Department of Neurology and Neurological Sciences, Stanford University School of Medicine, Stanford, California 94305-5122
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ABSTRACT |
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Sohal, Vikaas S. and John R. Huguenard. Long-range connections synchronize rather than spread intrathalamic oscillations: computational modeling and in vitro electrophysiology. J. Neurophysiol. 80: 1736-1751, 1998. A thalamic network model was developed based on recent data regarding heterogeneous thalamic reticular (RE) cell axonal arborizations that indicate at least two projection patterns, short-range cluster projections and long-range diffuse projections. The model was constrained based on expected convergence and the biophysical properties of RE and thalamocortical (TC) cells and their synapses. The model reproduced in vitro synchronous slow (3-Hz) oscillatory activity and the known effects of T-channel blockade and cholecystokinin (CCK) application on this activity. Whereas previous models used the speed at which 3-Hz oscillations propagate in vitro to infer the spatial extent of intrathalamic projections, we found that, so long as the
-aminobutyric acid-B synaptic conductance was adjusted appropriately, a network with only short-range projections and another network with both short- and long-range projections could both produce physiologically realistic propagation speeds. Although the
3-Hz oscillations propagated at similar speeds in both networks, phase differences between oscillatory activity at different locations in the network were much smaller in the network containing both short- and long-range projections. We measured phase differences in vitro and found that they were similar to those that arise in the network containing both short- and long-range projections but are inconsistent with the much larger phase differences that occur in the network containing only short-range projections. These results suggest that, although they extend much further than do short-range cluster projections, long-range diffuse projections do not spread activity over greater distances or increase the speed at which intrathalamic oscillations propagate. Instead, diffuse projections may function to synchronize activity and minimize phase shifts across thalamic networks. One prediction of this hypothesis is that, immediately after a collision between propagating oscillations, phase gradients should vary smoothly across the thalamic slice. The model also predicts that phase shifts between oscillatory activity at different points along a thalamic slice should be unaffected by T-channel blockers and decreased by suppression of synaptic transmission or application of CCK.
The thalamus participates in a variety of cortical rhythms including 7- to 14-Hz sleep spindles and absence seizures (for a review see Steriade et al. 1993 Model neurons
We studied a network model that included 64 TC and 64 RE neurons. These were based on earlier models of TC (Huguenard and McCormick 1992
INTRODUCTION
Abstract
Introduction
Methods
Results
Discussion
References
). The most direct evidence for thalamic involvement arises from lesioning studies. Sleep spindles can be demonstrated in the thalamus after decortication in cats (Morison and Basset 1945
) yet are absent from cortical tissue that was isolated from thalamus (Burns 1950
). Although it was shown that cortical feedback does play an important role in spindle coherence (Contreras et al. 1996
), it is clear that spindle generation is largely a thalamic phenomenon. In fact, the thalamus generates spindlelike oscillations in vitro (Bal et al. 1995a
,b
; Kim et al. 1995
; Ulrich and Huguenard 1997
; von Krosigk et al. 1993
), and the
-aminobutyric acid-A (GABAA) antagonist bicuculline transforms these oscillations into more synchronized
3-Hz oscillations (Bal et al. 1995a
,b
; Ulrich and Huguenard 1997
; von Krosigk et al. 1993
) that resemble the thalamic neuronal activity observed during experimental absence seizures (reviewed by Huguenard and Prince 1997
). These observations suggest that thalamic circuitry may contribute to the generation of pathological oscillations in vivo.
) and in vitro recordings (Bal et al. 1995a
,b
; Huguenard and Prince 1994
; von Krosigk et al. 1993
) and computational modeling (Destexhe et al. 1996a
; Golomb et al. 1996
) elucidated a probable mechanism for the spindlelike and more synchronized
3-Hz oscillations; bursts in thalamic reticular (RE) cells elicit inhibitory postsynaptic potentials (IPSPs) followed by rebound bursts in thalamocortical (TC) cells. These bursts in TC cells then elicit excitatory postsynaptic potentials (EPSPs) that produce bursts in RE cells, causing the cycle to repeat.
), could result from the recruitment of additional cells in each cycle by divergent connections between RE and TC cells (Destexhe et al. 1996a
; Golomb et al. 1996
). These models used the speed at which oscillations propagate in vitro to infer that connections between RE and TC cells contact cells within a region ~100 µm in radius (Destexhe et al. 1996a
) or that the sum of the radius of RE to TC connections and that of TC to RE connections is ~100 µm (Golomb et al. 1996
).
), consistent with the aforementioned predictions about connections from RE to TC cells (Destexhe et al. 1996a
; Golomb et al. 1996
). However, the same anatomic study also found an approximately equal number of RE cells whose axonal arborizations extend over radii of several hundred microns (diffuse projection cells), and another study found that certain physiological properties, such as synaptic strength, are correlated with the size of an RE cell's axonal arborization (Cox et al. 1997b
). The combined findings of widely distributed apparent synaptic contacts and the weak synaptic coupling strength suggested the diffuse cells could not powerfully influence individual TC cells but rather would only exert a weak ("tickler") effect. The relatively large distances over which tickler connections extend are not in accord with estimates based on propagation speeds observed in vitro, nor are the functional implications of tickler connections or of the physiological differences among RE to TC connections known.
3-Hz oscillations in the bicuculline-treated thalamic slice. We show that a network model of this preparation can exhibit similar, physiologically realistic propagation speeds in the presence and absence of long-range tickler connections. We also compare phase differences across the slice, during oscillations in vitro, and in network models with and without long-range tickler connections. We find that only network models that include long-range tickler connections can reproduce the relatively small phase differences observed in vitro. These findings suggest that the primary function of long-range tickler connections (Cox et al. 1996
) may be synchronizing activity and minimizing phase shifts across thalamic networks rather than spreading activity over greater distances (and at greater speeds).
