Theta Oscillations Gate the Transmission of Reliable Sequences in the Medial Entorhinal Cortex

Oscillatory and bistable dynamics of an MEC network motif

Layer II of the MEC consists of two distinct microcircuits with characteristic patterns of connectivity within and sparse connections across circuits (Witter et al., 2017; Nilssen et al., 2018). Stellate cells and fast-spiking parvalbumin positive (PV+) interneurons (Couey et al., 2013; Buetfering et al., 2014) form one circuit while pyramidal cells and 5HT3A interneurons form the other (Witter et al., 2017; Nilssen et al., 2018). This di-synaptic circuit motif, where principal neurons interact via an inhibitory intermediary, is prevalent throughout the EC (Couey et al., 2013; Fuchs et al., 2016; Nilssen et al., 2018). To understand how a network’s architecture affects its dynamics we simulated a simple network motif (Fig. 1B), a building block of the MEC, that consisted of biophysically detailed models of stellate cells (Dickson et al., 2000; Acker et al., 2003; Rotstein et al., 2006; Heys et al., 2010) and inhibitory interneurons (Wang and Buzsáki, 1996). Stellate cells generate characteristic subthreshold oscillations of their membrane potential in response to depolarizing inputs (Alonso and Klink, 1993). The frequency of subthreshold oscillations and the resonant frequency vary monotonically as a function of the magnitude and time scale of the h–current (I_h; Garden et al., 2008; Giocomo and Hasselmo, 2008; Heys et al., 2010). Here, we modeled these properties using a hyperpolarization activated depolarizing current (I_h; Dickson et al., 2000; Shay et al., 2016) and an amplifying persistent sodium current (I_NaP; Magistretti and Alonso, 2002) in addition to leak and spiking currents (I_L, I_Na, and I_K). We modeled interneurons using modified sodium and potassium currents that allowed them to spike at high frequency (Wang and Buzsáki, 1996). Both interneurons shown in Figure 1B received supra-threshold input. However, since they inhibit each other, only one of the neurons spiked while the other remained silent. Successive inhibitory spikes from an interneuron activated the depolarizing I_h current in the postsynaptic stellate cell. This eventually drove the stellate cell to spike. Excitatory drive from the stellate cell activated the other interneuron of the pair that, in turn, silenced the first one. The activity of this motif switched rhythmically. The interneurons alternated between episodes of spiking and quiescence, and oscillated out of step with each other. Rebound spikes by stellate cells marked every transition from spiking to quiescence (Fig. 1B) and may serve as a viable mechanism to generate periodic firing fields of grid cells (Shay et al., 2016). Excitation because of rebound spiking caused rhythmic switching. When we removed excitatory inputs from stellate cells to the interneurons, only one interneuron remained active (Fig. 1C). Transitions in the activity of inhibitory interneurons can drive other stellate cells to fire, potentially leading to a pattern of activity where different neurons are sequentially activated. As the input to inhibitory interneurons increased so did its spiking frequency (Wang and Buzsáki, 1996). At higher frequencies, the time between inhibitory spikes was too short for the postsynaptic neuron to fire a rebound spike and cause a switch in the activity pattern (Fig. 1D). In this parameter regime, the network acted as a bistable switch where one of the interneurons remained active until a transient external perturbation toggled the switch (Fig. 1D). When inhibitory input to one of the stellate cells ceased, it emitted a rebound spike that marked the transition from one state of the network to the other (Fig. 1D). Thus, depending on the parameter regime, the motif simulated here can act as an autonomous oscillator (Fig. 1B) or as a switch (Fig. 1D) whose state could be toggled between activity and quiescence by a transient external perturbation. Further, we found that systematic changes to the h–channel conductance could predictably shift the system between a bistable and an autonomous oscillatory regime (Fig. 1E–G). Bistability is contingent on two key parameters in the model: the amplitude of depolarizing input to the interneurons and the synaptic strength. When we varied the input to the interneuron, we found that the bistable regime persisted for a broad swathe of inputs. Increasing the inhibitory synaptic conductance broadened the extent of the bistable regime. As the input to the inhibitory interneurons increased in magnitude, the frequency at which they fired also increased and tended to increase the threshold strength of an external input required to switch it from an active to an inactive state. We found that the dynamics of the motif simulated here, a building block of the MEC network, is stable to variation in some key model parameters. Despite our model’s concurrence with experiments and stability along some dimensions in parameter space, it is quite possible that slight changes in other parameters can lead it into a qualitatively distinct regime. Mittal and Narayanan (2018) have looked at precisely this aspect in the MEC. They simulated 150,000 variants of a stellate cell with diverse values of the channel parameters and found that 449 of these simulated neurons, scattered across a 55-dimensional parameter space, possessed electrophysiological responses that matched experimentally recorded stellate cells. Other neuron and network models also show similar variations as a function of key parameters (Prinz et al., 2004).

