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The Journal of Neuroscience, November 15, 2001, 21(22):9053-9067

Self-Organized Synaptic Plasticity Contributes to the Shaping of  and  Oscillations In Vitro

Andrea Bibbig^{1, 4}, Howard J. Faulkner², Miles A. Whittington³, and Roger D. Traub^{1, 4}

¹ Department of Pharmacology, University of Birmingham School of Medicine, Edgbaston, Birmingham B15 2TT, United Kingdom, ² Imperial College School of Medicine, London SW7 2AZ, United Kingdom, ³ School of Biomedical Sciences, University of Leeds, Leeds LS2 9NQ, United Kingdom, and ⁴ Department of Physiology and Pharmacology, State University of New York Health Science Center, Brooklyn, New York 11203

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

(30-70 Hz) followed by  (10-30 Hz) oscillations are evoked in humans by sensory stimuli and may be involved in working memory. Phenomenologically similar oscillations can be evoked in hippocampal slices by strong two-site tetanic stimulation. Weaker stimulation leads only to two-site synchronized . In vitro oscillations have memory-like features: (1) EPSPs increase during ; (2) after a strong one-site stimulus, two-site stimulation produces desynchronized ; and (3) a single synchronized epoch allows a subsequent weak stimulus to induce synchronized . Features 2 and 3 last >50 min and so are unlikely to be caused by presynaptic effects. A previous model replicated the transition when it was assumed that K⁺ conductance(s) increases and there is an ad hoc increase in pyramidal EPSCs. Here, we have refined the model, so that both pyramidalpyramidal and pyramidalinterneuron synapses are modifiable. This model, in a self-organized way, replicates the transition, along with features 1 and 2 above. Feature 3 is replicated if learning rates, or the time course of K⁺ current block, are graded with stimulus intensity. Synaptic plasticity allows simulated oscillations to synchronize between sites separated by axon conduction delays over 10 msec. Our data suggest that one function of  oscillations is to permit synaptic plasticity, which is then expressed in the form of  oscillations. We propose that the period of  oscillations, ~25 msec, is "designed" to match the time course of [Ca²⁺]_i fluctuations in dendrites, thus facilitating learning.

Key words: Hebbian synapses; 40 Hz; synchronization; EEG; learning; memory

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The generation of  (10-30 Hz) oscillations in cortical structures appears to be inextricably linked with the generation of  (30-70 Hz) oscillations. Examples in human EEGs are as follows. (1) "Evoked" gamma, followed by , is induced by auditory stimuli (Haenschel et al., 2000). As in vitro (Doheny et al., 2000), the  component is strongest in response to novel stimuli, whereas the  component habituates. (2) , mixed with , appears after exposure to a visual stimulus that needs to be remembered briefly (Tallon-Baudry et al., 1999). (3) In response to pictures or words, there is increased temporal/parietal coherence of EEG activity in the 13-18 Hz band (von Stein et al., 1999).

In vitro models may provide clues to the mechanism and function of in vivo / oscillations. In the CA1 region of rat hippocampal slices, / oscillations, lasting seconds, are most readily induced by two-site stimulation (Whittington et al., 1997a).  requires that the stimulus be sufficiently strong, so that pyramidal cells and interneurons remain depolarized enough to fire (Faulkner et al., 1999). Interneurons then continue to fire a network , whereas pyramidal cells skip beats, switching to  frequency, because of increased afterhyperpolarizations (AHPs) (Whittington et al., 1997a). Pyramidal cells must skip, on average, the same beats of the underlying , which is favored by increasing recurrent EPSPs between pyramidal cells (Whittington et al., 1997a).

Given that  oscillations in vivo may play a role in working memory (Tallon-Baudry et al., 1998), it is interesting that in vitro / oscillations have two "memory-like" features, lasting for >50 min (Whittington et al., 1997a). (1) Strong stimulation at a single site, applied once, has a lasting interference on the ability of subsequent two-site stimulation to induce synchronous . (2) Two-site weak stimulation normally induces only two-site synchronized , not followed by synchronized . Nevertheless, if a single instance of strong two-site stimulation is delivered, which induces two-site synchronized /, then, subsequently, two-site weak stimulation also induces two-site synchronized /. The long duration of these effects makes it unlikely that they depend on presynaptic mechanisms.

In this paper, we postulate that excitatory synaptic conductances modify during the course of an oscillation, a reasonable assumption given the observed Hebbian "learning rules" for CA3 recurrent pyramidalpyramidal and pyramidalinterneuron synaptic connections in vitro (Debanne et al., 1994, 1998; Laezza et al., 1999). We use a two-threshold LTP/LTD learning rule (Cormier et al., 2001) embedded into a large network model of multicompartment neurons. Our learning rule does not depend on the time-order of presynaptic and postsynaptic spiking. Although this is important in two-cell experiments (Markram et al., 1997; Bi and Poo, 1998), it does not seem necessary in network oscillations under the conditions that we consider. The learning rule depends on presynaptically induced, and postsynaptic voltage-dependent g_Ca-induced, [Ca²⁺]_i signals, which decay with time constants in the tens of milliseconds up to 100 msec. The model can replicate both the structure of / and, with some additional testable assumptions, also the memory-like features.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Simulation methods

The general model principles are as in Traub et al. (1999), with the major difference that certain synapses (from pyramidal cells to other pyramidal cells, and to interneurons) are modifiable, in Hebbian fashion, on the time scale of the oscillations, i.e., tens to hundreds of milliseconds. Such a network property was motivated by earlier simulations (Bibbig, 1999, 2000) in networks of integrate-and-fire neurons, as well as by certain recent experimental data (see Results and Discussion). In addition, the present model has fewer neurons than before (to allow more simulations), and the connectivity is somewhat more realistic.

