Regulation of gamma-frequency oscillation by feedforward inhibition: A computational modeling study

Abstract

Throughout the brain, reciprocally connected excitatory and inhibitory neurons interact to produce gamma-frequency oscillations. The emergent gamma rhythm synchronizes local neural activity and helps to select which cells should fire in each cycle. We previously found that such excitation-inhibition microcircuits, however, have a potentially significant caveat: the frequency of the gamma oscillation and the level of selection (i.e., the percentage of cells that are allowed to fire) vary with the magnitude of the input signal. In networks with varying levels of brain activity, such a feature may produce undesirable instability on the time and spatial structure of the neural signal with a potential for adversely impacting important neural processing mechanisms. Here we propose that feedforward inhibition solves the latter instability problem of the excitation-inhibition microcircuit. Using computer simulations, we show that the feedforward inhibitory signal reduces the dependence of both the frequency of population oscillation and the level of selection on the magnitude of the input excitation. Such a mechanism can produce stable gamma oscillations with its frequency determined only by the properties of the feedforward network, as observed in the hippocampus. As feedforward and feedback inhibition motifs commonly appear together in the brain, we hypothesize that their interaction underlies a robust implementation of general computational principles of neural processing involved in several cognitive tasks, including the formation of cell assemblies and the routing of information between brain areas.

Introduction

Among the endogenous rhythms of the brain, the oscillations in the gamma-frequency band (30–90Hz) are the most ubiquitous (Buzsáki and Wang, 2012). Gamma-frequency oscillations have been associated with many different brain processes such as attention (Chalk et al., 2010; Bauer et al., 2014), decision-making (Castelhano et al., 2014) and working-memory (Howard et al., 2003; Yamamoto et al., 2014). Thus, it is likely that gamma waves reflect some general aspect of neural computation. Gamma oscillations have two main roles: synchronizing neurons to fire within a time-constrained window (Singer and Gray, 1995; Csicsvari et al., 2003) and selecting which cells should fire (de Almeida et al., 2009a). Together, these two functions allow the formation of assemblies of cells (Harris et al., 2003; Olufsen et al., 2003; Tort et al., 2007), i.e. groups of neurons that fire synchronously within a limited time window and that can be distinguished from other groups by a target network (König et al., 1996). Cell assemblies are thought to provide the means for the elementary representation and communication of percepts or memories in the organization of cognition (Hebb, 1949), and might, therefore, explain the association of gamma-frequency oscillations with many different brain processes. Importantly, however, gamma-frequency oscillations are frequently aberrant in neurological diseases (Lewis et al., 2005; Uhlhaas and Singer, 2006; Iaccarino et al., 2016) which indicates that the processes underlying gamma oscillations operate within restricted parameter ranges in the healthy nervous system. Here we sought to evaluate whether the neural circuits that produce gamma-frequency oscillations are robust to natural fluctuations of brain excitation and propose a regulatory mechanism by which these neural circuits can maintain the oscillatory activity within the operational limits.

We hypothesize that two features of a neural circuit that produces gamma-frequency oscillations are essential for the appropriate detection of cell assemblies in a target network: the firing of the cells must be synchronous within specific temporal precision, and only the right cells should be allowed to fire. The fact that several important biophysical time constants (e.g., decay time of GABA_A IPSPs (Buhl et al., 1995) and the membrane time constant (Routh et al., 2009)) match the time scale of a single gamma cycle supports the idea that gamma-frequency oscillation events should not drift considerably from a signature neurodynamic pattern (Lopes-dos-Santos et al., 2011). From the perspective of the assembly detection process in the target network, if the oscillatory events are too fast, there may be insufficient time for the target cells to integrate the input. If these events are too slow, the target layer might separate one single assembly into multiple ones. In addition, precise timing allows the temporal coordination of neural systems, supporting mechanisms of brain communication through coherence (Fries, 2015; McLelland and VanRullen, 2016) and local circuitry computation revealed by phase-amplitude coupling (Canolty and Knight, 2010; Scheffer-Teixeira and Tort, 2016). As an indication of the importance of temporal coordination, abnormalities in neuronal synchronization occur in different brain disorders (Uhlhaas and Singer, 2006).

Not only the timing of firing but also the identity of the cells that fire is relevant for the identification of a cell assembly in the target network. Theoretical models indicate that the level of the sparsity of the network influences aspects of mnemonic processes such as the duration and capacity of memory storage (Bogacz and Brown, 2002; de Almeida et al., 2007; Leibold and Kempter, 2008; Kammerer and Leibold, 2014). If too many cells are allowed to fire within a gamma cycle, the overlap between patterns can make it hard for a target network to separate the identity of assemblies. Indeed, networks known to produce very sparse activity, such as the dentate gyrus, are essential for the separation of patterns as expressed both in electrophysiology (Leutgeb et al., 2007; Schmidt et al., 2012; Berron et al., 2016) and behavior (Baker et al., 2016). On the other hand, if too few cells are active within a gamma cycle, it might be challenging for a target network to separate one pattern from noisy synaptic activity (Ganguli and Sompolinsky, 2012), making it necessary to use pattern completion mechanisms such as recurrent synapses (Rennó-Costa et al., 2014) for accurate pattern recognition. It is, therefore, important to understand whether the circuits that produce gamma-frequency oscillations are robust to variable changes in the brain activity patterns and produce assemblies with invariant timing and selection of cells.

