphology of A- and B-cells, and of the differences in the conductance of the leak ion channels of the apical membranes and the membrane capacitance between A- and B-cells, the A-receptor can mainly respond to the intensity of EOD stimulus, while the B-receptor is sensitive to the distortion of the EOD waveform caused by the capacitive component of object.

Anatomical experiments (Szabo and Wersall, 1970; Bell et al., 1989) showed that the apical membrane of the A-cell in contact with the outer sensory chamber is small in area. However, the apical membrane of the B-cell protruding into the inner sensory chamber is quite large in area. For the A-cell, suppose S₁/S₂ = 1/a with a > 1 and C₁ = C₂ = C_A, then we have the following potential equation

(17)

where I = I_Ca + I_K + I_L.

Experiments have shown that most of the B-cells protrude into the sensory chamber, and this exposed cell surface is covered with microvilli (Bell et al., 1989). So we suppose S₁/S₂ = a for the B-cell. Due to the microvilli, the physical properties of apical membrane of the B-cell are also different from those of the A-cell. As a result, the capacitance of the membrane is assumed to be smaller than that of the A-cell: C₁ = C₂ = C_B = C_A/(ab) with a > b > 1. Furthermore, let the conductance of the leak channel of B-cell be much smaller than that of A-cell: g_0(B) = g₀/a². Then, for B-cell

(18)

Now, using Eqs. 17 and 18, we discuss the origin of the functional difference between A- and B-electroreceptors. With the same potential stimulus V_stim, the A-cell actually responds to the signal

(19)

while the B-cell responds to

(20)

Due to the condition a > b > 1, a given stimulus has a larger effect upon the B-cell than the A-cell. As a result, the B-cell has a lower threshold than the A-cell.

With suitably chosen values of the parameters a, b, and g₀, it is possible for the A-cell to hold the relation while for the B-cell It results that the A-receptor can be more sensitive to the intensity of the EOD than to the variation of the EOD waveform, whereas the B-receptor is more sensitive to the latter than to the former.

The relationship between the maximum intensity of EOD stimulus and the maximum time variation of EOD holds roughly for a natural biphasic EOD (Shuai et al., manuscript in preparation). In order that the model satisfies the relation, we adjust the values of parameters for A-cell to obtain the condition C_A/g < 1.8 × 10⁵ s. If we let g₀ be of the order of 1000 µS/cm², then C_A must be of the order of 10² µF/cm². For the B-cell, we have C_B/g_0(B) = (a/b)C_A/g₀ < 1.8 × 10⁵ s. In fact, in order to let the A-receptor respond sensitively to the EOD intensity and let the B-receptor respond sensitively to the variation of the EOD waveform, we require only the fact that the A- and B-cells possess the structure shown in Fig. 1. It has little relation to the detailed dynamics of the sensory cells and the afferent fibers. As a result, the above discussion is rather general.

The parameter values of A- and B-cells given in Table 1 are typical ones that satisfy the conditions mentioned above. By using these values, we have

(21)

for the A-cell. Furthermore, letting b = 1.2, then S₁/S₂ = 10, g_0(B) = 30 µS/cm², and C_B = 2.5 × 10³ µF/cm² for the B-cell, we obtain

(22)

Response to the square waveform stimuli

To investigate experimental response properties of A- and B-receptors, square waves were used as stimuli to the receptors (Bell, 1990; von der Emde and Bleckmann, 1992b). Experimental results show that, for stimulation with outside positive square waves, the maximum numbers of spikes discharged by A-receptors was two to four, whereas the maximum numbers for B-receptors was four to eight spikes. A large increase in the threshold of A- and B-receptors was observed when the duration of the square wave stimuli was decreased from 1.0 to 0.1 ms. The threshold values for different A-receptors varied over a wide range, whereas various B-receptors appeared to have relatively homogeneous thresholds.

Now we consider the origin of these experimental results based on our receptor model. To simulate the transient process during the rising and falling slope of the square wave stimulus, the stimulus should change linearly within a very short time period t. Then, during the rising or falling slope of the square wave stimulus, the stimulus linearly increases or decreases with a variation of ±V_stim/t, respectively, where V_stim is the height of the square wave. During the plateau region of the stimulus, its intensity is constant and has the value V_stim. If t  10 µs, then during the rise and fall of the stimulus the inequality 10⁵dV_stim/dt > V_stim(t) holds. Therefore, the B-receptor and the A-receptor respond primarily to the variation of stimulus intensity, as seen from Eq. 19. Positive square waves were used in the experiments (Bell, 1990; von der Emde and Bleckmann, 1992b) to discuss how A- and B-receptors respond to the various intensities of stimuli. Our model implies that it is mainly due to the variation at the onset of the positive square wave, rather than its intensity, that the A-cells as well as the B-cells are depolarized. Therefore, the response properties of the receptors to the square wave stimulus are quite different from those to the natural EOD stimulus, especially for the A-receptor, whose response arises mainly from the stimulus intensity.

