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Introduction

The synchronous activity of neural populations results in voltage fluctuations observable in macroscopic (e.g., scalp electroencephalography) and mesoscopic (e.g., local field potential or LFP) recordings. Rhythmic voltage fluctuations—or oscillations—have been observed in the mammalian brain for over a century [1]. Although the purpose of these oscillations remains unknown, neural rhythms appear to temporally organize network activity patterns, and pathological changes in these rhythms often accompany disease [2],[3].

What mechanisms produce these neural rhythms? The complexity of the vertebrate brain—resulting not only from the sheer number of neurons (approximately 10⁹) and their connections (approximately 10¹¹ [4]), but also from the many different neuron classes (e.g., the diversity of inhibitory interneurons [5])—affords no simple answers. Yet, simple characteristic structural patterns appear fundamental to the brain's organization [5],[6]. From these elementary network building blocks (i.e., structural and functional motifs) more complicated structures may be generated in an efficient way [7]–[9].

Perhaps a similar strategy may be used to understand the rhythmic electrical activity of the brain. For example, the simplest (yet biophysical) model of the 40 Hz (gamma) rhythm involves only two interconnected cells (one excitatory and the other inhibitory) with reciprocal synaptic connections. In this “gamma –motif” the decay of inhibition paces the rhythm [10]–[12]. This motif, and its variations, may be used to construct more complicated gamma networks [13]–[15].

Here we consider how the elementary dynamic building blocks (or dynamic motifs [16]) of the gamma (~40 Hz) and beta2 (~25 Hz) rhythms may combine to generate a slower beta1 oscillation. To do so, we develop computational models motivated by a novel method of rhythm generation—period concatenation—as we now describe. We recently observed that application of 400 nM kainate to rat somatosensory cortex in vitro resulted, initially, in distinct rhythms in different cortical layers [17]. In the superficial cortical layers (layers II and III, or LII and LIII) we observed gamma frequency rhythms (38±3 Hz, n = 25 observations), and in the deep cortical layers (layer V or LV) nonsynaptic beta2 rhythms (24±2 Hz, n = 25 observations). The dominant mechanisms that regulate these two rhythms in secondary cortex are known; the decay time of GABA_A IPSPs dictates the gamma period [12], and an outward potassium current in LV pyramidal axons sets the beta2 period [18].

The initial interval of coexistent gamma and beta2 rhythms preceded the transition to a slower beta1 oscillation (15±2 Hz, n = 10). This new rhythm appeared in both cortical layers upon reduction of glutamatergic excitation with 2.5 µM NBQX (an AMPA receptor antagonist that also reduces kainate drive [19]). We note that the initial interval of kainate application—and its associated network activity at gamma and beta2 frequencies—was an essential prerequisite to the generation of the beta1 rhythm. If we bathed the slices in kainate and NBQX initially (not kainate alone) then we found no persistent rhythms. Both the population activity (observed in the LFP) and the spiking activity of individual neurons suggested that the beta1 rhythm resulted from period concatenation; the period of the slower beta1 rhythm (~65 ms) equaled the sum of the periods of the two faster rhythms (gamma ~25 ms and beta2 ~40 ms). The ordering of the neural activity in each layer supported the period concatenation hypothesis: in the population and single unit activity of both layers we found delays consistent with individual cycles of the gamma and beta2 rhythms during a single beta1 oscillation. In particular, during the beta1 rhythm, the LII activity followed the LV activity by approximately one gamma cycle (≈1/(40 Hz) or 25 ms), and the LV activity followed the LII activity by approximately one beta2 cycle (≈1/(25 Hz) or 40 ms). To help illustrate these relationships between the cortical layers, we show a cartoon representation of the activity in Figure 1.

Figure 1. Cartoon illustration of the period concatenation phenomena observed in vitro.

In both panels, each vertical line represents a peak in the population activity (or LFP) of the superficial (LII) or deep (LV) cortical layers. Upper panel: under high kainate conditions, oscillations in LII and LV occur at gamma and beta2 frequencies, respectively. Lower panel: following application of NBQX, a beta1 rhythm occurs in both layers. The peaks in oscillatory activity are offset between the layers; a LV peak precedes a LII peak by one gamma cycle (left vertical red line), and a LII peak precedes a LV peak by one beta2 cycle (right vertical red line). The faster rhythms (gamma and beta2) combine to produce the slower beta1 oscillation.

doi:10.1371/journal.pcbi.1000169.g001

The in vitro observations suggest that concatenation of the faster gamma and beta2 rhythms (through a summation of their periods) may generate the slower beta1 oscillation. How might such a concatenation occur? As a simple illustrative model we consider two excitable oscillators, one tuned to spike at gamma frequencies and the other to spike at beta2 frequencies. With enough excitation, both oscillators independently fire at their natural frequencies. If we reduce this excitation and connect the separate oscillators in a particular way, we can generate the slower beta1 rhythm as the concatenation of the gamma and beta2 periods. To do so, we connect the oscillators so that a spike in one resets and causes the other to fire one cycle later. For example, if the beta2 oscillator fires, it resets the gamma oscillator, which then fires 25 ms (one gamma cycle) later. Subsequently, the gamma oscillator spike resets the beta2 oscillator, which then fires 40 ms (one beta2 cycle) later. Connected in this way, both oscillators fire at beta1 frequency; the period (65 ms) is the sum of the natural periods (25 ms and 40 ms) of the excitable oscillators.