The main circuit component for the emergence of gamma waves is perisomatic inhibition (Soltesz et al., 1993; Mann et al., 2005). The interplay of feedback inhibition and feedforward excitation produces an oscillation event with well-defined stages: (1) principal cells integrate the input and fire; (2) the excitatory activity discharges the inhibitory cells that, in turn, inhibits the principal cells (as well as the interneurons themselves); (3) without the excitatory input (and in the presence of interneuron-interneuron inhibition), inhibitory cells decrease their firing and the blockade to principal cells fades. In such a circuit, the level of the excitatory gain influences these three stages and thus could modify the dynamics of the gamma-frequency oscillation. Indeed, optogenetic stimulation of principal cells in a closed-loop with the network oscillation alters features of the gamma-frequency oscillations depending on the phase of stimulation (Nicholson et al., 2018). Therefore, a mechanism for normalizing the input would be expected to be an important feature of neuronal circuits. A natural candidate is the feedforward inhibition motif that is also present in most of the circuits in which one can find feedback inhibition (Suzuki and Bekkers, 2012; Fino et al., 2013). The feedforward inhibition motif consists of a disynaptic circuit with an excitatory projection from the upstream areas to an inhibitory pool of neurons, which in turn projects to the principal cells which also receive an excitatory projection from the same source. Such a synaptic arrangement can normalize the input by producing an inhibitory signal proportional to the magnitude of the input signal (Pouille et al., 2009; Khubieh et al., 2016). Other proposed functions for the feedforward inhibition motif include high-pass filtering (Gutnisky et al., 2017), suppression of asynchronous activity (Moyer et al., 2014), output quantization (Ferrante et al., 2009) and output forwarding and gating (Zemankovics et al., 2013). Importantly, the different functions of feedforward and feedback inhibition occur concurrently with their effects on excitation (Isaacson and Scanziani, 2011; Kee et al., 2015). Therefore, it seems reasonable to speculate that feedforward inhibition can influence how feedback inhibition produces gamma-frequency oscillations.

Here, we aimed to evaluate how the changes in input magnitude affected oscillation events produced by feedback inhibition and unveil the details of the interactions between feedback and feedforward inhibition in normalizing this process. To that end, we employed a computational model of a cortical layer including the two types of inhibitory motifs and studied how the input-output transformations were affected by manipulations on inhibitory circuitry features.

Material and Methods

All simulations were performed using Python custom scripts. The scripts are available from the corresponding author upon reasonable request. Numerical simulations used Euler’s method with 0.1 millisecond time resolution. Simulations with higher temporal resolution did not lead to significant difference in the results. All data were collected on 10 seconds simulations. We disposed of all the data collected in the first second. Data were averaged over a total of 72 simulations for each task.

Microcircuit model

The microcircuit model was based on the model used by Atallah and Scanziani (2009) and included a population of principal neurons (n_E = 1000), a population of feedback interneuron (n_FB = 200) and a population of feedforward interneuron (n_FF = 200). All neurons were modeled as a single compartment, integrate-and-fire neurons as described in (Atallah and Scanziani, 2009) with parameters selected to reflect experimentally observed values of hippocampal neurons. Membrane resistance was set to 60 MΩ in excitatory cells and 40 MΩ in inhibitory cells, and capacitance was set to 100 pF in all neurons (Glickfeld and Scanziani, 2006). Resting membrane potential was set to −55 mV in excitatory cells and −62 mV in inhibitory cells (Atallah and Scanziani, 2009). Firing threshold was set to −48.9 mV for excitatory cells and −52 mV for inhibitory cells. All neurons had after-hyperpolarization current of 133 pA with an exponential decay of 17 ms (de Almeida et al., 2009a). Each excitatory neuron received a constant, non-oscillating and individually specified excitatory current with the current amplitude distributed uniformly from 0 pA to a maximum input current (I_MAX) that was set before each simulation. All feedforward inhibitory neurons received the same input current of I_MAX multiplied by a specific gain factor (gain^A). For some experiments, a sinusoidal modulatory signal oscillating at 2 Hz (Figure 5), with strengths within the range of 0.5x to 1.5x, was used to multiply the excitatory current (mean maximum value: 400 pA) to all cells.

Figure 5. Network dynamics with delta-frequency modulation of the excitatory input.

