Experimental observations

The goal of this study is to understand the mechanism that neurons in the ELL of the weakly electric fish employ to cancel spatially redundant signals with different contrasts. To do that, we first analyze experimental data from in vivo recordings. Fig. 1 shows the extracellularly recorded response of superficial pyramidal (SP) neurons in the ELL under different stimuli. As one can see, SP cells respond strongly and in a phase-locked fashion to local stimuli (Fig. 1A, black), whereas the response is much broader in phase, and smaller in amplitude, when the stimulus is global (Fig. 1A, gray). By considering peri-stimulus time histograms (PSTH), we confirm that, within the range of frequencies of the AM considered, the response to global signals is effectively cancelled (Fig. 1B). As previous studies have addressed, the cancellation is most pronounced within this AM frequency range [16], and it is achieved by the emergence of a negative image of the original signal, generated by the feedback pathway [15], [16]. More importantly, whereas previous studies [16], [24] characterized cancellation for a single stimulus strength, in this study we present the stimulus at different contrast levels (i.e. different strengths). We observe that the stimulus is cancelled efficiently for a wide range of input contrast levels, with cancellation values over in all cases. Cancellation was measured by the ratio in gain between the local and global responses (see Methods for details).

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Figure 1. In vivo electrophysiological observations.

(A) Response of SP cells to AM stimuli, for the case in which the signal is local (black) or global (gray). For each case, a single extracellular recording trial and a raster plot are shown. (B) Cancellation of global stimuli for different AM frequencies. Upper row corresponds to local stimulation while lower row corresponds to global stimulation. In each panel, the mean PSTH ( neurons from several fish) for contrasts of (red), (green), (blue) and (violet) is displayed. The inset in the lower left panel shows the extent of the typical error bars for one of these curves (). (C) Percentage of cancellation as a function of input contrast, for different AM frequencies. (D) Degradation of the signal cancellation, averaged over all AM frequencies considered, as a function of input contrast.

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Furthermore, the level of cancellation appears to be approximately the same for all contrasts, from very low () to very high () values (Fig. 1C), with a minor decay of cancellation levels observed for very high contrasts. Contrasts higher than were not considered in this study, since P-unit electroreceptors encode AMs in a nearly linear manner up to . After that, the activity of P-units saturates and biologically relevant information can not be processed in the same quasi-linear regime [26]. When averaging over all the frequencies considered, we can observe that the degradation of the cancellation process (defined as the complementary of the frequency-averaged cancellation, i.e. ) is restricted to a range between and , and therefore the cancellation of global signals only varies in about for all the range of biologically relevant contrast levels (Fig. 1D).

Modeling local stimulation

We first consider the response of the SP neuron to local stimuli. The dynamics of the neuron membrane potential is modeled following a leaky integrate-and-fire (LIF) formalism [27], [28] with an extra burst-inducing mechanism. The subthreshold dynamics of the membrane potential is given by(1)where denotes rectification of the input, is the input from the P-units encoding the sensory stimulus, is the burst-inducing mechanism needed to reproduce the behavior of in vivo SP cells [29], [30], and is a Gaussian low-pass filtered noise of mean and standard deviation to fit the model to baseline (also referred here as spontaneous) activity conditions (i.e. no AM). As in the standard LIF formalism, when reaches a certain fixed threshold, a spike is recorded, and after that remains at a certain resting value during the absolute refractory period of the neuron.

When stimulated by a sinusoidal input, the model SP neuron responds with a modulation of its firing rate. Fig. 2A shows the maximum firing rate (solid black line) as a function of the amplitude of the sinusoidal-like signal entering the SP cell from the P-units. In color lines, we see the maximum firing rates observed experimentally for different contrasts entering the P-units. By looking at the intersections of the black curve with the color lines, we can determine the relationship between input contrast to the P-unit and input modulation to the SP cell (i.e. the P-unit output). The resulting input-output relationship for the P-units is shown in Fig. 2B. As we can see, the P-units display some degree of saturation for high input contrasts (of about ). This agrees with previously known results [26] which show that P-units behave as linear encoders for relatively low contrasts (up to ), and beyond that point they start to saturate. We can now easily include such a saturation effect into our model (see section Methods). Once this nonlinearity has been considered, the response of our model agrees very well with the experimental observations for local stimuli, as one can see in Fig. 2C, for different AM frequencies and contrasts.