Of course, biophysical cells are more complicated than simple excitable oscillators. In what follows, we examine the cell types, intrinsic currents, and synaptic connections that support the beta1 rhythm in vitro. In an ideal model of period concatenation, the mechanisms that support the faster rhythms combine to generate the slower oscillation. We will show that this scheme is nearly—but not quite—met in the computational model. The phenomenon of rhythm generation through period concatenation could be modeled in many different ways (involving, for example, different cell types, ionic currents, and synaptic connections [20].) Here we implement the elementary dynamic building blocks of the gamma and beta2 oscillators (i.e., a gamma motif and beta2 motif) and combine these to generate the beta1 rhythm. In doing so we describe how in vitro observations guide construction of the model and test its predictions. We begin by discussing the coexistent gamma and beta2 activity observed in vitro and simulated in the model. We then describe the transition to beta1 and propose that this transition requires a shift in the deep layer pyramidal cells from antidromic to orthodromic activity. Analysis of the cross-frequency interactions during the beta1 oscillation (without access to the underlying mechanisms) might simply suggest that the slower rhythm modulates the faster activity. As we will show, the combined approach of in vitro recordings, data analysis, and mathematical modeling reveals more detailed information about the intrinsic currents and synaptic connections that generate the beta1 rhythm through period concatenation of the faster oscillations.

Results

In this section we describe a reduced model of rat somatosensory cortex and the observations that constrain and verify this model. The model consists of four cell types observed in vitro [17]: a regular spiking (RS) pyramidal cell, a basket cell, a low-threshold spiking (LTS) interneuron [21]–[23], and an intrinsically bursting (IB) cell. We show a cartoon representation of the three superficial layer cells and the four-compartment deep layer IB cell in Figures 2 and 3A, respectively. Model chemical and electrical synapses connect these neurons consistent with cortical physiology and observations from the in vitro preparation, as described below. We focus on the intrinsic currents that determine the time scales of each dynamic motif, and on the synaptic connections between cortical layers that disturb and promote each rhythm. In two cases the model predictions are tested in vitro, and in all cases we use the model to suggest more detailed information not observable in experiment. We begin with simulations of the high kainate conditions.

Figure 2. The gamma-motif and its disruption.

(A) Reciprocal synaptic connections ( excitatory in red, inhibitory in blue) between a basket cell (B) and RS cell create a single Pyramidal-Interneuron-Network-Gamma (or PING) oscillator. A spike in the RS cell (green trace, spikes truncated) induces a spike in the basket cell (blue trace, spikes truncated). The basket cell then inhibits the RS cell which recovers and fires after approximately 25 ms. The decay time of the inhibitory synapse determines the period of the gamma rhythm. (B) Inclusion of a superficial layer LTS interneuron, and deep layer excitatory inputs (red arrows) to the superficial inhibitory cells, disturbs the gamma-motif.

doi:10.1371/journal.pcbi.1000169.g002

Figure 3. The beta2 reduced model and control of the beta2 rhythm by the M-current.

(A) The model of a single IB cell consists of four compartments: an apical dendrite (d_a), a basal dendrite (d_b), a soma (IB), and an axon (a). (B) Traces of the voltage in the IB cell axon (orange) and soma (red), and the M-current in the axon (black). Bursts of action potentials in the axon produce spikes and spikelets in the soma, and increase the axonal M-current. Between bursts of action potentials, the M-current slowly decays. The vertical scale in this figure is arbitrary. (C) The normalized power spectra of the voltage in the axon compartment with different scaling of the M-current backward rate function; normal backward rate function (solid curve), decreased backward rate function (dotted curve), and increased backward rate function (dashed curve). The dynamics of the M-current control the interburst interval.

doi:10.1371/journal.pcbi.1000169.g003

Superficial Layer Model in High Kainate Conditions

During the high kainate condition in vitro, coexistent gamma and beta2 rhythms occur in the superficial and deep cortical layers, respectively [17]. To model the superficial layer activity, we implement a population of Pyramidal-Interneuron-Network-Gamma (or PING) type oscillators, each consisting of a basket and RS cell. We show the voltage dynamics of a single gamma oscillator (i.e., the gamma-motif) in Figure 2A. The gamma activity results from the interactions between these two cell types, and the decay time of the inhibitory basket cell synapse determines the period of the gamma rhythm [12]. We also include a population of LTS interneurons in the superficial layer [21]–[23]. Under the high kainate conditions, these interneurons fire infrequently and have a disorganizing influence on the gamma oscillator, as we show in Figure 2B. But, following application of NBQX, the LTS interneurons play a vital role in establishing the beta1 motif, as we discuss below. We replicate the RS-basket-LTS circuit (shown in Figure 2B) twenty times and connect the RS neurons with electrical synapses to create a population of superficial layer cells, as described in the Methods section.