We chose to use a network model with relatively detailed (multicompartment) models of neurons and synapses for these reasons. (1) Past experience (Traub et al., 1996b, 1999) has shown that detailed models are the most useful for making specific biological predictions. The results are easy to compare with electrophysiological recordings, and quantitative parameters, as they become available, are most readily incorporated into such models. (2) At the same time, it is our opinion that the conceptual complexity of the detailed models is not necessarily more extreme than that of simpler network models (although it can be), despite the vastly larger number of parameters that go into describing the properties of "detailed" neurons. Not only are many of these parameters known experimentally, but the physical principles in the collective behavior of a network are often no harder to grasp in the detailed model than in the simpler one. We do believe, however, that there is still an important role for network simulations using simpler neuron models, such as integrate-and-fire neurons; it was simulations of such networks that gave rise to many of the ideas explored in this paper. Even then, it was not obvious beforehand that learning principles that work in a network of integrate-and-fire neurons would also work in a more complex network of synaptically interconnected multicompartment neurons.

Condensed description of the present model

Overall network structure. The network contains 768 pyramidal cells (excitatory cells, or "e-cells") and 384 interneurons ("i-cells"). The pyramidal cells are arranged into a 96 × 8 array. The long axis has a lattice spacing of 20 µm, and so 96 × 20 µm = 1.92 mm of extent along CA1 stratum pyramidale is represented. The interneurons are arranged in four 96-cell rows, overlapping the e-cell array. Each row of i-cells represents a type of interneuron, distinguished by its postsynaptic connectivity. When, in Results, we speak of the "left half" and "right half" of the array, we mean that the array is split into two halves along its long (96-cell) axis.

Individual neuronal properties; addition of M-current. Each e-cell is a multicompartment object (64 soma-dendritic compartments and 5 axonal ones), with fast g_Na, high-threshold g_Ca, C-type K⁺ conductance (voltage and Ca²⁺-dependent), delayed rectifier g_K(DR), and a slow Ca²⁺-dependent AHP K⁺ conductance, as described in Traub et al. (1994). As in Traub et al. (1999), the density of g_Ca has been reduced twofold with respect to the original model, to suppress bursting, which is not seen during / oscillations. An M-type voltage-dependent K⁺ conductance has been added, because of its contribution to medium-duration AHPs and to firing accommodation (Madison and Nicoll, 1984; Dutar and Nicoll, 1988) and its block, along with slower AHPs, by metabotropic glutamate receptors (Charpak et al., 1990). M-conductance was confined to the soma and proximal basal and apical dendrites. In a particular compartment, the value of the M-conductance was described by this equation:

(1)

In Equation 1, m is a Hodgkin-Huxley-like activation variable. The forward rate function (V) was, in inverse milliseconds and with V referred to resting potential (defined as 0 mV), equal to 0.02/(1 + exp((40  V)/5)), and the backward rate function (V) was 0.01 exp ((17  V)/18). The scaling constant in Equation 1 was time dependent (see below). [All other kinetic parameters were as in Traub et al. (1994).]

Each i-cell is also a multicompartment object (46 soma-dendritic compartments and 5 axonal ones), with multiple ionic conductances, as for e-cells, although without g_K(M) (Traub and Miles, 1995). Dendrites were electrically active (Traub and Miles, 1995; Martina et al., 2000).

[Ca²⁺]_i dynamics in these model neurons follows a simple first-order kinetic scheme, with updating of the variables every 0.25 msec (i.e., every 100 integration steps, each of which is 2.5 µsec). Thus, in each compartment, expressing concentration in arbitrary units:

(2)

In dendritic compartments, _Ca was 20 msec (cf. Miyakawa et al., 1992; B. Sabatini and K. Svoboda, personal communication) (we call this time constant "_post"). A similar scheme was used to simulate [Ca²⁺]_i generated by presynaptic activity; in the case in which the postsynaptic cell is a pyramidal cell, this can be thought of as the synaptically mediated component of spine [Ca²⁺]_i, although the spines themselves were not modeled explicitly. Thus, the program checks when an axonal spike reaches the point of connection between each pair of presynaptic and postsynaptic cells, allowing for axonal conduction delays. If a spike just reaches this site, "I_Ca" was defined to be 1 (units arbitrary) and otherwise to be 0. Equation 2 was applied at each synaptic connection, using _Ca = "_pre" = 25 or 100 msec (Koester and Sakmann, 1998). Note that the model does not simulate, in an explicit way, the effects of metabotropic glutamate receptors, acting via second messenger pathways, on [Ca²⁺]_i (Nakamura et al., 1999, 2000; Pozzo-Miller et al., 2000), but simulates only voltage-dependent effects. We are assuming, as a first approximation, that the metabotropic influence produces a tonic background [Ca²⁺]_i on which voltage-dependent changes are superimposed.

Synaptic connectivity. Each pyramidal cell contacts exactly 30 others, forming a contact with a single compartment in basal dendrites; the probability of connection decreases exponentially with a space constant of 1 mm along the long axis of the array. Each pyramidal cell contacts interneurons, again with probability falling off with 1 mm space constant and with density so that any interneuron was excited by 150 pyramidal cells. Contacts were to single compartments in the dendrites. [In Traub et al. (1999), pyramidal cell connectivity was globally random and did not fall off with distance, but this scheme does not appear to be consistent with data of Csicsvári et al. (1998).]

As in Traub et al. (1999), each pyramidal cell receives input from 80 interneurons, 20 from the interneurons of the first row ("basket cells"), 20 from interneurons of the second row ("axo-axonic cells"), 20 from i-cells of the third row ("bistratified cells"), and 20 from i-cells of the fourth row ("o/lm cells"). Interneurons receive the same number of inputs from other interneurons, as do pyramidal cells, with the exception that interneurons are not contacted by axo-axonic cells. Basket cells contact uniformly the soma and most proximal dendrites of pyramidal cells and dendrites of interneurons. Axo-axonic cells contact the initial segment (most proximal axonal compartment) of pyramidal cells. Bistratified cells and o/lm cells contact the dendrites of pyramidal cells and interneurons. The axons of interneurons are constrained to run no farther than 25 cell diameters (500 µm) along the long axis of the array; within this domain, interneuron connection probabilities are uniform.