METHODS
Abstract
Introduction
Methods
Results
Discussion
References
; McCormick and Huguenard 1992
) and RE neurons (Destexhe et al. 1996b
). Each neuron was modeled as a single compartment. VT and VR, the membrane potentials of TC and RE cells, respectively, evolved according to
(1)
where the specific capacitance of the membrane, Cm, equals 1 µF/cm2, gL is the leak conductance, EL is the reversal potential of the leak current, IT and ITs are low-threshold calcium currents, Ih is the hyperpolarization-activated cation current, IK and INa are the potassium and sodium currents underlying action potentials, IGABA-B is the inhibitory postsynaptic current mediated by GABAB receptors, and IAMPA is the excitatory postsynaptic current mediated by
(2)
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors. Parameters and kinetics of intrinsic currents are described in APPENDIX A.
; McCormick and Huguenard 1992
) and 14,260 µm2 for RE cells (Destexhe et al. 1996a
).
AMPA currents
We used a first-order activation scheme (Destexhe et al. 1996a; Wang and Rinzel 1992
) for AMPA receptor-mediated synaptic currents. At each postsynaptic site, R, the fraction of activated AMPA receptors evolved according to
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(3) |
GABAB currents
Initially, to simulate GABAB receptor-mediated synaptic currents, we followed the assumption of Destexhe and Sejnowski (1995) that each K+ channel only opens after binding multiple G-protein subunits. In such a model, K+ current is a supralinear function of the concentration of G-protein subunits, so the concentration of G-protein subunits must decay much more slowly than does the K+ current. By allowing G-protein subunits to accumulate, this slow decay produces significant temporal summation between GABAB responses, making it difficult to reproduce the type of paired-pulse depression of GABAB responses observed in vitro (Otis et al. 1993
).
). In contrast, if there is a nonlinear step between GABAB receptor activation and the opening of K+ channels, then residual second messenger, activated GABAB receptors, etc., present during the second response should change the relative contributions of fast and slow components of decay. Second, the ratio of the evoked GABAA and GABAB responses in ventrobasal thalamus (VB) neurons is fixed over a range of stimulation intensities (Huguenard and Prince 1994
). These two experimental observations suggest that there exists a physiological regime in which the GABAB response can be accurately simulated with a linear model.
; Ulrich and Huguenard 1996a
). The amplitude of each GABAB response equaled the product of the maximum postsynaptic GABAB conductance (
GABA-B), the fraction of presynaptic sites that released transmitter, and the fraction of postsynaptic GABAB receptors that were unoccupied at the time of transmitter release.
Probabilistic GABA release and paired-pulse depression
Whenever a presynaptic RE cell spiked, each of its axon terminals released -aminobutyric acid (GABA) with a certain probability. Activation of presynaptic GABAB receptors on RE cells (Ulrich and Huguenard 1996a
) produces paired-pulse depression of GABAA IPSPs and may similarly affect GABAB responses, as observed in dentate gyrus (Otis et al. 1993
). To model this phenomenon, the probability of GABA release decreased as a result of previous release from the same synapse.
Network architecture
As mentioned in the INTRODUCTION, this model represents a thalamic slice that was treated with bicuculline and therefore does not include any GABAA synapses. Instead, network connectivity consists solely of GABAB synapses from RE cells to TC cells and AMPA synapses from TC cells to RE cells. For simplicity, we did not model GABAB synapses between RE cells, which are extremely weak (Ulrich and Huguenard 1996a).
; Golomb et al. 1996
), this network has a one-dimensional architecture in which neurons are distributed evenly along a straight line in two layers (one layer consists of RE cells and the other consists of TC cells). The network models a slice 3.2 mm in length, so the spacing between neighboring cells corresponds to 50 µm along a thalamic slice. On the basis of anatomic evidence that connections between RE and TC cells are topographic (Agmon et al. 1995
; Mitrofanis and Guillery 1993
; but see Jones 1985
), the model assumes that an RE cell located at position x along the line contacts TC cells located within some radius of x and vice versa.
Connections from TC cells to RE cells
Retrograde labeling suggests that the axons of mouse TC cells from different barreloids remain well segregated in the thalamic RE nucleus and that the axonal arborization from a single barreloid has a radius <50 µm in the thalamic RE nucleus (Agmon et al. 1995). TC cells appear to make synaptic contacts on the somata of RE cells (Ide 1982
), but, even if TC cells do contact the dendritic arbors of RE cells, these arbors typically have radii <100 µm (Cox et al. 1996
; Lübke 1993
; Scheibel and Scheibel 1966
; Spreafico et al. 1991
). On the basis of these observations, the model included connections from TC cells to RE cells within a 50-µm radius. Unless otherwise noted, the total postsynaptic AMPA conductance on each RE cell was
AMPA = 150 nS (although the fraction of AMPA receptors was never >0.2, which corresponds to an effective
AMPA of 30 nS). Therefore the conductance at each AMPA synapse was proportional to 1/n where n is the total number of AMPA synapses on the postsynaptic cell.
Connections from RE cells to TC cells
RE cells can be separated into groups that have anatomically and physiologically different projections to TC cells (Cox et al. 1996, 1997b
). One group has projections to TC cells that are strong and contain a high density of axonal swellings but are restricted to a relatively small region (cluster connections). In contrast, the other group has connections to TC cells that are weaker and contain a relatively low density of axonal swellings but extend over a much more widespread region (diffuse or tickler connections). Viewed on the basis of these properties, RE cells appear to fall into distinct categories rather than along a continuum, and these categories appear to contain approximately equal numbers of cells (Cox et al. 1996
, 1997b
).
). As a result, the model was able to represent the heterogeneity of axonal arborizations of RE cells among TC cells simply by including two sets of connections from RE cells to TC cells without having to include distinct populations of RE cells. Therefore each model RE cell made two sets of connections onto TC cells. One set, representing tickler connections, only made a few synaptic contacts (6) (Cox et al. 1997b
) with each postsynaptic TC cell and elicited relatively weak postsynaptic GABAB currents but contacted TC cells over a relatively large radius (500 µm) (Cox et al. 1996
). The other set, representing cluster connections, made many synaptic contacts (257) (Cox et al. 1997b
) with each postsynaptic TC cell and elicited relatively strong postsynaptic GABAB currents but only contacted TC cells over a relative small radius (100 µm) (Cox et al. 1996
).