Two networks, one without (left panel) and one with (right panel) feedforward inhibition, were fed with the same excitatory signal. (A) Overall excitatory drive modulated by a 2 Hz sinusoidal with maximum excitatory current varying from 200 pA to 600 pA. As a control, the network without feedforward inhibition was also excited with an attenuated input, varying from 110 pA to 510 pA (results shown in gray in panels A-C and F). (B) Sample compound potential signal aligned to the excitatory input in (A) (solid line). Compound potential signal filtered in the delta-frequency band (2 Hz) and its associated phase (dashed red lines) are shown. (C) Scatter-plot of the frequency of each oscillatory event as a function of the delta-frequency oscillation phase with data from single oscillation events (gray marks) and an average signal (solid line; gray dashed line for attenuated input). Two cycles are shown for better visualization. The probability distribution of the frequency of individual oscillation events. (D) Wide-band oscillation power time-locked to delta-frequency oscillation peaks. Normalized to maximum power in the window with maximum power color-coded in red and low power in blue. Average compound potential signal (white line) placed on top for reference. (E) Same as (D) but with the frequency-power distribution z-scored for each time bin to reveal gamma-frequency modulation. (F) Scatter-plot of the percentage of active cells in each event as a function of the delta-frequency oscillation phase with data from single oscillation events (gray marks) and an average signal (solid line; gray dashed line for attenuated input). Two cycles are shown for better visualization. The probability distribution of the percentage of active cells in each event.

Excitatory and inhibitory synapses were modeled as conductance-based synapses. The reversal potential of inhibitory synapses was −75 mV for excitatory neurons and −55 mV for inhibitory neurons (Vida et al., 2006; Glickfeld et al., 2009). The reversal potential for excitatory synapses was 0 mV for all cells (Atallah and Scanziani, 2009). Excitatory and inhibitory synaptic conductances had instantaneous rise-times and exponential decays of 5 ms for excitatory synapses and 8 ms for inhibitory synapses (Atallah and Scanziani, 2009). All synapses had a connection probability of 40% except for the recurrent excitatory connections on pyramidal cells that have a connection probability of 15% (Sik et al., 1995; Traub et al., 1997). Postsynaptic conductance change after a presynaptic spike was deterministic in inhibitory synapses but is stochastic with a 50% release chance in excitatory synapses (Atallah and Scanziani, 2009). No synaptic plasticity rules were applied over the synapses.

Excitatory synapses connected the excitatory cells to feedback inhibitory neurons and, recurrently, to other excitatory cells with the average conductance of 2.0 nS and 0.15 nS, respectively (Miles and Wong, 1986; Miles, 1990; Glickfeld and Scanziani, 2006). In the feedback inhibitory circuit, inhibitory synapses connected inhibitory neurons to excitatory cells and, recurrently, to other inhibitory cells from the same type with the average conductance of 1 nS (Bartos et al., 2002). In the feedforward circuit, the inhibitory connections from the inhibitory neurons targeting other feedforward inhibitory neurons and the excitatory neurons had the same average conductance of the equivalent connections in the feedback inhibitory circuit modulated, respectively, by gain^B and gain^C factors. Synaptic weight distributions were set randomly from the distribution of synaptic size measured experimentally (Trommald and Hulleberg, 1997; de Almeida et al., 2009b).

Data analysis

All analysis was performed with custom routines utilizing Python. Spike-time and membrane potential values were stored for each neuron in each simulation. The time average of the membrane potential voltage of all neurons produced an “LFP-like” signal used for frequency analysis (Bazhenov et al., 2001; Hill and Tononi, 2005), identified as compound potential. The construction of the excitatory and inhibitory components of the compound potential followed the same method but using exclusively excitatory or feedback-inhibitory neurons, respectively. Oscillatory events were identified out of the compound potential signal. First, the compound potential was convolved by a 2.1 ms square filter to remove high-frequency components. Peaks and troughs were initially detected using scipy.signal.argrelextrema function of Python with a comparator window of 5 ms. New peaks and troughs were included when two extreme values of the same type were detected successively. The amplitude of the oscillation event (AMP) was the differential amplitude between one peak and the next trough. The inter-event interval (IEI) was the time between one peak and the next. The frequency of oscillation events was computed as the median of the inverse of the IEI values in one simulation. The linear regression between AMP and IEI was computed with the scipy.stats.linregress function of Python. The lag between excitatory and inhibitory signals was the advance in time of the cross-correlation from excitatory and inhibitory components of the compound potential (Figure 1G). The level of selection was determined by the complement of the ratio between the current input applied to the least excited excitatory cell that fired in one simulation and the maximum input current provided to the population (de Almeida et al., 2009a; b; Rennó-Costa et al., 2010).

Figure 1. Feedback-inhibition motif produces gamma-frequency oscillation and selects which cells can fire.

(A) Illustration of a canonical cortical circuit with a feedback inhibition motif. A regularly-paced, non-oscillatory and cell-specific input signal excites each of the principal cells (E). The output of the principal cells triggers an interneuron network (I_FB) that inhibits the principal cells. Excitatory and inhibitory neurons also have recurrent synapses within their populations (not shown, see Methods). (B) Raster plot of the spikes of principal cells (blue) and feedback inhibitory cells (green) aligned with the compound potential signal (computed as the sum of the membrane potentials of all cells). Amplitude of gamma-frequency events (AMP, mV) and the time interval between subsequent events (IEI, ms) are extracted from the compound potential signal. (C) Input current assignment to each principal cell. The maximum input current in these simulations was 200 pA. (D) Distribution of IEI. Average event frequency computed from the median IEI (14.57 ms, equivalent to an oscillation of 68.58 Hz. (E) IEI values plotted as a function of the related AMP value. The red line illustrates the linear relationship between the two variables (r = 0.59, pvalue < 10⁻⁶⁴). (F) Excitatory and inhibitory components of the compound potential computed from the time average of membrane potentials of specific neuronal groups. (G) Lag between excitatory and inhibitory neurons extracted from the cross-correlation of excitatory and inhibitory components of the compound potential. (H) Firing rate of principal cells as a function of its specific input current. “Level of selection” was computed as the complement of the ratio between the input current of the least-excited active cell and the maximum input current in the population. In the example with maximum excitation current of 200 pA, for the level of selection of 37.5%, the first active cell receives an input of 62.5% (i.e., 100% - 37.5%) of 200 pA.