Input-driven sequences in the MEC network

The EC receives sensory inputs from multiple cortical sources (Witter et al., 2017). Neurons in the deeper layers of the MEC, many of which are also grid cells, project to superficial layers, and innervate both the excitatory principal cells and inhibitory interneurons of Layer II. Almost half of these connections are targeted at the inhibitory interneurons (Ohara et al., 2018). Here, we examine the output of a model MEC network in response to a transient input that sequentially stimulates different interneurons in the network. Input was modeled as a brief pulse that lasted 40 ms; 85 ms later, we stimulated a different, randomly chosen interneuron. The onsets of successive pulses occurred 125 ms apart, the period of an 8-Hz theta oscillation.

In Figure 2A, all the inhibitory interneurons were connected to each other. The stellate cell received inputs from five randomly chosen interneurons. Our simulations compared two scenarios, one where there was no recurrent excitation from the stellate cells to the inhibitory interneurons (Fig. 2A) and another where each stellate cell randomly connected to six interneurons (Fig. 2B). In the absence of recurrent excitation from stellate cells, the interneuron that received external input generated spikes that inhibited all the other interneurons. When the pulse moved to a different interneuron, the locus of activity also shifted (Fig. 2C) and a rebound spike in postsynaptic stellate cells marked this shift. Since the stellate cell received inputs from multiple inhibitory interneurons, successive shifts from one interneuron to another elicited multiple rebound spikes from the stellate cell (Fig. 2C, bottom raster in gray background). Sequential activation of inhibitory interneurons and activation of stellate cells occurred reliably over multiple trials (Fig. 2C). Different trials were distinguished by continuous noise trains to stellate cells and interneurons with a mean amplitude that was 10% of the amplitude of the input to interneurons. When we introduced random excitatory connections from stellate cells to inhibitory interneurons (Fig. 2B), the response of the MEC network did not consistently follow the external drive (Fig. 2D). Interneurons received competing depolarizing inputs, one from the transient external drive and the other because of stellate cell spikes. The background noise and the history of activation determined which interneuron won this competition and silenced all others. This led to unreliable switching and considerable trial-trial variability in the activity of inhibitory interneurons. Therefore, the firing of stellate cells, that mark the shifts in interneuron activity, was also unreliable across trials (Fig. 2D, bottom trace in gray background).

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Figure 2.

Stability of input driven sequences. A, Network with sequential suprathreshold depolarizing pulses (temporal order of the input is shown by the red arrows) driving a subset of interneurons that were connected to a single postsynaptic stellate cell (filled gray circle). The response of the interneurons over 10 trials is shown as a raster plot in C. The stellate cell (bottom raster shaded in gray) responded reliably over multiple trials. When feedback excitation from the stellate cell to a randomly selected subset of interneurons (B) was introduced, the response of the interneurons to the sequential input was perturbed and stellate cells did not spike reliably. The duration of the inputs is marked by the colored boxes in C, D.

