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1Department of Physiology and Zlotowski Center for Neuroscience, Faculty of Health Sciences, Ben-Gurion University, Be'er-Sheva, Israel; 2Department of Mathematics, University of Pittsburgh, Pennsylvania; and 3Department of Mathematics, University of Transilvania Brasov, Brasov, Romania
Submitted 7 September 2005; accepted in final form 10 October 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSACKNOWLEDGMENTSREFERENCES |
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10 Hz that appears in cortical tissues at low extracellular magnesium concentration [Mg2+]o. We hypothesize that this activity is persistent, namely coexists with the quiescent state and depends on slow N-methyl-D-aspartate (NMDA) conductances. To explore this hypothesis, we construct and investigate a conductance-based model of excitatory cortical networks. Population bursting activity can persist for physiological values of the NMDA decay time constant (100 ms). Neurons are synchronized at the time scale of bursts but not of single spikes. A reduced model of a cell coupled to itself can encompass most of this highly synchronized network behavior and is analyzed using the fast-slow method. Synchronized bursts appear for intermediate values of the NMDA conductance gNMDA if NMDA conductances are not too fast. Regular spiking activity appears for larger gNMDA. If the single cell is a conditional burster, persistent synchronized bursts become more robust. Weakly synchronized states appear for zero AMPA conductance gAMPA. Enhancing gAMPA increases both synchrony and the number of spikes within bursts and decreases the bursting frequency. Too strong gAMPA, however, prevents the activity because it enhances neuronal intrinsic adaptation. When [Mg2+]o is increased, higher gNMDA values are needed to maintain bursting activity. Bursting frequency decreases with [Mg2+]o, and the network is silent with physiological [Mg2+]o. Inhibition weakly decreases the bursting frequency if inhibitory cells receive enough NMDA-mediated excitation. This study explains the importance of conditional bursters in layer V in supporting epileptiform activity at low [Mg2+]o. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSACKNOWLEDGMENTSREFERENCES |
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); reviewed in (McCormick et al. 2003). A special type of persistent state, asynchronous firing, was studied using theoretical and computational methods (Amit and Brunel 1997; Hansel and Mato 2001; Wang 1999, 2001). Experiments in monkeys, however, have revealed partially synchronized activity during working memory tasks that have oscillatory (Cardoso de Oliveira et al. 2001) or nonoscillatory (Vaadia et al. 1995) characteristics. The mechanisms that generate persistent synchronized states in cortical networks are still unknown.
Here we study the mechanism for a persistent synchronized state in a specific neocortical system that was studied in vitro: discharges of rhythmic bursting activity at low extracellular Mg2+ concentration ([Mg2+]o) solution (Kawaguchi 2001; Sutor and Hablitz 1989). These synchronized oscillations depend on NMDA receptors and appear in tissue segments containing layer V only (Silva et al. 1991). In rats, this layer contains, among other cell types, pyramidal neurons that are quiescent at rest and burst rhythmically when constant, depolarizing current is applied, suggesting that this intrinsic bursting property could facilitate the generation of these waves (Steriade 2004). The oscillation frequency in the somatosensory cortex is 8–12 Hz. Blocking GABAA-mediated inhibition considerably increases the amplitude of the local field potentials and the duration of the oscillatory episode (up to 3 s) and only slightly increases the oscillation frequency, namely the peak of the power spectrum increases by 10–20% (Flint and Connors 1996). With inhibition intact, some inhibitory neurons burst in synchrony with the excitatory cells, whereas other fire almost continuously (Kawaguchi 2001). Recently, properties of these waves were studied using a combination of optical, voltage-sensitive dye imaging and local field potential techniques. Episodes of bursting oscillations can be evoked or can start spontaneously at a few confined initiation foci (Tsau et al. 1998). Correlation between local field potentials measured at two points did not decrease considerably with distance, but there were large phase shifts at a large distance (Fig. 5 in Wu et al. 1999). The phase relationship between the starting times of bursting period was not constant in time (Figs. 4 and 7 in Wu et al. 1999), hinting that the activity after the steady-state activity is a result of collective dynamics and not of periodic stimulation of the slice by a group of pacemakers located at a certain location.
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) of the population spiking activity. Then we explore the effect of inhibition. We hypothesize that the synchronized waves at low [Mg2+]o appear as synchronized persistent bursting activity. This hypothesis raises several questions. First, the bursting time period of the population (field) is of the order of 100 ms. Because in each burst the active (spiking) phase of the burst eventually terminates, there is a process that terminates it that is slow in comparison with the inter-spike interval within the burst. To generate a subsequent burst, the slow N-methyl-D-aspartate (NMDA) conductance should overcome this slow process and initiate the spiking activity. For that, the NMDA-mediated conductance should decay slowly enough. How small can this decay time constant be to still support the next active phase? Is this value within the physiological range?