Deep Layer Model in High Kainate Conditions: M-Current Determines Frequency

To model the deep layer activity, we simulate a population of IB cells, each consisting of four compartments: an apical dendrite, basal dendrite, soma, and axon. We show a cartoon representation of a single IB cell model in Figure 3A. Recent experimental and modeling work has shown that the axonal M-current controls the period of the antidromic beta2 oscillation [18]. The same is true in the reduced model presented here; during beta2 activity, bursts of action potentials occur in an IB cell axon and travel antidromically to produce bursts of full action potential spikes and subthreshold “spikelets” in the soma (Figure 3B). The dynamics of the M-current determine the interburst interval [18]. When the axon generates an action potential, the M-current increases. After enough sequential action potentials, the M-current becomes large enough that the outward, hyperpolarizing current prevents further spiking. The M-current then slowly decays and, after sufficient time, another burst of action potentials can begin (Figure 3B). We may increase (or decrease) the interburst interval by decreasing (or increasing) the backward rate function of the M-current (Figure 3C). A decrease in the backward rate function causes the M-current to decay more slowly and the interburst frequency decreases (dotted curve). Conversely, an increase in the backward rate function increases the interburst frequency (dashed curve).

Deep Layer Model: Anatomy and Connections of the Dendrites

We model the dendrites of an individual deep layer IB cell with two compartments: a basal dendrite and an apical dendrite (Figure 3A). This is, of course, a crude approximation to the true dendritic form. We utilize the two compartments to mimic changes in ionic currents and synaptic inputs that occur across the dendritic structure. In particular, we increase the conductance of hyperpolarization activated currents (h-currents) in the apical dendrite [24],[25], and will connect excitatory NMDA synapses from IB cell axons to IB cell basal dendrites [26],[27] to generate the beta1 rhythm, as we describe below. Two types of inhibitory input also target the IB cell dendrites; the superficial layer LTS interneurons target the ascending apical dendrites, and a Poisson source of IPSPs target the basal dendrites (we provide an interpretation of these random inhibitory inputs below.) Although a crude approximation to the actual pyramidal morphology, we find that this simple model captures the essence of the observed dynamics, as we now describe.

The Simulated Cortical Column Produces Coexistent Gamma and Beta2 Rhythms

We show a cartoon representation of the entire reduced model in Figure 4A. We note that ascending excitatory synapses (from the deep layer IB cells to the superficial inhibitory cells) and descending inhibitory synapses (from the LTS interneurons to the IB cell apical dendrites) connect the two layers. In Figure 4B we show a typical simulation result for the 80 cells in the model (twenty of each type). We plot the spiking activity of the three superficial layer cell types, and the spiking activity of the dendrites, somata, and axons of the IB cell population. We find that the individual IB cell dendrites generate action potentials infrequently and remain in a mostly inactive state. The IB cell axons generate bursts of action potentials at beta2 frequencies that are weakly synchronized across the population. The weakly organized beta2 activity results in a noisy synaptic input to the superficial layer inhibitory cells. The result is a disorganization of the dynamic motifs that define the gamma rhythm; the basket cells are perturbed directly by the deep layer excitatory inputs, while the the RS cells are perturbed indirectly through the deep layer excitation of the LTS interneurons. We illustrated the effects of these disorganizing deep layer inputs on the gamma rhythm in Figure 2B and will show below that removing the noisy synaptic inputs from the IB cells increases the superficial layer gamma power.

Figure 4. Population model of the high kainate activity.