Theta oscillations reduce trial-trial variability of MEC responses

Feedback excitation from stellate cells stimulated postsynaptic interneurons in a manner that interfered with its response to an external input. Here, we sought to understand whether theta oscillations can attenuate the effects of excitation on the response of inhibitory interneurons.

Constraints on MEC network structure

As in the previous section, we simulated a model MEC network that consisted of stellate cells and fast spiking inhibitory interneurons. In what follows, we describe some of the features of the responses of grid cells in vivo that provide clues to constrain the topology of the MEC network. The receptive fields of different grid cells form an overlapping patchwork that covers space (Hafting et al., 2005; Fig. 3A). Within a localized region of the MEC, grid cell receptive fields share the same spatial frequency and orientation, but are phase-shifted with respect to each other. We assumed, since stellate cells in our model were activated by inhibition, neurons with overlapping spatial receptive fields that can fire in close temporal proximity, and with similar response patterns, must also receive overlapping inhibitory input (Fig. 3B). What are the consequences of this assumption? If receptive field overlap is indeed a reflection of the overlap of input connections to stellate cells, it must also constrain the responses of the system in different, independent environments. Environmental and experiential changes to the receptive field of one grid cell must be correlated to similar changes in other cells such that the spatial phase relationships are preserved. Changes in distal environmental cues change the orientation of a grid cell’s receptive field. Changes in the shape of the enclosure can deform the receptive field (Krupic et al., 2015). As an animal becomes familiar with the environment, the scale of grid cell receptive fields shrink (Barry et al., 2012). Despite these transformations, the phase relationship between the receptive fields of two grid cells, remains invariant (Wernle et al., 2018), suggesting that the spatial contiguity of receptive fields must be a consequence of the topology of the underlying network. A recent study found a mapping between the spatial periodicity of the receptive fields of grid cells and their physical location in the MEC (Gu et al., 2018). Grid cells with similar phases were arranged close to each other. This phase pattern repeated anatomically in a lattice that resembled the grid-like receptive fields of individual neurons. Further, the direction of movement of the animal was accompanied by a directed spread of grid cell activity in a local anatomic neighborhood. A similar topographic mapping between the behavioral states (head direction) and neuronal network structure (ring attractor) is found in the Drosophila central complex (Seelig and Jayaraman, 2015). More recently, the activity of the mammalian head direction circuit has been mapped to a low dimensional ring attractor (Chaudhuri et al., 2019). These observations are consistent with the architecture of attractor models, a class of theoretical models used to explain how grid cell receptive fields emerge from a network of neurons. Attractor networks typically possess a periodicity (or symmetry) that allows the component neurons to fire in a spatially periodic manner. For example, periodic activity may arise because of the radial symmetry of the connectivity kernel that leads to a hexagonally symmetric stable solution (Burak and Fiete, 2009). In other models (Guanella et al., 2007; Navratilova et al., 2012), the periodic structure of network leads to periodic patterns of activity.

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Figure 3.

Theta mediated stability of an MEC network. A schematic representation of the overlapping grid-like receptive fields of four grid cells is shown in A, left panel. As the rat traveled along a primary grid axis (marked in the figure as straight-line paths), the grid field repeated periodically (A, middle panel). This can be mapped to a circle (A, right panel). Stellate cells in the network (B, gray filled circles) received inhibitory input from the interneuron layer (colored circles). The strength of the inhibitory input followed a Gaussian profile (distribution shown in B). Each stellate cell connected to a randomly selected group of interneurons. The response of the stellate cells in the absence of theta drive to the inhibitory interneuron layer is shown as a raster plot in C. When theta oscillations (8 Hz) were present, stellate cells evoked a reliable response over ten trials (D). The mean firing rate convolved with a moving Gaussian window is shown in E, when theta oscillations were present (solid line) or absent (dashed line). The schematic below shows the grid fields of a rat running at a uniform velocity. The temporally periodic spiking pattern of stellate cells appears as a spatially periodic response.

