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1Departments of Physiology and Pharmacology, and Neurology, State University of New York, Downstate Medical Center, Brooklyn, New York; 2Department of Neuroscience, University of Pennsylvania School of Medicine, Philadelphia, Pennsylvania; and 3School of Biomedical Sciences, University of Leeds, Leeds, United Kingdom
Submitted 20 September 2004; accepted in final form 3 November 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSAPPENDIX AAPPENDIX BAPPENDIX CACKNOWLEDGMENTSREFERENCESAPPENDIX A REFERENCES APPENDIX B...APPENDIX C... |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSAPPENDIX AAPPENDIX BAPPENDIX CACKNOWLEDGMENTSREFERENCESAPPENDIX A REFERENCES APPENDIX B...APPENDIX C... |
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The above difficulties and hazards are too real to be dismissed readily. In our opinion, the only way to proceed is through a state of denial that any of the difficulties need be fatal. The reader must then judge whether the results, preliminary as they must be, help our understanding.
Previous models of cortical or thalamocortical circuits have been developed, usually with specific applications in mind (Bal et al. 2000
; Bazhenov et al. 2002, 2004; Bush and Sejnowski 1996; Contreras et al. 1996; Destexhe et al. 1996, 1999; Douglas and Martin 1991; Golomb and Amitai 1997; Lytton et al. 1997; Pinto et al. 2003; Wang and Rinzel 1993). These previous models tend to use small numbers of cells and usually represent each cell with one or a few compartments. We are not aware of a previous model that has a multiplicity of cell types and firing behaviors in cortical cells, including regular spiking (RS), fast rhythmic bursting (FRB) (Gray and McCormick 1996; Steriade et al. 1998), and intrinsic bursting (IB) (McCormick et al. 1982; Nowak et al. 2003). It has been unusual to include electrical coupling, particularly between the axons of principal cells, a form of coupling that appears to be essential for persistent gamma and for very fast oscillations (>70 Hz) (Cunningham et al. 2004a, b; Traub et al. 2002, 2003a, b).
We have attempted here to construct a thalamocortical circuit model that has applicability to the study of a range of emergent behaviors in the thalamocortical network. Our attempt hinges on an effort to be faithful to a range of intrinsic properties in different neuronal types (Llinás 1988) and to include a variety of between-cell interactions, both chemical synaptic and gap junction-mediated. Although the model does indeed describe, predictively, several sorts of network behaviors (as will be seen below), it is insufficient to describe many others: for reasons that include, but are not limited to, the omission of many cell types, the requirement to make many guesses about structural details, the absence of mechanisms to simulate synaptic plasticity (either short-term or long-term), and the restriction of the model to a single column. In particular, the present model is best suited to address the physiology of network oscillations and epileptogenesis.
A major question to be addressed here concerns the role of axonal gap junctions (Schmitz et al. 2001) in epilepsy. A number of experimental studies suggest a role of gap junctions in epileptogenesis (Gajda et al. 2003; Jahromi et al. 2002; Köhling et al. 2001; Pais et al. 2003; Perez-Velazquez et al. 1994; Ross et al. 2000; Schweitzer et al. 2000; Szente et al. 2002; Traub et al. 2001, 2002). Earlier modeling studies of hippocampal pyramidal cell networks (Traub et al. 1999) predicted that gap junctions, if located between the axons of pyramidal cells, are expected to have 2 effects on epileptogenesis: 1) they would lower the extent of recurrent chemical synaptic excitation required for population synchronization of bursts of action potentials; 2) they would introduce a very fast oscillation (>70 Hz, "VFO") on top of the interictal field potential. [Earlier studies of such a superimposed very fast oscillation in hippocampus (Snow and Dudek 1984) had considered the possibility of field-effect–induced synchronization.] In the cortex, VFO is of interest not only because of its association with human epilepsy (Bragin et al. 1999; Staba et al. 2004), but also because of the appearance of VFO in somatosensory evoked potentials in rat barrel cortex (Jones and Barth 1999, 2002; Jones et al. 2000).
We have paid particular attention to electrical coupling between principal cell axons because of the necessity to include such an effect, to account for experimental recordings of kainate-induced persistent gamma in superficial layers of auditory cortex in vitro (Cunningham et al. 2004), and also because of clinical data documenting, in human epilepsy patients, the presence of VFO superimposed on seizure burst complexes, and on interictal spikes (Traub et al. 2001): earlier work in hippocampus (Traub and Bibbig 2000) had shown that the postulate of axonal coupling between hippocampal pyramidal cells could account for the occurrence of VFO on top of physiological sharp waves (Ylinen et al. 1995), even though the sharp waves are primarily mediated by a synaptically coupled network. Further details suggesting that axonal coupling could occur in cortex are discussed in APPENDIX B.
Topics covered in this paper include: kainate-induced persistent gamma oscillations and sleep spindles; then epileptogenesis, with illustrations of patterns resembling interictal spikes, fast runs, spike-wave, and polyspike-wave. These topics were selected because they are logically related: they are all associated with slow sleep oscillations, and its transition to seizures, in vivo (Steriade 2003). We include some experimental recordings that are consistent with some of the model predictions, particularly on the presence of very fast oscillations superimposed on epileptiform field potentials, and on firing patterns of layer 4 spiny stellate cells during seizurelike events. Further topics are considered in the appendices. APPENDIX A describes how individual cell types were modeled. APPENDIX B describes between-cell interactions, both synaptic and by electrical coupling. APPENDIX C deals with technical issues of how the large computations were carried out.