GABA-B) and the radii of tickler and cluster connections, determine the GABAB conductances at every tickler and cluster synapse on that TC cell. As described in the RESULTS, we tested many values of
GABA-B and choose the one for which simulated oscillations propagated at speeds similar to those observed in vitro. Then we compared results from simulations in which tickler connections had the radius measured anatomically (500 µm) to those from simulations in which tickler connections had the same radius as cluster connections (100 µm). Because all other parameters were physiologically constrained, we could then deduce possible functions of long-range (500 µm in radius) tickler connections.
Computational methods
All simulations were run with NEURON (Hines and Carnevale 1997) at a temperature of 37°C and with a time step of 0.1 ms. Every simulation began with a 500-ms delay period preceding stimulation of any neurons that allowed window currents, etc., to reach steady state. This delay period is not shown in any figures. Distributions of pseudorandom variables were always initialized with the same seed value, allowing us to compare results from networks with different parameters but the same "random" initial conditions. To check for finite-size effects, we repeated our simulations with a network of 128 each TC and RE cells representing a thalamic slice of the standard length (so connections from one cell contacted twice as many cells as in the standard network). We also checked for edge effects by repeating our simulations with a network of 128 each TC and RE cells representing a thalamic slice that was twice the standard length (so connections from one cell contacted the same number of cells as in the standard network).
In vitro slice recordings
Slice experiments were performed as previously described (Huguenard and Prince 1994). Rat pups of either sex, aged 14-18 days (p14-p18) were anesthetized (50 mg/kg pentobarbital) and decapitated, and the brains were rapidly removed and placed in chilled (4°C) low-Ca, low-Na slicing solution consisting of (in mM) 234 sucrose, 11 glucose, 24 NaHCO3, 2.5 KCl, 1.25 NaH2PO4, 10 MgSO4 and 0.5 CaCl2, equilibrated with a mixture of 95% O2-5% CO2. Horizontal slices 400 µm thick were obtained with a vibratome (TPI, St. Louis). The slices were incubated in 30°C oxygenated saline before being placed in a modified (partially submerged) interface recording chamber. Slices were maintained at 34 ± 1°C in the recording chamber and remained viable for several hours under these conditions. The superfusion solution consisted of (in mM) 126 NaCl, 26 NaHCO3, 2.5 KCl, 1.25 NaH2PO4, 0.63 MgCl2, 2 CaCl2, 0.01 bicuculline methiodide (BMI), and 10 glucose. Extracellular stimuli were delivered through a pair of 1- to 10-M
tungsten electrodes (FHC, Maine) glued together with a separation of about 100 µm. Extracellular unit and multiunit recordings were obtained with 5- to 10-M
tungsten electrodes and normally filtered between 100 and 3 KHz.
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RESULTS |
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Here we briefly describe the results of extracellular recording from thalamic slices and then describe in detail simulations of this and other experimental data with our network model. Finally, we quantitatively compare the experimental results with results obtained with two network models with different parameters. In the following section, we focus on results from the standard network, having 64 TC cells and 64 RE cells, because results from the networks with 128 TC cells and 128 RE cells were qualitatively and quantitatively similar to these.
Slow, synchronized intrathalamic oscillations in vitro
Figure 1, A and B, shows contour plots of extracellularly recorded activity from RE and TC cells, respectively, as functions of time (running horizontally) and distance along the slice (running vertically). These figures show that, as in previous experiments (Huguenard and Prince 1994), indirect excitation of RE cells via stimulation of internal capsule elicits
3-Hz oscillations. They also show that these oscillations remain highly synchronized across widely separated RE and TC cells.
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Interactions between RE and TC cells produce synchronized network oscillations
As in Fig. 1, in earlier in vitro studies (Bal et al. 1995a,b
; Huguenard and Prince 1994
; Kim et al. 1995
; von Krosigk et al. 1993
) and earlier models (Destexhe et al. 1996a
; Golomb et al. 1996
), interactions between RE and TC cells produce synchronized oscillations at
3 Hz in our network model. Figure 2A shows a contour plot of RE cell activity (i.e., the spike rate of each RE cell, computed with 10-ms wide bins) as a function of time (horizontal axis) and distance along the slice (vertical axis). Figure 2B shows an analogous contour plot of TC cell activity. Figure 2C plots membrane potential versus time for an RE cell at the center of the slice, and Fig. 2D plots the membrane potential versus time for the corresponding TC cell.
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,b
; von Krosigk et al. 1993
). For the network shown,
GABA-B = 31.5 nS (although the fraction of GABAB receptors activated was never >0.7, corresponding to an effective
GABA-B of 22.1 nS).
Depression of GABA release causes oscillations to terminate
The oscillations shown in Figs. 1 and 2 lasted~2 s, similar to the duration of slow, synchronous oscillations observed in vitro (Bal et al. 1995a,b
; Huguenard and Prince 1994
; Kim et al. 1995
; von Krosigk et al. 1993
). In the network model, IPSPs in TC cells gradually waned, as shown in Fig. 2D, because of the presynaptic suppression of GABA release, until they were no longer sufficiently hyperpolarizing to elicit rebound bursts, and the oscillation terminated. Figure 3, A and B, shows contour plots of activity in a network, identical to those depicted in Fig. 2, except that the probability of GABA release does not decrease. Whereas the oscillations shown in Figs. 1 and 2 terminated after~2 s, the network activity shown in Fig. 3 continued to oscillate for the duration of the simulation (3 s).
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).
