INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Sensory processing involves several brain areas, and within each area, several groups of neurons. It has been proposed that the neuronal assemblies representing different parts of one "object" are bound together by synchronous oscillatory activity in the  (30-70 Hz) and/or  (10-29 Hz) range (e.g., Gray 1994; Singer and Gray 1995; Tallon-Baudry et al. 1999). Stimulus-specific synchronized  activity has, for example, been reported in the visual cortex of the anesthetized cat (Eckhorn et al. 1988; Gray and Singer 1989; Gray et al. 1989) and in the awake monkey (Eckhorn et al. 1993; Frien et al. 1994; Kreiter and Singer 1996). Stimulus-specific synchronized  or / activity has also been shown in the human brain during perceptual and learning tasks (Tallon-Baudry et al. 1998, 1999, 2001; von Stein et al. 1999), and most interestingly, synchronization was reported despite separation between participating areas of several millimeters and up to several centimeters (Desmedt and Tomberg 1994; Miltner et al. 1999; Tallon-Baudry 2001). Areas separated by distances of several centimeters are likely to be connected by fast cortico-cortical connections, although the relevant axonal conduction delays are not known. However, axonal collaterals within the gray matter are reported to conduct with axonal conduction velocities as small as 0.1 mm/ms (Aroniadou and Keller 1993). Thus even spatial separations of 3-4 mm (common for extrinsic collaterals; Rockland and Drash 1996), or of 3-6 mm (common for intrinsic, i.e., within-area, collaterals; K. Rockland, personal communication) lead to large temporal separations, in principle up to 60 ms. Because synchronization has been reported over these distances, there must be a mechanism for synchronizing neocortical neuronal assemblies despite significant separation in space and/or very large conduction delays.

Kopell et al. (2000) showed that, while two assemblies separated by 10 ms or more could synchronize during a  oscillation, they could not stably synchronize during a  oscillation [by " oscillation" we mean an oscillation with inhibitory cells (i-cells) firing at  frequency while excitatory cells (e-cells) skip beats and thus only fire at  frequency]. This was shown mathematically and using simulations with conductance-based models, not including, however, synaptic plasticity. Other studies have reported that synchronization was unreliable if axonal conduction delays of more than 6 ms were involved (Ritz et al. 1994), or that synchronization could be achieved despite larger conduction delays if the carrier frequency was slow enough (König and Schillen 1991). Again, the synapses used in these studies were not plastic.

Preliminary modeling studies using networks of simple (Bibbig 1998, 1999, 2000; Bibbig and Traub 2000) and detailed compartmental neuronal models (Bibbig et al. 2001) showed that synchronization with axonal conduction delays of 10 ms and more can be achieved with the help of interneuron plasticity (the plasticity of reciprocal pyramidal/interneuron conductances, usually in this paper referred to as e  i and i  e synapses). In Bibbig et al. (2001), potentiation of i  e synapses is replaced by growing afterhyperpolarizations (AHPs) in pyramidal cells but not in interneurons. This is possible even though average axonal conduction delays 8 ms force the two assemblies to fire with large phase lags, leading to an almost anti-phase activity, shortly after the beginning of the oscillation. But with a so-called "synchronizing-weak-beat" ( beat with sparse i- and e-cell activity), the oscillation can switch from almost anti-phase to near-synchrony. In Bibbig (1999, 2000), Bibbig and Traub (2000), and Bibbig et al. (2001), we introduced the phenomenon of a synchronizing-weak-beat and named it "i-weak beat," referring to its sparse i-cell activity. However, we now call it a "synchronizing-weak-beat" to emphasize its function, and because not only i-cell, but also e-cell, activity is sparse during such a beat. We will explain in RESULTS and Fig. 11 exactly how a synchronizing-weak-beat can lead to synchronization.