(A) A cartoon representation of the model. The populations in both layers contain twenty replications of each cell, although we only draw three of each cell type in the figure. The deep layer consists of IB cells each with four compartments: d_a, the apical dendrite; d_b the basal dendrite; IB, the soma; and a, the axon. A Poisson source of IPSPs inhibits each basal dendrite, and we indicate this source with the label IPSPs. The superficial layer contains three cell types: RS cells (RS), basket cells (B), and LTS interneurons (LTS). We denote the termination points of excitatory and inhibitory synapses with red and blue filled circles, respectively. We connect all of the IB cell axons and RS cells with gap junctions (indicated by the horizontal red lines). (B) The spiking activity of the superficial layer cells and deep layer IB cell compartments: B (blue) basket cells; RS cells (green); LTS interneurons (purple); d_a (orange) apical dendrites; d_b (gold) basal dendrites; IB (maroon) somata; a (red) axons. Each colored dot represents a spike in a single compartment or cell. The horizontal black line indicates 20 ms. (C) The average cross-correlation between the RS cells and IB cell axons. No obvious correlation structure is apparent. (D) The average power spectra of the RS cells (green) and IB cell axons (red). Coexistent gamma and beta2 activity occur in the superficial and deep layers, respectively.

doi:10.1371/journal.pcbi.1000169.g004

We plot in Figure 4D the population average power spectra (see Methods section) of the RS cells (green curve) and IB cell axons (red curve) and find broad spectral peaks in the gamma range (40–50 Hz) in the superficial layer and in the beta2 range (20–30 Hz) in the deep layer. We also show the average cross-correlation (see Methods section) between the spike times of the IB cell axons and RS cells in Figure 4C. We find no obvious correlation between the activity of the two layers. Thus, although the layers interact through chemical synapses, these interactions are too weak to correlate the spike times of the two layers. In particular, the disorganizing beta2 frequency input from the IB cells to the superficial layer inhibitory cells does not phase lock (and therefore correlate) the rhythmic activity of the two layers. That the model generates coexistent gamma and beta2 rhythms is not surprising—we include in the model components necessary to produce these two rhythms and allow only weak interactions between the two layers.

Having established that the reduced model can generate gamma and beta2 oscillations, we now determine how these model rhythms respond to a series of experimental manipulations performed in vitro. In each challenge the experimental results test and constrain the computational model. In addition, we use the model to suggest more detailed information not accessible in experiment. Our manipulations focus on the synaptic connections and intrinsic currents that support (or disturb) each rhythm.

Separating Cortical Layers Boosts Superficial Gamma Power

Analysis of in vitro slice preparations revealed a statistically significant increase in the gamma power of the superficial layers (215±29%, n = 6, P<0.05) following a lesion through layer IV. We show example LFP recordings and power spectra for these in vitro results in Figure 5C. These observations suggested some functional connectivity between LV cells and superficial layer neurons involved in generating the gamma oscillation [17].

Figure 5. Population model of the high kainate activity with cortical layers separated.

(A) Cartoon representation of the model. The ascending synapses and the apical dendritic compartments of the IB cells have been removed. (B) The spiking activity of the superficial and deep layer cells in the model. We note the increased synchrony of the superficial layer activity compared to that observed with layers intact. (C) Power spectra of the intact and lesioned in vitro slice preparation. In the former, LFP data were recorded from intermediate cortical layers. In the latter, we show the power spectra of LFP data recorded from deep layers (LV) and superficial layers (LII). The power is expressed in units of V²/Hz. Below each power spectrum we show example LFP traces from intermediate (left) and superficial (right) cortical layers. The vertical and horizontal lines at the bottom left indicate 50 µV and 100 ms, respectively. (D) The average power spectra of the RS cells (dark green) and IB cell axons (red) in the model following the separation of layers. For comparison, we also plot the average RS power spectrum for the intact model (light green). The superficial layer gamma power increases following the separation of the layers.

doi:10.1371/journal.pcbi.1000169.g005

In the computational model, the superficial layer PING oscillators produce the gamma rhythm. We expect that ascending excitatory synapses from the IB cell axons perturb these superficial layer oscillators (both directly and through activation of superficial LTS interneurons) and therefore disturb the gamma rhythm. To test this expectation we separate the cortical layers in the model and observe the resulting activity. Specifically we remove the ascending excitatory synapses from the IB cells to the inhibitory cells, and we disconnect the apical dendrites from the IB cells (we assume that a slice through layer IV severs the ascending dendrites of the IB cells.) We suggest the associated parameter changes in Figure 5A. Following separation and elimination of the disorganizing inputs, the spiking activity of the superficial PING oscillators becomes more synchronized (Figure 5B). This increased synchronization boosts the superficial gamma power (Figure 5D), in agreement with the in vitro results.

In experiment, separating the superficial and deep cortical layers requires a resection through the intervening layer IV. This physical separation necessarily destroys all interlaminar connections. In the model, we may study specific interlaminar connections to determine those that most disturb the superficial gamma rhythm. To do so we remove, one by one, the three types of interlaminar connections: (i) excitatory synapses from the IB axons to the basket cells, (ii) excitatory synapses from the IB axons to the LTS interneurons, and (iii) the apical dendritic compartments of the IB cells. We find that eliminating the first two connections (but not the last) increases the superficial gamma power (data not illustrated). The excitatory input from the IB cells disturbs the gamma oscillators directly by depolarizing the basket cells, and indirectly by exciting the LTS interneurons (which spike and hyperpolarize the RS cells.) Without the disorganizing effects of the deep layer input, the RS cells drive the population of PING oscillators in synchrony.