Figure 3A shows the overlapping receptive fields of four stellate cells. The receptive fields have the same spatial period, but are phase-shifted with respect to each other. From the responses of these four neurons, one cannot identify a unique point where the animal would be located. The locus of points where the activity of the neurons is identical lie at the vertices of a triangular grid. Given the symmetry of grid cell responses, all equivalent locations in space can be mapped to a single point on a torus (Shilnikov and Maurer, 2016). Any straight-line trajectory along a primary grid axis generates a periodically repeating pattern of activity that can be mapped to a circle (Yoon et al., 2016; Fig. 3A). In keeping with our assumption that overlapping responses are generated by stellate cells with overlapping inputs, we modeled the stellate-interneuron network as a ring where “neighboring” stellate cells that fire in close spatial and temporal proximity received inputs from overlapping groups of interneurons (Fig. 3B). The strength of synaptic input from an interneuron to the stellate cell layer followed a Gaussian profile. The interneurons were all–to–all connected in this network (Fig. 3B). The neighborhood relationship between interneurons was therefore inherited from the inputs that drove the interneurons and their connections onto the stellate cell layer. In contrast to the structured and local connectivity from interneurons to stellate cells, feedback excitation onto the inhibitory layer was random. We modeled the trajectory of the animal by sequentially activating neighboring inhibitory interneurons with a transient input. In our simulations, we assumed that inhibitory interneurons received the same input every time the animal was at a particular location. Each vertex of the hexagonal receptive field of a particular grid cell showed the summed spiking activity of the neuron at that location. Therefore, the ability to reliably follow the input (how the cell responded in time) translated to the ability to respond in a grid like pattern. The raster plots (Fig. 3C) show the response of a subset of 20 stellate cells to a temporally varying input. Predictably, the response of the network was not reliable across trials as it did not always follow the input. The activity of one of the neurons, averaged across 10 trials, is shown in Figure 3E. Because of the ring like topology of the network, the neuron periodically received a supra-threshold input. The neuron was just as likely to fire when an input was present as when it was absent (Fig. 3E, dashed line).