In this paper, we extend the analysis of Bibbig et al. (2001) by examining more closely the mechanism with which interneuron plasticity leads to changes in synaptic conductances that can generate a synchronizing-weak-beat and how this synchronizing-weak-beat in turn can synchronize two assemblies that have been firing almost in anti-phase. For these "long-range synchronization after almost anti-phase activity" simulations, we use a different kind of network model than in Bibbig et al. (2001), namely a network of simple integrate-and-fire neurons with refractory mechanism. This is to show that the ability to synchronize is not dependent on "fancy" mechanisms but depends on simple mechanisms inherent in networks of virtually all types of neuronal models. And we show that purely local i  e conductances are sufficient to generate synchrony, meaning that not even a few between-assembly i  e conductances (as used in the detailed network model here and in Bibbig et al. 2001) are necessary. In addition to these simulations with the simple network model, we present new data showing that the synchronized oscillations generated here with long conduction delays share some characteristics with tetanically induced  oscillations: they are at  frequency and they show missed e-cell beats (see Traub et al. 1999). However, they also have a feature in common with synchronous  oscillations with short conduction delays: they are not dependent on e  e synapses (see also Kopell et al. 2000; Traub et al., 1999), although the synchronization process is accelerated if e  e synapses are present. (Thus in vivo, e  e synapses are probably involved in this synchronization process.) We show that long-range synchronization at  frequency depends, however, on e-cells of one assembly projecting to the other assembly's i-cells (i.e., e  i connections). We point out conditions under which synchronizing-weak-beats and synchronization can take place. In addition, we will examine which time-independent parameters influence the synchrony behavior, i.e., lead to immediate synchrony or to almost anti-phase behavior. Some important parameters in this regard are as follows: amplitude/duration of pyramidal cell inhibitory postsynaptic potential (IPSP) in relation to interneuron IPSP, or in more technical terms, i  e conductance relative to i  i conductance and relative to the delay, or alternatively, the e-cell AHP in relation to the i-cell AHP. Furthermore, we introduce the concept of a "long-interval doublet" (a pair of interneuron action potentialsseparated by approximate "delay milliseconds"in which the first spike is induced by tonic inputs and/or excitation from nearby e-cells, whereas the second spike is induced by (delayed) excitation from distant e-cells). And we show that "long-interval population doublets" (long-interval doublets of the i-cell population) are necessary for synchronized firing of two assemblies separated by long axonal conduction delays in our networks. Failure to produce them leads to almost anti-phase activity at  frequency. Finally, we will analyze why two assemblies, separated by a conduction delay of approximately one-half a  period (depending on the exact network model:  period 20-25 ms, average delay 8-17 ms), if they are firing at  frequency, will "like to fire" and thus be stabilized in an almost anti-phase  oscillation. Such an oscillation persists if no synchronizing-weak-beats can occur.

Some of these data have been published in abstract form (Bibbig 1998, 1999; Bibbig and Traub 2000).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

We used two network models in the simulations shown in this paper: a "detailed" model consisting of a large network of detailed (multicompartment) neurons (Fig. 1) and a "simple" model with a (relatively) small network of simple integrate-and-fire neurons (Fig. 2).

View larger version (19K): [in this window] [in a new window]   Fig. 1. A: detailed network model consists of 768 excitatory and 384 inhibitory compartmental neurons, arranged in 2 blocks, constituting assemblies 1 and 2, respectively. The 2 assemblies are separated by a minimum delay: 0 ms in simulations modeling nearby assemblies in hippocampal slices (as in Fig. 3A), 6 ms in simulations modeling long-range synchronization (10 and 15 ms in the simulations shown in this paper). Excitatory pyramidal cells ("e-cells") project globally with a probability exponentially decreasing depending on distance (illustrated by the distance-dependent arrow size), whereas inhibitory interneurons ("i-cells") project only locally to approximately one-half the array. B: learning rule for e  e and e  i (in some simulations, i  e) synapses is Hebbian depending on presynaptically induced and postsynaptic [Ca2+]i.

View larger version (16K): [in this window] [in a new window]   Fig. 2. Block diagram of the simple network model, learning rule, and simulation parameters. A: network structure of the simple model consisting of 120 excitatory cells (e-cells) and 120 inhibitory interneurons (i-cells). These cells are each modeled as simple integrate-and-fire neurons with a firing threshold and a refractory period. e-Cells and related i-cells are subdivided into 4 assemblies of 30 cells each. Usually, assemblies 1 and 3 were stimulated together with a tonic input. The goal was to have these assemblies (1 and 3) synchronize to form a "global cell assembly." e-Cell projections are far-reaching, whereas i-cells project only locally, i.e., i-cells of 1 group only form projections to some of the neighboring cells, so that i-cells of group 1 do not project to e- and i-cells of group 3 (this is why usually these 2 assemblies were stimulated together; we wanted to exclude synchronizing long-range i  e effects). Spike transmission of all neurons is via distance-dependent axonal conduction delays of 0.5 mm/ms. B: learning rule: at all synapses (e  e, e  i, and i  e), we used a Hebbian 2-threshold learning rule (Artola et al. 1990; Bienenstock et al. 1982) with the product of a "presynaptic" [excitatory postsynaptic postential (EPSP) or inhibitory postsynaptic potential (IPSP)] and the postsynaptic signal (spike rate) as a measure. The product should represent the supralinear effect of the EPSP/IPSP and the postsynaptic backpropagating action potential on [Ca2+] of the dendritic spine (Yuste and Denk 1995). If the product exceeded the LTD-threshold, the synaptic weight was decreased by a small amount, . If it was above the LTP-threshold, the weight was increased by 5 (e  e) or by about 2 (e  i and i  e).

Detailed network model

The detailed network model is almost identical to the model used and described in detail in Bibbig et al. (2001). Thus here we will concentrate on summarizing the most important principles of this network model and emphasizing the few differences between the two models.