A note on terminology: sleep spindles refer to a well-known in vivo population phenomenon (Steriade 2001, 2003), appearing in natural slow-wave sleep in addition to other states, and consisting of cellular oscillations (about 10 to about 15 Hz, depending on species) that involve thalamic relay cells, nucleus reticularis thalami (nRT) cells, and cortical cells. Network phenomena in vitro, which exhibit a similar appearance in terms of cellular oscillations, shall be called simply "spindles."
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSAPPENDIX AAPPENDIX BAPPENDIX CACKNOWLEDGMENTSREFERENCESAPPENDIX A REFERENCES APPENDIX B...APPENDIX C... |
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We confine ourselves here to general comments on our philosophy of modeling and the overall network architecture. Specific details on single-cell properties are described in APPENDIX A; on synaptic and gap-junctional interactions (connectivity, kinetics, and conductance amplitudes) in APPENDIX B; and on programming issues and the use of the parallel computer in APPENDIX C. [In addition, interested readers may obtain copies of the Fortran code and Linux compilation and execution scripts by writing to roger.traub{at}downstate.edu.]
The approach to modeling single neurons grew out of 2 earlier studies (Traub et al. 1994, 2003c ). The code described in the latter reference was the basis for simulating 2 of the cell types here. The approach is to use an electrotonic architecture containing dozens of compartments, but nowhere near the number of compartments used to model an anatomically reconstructed neuron. Network simulations, on a large scale anyway, are not practical with such detailed neurons. Dozens of compartments are sufficient to capture certain aspects of neuronal function, including differences in electrogenesis between axon, soma, and dendrites; action potential initiation in the axon; dendritic calcium spikes and bursts; spike backpropagation; and to allow for axons and/or dendrites to be electrically coupled between neurons.
The structure of a particular neuron is described by its compartmental topology; the values of electrotonic parameters such as specific membrane capacitance, membrane resistivity, and internal resistivity (some of which can be different in the axon compared with soma/dendrites); the densities of a fixed repertoire of ionic conductances, where the same repertoire of conductances was used, for the sake of simplicity, in all cell types; and by parameters describing the kinetics of [Ca2+] concentration in a thin submembrane shell. Again for the sake of simplicity, the first-order kinetic scheme for submembrane [Ca2+] concentration was the same for all cell types; only the particular parameters were different. Submembrane [Ca2+] concentration is used to gate the slow AHP conductance, and (along with membrane voltage) one of the fast K conductances—the "C" conductance. All neurons of a given type (e.g., layer 2/3 RS pyramidal neurons) have the same parameter set: heterogeneity can be introduced by the use of slightly different bias currents. A final simplification is to use, wherever possible, identical kinetics for voltage-sensitive channels between different neuron types. Exceptions to this latter rule include the use of different fast gNa and delayed rectifier gK(DR) kinetics in pyramidal cells versus cells with stellate or interneuron-like morphology; and the use of different T-channel kinetics in nRT versus thalamocortical relay (TCR) neurons. Our experience has been that using 50 to 100 or so compartments is sufficient to capture many detailed aspects of neuronal firing behavior (Traub et al. 1994, 2003c).
The "standard repertoire" of active ionic conductances are these: fast, transient, gNa; persistent gNa; K conductances of delayed rectifier, A (transient, inactivating), slow AHP, C (fast voltage- and calcium-dependent), "K2," and "M" types; high- and low-threshold gCa; and a relatively slow anomolous rectifier, or "h," conductance.
The cell types and cell locations of the 3,560-neuron model are shown in Fig. 1. The cortical portion of the model is one-dimensional, the dimension being cortical depth: dimensions parallel to the pia are not represented, so that the structure can be thought of as a column. Space is not defined within the thalamic portion. The reader should note the following: there is no layer 1; layers 2 and 3 are lumped together; a large variety of neuronal types are omitted, including but not limited to: neurogliaform cells, double bouquet cells, multipolar bursting neurons (Blatow et al. 2003), and numerous other sorts of interneurons; there are no pyramidal cells in layer 4; synaptic inhibition in layer 4 derives primarily from deep interneurons; there are no FRB cells in deep layers nor FRB interneurons (which were shown to exist by Steriade et al. 1998); there is homogeneity of cell structure within layers. Considerations in choosing the repertoire that we used were these: we began with a model of layer 2/3 circuitry that included RS and FRB pyramidal cells, as well as superficial fast-spiking (FS) and low-threshold spiking (LTS) interneurons (Cunningham et al. 2004a). We needed layer 4 stellate cells as the major recipient of thalamic inputs. Tufted pyramids in layer 5 are a major neuronal type, much studied, and important for cortical outputs not headed for the thalamus; and both IB and occasionally RS firing patterns have been described in these cells (Williams and Stuart 1999). Layer 6 pyramids were needed as an interface to the thalamus. Deep interneurons were necessary because, for among other reasons, we know that in vitro gamma/beta oscillations have different structure in deep versus superficial layers (A. Roopun and M. A. Whittington, unpublished data). Finally, both nRT and TCR thalamic neurons are essential for the understanding of thalamic oscillations, including sleep spindles, as presented here, but also for subsequent work including delta waves and the slow (<1 Hz) oscillation of sleep (Steriade et al. 1993).