Synaptic actions. AMPA- and GABA_A-receptor-mediated synaptic connections were simulated. A unitary synaptic conductance was switched on when (1) the most distal axonal compartment of the presynaptic neuron was depolarized 70 mV from the rest, and no such depolarization has occurred in the last 4 msec, and (2) a signal propagated, over a delay line representing axonal conduction delay, from the axonal compartment to the postsynaptic neuron. The general form of a unitary ee synaptic conductance was c_ee t exp (t/2), where t is the time in milliseconds, and c_ee is a scaling parameter; for unitary ei synaptic conductance it was c_ei t exp (t). The scaling parameters c_ee and c_ei depend on learning in a manner described below.

The general form of a unitary IPSC was c_i exp(t/10). Default values of c_i were as follows: basket cellpyramidal cell, 1.6 nS; basket cellinterneuron, 2.3 nS; axo-axonic cellpyramidal cell, 1.6 nS; bistratified or o/lm cellpyramidal cell, 1.6 nS; bistratified or o/lm cellinterneuron, 0.23 nS. In simulations of the effects of morphine, smaller values of c_i were used (Madison and Nicoll, 1988).

Stimulation conditions. As before (Traub et al., 1999), oscillations were evoked by applying tonic "metabotropic" conductances to dendrites of principal cells and interneurons (Whittington et al., 1997b). The reversal potential of this conductance was 60 mV positive to resting potentials. Interneurons received a conductance of 4.0-4.2 nS. For pyramidal cells, "low heterogeneity" and "high heterogeneity" conditions were used on different occasions. In the former, the maximum tonic conductance was 75.0-77.5 nS; in the latter, it was 75.0-82.5 nS. In some cases, stimulation was applied only to half of the array. Details are specified in Results. The tonic excitatory conductance to pyramidal cells was time dependent, starting at 0 at time 0, rising to its maximum over 100 msec (Whittington et al., 1997b), staying constant for the next 700 msec, and then declining linearly with time to 55% of the maximum value, agreeing qualitatively with experimental data (Whittington et al., 1997b).

Time-varying maximal K⁺ conductances. As in the previous study (Traub et al., 1999), and as motivated by experimental observations (Whittington et al., 1997a), certain K⁺ conductances are presumed to be suppressed at the beginning of the tetanically elicited  oscillation and then to recover during the course of  and into . In this study, time-varying K⁺ conductances were g_K(M) and g_K(AHP). To define the value of g_K(M) density in each compartment, the scaling constant in Equation 1 was varied as follows: it was 0.25 for time 250 msec, grew linearly with time to 1.3 over the interval 250-1000 msec, and then stayed at 1.3. To define g_K(AHP) density, the reference value as used in Traub et al. (1994) was used, but it was also multiplied by a time-dependent scaling constant. This scaling constant was 0.25 for time 250 msec, usually grew linearly with time to 1.25 over the interval 250-1000 msec, and then stayed at 1.25. In some cases, both of the scaling constants grew linearly with time to 1.3 and 1.25, respectively, over the interval 250-500 msec, and then stayed fixed.

It should be noted that metabotropic glutamate receptors depress K⁺ conductances via a G-protein-dependent pathway, whereas the metabotropic slow EPSP is mediated by a G-protein-independent process, involving an Src-family protein tyrosine kinase (Guérineau et al., 1994; Heuss et al., 1999); thus, these two parameters, EPSC and K⁺ conductances, in principle could be independently regulated.

Learning. As noted above, ee synapses and ei synapses are modifiable during the course of a simulated oscillation. The general rules for this modification, which is "Hebbian" in the sense of depending on correlations between presynaptic and postsynaptic activity, are as follows.

(1) c_ee and c_ei, the scaling constants for ee and ei synaptic connections, respectively, can assume independent values at each synaptic connection, not depending on values assumed at other connections (apart from the initial conditions).

(2) The program sets initial values and maximum values for the scaling constants. The minimum values are 0. Initial values are c_ee = 0.3 nS and c_ei = 1.0 nS. Maximum values are c_ee = 7.5 nS and c_ei = 3.0 nS.

(3) The signals used to "integrate" presynaptic and postsynaptic activity, and hence used to determine whether synaptic conductances increase, decrease, or remain fixed over some time interval, are [Ca²⁺]_i concentrations. The "presynaptic" signal can be thought of as a local [Ca²⁺]_i signal gated by a presynaptic action potential and might correspond (in the case of a pyramidal cell) to the [Ca²⁺]_i rise in the spine induced by presynaptic activity. The "postsynaptic" signal can be thought of as a localized [Ca²⁺]_i signal induced by voltage-dependent activity in the postsynaptic cell, in basal dendrites (for pyramidal cells), or in selected portions of the dendrites (for interneurons). The equations governing [Ca²⁺]_i dynamics were described above. The postsynaptic signal used was not [Ca²⁺]_i in the individual dendritic compartment on which the synapse was located; rather, the total value of [Ca²⁺]_i was used, summing over compartments on which excitatory synapses could be located. This spatial averaging was done to smooth over wide differences in peak [Ca²⁺]_i values that could occur at different dendritic locations. Consideration of each separate [Ca²⁺]_i signal would have introduced impractically many parameters into the system, because each dendritic compartment, in principle, might have needed its own values of the learning thresholds (see below). In addition, it should be noted that somatic spikes propagated, in our model, to all compartments in the basal dendrites with little decrement. We did not explicitly simulate the release of [Ca²⁺]_i from internal stores or the actions of metabotropic glutamate receptors on [Ca²⁺]_i dynamics.

(4) Learning began 175 msec into the simulation, to allow equilibration of the system.