Blocking IPSPs Boosts Deep Beta2 Power

Inhibition plays a vital role in pacing the gamma and beta1 rhythms [12],[17]. To determine the effect of inhibition on the beta2 oscillation, we applied 250 nM of gabazine to the in vitro slice preparation and found a statistically significant (n = 5 observations, p<0.05) increase in the deep layer beta2 power (Figure 6C) consistent with previous results [18].

Figure 6. Population model of the high kainate activity with IPSPs blocked.

(A) Cartoon representation of the model with all inhibitory synapses removed. (B) The spiking activity of the superficial and deep layer cells in the model. We note the increased synchrony of the deep layer activity compared to that observed under normal conditions. (C) Power spectra of the in vitro slice preparations under control conditions and following application of 250 nM of gabazine. The power is expressed in units of V²/Hz. Below each power spectrum we show example LFP traces; the vertical and horizontal lines at the bottom left indicate 50 µV and 100 ms, respectively. (D) The average power spectra of the RS cells (green) and IB cell axons (dark red) in the model following IPSP block. For comparison, we also plot the average IB power spectrum in the original model (light red). The deep layer beta2 power increases following the elimination of all IPSPs.

doi:10.1371/journal.pcbi.1000169.g006

In the model, IPSPs (from the superficial layer LTS interneurons and deep layer random sources) target the IB cell dendrites. We think of the random inhibitory inputs as representing a noisy motif, perhaps important for other deep layer rhythms, but disruptive to the beta2 activity. Therefore, we expect that blocking IPSPs will eliminate these inputs and reduce the orthodromic, disruptive influence on the antidromic, beta2 rhythm. We determine the effect of blocking IPSPs in the model by setting the conductance of all inhibitory synapses to zero, as we indicate in Figure 6A. We find that elimination of IPSPs in the model reduces the orthodromic disruption of the beta2 rhythm. The IB cell dendrites now burst in response to the antidromic propagation of axonal activity. Without the disruptive inhibitory input to the IB cell dendrites, the IB cell axons burst with greater synchrony (Figure 6B), and thus boost the beta2 power (Figure 6D) in agreement with the experimental results. We note that removing the basket cell IPSPs causes the electrically coupled RS cells to establish a positive feedback loop and fire rapidly (Figure 6B). In the simple model of the gamma -motif implemented here, the RS cells spike on each cycle of the gamma rhythm. In vitro, individual RS cells spike much more sparsely during gamma activity [17]. We do not model this sparse activity here, which would require a much larger population of RS cells [12],[13]. Therefore the model of superficial layer activity is not valid after removing the IPSPs that pace the gamma rhythm.

We also use the model to investigate the type of inhibitory synapse most disruptive to the beta2 rhythm. To do so, we determine the individual effects of removing inhibitory synapses from: (i) the basket cells alone, (ii) the LTS cells alone, and (iii) the random sources (to the IB dendrites) alone. We find that, of the three, removing the latter two increases the beta2 power (simulations not illustrated). Again we conclude that eliminating the disruptive activity of the IB cell dendrites (here by removing disruptive inhibitory inputs to these compartments) boosts the beta2 activity.

Blocking h-Current Boosts Deep Beta2 Power

The beta2 rhythm in the deep layer IB cell population originates in the axonal compartments (in fact, the axonal M-current sets the period of this rhythm [18]). Input from the IB cell dendrites disturbs this rhythm, and we expect that silencing the dendritic compartments would boost the deep layer beta2 power. We test this hypothesis in the mathematical model by blocking a current important to both the dendrites and their superficial layer inputs (the LTS interneurons): the h-current. We do so by setting the h-current conductance to zero in the RS cells, LTS interneurons, and IB cell dendrites. We shade the affected cells and compartments gray in Figure 7A and plot an example of the model spiking activity in Figure 7B. We find that blocking all h-currents eliminates the disruptive effect of the IB cell dendrites on the beta2 rhythm. The IB cell axons burst with increased synchrony and therefore the beta2 oscillations in the IB cell population increase in magnitude. The result is an increase in beta2 power, as we show in Figure 7D. We tested the effect of h-current block in vitro by applying 10 µM ZD-7288 to the slice preparation and found a dramatic increase in the beta2 power (Figure 7C). This increase was statistically significant (n = 5 observation, p<0.05) and verified the model prediction.

Figure 7. Population model of the high kainate activity following h-current block.