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The neurons were connected together 1) by chemical synapses, using AMPA and NMDA receptors, and 
-aminobutyric acid-A (GABAA; but not GABAB) receptors; and 2) gap junctions, that were nonrectifying and voltage-independent. Connections of both sorts were "wired up" randomly, subject to constraints on how many connections there were, and the possible locations of postsynaptic compartments. A given excitatory synapse activated both 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (NMDA) receptors. Gap junctions were located between dendrites of cortical interneurons, of nRT cells (Landisman et al. 2002), and of TCR cells (Hughes et al. 2002a). Gap junctions could also be located between the axons of 1) the pool of superficial pyramids, RS and FRB; and/or 2) the pool of spiny stellates; and/or 3) the pool of layer 5 tufted pyramids; and/or 4) the pool of layer 6 nontufted pyramids. It was a major assumption that only homologous sorts of glutamatergic neurons could be electrically coupled by their axons (see APPENDIX B).
We justified the use of axonal coupling as follows: 1) it is necessary in models for the occurrence of gamma oscillations (Cunningham et al. 2004a); 2) spikelets occur in cortical neurons (Cunningham et al. 2004a; Deschênes,1981; Thomson and Bannister 2004; however, the Deschênes study attributed the spikelets to synaptic activation); 3) there is staining for pannexin 2 [a putative component of the electrical coupling substrate between axons (Bruzzone et al. 2003)] throughout cortical layers 2–6 (Cunningham et al. 2004a); 4) very fast oscillations occur in the cortex (Traub et al. 2001; this paper).
Certain important state variables could not be included, such as fluctuations in extracellular ion concentrations. In addition, we did not allow for afferent inputs (coming from outside the model network), or for specific effects of neuromodulators on membrane properties, although we did depolarize selected neuronal subpopulations (including FRB neurons, and at times pyramidal neurons in layers 5 and 6), using steady bias currents. All collective behaviors simulated are thus essentially "autonomous" in the model network.
The effects of the many simplifications made here will become known as progressively more detailed models are constructed and their behaviors analyzed. It is to be hoped that—as the model incorporates further cell types, membrane currents, metabotropic effects, more accurate synaptic connectivity, and so forth—then it will be possible to study a broader range of network phenomena, including the slow oscillation of sleep, gamma oscillations in deep cortical layers, and cortical responses to thalamic activation.
In APPENDIX B, we list the set of "baseline" synaptic conductance scaling constants. These, and details of connectivity, were arrived at after extensive (dozens) of preliminary simulations. (Many dozens of preliminary simulations were also necessary for each individual cell model.) Then, for this paper, we can list modifications in synaptic conductances relative to the baseline values. APPENDIX B also describes between-cell connectivity (synaptic and gap junctional), methods for estimating field potentials, and other matters related to ensemble activity.
APPENDIX C describes computer science and numerical integration aspects of how our large calculations were performed on a parallel computer (Linux cluster). (For questions about these issues and copies of the code, or portions thereof, the reader can contact roger.traub{at}downstate.edu.)
The parameters of greatest interest in the RESULTS will be these: which populations of cortical principal cells are electrically coupled; how cortical inhibitory postsynaptic conductances (IPSCs) are scaled; steady depolarizing currents to particular subpopulations of neurons; the properties of AMPA and NMDA conductances at synapses between layer 4 spiny stellate cells; the effects on cortical activity of disconnecting the thalamus.
In vitro experimental methods
Horizontal slices (450 µm thick) were prepared from adult male Wistar rats (150–250 g). Neocortical slices containing primary and secondary auditory regions and secondary parietal regions were maintained at 34°C at the interface between warm wetted 95% O2-5% CO2 and artificial cerebrospinal fluid (CSF) containing (in mM): KCl 3; NaH2PO4 1.25; MgSO4 1; CaCl2 1.2; NaHCO3 24; glucose 10; NaCl 126. Extracellular recordings from primary auditory cortex were obtained by using glass micropipettes containing artificial CSF (resistance <0.5 M
). Intracellular recordings were obtained with sharp microelectrodes filled with potassium acetate (resistance 30–90 M), and, in some cases, with the addition of 2% biocytin. For identification of biocytin-filled cells, slices were immediately fixed in 4% paraformaldehyde (PFA) in phosphate-buffered saline, following the recording. Signals were analog filtered at 2 kHz and digitized at 10 kHz. Cells other than those in layer 4 were identified by physiological criteria (regular spiking, fast spiking, intrinsic bursting). Slices were bathed in 400 nM kainate (Tocris, Bristol, UK) and 40 µM picrotoxin (Tocris). In some cases, CGP55845A (10 µM, Sigma-Aldrich UK, Dorset, UK) was added as well, to block GABAB receptors. All procedures were carried out in accordance with the UK Animals (Scientific Procedures) Act of 1986.
