A Unifying Mechanistic Model of Excitatory-Inhibitory Interactions in the Auditory Cortex

Abstract The mammalian sensory cortex is comprised of multiple types of inhibitory and excitatory neurons, which form sophisticated microcircuits for processing and transmitting sensory information. Despite rapid progress in understanding the function of distinct neuronal population, the parameters of connectivity that are required for the function of these microcircuits remain unknown. Recent studies found that two most common inhibitory interneurons, parvalbumin- (PV) and somatostatin-(SST) positive interneurons control sound-evoked responses, temporal adaptation and network dynamics in the auditory cortex (AC). These studies can inform our understanding of parameters for connectivity of the excitatory-inhibitory cortical circuits. Specifically, we asked whether a common microcircuit can account for the disparate effects found in studies in different groups. We built rate and spiking models of the auditory cortex consisting of excitatory, PV and SST neurons, and searched the space of connectivity parameters to identify the set that can account for the experimental findings from multiple groups. We identified microcircuit parameters that accounted for the differential effects of PVs and SSTs in stimulus-specific adaptation, forward suppression and tuning-curve adaptation, as well as the influence of PVs on functional connectivity in the circuit. The unifying mechanisms of the model included a depressing synapse from PVs to excitatory neurons, and a facilitating synapse from excitatory neurons to SSTs. This approach brought together multiple findings from different laboratories and identified a unified circuit that can be used in future studies of upstream and downstream sensory processing. Significance Statement The mammalian auditory cortex is comprised of multiple types of inhibitory and excitatory neurons, which form sophisticated microcircuits for processing and transmitting sensory information. Distinct inhibitory neuron subtypes play distinct functions in auditory processing, but it remains unknown whether these phenomena are due to a unified microcircuit or require multiple circuits. Here, we built minimal rate and spiking models and identified a specific set of synaptic mechanisms and parameters that could best reproduce the broad set of experimental results in the auditory cortex. The simplicity of our model provides an understanding of inhibitory cortical processing at the circuit level, unifying the results from different laboratories, and provides for a novel computational framework for future studies of cortical function.

Introduction substantially more complex three-unit rate model and three-unit spiking models. The increased complexity of the 97 models was mitigated by using the single-unit rate model as a template with data from the literature to constrain

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The input function ( ) consists of blocks of inputs with stimulus duration and interval based on the 22 experimental paradigm. We show the stimulus duration and stimulus interval for each paradigm in Table 1 and

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Using the single-unit rate model as a template, we arranged copies into three units with lateral cortical 55 and thalamic connections (Figure 2A). This arrangement endowed our model with a gross tonotopy, which we 56 used to explore spectrally and temporally complex auditory inputs.

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The first, or leftmost unit, satisfies

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where d ( ) = # ( ) + d ( ) . Note that each set of equations are almost identical to the single-unit case,

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but with the addition of lateral terms along with facilitating and depressing terms ? and ? . The lateral terms are 65 between immediate neighbors and include lateral SST to Exc (facilitating), Exc to Exc, Exc to PV, and PV to Exc 66 (depressing). The facilitating terms ? increase from 0 to nonzero values as unit receives inputs, and the 67 depressing terms s ? decrease from 1 to lower values as unit s receives inputs. While the three-unit rate model 68 appears to be substantially more complex, the parameters are strongly constrained by the single-unit rate model.

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Each unit of the three-unit model is designed to mimic the excitatory and inhibitory currents of the single-unit rate   2). The functions e ( ) are time-dependent inputs with the strongest preference for unit , and the profiles of ! , 77 # , and d are shown in Figure 2F (these profiles are the same as the profile in the single-unit model, Figure 1B

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where h T and h U are as in Equation 3. We used the inputs ? ( ) as a proxy for the excitatory activity ? ( ) so

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and used the thalamic input as a proxy for excitatory activity to simulate the depression variable as ? = ? . All

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The activity of the model is shown in Figure 2, where three successive auditory stimuli were applied in 06 order of the frequencies ! , * , and # , stimulating the left, center, and right units, respectively ( Figure 2C-E).

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The center unit ( # ) responded equally well to both ! and # ( Figure 2D), which is a necessary response for SSA 08 paradigms. For simplicity, activation of an adjacent unit did not affect the thalamic variable, i.e., ! , # , and d 09 were left unaffected by ! , # , and d , respectively. We assumed that the frequency difference between ! and 10 # was great enough that auditory inputs at ! ( # ) did not affect units responsive to # ( ! ).