(A) Cartoon representation of the model with h-current removed from the cells and compartments shaded gray. (B) The spiking activity of the superficial and deep layer cells in the model. We note the increased synchrony of the deep layer activity compared to that observed under normal conditions. (C) Power spectra of the in vitro slice preparations under control conditions and following application of 10 µM ZD-7288. The power is expressed in units of V²/Hz, and we note the change of scale in the two figures. Below each power spectrum we show example LFP traces; the vertical and horizontal lines at the bottom left (right) indicate 50 µV (100 µV) and 100 ms, respectively. (D) The average power spectra of the RS cells (green) and IB cell axons (dark red) in the model following h-current block, and without h-current block (light red). The deep layer beta2 power increases following h-current blockade.

doi:10.1371/journal.pcbi.1000169.g007

The global blockade of h-current does not suggest which particular cell type (if any) is most important to this effect. Therefore, we consider in the model the effects of h-current block: (i) in the RS cells alone, (ii) in the LTS interneurons alone, (iii) in the IB cell basal dendrites alone, and iv) in the IB cell apical dendrites alone. Of these three, only the middle two increase the beta2 power. By eliminating the h-current in the IB cell basal dendrites, we essentially silence these compartments and reduce the effects of the random inhibitory synaptic inputs to them. By eliminating the h-current in the LTS interneurons, we silence these cells and their disturbing inputs to the IB cells. We conclude that eliminating the disruptive inputs to the IB cells enhances the deep layer (antidromic) beta2 rhythm.

Strengthening NMDA Synapses between IB Cells Preserves the Fast Rhythms

In vitro, the initial interval of coexistent gamma and beta2 rhythms must precede the slower beta1 activity. Without this initial interval (i.e., with immediate application of NBQX to the slice preparation) no beta1 activity occurs. We therefore expect that the coexistent fast rhythms change the network in a way that supports the slower beta1 oscillation. In addition, we know that NMDA receptor-mediated synaptic events support the beta1 activity—blocking NMDA before or after NBQX application prevents the beta1 oscillations (as we discuss in detail below). To model the change in network structure that occurs during the fast rhythms, we assume a strengthening of all-to-all NMDA synapses from each IB cell axon to all IB cell basal dendrites [26],[27]. We note that potentiation of NMDA synapses has been observed in LV pyramidal cells with bursting behavior [28].

We expect that these NMDA synapses between the IB cells strengthen gradually during the interval of coexistent gamma and beta2 activity. What effect does this gradual change have on the faster rhythms? In vitro we observed that the patterns of field potentials remained constant from the onset of gamma and beta2 activity to immediately before application of NBQX. Whatever changes occurred in the network to support the beta1 activity had no impact on the faster rhythms. We test this in the model by strengthening the NMDA synapses between the IB cell population under the high kainate conditions. In agreement with the experimental observations, we find no change in the gamma and beta2 activity of each layer; the power spectra in the deep and superficial layers match those shown in Figure 4D (results not shown.) We conclude that the strengthening NMDA synapses between the IB cells do not impact the network dynamics until a reduction of excitation (induced by NBQX) occurs in the model, as we now describe.

Low Kainate Drive: Synchronous Beta1 Rhythm

After an initial interval of coexistent gamma and beta2 rhythms in vitro, the transition to beta1 activity followed application of NBQX, resulting in a reduction of glutamatergic excitation via AMPA and kainate receptor subtypes. In both the deep and superficial layers, population activity (as observed in the LFP) oscillated at beta1 frequency and a consistent lead/lag relationship appeared between the two layers: LII activity preceded LV activity by ~40 ms, and LV activity preceded LII activity by ~25 ms (see Figure 1). We note that the timing of these lead / lag relationships suggests that the faster dynamics motifs (i.e., the gamma-motif [25 ms period] and beta2-motif [40 ms period]) collaborate to generate the beta1 rhythm [17].

We now consider whether such collaborative interlaminar interactions can produce the beta1 oscillation in the model. We first assume that reduction of glutamatergic excitation decreases excitation in the network and inactivates the gamma and beta2 motifs [29]. We approximate this reduction by decreasing the depolarizing input current to the superficial layer cells, and the deep layer IB cell dendrites and axons. We also halt the Poisson distribution of IPSPs to the IB cell dendrites, thereby eliminating these disruptive inputs. We indicate these changes in Figure 8A by shading blue the affected cells and compartments. As described above, we also include NMDA synapses (decay time constant 100 ms) within the deep layer IB cell population that target the IB cell basal dendrites; we indicate these slowly decaying excitatory synapses with red lines and filled circles in Figure 8A.

Figure 8. Population model of the low kainate drive activity following an interval of coexistent gamma and beta2 rhythms.

