Slow State Transitions of Sustained Neural Oscillations by Activity-Dependent Modulation of Intrinsic Excitability

Transition between slow bursting and tonic firing in a single cell

A brief increase in [K⁺]_o was used to initiate activity in an otherwise silent PY neuron. An isolated PY cell model (G_Ca = 0.015 mS/cm², G_h = 0.1 mS/cm²) with a spatially limited extracellular compartment responded to a brief [K⁺]_o elevation with oscillatory firing before returning to rest. Although [K⁺]_o decreased to its resting value (3.5 mm), the PY cell exhibited several oscillatory modes: first, slow-bursting oscillations and then tonic firing (Fig. 1A, transition in B). For significantly elevated [K⁺]_o >5.6 mm, bursts with spike inactivation, pronounced after-hyperpolarization, and strong calcium influx occurred (Fig. 1C, left, phase space plot of a single burst). Calcium influx during the depolarized state caused the calcium-activated potassium conductance to activate, which in turn mediated burst termination and subsequent after-hyperpolarization (supplemental Fig. 2, available at www.jneurosci.org as supplemental material). For [K⁺]_o = 5.6 mm, however, there was a transition to bursts with reduced after-hyperpolarization and with minor spike inactivation (Fig. 1C, middle). For [K⁺]_o = 5.5 mm, the cell switched to tonic firing (Fig. 1C, right, spike doublets) before it eventually returned to rest. Both during slow bursting and tonic firing, potassium efflux occurred mainly through the fast delayed rectifier (41 and 43% for bursting and tonic firing, respectively) and through the leak conductance (43 and 56%, respectively). These oscillatory firing patterns for elevated [K⁺]_o are not a mere consequence of the depolarization of the neuron because an injected depolarizing current step caused only tonic firing with spike adaptation (supplemental Fig. 1, available at www.jneurosci.org as supplemental material).

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Figure 1.

A, A single PY neuron exhibits bursting and then tonic firing after a brief increase in extracellular potassium concentration ([K⁺]_o) before returning to rest. B, Membrane voltage (top) and [K⁺]_o (bottom) time courses for the time interval underlined in A. The transition from bursting to tonic firing (spike doublets) at [K⁺]_o = 5.6 mm is shown. C, Phase space plots ([K⁺]_o, intracellular calcium [Ca²⁺]_i, and membrane voltage V_m) for the three intervals underlined in A. The arrow indicates the direction of time. A single burst with minor spike inactivation and after-hyperpolarization (left), a single burst without spike inactivation and reduced after-hyperpolarization (middle), and a spike doublet (right) are shown.

Because [K⁺]_o changed on a very slow time-scale, we next treated [K⁺]_o as a parameter to determine the firing behavior of the cell as a function of [K⁺]_o. Therefore, all of the mechanisms controlling [K⁺]_o evolution in our model were blocked and the neuron behavior was analyzed for different fixed values of [K⁺]_o within the physiological range. Specifically, we focused on the occurrence of different stable firing modes as a function of parameter [K⁺]_o (Fig. 2). For low [K⁺]_o, the neuron was at rest (Fig. 2A, left). For increasing [K⁺]_o, the resting potential became more depolarized as the driving force for potassium decreased. For [K⁺]_o = 4.85 mm, the neuron switched to tonic firing by means of a saddle-node bifurcation [type I neural oscillator (Rinzel and Ermentrout, 1989; Ermentrout, 1996)] (Fig. 2C, fixed point bifurcations). Tonic firing and slow bursting coexisted for [K⁺]_o between 5.45 and 6.35 mm. Slow bursting was the only stable state for [K⁺]_o between 6.35 and 9.45 mm. At [K⁺]_o = 9.46 mm, a new stable state corresponding to a depolarized state (V_m = −26.3 mV) appeared by means of a subcritical Hopf bifurcation (Fig. 2C). The depolarized state coexisted with the slow bursting regimen in a narrow bistable region before it became the only stable state at [K⁺]_o = 10.05 mm (Fig. 2A, right).

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Figure 2.