Second, should the single cells have a propensity to burst to obtain network bursting or can bursting be obtained as a network effect with the single cells only being regular spiking cells?
Third, does the fast, AMPA-mediated excitation support or disrupt persistent bursting? How does it affect the properties of this activity, such as bursting duration, frequency and level of synchronization?
Fourth, how large should [Mg2+]o be to prevent the persistent activity? How do the burst properties depend on [Mg2+]o in the parameter regime where they exist?
And fifth, how does intracortical inhibition affect the discharge frequency? Does the inhibitory effect depend on the type and conductance strength of excitatory receptors (AMPA and NMDA) on inhibitory cells?
To answer these questions, we construct and analyze a conductance-based model of one-dimensional networks with spatially decaying connectivity.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSACKNOWLEDGMENTSREFERENCES |
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EXCITATORY CELLS.
We model excitatory cortical cells that can fire either in trains of single action potentials (regular spiking, RS) or in fast bursts of action potentials (intrinsically bursting, IB) (Connors et al. 1982; McCormick et al. 1985). To evaluate the importance of the intrinsic bursting property, we use a model that can be transformed, by modifying a single parameter, from RS to IB with an increasing number of spikes within a burst (Golomb and Amitai 1997; D. Golomb and Y. Yaari, unpublished data). The single-cell dynamics is described by a single-compartment Hodgkin-Huxley type model with the use of a set of coupled differential equations
 | (1) |
). These currents are either calcium dependent, such as IAHP (Pinsky and Rinzel 1994; Sah 1996), or voltage dependent, such as IM (Bibbig et al. 2001; Mainen and Sejnowski 1996; Yue and Yaari 2004). The current IK-slow contributes the slow variable that is needed for adaptation or bursting in RS cortical cells (Rinzel and Ermentrout 1998). If the kinetics of INa and IKdr are fast enough, and with strong enough persistent sodium conductance (gNaP), the model may produce IB behavior in response to prolonged current injection. For example, for 
, the neuron bursts if gNaP > 0.091 mS/cm2. Neurons from this type are called "conditional intrinsically bursting" (cIB). Hence, the conductance gNaP can be viewed as an "intrinsic bursting parameter." The equations and parameters of the model are given in APPENDIX A.
This single-cell model for cortical pyramidal neurons was chosen for two reasons. First, it is a minimal model that can show both regular spiking and bursting behavior. Second, the role of gNaP in the intrinsic bursting property in the model is consistent with experimental observation showing the importance of this conductance for both single-cell bursting in cortical structures (e.g., Su et al. 2001) and for bursting in neocortical networks bathed in low [Mg2+]o (Castro-Alamancos and Rigas 2004).
INHIBITORY CELLS.
We model inhibitory cortical cells using the Wang-Buzsáki model of fast spiking (FS) neurons (Wang and Buzsáki 1996). The single-cell dynamics is described by a single-compartment model
 | (2) |
Synaptic models
Neocortical cells receive fast, AMPA-mediated, and slow, NMDA-mediated excitatory postsynaptic potentials (EPSPs) from neighboring excitatory cells (Gil and Amitai 1996), and GABAA-mediated inhibitory postsynaptic potentials (IPSPs) from neighboring inhibitory cells (Hansel and Mato 2003; Wang and Buzsáki 1996). A gating variable sAMPA for an AMPA receptor, representing the fraction of open channels, is modeled according to
 | (3) |
and 
(Golomb and Amitai 1997; Wang and Rinzel 1993), 
. The rise time of the AMPA synapses is assumed to be instantaneous, and decay time of those synapse is 
(Stern et al. 1992).
The rise time of NMDA receptors cannot be neglected, and therefore two differential equations are needed to model those synapses. We use the phenomenological equations of Golomb et al. (1996)
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 | (5) |
, 
, 
. The decay time of NMDA-mediated excitatory postsynaptic conductances (EPSCs) in cortex varies from 65 to 108 ms (Kumar and Huguenard 2003). For our reference parameter set, we use the value 
NMDA =100 ms, corresponding to the value of intracortically evoked NMDA-mediated EPSCs (Kumar and Huguenard 2003). The effect of this parameter is assessed in the following text. For simplicity, effects of synaptic depression and facilitation (Golomb and Amitai 1997; Tsodyks and Markram 1997) are not included in the model.
A gating variable sGABAA for a GABAA receptor, is modeled according to
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, 
(Hansel and Mato 2003; Wang and Buzsaki 1996).
The total synaptic conductance a neuron receives, gSin (for AMPA, NMDA, or GABAA) is calculated by summing the synaptic variable of each of its presynaptic neurons as described in the following text. The AMPA current is
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; Traub et al. 1991). The NMDA current is therefore
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NMDA. The value of NMDA is –
for 
and is –31 mV for NMDA-mediated intracortical synapses at physiological levels of [Mg2+]o (Kumar and Huguenard 2003); it increases logarithmically with [Mg2+]o (Jahr and Stevens 1990). Using the data for intracortical connections, we find that NMDA = 10.5 mV x ln([Mg2+]o/38.3 mM).