For the analysis of phase-amplitude coupling, we filtered through the convolution of the compound potential by a complex finite impulse response (FIR) filter built from a Hanning window envelope applied over three cycles of a 2 Hz complex sine signal (sine + i*cos). The phase of the delta-frequency component was extracted as the angle of the complex filtered signal. The phase of each oscillation event (see above) was associated with the delta-frequency phase at the time of the beginning of the event when the compound potential showed a local peak. The average frequency of oscillation events ($\stackrel{‒}{f}$, inverse of the IEI) of all oscillation events (e) as a function of the delta phase (p) was computed as:

$$\stackrel{‒}{f}(p)=\frac{{\sum }_e\left({\left(\left(1+\cos \left(p-p(e)\right)\right)∕2\right)}^{10}\ast f(e)\right)}{{\sum }_e\left({\left(\left(1+\cos \left(p-p(e)\right)\right)∕2\right)}^{10}\right)}$$

Where f(e) and p(e) are, respectively, the frequency and delta-frequency phase of oscillation event e. The average percentage of active cells in an oscillation event was computed in a similar fashion considering the percentage of active cells in each oscillation event. The delta-frequency time-locked signal was constructed as a multi-taper set, centered at every peak of the delta-frequency compound potential component, with duration of three delta-frequency cycles (1.5 seconds). For the power analysis, for each of the subject frequencies, the tapers were filtered by a FIR filter composed of three cycles of the frequency enveloped by a Dolph-Chebyshev window with 50 dB attenuation. The average values presented are the average of all tapers. In one analysis, to reveal the peak frequency, the power values of all frequencies were z-scored individually for each time bin of the multi-taper set.

Results

Feedback inhibition and gamma oscillations in the reference network

To elucidate the details of how the overall strength of input drive affects gamma-frequency oscillation, we built a model of the excitation-inhibition network. In our simulations, we considered a model circuit with two populations, one consisting of excitatory cells and the other of inhibitory cells, mutually connected in a feedback inhibition motif (Figure 1A; see Materials and Methods). The model was previously used to demonstrate a relationship between the amplitude of oscillatory events in the LFP and their duration in the hippocampus (Atallah and Scanziani, 2009), as experimentally observed during gamma frequency oscillations. In our simulations, the network produced oscillatory activity as expressed in the compound potential, which approximates an LFP signal as a population average of the membrane potential (Figure 1B). The main difference between our simulations and the previous report (Atallah and Scanziani, 2009) is that we injected a different current in each excitatory cell, according to a linear distribution of weights between zero and a maximum input current of 200 pA (Figure 1C). The interval between the detected events (IEI) followed a slightly skewed distribution with an average duration of 14.57 ms, which is equivalent to 68 Hz (Figure 1D). The interquartile range of event frequencies was between 59 Hz and 90 Hz. As in the previous report (Atallah and Scanziani, 2009), we also found a significant positive correlation between IEI and the variation in amplitude in the compound potential signal in each event (Figure 1E, r = 0.59). We computed the lag between the excitatory and inhibitory components of the LFP (Figure 1F) through the evaluation of the temporal lag of the cross-correlation signal (Figure 1G). The value we observe in simulation (2.6 ms) is comparable to the one observed in vivo (slightly above 2 ms) (Atallah and Scanziani, 2009) and in vitro (around 3 ms) (Miles, 1990). The results above indicate that our implementation is suitable as a reference model for the generation of gamma-frequency activity.

We also evaluated the “level of selection” in the simulated network (de Almeida et al., 2009a) (Figure 1H). A selection process emerges in the feedback circuit because the onset of inhibition establishes a global time limit for the principal cells to integrate and fire. The time required for a cell to accumulate excitatory input current and reach firing threshold is inversely proportional to the excitatory gain: cells with lower excitatory drive require more time to fire. The sum of the time of the first spike of the most excited principal cells and the delay of the feedback inhibition onset might be shorter than the time required for some principal cells to reach the threshold. Importantly, without the feedback inhibition triggered by the more excited principal cells, these ‘slower’ cells would eventually emit action potentials, which characterizes a process of competition for firing between neurons. We used the difference between the highest excitatory current in the population and the minimal excitatory current required for a cell to fire, normalized to the highest excitatory current, as a metric of the level of selection in the circuit (Figure 1F) (see also Materials and Methods). As an example, in the simulation, the observed 37.5% level of selection indicates that a cell that is excited with more than 62.5% of the strongest excitatory current, or 125 pA, will also be active. On the other hand, another neuron with less than 62.5% of the maximal excitatory current would not produce spikes in any cycle. Note that, as the feedback inhibition becomes stronger, fewer cells will be able to fire, so the value for the “% level of selection” will decrease (for example, if only the strongest excited cell fires and inhibits all other cells, the % level of selection value would be zero).

