Mutual inhibition between SOM and VIP neurons creates an amplifier

To gain a deeper understanding of the circuit mechanisms and the interplay of the interneuron types, we next studied a simplified microcircuit consisting only of the three interneuron classes expressing PV, SOM and VIP (Fig 1E, left). The advantage of this simpler model is that it bypasses the nonlinearities of PCs, thereby allowing an in-depth mathematical analysis of the parameter dependencies of the network. All results are later verified in the full circuit. In the simplified network, the local PC input onto the GABAergic interneurons is replaced by additional excitatory inputs to maintain realistic firing rates. Compartment-specific inhibition onto PCs is represented by the population firing rate of the respective interneuron type: the rate of SOM neurons reflects the strength of dendritic inhibition and the rate of PV neurons the strength of somatic inhibition. Similar to the PC rate in the full microcircuit, the difference of the PV and SOM neuron rates shows a sigmoidal dependence on the modulatory VIP cell input (Fig 1F), the slope of which quantifies the system’s sensitivity to changes in the modulatory input.

Again, we compared the circuit to a reference network without VIP neurons (Fig 1E, right), in which modulatory inputs impinged directly onto the SOM neurons. In line with the full model, we observed that the removal of the VIP neurons led to a prominent reduction of the sensitivity to modulatory inputs, i.e., a reduced slope of the somato-dendritic difference of inhibition (cf. Fig 1D and 1F). To quantify the effect of the SOM-VIP motif, we introduced an amplification index A, defined as the logarithm of the ratio of slopes in the two networks with and without VIP neurons (cf. Fig 1F solid and dashed lines, and see Methods for more details). An amplification index larger than zero indicates that the interneuron network amplifies weak input onto VIP neurons in comparison to the reference network.

The simplified circuit allows us to derive a mathematical expression for the amplification index, which shows that the amount of amplification depends critically on two circuit properties (see Methods for a detailed derivation). Firstly, it increases with the effective VIP→SOM connection strength, reflecting the monosynaptic effect of VIP inputs onto SOM neurons. Secondly, it depends in a highly nonlinear way on the product of the connection strengths from SOM→VIP and VIP→SOM. This second dependence is a consequence of the mutual inhibition between SOM and VIP neurons: an increase in VIP firing rate not only inhibits SOM neurons, but also further disinhibits the VIP neurons themselves, which in turn increase their rate, further inhibiting the SOM neurons etc. As this mutual inhibition approaches a critical strength, the amplification index increases rapidly. Beyond the critical strength, the circuit transitions into a competitive winner-take-all regime, in which either the SOM or the VIP neurons are silenced by the other population. Our mathematical analysis is confirmed by simulations, which also show a rapid increase of the amplification index as the mutual inhibition between VIP and SOM neurons increases (see Fig 1G for symmetric mutual inhibition strengths, and S1 Fig for asymmetric weights). Much stronger VIP→SOM connection strengths are required to achieve an amplification (A > 0) when the back-projection SOM→VIP is knocked out, effectively eliminating the mutual competition between VIP and SOM neurons (red line in Fig 1G). These results demonstrate that mutual inhibition is a key player in the amplification of weak inputs onto VIP cells.

Connectivity and short-term plasticity support the amplification

If the SOM-VIP motif were to serve as an amplifier for weak modulatory signals, other circuit properties should also support this function. A candidate mechanism that would further enhance the competition between SOM and VIP is synaptic short-term faciliation (STF). Although short-term plasticity between different types of GABAergic interneurons has received limited attention, STF has indeed been demonstrated for the mutual connections between SOM and VIP neurons [39]. We therefore enhanced the network model by a Tsodyks-Markram type model of short-term plasticity [40, 41] (see Methods for more details). For the sake of simplicity, SOM→VIP and VIP→SOM synapses had equal facilitation parameters (Fig 2A). The overall STF strength was varied by changing the initial release probability, while adjusting the synaptic weight in order to keep the initial postsynaptic response constant (see Methods for further details). As expected, STF causes an increase of the amplification index, such that smaller mutual inhibition strengths are sufficient to achieve an amplification index above the amplification threshold A = 0 (Fig 2A).

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Fig 2. Connectivity properties of the SOM-VIP motif support the amplification in the interneuron network.