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Axonal propagation delay and synaptic delays are neglected because the time scale of bursting oscillations is in the order of 100 ms, much larger than the time scale of those delays, which are in the order of a few milliseconds.
Network architecture
Propagation of activity along cortical slices has been modeled before in one-dimensional networks (Ermentrout 1998; Golomb 1998; Golomb and Amitai 1997; Golomb and Ermentrout 1999, 2002; Golomb et al. 2001), where the extensive intralaminar connectivity is neglected, assuming cortical columns are rapidly and fully recruited. In the same manner, our model consists of a one-dimensional system. Excitatory and inhibitory neurons are equally distributed along the interval 0 
x L, where L is the slice length and x is the neuron position. The cell density is 
, and the number of neurons from each type is 
. The position of the ith neuron, either excitatory or inhibitory, 1 i N, is 
. The interaction between neurons is assumed to decay with the distance between them. The "synaptic footprint shape," w(x), denotes the functional dependence of the synaptic connectivity on the distance between the preand postsynaptic cells. It is assumed here to be exponential with a characteristic delay length ("footprint length") 
, namely 
. The model is studied in the parameter regime 1/ << << L. The total synaptic conductance gSin affecting a neuron is
 | (10) |
, where and and denote the pre- and postsynaptic populations, respectively, that can be E (excitatory) or I (inhibitory). The coupling among neurons in the inhibitory population, either excitatory or inhibitory (Beierlein et al. 2003), is neglected here. To shorten the notation, we omit the superscripts if they are EE. We use open boundary conditions. The integro-differential equations (1–10), together with the differential equations for the auxiliary variables, are discretized on a grid, as described in APPENDIX A. Initial conditions
As in our previous work (Golomb 1998; Golomb and Amitai 1997), we initiate our model from a state at which all the neurons, except a group at the left edge, are in their resting state. A wave is initiated by depolarizing a group of neuron, both excitatory and inhibitory, within a length or /2 at the left edge, such that they generate action potentials. The firing neurons may recruit resting neurons through their synaptic connections to initiate a propagating discharge.
Definitions of states
We define the state of the system based on its long-time behavior. Therefore a state of transient activity, in which the network is silent after brief transient activity, is considered to be quiescent. A transient bursting state followed by a tonic state is considered to be tonic.
For a single-cell model, or a state of a disinhibited network that can be represented by a single cell coupled to itself, the notion of "bursting cell" means here "a spiking cell that is not firing periodically (tonically)."1 To determine whether the voltage time course of an isolated neuron is tonic or bursting, we run a simulation for a long time Tm after a transient time Ttransient. We find the minimal inter-spike interval Tisi,min and the maximal inter-spike interval Tisi,max. The cell is considered to be in a tonic mode if Tisi,min/Tisi,max > 0.9. Otherwise, the cell is in a bursting mode.
The situation is more complicated in a network, where neurons can fire in an irregular and aperiodic manner. Therefore a neuron in a network is defined to be "bursting" if there is a clear distinction between brief and prolonged inter-spike interval, namely Tisi,min/Tisi,max < 0.33. The properties of the network are determined by considering the middle group of excitatory neurons with length L/2 and ignoring the two edges of length L/4. A network is considered to be in a tonic mode if all those middle excitatory neurons fire tonically. Borders between regimes of activity are determined using the bisection method (Golomb and Ermentrout 1999).
For both single neurons and networks, we compute the average number over the excitatory population of spikes within a burst, Ns, and the bursting frequency, f. The burst duration Td is defined as the average time between the first spike and the last spike in a burst.
Correlation calculation.
The average membrane potential of a neuron is
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 | (12) |
Synchrony measure
For networks with all-to-all coupling, the synchrony measure 
of the excitatory population is defined as the fluctuation of the population-average membrane potential, normalized by the average fluctuations of the membrane potentials of single neurons (Golomb and Rinzel 1993, 1994; Golomb et al. 2001). We extend this definition for networks with spatially decaying synaptic coupling by calculating the population average over excitatory neurons in the neighborhood of a specific neuron in the middle of the chain, as explained in APPENDIX B. The measure varies between 0 (asynchronized state) and 1 (fully synchronized state).