In vivo experimental methods
Experiments were conducted in accordance with the ethical guidelines of the National Institutes of Health and with the approval of the Institutional Animal Care and Use Committee of the University of Pennsylvania. Adult male Sprague–Dawley rats (350–450 g) were anesthetized with pentobarbital (50 mg/kg intraperitoneally). Buprenorphine (0.03 mg/kg subcutaneously) was administered to provide additional analgesia. Animals were paralyzed with gallamine triethiodide and artificially ventilated. End-tidal CO2 (3.5–3.7%) and heart rate were continuously monitored. Body temperature was maintained at 37°C by servocontrolled heating blanket and rectal thermometer (Harvard Apparatus, Holliston, MA). The depth of anesthesia was maintained by supplemental doses of the same anesthetic to keep a constant heart rate and a constant high-amplitude, low-frequency electroencephalogram (EEG) as recorded from a bipolar electrode inserted into the cortex.
For cortical intracellular recordings, a craniotomy was made to expose the surface of the barrel cortex (1.0–3.0 mm posterior to bregma, 4.0–7.0 mm lateral to the midline). The dura was resected over the recording area and mineral oil was applied to prevent dessication. The stability of recordings was improved by drainage of the cisterna magna, hip suspension, and filling of the holes made for recording with a solution of 4% agar.
Intracellular recordings were performed with glass micropipettes filled with 3 M potassium acetate and DC resistances of 80–90 M. A high-impedance amplifier (band-pass of 0–5 kHz) with active bridge circuitry (Cygnus Technology, Delaware Water Gap, PA) was used to record and inject current into the cells. Data were digitized at 10 kHz and stored on a Nicolet Vision (Nicolet Instrument Technologies, Madison, WI). A computer operating Labview (National Instruments, Austin, TX) was used for the on-line averaging of responses. All data analysis was done off-line using routines written in Igor Pro (Wavemetrics, Lake Oswego, OR).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSAPPENDIX AAPPENDIX BAPPENDIX CACKNOWLEDGMENTSREFERENCESAPPENDIX A REFERENCES APPENDIX B...APPENDIX C... |
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Persistent, or pharmacologically induced, gamma oscillations occur in in vitro preparations and are called "persistent" because, once initiated, they continue as long as the slice remains healthy (Fisahn et al. 1998). In rat auditory cortex in vitro, kainate-induced gamma oscillations have their maximal amplitude in superficial layers (Cunningham et al. 2004a). Interestingly, two other sorts of in vitro gamma oscillations have maximal amplitude in the superficial layers: interneuron gamma evoked by stimulating metabotropic glutamate receptors, during pharmacological blockade of ionotropic glutamate receptors (Whittington et al. 1995); and thalamically evoked cortical gamma oscillations in thalamocortical slices in vitro (Metherate and Cruikshank 1999). [Not all gamma oscillations in vitro follow this rule, however: when gamma is evoked in somatosensory cortex in vitro, with carbachol plus a low concentration of kainate, then the gamma occurs in all cortical layers, with deep gamma 180° out of phase with superficial gamma (Buhl et al. 1998).]
Our earlier model of auditory cortex kainate-induced persistent gamma (Cunningham et al. 2004a) involved simulations of layer 2/3 only, with RS and FRB pyramidal cells, FS interneurons (basket and axoaxonic), and LTS interneurons. Electrical coupling, between axons, occurred within and between RS and FRB pyramidal cell populations; dendritic electrical coupling occurred between FS interneuron dendrites and between LTS interneuron dendrites. As in earlier models of persistent gamma in hippocampus (Traub et al. 2000, 2003a, b), the superficial neocortical model produced gamma as a result of axonal spiking percolating through the principal cell axonal plexus (in the neocortical case, having such spiking boosted by FRB bursting), with resultant bursts of orthrodromic activation of interneurons, and with the interneurons then interrupting the principal cell somata and axons for some tens of milliseconds, by GABAergic inhibition, thereby producing the gamma period.
Figure 2 demonstrates persistent gamma in a full-thickness model of neocortex (the thalamus being disconnected here), one that includes the cell types of the original superficial cortical model, but many other cell types as well: layer 4 spiny stellates, deep pyramids, deep interneurons. Different bias currents were applied to some of the neurons, particularly to superficial FRB neurons (see legend). (This applies to other simulations as well.) As before (compare Cunningham et al. 2004a), gamma is of highest amplitude in the superficial layers (Fig. 2A); and cells in the superficial layers have similar firing patterns to the previous model, and to in vitro experiment: sporadic somatic firing of superficial RS pyramids on a background of rhythmic synaptic potentials (Fig. 2B), superficial FRB pyramids discharging on approximately every other burst, superficial FS interneurons (e.g., the basket cell shown in bottom right panel) firing on a majority of the gamma waves, and superficial LTS interneurons firing less than the FS cells. This simulation, however, is not sufficient to explain why the deep layers are not generating their own gamma, or at least being driven more strongly by the superficial gamma. The explanation could lie in differences in gap junctional connectivity between the two regions, in properties of interlaminar synaptic connections, in the model's lack of deep FRB neurons, or in other structural features. It is important to note that in vivo (in recordings from cat pericruciate gyri, anterior and posterior suprasylvian areas, and area 18 of the marginal gyrus), gamma oscillations occur in both superficial and deep layers with comparable amplitude (Steriade et al. 1996). In addition, FRB cells have been recorded in the infragranular layers of cat pericruciate and suprasylvian gyri (Steriade et al. 1998) and of cat primary visual cortex (J. Cardin and D. Contreras, unpublished data).