Blocking T-channels reduces the duration and increases the period of oscillations
We simulated the effects of T-channel blockers by reducing PTs and PT, the permeabilities of T currents in RE and TC cells, respectively (the network was otherwise identical to that represented by Fig. 2). Figure 4 includes two contour plots that compare RE cell activity before and after a 25% reduction in PT and PTs. Figure 4 also plots the period, duration, and phase shift (this phase shift will be defined and discussed later) of 3-Hz oscillations versus the percent reduction in PT and PTs. These plots demonstrate that, as in experiments (Huguenard and Prince 1994
; Smith and Huguenard 1996
), modest T-channel blockade shortens the duration and reduces the period of the slow, synchronous oscillations. For example, oscillations in the control network had a period of 328 ms and lasted eight cycles, whereas after 25% T-current blockade the period was 363 ms and the oscillations lasted five cycles.
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) and shown in Fig. 2, C and D, most of a cycle is spent in the time it takes IPSPs in TC cells to decay, producing a rebound burst. When some T-channels are blocked, the threshold for a Ca2+ spike (a dynamic variable) (see Ulrich and Huguenard 1996b
) becomes more depolarized. As a result, TC cells reach threshold later, prolonging each cycle of the oscillation. Similarly, when some T-channels are blocked, stronger IPSPs are required to elicit rebound bursts. As a result, IPSPs fall below this critical strength, and the oscillation terminates earlier.
Suppression of synaptic transmission shortens both the duration and period of oscillations
We also simulated the suppression of intrathalamic synaptic transmission by reducing both synaptic conductances, GABA-B and
AMPA (the network was otherwise identical to that represented by Fig. 2). Figure 5 includes two contour plots that compare RE cell activity before and after suppressing synaptic strength by 25% and also plots period, duration, and phase shift versus percent suppression of synaptic strength. Suppression of synaptic strength shortened both the duration and period of oscillations in network activity. For example, after a reducing synaptic strength by 25%, the duration of oscillations fell from eight cycles to four cycles, and their period fell from 328 to 319 ms.
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Depolarization of RE cells by CCK has biphasic effects on oscillations
Cholecystokinin (CCK) depolarizes RE cells by decreasing a leak K+ conductance (Cox et al. 1995). CCK also has biphasic effects on
3-Hz thalamic oscillations in vitro; in some slices, low concentrations prolonged these oscillations, but higher doses suppressed these oscillations, whereas in other slices all doses of CCK tested suppressed oscillations (Cox et al. 1997a
). We simulated the application of CCK in network models as follows. First, we assumed that the leak current in RE cells was composed of K+ and Na+ currents. Assuming that ENa = +50 mV and EK =
105, we calculated gleak,K and gleak,Na. To simulate varying doses of CCK application, we then increased gleak,K by varying amounts (cf. Sohal and Huguenard 1998
).
Propagation of synchronous oscillations
Slow, synchronous oscillations originating at one point were observed to propagate across thalamic slices (Kim et al. 1995
Propagation of oscillations in the absence of long-range tickler connections
After reducing the radius of tickler cell connections from 500 to 100 µm, while holding constant both the total GABAB conductance on each TC cell and the ratio between GABAB conductance postsynaptic to tickler connections and that postsynaptic to cluster connections, focal stimulation still elicits a synchronous,
Long-range, but not short-range, tickler connections produce phase differences that are consistent with experimental measurements
We compared phase differences recorded during in vitro oscillations elicited by stimulation of internal capsule (such as those depicted in Fig. 1) to phase differences that occurred during simulations of these oscillations in networks with either long-range tickler connections and
Collisions among propagating oscillations
Initially out-of-phase oscillations that begin at opposite ends of a thalamic slice in vitro can propagate toward the center of the slice where they coalesce and come into synchrony (Kim et al. 1995
Through a combination of computational modeling and in vitro extracellular recording, we observed the following. 1) Activity in bicuculline-treated thalamic slices oscillates at Propagation speed depends on synaptic strength and the spread of intrathalamic connections
As noted previously (Golomb et al. 1996 Phase differences measured in vitro also constrain network models
Because there exist multiple combinations of two parameters Long-range connections synchronize, rather than spread, oscillations
The assumption that the range of intrathalamic connectivity determines the distance over which oscillations spread per cycle and hence the speed of propagation is implicit in models of intrathalamic oscillations that use the speed of propagation to infer the spread of intrathalamic connectivity (Destexhe et al. 1996a Effects of edges and finite size of the model
The edges and finite size of the model should not have created spurious conclusions because our conclusions were based on phase shifts and speeds at which oscillations propagated, neither of which was affected by the model's edges or finitude. First consider the phase shifts. We measured phase differences on the first cycle of oscillations after broad stimulation. As a result, the presence of edges could only affect TC cells <500 µm (the radius of tickler connections) from an edge. We excluded these cells when computing phase differences. Therefore the computed phase difference was unaffected by the presence of edges. In addition, the conclusions regarding phase shift were not dependent on model size as a 256-cell network (representing a standard-length slice) produced results similar to those observed in vitro only when long-range tickler connections were present.
Limitations of this model
Two shortcomings of this network model of intrathalamic oscillations are that it lacks the refractory period between oscillations observed in vitro (Kim et al. 1995 Predictions
We quantitatively compared the maximum phase difference in between the activity of TC cells after widespread stimulation of RE cells in a network model with long-range tickler connections, in a network model with short-range tickler connections, and in vitro. This comparison could be improved. The network models assumed values of We thank B. Yue for assistance with the extracellular recording experiments.
This work was supported by National Institute of Neurological Disorders and Stroke Grants NS-06477 and NS-34774 and the Pimley Research Funds.