Fast-slow analysis
We used the fast-slow method (Bertram et al. 1995; Hoppensteadt and Izhikevich 1997; Izhikevich 2000; Rinzel and Ermentrout 1998) to study the bursting mechanism of a single excitatory cell coupled to itself. This method has been applied successfully to analyze periodic bursting of various biophysical models of neurons and networks (e.g., Jung et al. 1996; Mandelblat et al. 2001; Tabak et al. 2000). The method makes use of basic dynamical systems theory (Strogatz 1994), which separates the variables of the system into two subsystems, "fast" and "slow," according to their overall timescales. In our system (the reduced model of an excitatory cell coupled to itself), the variables V (the membrane potential), h (inactivation of INa), n (activation of IKdr), and sAMPA (the fraction of open AMPA channels), with time scales on the order of 1 ms, are considered to be fast. The variables z (activation of IK-slow) and sNMDA (the fraction of open NMDA channels), with time scales on the order of 100 ms, are considered to be slow.2 In the first stage of the analysis, as will be described later, the variable sNMDA is considered to be constant, and the fast subsystem includes only the variable z. The bifurcation diagram of the fast subsystem is computed with the slow variable z considered as a parameter. Then the dynamics of the slow subsystem are computed using the time-averaged values of the fast subsystem. This is done by plotting two more curves on the bifurcation diagram. One is the activation curve of IM, 
(Eq. A15). The second curve is the equivalent voltage during the oscillatory state, Vequiv (Bertram et al. 1995; Mandelblat et al. 2001), which is computed to determine the slow evolution of z in the full model. The value of Vequiv for a periodic cycle with a specific z is the voltage that, for a rest state (fixed point), yielded a value of dz/dt that is equal to the value of dz/dt averaged over the cycle. It was defined implicitly by the equation
 | (13) |
–1(z) (where z–1 (z) means the inverse function of z), than z in the full dynamical system (fast + slow together) increases slowly with a rate of order z. If Vequiv(z) < z–1(z), z decreases slowly.
The type of bursting we study in this paper belongs to the class of "square-wave bursters" (Rinzel and Ermentrout 1998), where the fast subsystem exhibits bistability of a rest state and a periodic firing state. During the rest state, the slow variable z decreases slowly, until the rest state does not exist anymore, and the fast subsystem jumps to the periodic firing state. During this state, the slow variable z increases slowly, until the firing state does not exist anymore and the fast subsystem jumps back to its rest state. This mechanism generates periodic bursting.
The bursting dynamics in our system is different from the classical square wave bursting mechanism because the slow variable sNMDA is not really a constant. We examine the effects of this second slow variable below.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSACKNOWLEDGMENTSREFERENCES |
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. In a second step, we study how the properties of the states vary when [Mg2+]o is elevated. Finally, we explore the role of inhibitory interneurons. Synchronized persistent bursting for [Mg2+]o = 0
The disinhibited slice model with the reference parameter set 
exhibits persistent synchronized bursting activity in response to stimulation at the "left." The initial burst propagates as a traveling pulse at a constant velocity, similar to propagating single bursts in models of slices in which inhibition was reduced (Golomb and Amitai 1997) (Fig. 1A, left). Subsequent bursts propagate in a less orderly manner, especially near the edges of the slice model, and do not maintain a constant velocity of the left-to-right propagation. At long times after the initiation of the activity, cells burst with a frequency of 
, and the bursting activity is synchronized, although not fully synchronized (Fig. 1A, right). Neighboring cells tend to burst at similar times (Fig. 1B, left), but spikes within bursts are only weakly synchronized, if at all (Fig. 1B, right). Summarizing, the model with reference parameter set generates persistent synchronized behavior with properties similar to those seen experimentally (Flint and Connors 1996; Silva et al. 1991; Sutor and Hablitz 1989; Tsau et al. 1998; Wu et al. 1999).
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) of the membrane potential V(t) of one neuron (Eq. 12) in the middle of the slice (
, Fig. 2A) and the cross-correlation Cx,x'() of this neuron with its neighbor at 
(Fig. 2B). Both the auto- and cross-correlations oscillate on the bursting time scale, reflecting a periodic firing pattern and synchronization at the bursting level. The auto-correlation shows peaks at the spiking time scale for near zero (Fig. 2A, top right) but almost no peaks at the spiking time scale (and only one wide peak at the bursting time scale) for value around other integer multiplications of the time period 
(Fig. 2A, bottom right). The cross-correlation shows almost no peaks at the spiking time scale around 
(Fig. 2B, top right) and no peaks at the spiking time scale near other integer multiplications of Tper (Fig. 2B, bottom right). For all neurons within a footprint length around the neuron at 
, the amplitude of the cross-correlation is almost constant with the distance between the neurons, and the phase of the cross-correlation is zero. We conclude that the neurons are synchronized on the bursting time scale, at least within distances of order , but not on the spiking time scale. Note that noise, heterogeneity, and sparseness are expected to cause the cross-correlations to decay with space and time (Golomb 1998).
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3 s (Flint and Connors 1996). This termination can be explained by the effects of slow processes, such as slow inactivation of Na+ channels (Fleidervish and Gutnick 1996; Fleidervish et al. 1996), that are not included in the model.
