A Biophysical Basis for Learning and Transmitting Sensory Predictions

Homeostatic (anti-Hebbian) forms of synaptic are effective at eliminating “prediction errors” that signal the differences between predicted and actual sensory input. However, such mechanisms appear to preclude the possibility of transmitting the resulting predictions to downstream circuits, severely limiting their utility. Using modeling and recordings from the electrosensory lobe of mormyrid fish, we reveal interactions between axonal and dendritic spikes that support both the learning and transmission of predictions. We find that sensory input modulates the rate of dendritic spikes by adjusting the amplitude of backpropagating axonal action potentials. Homeostatic plasticity counteracts these effects through changes in the underlying membrane potential, allowing the dendritic spike rate to be restored to equilibrium while simultaneously transmitting predictions through modulation of the axonal spike rate. These results reveal how two types of spikes dramatically enhance the computational power of single neurons in support of an ethologically relevant multi-layer computation.

maintain a constant dendritic spike rate by enforcing cancellation while simultaneously inducing 81 modulations in axonal spike rate that transmit sensory predictions. Our modeling work is based 82 on a multi-compartment neuronal model, but the basic results are recapitulated in a model with 83 only somatic and axonal compartments. Thus the mechanism we describe does not require 84 dendritic computation, but relies instead on an electrotonically distant spike initiation site in the 85 axon-a common feature of neurons that, to our knowledge, has not been previously connected 86 to learning. 87 88 Results 89 90 Dendritic spikes are triggered by backpropagating axonal spikes 91 MG cells fire two types of sodium channel-dependent action potentials known as broad and 92 narrow spikes ( Figure 1B). Broad spikes are likely initiated in the proximal apical dendrites, 93 have a high threshold, and are emitted at low rates, while narrow spikes are likely initiated in the 94 axon, have a low threshold, and are emitted at high rates (Bell et al., 1997b;Engelmann et al., 95 2008; Grant et al., 1998). Broad spikes induce long-term depression at granule-MG cell synapses 96 (Bell et al., 1997c;Han et al., 2000). Given their critical role in plasticity induction, we sought to 97 determine what factors control broad spike firing in vivo. Confirming prior studies (Bell et al.,98 1997b; Grant et al., 1998;Sawtell et al., 2007), we found that broad spikes are invariably 99 preceded by a narrow spike at a characteristic interval of ~3 ms ( Figure 1B,D and Figure 1- figure  100 supplement 1A-B). This observation, by itself, does not indicate that narrow spikes play a causal 101 role in evoking broad spikes, as preceding narrow spikes could arise simply because broad spikes 102 have a higher threshold than narrow spikes. To further evaluate this question, we examined a 103 previously developed multi-compartment MG cell model that recapitulates critical features of 104 MG cell responses described above ( Figure 1A, right) (Muller et al., 2019). This model is 105 reduced, containing a minimal set of voltage-gated and synaptic conductances (Materials and 106 methods), which makes it amenable to detailed analysis. The model is tuned to produce observed 107 broad and narrow spike rates, but no fine-tuning of parameters is required to produce the results 108 we report. In fact, as we show later, the basic effects can be reproduced in a further reduced 109 model. 110 111 When input currents were adjusted in the model to evoke the ~50 Hz narrow spike firing and ~2 112 Hz broad spike firing seen in vivo, broad spikes in the model cell were always preceded by a 113 narrow spike at an interval of ~3 ms ( Figure 1C-D and Figure 1-figure supplement 1B). Blocking 114 narrow spikes by turning off active conductances in the axonal compartment abolished broad 115 spike firing over a range of input strengths ( Figure 1E), while injecting a brief spike-like 116 depolarizing current into the soma (with active conductances in the axon turned off) evoked 117 broad spikes after a similar delay (Figure 1-figure supplement 1C). These results confirm a 118 causal role for narrow spikes in evoking broad spikes in the model. Monitoring the voltage at 119 various locations revealed that even though axonal depolarization resulting from the narrow 120 spike is highly attenuated by the time it reaches the soma ( Figure 1F, open arrowhead), it 121 nevertheless spreads passively into the proximal apical dendrites where it activates voltage-gated 122 sodium and potassium channels to evoke a local dendritic spike ( Figure 1F, blue). Depolarization 123 from the local dendritic spike then propagates into other apical branches leading to additional 124 spike initiations at multiple sites throughout the apical dendrite. These local dendritic spikes sum 125 to produce a broad somatic spike after a delay of several milliseconds from the triggering narrow 126 spike ( Figure 1F,

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Each circle represents the average amplitude for the given baseline membrane potential.