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Figure 2. Fitting of the model to local response.

(A) Maximum firing rate (solid black line) reached by the model SP neuron as a function of the amplitude of the sine wave resembling the input from P-units. The AM frequency is , and similar responses were found for other frequencies. Colored lines indicate the maximum firing rate observed in the experiments for four different AM contrasts. (B) By considering the crossing points between the black line and the colored lines in panel A, we establish a dependence between AM contrasts (i.e. input to P-units) and amplitude of the signal arriving at the SP cell, , and thus obtaining an AM input-output function for the P-units. Points denote frequency-averaged quantities, while bars denote the standard deviation of each average. (C) Once this P-unit nonlinear feature is considered, the model (solid lines) is able to properly fit the experimental observations (points, shown previously in Fig. 1B) for different input frequencies and contrasts of the local stimulus. For panels A and C, the color code for contrast is red ( contrast), green (), blue () and violet ().

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Modeling global stimulation

We consider now the situation in which we have a spatially global stimulus in the system. The presence of the global stimulus activates the feedback pathway which projects onto the SP cells via the PFs. This implies considering an extra term in the dynamics of the membrane potential of the SP neuron, which is now(2)In the last term, is the strength of the feedback (which will depend nonlinearly on the contrast since the feedback pathway is also driven by P-unit activity), and the term mimics the effect of disynaptic inhibition driven by the PFs. More precisely, since the reversal potential of the inhibitory synapse (GABA-A receptors) is close to the resting potential of the SP cell [31], inhibition was modelled as an extra shunting conductance [32]. The quantity is the strength of the particular PF which is active (i.e. which is transmitting a burst arriving from a granule cell) at the phase segment of the signal cycle. For simplicity, we assume that only one PF is active at a given time (see section Methods for details).

At the SP cell, we distinguish between small and large bursts, since the characteristics of the learning rule will be different depending on the burst size [23]. Adopting the burst definitions which were explicitly characterized in [23], we will consider the 2-spike burst as the typical small burst, and the 4-spike burst as the typical large burst (see Fig. 3A, and section Methods for further details on burst definition). Small and large bursts have different roles in the cancellation process, as it was found in [16]: large bursts cover long periods associated with low-frequency input, while small bursts have a similar role for high frequencies and are also important in the timing of the plasticity for these input frequencies.

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Figure 3. Parallel fiber weights and learning contrast.

(A) Example of small and large bursts as defined in the text. Two spikes occurring within a window constitute a small burst, whereas four spikes within constitute a large burst. (B) Burst-driven learning rules for small and large bursts, as a function of the timing of the presynaptic and postsynaptic bursts. Points indicate data from in vitro recordings (taken from [23], each point being averaged over trials in the experimental plasticity protocol), and lines are the fit employed in the model. (C) PF weights as a function of the stimulus phase, for different learning contrasts . (D) Firing rate of the SP cell as a function of the stimulus phase, for local (gray) and global (colored) stimulation. The learning contrast was set at and the test contrast was (left panel) or (right panel). For both panels, lines correspond to model predictions and points to experimental data shown previously in Fig. 1B.

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For burst timing purposes, the temporal location of a given burst is identified as the temporal location of its first spike. The burst-STDP learning rule employed, based on in vitro experimental recordings [23], is shown in Fig. 3B.

Every time a pair of presynaptic-postsynaptic bursts occur, each PF weight is updated according to the following rule(3)where and are used if the burst of the SP cell is a small burst, and and are used if it is a large burst. The presynaptic burst is assumed to match the burst type occurring at the SP cell. Once again, symbolizes rectification, which means this rule is applied to all weights whose phase segment began at a time as long as . Beyond this range, the weights are unchanged.