Dependence of activity regimes on NMDA and gNMDA
The bursting activity of each neuron is terminated by the slow potassium current IK-slow, with activation kinetics constant 
. Once this current has decayed, a strong enough slow NMDA-mediated excitation is responsible for the generation of a new burst. Therefore the decay time constant of the NMDA-mediated synapses, NMDA, is an important parameter for generating a persistent synchronized bursting state. We expect that this state can appear only if NMDA is large enough in comparison with z. Furthermore, one may expect that the NMDA value that allows persistent bursting should be large enough in comparison with Tper and that increasing gNMDA may allow persistent bursting even for smaller NMDA. To examine the roles of NMDA and gNMDA, we computed the regimes in the NMDA–gNMDA plane where stimulation of the slice led to one of three states: persistent bursting activity, persistent tonic activity, or nothing but the quiescent state. The calculation was carried out without (Fig. 3A) and with (B) the inclusion of AMPA-mediated synaptic conductances (gAMPA = 0 and 0.08 mS/cm2 respectively). For both cases, bursting activity appear for large enough NMDA and moderate values of gNMDA. Interestingly, bursting activity is found even for NMDA values that are smaller than z, especially for 
. For larger values of gNMDA and for NMDA that is not too small, the activity is tonic, whereas for small gNMDA or small NMDA, the persistent activity does not prevail. The gNMDA interval where bursting activity exists increases with NMDA and extends toward low-gNMDA values. Increasing gAMPA shifts the boundary of the bursting regime toward larger NMDA values. This means that for moderate values of NMDA, increasing the fast, AMPA-mediated excitation may abolish the persistent bursting behavior. This network behavior stems from the fact that increasing gAMPA increases the number of spikes within a burst (see following text), and therefore increases the activation of IK-slow that terminates the burst and prevents the next one. Similarly, the transition from the tonic regime to the bursting regime occurs for larger gNMDA values as gAMPA increases, because the stronger IK-slow due to the increased number of spikes terminate the spiking activity and generates a quiescent period before the next firing period begins.
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The large synchronization on the bursting time scale among neighboring neurons in cases such as those shown in Figs. 1 and 2 allows the use of a simple approximation for the large dynamical system: a neuron coupled to itself with the same gNMDA and gAMPA values as in the network model (van Vreeswijk and Hansel 2001). This reduced model is amenable to mathematical analysis and is much faster to simulate to produce phase diagrams describing the bursting and tonic regimes as functions of various parameters. This simplified model is only an approximation, however, because the spikes within the bursts are not (or only weakly) synchronized. If the bursts themselves are less synchronized in the full network model, this approximation becomes even worse. Therefore we compare between the behavioral regimes of the reduced model (Fig. 3, C and D) and the full network model (Fig. 3, A and B). Qualitatively, the behavioral regimes of the two models are similar. Two main differences are found, however. First, for , the bursting regime extends toward low NMDA in the full model in comparison with the reduced one. This effect is caused by the appearance of states with low synchronization in the full model as will be shown in the following text. Second, for 
, there is a "tristable" regime where the bursting, tonic and quiescent states are stable states. Interestingly, the tristable regime has a tongue-like shape. Each tongue corresponds to a certain number of spikes in the burst, as written in Fig. 3D. Because the transition between a certain number of spikes to the subsequent one occurs through chaotic behavior (Terman 1992), it is not surprising that the tongue structure is complex. In our network simulations, the state is selected by the initial conditions, and therefore tristability is not seen. Overall, the reduced model can be used to describe the dynamical regimes of the whole network unless the network exhibits bursts that are not highly synchronized.
Dependence of bursting patterns on NMDA and gNMDA
We next examine how the main properties of the wave: the number of spikes within a burst Ns, the bursting frequency f, the burst duration Td, and the synchrony measure depend on the properties of NMDA synapses: gNMDA (Fig. 4, A and B) and NMDA (Fig. 4, C and D). The examination is carried out for parameters that lead to bursting, without () and with () AMPA excitation. Computation was performed for both the full network model (red) and the reduced model of one cell coupled to itself (black), to further compare their behavior. In the regimes where the two models yield persistent synchronized bursting, the bursting properties Ns, f, and Td, are, in general, very similar, showing that the reduced model mimics the full network model well. The number of spikes Ns is almost independent of gNMDA and NMDA. The bursting frequency f increases with gNMDA but only weakly with NMDA. To obtain physiological values of f, 12 Hz (Flint and Connors 1996), gNMDA should not be too large. The burst duration Td increases weakly with gNMDA and is almost independent of NMDA. A main determinant of the burst duration is the time constant of the slow potassium current, z (see the section on the fast-slow analysis).