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Several lines of evidence suggest a causal role for backpropagating narrow spike in evoking 156 broad spikes in vivo through a process similar to that described in the model. First, eliminating 157 narrow spikes with hyperpolarizing current or a sodium channel blocker in the recording pipette 158 revealed that peak somatic depolarization due to narrow spikes is much greater than that due to 159 subthreshold input alone ( Figure 1G). Second, in vivo (as in the model), the probability of 160 evoking a broad spike depends strongly on the membrane potential at the peak of the recorded 161 narrow spike ( Figure 1H and narrow spikes that immediately precede a broad spike not only arise from more depolarized 163 potentials (as would be expected based on the higher threshold for broad spikes) but also exhibit 164 larger amplitudes than narrow spikes not preceding a broad spike ( Figure 1I and Figure 1- figure  165 supplement 1H-K). It is difficult to see why this would be the case if the narrow spike were not 166 causal. We also observed that the amplitude of backpropagating narrow spikes depends on the 167 baseline membrane potential, an effect seen in other systems (Grace and Bunney, 1983), 168 presumably due to the voltage-dependence of the membrane conductance. 169 170 171 A biophysical model of negative image formation and transmission 172 We next examined how sensory input affects broad and narrow spike firing in the multi-173 compartment model. In vivo studies have revealed two sub-classes of MG cells: BS-cells, in 174 which the broad spike rate is decreased by sensory input, and BS+ cells, in which the broad spike 175 rate is increased (presumably through dis-inhibition). While for simplicity we focus on modeling 176 BS-cells, the mechanisms we describe can also explain responses in BS+ cells (Figure 2- figure  177 supplement 1). For clarity, we consider constant sensory input, but we have verified that all the 178 results we report apply to time-dependent sensory inputs matching those in vivo (Figure 2- figure  179 supplement 3). Adding constant inhibitory input to basilar dendritic compartments potently 180 reduces the broad spike firing rate with little effect on the rate of narrow spikes (Figure 2A supplement 2b), so that the broad spike threshold is rarely crossed ( Figure 2B, dashed line). The 185 increased conductance due to the inhibitory input reduces the peak membrane potential by 186 attenuating the passive spread of the narrow spike from the axon initial segment, as seen in the 187 small reduction in the amplitude of the narrow spike at the soma (~0.75 mv; Figure 2C, E, red 188 and these well-characterized plasticity dynamics on narrow and broad spike firing (rather than on the 197 plasticity mechanism itself), we simply reproduce the known effect of this plasticity in our 198 model. In other words, we set the strengths of excitatory conductances onto apical dendrites to 199 cancel the effects of inhibition on the broad spike rate (

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An equivalent computational explanation for this phenomenon can be constructed by expressing 229 the broad spike rate, R bs , as the product of two factors, the probability of a narrow spike evoking 230 a broad spike, p, and the rate of narrow spikes, r ns ; R bs = p ⋅ r . The factor p reflects the 231 functional coupling between backpropagating narrow spikes and broad spikes (similar to the 232 "safety factor" described in classical studies of initial segment-somatodendritic spike coupling 233 (Coombs et al., 1957a, b;Fuortes et al., 1957;Renshaw, 1942) . Sensory input selectively affects 234 the broad spike rate by reducing the value of p. (While narrow spike peak voltage is the 235 dominant factor affecting p ( Figure 1H), other factors may also contribute ( Figure 2-figure  236 supplement 4)). Specifically, suppose that the broad spike rate R bs = p ⋅ r is at its equilibrium 237 value in the absence of sensory input, with p = p 0 . Introducing inhibition due to sensory input 238 reduces p causing the broad spike rate to decrease. Synaptic plasticity restores the broad spike 239 rate by returning p ⋅ r and thus R bs back to its equilibrium value ( Figure 2F, dashed line). 240 However, through this process p is not restored to its previous value p 0 , but instead remains 241 smaller than p 0 . Thus, the restoration of the broad spike rate requires a compensatory increase in 242 r ns , the narrow spike rate ( Figure 2F). 243 244 Negative image formation and transmission in vivo 245 The model makes two key predictions regarding negative image generation and transmission that 246 we tested in vivo: (1) sensory input modifies narrow spike amplitude and (2)  To test prediction 2 concerning the changes in baseline membrane potential, we examined 258 narrow spike amplitudes during the learning of negative images induced by pairing an 259 electrosensory stimulus with the motor command that discharges the electric organ (Bell, 1981). 260 This analysis was only possible for BS+ cells because of the much faster time-course of 261 cancellation in these cells (Muller et al., 2019). As expected, cancellation of sensory-evoked 262 increases in broad spike firing was driven by a temporally-specific hyperpolarization of the 263 underlying membrane potential ( Figure 3D, inset). Importantly, sensory-evoked changes in 264 narrow spike amplitude were not reversed as negative images formed, a critical feature for our 265 model of negative image transmission ( Figure 3D). In fact, the amplitude of backpropagating 266 narrow spikes actually increased due to the prominent inverse correlation of the narrow spike 267 amplitude and the baseline membrane potential ( Figure 1I). This effect amplifies the mechanism 268 identified in the model, leading to even more robust negative image transmission by narrow 269 spikes (Figure 3-figure supplement 2). Defining Amp as the change in narrow spike amplitude 270 due to sensory input and S as the slope of the relationship between narrow spike amplitude and 271 the baseline membrane potential, the negative image is equal to -Amp/(1+S). Our data suggest