(A) Network of inhibitory PV, SOM and VIP neurons with short-term facilitation (STF) of SOM→VIP and VIP→SOM connections (left). STF increases the amplification index. Smaller mutual inhibition strengths are sufficient to achieve an amplification index above the amplification threshold A = 0 (right). Inset: The dynamics of the mutual inhibition strength (’weight’) after a step increase of the presynaptic firing rate from 0/s to 5/s for three different values of the initial release probabilities. STF parameters are equal for SOM and VIP neurons: U_s = 0.1 (strong STF), U_s = 0.5 (weak STF), U_s = 1 (no STF) and τ_f = 100 ms. (B) Network of inhibitory PV, SOM and VIP neurons with artificially introduced recurrent connections among both SOM and VIP neurons (left). Recurrence leads to a decrease of the amplification index (right). Recurrence strengths are equal for SOM and VIP neurons. Mutual inhibition strength .

https://doi.org/10.1371/journal.pcbi.1006999.g002

In contrast to PV neurons, which show strong inhibitory connections onto other PV cells [3, 13, 14], SOM and VIP neurons only very rarely inhibit other neurons of the same class [3, 13, 15]. To investigate whether this lack of recurrent inhibition supports the amplification properties of the network, we artificially introduced recurrent connections among both SOM and VIP neurons. We systematically varied their strength, while keeping the strength of mutual inhibition between the two populations constant. For simplicity, we considered a symmetric situation in which the strength of the recurrent inhibition is the same among VIP and SOM neurons (see Fig 2B), but similar results are obtained in asymmetric situations (see S2 Fig). We found that recurrent connections among SOM and VIP neurons lead to a strong reduction of the amplification index (Fig 2B), even for relatively weak recurrent connections (see Methods for a mathematical analysis). The strongest amplification was always observed for a connection strength of zero, that is, when recurrent inhibition is absent.

In summary, connectivity properties like short-term facilitation and the absence of recurrent connections among both VIP and SOM neurons support the effective translation of small stimuli onto VIP cells into large changes of somato-dendritic inhibition.

Spike-frequency adaptation introduces a frequency-selective amplification

Both SOM and VIP neurons show an absence of recurrent inhibition within the same population, but they make use of a different negative feedback mechanism: spike-frequency adaptation (SFA). SFA is a prominent feature observed in many cortical neurons [3, 42]. Upon stimulation, adapting cells decrease their firing rate gradually, and consequently exhibit a difference between steady-state and onset firing rate. SOM neurons feature salient SFA [3, 5, 6, 11, 16], whereas VIP neurons claim a broad spectrum from weak to strong adaptation. To study the effect of SFA, we augmented the rate dynamics of SOM and VIP neurons by an additional rate adaptation variable (see Methods). The adaptation process is governed by two parameters: an adaptation strength and a time constant. While the adaptation time constant controls the temporal evolution of the adaptation process, the adaptation strength controls the difference between onset and steady-state firing rate. An adaptation strength of one corresponds to a steady-state firing rate that is half the onset firing rate. For simplicity, we again assumed the same adaptation parameters for SOM and VIP neurons.

Adaptation and recurrence both generate a negative feedback on neuronal activity. For comparability, we parameterized the strength of recurrence such that the steady state activity in a population is the same when adaptation strength and total recurrence strength have the same value. While we expected both adaptation and recurrent inhibition to weaken the amplification, they differ with respect to the time scales on which they operate. Recurrent inhibition acts on the rapid time scale of synaptic transmission (e.g, of 5–10 ms for GABA_A receptor-based transmission). In contrast, adaptation operates on a wide range of time scales from tens to thousands of milliseconds [42]. Consequently, adaptation allows a gradual transition over time from amplification to attenuation. To characterize the time dependence of the amplification properties, we performed a frequency response analysis (Fig 3A and 3B), a common technique to analyze linear dynamical systems. The idea is that any input signal can be decomposed into oscillations of different frequencies, and that the system processes these different oscillations independently. In practice, this means that the response of the system to any signal can be understood as a superposition of the responses to its frequency components. To characterize the response of the circuit to different frequencies, we hence stimulated VIP neurons with oscillating inputs. The difference of the population firing rates of PV and SOM neurons—as a reflection of the somato-dendritic distribution of inhibition—oscillates in response to this stimulation with the same frequency, but phase-shifted. The logarithm of the ratio of the oscillation amplitudes in networks with and without VIP neurons (full vs. reference network, see Fig 1) yields a frequency-resolved amplification index that ranges from negative (attenuation) to positive (amplification) values. We found that increasing recurrent inhibition among both SOM and VIP neurons systematically reduces the frequency-resolved amplification index across all frequencies (Fig 3A), confirming the steady-state analysis (cf. Fig 2B). Note that to control for the overall reduction in activity when recurrent inhibition is increased, we adjusted the time-independent external inputs such that the mean firing rates are kept constant in spite of changes in recurrent strength. In contrast to recurrent inhibition, spike-frequency adaptation introduces a prominent frequency selectivity: For low-frequency oscillations, the frequency-resolved amplification index decreases with increasing adaptation strength. For high-frequency oscillations, it increases (Fig 3B and 3C). Furthermore, the circuit exhibits a preferred frequency (resonance frequency), for which it yields a maximal response. This resonance frequency arises from an interplay of the rate and adaptation time constants. For low-frequency oscillations, adaptation is fast enough to track the input changes and hence able to suppress the frequency-resolved amplification index. With increasing stimulation frequency, this suppression weakens because adaptation is too slow. As a consequence, the oscillation amplitude increases with stimulation frequency. For high-frequency oscillations that are faster than the time constant of the firing rate, the frequency-resolved amplification index declines, because the neuron cannot react sufficiently quickly. Neuronal adaptation hence introduces a frequency-selective amplification that preferentially transmits specific neuronal rhythms within the broad spectrum of oscillations in the brain [43].