States with low synchrony
For , the full model, but not the reduced model, shows bursting activity for low gNMDA values (Fig. 4A). These bursting states have low synchrony: is <0.2 in comparison to values >0.5 that are obtained for larger gNMDA values for which the bursts are highly synchronized. We find that for the value 
, increases sharply. For this same gNMDA, f of the full model shows a small deflection downward, and the reduced model starts to fire bursts instead of being quiescent. Similarly, states with low bursting synchrony are found for , the reference value of gNMDA and low NMDA, <60 ms (Fig. 4C). For , such low-synchrony states do not appear, and all the bursting states are highly synchronized.
To explore the nature of this state further, we present a rastergram obtained in response to stimulation of the slice model with 
and in Fig. 5A. Initially, the pattern of activity (Fig. 5A, left) resembles that of the reference parameter set (Fig. 1A, left). The firing pattern in the steady state, however, is more complicated (Fig. 5A, right). The bursting time of neurons is widely distributed as exhibited both in the rastergram and in the voltage time courses of two bursting neurons (Fig. 5B). To determine whether the network is asynchronous, we computed the synchrony measure for a local population of 2 + 1 neurons, as a function of 1/ where varies from 64 to 1,024 (Fig. 5C). The value of remains roughly constant near the value of 0.17, indicating that the state shows a low level of synchronization and is not asynchronized.
Dependence of activity regimes and bursting patterns on gAMPA and gNaP
The parameters gNMDA and NMDA control the slow dynamics of the system, on the order of 100 ms. Bursting behavior, however, depends also on the fast dynamics of the system (Bertram et al. 1995; Izhikevich 2000; Rinzel and Ermentrout 1998). In particular, fast, AMPA-mediated excitation was shown to induce bursting activity in networks of excitatory neurons with spike adaptation (van Vreeswijk and Hansel 2001). Even without AMPA excitation, our model neuron can generate endogenous bursting behavior in response to constant applied current, and the parameter gNaP represents the intrinsic conditional bursting property of the single neuron. Without gNaP, the neuron fires only tonically in response to a step current pulse. It responds by bursting to this current pulse for large enough gNaP, and the number of spikes within each burst increases with increasing gNaP (Golomb and Yaari, unpublished).
We study how persistent bursting in the network depends on gNaP and gAMPA by determining the various behavioral regimes as functions of these parameters using the reduced model. The phase diagram is plotted in Fig. 6 for two values of gNMDA: the reference value 
(A) and a value that is 50% larger, 
(B). At the lower gNMDA value, tonic firing is obtained for small values of gNaP and gAMPA (but not for 
). Bursts are obtained for moderate gNaP values and gAMPA values that are not too large. The behavior for the higher gNMDA value, 0.105 mS/cm2 shares some similarity with the behavior for 
but differs in the following aspects. First, the tonic regime extends toward larger gNaP values. Second, the bursting regime extends toward larger gAMPA values. Third, bursts are generated even for , though in a restricted regime. Fourth, a tristable regime emerges for moderate values of gAMPA and gNaP. For both values of gNMDA, increasing gAMPA or gNaP transfers the activity from the bursting to quiescent states. Reducing gAMPA may transfer the activity from bursting to tonic state but not to quiescence (except in the small areas near the tongues). Reducing gNaP, however, mostly transfers the activity from bursting to quiescent state. The intrinsic conductance gNaP and the synaptic conductance gAMPA display therefore some similarities, but also some differences, in their effects on the bursting activity.
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on gAMPA is shown in Fig. 7A for . Increasing gAMPA causes Ns to increase and f to decrease. These two phenomena are related because an increase in Ns leads to an increase in the activation variables of IK-slow, resulting in an elongation of the inter-burst interval. The burst duration Td increases only weakly with gAMPA. Larger Ns values that result from increasing gAMPA indeed prolong the burst duration, but when Ns is kept constant, the burst duration even decreases with gAMPA. The bursts are synchronized ( > 0.5), and gradually increases with gAMPA with many local peaks corresponding to minima in Td.
The effect of Ns and f on gNaP is similar to the effect of gAMPA, whether gAMPA is zero (Fig. 7B) or gAMPA = 0.08 mS/cm2 (Fig. 7C). The frequency f, however, depends only weakly on gNaP for , and Td is almost constant with some fluctuations. For , fluctuates and may reach low values representing states of low synchrony (Fig. 7B, bottom). Such low synchrony states do not exist for larger gAMPA (Fig. 7C, bottom). We conclude that the network exhibits persistent burst synchronization with substantial AMPA excitation. Without it, bursting states of low synchrony appear in many parameter regimes.
Fast-slow analysis of bursting activity
DESCRIPTION OF THE ANALYSIS.
