Conceived and designed the experiments: MAW. Performed the experiments: AKR LMC. Analyzed the data: AKR. Wrote the paper: MAK NJK. Performed the computational modeling: MAK. Developed the computational model: RDT. Conceived the mathematical modeling: NJK.
The authors have declared that no competing interests exist.
Rhythmic voltage oscillations resulting from the summed activity of neuronal populations occur in many nervous systems. Contemporary observations suggest that coexistent oscillations interact and, in time, may switch in dominance. We recently reported an example of these interactions recorded from in vitro preparations of rat somatosensory cortex. We found that following an initial interval of coexistent gamma (∼25 ms period) and beta2 (∼40 ms period) rhythms in the superficial and deep cortical layers, respectively, a transition to a synchronous beta1 (∼65 ms period) rhythm in all cortical layers occurred. We proposed that the switch to beta1 activity resulted from the novel mechanism of period concatenation of the faster rhythms: gamma period (25 ms)+beta2 period (40 ms) = beta1 period (65 ms). In this article, we investigate in greater detail the fundamental mechanisms of the beta1 rhythm. To do so we describe additional in vitro experiments that constrain a biologically realistic, yet simplified, computational model of the activity. We use the model to suggest that the dynamic building blocks (or motifs) of the gamma and beta2 rhythms combine to produce a beta1 oscillation that exhibits cross-frequency interactions. Through the combined approach of in vitro experiments and mathematical modeling we isolate the specific components that promote or destroy each rhythm. We propose that mechanisms vital to establishing the beta1 oscillation include strengthened connections between a population of deep layer intrinsically bursting cells and a transition from antidromic to orthodromic spike generation in these cells. We conclude that neural activity in the superficial and deep cortical layers may temporally combine to generate a slower oscillation.
Since the late 19th century, rhythmic electrical activity has been observed in the mammalian brain. Although subject to intense scrutiny, only a handful of these rhythms are understood in terms of the biophysical elements that produce the oscillations. Even less understood are the mechanisms that underlie interactions between rhythms; how do rhythms of different frequencies coexist and affect one another in the dynamic environment of the brain? In this article, we consider a recent proposal for a novel mechanism of cortical rhythm generation: period concatenation, in which the periods of faster rhythms sum to produce a slower oscillation. To model this phenomenon, we implement simple—yet biophysical—computational models of the individual neurons that produce the brain's voltage activity. We utilize established models for the faster rhythms, and unite these in a particular way to generate a slower oscillation. Through the combined approach of experimental recordings (from thin sections of rat cortex) and mathematical modeling, we identify the cell types, synaptic connections, and ionic currents involved in rhythm generation through period concatenation. In this way the brain may generate new activity through the combination of preexisting elements.
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
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
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
Here we consider how the elementary dynamic building blocks (or dynamic motifs
The initial interval of coexistent gamma and beta2 rhythms preceded the transition to a slower beta1 oscillation (15±2 Hz,
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.
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
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
(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.
(A) The model of a single IB cell consists of four compartments: an apical dendrite (
During the high kainate condition in vitro, coexistent gamma and beta2 rhythms occur in the superficial and deep cortical layers, respectively
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
We model the dendrites of an individual deep layer IB cell with two compartments: a basal dendrite and an apical dendrite (
We show a cartoon representation of the entire reduced model in
(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:
We plot in
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.
Analysis of in vitro slice preparations revealed a statistically significant increase in the gamma power of the superficial layers (215±29%,
(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.
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
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.
Inhibition plays a vital role in pacing the gamma and beta1 rhythms
(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.
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
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.
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
(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.
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.
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
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
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
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
(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).
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 (
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
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;
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
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
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
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.
To illustrate the interplay of the M-current and h-current during beta1, we plot in
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.
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:
In vitro separation of the deep and superficial cortical layers abolished the beta1 rhythm
In each case, we show the power spectra (in units of V2/Hz) computed during control conditions and: (A) after a lesion through layer IV, (B) after ISPSs blocked, (C) after NMDA blocked, and (D) after h-current blocked. To the right of each power spectrum we plot 500 ms of recorded data during the control (black) and manipulated (gray) conditions; the vertical black lines indicate 50 µV. In each case, the experimental manipulation reduces the beta1 power.