(5) The learning code was executed once per millisecond. It used a two-threshold rule formally similar to (but not identical to) that used by other authors (Bienenstock et al., 1982; Artola et al., 1990). Thus, fixed postsynaptic and presynaptic thresholds were set at the beginning of the program, T_post and T_pre, equal to 75 and to 1.0, respectively (units arbitrary). If both presynaptically gated and postsynaptic [Ca²⁺]_i signals were above their respective threshold values, then the appropriate scaling constant was increased by a preset "up" value. If one of the [Ca²⁺]_i signals, but not the other, was above its respective threshold, then the appropriate scaling constant was decreased by a preset "down" value. If both [Ca²⁺]_i signals were below the respective thresholds, then the scaling constant was not changed. Specific choices most often used for the up and down values were these: c_ee up, 18.75 pS; c_ee down, 1.875 pS; c_ei up, 6.0 pS; c_ei down, 0.6 pS. Other choices were also tried, particularly when the time constants for relaxation of [Ca²⁺]_i were varied. Suitable values for T_post and T_pre were found after extensive trial simulations.

(6) An alternative learning rule was sometimes used for ei synaptic modification, intended to emulate use-dependent removal of polyamine block of AMPA receptors (Rozov et al., 1998). In this case, opening of AMPA receptors on interneurons, induced by transmitter binding, leads to detachment of a molecule from the receptor, a molecule the presence of which would lower channel conductance. The binding of the transmitter, glutamate, is determined, of course, by presynaptic firing. Thus, the presynaptic signal to be used in the learning rule does not correspond to synaptically induced changes in [Ca²⁺]_i but rather to the extent of AMPA receptor-gated channel opening. The signal was constructed formally in the same way as described above (Eq. 2 and after), only with a decay time constant of 1 msec rather than 25 msec, approximating the kinetics of AMPA receptor-gated conductance in interneurons (Geiger et al., 1995). Specifically, let us call the presynaptic signal in the present case "AMPA-gated." Then:

(3)

As for the usual learning rule, I_Ca,pre equals 1 when a spike reaches the presynaptic terminal and otherwise is 0.

In such a case, when polyamine block at ei synapses is being simulated, the learning parameters were changed: T_pre was set to 0.1 (because the channel is open now only a small fraction of the time, and a low threshold is necessary for potentiation to occur at all), and T_post was set to 0, making synaptic modification solely dependent on presynaptic firing, i.e., on glutamate release (Rozov et al., 1998). c_ei up was 15 pS, and c_ei down was 0.3 pS. (Note that the presynaptic signal is not identical in time course to a unitary EPSC, but with synaptic modifications occurring only once per millisecond, it is accurate enough for our purposes.)

(7) We did not try an analogous presynaptic learning rule for synapses between pyramidal cells for two reasons. First, [Ca²⁺] imaging data (Yuste and Denk, 1995) are consistent with a Hebbian mechanism. Second, the effects of intense tissue stimulation on oscillations, effects that appear to be mediated at least in part by changes in synaptic strength, last for tens of minutes (Whittington et al., 1997a); this makes a presynaptic mechanism unlikely.

Some relevant characteristics of the learning algorithm, for the case _pre = 25 msec and _post = 20 msec, are shown in Figure 1. Of note are several features. First, the time constants of [Ca²⁺]_i decay in the model, 20-25 msec, are taken to represent what we presume to be the fastest decay time constant of this signal in dendrites (Miyakawa et al., 1992; Sabatini and Svoboda, personal communication); other, slower time constants are also present (Koester and Sakmann, 1998; Majewska et al., 2000; Schwartz and Alford, 2000). [In some cases, therefore, we also used _pre or _post (or both) = 100 msec at ee connections.] Second, the fraction of time that a [Ca²⁺]_i signal spends above threshold is influenced by firing rate (compare, for example, the  and  portions of the simulation); learning here thus is influenced by oscillation period, as well as by details of synchronization. Learning will be influenced, in addition, by "beat-skipping"; for instance, if the postsynaptic cell fails to fire on a peak of a particular  wave, then the postsynaptic dendritic [Ca²⁺]_i signal will be severely attenuated. This occurs because the model voltage-dependent calcium conductance, in basal dendrites, closely follows the depolarization induced by the somatic action potential; the latter readily and faithfully propagates into the basal dendrites of model pyramidal neurons. [The reader will recall that recurrent pyramidalpyramidal synaptic connections, in CA1, are largely in the basal dendrites (Deuchars and Thomson, 1996), and that is where pyramidal cell learning is presumed to occur during tetanic CA1 / (Faulkner et al., 1999; Traub et al., 1999).] Finally, in thinking about the results, the reader must constantly have in mind axonal conduction delays in the system, which in some of our simulations are >10 msec. It is the correlation between [Ca²⁺]_i signals at postsynaptic dendrites and presynaptic terminals (not presynaptic cell bodies) that controls synaptic conductance changes; depolarization at the presynaptic terminal can be delayed by more than half a  cycle from the action potential at the presynaptic soma. This important detail is different from the learning schemes used in many connectionist models.

View larger version (47K): [in this window] [in a new window]   Figure 1.   Example of fluctuations in [Ca2+]i signals during the course of a simulated network oscillation (same simulation as Fig. 3). The entire  and a portion of the  oscillation are shown. Top traces, Signals from a site in the basal dendrites of pyramidal cell 1. Thick trace is the local [Ca2+]i signal (units arbitrary), and thin trace is the postsynaptic voltage at the same site (action potentials truncated; the amplitude of action potentials at this site is ~60 mV depolarized from rest). Note the subthreshold voltage fluctuations between full action potentials, during . Middle trace, Total [Ca2+]i signal in pyramidal cell 1 (units arbitrary); horizontal line through this trace shows the postsynaptic learning threshold, Tpost. Bottom trace, A presynaptic [Ca2+]i signal at a synapse located at the same site in the basal dendrites of pyramidal cell 1; horizontal line through this trace shows the presynaptic learning threshold, Tpre. Decay time constants are post = 20 msec (dendritic voltage-induced signal) and pre = 25 msec (presynaptically induced signal) of the same order as the period of the  oscillation.