The reduced model of one cell coupled to itself describes the properties of the network discharges very well (Fig. 3, 4, and 7). Therefore we use this simple model to analyze the properties of the persistent synchronized activity, using the fast-slow method (Bertram et al. 1995; Izhikevich 2000; Rinzel and Ermentrout 1998). We separate the variables of the system (the reduced model) into two subsystems, fast and slow, according to their overall time scales. The fast subsystem includes the variables V, h, n, and sAMPA, and the slow subsystem includes the variables z and sNMDA. We analyze the behavior of the single neuron coupled to itself for the reference parameter set, gNMDA = 0.07 mS/cm2, gAMPA = 0.08 mS/cm2 (Fig. 8, A–C). To describe the effect of the two types of synapses on the burst properties, we carry out the analysis also for two more cases: AMPA excitation is blocked (gNMDA = 0.07 mS/cm2, gAMPA = 0; Fig. 8, D–F) and NMDA excitation is enhanced (gNMDA = 0.105 mS/cm2, gAMPA = 0.08 mS/cm2; Fig. 8, G–I). As a first step, we plot the voltage time course, V(t), and the time courses of the slow variables z(t) and sNMDA(t), computed by simulating the reduced model during one time period Tper, in Fig. 8, A, D, and G, for those three parameter sets.
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NMDA >100 ms affects the bursting properties only weakly, namely the behavior for physiological value of NMDA is quite similar to the behavior for very large NMDA. Therefore as a first step, we study the system in the limit NMDA>> z and fix the slow variable sNMDA at a constant, average value it obtains during the cycle: 0.85, 0.92, and 0.93 for the three parameter sets (Fig. 8, A, D, and G, respectively, bottom) and consider the slow variable z to be a parameter. We compute the bifurcation diagrams of the fast subsystem as a function of z (Fig. 8, B, E, and H). The steady state (fixed point; thin black line) is stable for large z. This stable rest state coalesces with an unstable state and ceases to exist in a saddle-node bifurcation. The rest state, this time a high plateau, is stable again for small values of z. At the z value where the high rest state gains its stability (a Hopf bifurcation), an oscillatory state (limit cycle) emerges, corresponding to periodic firing. This oscillatory firing state extends toward the right. We plot the minimal and maximal values of the membrane potential during the firing state (thick black lines). There is a narrow regime in which the continuity of the stable oscillatory state is disturbed by an unstable oscillatory state, and for large values of z the amplitude of the oscillatory state that gains its stability is larger than the oscillation amplitude for smaller z. The oscillatory state coalesces with another unstable oscillator state (saddle-node of periodics bifurcation) that immediately terminates by intersecting the unstable fixed point (homoclinic, or saddle-loop, bifurcation).
Bistability of firing (oscillatory) and quiescent states in the dynamics of the fast subsystem is necessary, but not sufficient, for allowing bursting activity with one slow variable. To further determine the conditions for bursting in the full (fast and slow) model of a cell coupled to itself, we plot two more curves: the activation curve of the current IK-slow, z(V) (Eq. A15), plotted in green, and the equivalent voltage during the oscillatory state of the fast subsystem, Vequiv (Eq. 13), plotted in blue. For values of Vequiv above this green line, z slowly increases, and for values of Vequiv below this green line, z slowly decreases. For the three parameter sets we investigate (Fig. 8, B, E, and H), the solid (stable) blue line is above the green line, and therefore z increases during the firing state until it terminates the oscillations. The thin black line is below the green line, and therefore z decreases during the silent state until the rest state disappears (at the saddle-node bifurcation). The full dynamical system exhibits, therefore periodic bursting ("square-wave bursting") (Rinzel and Ermentrout 1998) for the three parameter sets shown in Fig. 8.
To complete the analysis, the two variables should be considered as slow variables with similar time scale. The activity regimes: rest, bistability, and firing are shown in the sNMDA-z plane (Fig. 8, C, F, and I). The boundaries of those regimes are diagonal lines, and their z coordinate increases with their sNMDA coordinate. The projection of the trajectory of the bursting state is shown in green. It oscillates through the bistable regime between the silent, rest state and the firing state. Both intrinsic neuronal properties (Fig. 8E) and AMPA-mediated excitation (Fig. 8B) contribute to the generation of this bistable regime. Using this analysis, we now explain the dynamical behavior of the model.
EXPLAINING THE EFFECTS OF VARYING gAMPA. We first explain why Ns increases with gAMPA and f decreases with it, as shown in Fig. 7A. Comparison between the two values of gAMPA (Fig. 8, B and E) shows that the bistable regime is wider for larger gAMPA. The low rest states (both stable and unstable) are not affected by gAMPA because the AMPA activation curve is substantially larger than 0 only for very depolarized potentials (the dependencies of the rest voltages on z are slightly different for the two values of gAMPA because of the different values of the constant sNMDA). The conductance gAMPA, however, affects the periodic action potentials by increasing their frequency and reducing their amplitude. This happens because after the self-coupled neuron fires an action potential, the depolarizing conductance advances the firing of the subsequent one, the sodium current INa is still partially inactivated, and therefore the action potential overshoot is lower, the activation of IKdr is smaller, and the afterhyperpolarization that follows the action potential is less pronounced. Hence, for the same z, the minimal voltage during the action potential is more depolarized for larger gAMPA. The value of z, for which the minimal value of the action potential nearly coincides with the voltage of the unstable rest state, is larger and the bistable regime is wider when gAMPA increases. As a result, Ns is larger and f is smaller for larger gAMPA.