The most important characteristic of our network is that we use self-organized Hebbian plasticity for modifying synaptic conductances. This means that, as in Bibbig et al. (2001), synapses from excitatory pyramidal cells to other pyramidal cells and to inhibitory interneurons (i.e., e  e and e  i synapses) are modifiable, in Hebbian fashion, on the time scale of the oscillations, i.e., tens to hundreds of milliseconds. In addition, in some simulations (e.g., Fig. 8) i  e synapses are modifiable according to a Hebbian learning rule. The plasticity of excitatory and inhibitory synapses was motivated by earlier simulations (Bibbig 1998, 1999, 2000) in networks of integrate-and-fire neurons, as well as by recent experimental data of plasticity of excitatory and inhibitory synapses in hippocampal slices during tetanus-induced  and  oscillations (e.g., Whittington et al. 1997b for e  e; Bibbig et al., 2001 for e  i; M. A. Whittington, unpublished data, for potentiation of i  e conductance; note, however, that changes in i  e synaptic connections are difficult to isolate and document experimentally during network  or  oscillations: this is the case because isolation of IPSPs requires blockade of AMPA receptors; but block of AMPA receptors prevents  and  portions of the oscillation from occurring in a normal way; Traub et al. 1999). There are also data that document interneuron plasticity in hippocampus and in neocortex, generated by different stimulation paradigms (e.g. Ouardouz and Lacaille 1995; Perez et al. 2001; Rozov et al. 1998 for plasticity of e  i synapses; Holmgren and Zilberter 2001; Perez et al. 1999 as examples of i  e plasticity in hippocampus and in neocortex, respectively).

There are three differences with the large, detailed model used in Bibbig et al. 2001.

1.) The neuronal network was split into two "blocks," separated by 10-ms conduction delay throughout all simulations shown in this paper, to examine long-range synchronization. A minimum delay of 10 ms is equivalent to an average delay of 12 ms between the two assemblies. This average delay is the value usually provided in RESULTS and Fig. 1. In addition to the simulations shown in this paper (12- and 17-ms average delay), we performed numerous additional simulations with many other delays. The network behavior described in this paper, i.e., almost anti-phase activity at  frequency and synchronous activity at  frequency showing missed e-cell beats, is produced for average conduction delays  8 ms, i.e., a minimum delay 6 ms (Fig. 1A).
2.) A fixed i  e conductance was varied from simulation to simulation, or else i  e plasticity was incorporated (as already mentioned above), to investigate the role of i  e conductances in long-range synchronization.
3.) g_K(M) and g_K(AHP) were not altered during the oscillation, to isolate i  e conductance effects.

All other parameters are as described in detail in Bibbig et al. 2001. The most important features of the model (including the above mentioned 3 differences) are summarized in the following paragraphs.

Condensed description of the most important features of the model

OVERALL NETWORK STRUCTURE. The network contains 768 excitatory pyramidal cells ("e-cells") and 384 inhibitory interneurons ("i-cells"). As seen in Fig. 1, the pyramidal cells are arranged into two 48 × 8 blocks representing two stripes of neuronal tissue. The interneurons are arranged into two blocks of 48 × 4 cells, overlapping the e-cell arrays. The two blocks of e- and i-cells represent two local cortical areas (each of them approximately 1 mm wide), which are separated by a long distance (we conducted simulations with "minimum" long axonal conduction delays between the "innermost ends of the blocks" ranging from 6 to 17 ms; most simulations shown here have minimum conduction delays of 10 ms; in Fig. 4, C and D, the minimum conduction delay was 15 ms).

INDIVIDUAL NEURONAL PROPERTIES. Each e-cell is a multicompartment object (64 soma-dendritic compartments and 5 axonal ones) containing Na⁺, Ca²⁺, and different sorts of K⁺ conductances as originally described in Traub et al. (1994), with the exact values and densities (including the addition of an M-type voltage-dependent K⁺ conductance) given in Bibbig et al. (2001).

SYNAPTIC CONNECTIVITY. Each pyramidal cell is contacted by 30 other pyramidal cells and by 80 interneurons (20 "basket cells" from the 1st row, 20 "axo-axonic cells" from the 2nd row, 20 "bistratified cells," and 20 "o/lm 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 of interneurons. Each interneuron was excited by 150 pyramidal cells. Interneurons receive 60 inputs from other interneurons, 20 of each sort, with the exception that interneurons are not contacted by axo-axonic cells. For further connectivity details see Bibbig et al. (2001).

SYNAPTIC ACTIONS. Only AMPA- and GABA_A-receptor-mediated synaptic connections were simulated, not N-methyl-D-asparatate (NMDA)- or GABA_B-receptor-mediated actions. The general form of a unitary e  e synaptic conductance was c_ee t exp(t/2), where t is the time in milliseconds and c_ee is a scaling parameter; for a unitary e  i synaptic conductance, it was c_ei t exp(t). The general form of a unitary IPSP was c_ie exp(t/10) or c_ii exp(t/10), respectively. Default values (in RESULTS they are often referred to as "usual" values, because these are the values used in former publications) of c_ie and c_ii were as follows: c_ie i-cell  pyramidal cell, 1.6 nS, for all types of i-cells; c_ii basket cell  interneuron, 2.3 nS; and c_ii bistratified or o/lm cell  interneuron, 0.23 nS. As indicated in the respective paragraphs, c_ie and c_ii are set in some simulations to higher values. In a few other simulations c_ie and c_ii are only enhanced if the presynaptic i-cell is a basket cell. The scaling parameters c_ee and c_ei, and in some simulations, c_ie and c_ii, depend on "learning" in a manner described below.