In the model, beta1 activity involves interactions between the deep and superficial layers. We therefore expect that removing interlaminar connections should destroy the rhythm. To test this we mimic a lesion through layer IV in the model by eliminating the ascending excitatory input from the IB cells to the superficial inhibitory cells, and disconnecting the apical dendritic compartments of the IB cells. We find that this separation does indeed destroy the beta1 rhythm in agreement with the in vitro results. After this change we find no activity in either cortical layer of the model. (Here, and in what follows, we do not show the simulation results; the manipulations eliminate or greatly reduce the beta1 activity in the model, and the associated figures—of limited spiking activity or flat power spectra—are not informative.) The beta1 activity stops for one primary reason in the model: the rhythm cannot propagate between the cortical layers without interlaminar connections.
We also use the model to examine the effects of individual interlaminar interactions during the beta1 rhythm. We find that eliminating the ascending synapses from the IB cells to the superficial inhibitory neurons reduces the beta1 activity, and that disconnecting the IB cell apical dendrites destroys the beta1 rhythm. We note that these apical dendrites connect the superficial to deep layer through the inhibitory LTS synapses, which act to depolarize the apical IB dendrites (through activation of an h-current). Without these sources of excitation to the IB cells, the beta1 rhythm cannot continue.
The IPSPs serve a fundamental role in generating the beta1 rhythm
In the model the beta1 rhythm propagates from the superficial to deep cortical layer through inhibitory synapses (from the LTS interneurons to the IB cell apical dendrites). We therefore predict that blocking IPSPs should eliminate the descending inputs and disrupt the beta1 rhythm. We test this prediction in the model by setting the conductance of all inhibitory synapses to zero, as we indicated in
We find that, in the model, removing only the basket cell synapses reduces the beta1 activity, and that removing only the LTS interneuron synapses destroys the beta1 activity. In the former case, loss of the basket cell synapses removes the mechanism pacing the beta1 rhythm in the superficial layer. Without this mechanism, cells in both layers continue to spike, but in a disorganized manner, therefore reducing the beta1 power. In the latter case, the LTS interneurons do not form inhibitory synapses on the IB cell apical dendrites and the dendritic h-currents—necessary to depolarize the IB cell dendrites—remain inactive. The IB cells no longer burst and, without their excitatory postsynaptic effects, the entire network remains inactive.
In vitro the transition to beta1 required an initial interval of coexistent gamma and beta2 activity
In the model we proposed that strengthening NMDA receptor-mediated synapses between the IB cell population supported the beta1 rhythm. It did so by providing a slow depolarization—and increased excitability—to IB cell basal dendrites. To test the role of this current in the model we remove these excitatory synapses from the IB cell axons to the IB cell basal dendrites. The effect of this parameter change is to hyperpolarize the IB cell basal dendrites and dramatically reduce the spiking activity in the IB cell population. This hyperpolarization and reduced connectivity also increases the interburst interval of the IB cell axons and desynchronizes their output. The desynchronization results in weak synaptic input to the superficial layer. Both the superficial and deep layers continue to spike at a slower rate and in a more disorganized way (simulations not illustrated), eliminating the beta1 power in both layers, in agreement with the in vitro results.
We proposed above that the h-current plays a vital role in maintaining the beta1 rhythm; the LTS interneurons inhibit the IB cell apical dendrites and activate this depolarizing current to continue the beta1 activity. We therefore predict that, in contrast to the potentiating effect of blocking the h-current on the beta2 rhythm (
We find the same result in the model if we block the h-current in the RS cells alone, in the LTS interneurons alone, in IB cell basal dendrites alone, or in the IB cell apical dendrites alone. We note that the h-current block results in a hyperpolarization—and therefore inactivation—of the affected cells or compartments. We conclude that inactivation of either the IB cell dendrites, RS cells, or LTS interneurons eliminates the beta1 rhythm. The former two populations provide the excitatory drive necessary to continue the beta1 rhythm through their postsynaptic effects. The latter population depolarizes the IB cell apical dendrites through activation of an h-current.