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Individual model thalamic relay cells, and model reticular neurons, fire in bursting and tonic modes, as occurs physiologically (Bal and McCormick 1993; Jahnsen and Llinás 1984a 1984b; Contreras et al. 1993; Deschênes et al. 1984) (APPENDIX A); model thalamic cells can generate rhythmic bursts at approximately 5 Hz during injection of a steady current (not shown). Figure 3 shows a simulated thalamic network spindle and its influence on the cortex. The spindle is initiated by a spontaneous burst in the reticular neurons (Fig. 3A). The spindle has a frequency of about 16 Hz (slightly above the frequency range for sleep spindles in cats, but at the upper limit for humans), and has a waxing/waning course (seen in the TCR average in Fig. 3A). The relatively fast spindle frequency shown here may be related to the relatively rapid time constants used for the decay of nRT cell-induced GABAA receptor-mediated IPSCs in TCR cells: 3.3 and 9 ms for the fast and slow components, respectively (APPENDIX B).
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). Note that individual TCR neurons exhibit the expected sawtooth-like voltage fluctuations (resulting from rebound low-threshold calcium spikes), and each neuron fires on only a fraction of the waves. In contrast, nRT neurons discharge a burst on each spindle wave, as occurs in both in vivo (Contreras et al. 1993) sleep spindles and in vitro (Bal et al. 1995a, b) spindles. The nRT neurons do not, however, show a tonic depolarization as seen in vivo [perhaps attributable in part to persistent gNa (Contreras and Steriade 1993)]; the model nRT neurons instead exhibit the slight tonic hyperpolarization that usually occurs in vitro in ferret slices (Bal et al. 1995a, b). [There are exceptions, however: sometimes nRT neurons in vitro do exhibit an underlying depolarization during a spindle oscillation, sustained by a persistent sodium conductance (Kim and McCormick 1998).] This spindle is generated entirely within the thalamic portion of the model, as corticothalamic excitatory postsynaptic conductances (EPSCs) (AMPA) in TCR (after the initial wave) are at most 0.2 nS.
Figure 3B illustrates the effects of the thalamic spindle on spiny stellate neurons: a series of synaptic depolarizations, sometimes with action potentials. Note further the coherent depolarizations in the spiny stellate cells, as shown in the middle trace in Fig. 3B, an inverted average of the somatic potentials of all layer 4 spiny stellates; this coherence is aided by the electrical coupling between spiny stellate axons used in the simulation. In addition, a small burst occurs (Fig. 3, B and C, *); this burst results because of the strong recurrent chemical synaptic excitation between the spiny stellates; when the coupling is weakened 8-fold, the burst does not occur (not shown). Multiphasic waves similar to the asterisk-marked burst in Fig. 3C, are on occasion observed in in vivo sleep spindles (Contreras and Steriade 1996; see also Beierlein et al. 2002).
Figure 3C shows (top portion) that cortical neurons, layer 5 tufted IB pyramids in particular, actually "see," at spindle frequency, a superimposition of synaptic excitation and inhibition; the inhibition results in the model because of strong feedforward excitation of interneurons by thalamic afferents (Swadlow 2003). Trains of IPSPs have been observed experimentally in vivo on the depolarizing phase of the slow oscillation in cortical neurons (Steriade et al. 1993).
Finally, Fig. 3C illustrates the behavior of superficial layer 2/3 RS pyramids, including the average of all of the layer 2/3 pyramids (note that the average signal is inverted, so as to approximate a local field potential). During the spindle itself, there is a mixture of spindle intervals (about 16 Hz) and gamma intervals (about 30 Hz); gamma is possible in the superficial layers because, in this simulation, superficial pyramids are electrically coupled by their axons. As the spindle ends, gamma alone is present. In vivo as well (in cats) sleep spindles are often followed by a run of gamma oscillation (Steriade et al. 1996), although the gamma in vivo is in both cortex and thalamus. In our model, the gamma is only in the cortex. This difference from in vivo results may arise because the model does not include a mechanism for sustained depolarization of nRT and TCR neurons, such as occurs during the slow oscillation in vivo (Contreras and Steriade 1995; their Figs. 4 and 8), and which may be mediated by metabotropic glutamate receptors (Blethyn et al. 2003; Hughes et al. 2002), and/or by a persistent sodium conductance (Kim and McCormick 1998).
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), but is different from what has been described in vivo, wherein an isolated portion of nucleus reticularis appears to spindle on its own (Steriade et al. 1987). The model as now constituted appears rather to be consistent with the notion of Ulrich and Huguenard (1997) that within-nRT synaptic inhibition subserves some purpose other than spindling, perhaps what they refer to as "lateral inhibition."