Effects of input excitation on oscillation frequency and level of selection

With the reference network (Figure 1) in place, our first goal was to evaluate how the level of input excitation in a neural network with a feedback inhibitory motif affects two distinct features of gamma frequency oscillations: the frequency of oscillatory cycles and the level of selection. We ran simulations in which we varied the maximum excitatory input current to the network (Figure 2A). The magnitude of the excitatory current impacts both the maximum firing rate in the principal cell population and the percentage of neurons that are active in one simulation (Figure 2B). We found that the maximum input current had a modulatory effect on the frequency of oscillatory events (Figure 2C). With an excitatory input current close to the minimum required for the network to produce spikes (110 pA), the network produced oscillations in the range between 40 Hz and 60 Hz, whereas for an excitatory current about 6 times of this value (600 pA), the network oscillated between 80 Hz and 120 Hz. The overall level of the excitatory drive also modulated the competition level in the network (Figure 1F). Raising the maximum excitatory current from 110pA to 600 pA increased the level of selection from about 20% to above 40%.

Figure 2. The frequency of oscillatory activity and the level of selection are dependent on the magnitude of the excitatory input to the feedback circuit.

(A) Illustration of a canonical cortical circuit with a feedback inhibition motif. Principal cells receive excitatory current drive with a linear distribution characterized by a maximum input current. Recurrent connections not shown. (B) (Left) Maximum firing rate in the principal cell population and (Right) the percentage of cells that produced spikes in the simulation. (C) Median and quartile frequencies of oscillatory events in one simulation. (D) Level of selection. All plots shown as a function of the maximum input current with the mean (solid lines) and range of values (shaded area) among multiple simulations. Dashed lines in (C) show the mean quartiles.

Feedforward inhibition reduces the effects of excitatory inputs on the oscillation frequency and level of selection

Next, we tested the hypothesis that the feedforward inhibition motif can reduce the effect of the excitatory current on the frequency of oscillation and the level of selection. We incorporated a feedforward inhibition motif to the canonical circuit (Figure 3A). We introduced three modulatory gains to this circuit to allow a fine tuning of the feedforward inhibitory dynamics. gain^A modulates the overall excitation drive of feedforward inhibitory neurons with a direct positive impact on the firing rate of such neurons (Figure 3B, left panel). gain^B modulates the strength of recurrent inhibition within the feedforward inhibitory neuron population and has a negative effect on the firing rates of these inhibitory neurons (Figure 3B, right panel). gain^C modulates the inhibitory synapses from the feedforward inhibitory neurons to the principal cells. To simplify the evaluation of the impact of the feedforward inhibition on the population oscillatory activity, we fixed gain^A and gain^B and varied gain^C in the next series of simulations. Note that a high value of gain^A (200%) was chosen to allow the feedforward inhibitory circuit to be effective for low excitatory input values (maximum excitation of 150 pA) and was counterbalanced by a high value of gain^B (200%). Interestingly, the introduction of feedforward inhibition in the network had little effect on the impact of the excitatory drive on the maximum firing rate of principal cells (i.e., the rate of firing of the most active cell, Figure 3C). The result is intriguing if one considers that the role of feedforward inhibition is to regulate the input excitation’s effect on the principal cells and is discussed further in this section. Contrary to maximum firing rate, the relation between the maximum input current and the percentage of active principal cells changed dramatically with different levels of feedforward inhibition (Figure 3D) (for further analysis of the results in Figure 3C and 3D, see below).

Figure 3. Feedforward inhibition can normalize the frequency of events oscillations and level of selection in a feedback inhibition circuit.

(A) Illustration of a canonical cortical circuit with feedback and feedforward inhibition motifs. Three different modulatory gains (gain^A, gain^B and gain^C) allow the configuration of the feedforward inhibition impact onto the network. (B) Average firing rate of the feedforward inhibitory cells as a function of the maximum excitatory current, shown for (Left) varying values of gain^A with gain^B fixed to zero, and (Right) varying values of gain^B with gain^A set to 200%. (C) Maximum firing rate in the principal cell population and (D) the percentage of cells that produced spikes in the simulation (mean values considering all simulations. (E) Frequency of oscillatory events (average median among different simulations). (F) Level of selection (mean among different simulations). (G) Trace of the membrane potential of the most excited cell (600 pA) in the population in simulations without (top) and with (bottom) feedforward inhibition. Compound potential (blue line) and the computed oscillatory event onsets (negative peaks in blue line, red lines) are shown for reference. For panels C to G, gain^B was set to 200%.