Note that the burst-induced depression found in vitro is purely depressing and would eventually decrease all PF weights to zero. To avoid that, we include a non-associative potentiating rule so that the weights slowly relax back to with a time constant of according to(4)This rule maintains the independence of synaptic weights and is biologically plausible, since Lewis and Maler [33] demonstrated a presynaptic form of synaptic enhancement in PFs. This enhancement was elicited when PF discharge occurred, without the need of a concomitant pyramidal cell burst response. Such a form of potentiation lasted for many minutes, and a weak potentiation with this or larger would have been difficult to detect experimentally. Furthermore, similar potentiation rules have been experimentally observed in mormyrid fish, and it has been shown to play an important role in cancellation in these fish [34], in which there is a corollary discharge.

Below we will see that the homeostatic time constant may play an interesting role in the learning dynamics. The response of the SP neuron model was always recorded after a certain learning period, during which all PF weights reached their equilibrium state.

One of the main points to take into account is that the strength of the PF synapses will depend on the stimulus contrast employed during the learning period (we denote such contrast level as learning contrast, ). This is due to the fact that the occurrence of SP bursts, which are the driving events of PF plasticity, depends strongly on the stimulus contrast. The effect of the contrast on PF weights is shown in Fig. 3C, where one can see that the distribution of PF strengths is similar for different learning contrasts , although not exactly the same. In all cases, the sinusoidal stimulus shapes the PF weights to form a negative image of the signal, which constitutes the basis of the cancellation phenomenon. The weights for different learning contrasts are almost identical around the peak of the stimulus (corresponding here to a phase of ), where the SP neuron is mainly driven by the stimulus (and noise plays a relatively minor role), and bursts are more likely to occur. The variability of PF weights with the learning contrast is higher for the signal trough (around phase of ), where bursts occur scarcely and are not able to efficiently shape the weight distribution. As we can see, PF weights around the signal trough are higher for high learning contrasts. This is due to the fact that a high-contrast stimulus induces a strong hyperpolarization in the SP membrane potential at the stimulus trough, lowering the chances of bursting for that stimulus phase and preventing the depressing LTP rule to decrease the weights.

Interestingly, even though both the feedforward and the feedback inputs are sinusoidally driven, the distribution of PF weights significantly deviates from a sinusoidal function, as one may clearly observe from Fig. 3C. Such a deviation has its origin in the highly nonlinear nature of burst dynamics in SP cells. Indeed, it is known that bursting rate displays a highly nonlinear, exponential-like relationship with input (see Fig. 5B in [24]). Since bursts are the main events driving the PF learning, the nonlinear input-burst relationship is translated into a nonlinearity in the PF weight distribution emerging from learning. This, successively, turns the feedback input to the SP cell into a highly nonlinear contribution that prevents treatment of the cancellation phenomenon as a trivial linear summation of sine waves that are out of phase with one another.

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Figure 5. Cancellation of global stimulus with feedback saturation.

(A) Model predictions of the level of cancellation of global signals as a function of contrast, for different signal AM frequencies (colored lines) and different learning contrasts considered (different panels). We have introduced here a factor which takes into account the saturation of the feedback pathway for high stimulus contrasts. The gray line indicates the frequency-average cancellation levels measured experimentally (from data in Fig. 1C). (B) Degradation level (defined as the quantity , once frequency is averaged) as a function of the contrast for different learning contrasts (red , green , blue , violet ). Symbols denote experimental data. As we can see, the optimal learning contrast is around .

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In Fig. 3D, two examples of cancellation of a global signal of a frequency of , a learning contrast of , and different stimulus contrasts are shown (in each panel, the corresponding SP neuron response to same-frequency, same-contrast local signals is displayed in gray for comparison purposes). As we can see, the cancellation is very good in both cases, although the model overestimates the degree of cancellation for the contrast case (violet line in right panel). In both panels, experimental data are plotted with points and model results are displayed with lines.