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Fig 3. Spike-frequency adaptation enables frequency-selective amplification and preserves co-activity.

(A) Frequency response analysis of the interneuron network with recurrent connections among both SOM and VIP neurons. Increasing recurrence reduces the frequency-resolved amplification index across all stimulation frequencies. (B) Same as in (A), but with spike-frequency adaptation instead of recurrence. The circuit yields a maximal response at a resonance frequency. With increasing adaptation strength, this resonance frequency increases. (C) The frequency-resolved amplification index decreases with increasing adaptation strength for low-frequency oscillations, but increases for high-frequency oscillations (frequency-selective amplification, cf. markers in (B)). (D) Neurons show stronger correlations (positive or negative) with spike-frequency adaptation (left, upper panel) than with recurrent inhibition among SOM and VIP neurons (left, lower panel). Mean SOM-SOM neuron correlation is more sensitive to increasing recurrence strength than to adaptation strength (right). (E) While strengthening mutual inhibition leads to an increase of the maximal frequency-resolved measure of amplification, it does not change the resonance frequency (top, adaptation strength b = 0.8). In contrast, increasing the adaptation strength allows to adjust the resonance frequency, with a weak impact on the maximum of the frequency-resolved amplification index (bottom, recurrence strength ). (F) Schematic representation of both separate “knobs” (mutual inhibition and adaptation) and their independent control of the amount and frequency-selectivity of the amplification. Parameter (A-E): Mutual inhibition strength .

https://doi.org/10.1371/journal.pcbi.1006999.g003

Spike-frequency adaptation and recurrent inhibition also have distinguishable consequences for the correlation structure of the interneuron network. Karnani et al. (2016) [39] demonstrated that both SOM and VIP neurons are cooperatively active as populations rather than individually. We studied this co-activity by stimulating both interneuron populations with shared and individual noise on top of a constant background input. The shared noise between members of the same interneuron class introduced strong correlations between both VIP/VIP and SOM/SOM neurons as described by Karnani et al. [39]. We then studied how recurrent inhibition and adaptation differentially affect the co-activity of the populations, quantified by the averaged pairwise correlation coefficient. We found that recurrent inhibition strongly decreases the correlation between members of the same interneuron class [44] (Fig 3D), while adaptation introduces only a marginal reduction, and thereby preserves the high correlations seen by Karnani et al. [39].

As for any amplifier, it would be useful to allow a dynamic adjustment of the amplification in the circuit. Very strong amplification would lead to small ranges of effective modulatory signals and a rapid saturation of the system, and hence to potential distortions in the translation of the modulatory signal into somatic and dendritic inhibition. Similarly, it could be beneficial to allow an adjustable frequency-based selection of the modulatory input. A neuronal mechanism that is suitable to tune these two properties could be a neuromodulatory control of circuit parameters [18, 35–38, 45]. In simulations, we found that a strengthening of mutual inhibition increases the overall amplification, while leaving the resonance frequency largely unaltered (Fig 3E, top). At the same time, changes of the adaptation strength allow to tune the resonance frequency, while leaving the maximum of the frequency-resolved measure of amplification largely unchanged (Fig 3E, bottom). In summary, the circuit seems to display separate “knobs”, which offer an independent control of the amount and frequency-selectivity of the amplification through separate neuromodulatory channels (Fig 3F).