As noted above, experimentally, ei potentiation can also occur by activity-dependent removal of polyamine block, in AMPA receptors lacking GluR-B (GluR2), i.e., in AMPA receptors of a sort often found in interneurons (Geiger et al., 1995). This phenomenon is Hebbian in that it requires presynaptic activity, whereas it is being expressed postsynaptically (Rozov et al., 1998). On the other hand, removal of polyamine block occurs faster at hyperpolarized membrane potentials in the postsynaptic neuron than at depolarized potentials (Rozov et al., 1998). Therefore, it was considered necessary to examine this type of learning in the model as well. Note that Rozov et al. (1998) described conductance increases, with 33 Hz stimulation, up to ~38% in homomeric GluR-B(Q) channels expressed in human embryonic kidney cells.

We did not use learning at inhibitory synapses in this model [see, however, Bibbig (1999, 2000)]. [Changes in ie synaptic connections are difficult to document experimentally during the course of /. For example, isolation of IPSPs requires blockade of AMPA receptors, but block of AMPA receptors prevents the  portion of the oscillation from occurring in a normal way (Traub et al., 1999).] We did not attempt to model the decay of synaptic potentiation. On the one hand, we lack the quantitative data to do so; on the other hand, the decay time constants are likely to be minutes or longer, and it is only practical for us to simulate a few seconds of neuronal activity, at most.

Axon conduction delays. Pyramidal cell axons conducted at 0.5 mm/msec, and interneuron axons conducted at 0.2 mm/msec. Thus, the maximum conduction delay for excitation across the array was 3.84 msec. In some simulations, the array was split in two, imposing an extra conduction delay (up to 20 msec) for axons crossing the midline (cf. Kopell et al., 2000).

Noise. Noise was simulated, as before (Traub et al., 1999), with ectopic spontaneous axonal action potentials, originating by independent Poisson processes, with the average interval at 10 sec in e-cell axons and 5 sec in i-cell axons.

Signals saved and data analysis. The program saved voltages of selected cells (soma, dendrites, terminal axon), [Ca²⁺]_i signals, and synaptic input conductances. It saved, in addition, e-cell spatial averages (56 cell somata) and i-cell spatial averages (28 cell somata), one average from either end of the array. The average signals are presented both as raw data, and in auto- and cross-correlations, the latter using 200-800 msec of data. Average values of synaptic scaling constants, c_ee and c_ei, were also saved. These data were saved either as averages regardless of relative position of presynaptic and postsynaptic cells or as averages over the four cases in which presynaptic or postsynaptic cells lay in left versus right halves of the array.

Data base, run times, programming, and systems aspects. After numerous preliminary simulations, mostly aimed at defining parameters of the learning rule, a data base of >120 simulations was accumulated. Code was written in FORTRAN augmented with extra instructions for a parallel computer and run on an IBM SP2 machine with 12 processors. A typical 2 sec simulation took ~6 hr to run. For details on programming aspects, contact rtraub{at}netmail.hscbklyn.edu.

Experimental methods

Transverse dorsal CA1 hippocampal slices 400-450 µm thick were prepared from brains of Sprague Dawley rats (200-250 gm), which were killed by cervical dislocation followed by decapitation. Slices were maintained at 34-35° at the interface between warm, wetted 95% O₂-5% CO₂ and artificial CSF (ACSF) containing (in mM): NaCl 135, KCl 3, NaHCO₃ 16, NaH₂PO₄ 1.25, CaCl₂ 1.5-2, MgCl₂ 0.8, D-glucose 10.

Oscillations were evoked with tetanic stimuli delivered to proximal stratum radiatum at two sites simultaneously (CA1a and CA1c; separation 1.5-2.5 mm). Two types of experiments were performed. In the first, fast-spiking interneurons were impaled at the level of stratum pyramidale at one site, with tetani consisting of eight stimuli delivered at 100 Hz. Recording electrodes (40-90 M) were filled with 2 M potassium acetate or potassium methylsulfate. In the second, both stratum pyramidale and distal stratum oriens field potentials were recorded at one or both sites, with tetani consisting of 20 pulses at 100 Hz. Recording electrodes (0.5-1 M) were filled with 2 M sodium chloride.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

EPSP amplitudes increase in interneurons during the course of tetanically elicited  oscillations

In Whittington et al. (1997a), two cellular phenomena were documented as taking place, simultaneously, during the transition from  to  frequencies in tetanically induced oscillations: an increase in apparent spike AHPs and an increase in AMPA-receptor-mediated EPSPs in pyramidal cells. Figure 2, A and B, shows that compound EPSPs also increase in interneurons under the same conditions. The EPSPs in interneurons during  are broader, with an apparent multicomponent structure, than the EPSPs during ; in control simulations, phasic AMPA receptor-mediated excitations also become broader during  as compared with . The increase in interneuronal EPSPs in the oscillating slice stands in contrast to the depression of interneuronal EPSPs often seen in the resting slice, when a single presynaptic pyramidal cell is induced to fire repetitively (Ali et al., 1998).

View larger version (29K): [in this window] [in a new window]   Figure 2.   Compound EPSPs grow in both interneurons and pyramidal neurons during the course of tetanically evoked /. A, Intracellular recording of EPSPs in an interneuron, hyperpolarized to 70 mV by current injection (0.3 nA). Example traces show pattern of EPSPs during the initial  component of the post-tetanic response and the later  component. Calibration: 100 msec, 5 mV. B, Example trace of EPSPs recorded from the beginning of the post-tetanic response to the beginning of the  oscillation, in an interneuron held at 70 mV. Calibration: 100 msec, 5 mV. Graph shows pooled data from five interneurons with mean (±SEM) EPSP amplitudes for each of the first 20 periods of the post-tetanic response. C, Example of field EPSPs recorded in stratum oriens. Trace shows response from the end of the tetanic stimulation to the beginning of the  oscillation. Graphs shows pooled data, expressed as mean (±SEM; n = 4) amplitude of the field EPSP for each of the 20 periods of oscillation. D, Growth of field EPSPs is not an artifact of growth of population spikes. Traces recorded concurrently in stratum oriens (top traces) and stratum pyramidale (bottom traces) during the initial -frequency oscillation and the later -frequency oscillation. Calibration: 100 msec, 2 mV.