STIMULATION CONDITIONS. As in Traub et al. (1999), oscillations were evoked by applying tonic "metabotropic" conductances to dendrites of principal cells and interneurons (Whittington et al. 1997a). The reversal potential of this conductance was 60 mV positive to resting potentials. Interneurons received a tonic conductance of 4.0-4.2 nS; pyramidal cells received a maximum tonic conductance of 75.0-82.5 nS. The tonic excitatory conductance to pyramidal cells was time-dependent, starting at 0 at time 0, rising to its maximum over 100 ms (Whittington et al. 1997a), staying constant for the next 700 ms, and then declining linearly with time to 55% of the maximum value, agreeing qualitatively with experimental data (Whittington et al. 1997a).

K⁺-AHP AND M-CURRENT CONDUCTANCES. Unlike Traub et al. (1999) and Bibbig et al. (2001), g_K(M) and g_K(AHP) were kept constant throughout all simulations with a scaling constant of 0.25. Even though experimental data indicate their tetanus-induced time-dependent variation (Whittington et al. 1997b), this was done to concentrate on i  e conductance variations and isolate the resulting effects which might be similar to effects of varying g_K(M) and g_K(AHP). Note that it is conductance amplitude that is manipulated in our simulations, but it is duration that is of greatest functional importance.

(1)

"LEARNING." As mentioned at the beginning of METHODS, e  e synapses, e  i synapses, and in simulations shown in Fig. 8, i  e synapses, are modifiable during the course of a simulated oscillation. In general, these synapses modify according to a Hebbian learning rule, in the sense of depending on correlations between pre- and postsynaptic activity. Modification is governed by the following rules.

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AXON CONDUCTION DELAYS. Pyramidal cell axons conducted at 0.5 mm/ms, and interneuron axons conducted at 0.2 mm/ms. Thus if the minimum conduction delay for excitation between the two blocks shown in Fig. 1 were 10 ms, then the maximum conduction delay would be 13.84 ms, and the average conduction delay was approximately 12 ms. These values were typical, but in the simulation shown in Fig. 4, C and D, the minimum conduction delay was 15 ms. In analyzing how cells of one site can influence cells of the other site, it must be taken into account that there is a spike generation time: This time very much depends on the condition of the cell, the size and form of the excitatory postsynaptic potential (EPSP), and where on the dendrite the EPSP is generated; this can be as short as a fraction of a millisecond and up to many milliseconds. On average it is approximately 2 ms.

NOISE. Noise was simulated, as before (Traub et al. 1999), with ectopic spontaneous axonal action potentials, originated by independent Poisson processes, with an average interval of 10 s in e-cell axons and of 5 s 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 ms of data. Average values of synaptic scaling constants, c_ee, c_ei, and c_ie, were also saved.

DATABASE, RUN TIMES, PROGRAMMING, AND SYSTEMS ASPECTS. More than 70 simulations were performed with the large, detailed model with different e  e, e  i, and i  e conductances, different axonal conduction delays, plasticity at different synapses, etc. 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-s simulation took about 6 h to run. For details on programming aspects, contact roger.traub{at}downstate.edu.

Simple network model

OVERVIEW OF NETWORK STRUCTURE, NEURONAL PROPERTIES, SYNAPTIC CONNECTIVITY, PLASTICITY AND STIMULATION PARAMETERS. Figure 2A shows a block diagram of the network structure of the simple model consisting of 120 excitatory neurons (e-cells) and 120 inhibitory cells (i-cells), each modeled as a leaky integrate-and-fire neuron with refractory period and noise term. e-Cells and related i-cells are organized in two chains, which were subdivided into four assemblies of 30 cells each. Usually, assemblies 1 and 3 were stimulated together with a tonic input. The goal was to have assemblies 1 and 3 synchronize to form a "global" cell assembly (as was the goal for the 2 assemblies, 1 and 2 in the detailed model; Fig. 2B). As with the detailed model, this synchronization could be reached with the help of interneuron plasticity, that is to say, plasticity of e  i and i  e conductances. e-Cell projections are far-reaching; that is, e-cells from assembly 1 project to e-cells and i-cells of all other assemblies (1-4). In contrast, i-cells project only locally; that is to say, i-cells of group 1 can maximally project to e- and i-cells of assembly 2 (as well as assembly 1), but do not project to e- and i-cells of group 3 (which is simultaneously stimulated and should be synchronized with assembly 1; see goal above) and assembly 4. This means that, in the case of the simple network model, the i  e and i  i connectivity is "functionally local," whereas in the detailed model it is not. We chose this functionally local i-cell connectivity here in the simple model to show that it is not necessary to have some i  e connections projecting to the respective other assembly for the two assemblies to be synchronized (as we had in the simulations with the detailed model shown here). We also conducted simulations with functionally local i-cell connectivity with the detailed model that shows the same result (not shown here): large enough local i  e projections, not reaching the other assembly, are sufficient for synchrony to occur (if e  i connections are far reaching; see RESULTS). Action potential transmission of all neurons occurs via distance-dependent axonal conduction delays of 0.5 mm/ms.