Responses of A- and B-receptors to a 10-ms outside positive square wave stimulus are calculated as a function of the stimulus intensity, and are given in Fig. 8. Here t = 10 µs is used and the stimulus is applied at t = 0.25 ms. For A- and B-receptors, as the stimulus intensity increases, the spike latencies decrease, and additional spikes are added to the response. The threshold stimulus intensity for the A-receptor is 2.83 mV with the spike latency of 4.35 ms. At the stimulus intensity of 9 mV, the first spike latency decreases to 2.69 ms, and the number of spikes becomes 5. For the B-receptor, the threshold value is 1.84 mV with a spike latency of 4.43 ms. At the stimulus intensity of 3.3 mV the first spike latency is 2.63 ms and the number of spikes increases to eight. Fig. 9 shows the latencies of each spike responding to the various stimulus intensities for A- and B-receptors. Here, the latency indicates the time duration between the onset of stimulus and the peak of each spike in the afferent fiber. For the A-receptor, when the stimulus intensity increases from 2.83 to 10.3 mV, the latency of the first spike decreases from 4.35 to 2.65 ms. For the B-receptor, in contrast, the latency of the first spike decreases from 4.43 to 2.62 ms when stimulus intensity increases from 1.84 to 3.32 mV. The variability of the first spike latency is large when the stimulus intensity is a little higher than the threshold, and becomes rapidly small as the stimulus is increased. This is in good agreement with the observed results (Bell, 1990), although the first-spike latency obtained for the threshold intensity is short compared with the experimental results (Bell, 1990).

View larger version (25K): [in this window] [in a new window]   FIGURE 8   The responses of afferent fiber innervating A- and B-cells induced by the 10-ms outside positive square wave stimuli for various stimulus intensities. (A) For the A-receptor, the increase in the stimulus intensity above the threshold causes a reduction in latency of the response and an increase in the number of spikes from 1 to 5. (B) For the B-receptor, the increase in the stimulus intensity above the threshold causes a reduction in latency of the response and an increase in the number of spikes from 1 to 8.

View larger version (15K): [in this window] [in a new window]   FIGURE 9   The latencies of the first several spikes in A- and B-receptors generated by 10 ms outside positive square stimuli with various intensities. Here, the voltage of the stimulus is shown on the y axis, while latency is shown on the x axis. (A) The latencies of five spikes for the A-receptor. (B) The latencies of eight spikes for the B-receptor.

It has been shown (Bell, 1990) that the time interval between two adjacent spikes increases gradually for the later spikes. This property arises from the following reason. With the onset of the stimulus, the cells are rapidly depolarized. The basal membrane potential approaches its maximum value in a short time interval, then it slowly decreases because of the slow increase of the free intracellular Ca²⁺ in the submembrane space. A similar change is caused for the release of transmitter at the afferent fiber synapse (i.e., I_ps). Thus the time intervals between the spikes increase gradually with time, as shown in Fig. 9.

Higher square wave stimulus leads to a larger depolarization of the cell with a longer time duration. This causes the shorter latency of each spike and a higher spike number for the afferent fiber. An increase of duration of the stimulus induces a similar effect. After the depolarization of the cell, the remaining outside positive stimulus always has a lasting depolarization effect on the basal membrane potential of the cell and can delay the return of the cell to the resting attraction region. Thus, although the increase of stimulus duration has little effect upon the latency of the first spike, it affects the following spikes so that the latencies of the following spikes decrease and the spike number increases.

Now we consider the effect of the duration of a square wave stimulus on the threshold of the receptors. The calculated results for the dependence of threshold stimulus intensity on the duration of outside positive square wave stimuli are given in Fig. 10 for A- and B-receptors. There are two typical transition time intervals in the cell when it is depolarized by a stimulus. One is the transition interval of the sensory cell from the resting equilibrium state to the greatly depolarized state just after the onset of the square wave stimulus. This time interval is very short. In our model, it is ~0.3 ms. The second interval is the transition time of the afferent fiber from the onset of transmitter release to the appearance of the first spike. Because of the short transition interval of the sensory cell, the time interval of the afferent almost equals the first spike latency, which is ~4.3 ms. If the duration of the stimulus is so long that its downfall comes after the first spike, the stimulus duration has no effect upon the first spike. Therefore, the threshold does not depend on the duration, as shown in Fig. 10. If the downfall of the stimulus occurs within the transition time interval of the afferent (~0.3-4.3 ms), the negative variation of the stimulus can rapidly cause the cell to go back to the resting attraction region. In this case, the release of transmitter decreases a little, and then a little stronger stimulus is required to obtain a spike. If the downfall of the stimulus occurs within the transition interval of the cell (<0.3 ms), the basal membrane potential of the cell is drawn back to the resting attraction region before the transition to the depolarization state occurs. Then no spikes can be achieved in the A- and B-afferents. Thus, a high intensity of the square wave stimulus is required to depolarize the cell greatly enough to overcome the negative effect of the early downfall of the stimulus.