Bifurcation analysis of PY cell dynamics. A, Maximum and minimum of membrane voltage as a function of [K⁺]_o. Solid lines, Stable fixed points. Circles, Stable limit cycles. Four stable states are found: rest, tonic firing, slow bursting, and a depolarized state, with bistability occurring between tonic firing and slow bursting and between slow bursting and the depolarized state (G_h = 0.0 m S/cm²). B, Poincare cross section. Intracellular calcium concentration for V_m = −20 mV as a function of slowly increasing (top) and decreasing (bottom) [K⁺]_o is shown. Inset, Enlargement of transition from tonic firing to slow bursting (G_h = 0.05 m S/cm²). C, Stable fixed points corresponding to rest and the depolarized state are connected via unstable fixed points. Transition from silent to tonic firing is a saddle-node bifurcation. Transition from slow bursting to depolarized is a Hopf bifurcation. Insets illustrate eigenvalues at the bifurcation point. D, Three-dimensional representation of inset from B showing transition from tonic firing to frequency-modulated fast firing. Neimark–Sacker bifurcation of limit cycle leads to slow spike frequency modulation for [K⁺]_o > 6.33 mm. Color scheme, From blue to red for low to high values of [K⁺]_o.

To further characterize the transition between tonic firing and slow bursting, we computed a Poincare cross section by calculating the values of intracellular calcium concentration [Ca²⁺]_i each time the membrane potential crossed the threshold V_m = −20 mV; these values were plotted as a function of [K⁺]_o (Fig. 2B, Poincare cross section). In such a representation, periodic oscillations (corresponding to limit cycles) are represented as points defined by a threshold crossing of a trajectory. This approach allows the graphical representation of changes in the nature of an oscillatory (firing) behavior as a function of a parameter. For a given value of [K⁺]_o, a tonic spiking regimen is represented on this Poincare plot by a single point because [Ca²⁺]_i assumes the same value at V_m = −20 mV for every spike. During a burst, however, [Ca²⁺]_i increases after each spike of a given burst. Therefore, a burst appears as a group of points each representing a single spike. In other words, a set of parallel lines in the Poincare plot (Fig. 2B, Poincare cross section) illustrates a range of [K⁺]_o values for which bursting occurred. We gradually increased and decreased [K⁺]_o to reveal the complete tonic firing and slow bursting region, respectively. The bistability between tonic firing and slow bursting was associated with a hysteresis (Fig. 2B, compare top and bottom plots). For increasing [K⁺]_o, the cell stayed in tonic firing until slow bursting with spike inactivation became the only stable state at [K⁺]_o = 6.40 mm. Decreasing [K⁺]_o caused the cell to stay in slow bursting mode until tonic firing was the only stable state at [K⁺]_o = 5.75 mm (different from Fig. 2A because now non-zero G_h = 0.05 mS/cm²). For increasing [K⁺]_o, the tonic firing region consisted of three subregions with single spikes, spike doublets, and single spikes, respectively (Fig. 2B, top). Time courses for spike doublets ([K⁺]_o = 5.00 mm) and single spikes [K⁺]_o = 6.00 mm) are shown in Figure 3, A and B, respectively. Note that the left region with single spikes existed only in the case of a nonzero h conductance (G_h = 0.05 mS/cm²). At [K⁺]_o = 6.39 mm, tonic firing started to exhibit slow modulation of the membrane voltage and the spiking frequency (Fig. 3C,D, spectrogram). Detailed bifurcation analysis revealed that at [K⁺]_o = 6.39 mm, the limit cycle corresponding to the tonic spiking lost its stability by a supercritical Neimark–Sacker bifurcation (Kuznetsov, 2004), leading to a stable invariant torus in the phase space representation of the dynamical system (Fig. 2D). This type of behavior was not found for decreasing [K⁺]_o (Fig. 2B, bottom). Instead, two distinct bursting regimens, with and without spike inactivation, were found (Fig. 3E,F). For these bursts, transition from silent state to spiking (burst onset) occurred through a saddle-node bifurcation of the fixed point corresponding to the silent (hyperpolarized) state. Return to the silent state (burst offset) occurred through a Hopf bifurcation of the fixed point corresponding to the depolarized (upper) state. Activation of I_K(Ca) (slow variable) controlled the transitions between silent and oscillatory states during bursts (data not shown).

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Figure 3.