Intrinsic currents
As in earlier studies (Hodgkin and Huxley 1952 Time course of postsynaptic GABAB currents
Like voltage-dependent intrinsic currents that were modeled by using Eq. A1, the time course of postsynaptic GABAB currents was modeled as the product of an activation function, m(t), and an inactivation function, h(t). On the basis of in vitro measurements made in the dentate gyrus at 34°C (Otis et al. 1993 Postsynaptic GABAB receptor occupancy
To calculate the fraction of occupied postsynaptic GABAB receptors at a synapse, we assumed that the fraction of GABAB receptors occupied because of an earlier GABAB response equaled the number of GABAB receptors activated on that response up until that response's peak and thereafter decayed with the same time course as the GABAB current, i.e.,
3-Hz oscillations in the network model. This result reproduces the biphasic effect of CCK on the duration of intrathalamic oscillations in vitro. It also suggests that slices that have a monophasic response to CCK (shortening of oscillations at all doses) have a lower average gleak,K than do slices that have a biphasic, dose-dependent response to CCK and that the network model falls in the former category. Furthermore, as observed in vitro (Cox et al. 1997a
), low gleak,K (produced by high doses of CCK) increases the tonic, nonrhythmic, activity of RE cells. Figure 6 also suggests that CCK application should decrease the period of
3-Hz oscillations slightly. Although an earlier experiment did verify that oscillations remain in the 2- to 4-Hz range after CCK application (Cox et al. 1997a
), small changes in the period were specifically measured. Subsequent analysis of those results indicates that the period is slightly decreased, especially for modest CCK concentrations (e.g., see Fig. 4B of Cox et al. 1997a
).
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FIG. 6.
Effects of cholecystokinin (CCK) on simulated oscillations. On the basis of experimental evidence, we simulated CCK application by reducing the leak potassium conductance, gleak,K, in RE cells. A-C: contour plots of RE cell activity for 3 values of gleak,K. D: plots of the maximum phase shift, period of oscillations, and duration of oscillations vs. relative gleak,K (relative gleak,K = 1 corresponds to the value used in the network model). All values are normalized by the values in the control network (the network with gleak,K = 1). Details for contour plots and synaptic conductances are given in Fig. 2.
) and one-dimensional network models (Destexhe et al. 1996a
; Golomb et al. 1996
). Similarly, in this model stimulating a few RE cells located around the center of the model slice created an oscillation that propagates outward in both directions. Figure 7, A and B, shows the propagation of activity through RE and TC cells, respectively. As described in METHODS, we measured the speed of propagation for many values of
GABA-B. In Fig. 7, A and B, as in earlier simulations,
GABA-B = 31.5 nS so that the speed of propagation was 0.28 mm/s, which is within the range of values observed experimentally. APPENDIX A describes how the speed of propagation was calculated.
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FIG. 7.
Propagating oscillations in a network model with long-range tickler connections. Contour plot of concurrent RE cell (A) and TC cell (B) activity as a function of time and distance along the model slice. An oscillation is elicited by focal stimulation near the center of the slice and subsequently propagates outward at a physiologically realistic speed of 0.28 mm/s. GABA-B was 30.5 nS, [
GABA-B(tickler) = 21.8 nS and
GABA-B(cluster) = 8.7 nS] and
AMPA was 150 nS, the radius of connections from TC cells to RE cells was 50 µm, the radius of tickler connections was 500 µm, and the radius of cluster connections was 100 µm. To initiate oscillations, RE cells near the center of the model slice were probabilistically excited above the burst threshold.
3-Hz oscillation at the center of the slice. However, when
GABA-B = 31.5 nS, this oscillation no longer propagates outward (not shown). However, when the total GABAB conductance on each TC cell,
GABA-B, is increased to 42 nS, then even with these "short-range" tickler connections, oscillations elicited by focal stimulation propagate outward from the center of the slice at a speed of 0.24 mm/s. Figure 8, A and B, displays RE and TC cell activity, respectively, in this network. Thus, we found that by choosing
GABA-B appropriately, both short-range and long-range tickler connections produce synchronous,
3-Hz oscillations that propagate at speeds similar to those observed in vitro. However, comparison of Figs. 7 and 8 reveals that the synchronous,
3-Hz oscillations are not identical under the two sets of conditions, long-range tickler connections with
GABA-B = 31.5 nS and short-range tickler connections with
GABA-B = 42 nS. Phase differences between distant RE or TC cells seem to be much larger in the latter network (shown in Fig. 8) than in the former (shown in Fig. 7). We sought to quantify this discrepancy to determine which network better matched experimental data.
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FIG. 8.
Propagation of oscillations in a network model without long-range tickler connections. Contour plot of concurrent RE cell (A) and TC cell (B) activity as a function of time and distance along the model slice. An oscillation is elicited by focal stimulation near the center of the slice (equivalent to that used in Fig. 7) and subsequently propagates outward at a physiologically realistic speed of 0.24 mm/s. Note that, although the speed at which oscillations propagate is similar to that in a network model with long-range tickler connections (shown in Fig. 7), in this case oscillatory activity is much less synchronized across the network. GABA-B was 42 nS [
GABA-B(tickler) = 30 nS and
GABA-B(cluster) = 12 nS], the total postsynaptic conductance on each RE cell was
AMPA = 150 nS, the radius of connections from TC cells to RE cells was 50 µm, and the radius of both tickler and cluster connections was 100 µm.
GABA-B = 31.5 nS (shown in Fig. 2) or short-range tickler connections and
GABA-B = 42 nS (shown in Fig. 9). All other parameters in the two simulated networks were equal. For each simulated or in vitro oscillation, we calculated the phase difference between TC cell activity at every pair of locations during the first cycle of the oscillation by cross-correlation and found the maximum phase difference. Because extracellular recordings were not made at the edges of the slice and to minimize edge effects, in the network model we only calculated phase differences for cells
500 µm away from the edges. During two in vitro oscillations the maximum phase differences were 50 and 60 ms. Simulation of the network with long-range tickler connections resulted in a maximum phase difference of 69 ms, whereas a simulation with short-range connections yielded a maximum phase difference of 151 ms. To provide a sense of just how different these two values were, we found that the maximum phase differences were 68 and 62 ms in the networks with 25% blockade of T-channels and with 25% suppression of synaptic transmission, respectively. This shows that changing the range of tickler connections has a much larger effect on the maximum phase difference than do other manipulations, suppression of synaptic transmission or T-channel blockade, which significantly alter network dynamics. Thus, in response to focal excitation at the center of the slice, both short-range and long-range tickler connections produce oscillations that propagate with approximately equal speeds. However, during simulations of in vitro oscillations elicited by broad stimulation of RE cells, only the network with long-range tickler connections produces phase differences that are similar to those observed in vitro. We also checked for finite-size effects on the maximum phase shift after broad stimulation, with the 256-cell network representing a standard-length slice. In this network the maximum phase shift after stimulation of RE cells (71 ms) was quantitatively similar to that observed in vitro when long-range tickler connections were present but was much larger (123 ms) when they were absent.