DETAILS OF THE NETWORK STRUCTURE. Connection probability of e  e synapses was 0.3 within a radius of 19 neurons (both to the rightward and the leftward site of a given e-cell), and it was 0.1 to all other e-cells. Connection probability of e  i synapses was 0.6 within a radius of 19 neurons of a given e-cell, and it was 0.3 to all other i-cells. The local connection probability of i  e connections was 0.7 within a radius of 13 neurons (to the right and the left side of a given i-cell), and the connection probability of i  i connections was 1.0 within this radius of 13 neurons. Connection probability of i  e and i  i connections was 0 outside this radius of 13 neurons.

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LEARNING RULE. At all synapses (e  e, e  i, i  e, and sometimes i  i) we used a Hebbian two-threshold learning rule (Artola et al. 1990; Bienenstock et al. 1982) with the product of a presynaptic, or more precisely, presynaptically induced signal (EPSP or IPSP) and the postsynaptic signal (spike rate) as a measure (Fig. 2B). The product of both signals should approximately represent the intracellular [Ca²⁺] of a postsynaptic neuron at a real synapse. It should also reflect pre- and postsynaptic effects at all these synapses, e.g., the supralinear effects of the EPSP and the postsynaptic backpropagating action potential on [Ca²⁺] values at e  e synapses (Yuste and Denk 1995), the dependency on presynaptic activation and postsynaptic depolarization at e  i synapses (Perez et al. 2001), and pre- and postsynaptic activity influencing long-term changes at i  e synapses (Holmgren and Zilberter 2001). If the product exceeded the LTD-threshold (long-term depression), the synaptic scaling parameter was decreased by a small amount (). If it was above the LTP threshold (long-term potentiation), this scaling parameter was increased by 5 (e  e) or by about 2 (e  i and i  e). This learning algorithm was performed every time-step of the simulation, i.e., twice per millisecond.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

In this paper, we consider patterns of oscillations that involve e-cells and i-cells at two separate regions in neural tissue. Only a finite number of patterns occur in our simulations. It is therefore possible to give each pattern a name (Fig. 3): with short delays (<8 ms), we can get a synchronous  (Fig. 3Aa), a  oscillation with a phase-shift between the two sites (Ab), a synchonous  (Ac), and a  oscillation with a large (almost anti-phase) phase-lag between the two sites (Ad). With long conduction delays between the two sites (8 ms), a synchronous  oscillation generated by network interactions between the two sites seems impossible (we will argue in Fig. 7 why). However, we can generate a  oscillation with a large, almost anti-phase phase-shift between the two sites (Bb), a synchonous "" (Bc), and a slow oscillation at  or even  frequency with a large (almost anti-phase) phase-lag between the two sites (Bd). ("", because it shares some characteristics with the  oscillations generated with short delays, i.e.,  frequency and beat skipping activity of e-cells, whereas i-cells fire at  frequency; but lacking other characteristics, such as dependence on e  e synapses.) In this paper, we will concentrate on simulations with long axonal conduction delays between the two assemblies, i.e., Fig. 3B.