Feedforward inhibition reduces the influence of the level of the excitatory drive on the frequency of gamma oscillation (Figure 3E) and the level of selection (Figure 3F). Without feedforward inhibition, raising the maximum excitatory current from 110 pA to 600 pA increases the average oscillatory frequency from ~50 Hz to ~100 Hz. With effective feedforward inhibition, the same manipulation leads to an average oscillatory frequency ranging from ~50 Hz to ~60 Hz. Also, the level of selection that changes from 20% to ~40% without feedforward inhibition remains close to 20% with the activation of the feedforward inhibition. These results are particularly important as they indicate that a major function of feedforward inhibition is that it stabilizes the frequency of gamma oscillations and the level of selection in the presence of changes in the excitatory drive (see also below).

The fact that feedforward inhibition did not affect the maximum cell firing rate in the population is surprising because as feedforward inhibition is strengthened, the resultant excitation (feedforward excitation – feedforward inhibition) becomes weaker. Such an effect could be a result of the probabilistic network connectivity with some cells not receiving feedforward inhibition. However, we verified that all principal cells received feedback inhibition in our network. In addition, the most excited cell still received inhibition (e.g., approximately 40% of the inhibitory current of the most inhibited cell when the maximum excitatory drive was set to 600 pA; or 15% when the drive was set to 200 pA). Therefore, another explanation is required. As the frequency of oscillation events is reduced, if the number of times that a cell fires within each oscillation cycle does not change, the number of spikes should decrease. However, the changes in the level of selection and the percentage of active cells help to explain our observation of the maximum firing rate. Without feedforward inhibition, the average most excited cell fired in about 55±8% of the oscillation cycles with an average of 1.11±0.03 spikes per event considering a maximum excitation current of 600pA (Figure 3G, top panel). A given cell does not fire in all oscillation events because, although the excitation of a cell may be above the others, the initial membrane potential of the cell might be low due to bursting (multiple spikes in the same event) in the previous event and also the feedback inhibition might be relatively high during the previous event for this particular cell. As a result, a cell with a lower excitatory drive might fire first because of a higher initial membrane potential at the beginning of the event. With feedforward inhibition, the most excited cell fired in 77±5% of the oscillation events with an average of 1.11±0.03 spikes per event considering the same maximum excitation current as above (Figure 3G, lower panel). The augmented number of events in which the most excited cell fired was a result of the lower number of active cells, which reduced the probability that some other cell would fire before the most excited cell. As the number of spikes per event remained unaltered, the fact that the most excited cell fired in more oscillation events counterbalanced the reduction in the frequency of oscillation events and kept the firing rate of the most excited cell unchanged.

Since the feedforward motif can normalize the frequency of gamma oscillatory events and the degree of competition to stable levels, the question arises whether different network configurations can promote oscillatory activity at distinct frequencies and competition levels. Indeed, in the next series of simulations we found that different sets of values for the modulatory gains can produce network activity with similar principal cell firing rate values (Figure 4A) that would stably oscillate at distinct oscillatory bands (Figure 4B) and promote different levels of competition (Figure 4C). Therefore, importantly, the features of the feedforward inhibition motif can determine different oscillatory patterns with the exact same excitatory input.

Figure 4. Stable oscillatory activity at different frequencies and competition levels.

(A) Maximum firing rate, (B) Median frequency of oscillatory events and (C) Level of selection in the principal cell population for two different setups of the feedforward inhibition motif (red: gain^C = 20% and gain^A = 200%; purple: gain^C = 40% and gain^A = 80%; gain^B = 200% for both). All plots shown as a function of the maximum excitatory input current. Solid lines and shaded area represent the mean value and the range of values among different simulations. Dashed lines in (B) represent the mean interquartile range of frequencies of events.

Effects of feedforward inhibition in the presence of high excitatory input variability