On the other hand, if we repeat the simulation of Fig. 4, isolating the nucleus reticularis, but now (Fig. 5) further depolarize nRT neurons (bias currents in Fig. 4 = 0.17 to 0.18 nA, in Fig. 5 bias currents = 0.27 to 0.28 nA), then a synchronized reticularis oscillation does occur (Fig. 5), at approximately 6 Hz. This simulated isolated nRT oscillation requires gap junctions to remain synchronized (Fig. 5B), but not within-nRT synaptic inhibition (Fig. 5C). Still further depolarization of the nRT neurons in the model resulted in rapid tonic firing; on the other hand, metabotropic effects such as reducing one or more K+ conductances might have allowed a 10-Hz oscillation, a matter not further explored here. Landisman et al. (2002) observed oscillations in the reticular nucleus in vitro, that required gap junctional communication, but not synaptic transmission; their oscillations could occur at frequencies around 10 Hz. An in vivo study has also found evidence for electrical coupling between nRT neurons, in the form of halothane-sensitive spikelets, and simulations in that study showed that such coupling could contribute to the synchronization of sleep spindle oscillations (Fuentealba et al. 2004).
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In the monograph of Traub and Miles (1991), some of the basic principles of epileptogenesis in the disinhibited hippocampal CA3 region in vitro were analyzed. CA3 pyramidal cells are intrinsically bursting neurons, and are synaptically connected in such a way that an intrinsic burst in a single presynaptic neuron can evoke, with latency of tens of milliseconds, a burst in a monosynaptically connected postsynaptic neuron (Miles and Wong 1986, 1987). In addition, there is enough recurrent excitatory connectivity, even in vitro, so that on average bursting in one presynaptic neuron will actually evoke bursting in more than one postsynaptic neuron. Thus by a chain reaction, bursting in a single neuron can lead to bursting throughout the whole population, with latency from initial burst to peak number of cells firing dependent on the latency for bursting to spread from cell to cell, and on the density of connections. Although some of the excitatory synaptic connections between neocortical layer 5 tufted pyramids are extremely powerful (see APPENDIX B), we are not aware of data documenting the transmission of a burst from one neuron directly to another in neocortex, either for the case of layer 5 pyramids or for other pairs of neocortical neurons (either of homogeneous cell type or not). Thus it is not clear whether the above analysis of hippocampal bursts applies to cortex. It is certainly the case that during a synchronized burst in neocortex, each principal neuron "experiences" a very large EPSP (Gutnick et al. 1982), consistent with the synchronized discharge of many neurons, although this information is not sufficient to define how the synchrony comes about.
In Traub and Miles (1991), we also considered the case in which a homogeneous population of neurons were all regular spiking, and synaptic connections were not strong enough to transfer firing from a single neuron to another neuron (Fig. 6. 11 of that monograph); we asked what sort of stimulus was necessary to synchronize the population. Clearly, firing in a single neuron will no longer suffice. It turns out that a threshold number of cells needs to be discharged together; the value of the threshold number depends on parameters, of course, but in general it can be much smaller than the total number of cells, even if much larger than one. What we did not consider at that time, however, was the possibility of electrical coupling between axons (Schmitz et al. 2001), which constitutes another pathway (besides excitatory chemical synapses) whereby action potentials might cross from neuron to neuron.
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SYNCHRONIZED AND PARTIALLY SYNCHRONIZED INTERICTAL BURSTS IN MODEL NEOCORTEX (THALAMUS DISCONNECTED), WITH INCOMPLETE DISINHIBITION, WEAK RECURRENT EXCITATION BETWEEN LAYER 4 SPINY STELLATES, AND ELECTRICAL COUPLING.
Figure 6 illustrates varieties of "interictal" behavior when IPSCs are reduced (to 1/10 their baseline values), recurrent excitation between layer 4 spiny stellates is small (EPSCs at 0.25 x baseline; see APPENDIX B), and axonal gap junctions are "open" between superficial pyramids, between spiny stellates, and between layer 6 RS pyramids. (By "open" we mean that the gap junction has a high enough conductance that a spike can cross from axon to axon.) Figure 6A illustrates a "classical-appearing" interictal burst, which produces fields in superficial and deep layers and bursts in all cortical principal cell types. The bottom traces are the total AMPA conductance (green) and GABAA conductance (orange) developing in the layer 6 pyramid. Note the large "PDS" (paroxysmal depolarization shift) AMPA conductance, >100 nS (Matsumoto and Ajmone Marsan 1964; Prince 1968; Sawa et al. 1963). The inhibitory conductance during the PDS is small, even though inhibition is not completely blocked, and interneurons are firing robustly (not shown). Of additional note is the very fast oscillation (VFO) superimposed on the fields, especially at 1 mm (topmost trace), a topic to be considered further later on. Finally, there is a low-amplitude oscillation in the field at 1 mm (about 20 Hz), which results from the activity of the electrically and synaptically coupled superficial FRB pyramidal neurons.
In addition to the obvious synchronized burst in Fig. 6A, partially synchronized bursts can also occur; one of these is shown in Fig. 6B. The partially synchronized burst is most obvious (in terms of the fields and firing of individual neurons) in the more superficial layers, although a clear EPSC is seen in the illustrated layer 6 pyramid (bottom trace). Three layer 5 pyramids did not fire in the simulation of Fig. 6B, and the mean synaptically induced depolarization was 10 mV. The occurrence of partially synchronized bursts in the model is reminiscent of localized small-amplitude bursts during epileptogenesis in the cat in vivo (Steriade and Contreras 1998).