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FIG. 9.
Synchronous, slow ( 3-Hz) oscillations in a network model without long-range tickler connections. Contour plot of concurrent RE cell (A) and TC cell (B) activity as a function of time and distance along the model slice. At time 0, to simulate the indirect excitation of RE cells via stimulation of internal capsule, 1/3 of the RE cells were excited above the burst threshold. Note that the resulting oscillations are much less synchronized across the network than in a network with long-range tickler connections, shown in Fig. 2. In this network, synaptic conductances were the same as in Fig. 8.
(4)
1 so that we are far from saturation, the main source of variation in the phase of TC cell bursting, tburst, results from the second term on the right hand side of Eq. 4. This term can depend on
and n. Because n is a random variable that represents the number of RE cells presynaptic to a given TC cell that burst, it depends on the spread of connections from RE cells to TC cells and the probability that an RE cell reaches the burst threshold during the initial stimulation. Thus to a first approximation, the variation in the phase of TC cell activity depends in a relatively straightforward way on the synaptic conductance,
, and on the radius of connectivity from RE cells to TC cells but does not depend on any other network parameters. This property makes phase shifts during the first cycle of TC cell activity a useful measurement with which to constrain the radius of RE to TC connectivity and the strength of synaptic transmission in our network model.
). On the basis of our earlier results, we hypothesized that long-range tickler connections may help to synchronize colliding oscillations. To test this hypothesis, we simulated collisions among propagating oscillations in networks with long-range and short-range tickler connections (
GABA-B was 31.5 and 42 nS, respectively, in these 2 networks). To simulate collisions among oscillations propagating from opposite ends of a thalamic slice (Kim et al. 1995
), we initially stimulated two groups of RE cells on opposite ends of a network, 100 ms out of phase.
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FIG. 10.
Colliding oscillations in a network with long-range tickler connections. Contour plot of concurrent RE cell (A) and TC cell (B) activity as a function of time and distance along the model slice. Two oscillations are initiated by focal excitation of RE cells on opposite ends of the network and 100 ms out of phase. As they propagate outward, the 2 oscillations collide, coalesce, and come into synchrony. Note that, on the cycle of activity immediately after the collision, the phase gradient varies smoothly across the network. Synaptic conductances were the same as in Fig. 7.
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FIG. 11.
Colliding oscillations in a network without long-range tickler connections. Contour plot of concurrent RE cell (A) and TC cell (B) activity as a function of time and distance along the model slice. Two oscillations are initiated by focal excitation of RE cells on opposite ends of the network and 100 ms out of phase. As they propagate outward, the 2 oscillations collide, coalesce, and come into synchrony. Note that, in contrast to Fig. 10, on the cycle of activity immediately after the collision, the phase gradient changes sharply where the 2 oscillations collided. Synaptic conductances were the same as in Fig. 8.
DISCUSSION
Abstract
Introduction
Methods
Results
Discussion
References
3 Hz, and these oscillations are synchronized with small phase differences over relatively large distances in VB and RE. 2) Similar oscillations result in a network model through the previously published cycle of events; RE cells' bursts elicit slow GABAB IPSPs in TC cells, these IPSPs deinactivate IT and are therefore followed by rebound bursts in TC cells, and TC cells' bursts elicit fast EPSPs in RE cells, producing bursts. 3) In the network model, modest T-channel blockade reduces the duration and increases the period of these oscillations, whereas modest synaptic suppression shortens both the duration and the period of these oscillations. For some initial values of gleak,K, CCK prolongs oscillations at low doses but curtails these oscillations at higher doses. For lower initial values of gleak,K, all doses of CCK curtail oscillations. CCK always shortens the period of these oscillations. 4) Focal excitation of RE cells elicits oscillations that propagate outward. By choosing
GABA-B appropriately, we were able to obtains propagation speeds similar to those observed in vitro (Kim et al. 1995
) in networks with either short-range or long-range tickler connections. 5) However, by using the
GABA-B values that produce physiologically realistic propagation speeds, the network model with short-range tickler connections and the model with long-range connections exhibit radically dissimilar phase differences between activity at distinct locations. Only the relatively small phase differences produced by the network with long-range tickler connections are similar to those observed in vitro. Furthermore, although colliding oscillations coalesce in both networks, the colliding oscillations come into synchrony much more rapidly in the network with long-range tickler connections.
3-Hz intrathalamic oscillations in vitro (Bal et al. 1995a
,b
; Huguenard and Prince 1994
; Kim et al. 1995
; von Krosigk et al. 1993
). Observations 2 through 4 demonstrate that our network model is consistent with a wide body of experimental observations. The final observation constitutes our main finding that, when
GABA-B is chosen to reproduce the propagation speed observed in vitro (Kim et al. 1995
), the network without long-range tickler connections does not reproduce the maximal phase differences we found during
3-Hz oscillations in vitro.
) the speed with which synchronous
3-Hz oscillations propagate depends on both the synaptic conductance and the radius of connections. This has important consequences for conclusions drawn from computational models. For example, other models assumed a value for synaptic conductances and, based on that value, estimated the spread of intrathalamic projections for which model oscillations propagate at physiologically realistic speeds (Destexhe et al. 1996a
; Golomb et al. 1996
). In contrast, we emphasize that the GABAB synaptic conductance can be adjusted (e.g., networks with only short-range connections have
GABA-B = 42 nS whereas those with both short and long-range connections have
GABA-B = 31.5 nS), so that the extent of intrathalamic connectivity could increase fivefold without significantly changing the speed at which oscillations propagate. Thus we could not distinguish networks with only short-range connections from those with both short- and long-range connections just on the basis of propagation speed.