View larger version (25K): [in this window] [in a new window]   Fig. 3. Examples of oscillation patterns found in detailed network simulations modeling 2 assemblies (each comprising e-cells and i-cells) separated by short (2 ms average) or long axonal conduction delays (12 ms on average). Toptraces: average e-cell voltages from either end of the array [designated here as Ve1 (thick line) and Ve2 (thin line)], for left and right, respectively. Bottom trace: average i-cell voltage from the first site (Vi1). The time scale bar is the same for all recordings. A: different oscillation types found with short axonal conduction delays between the 2 assemblies. Aa: synchronous oscillation at  frequency (usually 40-50 Hz in the simulations with the detailed and the simple network model, i.e., 20- to 25-ms period). In this type of oscillation the e-cells of the 2 assemblies fire almost synchronously (Ve1, Ve2), with an average phase lag of less then 2 ms between the 2 sites. The i-cells generate 2 spikes most of the time, i.e., they fire doublets, which stabilize synchrony between the 2 sites. This type of oscillation is seen, for example, in hippocampal slices stimulated at 2 sites with a tetanic stimulus (Traub et al. 1996; Whittington et al. 1997a). Ab:  frequency oscillation with a phase-lag between the 2 sites. Here (most of) the i-cells only generate 1 spike/wave, seen in the bottom trace, showing the average i-cell signal of assembly 1. This kind of oscillation is seen in hippocampal slices if 2 sites are stimulated simultaneously after only 1 site was stimulated strongly before (Bibbig et al. 2001; Whittington et al. 1997b). Ac: synchronous oscillation at  frequency (usually 10-25 Hz, i.e., 40- to 100-ms period in simulations). Here, the 2 e-cell assemblies fire (almost) synchronously during the so-called strong beats (3 of them are shown in the top e-cell traces), whereas they do not fire in between in the 2 missed beats in the middle, where only i-cells are active (see bottom trace). Due to longer-lasting e-cell afterhyperpolarizations (AHPs; compared with i-cell AHPs), the i-cells actually fire just before the e-cells would fire their next beat, and thus inhibit the e-cells so that the e-cell beat is skipped. Note that the i-cells fire doublets during the strong beats and only singlets during the missed beats due to the lack of e-cell input, so that only the tonic input is available. Synchronous  is seen in hippocampal slices after a strong tetanic 2-site stimulation (Traub et al. 1999; Whittington et al. 1997b). Ad: almost anti-phase  oscillation. Here the 2 e-cell assemblies fire alternately and the i-cells only generate singlets. (Nearly) anti-phase  is sometimes experimentally observed with tetanic 2-site stimulation in the presence of diazepam (Faulkner et al. 1999). B: different oscillation patterns found with long axonal conduction delays between the 2 assemblies. Because there are no slices available with average axonal conduction delays this large, only oscillation types found in simulations are mentioned. These types of oscillations might, however, occur in vivo. Ba: synchronous oscillations at  frequency are not seen in simulations with detailed or simple models if the 2 assemblies are separated by long axonal conduction delays (Bibbig 2000; Kopell et al. 2000). Bb: almost anti-phase  frequency oscillation. Here the i-cells more often generate singlets than they generate doublets, although in every beat some i-cells do generate a doublet (seen as second small "population spike" in the average i-cell trace of the first assembly; see e.g., Fig. 7). Bc: synchronous oscillation at  frequency (usually approximately 20-30 Hz, i.e., 35- to 50-ms period in simulations). This "" has different characteristics than have  oscillations with short conduction delays (see Fig. 8). Note that Tallon-Baudry et al. (2001), in human memory tasks, only record  oscillations that are not synchronized between the 2 areas (separated by several centimeters, the conduction delay not being directly accessible in humans), whereas, in contrast,  oscillations are synchronous. This is consistent with our simulations shown in Fig. 3B, a-c. Bd: almost anti-phase slow oscillation (usually approximately 7-10 Hz, 100-130 ms), i.e., the frequency is low, in the  or  range. The oscillation is almost anti-phase, and like  oscillations, shows strong and missed e-cell beats. In addition, the missed e-cell beats here are not synchronous but come with a phase lag of approximately "delay + spike generation time" milliseconds after the strong beat of the respective other site. Remark: similar oscillations as shown in Fig. 3, A and B, can also be obtained with the simple network model.

During these oscillations, in each period, e-cells (respectively, i-cells) fire in approximate synchrony with nearby e-cells (respectively, i-cells), or else are silent. In addition, in all the cases we consider, e-cell populations at the two sites fire at nearly the same frequency; likewise, i-cell populations at the two sites fire at nearly the same frequency. (i-Cells, however, may have a different mean frequency than e-cells.) Finally (in almost all cases), e-cells fire 0 or 1 action potentials per period (wave) while i-cells fire 0, 1, or 2 spikes during this time interval. A beat is a population spike of both e- and i-cells, and an e-cell beat (respectively, i-cell beat) is a population spike of the e-cell (i-cell) population. A doublet (or synonymously i-cell doublet or i-doublet) consists of two i-cell action potentials per i-wave, which follow each other in close succession (5 ms). With tetanic two-site stimulation and short conduction delays (1-3 ms between the two assemblies), the first action potential of the doublet is generated by tonic input combined with synaptic excitation from local e-cells, whereas the second action potential is generated mainly by the distant e-cells and thus follows the first one after approximate "delay milliseconds" (Traub et al. 1999).

We generalize this idea to larger conduction delays using the term long-interval doublet to refer to a pair of interneuron action potentials, also temporally separated by approximately "delay milliseconds" (which can be a large portion of the oscillation period when the two regions are, as in this paper, separated by a delay 8 ms). Long-interval doublets are generated by the following two principles: 1) the first action potential is induced by tonic input and excitation from nearby e-cells, while 2) the second action potential is induced by (delayed) excitation from distant e-cells. Long-interval population doublets are long-interval doublets in all or almost all i-cells of the population. The second beat of an interneuron long-interval population doublet can then inhibit nearby e-cells (which would be expected to fire after the i-cell beat assuming inhibition in e-cells is larger/longer than in i-cells; see Figs. 6 and 7). This leads to a firing pattern characterized by e-cells firing on every second period of the underlying i-cell rhythm (i.e., they skip alternate beats; e.g., Fig. 3Bc). This means that in oscillations with skipped e-cell beats and long-interval doublets/long-interval population doublets, i-cells generate two action potentials per e-cell period but only one action potential per i-cell period. Long-interval doublets thus do not look like doublets, as previously defined for short inter-areal delays (Traub et al. 1996), but they share a common underlying mechanism.

Summarizing the last paragraphs, oscillation patterns are defined by the following parameters: 1) frequency of the e-cells (e.g.,  = 30-70 Hz,  = 10-29 Hz); 2) frequency of the i-cells; 3) phase angle of the e-cells at one site relative to e-cells at the other site; 4) phase angles of e-cells at one site relative to i-cells of that site (phase anglespluralbecause i-cells may fire more times per cycle than e-cells); 5) firing pattern of i-cells, e.g., whether or not doublets occur; and 6) wave-to-wave variations in the number of cells at each site firing per beat. Figure 3 illustrates examples of different oscillation patterns, with references to the literature for those patterns that have been observed experimentally.