Time course of membrane voltage (1 s duration) for different values of [K⁺]_o. A, Spike doublets for [K⁺]_o = 5.00 mm. B, Fast tonic firing for [K⁺]_o = 6.00 mm (tonic firing branch, [K⁺]_o increasing). C, Fast tonic firing modulated by slow oscillation corresponding to Neimark–Sacker bifurcation of limit cycle and consecutive torus bifurcation with period doubling for [K⁺]_o = 6.39 mm (tonic firing branch, [K⁺]_o increasing). D, Spectrogram of C; normalized arbitrary units. E, Bursting with spike inactivation for [K⁺]_o = 6.00 mm (slow bursting branch, [K⁺]_o decreasing). F, Bursting without spike inactivation for [K⁺]_o = 5.75 mm (slow bursting branch, [K⁺]_o decreasing).

The biophysical mechanism for the bistability with hysteresis between tonic firing and slow bursting was examined by (1) comparing the ionic currents in both regimens for [K⁺]_o = 6.00 mm (supplemental Fig. 2, available at www.jneurosci.org as supplemental material) and (2) systematically varying the intrinsic conductances involved in burst generation (supplemental Fig. 3, available at www.jneurosci.org as supplemental material). We found that the different levels of deinactivation of the high-threshold calcium current I_Ca explain the hysteresis between tonic firing and bursting. I_Ca reinforced bursting and therefore increased the range of [K⁺]_o values for which the neuron stayed in bursting mode. In contrast, when the neuron was in tonic firing mode, insufficient I_Ca deinactivation between spikes prevented switching to the bursting mode unless [K⁺]_o was strongly elevated (supplemental Fig. 2, bottom panels, available at www.jneurosci.org as supplemental material). To confirm the role of I_Ca and to test the robustness of our model, we separately varied the three intrinsic conductances G_Ca, G_NaP, and G_KCa, involved in initiating, sustaining, and terminating bursts, to study their impact on the bistability with hysteresis between tonic firing and bursting. Further, the effect of introducing a hyperpolarization-activated depolarizing conductance G_h was examined. As anticipated, the width of the hysteresis between tonic firing and bursting varied with G_Ca, whereas changes in the other conductances had minimal impact on the qualitative nature of the hysteresis (supplemental Fig. 3, available at www.jneurosci.org as supplemental material). These findings indicate the pivotal role of the high-threshold calcium conductance G_Ca in mediating bistability between the two oscillatory firing regimens.

Slow transitions between oscillatory states in small network

To determine whether the hysteresis between tonic firing and slow bursting was maintained in a network of neurons, we studied a network consisting of five PY cells with all-to-all excitatory coupling and one IN, which mediated global inhibition. When [K⁺]_o was held at various fixed values, four different states equivalent to those in the single cell model were found: silence, fast run, slow bursting, and a depolarized state (Fig. 4A, left, transition from fast run to slow bursting). Fast run (fast bursting with 2–4 spikes) corresponded to tonic firing found in the single cell model. Each of the three transitions between neighboring states was associated with a hysteresis, confirming that the hystereses found in the single cell model were indeed maintained in the small network.

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Figure 4.

Globally connected network with five PY cells and one IN. A, Left, Bifurcation analysis (minimum of membrane voltage): hysteresis between fast run and slow bursting for [K⁺]_o between 5.0 and 5.4 mm. Circles denote stable limit cycle oscillations. Top right, Schematic of slow periodic network dynamics. Bottom right, Averaged [K⁺]_o change rate as a function of [K⁺]_o. B–D, Slow transitions after initial brief increase in [K⁺]_o. B, Network activity of PY cells (40 s duration) shows alternating epochs of fast run and slow bursting. C, [K⁺]_o increased during fast run and decreased during slow bursting. The upper switching point for transition from fast run to slow bursting and the lower switching point for transition from slow bursting to fast run correspond to hysteresis endpoints in A. D, Membrane voltage time course of PY cell (top trace) and of IN (bottom trace).