Emergence of contrast invariance

The PF weights are modified via long-term plasticity mechanisms, which operate in the order of minutes to hours. Since changes in stimulus contrast associated with behavior (i.e. tail bending becoming narrower) may occur on the scale of milliseconds to seconds, one can not expect that the weights will be able to adapt fast enough to new presented contrasts in realistic situations. More likely, PF weights will reach some stationary level (as a result of some time-averaged contrast level provided by day-to-day natural stimuli), and then such an equilibrium level will be used to cancel any particular contrast level that the fish receives. In such a situation, we could expect certain differences in the quality of the cancellation depending on the learning contrast assumed in the simulations.

The model prediction of the cancellation level for different frequencies and contrasts, and assuming different learning contrasts, is shown in Fig. 4. In all cases, the cancellation levels are maintained on values over for different contrast levels, as in the experimental observations (see Fig. 1C), and thus indicate the emergence of contrast invariance in the cancellation of global stimuli in the model. It is particularly interesting to note, from the model results, that the specific learning contrast chosen has little impact on the results, contrary to what was expected, and that the levels of cancellation are broadly the same for all AM frequencies (with high frequencies having slightly lower values, as seen also in the experimental data).

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Figure 4. Cancellation of global stimulus without feedback saturation.

Model predictions of the level of cancellation of global signals as a function of contrast, for different signal AM frequencies (colored lines) and different learning contrasts considered (different panels).

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Note that, in addition to this counterintuitive lack of impact of the learning contrast in the cancellation, some qualitative differences appear with respect to the experimental data. Concretely, for all frequencies and learning contrasts, the level of cancellation slightly increases with the stimulus contrast according to the model predictions, while in the experiments it slightly decreases. The origins of such model/experiment discrepancy may be diverse, but it is reasonable to assume that they could be mainly due to the lack of a key ingredient in the model's feedback pathway, since the response of the model and experiment for the local signal were in good agreement both qualitatively and quantitatively (see Fig. 2).

Since the discrepancy is mostly evident for large contrast values, one might want to consider, as a first approach, possible features of the real system that may be particularly relevant at those conditions. A relevant factor to consider here is the existence of saturation effects along the feedback pathway. Saturation is inherent to all spiking neurons, and high contrast input to granule cells might cause saturation in two, not mutually exclusive ways: higher contrasts might evoke discharge in a greater number of granule cells and/or it might evoke a higher frequency discharge in granule cells. The first situation was examined by Berman and Maler who used stronger PF stimulation to activate greater numbers of PFs; clear saturation of the PF response was observed (Fig. 5B in [32]). The second scenario was studied by Lewis and Maler [35]; this study demonstrated a saturating SP response to increasing PF stimulation frequency (Fig. 5 in [35]). The presence of saturation in the PF-SP synapses imposes a limit in the feedback strength for increasing stimulus contrasts, which would naturally lead to a devaluation of the cancellation quality for high contrasts as observed in vivo. Furthermore, it is reasonable to think that, in addition to this PF saturation, the bursting activity of granule cells might as well saturate for high enough contrast values (since only a limited number of granule cell bursts can be generated within one stimulus cycle), adding an extra layer of saturation to the feedback pathway.

To take into account this saturating behavior in the model, we consider a small correction in the feedback gain for high contrast values by introducing a factor in the last term of Eq. 2. This extra factor will be one for low contrasts (i.e., and ) and less than one of higher values (we chose for contrast and for contrast, although other values are possible without qualitatively varying our conclusions). The cancellation levels with this new assumption are shown in Fig. 5A, for different frequencies and learning contrasts. As we can see now, the slight decrease in cancellation levels with increasing contrasts is present for all learning contrasts, in agreement with experimental data (shown as a gray line in panels of Fig. 5A for a direct visual comparison). Therefore, we have identified, via a computational model, the saturation of the feedback pathway as a plausible origin of the slight cancellation decrease with contrast. Our model also provides some insight into the plausible saturating dynamics of these granule cells, which have not been recorded in vivo to date.