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Axonal, but not dendritic, compartmentalization is required for MG cell function 299 The differential effect of sensory input on broad and narrow spikes might suggest that spatial 300 targeting of synaptic inputs onto MG cells is essential for learning and transmitting negative 301 images. We tested this by varying the location of the sensory input in the model. Surprisingly, 302 inhibition onto the proximal apical dendrites ( Figure 4A) or soma ( Figure 4B) yielded similar 303 model performance as inhibition onto basilar dendrites (Figure 2A). In both cases, sensory input 304 robustly decreased broad spike firing with little effect on narrow spike firing ( Figure 4A, B,  305 inhibition), and the addition of excitatory input to the apical dendrites cancelled the effects of 306 sensory input on the broad spike rate while simultaneously modulating narrow spike output 307 ( Figure 4A, B, cancellation). Furthermore, if a mixture of excitatory and inhibitory input is 308 delivered to the basilar dendrites, narrow spike firing rate is also increased while broad spike 309 firing decreased ( Figure 4C), matching prior in vivo observations (Muller et al., 2019). These 310 results suggest that neither spatially segregated synaptic inputs nor dendritic 311 compartmentalization are strictly required for differential control over broad and narrow spikes. 312 313 To test this further, we constructed a simple conductance-based integrate-and-fire model with 314 only two compartments representing an axon and soma. Remarkably, the same qualitative results 315 described for the morphologically realistic multi-compartment model were reproduced by the 316 attenuation of the backpropagating axonal narrow spike in the somatic compartment ( Figure 4D-317 F). While this result in no way excludes important functional roles for the numerous 318 morphological, synaptic, and biophysical specializations of real MG cells, it suggests that the 319 essential biophysical requirements for continual learning and signal transmission are surprisingly 320 minimal.

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Discussion 323 Learning is typically associated with Hebbian forms of plasticity with homeostatic plasticity 324 playing a stabilizing role by enforcing a return to equilibrium. Here we identified biophysical 325 mechanisms that allow homeostatic synaptic plasticity to transmit learned signals. Using in vivo 326 recordings and biophysical modeling we showed that sensory input modulates the rate of 327 dendritic spikes by adjusting the amplitude of backpropagating axonal action potentials. 328 Homeostatic plasticity counteracts these effects through changes in the underlying membrane 329 potential, allowing the dendritic spike rate to be restored to equilibrium while simultaneously 330 transmitting predictions through modulation of the axonal spike rate. The core requirements of 331 the mechanism we describe--separate axonal and somatodendritic action potentials and an 332 electronically distant site of axonal spike initiation-are found in many classes of neurons 333 (