Mechanism of theta-induced reliability

How do theta oscillations affect the network such that it responds selectively to an external input and not to distractors, namely, competing excitatory spikes from stellate cells? The networks simulated in Figure 3 operated in a regime where a transient drive or an excitatory spike can cause a perturbation that toggles the activity of the interneurons (Fig. 1C). To understand the effect of theta oscillations on the network, we first simulated a simple network (Fig. 4A) where the two interneurons received different constant depolarizing inputs. The neuron receiving the higher input continually spiked while the other neuron was quiescent. In order to toggle the activity of this network, we stimulated the quiescent neuron with a transient pulse. For sufficiently strong pulses, the neuron’s activity switched for the duration of the input (Fig. 4B, bottom traces). Weaker pulses, on the other hand, did not evoke a transient switching response (Fig. 4B, top traces). Here, the network motif was simulated in the bistable regime (Fig. 1D). Stellate cells can also generate autonomous periodic spiking (for a detailed phase response analysis, see Acker et al., 2003). Next, we stimulated the interneurons with a periodic theta drive that modulated the firing rate of the interneurons. Interneurons were maximally depolarized at the trough of theta and were most likely to spike then. When a weak pulse arrived at the quiescent neuron during the depolarizing phase of the theta oscillation, it successfully toggled the network. However, inputs that arrived at other phases did not cause a switch (Fig. 4C). Therefore, theta created periodic temporal windows where the activity of the network could be switched from one interneuron to another. In a larger network, we found that when the input arrived at the receptive phases of theta, the stimulated interneurons responded reliably to the input and the locus of activity of the network followed the input (Fig. 4D). However, merely creating periodic windows where external inputs can drive the network is not sufficient to generate reliable activity. Excitatory spikes from stellate cells can also occur during this receptive phase and perturb the response of the network to an external input. We found that when theta oscillations were present, the stellate cell spikes were tightly synchronized (Fig. 4D). These spikes always followed a burst of activity in the presynaptic interneurons (Fig. 4D, inset, F). The interneurons were locked to the trough of theta oscillations while stellate cell spikes occurred at a later phase because of the time taken to generate a rebound spike when released from inhibition. The stellate cell spikes occurred when the interneurons were hyperpolarized and were impervious to feedback excitation from stellate cells (Fig. 4D,F). The input-triggered response of the network segregated inhibitory interneuron spikes from the stellate cell responses (Fig. 4D). When theta was present and the external input occurred at a receptive phase, it was followed by a reliable burst of interneuron spikes, followed by the activity of stellate cells. In the absence of theta, stellate cells fired after each burst of interneuron spikes (Fig. 4E, inset). However, the stellate cell spikes were broadly distributed throughout the time between successive inputs to the network (Fig. 4E,G). Thus, in the absence of theta, stellate cells could effectively perturb the activity of the network to act as a “distractor” that derailed network’s response to an external drive. Earlier studies have looked at the oscillatory mechanisms that separate target network responses from distractors (for review, see Lengyel et al., 2005). For example, Hopfield (1995) used oscillations to create concentration-invariant odor representations that could be read out by follower networks using time-delay lines to separate the target pattern from distractors. Work by Li and Dayan (1999) consider the case of pattern recognition by selective amplification. They showed that excitatory-inhibitory networks tended to fall into spurious attractors termed “hallucinations.” The ability of the system to elude these spurious attractors improved when the network autonomously generated oscillations. These studies, like ours, use oscillations to construct noise-free representations. However, the mechanism we employ is different from that used in earlier studies. We show that inputs that arrive within a window of theta oscillations can elicit a response from stellate cells, while those that do not arrive within the receptive phase window, have a higher threshold to elicit a similar response. Distractors (recurrent excitation within the MEC network) get relegated to the hyperpolarized phase of the theta oscillations. This temporal organization of stellate cell and inhibitory interneuron spikes with respect to a theta clock ensure that the network listens to external inputs while ignoring its own recurrent excitatory activity.

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Figure 4.

Mechanism of theta-induced reliability. A, Membrane potential of inhibitory interneurons when they received different suprathreshold inputs. The neuron receiving the higher input fired continually (bottom trace) while the other remained silent (top trace). B, The same network motif as (A; stellate cells not shown) was simulated. A weak transient pulse was given to the interneuron receiving the lower DC (Direct current) input (B, top panel). The stimulated interneuron (top trace) did not switch. A strong pulse caused a successful switch (bottom two traces). C, Theta rhythmic drive was given to both the interneurons. A weak pulse (duration of the pulse is marked by the gray bar) caused a switch when it occurred in some phases of the theta oscillation (bottom gray trace) but not during others. D, Raster plot showing the reliable response of a subset of 20 interneurons (red line) and 20 stellate cells (black lines) from a larger network of 80 neurons, to an external input (marked as a gray bar in the magnified raster plots). The response of the same network in the absence of theta oscillations is shown in E. Each row in D, E shows the response of a neuron across 10 trials which are plotted between the horizontal lines in the raster plots. F, left, Raster plot shows the phase (with respect to the theta oscillation) at which stellate cells (black lines) and interneurons (red lines) spiked. The gray area marks the phase of the oscillation when the stimulus was present. Right, Polar plot showing a histogram of the phases where spikes occurred. The external input is marked in gray. G, Response of the network when theta oscillations were absent. The phase was defined in terms of the input pulse that sequentially and periodically stimulated neighboring interneurons.