(A) To represent the reduced excitation in the model, we hyperpolarize the cells and compartments shaded blue. We include NMDA synapses extending from each IB cell axon to all IB cell basal dendrites; we show these synapses in red. (B) The spiking activity of the three superficial layer cells, and of the dendrites, somata, and axons of the IB cell population. The numbers within yellow circles and gray lines indicate the steps in the beta1 rhythm as it propagates from the deep to superficial layer and back. (C) The average cross-correlation between the RS cells and IB cell axons. (D) The average power spectra of the RS cells (green) and IB cell axons (red).

doi:10.1371/journal.pcbi.1000169.g008

Mechanisms of the Beta1 Rhythm: Simulation Results

Producing the beta1 rhythm requires many mechanisms that support the superficial gamma and deep beta2 activity. To describe these mechanisms, we follow a cycle of the simulated oscillation as it propagates from the deep to superficial layer and back (Figure 8B). We start at a burst of activity in the IB cell axons (red dots, see Label 1 in Figure 8B). This burst delivers strong excitatory input to the superficial layer inhibitory cells. The basket cells spike (blue dots, Label 2 in Figure 8B), thus inhibiting the RS cells and the LTS interneurons. We note that the basket cells fire and inhibit the LTS interneurons before the ascending NMDA synapses can depolarize these interneurons. This occurs in the model because we make the rise time of the NMDA synapse longer for the LTS interneurons than for the basket cells (see Methods section).

After receiving inhibitory input, the h-currents of the RS cells activate and depolarize these cells on a slow time scale. The RS cells recover from inhibition and spike (green dots, Label 3 in Figure 8B), inducing the population of LTS interneurons to fire. Only a weak excitatory input from an RS cell is required to push an LTS interneuron past spike threshold; the slowly decaying excitatory input from the deep layer and the intrinsic h-current have steadily depolarized each LTS interneuron. Upon spiking (purple dots, Label 4 in Figure 8B), the LTS interneurons inhibit the RS cells and the IB cell apical dendrites, temporarily halting the slow depolarization of the deep layer cells due to NMDA synaptic input. The slow depolarization of the dendrites continue—due to both h-currents and NMDA synapses—and the IB cells spike (Label 5 in Figure 8B), inducing a synchronous burst in the deep layer population and restarting the beta1 cycle.

Computing the population average power spectra of the deep and superficial layer cells, we find peaks near 13 Hz and higher order harmonics (e.g., 26 Hz, 39 Hz, 52 Hz, and so on; Figure 8D). For the RS cell, we note that the largest peak occurs near 26 Hz. The power of this peak includes two contributions: (1) harmonic power of the 13 Hz oscillation (note 2×13 = 26 Hz), and (2) power resulting from the interval between an RS cell spike and the subsequent inhibitory input (one beta2 cycle).

The model of beta1 activity agrees with the observed rhythm in a fundamental way: this rhythm results from period concatenation. To show this we plot in Figure 8C the population average cross-correlation between the spiking activity of the RS cells and the IB cell axons. We find two intervals of increased correlation: between approximately −35 ms and −20 ms, and between approximately 45 ms and 60 ms. Thus, the deep layer population activity precedes the superficial layer activity by approximately 30 ms. The reason for this delay is that the bursting IB cell axons activate the superficial basket cells that inhibit the superficial RS cells. We also conclude that the superficial pyramidal activity precedes the deep layer activity by approximately 50 ms. The reason for this delay is that the RS cells activate the superficial LTS interneurons which inhibit the deep IB cell apical dendrites. These correlation results show that the temporal distributions of the superficial and deep layer activities in the model agree with those of the experiments. In both the model and experiments, during the beta1 rhythm the LII activity follows the LV activity by approximately one gamma cycle (30 ms or ≈33 Hz), and the LV activity follows the LII activity by approximately one beta2 cycle (50 ms or ≈20 Hz) [17].

Mechanisms of the Beta1 Rhythm: Role of Intrinsic Currents

We note that the rhythmic activity of the model IB cells differ during the high kainate and low kainate drive conditions. During high kainate conditions, antidromic activity generates the beta2 rhythm. The M-current in the axons sets the period of the IB cell bursting [18], and the dendritic activity interferes with this rhythm. During low kainate drive conditions, following potentiation of the NMDA synapses, orthodromic activity generates the beta1 rhythm. The h-current contributes to the dendritic depolarization following inhibitory input from the superficial layer and induces the axons to fire, thus continuing the beta1 oscillation.