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We incorporated paradigm-dependent baseline states in the three-unit rate model. The parameters 12 switch between weak and strong baseline inhibition, where weak inhibition corresponds to high thalamic activity,

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and strong inhibition corresponds to relatively low thalamic activity. This idea is captured more precisely by the 14 facilitating variable, 16 where ( ) is the sum of all thalamic inputs (independent of the tonotopic arrangement), i T = 1500, and i U = 17 100. As the experimental paradigm progresses, grows and eventually saturates (over the course of 18 approximately 15 seconds). A simulation of Equation 7 is shown in Figure 1C for the various auditory paradigms.

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If is above the threshold LM = 0.22, the system exhibits weak baseline inhibition, and the synapses

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. CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint and the sum ∑ z in the synaptic current iterates over the presynaptic neurons, ∈ { , , }. If the presynaptic 50 neuron is excitatory (inhibitory), then z = 0mV (−67mV). If a synaptic connection existed from PV to Exc, we

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Following a presynaptic spike from neuron , the postsynaptic effect on neuron appears as an 69 instantaneous spike in the postsynaptic conductance ?@ → ?@ + ?@,max / X , where ?@,max is given by Equation

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10, and X stands for the presynaptic neuron type (Exc, PV, or SST

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In the absence of presynaptic spikes, the conductances ?@ decay exponentially to zero:

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. CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint The term is a dimensionless slow timescale facilitation variable that depends on the thalamic drive, and

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The dynamics of the dendritic (stick) compartment obey

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where the parameter = 0.3 is the ratio of somatic to total surface area [23].

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For PV and SST interneurons, the equations are the same as Exc except that there is no dendritic 91 component, and parameters differ marginally (see Table 2). SSTs, unlike PVs, have no incoming synaptic    The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint

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Differential effects of interneuron suppression in stimulus-specific adaptation 10 Almost all neurons (95%) in AC exhibit stimulus-specific adaptation, a phenomenon in which neurons 11 reduce their response selectively to the inputs that is presented frequency in the stimulus (standard tone in an 12 oddball), while preserving the initial strong response to the less frequent input (deviant tone) [26]. Previous 13 studies found that following a presentation of the deviant tone, the excitatory neurons adapt over successive 14 presentations of the standard [26,27]. This phenomenon was largely attributed to feedforward thalamo-cortical 15 depressing synapses [14,28], but such models could not account for the full range of the effects that were 16 observed [9]. A recent study found that inhibitory neurons exhibit differential control over the stimulus-specific

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We tested whether our model could reproduce the differential effects of suppressing PVs and SSTs on   The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint

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Although the single-unit rate model provided an important insight, the model lacked the ability to 47 distinguish between auditory frequencies, which is a necessary aspect of stimulus-specific adaptation and many 48 other optogenetic paradigms. To further test the inhibitory mechanisms discovered using the single-unit model, 49 we extended the model to a rate and spiking model with three iso-frequency units, in which each microcircuit 50 received inputs of specific preferred frequencies. As mentioned above, the three-unit circuitry was based on the 51 single-unit model and the parameters chosen to reproduce the inhibitory and excitatory currents. For example, 52 an auditory input to the left unit caused lateral excitatory and inhibitory currents to enter the center unit. These 53 currents to the center unit were designed to be similar to the currents in the case of the single-unit. Due to 54 symmetry, we easily performed the same procedure for auditory inputs to the right unit: lateral excitatory and 55 inhibitory currents from the right were designed to enter the center unit in a manner similar to the single-unit 56 case. Using this procedure, we were able to extend the differential SST and PV inhibition in the single-unit model 57 to work in the case of a tonotopy without the need for exhaustive parameter fitting.

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The procedure for reproducing SSA is as follows: record from the center unit and apply the standard 59 tone to the right unit using the stimulus interval and duration in Table 1 ( Figure 4A). The synaptic depression in 60 the thalamus then adapts the responses, and the center unit responds with a mean firing rate similar to Figure   61   The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint 3D, tone number 4. With a 10% chance, apply the oddball tone to the left unit. At this event, the center unit 62 responds with a firing rate similar to Figure 3D, tone number 1 because the thalamic input of the left unit has not 63 adapted. In this brief time, the stimulus has skipped the right unit, and the depression variable of the right unit 64 recovers more than usual. The next several standard tones applied to the right unit produce center-unit 65 responses similar to Figure 3D tone numbers 2-4, because the depression variable has had a little time to 66 recover, but not so much that the first response is like the oddball response. Continued tones to the right unit 67 evoke center-unit responses similar to Figure 3D

Change in excitatory neuron responses due to PV activation (middle), and SST activation (right). PV activation 81
resulted in a near-uniform decrease in FRs, whereas SST resulted in an increase in adaptation.