To better understand the impact of feedforward inhibition on the electrophysiology of the network in conditions of high excitatory input variability, we simulated the conditions of delta-gamma phase-amplitude coupling (Andino-Pavlovsky et al., 2017). We considered two otherwise identical networks, one with feedforward inhibition and another without, and analyzed characteristic features of the compound potential with the excitatory input current modulated in the delta-frequency range (2 Hz, Figure 5A). The delta-frequency modulation of the input modulated the compound potential signals of both networks (Figure 5B). For the analysis, we extracted the delta-frequency component of the compound potential signals and its phase (see Methods). The network without feedforward inhibition produced gamma-band oscillation events of higher frequency (mean and std, 92±29Hz, median and interquartile, 90Hz [80Hz, 112Hz]) compared to the network with feedforward inhibition (mean and std, 65±30Hz, median and interquartile, 58Hz [43Hz, 81Hz]) (Figure 5C). The modulation of the gamma oscillation event frequency by the phase of delta-component was also higher in the network without feedforward inhibition (27 Hz compared to 20 Hz). In the network without feedforward inhibition, reducing the average level of excitation also led to a reduction in the frequency of gamma-band oscillation events (mean±std: 86±30 Hz; median and interquartile: 83Hz [63Hz, 106Hz]) but did not reduce the modulation of the gamma oscillation event frequency (40 Hz difference between highest and lowest frequencies). We also evaluated the average power of different frequency bands time-locked to the peak of the delta-frequency component (Figure 5D). The power of the high-frequency events was modulated by the delta-frequency band phase, indicating the occurrence of phase-amplitude coupling. Normalizing the time-locked power distribution of the high-frequency events in each time bin (z-score, see Methods) revealed that the modulation of the peak oscillation frequency by the delta-frequency phase was stronger in the network without feedforward inhibition (Figure 5E). We also evaluated the percentage of active cells in each gamma cycle (Figure 5F), a measure that is directly related to the sparseness of the neural code. Without feedforward inhibition the network had on average more active cells (mean±std: 1.9±0.8%; median and interquartile: 1.9% [1.4%, 2.4%]) compared to the network with feedforward inhibition (0.6±0.3%; 0.6% [0.4%, 0.8%]). The average number of active cells was modulated more strongly by the delta-frequency oscillation in the networks without feedforward inhibition (mean value oscillated between 1.5% to 2.3%, with a dynamic range of 0.8%) than in the complete network (mean value oscillated between 0.5% to 0.7%, with a dynamic range of 0.2%). Reducing the average level of excitation in the network without feedforward inhibition also reduced the percentage of active cells (mean±std: 1.7±0.8%; median and interquartile: 1.7% [1.2%, 2.3%]) and did not reduce the dynamic range of average values as a function of input phase (1.2%). Taken together, these results show that feedforward inhibition in the presence of high excitatory input fluctuation tends to reduce the average gamma frequency (in agreement with the results in Fig 3E, compare gain^c = 0% versus gain^c = 20%) and increases sparseness of firing (in agreement with Fig 3D), while simultaneously decreasing the modulation of the gamma oscillations by the delta waves. Such results cannot be reproduced by a reduction of the excitatory drive, which also leads to lower gamma frequency and increased sparseness of firing but does not reduce the modulation of the gamma oscillations by the delta waves.

One might suggest that the regulation of gamma frequency oscillations described above can be explained simply by the fact that principal cells in a network with feedforward inhibition receive stronger inhibitory conductance due to the addition of the feedforward system. However, the effect of adding the feedforward system on the overall level of conductance can be counterintuitive. For instance, in the example shown in Figure 5, the addition of feedforward inhibition reduced the overall level of conductance by about 20%. The added conductance of feedforward inhibition reduces the overall firing rate of principal cells, which, in turn, reduces the excitation of feedback interneurons. As the connectivity between feedback interneurons and principal cells is stronger than the inhibitory connectivity in the feedforward circuit, the net effect of the addition of the feedforward motif in our model is a counterintuitively smaller overall level of inhibitory conductance experienced by the principal cells.

Discussion

Here we propose that a primary role of the feedforward inhibition motif, when in concert with a feedback inhibition motif, is to modulate a basic property and address a potential limitation of a canonical excitatory-inhibitory circuit in producing gamma-frequency oscillations: the frequency of the population oscillation and the level of selection are both affected by the magnitude of excitation in the input signal (Figure 2). Feedforward inhibition can produce an inhibitory current that increases proportionally with the excitatory input current magnitude. Our model demonstrates that such an inhibitory current is sufficient to maintain the frequency of population activity and the level of selection in a way that they become largely invariant to the overall magnitude of the excitatory input signal (Figure 3).

Robustness of gamma oscillations and circuit function

The proposed robustness of gamma-frequency oscillation might be essential for a proper function of the brain. For instance, precise oscillatory frequency might be important for signal routing and multiplexing (Akam and Kullmann, 2010, 2014). In such a framework, interneurons in a receiving network would preferentially resonate at a specific gamma-frequency range, establishing a communication channel not prone to interference of other inputs. Indeed, in agreement with the latter notion, multiple gamma-frequency channels have been observed in the hippocampus (Colgin et al., 2009; Butler et al., 2018; Lopes-dos-Santos et al., 2018). Independent gamma frequency channels could arise from having different interneuron types mediating gamma-frequency oscillations (Middleton et al., 2008). The mechanism we propose here based on our simulation results is an alternative solution that allows precise selection of gamma frequency based on feedforward inhibition parameters with the same type of feedback interneuron cell type (Figure 4). Such a system would still require a mechanism for integrating the different independent channels within a longer oscillatory cycle (e.g., theta-frequency oscillations) which could be potentially implemented by a separate theta-frequency mediated gating system (Leão et al., 2012).

Robustness in the level of selection has direct impact on mechanisms dependent on sparse activity. Our results indicate that feedforward inhibition enhances the level of selection, enforcing sparseness (note that augmenting the level of selection leads to fewer cells firing, which, in turn, manifests itself as an increasing level of sparseness). Therefore, a possible effect of the impairment of the feedback inhibition would be observed in tasks in which sparseness of activity is important, such as pattern separation (Leal and Yassa, 2018) or memory storage (de Almeida et al., 2007). Such effects would be most likely to take place in brain structures known to have sparse activity such as the hippocampus (Wixted et al., 2014), especially the dentate gyrus (Leutgeb and Leutgeb, 2007; Leutgeb et al., 2007; de Almeida et al., 2009b; Rennó-Costa et al., 2010; Berron et al., 2016). Other mechanisms that produce gamma-frequency oscillations, such as gap junctions (Gibson et al., 1999; Chu et al., 2003), interneuron-interneuron networks (Wang and Rinzel, 1992; Cardin et al., 2009) and resonant properties of basket cells (Bartos et al., 2007), could still contribute to the regulation of robust oscillatory frequencies but provide no mechanism to control the effect of varying input amplitude in the level of selection and sparseness.