When [K⁺]_o was released and continuously updated based on K⁺ currents, K⁺ pumps, and glial buffering (Eq. 5), a major difference between the single-cell model and the network model was found. For a single PY cell surrounded by an extracellular compartment, [K⁺]_o decreased steadily during both tonic spiking and bursting. Thus, the hysteresis between tonic firing and slow bursting had no effect on neural dynamics triggered by elevated [K⁺]_o because the cell never switched back from tonic firing to bursting. In the small network, however, the cells did not return to rest after the initial [K⁺]_o injection but instead displayed self-sustained oscillations with periodic alternations between fast run and slow bursting (Fig. 4B, activity of all five PY cells as a function of time for 40 s). [K⁺]_o increased during fast run, which caused a transition of the network to the slow bursting regimen as predicted by the upper endpoint of the hysteresis found by bifurcation analysis (Fig. 4A, arrow 1, top right). If there were no hysteresis, this would lead to an immediate return to fast run because [K⁺]_o decreased during slow bursting. Because of the hysteresis, however, slow-bursting mode was maintained until the lower transition point for [K⁺]_o back to fast run was reached, where the cycle restarted (Fig. 4A, arrow 2, top right). By averaging over many periods during fast run and a single period during slow bursting, we determined the time derivative of extracellular potassium concentration as a function of [K⁺]_o if it were free to change. This further illustrates the fact that [K⁺]_o increases during fast run and decreases during slow bursting (Fig. 4A, bottom right, dashed line represents d[K⁺]_o/dt = 0 “nullcline”). The dependence of the lower and upper transition points on whether [K⁺]_o was increasing or decreasing matched the prediction from bifurcation analysis (Fig. 4A, end points of hysteresis). Thus, synaptic interaction stabilized the network in the hysteresis region between fast run and slow bursting ([K⁺]_o between 5.0 and 5. 4 mm). Synchrony is not required for this mechanism to work as long as the time derivatives of potassium follow this general rule. Rather, overall excitation mediated by network interaction needs to provide sufficient excitatory drive for firing frequencies, which enable potassium to increase during tonic firing (fast run) without causing a switch to the bursting regimen. Note that a single PY cell with a self-excitatory synapse did not exhibit these transitions (data not shown); it displayed either bursting (for strong coupling) or tonic firing (for weak coupling). Because the inhibitory interneuron was active during epochs of fast run, the spike structure of the fast run was different between the single cell and small network models. PY neurons from the network displayed fast oscillations with groups of spikes separated by short hyperpolarizing events mediated by IPSPs. When excitatory input to the inhibitory interneurons was reduced such that they became silent during fast run, PY neurons displayed tonic spiking activity similar to the single-cell model (supplemental Fig. 4, available at www.jneurosci.org as supplemental material).

Single-cell model analysis revealed that high-threshold calcium current I_Ca inactivation was critical for determining the oscillatory mode (tonic spiking or bursting). In an additional set of simulations, we extended this finding to the network model by perturbing inactivation variable h of I_Ca for all neurons in the network to induce switching between the two oscillatory states (supplemental Fig. 5, available at www.jneurosci.org as supplemental material). The network switched from fast run to slow bursting when enforcing a value of h = 0.53 (mean value between bursts) for 0.25 s. Similarly, imposing a value of h = 0.36 (mean value during fast run) for 1 s during slow bursting caused the network to switch to fast run.

Role of synaptic conductances

Network interaction was critical for the occurrence of alternating epochs of fast run and slow bursting. Therefore, we systematically varied the synaptic strength to uncover its specific effect on the network dynamics. Increasing excitation (PY–PY coupling) without changing inhibition (IN–PY = 1.1) widened the [K⁺]_o hysteresis (from 0.24 to 0.62 mm for PY–PY = 0.8 and 1.2, respectively) (Fig. 5A, left, dashed line) and prolonged the episodes of slow bursting at the expense of fast run (solid lines). The increase in hysteresis width was mainly caused by a lower value of [K⁺]_o for which the network switched from slow bursting to fast run (from 5.38 to 4.77 mm for PY–PY = 0.8 and 1.2, respectively) (Fig. 5A, right). In contrast, increasing inhibition (IN–PY coupling) without changing excitation (PY–PY = 0.9) shortened epochs of slow bursting and prolonged epochs of fast run, respectively (Fig. 5B, left, solid lines). The [K⁺]_o hysteresis was narrowed (from 0.75 to 0.28 mm for IN–PY = 0.8 and 1.2, respectively) (Fig. 5B, left, dashed line), mostly by an increase in the range of [K⁺]_o for which fast run was the only stable state (from 4.86 to 5.36 mm for IN–PY = 0.8 and 1.2, respectively). Thus, during fast run, [K⁺]_o increased more slowly because of the reduced firing frequency. Additionally, increased inhibition forced the network back to fast run for higher [K⁺]_o (Fig. 5B, right). Examples of nonalternating regimens resulted from major changes in the balance between excitation and inhibition (Fig. 5C, right, areas in dark blue). For strong excitation and weak inhibition, the network returned to the silent state after a single epoch of bursting. Conversely, strong inhibition and weak excitation led to sustained fast run (supplemental Fig. 6, time courses, available at www.jneurosci.org as supplemental material). Combinatorial variation of both excitatory (PY–PY) and inhibitory (IN–PY) synaptic coupling strength showed that the existence of the transitions between fast run and slow bursting is robust to changes in synaptic connectivity as long as the balance between excitatory and inhibitory coupling was maintained. Both the logarithmic ratio of the duration of fast run and slow bursting episodes and the width of the hysteresis were only slightly affected by changes in synaptic connectivity strength as long as excitation and inhibition was scaled accordingly (Fig. 5C, diagonal structure).