Optimal learning contrast

A second conclusion that we can make from our modeling results concerns the level of cancellation for different learning contrasts. As we can see in Fig. 5A, considering different learning contrasts has now a clear effect on the cancellation properties, as opposed to the case in which saturation of the feedback pathway was not considered. Indeed, high learning contrasts shift the cancellation curves to higher values, leading to higher cancellation levels for all contrasts and frequencies considered. The closest agreement with the experimental data is obtained with a learning contrast of , as we can see in Fig. 5A and more clearly in Fig. 5B. Lower learning contrasts lead to low values of overall cancellation, mainly because PF weights are tuned to cancel only weak modulations and are not able to overcome a large-amplitude signal completely. On the other hand, higher learning contrasts produce an over-cancellation at low input contrasts (that is, the SP cells have a peak of firing rate where the stimulus displays a trough, and vice versa) which is not observed experimentally. Due to the saturation of the feedback pathway with contrast, the PFs have to span a wider range of weight values in order to obtain a proper cancellation, and this has a negative effect when trying to cancel low contrast signals. Therefore, the optimal learning contrasts are those situated just below the appearance of a strong saturation in the feedback pathway, but strong enough to allow cancellation for the whole regime of linear encoding of P-units, around . For such an optimal learning contrast, individual firing rate responses are also shown in Fig. 6 as a function of the global stimulus phase for different contrasts and frequencies. The figure also shows the corresponding experimental SP response to local stimulation for comparison purposes.

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Figure 6. Cancellation at different frequencies and contrasts.

Comparison between experimental data (points) and model predictions (lines) for the cancellation of a global signal for different AM frequencies (columns) and contrasts (red contrast, green contrast, blue contrast, violet contrast). In each panel, the corresponding experimentally measured SP response to local stimulus is also shown in gray. The learning contrast chosen for the model was , which optimizes the agreement between model predictions and experimental data.

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It might be argued that is only optimal when compared to the few other values of the learning contrast considered here. To better characterize the optimal learning contrast and its robustness, we have employed our model to extend our study and to consider other learning contrasts. We consider now a range of possible learning contrasts (around eleven values from to ) and we compute, for each one of them, the degradation of cancellation as a function of the stimulus contrast, as in Fig. 5B. For learning contrasts not considered in the experiments, such as , values for the P-unit adaptation and feedback saturation in the model were obtained by linear interpolation between known values. We also define a function which quantifies the discrepancy between the model prediction for a given and the experimental data. The error function is given by(5)where runs over all stimulus contrasts considered (up to for an easier comparison with experimental data), and are, respectively, the model and experimental degradation for the stimulus contrast . As we can see in Fig. 7A, the error function is minimal for . Considering surrounding contrast levels with similar error function values would give us a range of optimal learning contrast of contrast levels, which correspond to reasonably low error levels in the figure. Interestingly, contrast levels around this range are commonly found within the natural environment of the weakly electric fish (Yu et al., personal communication). In particular, it has been experimentally shown that the presence of free-swimming conspecifics induces a certain range of contrast levels in the electric fish, being these levels centered and more common around (see Fig. 3B in [25]), in strong agreement with the optimal contrast level predicted by our model.

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Figure 7. Optimal learning contrast and potentiation time constant.

(A) Error function, defined as the sum of the squared distances between experimental and model points, as a function of the learning contrast. Dashed lines, located at and , enclose the region of learning contrast levels which give a reasonably good fit (Error) between data and simulations. (B) Same as panel A, but for different time constant values of the potentiation learning rule for comparison. (C) Minimum of each error curved displayed in panel B, as a function of the potentiation time constant. The lowest error is obtained for potentiation time constants of about seconds.

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In addition to identifying an optimal learning contrast around , our model gives us insight into the dynamics of weak PF potentiation. As it has been argued, the potentiation rule is hard to find experimentally, since it is expected to work at very long time scales and would therefore have a hardly appreciable effect during in vitro recordings [23] (associative potentiation rules have been, however, found in mormyrid fish [34]). To study the potentiation time scale in detail, we evaluate the error measurement defined above as a function of the learning contrast, for different values of the potentiation time constant . As Fig. 7B shows, different values lead to different error curves. Time constants of or above yield mainly the same results, i.e. the optimal learning contrast lies around . Smaller time constants significantly deviate from this value, which is to be expected since a small time constant would lead to more rapid forgetting of the phase-specific synaptic strength and thus significantly modify the learning rule and the PF weight distributions. The model learning rule would thus not fit the experimental data anymore. For time constants of or lower, the optimal learning contrast is found to be higher, which makes intuitive sense because the system forgets faster, but the minimal error reached in these cases is substantially larger than for larger time constants. This can be clearly seen in Fig. 7C: the minimal error decreases as increases, until a global minimum is reached for (corresponding to an optimal learning contrast of about ). This value for the potentiation time constant is therefore a strong prediction of our model. In vitro experiments that pharmacologically eliminate the confounding effects of postsynaptic depression [23] and disynaptic inhibition [35] should be able to precisely estimate and therefore test our prediction.