In order to experimentally test the hypothesis that feedforward inhibition regulates gamma-frequency oscillation, one would need to selectively impair feedforward inhibition but not feedback inhibition. Such a manipulation may be obtained to some extent with mutant mice (Sasaki et al., 2006) or with optogenetic approaches (Valeeva et al., 2016), however, the selectivity of modulation of feedforward versus feedback inhibition is not always clear (Elfant et al., 2007). From the results of our simulations, we can predict some of the effects of impairing feedforward inhibition in isolation. First, with impaired feedforward inhibition, we expect an increase in the average oscillatory frequency (Figure 3E and5C, D, E). Second, we presume a higher variation in the duration of oscillation cycles (Figure 5C, E). Since the feedforward inhibition current is proportional to the feedforward excitation, any upswing in the input excitation is quickly balanced by a comparably strong inhibition current. Without feedforward inhibition, the fast changes in the feedforward excitation won’t have a counterbalance signal and will be integrally transmitted to the cells with direct effects on single cycle duration. An implication of this second effect is that instantaneous gamma-frequency will follow the phase of lower-frequency oscillations of the input signal, such as delta-frequency, producing a ramp on the gamma-frequency if plotted against the low-frequency phase (Figure 5C,E). This effect might be evident in other phase-amplitude coupling observations, such as theta-gamma rhythms (see page 3 in Tort et al., 2008). The feedforward inhibition current will help to ensure that the gamma-frequency produced by the network will stay fairly constant. Third, we predict reduced sparseness in population activity (Figure 3D, 5F) resulting from the loss of the feedforward inhibitory current.

Level of abstraction and limitations of the network model

Since this is a computational simulation study, it is important to note the level of abstraction that was employed in the model. Our model was based on a previously published model (Atallah and Scanziani, 2009) that was shown to be able to capture the experimentally observed relationship between the amplitude and duration of oscillatory events in the LFP in the hippocampus during gamma frequency oscillations. While the model had a level of biological realism appropriate for studying key aspects of gamma oscillations while preserving relative simplicity and tractability, it also had some potentially important limitations. First, the model network used in this study relied on the fast GABA_A receptor-mediated IPSCs, while slower GABA_B receptor-mediated events (Li et al., 1996) were not considered. These model features are broadly in line with the key role attributed to the fast, phasic GABA_A inhibition for the generation of gamma oscillations, but it should also be noted that in vitro and in vivo experimental evidence indicates the existence of modulatory roles for GABA_B receptor-mediated processes in gamma oscillations (Leung and Shen, 2007; Johnson et al., 2017; for a review, see Kohl and Paulsen, 2010). Second, our model considered two distinct populations of feedback and feedforward interneurons. Indeed, there are interneuronal subtypes that are thought to play primarily feedback (e.g., oriens lacunosum-moleculare cells in the CA1) or primarily feedforward roles (e.g., neurogliaform cells that receive no recurrent excitation from CA1 pyramidal cells). There are also interneuronal subtypes that seem to participate in both. The functional impact of such synaptic and cellular-level model simplifications will need to be investigated in future studies using large-scale, biologically highly realistic, complex models (Bezaire et al., 2016).

Summary and outlook

The inherent complexity and heterogeneity of the neural circuit wiring makes it difficult to identify computational principles underlying neuronal information processing (Maass, 2016). Understanding the rules of local circuits with inhibitory and excitatory components is particularly challenging given the notable the diversity of interneuron cell types and connectivity motifs in the brain (Soltesz, 2006; Soltesz & Losonczy, 2018). The baseline circuit with feedforward and feedback inhibition discussed in this work is commonly found throughout the brain and it implements a simple canonical computational principle, known as the E%-MAX winner-take-all (de Almeida et al., 2009a). Models based on such competition principle have shown important neuronal computation features such as sharpening of orientation tuning curve in the visual cortex (de Almeida et al., 2009a; Lisman, 2014), formation of place fields (de Almeida et al., 2009b, 2010), hippocampal remapping (Rennó-Costa et al., 2010, 2014; Rennó-Costa and Tort, 2017) and memory restructuring during sleep (Blanco et al., 2015). All these computational models consider a minimal level of robustness that was only accomplished with the inclusion of feedforward inhibition, and our results presented in this paper provide novel mechanistic insights into how the interactions between feedback and feedforward inhibition help to normalize the effects of fluctuating excitatory inputs on gamma-frequency oscillation events. Future research will be necessary to fully understand the mechanistic reasons for the particular implementations of feedforward inhibitory motifs in distinct circuits serving different information processing functions and to develop novel interventions for psychiatric and neurological disorders when such inhibitory circuits are persistently compromised.