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In the rate and spiking model, the firing rates increased uniformly across all post-deviant tones ( Figure   84 4D,E). In the rate and spiking model, the firing rates exhibited an increase in disinhibition as a function of post-

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We performed a parameter sweep with four key parameters of circuit connectivity ( Figure 5). For the first 93 parameter, we chose recurrent excitation ( 22 , Figure 5A

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We reiterate that the three-unit model was developed to reproduce the compensating mechanisms of the 33 single-unit model: PV suppression results in constant disinhibition for repeated tones, and SST suppression 34 results in a compensating effect from PVs before adaptation that weakens as adaptation strengthens. As we 35 have seen, these differential roles explain experimental data in the SSA paradigm to a remarkable degree. We 36 then asked whether this simple mechanism is sufficient to reproduce additional optogenetic experiments. For 37 the remainder of the paper, we use the three-unit rate model with no parameter modifications except for the 38 changes in the inhibition modes and the auditory inputs that depend on the experimental paradigm (Table 1   The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint We used the same parameters for connectivity within the circuit as with SSA to reproduce the 62 experimental findings, with only slight changes to the input strength ( = 1.3). The stimuli used in the forward 63 suppression paradigm place the baseline state in the strong inhibitory regime ( Figure 1C). Both the rate ( Figure   64 6A middle, 6B middle) and spiking models ( Figure 6B bottom 6B bottom) yielded the experimentally measured 65 differential effects for PV ( Figure 6A) and SST inactivation ( Figure 6B): PV inactivation drove a selective 66 decrease in responses whereas SST inactivation drove a suppression of excitatory neuronal responses.

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At first glance, this result seems paradoxical given that PV suppression generally results in excitatory 68 disinhibition as shown in the adaptation and SSA results (Figures 3,4), but the underlying mechanism is 69 straightforward to understand. Following PV suppression, excitatory activity is indeed disinhibited, but this 70 disinhibition forces the thalamic variable g to decrease more compared to the control case. Upon receiving the 71 second tone, the input received by the excitatory population is weaker, in turn weakening the excitatory response.

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This weakening in the light-on trial is proportionally greater compared to the control case, so forward suppression 73 is strengthened. In the case of SST suppression, PVs compensate for the loss of inhibition in the first tone, but 74 lose the ability for compensation in the second tone, so Exc are able to respond more strongly relative to the 75 control case. Thus, forward suppression is weakened.

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Next, we tested the effects of activating PVs or SSTs (as could be done with ChR2 experimentally) on 77 model responses. The model predicted that both PV and SST activation will result in an increase of forward 78 suppression across preferred and sideband frequencies ( Figure 6D,E).

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To understand how inhibitory neurons affect adaptation across different frequency-tuned inputs, we presented a 87 sequence of 8 tones at each frequency to generate adapting tuning curves ( Figure 7A), and repeated this process 88 with PV and SST inactivation for the model circuit. We found that this auditory paradigm resulted in a below-89 threshold integration of , so the system switched to a state of strong baseline inhibition (and importantly, the 90 model did not respond in precisely the same way as in SSA and forward suppression).

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Our model reproduced the differential experimental effects of PV and SST suppression (Figure 7). In the 92 rate model before adaptation, PV and SST inactivation resulted in sideband disinhibition with little to no in Figure 7C, E, G and I represent the peak excitatory responses from the first and last simulations taken directly 99 from the simulations, whereas thinner lines are linear extrapolations to assist the visual comparison to the control 00 line (blue).

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The mechanisms behind these results involve the synaptic dynamics and the compensating mechanism 02 discussed in the earlier sections for SSA and adaptation. In the case of PV suppression, SSTs were the only 03 interneurons capable of contributing to Exc inhibition, so only Exc-SST synaptic dynamics drove the observed 04 effects. In particular, lateral SST to Exc synapses suppressed the center unit over each tone, and facilitation 05 allowed this suppression to persist throughout adaptation. Note that this preferred-frequency effect was not 06 observed in SSA because we never directly stimulated the center unit. Next, in the case of SST suppression, 07 the increasing disinhibition with adaptation at the preferred frequency was a consequence of the same 08 compensating mechanism as in SSA. Our model is the first to provide a microcircuit mechanism for the observed 09 experimental results.

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Our model predicted that before adaptation, PV activation resulted in a slight decrease at the preferred 11 frequency, whereas SST inactivation reduced overall firing rates across all frequencies ( Figure 7J, L). After

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Cortical neurons in AC receive inputs from the thalamic auditory nuclei. As the result, neuronal responses 29 in the cortex are correlated with neuronal firing in the thalamus. These interactions can be captured using an 30 Ising model to measure the connection from the thalamus to the cortex. When PVs were activated, the functional 31 coupling between cortical and thalamic responses [6] became stronger. The specific mechanism underlying this 32 change is unknown.

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Using the three-unit model, we identified a candidate mechanism for the enhanced thalamo-cortical 34 correlation following PV activation. We assumed that the functional connection from the thalamus to the cortex 35 is the same as the anatomical connection, so thalamic inputs directly modulated cortical responses in our model.