Ethics statement

All experimental procedures were approved by the University of Ottawa Animal Care Committee.

In vivo electrophysiology

Experimental recordings were performed as in [16]. Briefly, craniotomy is performed under general anesthesia. During the experiment the fish is awake but paralyzed with curare and locally anesthetized. Single-unit extracellular recordings from superficial pyramidal E cells of the centro-lateral segment of the electrosensory lateral line lobe were performed. These cells can be easily identified based to their location (depth and centro-medial position), their receptive field, their baseline firing rate and response properties. Stimuli consisted of amplitude modulations of the fish's own electric field. The stimulus was delivered through two large global electrodes placed on each side of the fish thereby achieving a global stimulation. For local stimulation, a small dipole was placed in the center of the cell's receptive field; the distance between the dipole and the skin was adjusted to maximally stimulate the whole receptive field of the cell while avoiding stimulation of receptors outside the classical receptive field.

Network architecture

To understand the biological mechanisms responsible for contrast-invariant cancellation of global stimuli, we consider a simplified model of the ELL and the indirect feedback pathway (see Fig. 8). AMs of the sensory input are encoded in firing rate modulations of the P-units. If the stimulus is spatially local, P-units transmit these modulations to the SP neuron, which projects to other higher brain regions. If the stimulus is spatially global, the feedback pathway is also activated (in addition to the feedforward pathway sketched above) and a population of EEL neurons called deep pyramidal (DP) cells transmit the signal from the P-units to a granule cell population, the eminential granularis posterior (EGp), via the Nucleus praeminentialis (nP). Each granule cell projects through parallel fibers onto the SP neuron, closing the feedback loop.

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Figure 8. Schematic diagram of the model considered.

The network architecture considered in the model involves the feedforward circuit (in green) and the indirect feedback pathway (in blue), which is active only for global stimulus.

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In the real system, the cerebellar feedback pathway to the ELL is bilateral [18], [55] and can take several routes before returning to the ELL. Furthermore, DP cells, which constitute the origin of the feedback pathway, display a variety of phase relationships with the stimulus depending on its location (i.e., the side of the body) and the specific cell type (E-cells, which fire at the signal peak, or I-cells, firing at the trough) [56], [57]. Finally, each granule cell will be located at a certain position in space and therefore it will be characterized by a particular distance to the target SP cell. All these elements together produce a wide range of feedback temporal delays, suggesting that the PFs can provide feedback to SP neurons at all possible phases of the global periodic stimulus. In addition to this, granule cells have been reported to phase-lock to periodic signals and to burst to sensory stimuli and be silent elsewhere [43], [45], [58], [59].

To include these features in our simplified model, we assume that (1) the array of PFs provide feedback to SP neurons at all possible phases of the sensory stimulus, and (2) granule cells in the feedback pathway respond in a bursty fashion, phase-locked to the AM frequency signal. We also take into account in the model that PFs also synapse onto inhibitory interneurons, which provide some level of inhibition to the SP neuron.

It has also been shown that certain long-term plasticity rules may adjust the weights of the parallel fibers. According to recent in vitro experiments, PFs projecting onto SP cells display a long-term depression (LTD) rule that depends on the timing of presynaptic and postsynaptic bursts [23]. Such a burst-driven learning rule, combined with the presence of PFs displaying a wide set of temporal delays, is thought to be responsible for the generation of a negative image of the input AMs, providing the substrate for signal cancellation [16].