These results demonstrate that spike-frequency adaptation, though similar in its steady-state properties to recurrent inhibition within SOM and VIP populations, enables a frequency-selective amplification with well-separated target parameters for neuromodulatory control. While a neuromodulation of synaptic and cellular properties has been studied both for pyramidal cells [45, 46] and interneurons [18, 35–38, 47], it is not clear how strong the various forms of modulation are in vivo and how they interact on the circuit level. It therefore remains to be seen whether interneurons support a sufficient degree of neuromodulation of spike-frequency adaptation and mutual inhibition to dynamically tune the frequency selectivity of the circuit.

The computational repertoire of the SOM-VIP motif

Our simulations indicate that the SOM-VIP subnetwork supports different computational functions, ranging from signal amplification and frequency selection to switching behavior for strong mutual inhibition between the populations. To understand how these computational states are determined by the parameters of the system, we ran extensive simulations of the SOM-VIP motif alone, accompanied by mathematical analyses of a linearized network without rate rectification.

We first investigated a network of VIP and SOM neurons that is consistent with experimentally observed characteristics, i.e., with adaptation and inhibitory connections exclusively between neurons of different type. Simulations reveal four operating modes for such a network (Fig 4A). For weak mutual inhibition, the two interneuron populations can be active at the same time, and modulatory VIP signals are attenuated (Fig 4A, region (a)). As the inhibition between the two populations increases, the amplification index increases and leads the circuit into an amplification domain (Fig 4A, region (b)). Beyond a critical strength of mutual inhibition, the network then transitions into a switch-like winner-take-all domain (Fig 4A, regions (c) & (d)), in which one of the two populations silences the other. Notably, this state comes in two variants: For weak adaptation, the winning population silences the other permanently, or until an external event switches the network to a different winner (Fig 4A, region (c)). We will show later that this allows transient VIP inputs to persistently switch the operating mode in the full microcircuit. For strong adaptation, the network shows oscillations (Fig 4A, region (d)), because adaptation gradually decreases the firing rate of the winning interneuron population. This releases the other population from inhibition until it can no longer be silenced and becomes the new winner and in turn starts to adapt. A mathematical analysis of the linearized network predicts the parameter ranges of the four computational states almost perfectly (see black lines in Fig 4A, and Methods for more details). The observed oscillations comprise a wide spectrum of frequencies that depend non-linearly on the strength and time constant of adaptation and on the strength of mutual inhibition (Fig 4B, see also S3 and S4 Figs for networks with asymmetric adaptation parameters for SOM and VIP neurons). Deviations between the frequencies observed in simulations and those predicted by the mathematical theory are caused by the omission of the rate rectification in the theory. The four computational states of the network are also observed when short-term plasticity is introduced into the network, although the transition boundaries change such that the switch-like state is reached for weaker mutual inhibition and the oscillation regime requires stronger adaptation (Fig 4C).

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Fig 4. Dynamical states of the SOM-VIP motif.

(A) Left: Bifurcation diagram reveals distinct operation modes: all interneurons are active (divided into amplification (a) and attenuation regime (b)), a winner-take-all (WTA) regime implementing a switch (c) and an oscillation regime (d). Regime boundaries (black lines) are obtained from a mathematical analysis (see S1 Appendix). Right: Example firing rate traces for all SOM (blue) and VIP (green) neurons for four network settings taken from the bifurcation diagram (cf. markers). Adaptation time constant τ_a = 50 ms. (B) The oscillation frequency in the oscillation regime depends on the adaptation strength (left), the adaptation time constant (right) and the total mutual inhibition strength (black: , gray: ). Left: τ_a = 50 ms, right: b = 1. The frequencies cover a broad range from Delta to Alpha oscillations. (C) When short-term facilitation (STF) is present, the WTA (switch) regime is enlarged and the oscillation mode requires stronger adaptation. Initial synaptic efficacy U_s = 0.1, facilitation time constant τ_f = 100 ms.