The intrinsic conductance gNaP also increases the bistable regime (not shown), and therefore Ns increases with gNaP (Fig. 7, B and C).
EXPLAINING THE EFFECTS OF VARYING NMDA.
We now turn to explain the dependence of Ns on NMDA and why activity stops at small NMDA values. The analysis becomes somewhat more complicated by the fact that the value of NMDA is not much larger than z. If the kinetics of sNMDA were very slow, the direction of the projection of the neuronal dynamics onto this plane would be parallel to the ordinate (z-axis; Fig. 8F). Because these kinetics are not much slower than that of z, the value of both z and sNMDA increase during the firing phase and decrease during the silent phase. Because the boundaries of the bistable regime are diagonal, the z interval that the neuronal trajectory covers during the bursting cycle is larger for faster NMDA kinetics. As a result, the number of spikes decreases somewhat with NMDA, for small NMDA values, as shown in Fig. 4, C and D.
If NMDA becomes smaller in comparison to z, the trajectory of the dynamical system in the sNMDA-z plane becomes more and more parallel to the abscissa. After firing several spikes, the system is stuck at the rest state (on the left), the bursting process is terminated, sNMDA decreases to 0 and z decreases to a very small value. This explains why persistent bursts cannot occur for too small NMDA (Fig. 3).
EXPLAINING THE EFFECTS OF VARYING gNMDA.
The numerical simulations (Figs. 4, A and B, and 3) show that, first, in the persistent bursts regime, f increases with gNMDA, whereas Ns and Td are almost independent of gNMDA; second, for large enough gNMDA, the activity becomes tonic instead of bursting. To explain these two numerical observations, we note that the slow currents IK-slow and INMDA for the self-coupled neuron at are (Eqs. A13 and A20)
 | (14) |
 | (15) |
If gNMDA is large enough, the solid blue curve, representing the effective potential Vequiv for the firing state, is shifted enough to the right such that it intersects with the green curve. In this case, the firing state is stable (in the full system) and the system does not burst.
If gNMDA is large, but still below the value for the tonic behavior of the full system, the neuron still bursts but the bistable regime is narrower. The value of dz/dt in the full system depends on [z(V) – z)/z (Eq. A14), namely on the distance between the thin solid black line and the green line for rest states, or the solid blue line and the green line for the tonic states. As shown in Fig. 8H, the impact of the shift to the right of the steady state curve is weak, because the value z(V) – z remains about 0.1. The bistable regime is narrower, and therefore the silent phase of the burst decreases as gNMDA increases. The shift of the solid blue curve to the right is much more important because the solid blue curve and the green curve almost intersect. Therefore the value of dz/dt during the firing phase of the burst is smaller for larger gNMDA. This compensates for the decrease in the z regime where the system is bistable, and therefore the burst duration Td, as well as Ns, are almost independent of gNMDA.
Effects of nonzero [Mg2+]o
Raising [Mg2+]o leads to several physiological effect. The main effect is on the response of NMDA receptors. Under physiological conditions, when the membrane potential is close to its resting values, NMDA receptors are blocked by Mg2+ ions. Increasing [Mg2+]o corresponds to shifting the sigmoid function fNMDA(V), the voltage dependency of the NMDA synaptic conductance on the postsynaptic voltage, toward the right. This means that NMDA, the half-maximum value of fNMDA(V), is depolarized as [Mg2+]o increases from – for to around –30 mV for physiological value (Jahr and Stevens 1990; Kumar and Huguenard 2003
. A: rastergrams; 
. Only firing times from every 32th cell are shown. Left: response to a local stimulation. Initially cells at left (x 
. These, as well as the rastergrams, show that the bursting activity of one neuron is shifted in time in comparison with the bursting activity of the 2nd one. C: synchrony measure 
(A and B) and on 
. Black lines denote results of simulating the reduced model of 1 cell coupled to itself.
; C, gAMPA = 0.08 mS/cm2. Red lines denote results of simulations of the full network model with 
. A: rastergrams. Only firing times from every 32th cell are shown. Each spike is represented by a solid circle. Because the circles representing spikes within a burst are adjacent, each burst of spikes looks like a continuous line in the rastergram. Left: response to a local stimulation. Initially cells at left (x 
and 
, located in the middle of the slice model. The traces on the left, as well as the rastergrams, show that bursts of neighboring neurons are synchronized, although not fully. The same traces plotted at a magnified time scale (right) demonstrates that spikes within the bursts are synchronized very weakly, if at all. The vertical dotted lines are plotted at the times when the upper neuron is firing.
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