Analysis of the in vitro data allowed us to constrain the computational model in many ways, but not completely. In the model of beta1 activity described above, we considered a strengthening of NMDA synapses between the population of deep layer IB cells. As a second model for the change in network connectivity that supports the beta1 rhythm, we consider NMDA synapses that
The essential ideas of the previous model can be captured in a simpler model, but at the expense of becoming more abstract. To construct a simpler model, we do not replicate the cells and dynamic motifs to create neural populations. Instead, we employ single copies of the gamma-motif, the beta2 -motif (a single IB cell), and the LTS interneuron. In addition, we represent the IB cell dendrite as a single compartment (and do not distinguish between the apical and basal dendrites.) In each cell and compartment we implement the same currents and synaptic connections utilized in the more detailed model (e.g., an M-current in the IB cell axon and ascending excitatory synaptic connections from the IB cell to the superficial inhibitory cells.) The goal of creating such a simple model is to understand the fundamental mechanisms that support the beta1 oscillation.
We show a cartoon representation of the simple model in
(A) The deep layer consists of a single IB cell with three compartments and a Poisson source of IPSPs. The superficial layer consists of three cells: RS, the RS cell; B, the basket cell; LTS, the LTS interneuron. (B) To represent the low kainate drive conditions, we hyperpolarize the cells and compartments colored blue. To represent the strengthening of NMDA synapses, we depolarize the dendritic compartment of the IB cell (yellow) and increase the strength of the ascending synapses (drawn with thickened black lines).
We have constructed a computational model, consisting of four cell types, and compared it with in vitro observations from rat somatosensory cortex
In this manuscript, we showed that a biologically realistic, yet simplified, computational model could reproduce the coexistent gamma and beta2 rhythms, the transition to beta1, and numerous experimental manipulations. Moreover, we used the model to suggest the specific mechanisms important for the generation and modification of each rhythm. In the superficial layer, reciprocal synaptic connections between an RS and basket cell created the inhibition-based gamma rhythm; the decay time of the basket cell inhibitory synapse determined the period of this oscillation. In the deep layer, the M-current in the IB cell axons paced the beta2 rhythm. We proposed in the model that these dynamic motifs combined to create the slower beta1 oscillation. During beta1 activity, the LTS interneurons (initially disruptive to the coexistent gamma and beta2 oscillations) served a vital role, and the rhythm propagated between the cortical layers. Thus, unlike the coexistent gamma and beta2 rhythms, the beta1 rhythm represented a state in which activity in both deep and superficial layers of neocortex temporally combined to produce oscillations in which interlaminar interactions were vital.
We considered four manipulations of the model and in vitro preparation during beta1 activity. The outcome of each manipulation—separating the cortical layers, blocking the h-current, blocking IPSPs, or blocking the NMDA synapses between IB cells—was the same: elimination of the beta1 rhythm. We noted that the loss of beta1 activity occurred when the rhythm could not propagate between the cortical layers; thus, connections between layers were essential to the beta1 rhythm. We also noted that the beta1 rhythm is not quite a concatenation of the gamma and beta2 rhythms. Instead, we found that two different mechanisms support the beta2 and beta1 oscillations. In the high kainate condition, the beta2 rhythm resulted from antidromic activity in the IB cell axons. In the subsequent low kainate drive condition, the beta1 rhythm resulted from orthodromic activity in the IB cell dendrites.