LIF model

For local stimulation, the SP neuron is modelled following a leaky integrate-and-fire (LIF) formalism (Eq. 1) with an extra term accounting for the bursting dynamics (DAP). Noise was introduced in the system via a low-pass filtered (with cut-off frequency ) Gaussian noise of zero mean and variance one. The variance is later adjusted via the parameter , and a constant bias is introduced to fit the experimental firing rate in spontaneous (i.e. no AMs) conditions. As in the standard LIF model, a spike is recorded when reaches the threshold , and after that remains at a certain resting value during the absolute refractory period of the neuron.

The EOD signal arriving at electroreceptors can be described, as a first approach, as a sinewave of amplitude and high frequency (). The presence of stimuli induces EOD amplitude modulations of frequency and contrast , so that the EOD amplitude is also a sinewave given by . Since electroreceptors encode AMs by modulating their firing rate, the AM frequency and the contrast are enough to characterize their behavior. The output firing rate of the P-unit population, which is driven by this input, is given by(6)P-units are known to display some level of saturation for high contrasts [26] (this effect is denoted in the above equation as ), and we observe such saturation in our in vivo recordings via the nonlinear response of the SP cell to different stimulus contrasts. By fitting the model SP cell response to the in vivo measured SP cell response, we determine the input-output amplitude relationship of the P-units (see Fig. 2A and B). The corresponding values are and . When using values other than these ones, linear interpolation was employed to estimate the new values of . To incorporate the effect of P-unit adaptation at low frequencies (as in [16]), we multiply the signal by a small constant factor of for .

In addition, as electroreceptor input is strictly excitatory, the input to the SP neuron is rectified and in Eq. 1 symbolizes rectification (that is, if , and otherwise). This aided us in incorporating the rectified nature of the pyramidal cell activity into the model. Values for these parameters are displayed in Table 1.

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Table 1. Parameter values for the equations of the model which approximate experimental data.

https://doi.org/10.1371/journal.pcbi.1003180.t001

Bursting mechanism

Superficial cell bursting drives the long-term plasticity rules operating in the PFs of the feedback pathway. The term in Eq. 1 models the depolarizing after-potential (DAP), an injection of current into the soma of the neuron after an action potential is fired due to presence of active channels in the cell's dendrites. This effect has been modeled previously in superficial cells [29], [30], and we adapt the parameter values used in these works to match the bursting rate of the model to our experimental observations.

The mechanism responsible for the generation of bursting is the following: after the cell fires () at time , it will receive a DAP (i.e. a small current injection) a short time later, as long as the previous time the cell fired is not too recent. This additional stimulation is modeled as a difference in alpha functions (Eq. 8), one generated by the soma voltage, and the other by some mean dendrite voltage. However, if the time interval between this spike time and the previous spike time is less than the refractory period of the dendrite, , then the DAP is inactive for the current spike. The refractory period is modeled as a dynamic variable that changes according to a secondary variable, , which is updated for each spike. The time just after the most recent spike was fired is referred as . The equations governing the DAP [30] are(7)(8)(9)(10)The parameters used in the above equations are listed in Table 2.

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Table 2. Parameters used in the DAP model.

https://doi.org/10.1371/journal.pcbi.1003180.t002

Parallel fibers

The feedback pathway is initiated by DP cells, which do not exhibit global cancellation behavior since they do not receive feedback, and these cells project onto the nucleus praeminentialis (nP) which in turn projects onto EGp granule cells. Finally, granule cells project massive numbers of excitatory PFs to the ELL, where they provide input to SP cells as well as local interneurons (stellate cells). The stellate cells in turn inhibit the SP cells via shunting GABA-A receptor channels.

Due to difficulties in recording them, the firing activity of EGp granule cells in the electric fish is not known. Similar granule cells in mammals, however, have been shown to respond to sensory input [43], [58] and to phase-lock their bursting to sinusoidal input [45]. We assume here, therefore, that the activity of each PF is one burst per stimulus period.