Figure 2, C and D, again documents the increase of field EPSPs (extracellular potentials corresponding to synchronized EPSPs in many nearby neurons, measured in stratum oriens) during the course of /. Figure 2 emphasizes that this increase occurs despite the fact that population spikes, in stratum pyramidale, are not larger in  compared with ; thus, the growth in field EPSPs is not a simple byproduct of increased synchronization, or increased firing, of pyramidal neurons (cf. Faulkner et al., 1999; Traub et al., 1999).

Categories of simulations

Most of the simulations performed fell into two categories. In the first category ("Category 1"), we fixed _pre at 25 msec, for connections onto either e-cells or i-cells; T_post was a positive number (75.0) in both e-cells and i-cells. This is, so to speak, "strictly Hebbian." In the second category ("Category 2"), we set _pre at 100 msec for ee connections (Koester and Sakmann, 1998), and we set it at 1 msec for ei connections (Rozov et al., 1998). T_post in pyramidal cells was the same as in the first category, but was set to 0 in interneurons. Thus, simulations in the second category use learning at excitatory connections onto pyramidal cells that is formally Hebbian, but with a longer presynaptic time constant than in the first category. Additionally, in the second category, excitatory connections onto interneurons learn by a process resembling removal of polyamine block. Learning rates at the different sorts of synapses were adjusted accordingly. The first category of model can replicate virtually all of the experimental data, and for most of the figures below, we illustrate examples chosen from the first category of model. Nevertheless, as noted below, the second category of model can also replicate many of the data.

We also used further variations of these cases, by altering _pre or _post in different combinations. These variations are noted in the text.

Learning can lead, in cooperation with increasing K+ conductances, to an organized transition

The mechanistic idea on the transition proposed in Traub et al. (1999) was this: tetanic stimulation evokes  oscillations, in which both pyramidal cells and interneurons participate and in case two sites are stimulated, with interneuron doublets stabilizing the synchrony between sites (Traub et al., 1996b; Whittington et al., 1997b). The oscillation is primarily gated by IPSPs, both in pyramidal cells and in interneurons. If the stimulus is strong enough to produce a long-lasting depolarization (Faulkner et al., 1999; Whittington et al., 2001), interneurons remain excited enough to generate a long-lasting  oscillation, whether or not the pyramidal cells are firing (Whittington et al., 1995; Traub et al., 1996a), so-called "ING" (interneuron network ). As AHP conductance(s) increase in pyramidal neurons, the latter become unable to follow the interneuron oscillation cycle by cycle and skip beats, so that pyramidal cells fire at  frequency, even as interneurons continue to fire at  frequency (more accurately, to fire singlets, doublets, or brief bursts at  frequency). Hence, increases in AHPs can account for  phenomenology, at least in individual pyramidal cells. Nevertheless, AHP increases, by themselves, do not account for what happens in the whole system: without coupling between the pyramidal cells, different pyramidal cells would tend to skip beats in a manner only loosely coupled together, a "disorganized" . The increases in pyramidal cell EPSPs that also occur, however, introduce correlations between which ING cycles are skipped by the different pyramidal cells and help to lead to an organized . This idea works, so far as it goes, both in detailed network simulations (Traub et al., 1999) and in reduced models that can be analyzed more rigorously (Kopell et al., 2000).

There is no obvious reason to think that the AHP increases are "self-organized" (that is, dependent on communication between cells in the network). It is possible, however, that the EPSP increases are self-organized, given that (1) Hebbian-type synaptic plasticity exists between hippocampal and cortical neurons (Stanton and Sejnowski, 1989; Debanne et al., 1994, 1998; Ouardouz and Lacaille, 1995; Markram et al., 1997; Laezza et al., 1999; Dragoi et al., 2000) and (2) metabotropic receptors (known to be critical for inhibition-based  rhythms (Whittington et al., 1997b, 2001), intrinsic membrane properties, and phasic synaptic inputs together interact to give supralinear increases in dendritic [Ca²⁺]_i signals, thus providing a possible physical substrate for Hebbian synaptic plasticity (Christie et al., 1996; Emptage et al., 1999; Nakamura et al., 1999, 2000; Normann et al., 2000; Perez et al., 2000, 2001). There is also in vivo evidence that metabotropic glutamate receptors are important for functional learning, although the cellular mechanisms are not clear (Balschun et al., 1999).

For these reasons, we allowed maximum g_K(M) and g_K(AHP) conductances to increase in a prespecified time-dependent manner (see Materials and Methods), similar to our earlier study (Traub et al., 1999), but, we let ee and ei synapses modify, cooperatively, as described in Materials and Methods. Figure 3 demonstrates that, at least with certain initial conditions and learning parameters, a realistic-appearing / oscillation still can occur. In particular, e-cell  occurs when i-cells continue to form an oscillation at  frequency, so that beat-skipping during  takes place (Fig. 3A, asterisk). Both  and  portions of the oscillation are synchronized (Fig. 3B). The slowing of  before the "switch" to  at time ~900 msec (Fig. 3C) is seen experimentally (Whittington et al., 1997a). The increase in ee and ei conductances, depicted in Figure 3D, allows interneuron singlets to switch to (mostly) doublets (Fig. 3A, d) during , as often occurs experimentally (M. A. Whittington, unpublished data), and the increase in conductances provides enough coupling between pyramidal cells to allow long-range synchrony of . Once  is established, further increases in synaptic conductances stop, in this simulation. This is a result of the relatively reduced e-cell firing rate during , with consequent reduction of [Ca²⁺]_i signals to levels that are most often below threshold for synaptic conductance increases. Note that spike AHPs increase from  to  (Fig. 3A, arrowheads). Analysis of the GABA_A conductance to this pyramidal cell (data not shown) indicates, however, that part of this increase is actually attributable to rises in synaptic inhibition: not growth of unitary IPSCs, an effect not included in this simulation, but rather a reflection of interneuron doublets and, during , an occasional triplet.