Theta-induced reliability persist despite qualitative changes in topology of feedback excitation

Stellate cells excite inhibitory interneurons (Couey et al., 2013; Buetfering et al., 2014). We modeled feedback excitation as random inputs from stellate cells to the inhibitory layer. Here, we ask the following: are the disruptive effects of feedback excitation a consequence of the the specific network topology? To address this question, we simulated a network of stellate cells and interneurons arranged on a ring as we did in previous simulations. However, instead of introducing broad random feedback excitation to the inhibitory interneurons, we chose local asymmetric inputs from each stellate cell (Fig. 5A). Figure 5B shows the response of the network to sequential input. In the absence of theta rhythmic drive, the network activity reflected influences from both the sequential input pattern and the intrinsic dynamics of the network. The asymmetry of connections dictated the direction along which the activity of the network spread. However, it did not reliably represent the input. In contrast, when theta oscillations were present, and the inputs arrived in a theta phase-locked manner, the output of the network faithfully followed the input (Fig. 5B, left panel). We show, regardless of the specific form of recurrent excitation, patterns of activity were disrupted when theta oscillations were absent. Therefore, recurrent excitation perturbs the response of the network to dynamic input and its effects are not an artifact of the specific connectivity profile.

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Figure 5.

Theta-induced reliability persists despite qualitative changes in topology of feedback excitation. A, Network topology. The stellate cells are arranged in the top layer and interneurons in the bottom layer. Each interneuron projects to its postsynaptic stellate cells with the weights that follow a Gaussian profile. Each single stellate cell projects to local postsynaptic interneurons with a Gaussian connectivity profile instead of random connections used in the previous simulations (shown in Fig. 3B). The Gaussian connectivity profile of the stellate neuron was asymmetrically shifted with respect to the central position occupied by the neuron in the ring. B, Raster plot showing the response of a subset of neurons across trials in the presence (left) and absence (right) of theta rhythmic input to interneurons.

Theta-induced reliability persists over a range of frequencies

Theta oscillations span a range of frequencies from 6 to 12 Hz in rodents. Do changes in theta frequency compromise the reliability of the network responses? As theta frequency increased, the time between inhibitory bursts and rebound spikes remained nearly the same since this was determined by the time scale of the rebound kinetics of stellate cells. Therefore, the phase at which stellate cell spikes occurred started to shift systematically as the frequency increased (Fig. 6A). As the distribution shifted rightward (Fig. 6A), it became more likely that stellate cell spikes would invade the receptive phases of the theta oscillation (Fig. 4C) and interfere with the ability of the network to follow an external input. We calculated the reliability across trials measured as the pairwise dissimilarity between spike trains (see Materials and Methods). This measure showed greater reliability (Fig. 6B, lower spike distance, circles) for theta between 6 and 12 Hz compared with a network that was not driven by theta oscillations (Fig. 6B, triangles in the box plot). Here, we simulated the network for a particular value of the time scale of the h–current, τ, that varies along the dorsoventral axis of the MEC. Given the broad range of theta over which the network remains reliable, we anticipate that other values of τ would also give us a reliable response. However, dorsoventral gradients in τ can shift the regime where the network is reliable. Changes in theta frequency can signal changes in the animal’s behavior. For example, theta frequency is linearly related to the animal’s velocity (Jeewajee et al., 2008) and can change dynamically with changes in velocity. As the animal’s velocity increased, it encountered grid fields more rapidly. This translated in our model as an increased rate at which external pulses arrived at neighboring interneurons. We found that the network responded reliably to external inputs despite velocity dependent changes in frequency (Fig. 6C).

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Figure 6.

Reliability across a range of theta frequencies. A, Histogram showing the distribution of phases of stellate cell spikes for a range of theta frequencies from 6 to 15 Hz. The green marker locates the mean of the distribution (B) box plot of the SPIKE distance (Kreuz et al., 2013) in the presence (filled circles) and absence (triangles) of theta as a function of the rate at which input pulses stimulated successive interneurons. C, Raster plot showing the response of the neurons (40 stellate cells, black lines; 40 interneurons, red lines) across ten trials (as in Figs. 4D,E, 5B). The instantaneous frequency of the theta oscillation was varied from 6.0 to 12.0 Hz in a periodic manner (blue trace shows the instantaneous frequency of the theta oscillation). A magnified version of the raster plot on the left shows spikes against a background of theta oscillation. Notice the theta frequency decreases in a manner indicated by the blue trace above.