Considering the natural distribution of temporal PF delays (see details on network architecture above), the total bursting PF input was assumed to be continuous in time. In the model, the feedback cycle associated with the stimulus cycle was discretized into segments of each. This implies that the number of segments changes with the AM frequency considered (for instance, we would have feedback segments for a stimulus, and segments for an stimulus). Each segment, labeled , becomes active at time , has a global strength (common for all PFs) and a synaptic weight (particular for each PF, see Eq. 2), and then becomes inactive at . Each segment is associated with the activity of a given PF for simplicity, although it could be associated with the activity of a certain set of coincident PFs as well. The total excitatory feedback input is therefore a step-wise continuous and periodic signal given by , for each segment as time moves from segment to segment during a period. Disynaptic inhibition, which is also modulated by PF activity, is modeled as an extra shunting conductance .

The global strength of the feedback is driven by DP cells, which in turn are driven by P-units. This implies a dependence of the feedback strength on the P-unit response which is modeled as a dependence of with the contrast :(11)with being the input/output relationship of the P-units. The parameter is set to for the global stimulus (in order to fit the mean firing rate measured experimentally at global stimulation of and contrast), and to zero for the local stimulus (since this type of stimulus does not activate the feedback pathway). The saturation of the feedback pathway is being taken into account in the parameter , which will be one for low contrasts (i.e., and ) and less than one of higher values (we chose for contrast and for contrast). When feedback saturation is not being considered in the model, we just set for all contrasts. Again, since P-units drive feedback, the P-unit adaptation at low AM frequencies discussed above will affect as well the feedback input. Therefore, we follow the same criterion as with and we multiply the whole feedback function by a small factor if to account for P-unit adaptation.

A necessary condition for cancellation is to have a stable phase relationship for each segment and, hence, each weight. Such a requirement is fulfilled by considering that there is a particular set of PFs responsible for the cancellation of a given AM frequency. This has been corroborated with in vivo electrophysiological measurements [16], and therefore we assume here that our feedback pathway is frequency-specific.

As also observed experimentally in [16], cancellation starts to decay at high AM frequencies, hypothetically due to failures of granule cells for bursting at least once per cycle under global stimulation (and thus failing to drive learning properly). In agreement with these observations, we notice a slightly consistent decay of cancellation for , which can be easily taken into account by reducing to a value of for this frequency, to improve the fitting of experimental data.

Burst definition and cancellation measurement

Following the definition of a burst that induces plasticity [23], the spike train of the model SP cell was constantly monitored for small (2 spikes within ) and large (4 spikes within ) bursts. These particular definitions of burst are only adopted here to simplify the online computations, as the quantitative behavior of our model does not depend sensitively on such assumptions, or even on the presence of strong intrinsic mechanisms for bursting generation (see Discussion for details).

It is worth mentioning here that spikes in each SP burst must be independent (i.e. there cannot be a small burst in a large burst, or a large and a small burst in 5 spikes). Since each PF segment produces a presynaptic burst arriving at the apical dendrite, there is one PF burst at every time in the model, and thus PF bursts are spaced apart. When the SP cell bursts under global stimulation at time , the burst learning rule identified in vitro (Eq. 3) is immediately invoked for all PF segments.

To quantify the level of cancellation of the global stimuli, we follow [16] and employ the following criterion:(12)where are, respectively, the amplitude of SP response (measured in spikes per second) for global and local stimulation. More precisely, the PSTH (i.e. firing rate as a function of stimulus phase) of the SP cell was fitted to a sinusoidal function (plus baseline level) for the global stimulation case, and the amplitude of such sinusoidal was taken as . Because of the rectification, the same fit could not be applied to the local stimulation case, since the response clearly deviates from a sine wave. The SP response to local stimulation was then fitted to a Gaussian distribution (plus baseline level) and the height of such a Gaussian was taken as . This criterion was followed for both experimental data and model predictions.

The degradation measurement is the complementary of the frequency-averaged cancellation, and it was employed to show, in a clear manner, how much the cancellation level is degraded when increasing the stimulus contrast (Fig. 1D). It is defined as(13)with is the number of AM frequencies considered in the study (which are and ) and the sum runs over all these frequencies.