https://doi.org/10.1371/journal.pcbi.1006999.g004

Two inhibitory populations that mutually inhibit each other may well be a common network motif in cortical circuits, and the absence of recurrent inhibitory connections within the two populations—as observed in the SOM-VIP motif—may not always hold. We therefore also performed an analysis of the computational states of a network with recurrent inhibition. Simulations reveal five operating modes for such networks (see S5 Fig). When mutual inhibition and recurrent inhibition is weak, we again found that both interneuron populations can be active at the same time. Depending on the strength of the mutual inhibition, we again observed attenuation and amplification, respectively. For sufficiently strong mutual inhibition between SOM and VIP cells, the amplification regime transitions into the switch-like state where only one population is active. In contrast to adapting neurons, the network did not show an oscillatory state. Instead, very strong recurrent inhibition introduces strong competition between the neurons within the interneuron populations, leading to pathological states where either one single cell per cell type is active (if mutual inhibition is weak) or only one single neuron at all is active (if mutual inhibition is strong). Again, these dynamical states and their transitions are predicted almost perfectly by a mathematical eigenvalue analysis (see black lines in S5 Fig, and Methods for derivation). The mathematical analysis also unveils that for sufficiently large populations, the pathological states require very strong synapses (ultimately, a single cell must silence all others) and are hence unlikely to be observed in the nervous system.

In summary, the SOM-VIP network motif allows different computational states, covering attenuation, amplification, switching and—for adapting neurons— oscillations in a frequency range of Delta (1-4 Hz), Theta (4-8 Hz) or Alpha (8-12 Hz) oscillations.

Switch between distinct processing modes in local microcircuits

To investigate the computational consequences of the SOM-VIP circuit, we returned to the full microcircuit comprising PCs and inhibitory PV, SOM and VIP cells (S6 Fig). We first verified that all results observed in the simplified interneuron networks still hold for the larger circuit. Again, stronger mutual inhibition and the presence of STF increase, while negative feedback mechanisms like recurrent inhibition or adaptation decrease the system’s sensitivity to the modulatory input (S6 Fig). Furthermore, adaptation leads to a frequency-selective amplification for which the processing of rapidly changing input signals benefits from powerful spike-frequency adaptation in SOM and VIP neurons (S6 Fig). Finally, the experimentally observed elevated correlation between members of the same interneuron class is also preserved for adaptation and decreases strongly for recurrent inhibition (S6 Fig). In summary, the results obtained in the simplified interneuron network also hold in the full circuit.

What is the computational impact of a somato-dendritic redistribution of inhibition on PCs? It is well established that on their apical dendrites, many pyramidal cells receive top-down input from higher cortical areas [22, 23] and matrix thalamic nuclei [24]. On their basal dendrites and the perisomatic domain, they receive bottom-up input from lower cortical areas and core thalamic nuclei [25]. Although inputs at the electrically distant apical dendrites have a small impact on initiating spikes at the axon initial segment [48, 49], they can initiate long-lasting calcium spikes when they coincide with back-propagating action potentials from the soma [50–52], leading to a significant gain increase of L5 pyramidal cells [53]. How the two streams of information are integrated is not fully resolved [54]. In particular, it is conceivable that top-down inputs are only used when they provide useful and reliable information, and are ignored otherwise. Given that calcium spikes in apical dendrites are very sensitive to inhibition [55], dendrite-targeting interneurons are well suited to control the integration of top-down inputs, and, consequently, the switch between distinct modes of operation. We therefore simulated the full microcircuit with top-down and bottom-up inputs, in a “switch” configuration of strong mutual inhibition between VIP and SOM, and without spike-frequency adaptation. For illustration, we chose as inputs two sinusoidal oscillations with different frequencies (Fig 5A & 5C), and determined how strongly these two input streams are represented in the firing rate of PCs (Fig 5A & 5B). When the network is in the “switch” regime, we found that small changes in VIP input are sufficient to switch the network between two computational states in which top-down inputs are either transmitted or cancelled entirely (Fig 5B). Interestingly, transitions between those two computational states can also be triggered by transient modulatory pulses. The evoked change of the processing mode persists even after the end of the pulse (Fig 5C), reflecting a bistability of the system. As a consequence, the current processing mode of the network depends on the recent history of the input to the SOM-VIP motif (S7 Fig).

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Fig 5. Integration or cancelation of top-down signals by modulatory VIP input.