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An action potential that arrives at the soma highly attenuated might seem an unlikely candidate 350 for impacting dendrites. We find, to the contrary, that the small size of backpropagating axonal 351 spikes in MG cells makes them susceptible to modulation and therefore an ideal candidate for 352 flexibly controlling dendritic events. Importantly, the amplitude of backpropagating action 353 potentials is highly sensitive to synaptic input (Llinas et al., 1968;Renshaw, 1942;Tsubokawa 354 and Ross, 1996), much more sensitive than rates of action potential generation. This provides a 355 mechanism for precise and, importantly, differential control of axonal and dendritic spikes that For surgery to expose the brain for recording, fish were anesthetized (MS:222, 1:25,000) and 416 held against a foam pad. Skin on the dorsal surface of the head was removed and a long-lasting 417 local anesthetic (0.75% Bupivacaine) was applied to the wound margins. A plastic rod was 418 cemented to the anterior portion of the skull to secure the head. The posterior portion of the skull 419 overlying the ELL was removed and the valvula cerebelli was reflected laterally to expose the 420 eminentia granularis posterior (EGp) and the molecular layer of the ELL, facilitating whole-cell 421 recordings from the ventrolateral zone of the ELL. Gallamine triethiodide (Flaxedil) was given 422 at the end of the surgery (~20 g/cm of body length) and the anesthetic was removed. Aerated 423 water was passed over the fish's gills for respiration. Paralysis blocks the effect of 424 electromotoneurons on the electric organ, preventing the EOD, but the motor command signal 425 that would normally elicit an EOD continues to be emitted at a rate of 2 to 5 Hz. In contrast to broad spikes, narrow spike amplitude varied across recordings from ~15 mV 445 (similar to values obtained from somatic recordings in vitro) to indistinguishable from 446 subthreshold synaptic events. The latter, which were typically obtained at more superficial 447 recording depths corresponding to the ELL molecular layer, were classified as putative apical 448 dendritic recordings (see figure supplement 1D). 449 450

Electrosensory stimulation 451
The EOD mimic was a 0.2 ms duration square pulse delivered between an electrode in the 452 stomach and another positioned near the electric organ in the tail. The amplitude was 25-50 µA at 453 the output of the stimulus isolation unit (stomach electrode negative). Recordings from 454 ampullary afferents showed that firing rate modulations evoked by this mimic are within the 455 range of those induced by the fish's natural EOD (Bell and Russell, 1978). We use the terms 456 sensory input or sensory response to refer to the effect of the mimicked electric field on the ELL. 457 Because we do not include prey-like electric fields the sensory input we discuss is entirely 458 predictable on the basis of the EOD command signal and is therefore entirely uninformative to 459 and 'unwanted' by the fish. inhibition was inserted into proximal apical compartments (11 compartments (19 segments)) 504 defined as those whose center is within 100 of the center of the soma. 505 506

Two compartment model 507
Conductance based integrate-and-fire model was used for the two compartment model (Fig 4d). 508 The equations for somatic and axonal membrane potential are: 509 510 511 512 Where g l is the leakage conductance, g c is the intercompartment conductance and g i and g e are 513 the inhibitory and excitatory conductances respectively. I e is external current (with Gaussian 514 noise) and is set to produce ~50 Hz narrow spike and ~2 Hz broad spike rates. When the axon 515 reaches the threshold for axonal spike, a spike shape plus a refractory period is imposed in the 516 axon. Broad spike rate was determined by the number of times the backpropagating axonal spike 517 reached a high threshold in the soma (this threshold was defined as the 97th percentile of the 518 backpropagating spike-peak in the initial period). 519 520 Measuring narrow spike amplitude differences 521 Quantifying narrow spike amplitude differences induced by sensory input is complicated by the 522 strong dependence of narrow spike amplitude on baseline membrane potential observed in vivo 523 (negative slope in Figure 1I). To account for this effect, we fit the slope of the relationship 524 between narrow spike amplitude and baseline membrane potential and report the difference 525 across conditions in the bias of these slopes. Similarly, to measure difference between expected 526 and actual amplitude (figure supplement 6A) we first fit a slope to the relationship between 527 amplitude and baseline membrane potential and then measure the distance from the fit. 528 529 We hypothesize (figure supplement 1H-I) that the attenuation of backpropagating narrow spike 530 amplitude is linearly proportional to the amplitude: 531 532 1 533 Then if we divide by average recorded mean we have the following equality: 534 535 536 537 The average narrow spike amplitude differs widely across recordings (see figure supplement 538 1D), presumably due to recording location in the soma versus the proximal apical dendrites. 539 Hence, to compare differences in narrow spike amplitude evoked by sensory stimuli across 540 recordings we report the percentage change in narrow spike amplitude relative to the average 541 narrow spike amplitude for each cell. The same reasoning applies to analysis of the relationship 542 between amplitude and baseline membrane potential across different cells (figure supplement 543 7A). 544 545

F-I curve for narrow spikes 546
The fit between membrane potential and spike rate is approximately linear (see example in 547 Figure 3E). To minimize the effect of outliers we quantify the change in rate/mV as: 548 549 550 551 552 Measuring effect of sensory input on amplitude 553 Recorded amplitude of in vivo narrow spikes may change over the course of the recording (as 554 the quality of the recording changes). Therefore, we meausred amplitude differences in relation 555 to a control window within same recording period. Changes in amplitude from the first to the 556 second half of the recorded period (Figure 3d) was measured as the change in amplitude 557 relative to the control window within each half of the recording.