WITH STRONG RECURRENT EXCITATION BETWEEN LAYER 4 SPINY STELLATES, AND WITH EXTENSIVE ELECTRICAL COUPLING BETWEEN CORTICAL PRINCIPAL NEURONS, DISINHIBITION LEADS TO "POLYSPIKES." Figure 7 illustrates a series of simulations of the cortical portion of the model (thalamus disconnected), wherein recurrent synaptic connections between layer 4 spiny stellates are "strong" (AMPA conductances at 2 x baseline value, peak conductance 1.47 nS); in addition, axonal gap junction conductances were "high" within the following neuronal populations: superficial pyramids; spiny stellates; layer 5 pyramids; layer 6 pyramids. Each panel shows the effects of scaling cortical IPSCs by some value (x 0.05 in Fig. 7A, x 0.1 in Fig. 7B, etc.).
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) occurs, at about 10 Hz, consisting of 17 burst complexes that terminate spontaneously; the last 5 of the bursts are shown in the panel. Of note is the continuous VFO in the fields (*), and the near continuous firing of the spiny stellate cell (black trace in bottommost portion of the panel); examination of the average behavior of spiny stellates (not shown) indicates that this near-continuous firing is typical. Even with such near-complete disinhibition, with the model parameters used, axonal coupling is necessary for epileptogenesis: when none of the cortical principal cells has such coupling, there is no synchronized bursting, after an initial transient (not shown). When layer 4 spiny stellates are the only principal cell population to be electrically coupled (parameters as in APPENDIX B), then a 4-burst polyspike occurs, complete with VFO (not shown). If layer 5 tufted IB pyramids were the only principal cells to be electrically coupled, a single synchronized burst occurred, with VFO much attentuated compared with Fig. 7A. When superficial pyramids were the only principal cells to be electrically coupled, then a 15-Hz oscillation occurred in superficial layers only, that appeared to be driven by the superficial FRB pyramidal cells (not shown). If, in this latter situation, IPSCs were suppressed completely, then a single synchronized burst did occur, but with attenuated VFO (not shown). Thus recurrent synaptic connections and electrical coupling between layer 4 spiny stellates seem to be, in this model, critical factors in producing runs of synchronized bursts, and in generating VFO between, and superimposed on, the burst complexes.
When parameters are as in Fig. 7A, but IPSCs are somewhat larger (Fig. 7B), then double bursts occur, with a separation of about 100 ms between bursts. Again there is VFO (*) between and on top of the bursts. The spiny stellate cell fires throughout the double burst (as do other spiny stellates; not shown), whereas layer 5 IB cells fire 2 separate bursts (red trace in bottommost part of the panel, Fig. 7B). Layer 5 tufted RS pyramids and layer 6 nontufted pyramids also fire in discrete bursts, in a pattern similar to that of layer 5 IB cells, both during the fast run of Fig. 7A and during the double burst in Fig. 7B (not shown).
A further increase in IPSC size (Fig. 7C) abolishes epileptiform activity, as well as population oscillations visible at the field level. A still further increase in IPSC size (Fig. 7D) returns the system to state in which there are gamma oscillations (
30 Hz) in superficial layers alone, a state similar to that illustrated in Fig. 2. Spiny stellates are mostly silent during the conditions of Fig. 7, C and D, as are deep pyramidal cells.
Epileptiform double bursts occur in rat auditory cortex in vitro, in kainate plus blockade of GABAA and GABAB receptors
Figure 8 shows (on the left) data from the simulation of Fig. 7B, in which GABAA conductances have been reduced by 90% and GABAB conductances are absent. The simulation illustrates a double burst in the field (bottom trace),with all cell types also exhibiting double bursts, with the exception of the spiny stellate, that fires almost continuously throughout the double burst. The right side of Fig. 8 illustrates experimental recordings (not simultaneous) from double bursts recorded in rat auditory cortex in vitro, in the presence of kainate (400 nM) and blockers of GABAA and GABAB receptors (picrotoxin, 40 µM, and CGP55845A,10 µM, respectively). The experimental interburst interval is somewhat longer than the simulated interburst interval. Again, in the experiment, all of the recorded cell types exhibited double bursts as well, approximately in phase with the major field deflections, with the exception of layer 4 neurons identified as spiny stellates. These cells fired throughout the double bursts (n = 5); one of the spiny stellates is shown in Fig. 8. However, this cell does show some degree of spike adaptation toward the end of its firing. Particularly striking in this Fig. 8 are the similar appearances of the fields in model and experiment, with prominent VFO in both.
The simulated layer 2/3 cells exhibit bursts that are less depolarized than the corresponding experimental bursts, perhaps a consequence (in part) of the incomplete disinhibition in the simulation. In addition, the simulated layer 2/3 cells generate a few action potentials between the larger bursts, unlike the experiment; this could be a result of the more intense firing of the model layer 4 stellate neurons than in the experiment, combined with the strong synaptic activation of superficial neurons by layer 4 neurons.
VOLTAGE-INDEPENDENT NMDA RECEPTORS AT BETWEEN-SPINY STELLATE CONNECTIONS, COMBINED WITH ELECTRICAL COUPLING, CAN ALSO LEAD TO EPILEPTIFORM BURSTS WITH PROLONGED SPINY STELLATE FIRING, AS WELL AS VFO.