View larger version (43K): [in this window] [in a new window]   Figure 3.   Simulated / oscillation (category 1 model). A, Simultaneous traces showing local averages of e-cell voltages (from either end of the array), an e-cell (pyramidal cell) soma, the total AMPA conductance delivered to that e-cell, an i-cell (interneuron), and the total AMPA conductance delivered to that interneuron. Asterisk indicates underlying subthreshold  during ; arrowheads under the e-cell trace emphasize the growth of pyramidal cell AHPs (partly reflecting synaptic conductances) from  to . d is the first doublet generated by the interneuron, and s indicates a singlet amid the doublet firing. B, Superimposed auto- and cross-correlations of average e-cell signals, from  and  portions of the oscillation. Note the presence of low-amplitude  activity in the  correlations. C, Instantaneous frequency plot, calculated from local average e-cell signal at one site (cf. Whittington et al., 1997a). D, Average unitary synaptic scaling factors, for ee and ei synaptic connections, showing the time course of learning through the evolution of the oscillation. Virtually all of the learning takes place during . These signals were averages of excitatory synaptic connections on 64 e-cells and 32 i-cells.

Figure 3 illustrates another interesting feature: there is a period of a few hundred milliseconds in which interneurons do not fire doublets (doublet firing begins at approximately t = 275 msec), and yet  is synchronized, at least transiently [because  synchrony without doublets is not stable (Ermentrout and Kopell, 1998)]. The  cross-correlation of Figure 3B uses data from t = 100-300 msec. Once doublet firing begins, the oscillations slows, as expected, and remains synchronized (data not shown), also as expected (Traub et al., 1996b).

ei plasticity influences the number of interneuron doublets during  and the tightness of synchronization, whereas ee plasticity has little effect

With the same parameters as in Figure 3, but without ei plasticity (data not shown), the period of  from 100 to 300 msec had only a small (1.5 msec) lag in the cross-correlation of e-cell signals, but with few doublets occurring during any portion of ; the period of  from 300 to 500 msec had a cross-correlation that, although possessing a central peak near 0 (0.6 msec), had multiple small side peaks. (The corresponding cross-correlation for the simulation of Fig. 3 was narrow and had a single small side peak.)  phenomenology was similar to the case of blocked ei plasticity, when ee plasticity was also blocked (data not shown).

Simulations with the second category of model (defined above) were also able to replicate / oscillations, which had an appearance quite similar to that in Figure 3 (Fig. 4).

View larger version (47K): [in this window] [in a new window]   Figure 4.   The / transition can also be replicated when ei learning simulates the removal of polyamine block (category 2 model) (cf. Rozov et al., 1998). The conditions of this simulation are the same as in the run of Figure 3, except that some of the [Ca2+]i dynamics and learning parameters were altered: pre for ee connections was increased from 25 to 100 msec, and for ei connections was reduced from 25 to 1 msec (corresponding approximately to the open time of interneuron AMPA receptors). In interneurons, Tpost, the postsynaptic learning threshold, was set to 0, making learning at ei connections entirely dependent on presynaptic activity, even as the learning is expressed postsynaptically. Finally, learning rates at ei connections were adjusted, as described in the learning section of Materials and Methods. The traces in A-D correspond to those in Figure 3.

Effects of altering _pre or _post

We tried the second category of model (that is, ei learning depends on polyamine unblocking) with different combinations of pyramidal cell _pre and _post. With _pre = 100 msec and _post = 20 msec, we have the usual second category case, described above. Setting both parameters to 100 msec led to a case in which there was a large slow envelope of the [Ca²⁺]_i signal in basal dendrites, making it impossible to select a fixed learning threshold, so that increases in EPSC amplitudes would not take place continuously. As a result, pyramidal cell doublets would occur quickly (data not shown), contrary to experimental observations. On the other hand, with _post = 100 msec and _pre = 25 msec, / occurred that resembled the case of Figure 3, but with certain important details disagreeing with the experiment. For example, it was not possible to chose the learning threshold T_pre so that no pyramidal cell doublets would occur during  and still have enough learning for synchronized  to occur (data not shown). These data suggest, therefore, that one or the other [Ca²⁺]_i decay time constants, but not necessarily both, should have a value similar to the  oscillation period; furthermore, at least in our hands, the model results were most realistic when it was _post that had a value close to the  oscillation period. Recall that the time constants of postsynaptic [Ca²⁺]_i decay in the model, 20-25 msec, are taken to represent what we presume to be the fastest decay time constant of this signal in dendrites (Miyakawa et al., 1992; Sabatini and Svoboda, personal communication).

Initial synaptic conditions could be important in determining whether organized  occurs

There is an interesting experimental observation on  that may be related to memory: smaller tetani evoke , which can be synchronized between two sites, but not , or at least not  that is synchronized between sites (Whittington et al., 1997a; Traub et al., 1999). Nevertheless, despite this, a single strong stimulus, evoking two-site synchronized , allows future weaker stimuli to evoke two-site synchronized . This observation might be explained by two assumptions, both testable in principle: (1) the learning rates in the system increase with the strength of stimulation, an idea consistent with observations that metabotropic glutamate receptors both influence synaptic plasticity (Bortolotto et al., 1999) and also exert a cooperative effect with phasic synaptic inputs and dendritic g_Ca in regulating [Ca²⁺]_i (Nakamura et al., 2000), and (2) a single two-site-synchronized  epoch leaves excitatory synapses potentiated above their baseline values. The simulations in Figure 5 were undertaken to test the feasibility of this idea.