(A) PCs of the full microcircuit are stimulated at the soma and the apical dendrites with two oscillations with different frequencies, emulating bottom-up (orange, x_E) and top-down (blue, x_D) input, respectively. PC rate reflects the two inputs x_E and x_D with different coefficients α and β, depending on the modulatory input. (B) In an amplification regime (), weak, permanent modulatory VIP input is sufficient to switch between two operation modes, in which top-down input is either integrated (β > 0) or cancelled (β = 0). In the WTA regime, the network exhibits hysteresis, that is, the level of modulatory input needed to cause a switch depends on the network state. For a range of inputs, the circuit is bistable. (C) In the bistable regime (), persisting transitions between the states can be triggered by strong, short pulses delivered to VIP neurons (10 ms duration, amplitude 8.4/s, timing denoted by green arrows). Parameters (A-C): Weight between SOM neurons and dendrites , External stimulation x_E = 25/s + 0.5 sin(5 t)/s, x_D = 7/s + 0.1 sin(30 t)/s, x_PV = 12/s, x_SOM = x_VIP = 3.5/s.

https://doi.org/10.1371/journal.pcbi.1006999.g005

In summary, we demonstrate that the integration of top-down feedback from higher cortical areas can be induced or prevented by persistent, weak input or short, strong input pulses onto VIP cells. As the network exhibits hysteresis, the switching depends on the collective state of SOM and VIP neurons.

Amplification of small mismatch signals

The SOM-VIP circuit can be interpreted as an amplifier for small differences between two inputs that impinge onto SOM and VIP neurons. We therefore simulated a network with two modulatory signals targeting SOM and VIP neurons. By systematically varying these inputs in an amplification regime, we verified that the somato-dendritic distribution of inhibition is determined by the mismatch between the two inputs (Fig 6A).

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Fig 6. Mismatch detection by SOM/VIP-dependent amplification.

(A) Additional modulatory input onto SOM neurons shifts the transition point (green) of the PC rate as a function of modulatory VIP input (top). When plotted as a function of the difference of SOM and VIP input, the transition points align, indicating that the circuit amplifies differences between two input streams (bottom). Additional input onto SOM neurons ranges from −2/s to 0/s. (B) Model for the integration of visual inputs and motor-related predictions [56]. SOM neurons, PV neurons and the somatic compartment of PCs receive external visual input, VIP neurons and the apical dendrites of PCs receive an internal (motor-related) prediction of the expected visual input. The connection strengths from PV neurons to the somatic compartment of PCs and the SOM→PV connection were chosen to ensure a response only when the visual input is switched off and the (motor-related) prediction is switched on (see S1 Appendix for details). (C) PCs respond with an increase in firing rate when visual input is off and the motor-related input is on (mismatch), but show a negligible increase in activity above baseline when both input streams are on. (D) Also, only negligible responses above baseline are evoked when motor-related input is permanently off (playback session). , . (E) The mismatch-induced increase in firing rate is more pronounced in an amplification regime (, , dark red) in comparison to an attenuation regime (, , gray). Parameters (B-E): Motor-related and visual input on corresponds to an additional input of 10.5/s and noise drawn from a Gaussian distribution with zero mean and SD = 3.5/s. Background stimulation x_E = 28/s, x_D = 0/s. Time constant of PCs increased by factor 6 to reduce onset responses.

https://doi.org/10.1371/journal.pcbi.1006999.g006

This observation is interesting in the context of a recent study of Attinger et al. (2017) [56]. The authors suggested a conceptual model for layer 2/3 of mouse V1, in which SOM neurons receive visual inputs, while VIP neurons and the apical dendrites of the PCs receive an internal (motor-related) prediction of the expected visual input. When properly tuned, the excitatory top-down input to the PC dendrites is then cancelled by SOM inhibition, as long as the internal prediction matches the sensory data. Deviations between sensory inputs and internal predictions, however, change the level of dendritic inhibition and thereby generate mismatch responses, as observed in a subset of PCs in V1 [56, 57] and in other systems [58, 59].

To test this hypothesis in silico, we stimulated our full circuit model with visual inputs—impinging onto PV and SOM neurons and the somata of PCs—and motor feedback—impinging onto VIP neurons and the apical compartment of PCs (Fig 6B). Similar to the findings of Keller et al. (2012) [56], we found network configurations in which we observed selective responses in PCs when motor feedback was present in the absence of visual stimulation (Fig 6C), but not when visual stimuli were presented in the absence of motor feedback (Fig 6D). These responses were reduced when the circuit was brought into an attenuation configuration by weakening mutual inhibition between VIP and SOM neurons (Fig 6E).

We therefore suggest that the amplification brought about by the competition between SOM and VIP neurons could serve to amplify small deviations between different input streams, such as sensory signals and internal predictions. Because such deviations are powerful learning signals for the internal prediction system [60], an amplification may be beneficial for learning highly accurate predictions.