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Figure 5.

Quantification of periodic slow transitions between slow bursting and fast run as a function of excitatory and inhibitory coupling. Balanced excitation and inhibition causes alternating epochs of slow bursting and fast run. Left, Duration of epochs of slow bursting and fast run (blue and green solid lines, respectively), width of hysteresis (dashed red line), and upper and lower endpoints of hysteresis (right) for changing excitatory coupling (A) (PY–PY) and inhibitory coupling (IN–PY) (B). Values determined from simulation of 400 s of activity are shown. Error bars indicate SEM. C, Overview plot showing logarithmic ratio of duration of fast run and slow bursting (left) and width of hysteresis (right) as a function of synaptic excitation and inhibition. The top left corner corresponds to the regimen with exclusive fast run, whereas the bottom right corner denotes the regimen with exclusive slow bursting.

Bursts differed in their characteristics depending on the balance between excitation and inhibition (Fig. 6). In the case of reduced excitation (PY–PY = 0.9) and inhibition (IN–PY = 0.8), bursts exhibited a gradual decline in spike amplitude, spike inactivation, and a pronounced after-hyperpolarization (Fig. 6A, first row). Excitatory AMPAergic input to PY cells built up during spiking at the onset of the burst. NMDA current was active until burst termination (Fig. 6A, second row). For increased inhibition (IN–PY = 1.2) (Fig. 6B), bursts showed decreased spiking frequency and lacked both spike inactivation and after-hyperpolarization. Excitatory synaptic currents mediated by AMPA and NMDA receptors were of the same order of magnitude and similar duration. Balancing increased inhibition (IN–PY = 1.2) by strengthening excitation (PY–PY = 1.2) increased both firing frequency within the burst and the total number of spikes per burst (Fig. 6C). Fast run spiking structure was less sensitive to moderate changes in the balance between excitation and inhibition, as long as the inhibitory interneuron remained active during periods of fast runs (supplemental Fig. 4, available at www.jneurosci.org as supplemental material).

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Figure 6.

Time course of burst and underlying synaptic currents for three different synaptic configurations. First row, Membrane voltage time course. Second row, Total synaptic current on PY cell. A, Weak excitation and weak inhibition (scaling factors: PY–PY = 0.9 and IN–PY = 0.8). B, Weak excitation and strong inhibition (scaling factors: PY–PY = 0.9 and IN–PY = 1.2). C, Strong excitation and strong inhibition (scaling factors: PY–PY = 1.2 and IN–PY = 1.2).

Slow transitions of oscillatory states in large network

In a larger network consisting of 60 PY cells and 15 INs with local synaptic connectivity, each PY neuron switched between fast run and slow bursting at a similar time scale, as in the case of the small, compact network (Fig. 7A, top: network activity as a function of time, bottom: single membrane voltage trace). In contrast to the small network, however, state transitions did not occur synchronously across the network. Rather, a given PY cell that reached the [K⁺]_o required for a state transition first switched its oscillatory firing mode and induced an according transition in the neighboring PY cells by synaptic coupling. Epochs of slow bursting or fast run therefore spread in neighboring cells. In other words, state transitions were induced according to the hysteresis in the leading neurons, which were the cells with the highest [K⁺]_o (Fig. 7B, top). Transitions in the neighboring cells were then promoted by the synaptic coupling. This resulted in a multitude of switching points for each neuron (Fig. 7B, bottom). The behavior of the neurons initiating network transitions was governed according to the bistability described before. Intracellular calcium [Ca²⁺]_i was highly elevated during paroxysmal activity (>0.004 mm), in comparison to the equilibrium value at the resting potential (0.0001 mm). [Ca²⁺]_i remained more or less constant during epochs of fast run and oscillated during bursting (Fig. 7C, top) because I_Ca was the major contributor to the burst depolarization. In any given PY neuron, peak value of [Ca²⁺]_i almost doubled during each burst (Fig. 7C, bottom). Including lateral K⁺ diffusion did not change the global structure of alternating epochs of fast run and slow bursting (data not shown).