View larger version (12K): [in this window] [in a new window]   FIGURE 10   The effect of duration of the positive square wave stimuli on the threshold stimulus intensity of A- and B-receptors. The upper line is for the A-receptor, while the lower is for the B-receptor.

Experimental results (Bell, 1990) have shown that different A-receptors possess different threshold values distributing over a wide range when they are stimulated with a 10-ms outside positive stimulus, while different B-receptors appear to have relatively homogeneous values of threshold. The microscopic origin of this difference arises from the following property of A- and B-receptors. It seems reasonable for actual A- and B-receptors that the values of parameters included in each receptor vary slightly for every receptor. Therefore, we investigated the effect of slight changes in parameter value on the response property of A- and B-receptors using the present model.

Stimulated by the threshold stimulus, the cell produces the threshold Ca²⁺-current density I_Ca^thr and the threshold synaptic current I_ps^thr that can fire the afferent fiber. Now, if one of the parameters in the cell model changes a little, it usually leads an increase or a decrease of I_Ca. To obtain the fixed threshold synaptic current I_ps^thr, it is required to decrease or increase the stimulus intensity. Then the intensity obtained is the new threshold intensity. From Eqs. 17 and 18 one can see that the change of A_stim required for the A-receptor is almost equal to that of B_stim required for the B-receptor. However, it can be seen from Eqs. 19 and 20 that the equivalent amount of increase in actual signal A_stim and B_stim requires different amounts of increase in stimulus intensity V_stim for A- and B-receptors, respectively. The A-receptor requires a larger increase in stimulus intensity than the B-receptor. For example, the value of the half-maximal activation potential V in Eq. 9 is changed a little around the value 5.6 mV. The simulation shows that, when the value V is changed to 5.61 or 5.59 mV, the threshold value of V_stim for the A-receptor changes from 2.83 to 2.77 or 2.90 mV; that is, the changes are ~2.3%; while for B-receptor it changes from 1.84 to 1.83 or 1.85 mV; that is, the changes are ~0.54%. A similar result can be obtained for the change of parameter values in the afferent fiber. For example, the simulation shows that, when the value of synaptic strength  is changed from 24 to 23 µA or to 25 µA, the threshold for the A-receptor changes to 2.75 or 2.95 mV; that is, the changes are ~2.3%; while for the B-receptor it changes to 1.83 or 1.85 mV; that is, the changes are ~0.54%.

The responses of A- and B-receptors to an outside negative square wave stimulus were also investigated in experiment (Bennett, 1965; Bell, 1990). At first glance, it seems strange that the receptor can be excited by an outside negative stimulus; with such a stimulus the receptor should be hyperpolarized, while in fact it is depolarized. As mentioned above, A- and B-receptors mainly respond to the variation part of square wave stimulus. For an outside negative stimulus, the variation is negative at its onset, while it is positive at its switchoff. The latter variation can depolarize both A- and B-cells. As a result, the responses of A- and B-receptors to outside negative square waves occur at the switchoff, while the responses to outside positive square wave stimuli occur at the onset of stimulus (Bennett, 1965; Bell, 1990).

We investigated the response of the present model to the outside negative square wave stimuli. When the stimulus duration is long enough, A- and B-cells are stabilized in a hyperpolarized state that is very close to the resting equilibrium state. Then the value of the activation threshold becomes almost the same as that for the outside positive stimulus, as it should be (Bell, 1990). By using a stimulus with short duration, the depolarization at the switchoff begins from the deeply hyperpolarized state generated by the onset of the stimulus. As a result, a higher threshold potential is required to depolarize the cell. This has been observed for the B-receptors (Bell, 1990). However, the increase of the threshold potential induced by a short negative square stimulus was not observed for the A-receptors. It seems to be one of the reasons for the disagreement that the stimulus duration used in the experiment was not short enough.

	CONCLUDING REMARKS

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To understand how the electrosensory system of the weakly electric fish G. petersii distinguishes the object, we presented an electroreceptor model. The m