Fleidervish et al. (1998) described, in mouse somatosensory cortex in vitro (using tangentially cut slices to isolate barrels in layer 4), a system of recurrent excitatory connections between spiny stellates that was in large part mediated by NMDA receptors. This recurrent system was powerful, in that epileptiform activity could occur in disinhibited preparations even during blockade of AMPA/kainate receptors. Furthermore, perhaps because the NMDA receptors contained the NR2C subunit, NMDA-mediated currents were at least partially independent of membrane potential and of [Mg2+]o: NMDA EPSPs could be detected at or near resting potential without lowering [Mg2+]o. The observed NMDA conductances were brief, having time constants <20 ms.
We thus ran some simulations in which AMPA receptor conductances at between-spiny stellate connections were "low" (peak conductance 0.18 nS), whereas NMDA conductances were made completely voltage-independent (for the usual scheme of voltage-dependence of NMDA conductances, see APPENDIX B). Figure 9 illustrates an example simulation with spiny stellate NMDA conductance at 1.25 x baseline value, 
NMDA = 15 ms, and with complete disinhibition. The thalamic portion of the network was disconnected, and electrical coupling was present between the axons of superficial pyramids, of spiny stellates, and of layer 6 pyramids. Although a double burst does not occur, the synchronized burst is prolonged and has prominent VFO. Additionally, the firing of spiny stellate neurons is prolonged compared with the firing of other principal cell types. An experimental epileptiform burst [rat auditory cortex bathed in kainate (400 nM) and picrotoxin (40 µM), but without block of GABAB receptors] is shown for comparison, in Fig. 9, to emphasize the similarity in cellular firing patterns, and in the occurrence of VFO in the field. [The precise experimental nature of the glutamate receptors at within-layer 4 spiny stellate connections in rat auditory cortex is (to our knowledge) not known, however.]
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Next, we shall consider 2 types of epileptogenesis in vivo: approximately 3-Hz spike-wave, and approximately 10-Hz fast runs.
THE CORTICAL NETWORK CAN GENERATE A SPIKE-WAVE-LIKE PATTERN IF—IN ADDITION TO PARTIAL DISINHIBITION AND ELECTRICAL COUPLING—LAYER 5 AND 6 PYRAMIDAL CELLS ARE DEPOLARIZED.
Synchronized epileptiform bursts shown in simulations in the previous figures were sporadic (at least so far as could be determined in simulations that usually lasted 1.6 s, and at most 2.5 s), even with GABAA receptors largely blocked. In particular, we did not see synchronized bursts at frequencies >1 Hz when deep pyramidal cells were depolarized with small currents of <0.1 nA, even in disinhibited conditions. [The assumption that >2–3 Hz spike-wave patterns in vivo require some degree of cortical disinhibition seems reasonable for several reasons: 1) there is an in vivo genetic rat model of spike-wave epilepsy in which intracortical inhibition is impaired (Luhmann et al. 1995). In addition, diffuse cortical application of a penicillin solution to cat cortex can elicit a spike-wave-like epileptic pattern (Avoli and Gloor 1982; Gloor et al. 1977). 2) A spike-wave-like pattern (but at very low frequencies, about 0.1 Hz) has been observed in vitro during blockade of GABAA and GABAB receptors (Castro-Alamancos and Rigas 2002). The question is whether cortical disinhibition is sufficient for a spike-wave-like pattern.] We thus wondered whether simply depolarizing pyramidal cells, in addition to having electrical coupling and partial disinhibition present, could generate a spike-wave-like pattern at about 3 Hz.
Figure 10 demonstrates that, at least in our model (in this case with the thalamic portion disconnected), simply depolarizing deep (layers 5 and 6) pyramids is sufficient to speed up, and make more regular, the occurrence of synchronized bursts, so that a 3-Hz network oscillation can occur: the depolarizing currents serve to overcome the tendency of pyramidal cell AHPs to slow the burst frequency [deep pyramidal cells have an afterhyperpolarization (AHP) decay time constant of 1 s; see APPENDIX A ]. In this figure, layer 5 and 6 pyramids were depolarized with 0.35–0.45 nA currents; whereas IPSC conductances were 0.2 x their baseline values; axonal gap junctions were open between superficial pyramids, between layer 4 spiny stellates, and between layer 6 pyramids; and AMPA receptors at connections between layer 4 spiny stellates had a "low" value (0.25 x baseline). The cellular firing patterns are similar to those previously illustrated, although spiny stellate bursting is brief (attributable to the limited recurrent excitation in layer 4), and layer 6 pyramidal cell bursting is more prolonged (attributable to the induced depolarization in this cell population). The only current in the neurons with time course appropriate for gating an oscillation at about 3 Hz is the slow calcium-mediated AHP current (compare Timofeev et al. 2004).
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, and also data below].
ORIGINS OF VFO IN THE SIMULATED FIELD.
Figure 11A shows the power spectrum of somewhat over 1.6 s of simulated field data (at 2 mm), encompassing 6 "spikes" (same simulation as in Fig. 10). Of note is the peak near 100 Hz, quite similar to the in vivo cat data of Grenier et al. (2003; their Fig. 1). The spectrum in the model case, however, is more complex than the in vivo data, in that there are several additional regions of energy (e.g., around 200 Hz, and 300–400 Hz); the latter might correspond to what has been called "fast ripples" in some of the human epilepsy literature (Bragin et al. 1999; Staba et al. 2004).
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nRT AMPA x 3; layer 6 pyramids 