Theta oscillations gate the transmission of competing inputs

When theta oscillations were present the network reliably followed external inputs that arrived at the receptive phase of theta. Here, we examine the response of the network when competing external inputs occur at different phases of the theta cycle. We simulated a network consisting of two groups of interneurons connected to the same pool of stellate cells. All the interneurons were coupled to each other. Each group of interneurons extended inhibitory connections to the stellate cells forming two separate ring networks (Fig. 7A). The upper ring of interneurons received a transient pulse that traveled in a counterclockwise direction while the lower ring received an input that traveled clockwise. The two rings were chosen to distinguish different input streams. The clockwise and counterclockwise streams competed to elicit a response in the same pool of stellate cells. We kept the amplitude of both the inputs constant and varied their phase relationship with respect to an external common theta oscillation.

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Figure 7.

Theta gates transmission of competing inputs. A, Topology of the network. Two sets of interneurons arranged on different rings (top and bottom empty circles) inhibit the same stellate cell population (middle ring with filled circles). Two separate input pulse trains (red and green bars) are given to the two inhibitory population. The input to upper ring followed a counterclockwise activity pattern while input to the lower ring followed a clockwise sequence. B, The phase of each input within a single cycle of theta for three different cases. The response of the network to these patterns of input are shown in C–E. The peak of the oscillation corresponds to the maximally hyperpolarized phase of theta. C, The input arriving at the depolarizing phase (red) elicited a spike (gray line) in the postsynaptic stellate cell while the one arriving during the hyperpolarized phase elicited none (this case corresponds to the top trace in B). D, Both the inputs arrive at the depolarizing phase. The input that caused a maximum depolarization (red) elicited a successful response (this case corresponds to the middle trace in B). E, Inputs to both the clockwise and the counterclockwise rings were shifted. This switched the stellate cell from following the counterclockwise input to following the clockwise input.

In the first of the cases tested (Fig. 7B), one of the inputs (clockwise) arrived when the interneurons were near their most hyperpolarized phase while the second input (counterclockwise) arrived at the depolarizing phase. Predictably, the second input succeeded in eliciting a sequence of spikes in the upper ring of interneurons that entrained the stellate cells and inhibited all the other neurons. We then varied the phase of the clockwise input such that the pulse occurred progressively closer to the phase of the counterclockwise input (Fig. 7B,D,E). When the input crossed a particular phase of the theta oscillation (Fig. 7B, bottom row, E), it elicited spikes in inhibitory interneurons of the lower ring. This inhibited all the other interneurons including the neurons on the upper ring of the network. Therefore, although the counterclockwise inputs occurred during a depolarized phase of theta, the response of the network followed the clockwise inputs because it occurred earlier and also during the depolarized phase of the theta cycle. Thus, the mechanism whereby the network selects between inputs is determined, not only by coherent theta gain modulation of inputs, but also by the temporal order and the phase at which the inputs occur. Thus, theta oscillations can serve as a gate that permits a particular temporal ordering of stellate cell responses while prohibiting a different temporal ordering in the same group of cells.

Several studies have shown that theta oscillation coherence plays an important role in interregional interactions. However, few studies have looked at how this can be implemented in different brain networks. Akam and Kullmann (2010) was the first study that provided a mechanistic understanding of how oscillations can be used to separate different streams of inputs in brain networks (termed demultiplexing). The form of demultiplexing implemented in our model is termed time division multiplexing by coherent gain modulation (Akam and Kullmann, 2014), where a periodic drive to the decoding circuits may be used to separate the inputs arriving at different phases of the drive. We used theta oscillations to modulate the gain and amplify inputs arriving at some phases while attenuating the effect of others.