GABA-B and the radius of tickler connections
that produce propagation speeds similar to those observed in vitro, we looked for an additional observable that would constrain the values of these two parameters in our network model. Casual comparison of Figs. 7 and 8 suggested that different values of these parameters that produced similar speeds could be distinguished based on the phase differences they produced during
3-Hz oscillations.
3-Hz oscillations were studied before (Golomb et al. 1996
). This study concluded that, away from edges and the wavefront, phase shifts grow linearly with distance; however, this would be true for any translation-invariant system. Furthermore, this study dealt with phase shifts that develop as the wavefront of the oscillation propagates. Because these phase shifts develop over time, they are affected by many aspects of network activity. Thus these phase shifts depend sensitively on network parameters (Golomb et al. 1996
). Such complexity would make it difficult to constrain our network model based on these phase shifts.
), we measured the phase shifts on the first cycle of activity after far-reaching stimulation of RE cells. As shown in RESULTS, these phase differences depend primarily on synaptic conductance and the radius of connectivity from RE to TC cells and are therefore particularly useful measurements with which to constrain the radius of RE to TC connectivity and the strength of synaptic transmission in our network model.
; Golomb et al. 1996
). Our observation that two networks in which the ranges of tickler cell connections differ by a factor of five may exhibit similar propagation speeds but radically different phase shifts dispels this assumption and suggests that long-range tickler connections (Cox et al. 1996
) function to synchronize activity and minimize phase shifts across the network rather than to spread activity over greater distances.
). However, there are also regions containing exclusively long-range, diffuse RE-to-TC cell connections such as the rostral pole of the RE complex (Steriade et al. 1984
). Our results may illustrate ways in which these regions can synchronize thalamic activity.
) and that in the model the period of oscillations become slightly shorter as the oscillation progresses, whereas in vitro (unpublished observations) and during clinical absence seizures (Panayiotopoulos 1994
) the period lengthens significantly. The first of these shortcomings might be corrected by inclusion of Ca2+-dependent persistent activation of Ih in TC cells. (In addition, the progressive hyperpolarization of RE/PGN cells observed during in vitro oscillations might influence oscillation period) (Bal et al. 1995a
). The h-current modulation mechanism was shown to reproduce the refractory period in earlier models (Destexhe et al. 1996a
), and there is strong evidence that it contributes to refractoriness in vitro (Bal and McCormick 1996
). To study how persistent activation of Ih would affect the period of oscillations, we simulated oscillations with the Ih model from a previous study (Destexhe et al. 1996a
) that included Ca2+-dependent persistent activation. We found that, as this persistent activation of Ih increased, the period of oscillations decreased. Thus, during an oscillation, as activation of Ih increases, the oscillation's period would gradually decrease, contrary to experimental observations.
). This suggests that the evoked IPSCs are dominated by input from only one or a few bursting RE cells, i.e., only a few of the many RE cells presynaptic to one TC cell burst after stimulation of internal capsule. If this is true, then the probabilities of RE cells bursting after stimulation of internal capsule have a Poisson distribution, which has a relatively low mean and high variance, and would therefore be similar to the distribution of RE cell firing assumed in the model.
GABA-B chosen to produce propagation speeds similar to those observed in vitro. However, the in vitro measurements of phase differences and propagation speeds were made in different thalamic slices, taken from different animals, by different researchers. A more rigorous test of this model would be to reproduce the propagation speed and phase differences of a single slice and any covariance of propagation speed and phase differences between slices. Predictions about how certain pharmacological manipulations should affect both the propagation speed and phase differences are as follows.
3-Hz oscillations in the network model, consistent with the effects of T-channel blockers measured in vitro (Huguenard and Prince 1994
; Smith and Huguenard 1996
). Because T-channel blockers increase the threshold for rebound bursts in TC cells, which should reduce the number of TC cells recruited by propagating oscillations and lengthen the period of these oscillations, T-channel blockers should reduce the speed at which oscillations propagate. We can also use Eq. 4 and simulations to predict how T-channel blockers will affect the maximum phase difference among TC cells (measured on the first cycle of activity after widespread stimulation of RE cells). From that equation we see that the maximum phase difference,
, should depend on the values of nmin, the minimum number of IPSPs that occur in a TC cell and elicit a rebound burst, and nmax, the maximum number of IPSPs that occur in a TC cell, in other words
(5)
after decreases in synaptic strength. Because the minimum IPSP required to elicit a rebound burst is independent of
(synaptic strength), the quantity nmin
should remain constant so that
depends on nmax
(6)
. Both of these effects can be explained by assuming that depolarizing RE cells reduces the deinactivation of ITs between bursts. Subsequently, RE cells' burst become shorter and evoke weaker IPSPs in TC cells. As a result, the effects of CCK on the period of oscillations and
are analogueous to the corresponding effects of suppressing synaptic strength.
. If it were possible to pharmacologically inhibit presynaptic but not postsynaptic GABAB receptors, we would predict that such a manipulation would prolong oscillations. At this point there are no such specific GABAB antagonists currently available.
) by using physiological methods. This could be done by focally stimulating a small group of RE cells (e.g., by blocking excitatory transmission and electrically stimulating RE cells or direct application of glutamate onto a small group of RE cells) and measuring the responses of TC cells at different locations. On the basis of the RE to TC projections assumed by the model, we would predict that focal stimulation of RE cells should produce strong inhibition within a central radius of
100 µm and weaker inhibition within a larger radius of
500 µm.
ACKNOWLEDGEMENTS
APPENDIX A: A MODEL PARAMETERS
; Huguenard and McCormick 1992
), we assumed that each intrinsic current was composed of many equivalent ion channels, each channel was comprised of at least one gate, and all of the gates comprising an individual channel must be open for the channel to conduct current. This leads to the same generic formalism for all voltage-dependent intrinsic currents
where
(A1)
is the current's condutance or permeability as a fraction of its maximum conductance or permeability.
where Vm is the membrane potential and gmax and Eeq are the current's maximum conductance and equilibrium potential, respectively. For the low-threshold Ca2+ currents, IT and ITs, we used the constant field equation
(A2)
where P is the maximum permeability, z is 2 (the valence of Ca2+), [Ca2+]i = 240 nM and [Ca2+]o = 2 mM are the concentration of Ca2+ inside and outside of the cell, respectively, F is Faraday's constant, and R is the universal gas constant.