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Figure 7.

Large network (60 PY cells and 15 INs) with local synaptic connectivity. A, PY cell activity as a function of time (top). The time course of V_m for PY 30 is shown (bottom, arrow in top). Cells switched between bursting and fast run as in the case of the small network. Because of the local synaptic connectivity, the activity pattern exhibited complex spatial structure. B, [K⁺]_o in extracellular compartments surrounding PY cells as a function of time (top). The time course of [K⁺]_o for PY 30 is shown (bottom, arrow in top). C, Intracellular calcium [Ca²⁺]_i in PY cells as a function of time. The time course of [Ca²⁺]_i for PY 30 is shown (bottom, arrow in top). Color scheme in B and C, From blue to red for low to high values of ionic concentration.

Slow transitions of oscillatory states in vivo

We recorded paroxysmal activity with similar patterns of sustained transitions between fast run and slow bursting from association areas 5 and 7 in anesthetized cats (Fig. 8). Cats under ketamine-xylazine anesthesia reveal a sleep-like slow oscillatory pattern with frequency 0.8–0.9 Hz (Steriade and Contreras, 1998). In 31 cats (n = 45), we found spontaneous paroxysmal discharges consisting of spike–wave (polyspike–wave) complexes (slow bursting, 1.5–3 Hz) and epochs of fast run (fast spiking, 8–15 Hz). In average, 2.4 (SD 1.4) episodes of fast run occurred during a seizure. Episodes of slow bursting and fast run lasted 37.2 s (SD 22.2) and 4.9 s (SD 5.7), respectively. The mechanism of these slow transitions is unknown. Our modeling work reproduces the qualitative features of these transitions between slow bursting and fast run and therefore suggests activity-dependent modulation of excitability (e.g., extracellular potassium dynamics) as a candidate mechanism.

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Figure 8.

Sample trace of electrographic seizure composed of alternating epochs of spike-wave complexes (slow bursting) and fast run. Top, Depth EEG and intracellular activity during normal slow oscillation and its transformation to paroxysmal activity composed of fast runs and spike-wave complexes. Thick gray lines indicate periods of fast runs. The other periods of the seizure are spike-wave complexes. Three expanded fragments indicated by horizontal bars and arrows show (from left to right) a period of slow oscillation, a paroxysmal fast run, and a period of spike-wave discharges (slow bursting).

Bistability and hysteresis

In our neocortical network models, we found a bistability with hysteresis between tonic firing and slow bursting under conditions of balanced excitation and inhibition. Thus, the hysteresis causes a sustained pattern of alternating episodes of fast and slow oscillations in the absence of external input. Network connectivity was critical to maintain oscillations. Although the hysteresis between several oscillatory states was found already in a single cell model, those states appeared only transiently after initial stimulation. Balanced excitatory and inhibitory connections between pyramidal cells and interneurons were essential to stabilize the network in the regimen of alternating modes. After an initial increase in [K⁺]_o, a small globally connected PY–IN network exhibits periodic transitions between fast run and slow bursting every few seconds. During fast run, the spiking frequency is sufficiently high for extracellular K⁺ to accumulate. When [K⁺]_o reaches the upper endpoint of the hysteresis, the network switches back to slow bursting, during which [K⁺]_o decreases until it reaches the lower endpoint of the hysteresis for transition to fast run. We showed that a similar mechanism also occurs in larger networks with more realistic synaptic connectivity where oscillatory regimens are initiated and driven by [K⁺]_o dynamics.

Various types of potassium-mediated bistabilities have been described in both experimental and theoretical studies. Elevated [K⁺]_o led to a bistability between tonic firing and quiescence in a model of a hippocampal pyramidal cell (Hahn and Durand, 2001). Exposing cardiac Purkinje fibers to changing K⁺ concentration revealed two stable levels of resting potentials; switching between these two stable states was associated with a hysteresis (Gadsby and Cranefield, 1977). A simplified model of cerebellar Purkinje cells comprised of high-threshold calcium and delayed-rectifier potassium channels reproduced an experimentally observed bistability between resting potential and a depolarized state (Yuen et al., 1995). Myocardial cells exhibited two stable levels of diastolic potential when exposed to 4 mm K⁺ in vitro (McCullough et al., 1990).