View larger version (40K): [in this window] [in a new window]   Figure 5.   can evolve into an "organized" (i.e., long-range synchronized)  rhythm, if the learning rate is fast enough, or with a slower learning rate, if combined with elevated initial excitatory conductances. Traces on the left are, respectively, average e-cell voltages from either end of the array (designated here V1 and V2) and the total AMPA conductance received by a selected e-cell. Traces on the right are cross-correlations of the last 800 msec (in 2 sec runs) of V1 and V2, time periods during which  would be expected to occur. A, Case in which learning rates are slow for ee and ei conductances, that is, half the usual values (see Materials and Methods). There is some rise in AMPA conductances during , and each side develops its own  rhythm, but in the absence of sufficient excitatory coupling, the sides produce  that is out of phase (double-headed arrow) (cf. Traub et al., 1999). The cross-correlation has its major peaks at  frequency. B, With the usual learning rate, AMPA conductances increase more than in A.  between the two sides is correlated, and the major side peak in the cross-correlation is at 77 msec (13 Hz). C, The learning rates are slow, as in A, but initial values of ee and ei conductances are elevated above their usual values (but not so much that e-cell doublets occur during  or that i-cell doublets occur immediately). Note the horizontal arrows at the start of the AMPA signals. Again,  between the two sides is correlated, and the cross-correlation is similar to that in B.

Figure 5 illustrates effects of learning parameters and of initial synaptic conductances. Figure 5A shows a simulation identical to that of Figure 3, except that the learning rates have been reduced 50%; that is, the up and down increment values have all been cut in half. In this case, ee connections do not become as strengthened as in Figure 3. Compare the AMPA conductance in an e-cell in Figure 5A with the AMPA conductance in the same e-cell in Figure 5B, which uses the same data as Figure 3. (Note also that the AMPA conductance stays smaller in Fig. 5A than in 5B, even before  has started, so that the reduced AMPA size in the  portion of Fig. 5A is not simply a consequence of reduced synchrony.) The  part of the simulation in Figure 5A is synchronized (phase differences <1.5 msec; data not shown), and  can develop at each individual site, because of the time-dependent growth in K⁺ conductances. Nevertheless, because of the small ee coupling present when  is starting, the two ends of the array do no oscillate in stable synchrony; indeed, our simulations showed an anti-phase oscillation (double-headed arrow).

The simulation of Figure 5A, using the slowed learning rates, was then repeated (Fig. 5C), but now with higher initial unitary ee conductance (5×; note horizontal arrows on the left of the figure showing initial excitatory conductances "seen" by a selected e-cell) and higher initial unitary ei conductance (1.25×). In this case, excitatory conductances become potentiated enough that  can synchronize between the two sides, as shown in the cross-correlation on the right. The higher initial excitatory conductances come at a price, however:  is now not organized as well, the cross-correlation (data not shown) containing a split peak instead of a single sharp peak near 0 msec. Maxima of this split peak were at 6.6 and +3.4 msec, for data from 100 to 300 msec after the start of the oscillation. Such a "disorganizing" effect on  appeared to be caused by the appearance of interneuron doublets at one site before the other site, followed by alternating singlets and doublets at each site, with the patterns out of phase between the two sites. Other data (Fuchs et al., 2001) suggest that excessively prolonged AMPA receptor-mediated excitation of interneurons can actually be detrimental to  synchrony. When initial unitary ee conductances were too large, pyramidal cell doublets occurred during  (data not shown), something observed only rarely experimentally (H. J. Faulkner and M. A. Whittington, unpublished data). Therefore, there is a constraint on how large initial excitatory conductances can be.

In summary, Figure 5 shows that lasting effects on EPSPs, produced by an oscillatory epoch containing , along with stimulus-dependent learning rates, could explain the experimental observations outlined at the start of this section. There is, however, another means to produce the results shown in Figure 5. This is to suppose that a weak stimulus allows the AHP conductance to return to baseline faster than does a strong stimulus; such an idea is also consistent with experimental observations on the suppression of the AHP conductance by metabotropic glutamate receptor activation (Charpak et al., 1990), although to our knowledge a dose-response curve has not been determined for the duration of AHP suppression versus metabotropic activation. Thus, when the simulations of Figure 5, A and C, were repeated, with the AHP recovering over the interval from 250 to 500 msec, as compared with the usual 250-1000 msec (used in Fig. 5A, and other simulations in this paper), then we observed the following. With synaptic conductances starting at their baseline values, as in Figure 5A, having the AHP recover rapidly, as might be expected with weak stimulation, led to  that was not synchronized between the two sites, analogous to the behavior shown in Figure 5A: not enough learning took place for  synchrony to occur. On the other hand, when using rapid recovery of the AHP, but with higher initial values of excitatory synaptic conductances (just as in Fig. 5C), then organized and synchronized  did occur, also as illustrated in Figure 5C (data not shown). In summary, there are two possible explanations for the ability of a single -inducing stimulation to allow subsequent weaker stimulations to induce synchronized : a dependence of learning rates on stimulus intensity and a dependence of AHP recovery kinetics on stimulus intensity. These explanations are not mutually exclusive.

In addition, the data of Figure 5 could also be replicated using a category 2 model, using the same manipulations as in Figure 5. Reducing the learning rates prevents organized  from occurring, whereas using reduced learning rates, along with increased initial values of the starting conductances, does allow organized  to occur (data not shown).

Because experimental observations (Whittington et al., 1997a, their Fig. 2), and also our model (Fig. 2), indicate that excitatory synapses become strengthened during  and during the transition rather than during  itself, the model therefore predicts the following: that strengthened excitatory synapses do not decay all the way back to their baseline conductances during  or during the subsequent "quiet" period before the next stimulus. A further testable prediction of the model, as discussed above (Fig. 5C), is that  evoked after an episode of