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Following an increase in inhibition, cortical responses became sharper, thus aligning more closely with thalamic 37 inputs and improving functional connectivity (Figure 8).  The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint PV activation (green) in the rate model resulted in an increase in the Pearson correlation between the 46 control (blue) and thalamic inputs (red), from 0.77 and 0.83 ( Figure 8B). Thus, whereas inhibitory activation 47 decreased the overall firing rate, the response became more synchronized to the thalamic inputs, resulting in an 48 increase in functional connectivity. In the spiking model, PV activation resulted in a delayed response of 49 excitatory activity, but we were interested in tested whether PV-activated Exc response profile resembled the 50 thalamic activity more than the control Exc response. To make this comparison, we shifted the PV trace so that 51 the onset of PV-activated Exc activity (green) coincided with the onset of the control curve (blue) ( Figure 8C. An 52 equivalent approach would be to measure the peak value of the cross-correlation between excitatory and 53 thalamic activity, but we shifted the data for simplicity). We observed an increase in the Pearson correlation from   neurons, such as SSTs and PVs, play a differential role in auditory processing, controlling adaptation at different 67 time scales and contexts, as well a functional connectivity. Our goal was to integrate the results of these studies 68 to understand whether the observed effects are due to a specific unified circuit, or to different modes and 69 connectivity patterns potentially found across AC.

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We built a unifying rate and spiking model that reproduced multiple key results from studies that tested  The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint successfully used a multi-unit rate model arranged in a coarse tonotopy consisting of inhibitory and excitatory 83 populations to reproduce general deviance detection, but model has not yet been adapted to explain differential 84 interneuron modulation. Another existing model of SSA including differential inhibitory modulation demonstrating 85 similar differential inhibitory effects as in our SSA result (Figure 4), but did not include a tonotopy [3]. These 86 models only included one type of inhibitory neuron type or did not include tonotopy, and therefore could not 87 account for the observed differential effects of suppression of SSTs and PVs on SSA across multiple frequencies.

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In the present study, we developed a simple rate and spiking model that accounted for multiple inhibitory cell 89 types and which faithfully reproduced the differential effects of SST and PV inactivation in SSA (Figure 4). In   . CC-BY-NC-ND 4.0 International license is made available under a The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint Existing models that reproduce the enhanced forward suppression from PV inactivation and the reduced 99 forward suppression from SST inactivation ( Figure 6) include multiple layers that require both depression and

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We incorporated depression and facilitation in the model synapses and reproduced the former results with only 02 a single layer, suggesting a surprisingly simple mechanism supporting forward suppression. Furthermore, the 03 models in the present study reproduced tuning-curve adaptation effects previously observed: SSTs exhibited 04 strong preferred-frequency disinhibition following adaptation, while PV disinhibition is independent of the degree 05 of adaptation (Figure 7) [4]. These results suggest that the underlying mechanism(s) of the model, namely the 06 PV/SST compensation effect, combined with the facilitating SST-to-Exc synapse and depressing PV-to-Exc 07 synapse, may serve as a unifying mechanism of adaptation.

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Finally, our models reproduced changes in functional connectivity in the cortex (Figure 8). By increasing 09 PV activity in the models, excitatory activity decreased but became more time-locked to thalamic inputs. This

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The effects of inhibition on sharpening cortical responses have been well-established, thus our models serve as 12 plausible mechanisms for this change [31-33].

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One drawback of the models is that they do not feature population spikes, which explain many 14 fundamental cortical responses in AC [31]. In future work, we will seek to reconcile the differences between our 15 models and the population spike model of SSA [30]. Establishing the importance of depression and facilitation 16 in different synapses and extending our model to include population spikes warrants further study.

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Our models show evidence of operating as a balanced network: as we vary the input strength to the rate 18 and spiking models ( Figure 9C,F), the ratio of excitatory to inhibitory inputs to the excitatory population ( Figure   19 9B,E) remains constant: the rate model has an excitatory/inhibitory ratio of 0.37 ( Figure 9B), and a ratio of 2.5 20 for the spiking model ( Figure 9E). These results demonstrate the potential importance of inhibitory-excitatory 21 balance in the cortex. In future studies, we will explore whether inhibitory-excitatory balance is a necessary or 22 sufficient condition for the results in this study.

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Although we do not explore simultaneous auditory stimuli in this study, it is worth mentioning the response

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Multiple studies from different laboratories revealed the differential effect of distinct inhibitory neurons in 29 auditory processing. Strikingly, a minimalistic model, built on simple mechanisms, produced differential 30 information processing by various subtypes of inhibitory neurons, and has unified these disparate studies. As  The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It . https://doi.org/10.1101/626358 doi: bioRxiv preprint