Neural network model

We simulated a rate-based network of excitatory pyramidal cells (N_PC = 70) and inhibitory PV, SOM and VIP cells (N_PV = N_SOM = N_VIP = 10). All neurons are randomly connected with connection probabilities (see Table 1) consistent with the experimental literature [12, 13, 15]. If not stated otherwise, all cells of the same neuron type have the same number of incoming and outgoing connections, respectively. This assumption is made merely for purposes of mathematical tractability and does not qualitatively alter the results.

The excitatory pyramidal cells are simulated by a two-compartment rate model taken from [34]. The steady-state firing rate r_E,i of the somatic compartment of neuron i obeys (1) where [x]₊ = max(0, x) is a rectifying nonlinearity and τ_E denotes a rate time constant (τ_E = 10 ms, unless stated otherwise). Θ denotes the rheobase of the neuron and I_i is the total somatic input generated by somatic and dendritic synaptic events and potential dendritic calcium spikes, (2) and are the total synaptic inputs into dendrites and soma, respectively, and c_i denotes the dendritic calcium event. λ_D and λ_E are the fraction of “currents” leaking away from dendrites and soma, respectively. The synaptic input to the soma is given by the sum of external bottom-up inputs x_E and PV neuron-induced (P) inhibition, (3) is the sum of top-down inputs x_D, the recurrent, excitatory connections from other PCs and SOM neuron-induced (S) inhibition: (4)

The weight matrices w_EP, w_DS and w_DE denote the strength of connection between PV neurons and the soma of PCs (w_EP), SOM neurons and the dendrites of PCs (w_DS) and the recurrence strength between PCs (w_DE), respectively (see Table 2). Note that all existing connections between neurons of type X and Y have the same strength, w_XY,ij = w_XY, X, Y ∈ {E, D, P, S, V}. All weights are scaled in proportion to the number of existing connections (i.e., the product of the number of presynaptic neurons and the connection probability), so that the results are independent of population sizes. The input generated by a calcium spike is given by (5) where c scales the amount of current, H is the Heaviside step function, Θ_c represents a threshold that describes the minimal input needed to produce a Ca²⁺-spike and denotes the total, synaptically generated input in the dendrites, (6)

Unless stated otherwise, parameters were taken from [34] (see Table 2). Note that we incorporated the gain factor present in Murayama et al. (2009) [34] into the parameters to achieve unit consistency for all neuron types.

The firing rate dynamics of each interneuron is modeled by a rectified, linear differential equation [33]: (7) where a_i represents an adaptation variable, w_ij denotes the relevant synaptic weight onto the neuron, u_ij describes a synaptic facilitation variable and x_i denotes external inputs. The rate time constant τ_i was chosen to resemble the GABA_A time constant of approximately 10 ms for all interneuron types included. The weight matrix W (see Table 3) was chosen such that the relative connection strengths are consistent with experimental findings [12, 13, 15, 17, 29]. When we simulated mismatch neurons (see Fig 6, main text), we set the total SOM→PV connection strength to , the total PV→PV weight to , and tuned the total synaptic strength from PV neurons to the somatic compartment of PCs in order to ensure a response only when the visual input is switched off and the (motor-related) prediction is switched on: . Note that the total weight is given by the product of the number of existing connections between two neuron types and the strength for individual connections.

In contrast to PV neurons, both SOM and VIP cells show pronounced spike-frequency adaptation [3, 5, 6, 11, 16], which is described by an adaptation variable a_i, (8)

At constant neuronal activity r_i, the adaptation variable a_i exponentially approaches the steady-state value b r_i with time constant τ_a. For simplicity, if not otherwise stated, the adaptation strength b and time constant τ_a are the same for both cell types (b ∈ [0, 2], τ_a = 100 ms). If adaptation is not present, we set the parameter b to zero.

Short-term facilitation is only modeled for SOM→VIP and VIP→SOM connections. The facilitation variable u_ij between neuron j and neuron i evolves according to the Tsodyks-Markram model [40, 41]: (9)

The facilitation variable u_ij ranges from 0 to 1 and represents the release probability, which changes according to the availability of calcium in the axon terminals. In the absence of presynaptic activity, the facilitation variable u_ij relaxes exponentially with time constant τ_f to a steady state U_s, which represents the initial release probability. Presynaptic activity increases the facilitation variable u_ij by an amount proportional to U_s. If not stated otherwise, the initial release probability U_s and facilitation time constant τ_f are equal for SOM and VIP neurons (U_s ∈ [0, 1], τ_f = 200 ms). When STF is not present, u_ij = 1 (or, equivalently, U_s = 1). In simulations where the strength of short-term facilitation is varied, we ensured comparability by scaling the weights w_ij by U_s, thereby keeping constant the initial synaptic response after a long period of inactivity.