The main goal of this paper is to formulate conditions under which two assemblies separated by large conduction delays (10 ms and morea delay easily reached between neurons in the neocortex) can fire synchronously, either from the beginning of the oscillation or after synaptic plasticity has taken place. The conduction delay value of 10 ms is based on the following experimental data: 1) axonal conduction velocity is <5 m/s (Swadlow 2000), and 2) synchrony is observed over distances of 5 or even 9 cm (Desmedt and Tomberg 1994; Tallon-Baudry et al. 2001). Even if areas separated by distances of several centimeters are likely to be connected by relatively fast cortico-cortical connection (although the relevant axonal conduction delays are not known), there are axonal collaterals within the gray matter reported to conduct with axonal conduction velocities as small as 0.1 mm/ms (Aroniadou and Keller 1993). Thus even spatial separations of 3-4 mm (common for extrinsic collaterals; Rockland and Drash 1996), or of 3-6 mm (common for intrinsic, i.e., within-area, collaterals; K. Rockland, personal communication) lead to large temporal separations, in principle up to 60 ms. Thus a 10-ms delay seems quite plausible. As synchronization has been reported over these distances, there must be a mechanism for synchronizing neocortical neuronal assemblies despite significant separation in space and/or very large conduction delays. A further comment on terminology: If we use terms like initial synchrony or capability of developing synchrony, we are interested, respectively, in whether or not the two assemblies fire synchronously from the beginning of the oscillation or whether or not the two assemblies are able to synchronize their activity after some period of time dependent on characteristics of plasticity.

In all simulations with the detailed network model shown in this paper, if not explicitly mentioned otherwise, e  e and e  i conductances were plastic according to a Hebbian learning rule, which takes two [Ca²⁺] values as measures (see Methods and Bibbig 2001): 1) [Ca²⁺] fluctuations induced by presynaptic activity and 2) [Ca²⁺] signals induced by postsynaptic voltage-dependent g_Ca. In simulations with the simple network model, these two kinds of synapses (e  e and e  i) were also plastic following a Hebbian learning rule, dependent on the EPSP (as a presynaptically induced measure) and postsynaptic instantaneous firing rate. We made numerous further simulations to show that plasticity of the e  e and e  i conductances per se is not essential for synchrony. That is, we could have shown instead nonplastic simulations, with certain fixed values for the e  e and e  i conductances; such nonplastic networks can also generate synchronous oscillations, exhibiting the same parameter-dependences, for synchronization of two assemblies separated by large conduction delays, as the plastic networks shown here in this paper. We chose, however, to show simulations with plastic e  e and e  i conductances, because there are experimental data demonstrating that these synapses are indeed plastic during tetanus-induced  oscillations: in vitro experiments using tetanic stimulation to yield synaptic potentiation or depression (Bibbig et al. 2001; Whittington et al. 1997b). In addition, there are other experimental data about e  e and e  i plasticity under different experimental conditions (see e.g., Geiger et al. 1999 for a review of e  i plasticity). In vivo, there are experiments in freely moving rats, where stimulation of the perforant path with LTP- or LTD-inducing tetanic stimuli leads to changes in the network behavior of CA1 pyramidal cells, suggesting that plasticity at least of e  e synapses plays a role there (Dragoi et al. 2000) and should be considered in our models.

In former papers we have shown the effect of growing e-cell AHPs on    oscillations (Traub et al. 1999; Whittington et al. 1997b). However, it has now been shown that i  e conductances also grow during  frequency activity induced by tetanic or theta-patterned stimulation (see Jensen et al. 1999 for short-term effects and Chapman et al. 1999; Grunze et al. 1996; Perez et al. 1999 for long-term potentiation). i  e Conductances also appear to grow by approximately a factor of two during tetanically induced two-site    oscillations (Whittington, unpublished data). To elucidate a possible site for synaptic plasticity in these oscillations, we removed any changes of the intrinsic e-cell AHP and concentrated on the plasticity of i  e conductances. This does not mean that we believe AHPs do not play a role in synchronizing two assemblies separated by large conduction delays. Rather, it means that i  e conductances can, in principle, do the job. This is important to know, because there are, to our knowledge, no experimental data on synchronizing mechanisms with such long delays available yet.

One of the major issues of this paper will be the dependence of synchrony of two assemblies separated by large conduction delays on the i  e conductance. We first examine how synchrony changes with the variation of a fixed i  e conductance (Fig. 4, A and B, C and D). Then we further analyze, for a given fixed i  e conductance, how the synchrony behavior changes with other parameters such as delay (Fig. 4, B and C) or i  i conductance (Fig. 5), and we try to explain how and under which conditions the two assemblies fire in-phase (Figs. 6 and 7), or develop an almost anti-phase oscillation (Fig. 7). Later on (starting in Fig. 8), we shall use plastic i  e synapses to examine under which conditions two formerly asynchronous firing assemblies are able to synchronize their activity.