The intrinsic currents of, and synaptic inputs to, the IB cell dendrites play an important role in the beta1 activity. We illustrate the dynamics of these currents and inputs within the dendritic compartments of a single IB cell in Figure 9. During a burst of activity in the IB cell axons (e.g., a strip of red dots in Figure 8B) the slowly-decaying NMDA current (red in Figure 9) activates and depolarizes the basal dendrite (near t = 25 ms in Figure 9). Approximately 30 ms later, inhibitory synaptic input from a superficial LTS interneuron hyperpolarizes the apical dendrite (arrow, upper figure) and activates the apical dendritic h-current (blue). Both the h-current and NMDA current continue to depolarize the IB cell until a fast sodium current quickly activates (gray) and the neuron fires a burst of spikes near t = 100 ms. During spiking, dendritic calcium and potassium currents activate (not shown), the h-current and NMDA current reset, and the beta1 cycle restarts. We note that the slowly decaying NMDA input depolarizes the basal dendrites, and that the population of dendrites enters a more active state than during the high kainate conditions (compare the dendritic activity shown in Figures 4 and 8). The synchronous bursts of activity strengthen the postsynaptic effect of the IB cells and effectively strengthen the ascending synapses from the deep to superficial layer.

Figure 9. Voltage of the apical dendrite (upper) and intrinsic or synaptic currents (lower) in the basal and apical dendrites of an IB cell during beta1.

In the lower figure we show the h-current (blue) and the fast sodium current (gray) of the apical dendrite, and the NMDA synaptic current (red) to the basal dendrite. Negative values indicate inward currents. Inhibitory input from an LTS interneuron arrives at the arrow (upper panel) and activates the h-current of the apical dendrite (lower panel) but has little effect on the NMDA current of the basal dendrite.

doi:10.1371/journal.pcbi.1000169.g009

To illustrate the interplay of the M-current and h-current during beta1, we plot in Figure 10 examples of the apical dendritic voltage (yellow) and axonal voltage (orange) of an IB cell, and the gating variables for the h-current (dotted curve) and M-current (dashed curve). When the apical dendrite receives an IPSP (from a superficial LTS interneuron), the h-current gating variable opens and depolarizes the dendrite, thus promoting spiking. When the IB cell generates an action potential, this gating variable closes and removes this depolarizing influence. The M-current in the axon behaves in an opposite way. When the axon bursts, this gating variable opens and hyperpolarizes the axon, thus preventing further spiking. The axon may spike again only after the M-current decays sufficiently. This example illustrates the complementary effects of the M-current and h-current in the IB cell model. The M-current acts to prevent spiking in the axon, while the h-current acts to promote spiking in the dendrite.

Figure 10. Interplay of the M-current and h-current during beta1.

Traces of the voltage in the IB cell apical dendrite (yellow) and axon (orange), the h-current gating variable in the apical dendrite (dotted curve), and the M-current gating variable in the axon (dashed curve). Vertical scale arbitrary.

doi:10.1371/journal.pcbi.1000169.g010

In an ideal model of period concatenation, the mechanisms that generate the two fast rhythms would combine to produce the slow oscillation. This statement is nearly—but not quite—appropriate for the model proposed here. In the model of superficial layer activity, the inhibitory synapse from the basket cell to the RS cell paces the gamma rhythm. Increasing the decay time of this synapse slows the gamma rhythm. A population of these basket cell synapses participates in the beta1 rhythm, and if we increase the decay time of these synapses, then we also slow the beta1 oscillation. Thus, a fundamental mechanism pacing the fast gamma rhythm—namely the basket cell inhibitory synapses—also paces the beta1 rhythm. A more complicated relationship exists between the beta2 rhythm and its contribution to beta1. During beta2 activity, M-currents in the IB cell axons pace the deep layer rhythm. But, during beta1 these M-currents have less influence on the frequency of the deep layer activity. More important are the h-currents, excitatory synaptic inputs, and inhibitory synaptic inputs to the IB cell dendrites. The transition from beta2 to beta1 involves a switch in the IB cell from antidromic to orthodromic activity. Therefore the beta2 component of the beta1 rhythm is dominated by mechanisms in the IB cell dendrites, not the IB cell axon. Thus, in the model, the concatenation depends on having two mechanisms (an M-current and an h-current combined with excitatory and inhibitory synaptic inputs) with the same time scale. In the model, the same cells generate the coexistent fast rhythms and combined slow oscillation, although the biophysical mechanisms important to each rhythm change in the deep layer IB cells. Simpler period concatenation models may be developed, but we do not believe that these models would agree in all ways with the in vitro data.

In what follows, we compare the model with an additional set of experimental manipulations. We show that the two are consistent and verify a model prediction of the fundamental role for the h-current in vitro. We also use the model to suggest specific mechanisms important to the beta1 activity and its propagation between cortical layers. We begin with the observation that:

Separating cortical layers destroys beta1 activity.

In vitro separation of the deep and superficial cortical layers abolished the beta1 rhythm [17]. We show these results here in Figure 11A, where we plot example LFP recordings and power spectra from the slice preparation. A lesion through layer IV in vitro revealed a statistically significant decrease in the beta1 power (n = 5 observations, p<0.05).

Figure 11. Results from in vitro slice recordings during low kainate drive conditions.

In each case, we sh