External stimulation

To achieve physiologically reasonable activity levels, all neurons are stimulated with a time-independent background rate x_i. PCs receive constant bottom-up input x_E at the soma and top-down feedback x_D at their dendrites. Additionally, VIP (in the full network setting) or SOM neurons (in the reference network) receive an external stimulus x_mod that was varied systematically to investigate the amplification properties of the microcircuit. If not indicated differently, all cells of the same neuron type are presented with an identical stimulus.

In the interneuron network, for the sake of comparability across different parameter settings, we always adjust the background inputs x_i such that the spontaneous activity (that is, at x_mod = 0) is equal to r₀ = 3/s for all interneurons. In the non-WTA regime this can be achieved by (10) with u_ij = 1, if short-term facilitation is not present, or (11) otherwise. In the full microcircuit comprising PC, PV, SOM and VIP cells, the external stimulation is set to x_PV = x_SOM = x_VIP = 3/s for the interneurons and x_E = 17.5/s and x_D = 21/s for the PCs, if not otherwise stated.

To characterize the dynamics of the system, we perform a frequency response analysis that measures the amplitude of the output signal as a function of the frequency 1/T. In that case, the weak external stimulus x_mod is expressed by a sine wave (12) where T ∈ [50, 600] ms. The frequency-resolved amplification index is then given by the logarithm of the ratio of the oscillation amplitudes in full and reference networks.

To investigate correlations, we stimulated SOM and VIP neurons with an input consisting of i) a constant component (calculated as before, see Eq (10)), ii) individual noise and iii) noise that it shared among the neurons of the same type: (13) where the noise terms ξ_shared and ξ_i are drawn at each time t from Gaussian distributions with zero mean and unit variance. The shared component of the noise accounts for the strong correlations seen by Karnani et al. (2016) [39]. Furthermore, the number of cells per neuron type is increased in these simulations by a factor 5 to obtain reliable statistical estimates.

For the example firing rate traces (in Fig 4 and S3 & S4 Figs), we stimulated SOM and VIP neurons with an input consisting of i) a constant component of 25/s and ii) individual noise drawn at each time t from a Gaussian distribution with zero mean and SD of 5/s.

Definition and mathematical derivation of the amplification index

To quantify the strength of amplification in our neural microcircuit, we introduce the amplification index, (14) where m_full and m_ref denote the slope of the sigmoid function of the full and the reference network (see Fig 1D or 1F), respectively. These slopes represent the redistribution of somatic and dendritic inhibition upon a change in modulatory VIP input. Hence, the amplification index measures how much stronger the redistribution is when weak input passes through the SOM-VIP motif instead of directly through the SOM neurons.

The amplification index can be calculated analytically for the simplified linear network without PCs and short-term facilitation. To this end, we first derive the slope m_full from the mean-field population dynamics, (15) (16) (17) where denotes the total weight from neuron type n onto m, and the rates r_n denote the mean-field population rate of neuron type n. For the sake of generality, we included the possibility of recurrent connections among both SOM and VIP neurons. Solving this set of equations for the steady-state population activity yields (18) (19)

The slope m_full is given by the derivative of r_PV − r_SOM with respect to x_mod, (20)

Similarly, the slope m_ref in the reference network can be derived from the mean field equations without VIP neurons and PCs, (21) (22) and is given by (23)

Finally, the amplification index is given by the ratio of the slopes, (24)

In the absence of recurrent inhibition within SOM and VIP populations, this expression simplifies to (25)

Mathematical analysis of the computational repertoire of the SOM-VIP network

To analyze the computational repertoire of the SOM-VIP motif, we considered a network composed of SOM (hereafter only S) and VIP neurons (hereafter only V) that are mutually and fully connected. Again, the rectifying nonlinearity of the neurons is neglected.

Simulation details and code availability

All simulations were performed in customized Python code written by LH. Differential equations were numerically integrated using a 2^nd-order Runge-Kutta method with a maximum time step of 0.05 ms. Neurons were initialized with r_i(0) = 0 Hz, a_i(0) = 0 Hz (if adaptation was modeled) and u_ij(0) = U_s (if STF was present) for all i. Source code will be made publicly available upon publication.