We noted that the initial interval of coexistent gamma and beta2 activity must precede the beta1 rhythm. If we start the in vitro slice preparation in the low kainate drive condition (i.e., with kainate+NBQX) we find no oscillatory activity. Therefore, the initial interval of fast rhythms must change the network to support the slower beta1 oscillation. In the model, we assumed that the coexistent fast rhythms facilitated a potentiation of NMDA synapses between the population of IB cells
In the population models proposed in this work, we considered the activity generated in a single cortical column of somatosensory cortex. An improved model would describe the activity of multiple, interacting cortical columns. Including interactions between columns would permit synaptic connections not present in the single-column model (for example, excitatory synapses from deep layer IB cells to superficial RS cells observed in rat motor cortex
What computational roles might the separate gamma and beta2 rhythms, and the transition to a slower beta1 oscillation, serve? In the in vitro observations and computational models considered here, the initial faster rhythms coexisted in different cortical layers. The transition to the slower beta1 oscillation established a common rhythm that propagated between both layers. We might therefore interpret the transition to the beta1 oscillation as binding the superficial and deep layers of a cortical column. This binding organizes subnetworks of neurons within the input (superficial) and output (deep) layers of a cortical column
Recent observations suggest that rhythms within distinct frequency intervals interact, and that these interactions may occur in different ways
Slices of parietal neocortex (450 µm thick), were prepared from adult male Wistar rats and maintained in an interface chamber. Details of artificial cerebrospinal fluid composition and recording techniques are in
The model consists of four cell types in two cortical layers. We describe the cells, synapses, and analysis methods in this section (detailed equations may be found in the
A simple model of the superficial layer gamma rhythm consists of two neurons—one excitatory and one inhibitory—interacting through reciprocal synapses (the so-called Pyramidal-Interneuron-Network-Gamma or PING rhythm
We connect the RS and basket cells with reciprocal synapses. The equations governing the synaptic dynamics are similar to those described in
We did not include fast rhythmic bursting (FRB) neurons in the reduced gamma model. Recent experimental and modeling results suggest that these cells provide excitation via axonal plexus activity to drive neocortical gamma rhythms
The final cell type we include in the superficial layer is a low threshold spiking (LTS) interneuron
To establish a population model of the superficial layer gamma activity, we create twenty replications of the RS-basket-LTS cell circuit described above and shown in
To model the deep layer beta2 rhythm, we implement a reduction of the LV tufted intrinsically bursting (IB) pyramidal cell
We connect the dendritic and axonal compartments to the soma through electrotonic coupling. The coupling conductances between the axon and soma were identical for both compartments. We set the coupling conductance from the dendrites to soma to exceed the coupling conductance from the soma to dendrites (so that the soma voltage has a weaker effect on the dendritic compartment's dynamics
To establish a population model of the deep layer activity we create twenty replications of the IB cell. Each member of the IB cell population consists of the same (four) compartments and currents described above. We make all parameters the same in each cell, except for the depolarizing currents to each compartment. We vary these inputs from cell to cell to establish heterogeneous bursting activity in the cell population; the interburst frequency of the twenty IB cells ranges from 20 Hz to 30 Hz. In the model, consisting of only twenty IB cells, we cannot represent a sparsely connected axon plexus which involves a large number of cells
From the superficial to deep layer we include a synapse from a single LTS interneuron to a unique apical dendrite of an IB cell (we note that the axons of superficial LTS interneurons extend up to layer I—a lamina rich in LV pyramidal cell apical dendrites
We use the Interactive Data Language (IDL) to compute numerical solutions to the model equations. We implement a second-order difference method with a time step of 0.01 ms, and follow the Euler-Maruyama algorithm to include stochastic inputs. Readers may obtain the simulation code by contacting the authors.
To analyze the simulation results we compute two measures: the power spectrum and the cross-correlation. We briefly describe each measure here (detailed discussions of these measure may be found in the literature, for example
To compute the cross-correlation between the spike times of the RS cells and IB cell axons, we first simulate the model to create 1 s of data. We then locate the spike times and create new “binary” time series for each (of the twenty) RS cell and IB cell axon. We make the binary times series 0 everywhere except at the spike times where we set the time series to 1. We then compute the cross-correlation between the binary time series of each IB cell axon and the corresponding RS cell. By corresponding, we mean the RS cell in the unique triad whose LTS interneuron forms a synapse on a particular IB cell dendrite. We compute the cross-correlation in this way to avoid the effects of correlations in the subthreshold membrane potentials (e.g., correlations in hyperpolarizations of two cells.) We compute these cross-correlations for all twenty pairs of IB and RS cells, and average the results over the twenty cell pairs.
Mathematical equations.
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The authors would like to thank three anonymous reviewers for their careful consideration of this manuscript. MAK thanks the Scientific Computing and Visualization group at BU for providing computational resources. AKR and MAW thank S. Baker for discussion of data.