(A3)
, and the maximum conductances were adjusted to give action potentials of realistic height and duration.
, in which all rate constants were temperature corrected by using a Q10 = 2.5. We set the maximum permeability of the T current, PT, to 50 × 10
9 cm3/s (at the simulation temperature of 37°C). This permeability falls within the range of values observed in intact cells (McCormick and Huguenard 1992
) and was chosen so that both the number of spikes per burst during bicuculline-induced synchronized oscillations and the change in the duration of these oscillations after partial blockade of T-channels were similar in simulations and experiments.
, in which all rate constants were temperature corrected with a Q10 = 3. We set the maximum h-current conductance to 20 µS/cm2. This was the same value used in a previous model (Destexhe et al. 1996a
) and was sufficient to produce characteristic oscillations in isolated TC cells (not shown) (see McCormick and Huguenard 1992
). As in previous simulations (Destexhe et al. 1996a
; Huguenard and McCormick 1992
; McCormick and Huguenard 1992
) the h-current equilibrium potential was
40 mV.
, characteristic of TC cells in a deafferented slice. At a typical resting membrane potential of
63 mV, T-current activation and inactivation reached a steady state, producing a significantly depolarizing "window current." To balance this depolarizing window current, the leak current's equilibrium potential had to be hyperpolarized relative to the resting membrane potential. We found that setting Eleak to
75 mV produced a resting membrane potential of
63 mV. Therefore the Eleak of each TC cell was drawn from a normal distribution with a mean of
75 mV and SD of 2 mV. We used a distribution of Eleak values to approximate the variation of cellular excitability in vitro and eliminate some spurious solutions that might have been produced by nonphysiological symmetry in the network.
) and temperature corrected all rate constants with Q10 = 2.5. As for IT, the maximum permeability of ITs, 20 × 10
9 cm3/s (at 37°C), was chosen so that both the number of spikes per burst during bicuculline-induced synchronized oscillations and the change in the duration of these oscillations after partial blockade of T-channels were similar in simulations and experiments.
85 mV in an RE cell was sufficiently hyperpolarized that ITs would deinactivate between bursts and the cell would not burst spontaneously. Therefore the Eleak of each RE cell was drawn from a normal distribution with a mean of
85 mV and SD of 2 mV.
), m(t) has a monoexponential rise (time constant = 45.2 ms) and is raised to the fourth power, whereas the inactivation function, h(t), consists of a biexponential decay (fast decay: time constant = 110.2 ms, relative weight = 84%; slow decay: time constant = 516.2 ms, relative weight = 16%) and is raised to the first power. The time constants were temperature corrected by using a Q10 of 2.2.
where the sum is taken over all previous GABAB responses at the same synapse (indexed by i), Ri is the fraction of receptors that were activated by GABAB response i, t is time, GABAB response i occurred at time ti, and
(A4)
(t) is m4(t)h(t) if t > tpeak and 1 otherwise [m(t) and h(t) are the activation and inactivation functions, respectively, for the GABAB current].
Probabilistic release and paired-pulse depression
Whenever a presynaptic RE cell spiked, it released GABA from a release site located on synapse i with probability pi. In accordance with paired-pulse depression of GABAB responses, pi at time t was given by
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(A5) |
Postsynaptic GABAB conductance
The total postsynaptic GABAB conductance on a TC cells, GABA-B, was distributed between tickler and cluster connections based on the following observations:
).
).
).
). Therefore, assuming that this study sampled equal numbers of cluster and tickler cells (reasonable given the first observation), the third observation provides indirect evidence that, within a small region of VB, a cluster cell and a tickler cell contact similar numbers of postsynaptic cells. Assuming that this is true, because tickler connections extend over a much greater distance than do cluster connections, one TC cell should receive many more tickler connections than cluster connections; the ratio of tickler to cluster connections on a postsynaptic cell will be approximately equal to the ratio between the area of the axonal arborization of a tickler cell and that of a cluster cell.
(A6)
Total synaptic input conductances
The total synaptic conductance on a TC cell is given by
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(A7) |
where N is the total number of AMPA synapses onto the RE cell, i indexes TC cells presynaptic to the RE cell, and Ri(t) is the fraction of AMPA receptors postsynaptic to TC cell i that are active at time t evolves according to Eq. 3.
(A8)
Calculating the speed of propagation in the network model
We used the same method as an earlier study (Golomb et al. 1996) to calculate the speed at which oscillations propagated in the network model. To calculate the speed of propagation in a particular direction, e.g., to the right, consider the subset of TC cells that spiked before any of their neighbors to the right spiked. We performed linear regression on the locations of these TC cells and the times of their first spikes and used the slope of the resulting best-fit line as the speed of propagation for oscillations in the network model.
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APPENDIX B |
---|
Maximum phase shift
Assume that every RE cell burst activates the same conductance in a postsynaptic TC cell. Let gGABA-B represent this conductance, normalized by the passive conductance of the TC cell, and n represent the number of active RE cells presynaptic to the TC cell. Then the total GABAB conductance activated in the TC cell will be ngGABA-B. Assume also that all presynaptic RE cells burst simultaneously and that IPSPs decay monoexponentially with a time constant of GABA-B. During an IPSP, the time constant for the h current,
m, is >600 ms (Huguenard and McCormick 1992
). Because
m and
GABA-B are both at least an order of magnitude longer than the membrane time constant, we can assume that changes in the membrane potential, Vm, are "slaved" to changes in the h current or IPSC. Thus, as the GABAB postsynaptic potential decays, Vm will be approximately
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(B1) |
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(B2) |
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(B3) |
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(B4) |
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(B5) |
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FOOTNOTES |
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Address reprint requests to J. R. Huguenard.
Received 6 October 1997; accepted in final form 16 June 1998.
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REFERENCES |
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