View larger version (35K): [in this window] [in a new window]   Fig. 4. Whether 2 assemblies separated by large conduction delays are able to synchronize depends on the delay and on the basket cell  pyramidal cell conductance. Left traces: average e-cell voltages from either end of the array [designated here as V1 (thin line) and V2 (thick line)] and the total GABAA conductance received by a single e-cell at the first site, respectively. Voltage traces and GABAA conductances show 500-ms runs. Right traces: auto- (thin line) and cross-correlations (thick line) of the 200-ms interval [300 ms, 500 ms] of V1 and V2, respectively. Scale bars are the same for all voltage traces, total GABAA conductances, and time in A-D. A: case with an average conduction delay of 12 ms between the 2 assemblies and the usual (fixed) basket cell  pyramidal cell conductance (see METHODS). The 2 sites are not able to synchronize, but stabilize in an almost anti-phase oscillation, as seen in the voltage traces and the cross-correlogram (oscillation period: 25.7 ms). B: increase of the fixed basket cell  pyramidal cell conductance by a factor of 3 synchronizes the 2 assemblies (again separated by 12 ms on average), from the beginning of the oscillation (oscillation period: 36 ms). As described in METHODS, there are 4 different kinds of i-cells: basket cells, axo-axonic cells, bistratified cells, and o/lm-cells. The synaptic coefficients are cie = 1.6 nS (see METHODS). Because only the basket cell  e-cell conductance is enhanced to 3 times the usual value and not all the other i  e conductances, the total i  e conductance is only enhanced by a factor of 1.5, an elevation that is hardly seen in the trace of the "total GABAA conductance received by a single e-cell" shown here, but it is measurable. C: increase of the average delay from 12 to 17 ms again makes the synchrony disappear, if the same basket cell  pyramidal cell conductance is used as in B. Voltage traces V1 and V2 and the cross-correlation show poorly correlated activity (peak of the auto-correlogram at 23.1 ms). D: the 2 assemblies synchronize despite the large average delay of 17 ms, following a further increase of the basket cell  pyramidal cell conductance to 20 times the usual value (see METHODS), with an oscillation period of 48 ms.

View larger version (42K): [in this window] [in a new window]   Fig. 5. For a given i  e (more precisely, basket cell  e-cell) conductance the synchrony depends on the i  i conductance. Left traces: 500 ms of the average e-cell voltages from either end of the array [top traces: designated as Ve1 (thin line) and Ve2 (thick line)] and the same time span of the average i-cell averages from the respective sides (Vi1, Vi2). Right traces: auto- (thin line) and cross-correlations (thick line) of the interval [300 ms, 500 ms] of V1 and V2. Scale bars are the same for all voltage traces and time scales in A and B. A: simulation of 2 assemblies separated by an average delay of 12 ms and a basket cell  e-cell conductance of 5 times the "usual" value. The i  i conductance was at its usual value (used in all simulations with the detailed model shown here other than B). The 2 assemblies fire synchronously with an oscillation period of 38 ms. Note that the i-cells fire twice during an e-cell oscillation period, i.e., they fire long-interval population doublets (please see Fig. 6 for explanation). B: another simulation of 2 assemblies separated by an average delay of 12 ms and a basket cell  e-cell conductance of 5 times the "usual" value, but this time the i  i conductance was also increased to 5 times the "usual" value. Now the two assemblies stabilize in an almost anti-phase oscillation with 1 site leading the other by 10.3 ms, as most clearly seen in the cross-correlogram of the last 200 ms of the voltage traces. The oscillation is at  frequency with a period of 23 ms. Note that the i-cells here only fire once per e-cell oscillation period.

View larger version (16K): [in this window] [in a new window]   Fig. 6. A: 2 assemblies separated by a large conduction delay can fire synchronously at  frequencywith skipping of alternate e-beats under the following 2 conditions: (a) e-cells of assembly 2 are still inhibited when activation from assembly 1 e-cells reaches i-cells (and e-cells) of assembly 2, i.e., after approximately "delay + spike generation time milliseconds" (indicated here as delay*) and (b) the e-cell IPSP must last longer than the i-cell IPSP, so that assembly 2 e-cells do not fire before assembly 2 i-cells. For similar IPSC time constants, i  e conductance must be larger than i  i conductance. Under these conditions, long-interval population doublets can suppress e-cells on alternate waves (as seen here in the second wave); [ oscillation, because the oscillation is at  frequency, but contrary to a  oscillation with short conduction delays (Traub et al. 1999) is not dependent on e  e synapses.] B: simulation confirming the hypotheses of A. Shown here are the intervals 160-230 ms of a simulation already shown in Fig. 4B. Top traces: (Ba) are average e-cell voltages from either end of the array (Ve1 and Ve2). Middle traces: the average e-cell voltage (Ve2, thick line, an identical copy of the top) and the average i-cell voltage (Vi2, thin line) from one end of the array (Bb). Bottom traces: "total AMPA conductance minus the basket cell GABAA conductance onto a particular e-cell" (Bc), and onto a particular basket cell (Bd), both from the same end of the array as the voltage averages in the middle trace. This difference signal provides an estimate of the net excitation to the neuron. Traces in Ba show that the two assemblies fire at  frequency with missed e-cell beats, and that they fire almost sync