In contrast to these studies, we investigated here a novel bistability between two different oscillatory firing regimens. In our model, [K⁺]_o dynamics not only created bistability but also forced periodic transitions between two metastable oscillatory states. The rate of [K⁺]_o change depends on the balance between neuronal K⁺ currents, K⁺ pumps, and glial buffering and, therefore, can be arbitrarily slow. Because of this, the time scale of slow transitions between oscillatory states in the model was not explicitly restricted by the time constants of any intrinsic or synaptic conductances. The relative duration of the different oscillatory modes depended on the width of hysteresis that was controlled by the combination of intrinsic and synaptic currents. Thus, mechanisms proposed in this work can potentially explain a wide range of oscillatory phenomena different by the time scale of oscillations and duration of alternating modes.

Other mechanisms may create multistability in synaptically coupled networks. Synaptic depression mediated bistability between two modes of oscillations in networks with recurrent inhibitory connectivity (Manor and Nadim, 2001). Bistability between silence and firing mediated by nonlinear firing characteristics of neurons and by feedback between the two neurons was found in small biological circuits in vitro (Kleinfeld et al., 1990). Although all of these mechanisms may potentially explain periodic transitions between different network states, an explicit assumption about the time constant of plasticity that should match the time constant of the slow transitions between modes is required. Our model is free from this limitation. As we showed here, interaction in a population of neurons (or neuronal circuits) mediated by extracellular ionic concentrations may lead to complex oscillatory regimens with continuous and arbitrarily slow transitions between individual stable modes.

Balanced excitation and inhibition

In our model, extracellular potassium dynamics caused sustained neural oscillations with alternating epochs of slow bursting and fast run, provided synaptic excitation and inhibition was balanced. Substantial change of the excitatory PY–PY or inhibitory IN–PY coupling away from the balanced state created regimens with exclusive fast tonic oscillations or slow bursting. The concept of balanced synaptic excitation and inhibition plays a central role in many neurobiological processes. Experimental in vivo, in vitro, and modeling studies of transitions between cortical active and silent states showed that periods of high activity are accompanied by a decrease in input resistance mediated by a simultaneous increase in both excitation and inhibition (Steriade et al., 2001; Bazhenov et al., 2002; Compte et al., 2003). Self-sustained activity mediated by balanced synaptic inputs might underlie short-term memory (Durstewitz et al., 2000) and the modulation of neuronal excitability with attention (Mehta et al., 2000). We find it notable that potentially similar conditions for the balance between excitation and inhibition can allow different forms of self-sustained activity and state-dependent transitions between different activity regimens.

Our model sheds a new light on the interaction between excitatory and inhibitory synaptic coupling during cortical oscillations. Traditionally, the role of inhibitory interneurons is to maintain reasonable firing frequency levels, and a shift in the balance between excitation and inhibition toward excitation is often associated with epileptic states (Dichter and Ayala, 1987; Tasker and Dudek, 1991; Galarreta and Hestrin, 1998; Nelson and Turrigiano, 1998). A number of studies conclude, however, that synaptic inhibition remains functional in many forms of paroxysmal activities (Higashima, 1988; Davenport et al., 1990; Traub et al., 1996; Esclapez et al., 1997; Prince et al., 1997; Cohen et al., 2002; Timofeev et al., 2002; Engel et al., 2003; Topolnik et al., 2003). In agreement with these findings, the role of inhibition is rather subtle in our model. If the balance of excitation and inhibition is shifted toward excitation, the PY cells were sufficiently active to stay in the bursting mode and fast run never occurred. Because [K⁺]_o decreased progressively during bursting, the network switched to the silent state. Thus, contrary to what was expected, decrease of inhibition in the model terminated oscillations because a single epoch of slow bursting was followed by silence (supplemental Fig. 6, available at www.jneurosci.org as supplemental material).

Synaptic plasticity can change the balance between excitation and inhibition and therefore move the network away from the region of sustained oscillations. These dynamic changes in synaptic weights could be especially prominent during high frequency synchronized paroxysmal oscillations. Therefore, the mechanism described in our model may potentially explain termination of the paroxysmal activity after seizures.