Cortical oscillatory activity as measured by electroencephalogram (EEG) is a clear signature of the general state of the brain. The waking state and the rapid-eye-movement (REM) phase of sleep are characterized by low-amplitude fast oscillations (Gray et al. 1989; Steriade et al. 1996) of a generally low spatiotemporal coherence (Destexhe et al. 1999). In contrast, during slow-wave sleep and anesthesia, the brain shows pronounced oscillatory activity at a variety of frequencies often with remarkable long range synchrony (Bullock and McClune 1989; Destexhe et al. 1999; Steriade et al. 1993a, 1996). During slow-wave sleep, low-frequency (<1 Hz) oscillations are visible both in the EEG, and in extracellular and intracellular recordings (Achermann and Borbély 1997; Lampl et al. 1999; Steriade et al. 1993b,c, 1996; Stern et al. 1997). Lesion studies have shown that this type of rhythmic activity originates in the cortex and is then reflected in subcortical structures (Amzica and Steriade 1995; Steriade et al. 1993c). Intracellular recordings in vivo showed that the slow oscillation is mediated by two phases: a period in which nearly all cell types within the cerebral cortex are depolarized and generate action potentials at a low rate (the so-called up state) interdigitated with a period of hyperpolarization and relative inactivity (the down state). The transition from the up to down states has been proposed to occur either in response to synaptic "fatigue" or depression (Contreras et al. 1996) or to the build-up of activity-dependent K⁺ conductances (Sanchez-Vives and McCormick 2000). A gradual increase in input resistance of pyramidal cells during the up state in vivo has been taken to indicate a steady decrease of a specific ionic conductance, suggesting a stronger role of depression of excitatory synapses over the activation of K⁺ conductances in the transition from the up to the down state (Contreras et al. 1996; Timofeev et al. 2000b).

Recently, spontaneous activity similar to the slow oscillations (<1 Hz) recorded in vivo has been described in an in vitro slice preparation of cerebral cortex when maintained in an ionically modified artificial cerebral spinal fluid (ACSF) solution that mimics ionic concentrations in situ more closely than the solutions traditionally used for cortical slice preparations (Sanchez-Vives and McCormick 2000). This helped to identify candidate cellular and circuit mechanisms underlying the generation of slow oscillations and wave propagation in ferret visual cortex slices. For example, pharmacological manipulations show that this activity depends on excitatory transmission through AMPA and N-methyl-D-aspartate (NMDA) receptors, suggesting a critical role of recurrent excitatory connections. Open issues remain in relation to the two main aspects of slowly oscillating cortical activity: the membrane potential sudden transition between and up state and a down state and the propagation of activity across the cortical network. Evidence suggests that the transition from the up state to down state is induced by the opening of a K⁺ conductance (Sanchez-Vives and McCormick 2000), whose time course has led to the hypothesis that it is a slow Na⁺-dependent K⁺ conductance g_KNa known to exist in these neurons (Sanchez-Vives et al. 2000). This raises the question of how an increase of g_KNa in a pyramidal cell could be compatible with an increase of its input resistance observed during the course of an up state. Another intriguing aspect of these membrane fluctuations is the sharpness of the transitions between the up and the down states, where the relative contribution of intrinsic and network mechanisms remains to be established. On the other hand, as observed experimentally, an up state episode consists of barrages of synaptic activity that initiate earlier in infragranular laminae and occur in all cortical layers. This reverberatory network activity propagates across the slice at ~10 mm/s and is followed by a silent period of 2-4 s. Blocking GABA_A-mediated inhibition results in epileptiform discharges that propagate along the slice at ~100 mm/s. The ability of a cortical network to sustain two propagation velocities has been studied mathematically by Golomb and Ermentrout (2001) for a simple model where each cell is an integrate-and-fire neuron and is allowed to fire only one spike. It remains to be examined how these two wave propagation modes can be realized in a biophysically realistic network model of conductance-based neurons.

The combined results obtained from the in vivo and in vitro preparations provide a framework to build a physiologically realistic network model of the slow oscillation in a cortical slice. We present here this biologically realistic network model, and we use it to address the aforementioned questions about the rhythmogenesis and wave propagation. We then speculate about how our model could relate to slow oscillations during natural slow-wave sleep and activity in the waking state in vivo.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION REFERENCES

The network model consists of a population of 1,024 pyramidal cells and 256 interneurons equidistantly distributed on a line and interconnected through biologically plausible synaptic dynamics. Some of the intrinsic parameters of the cells are randomly distributed, so that the populations are heterogeneous. This and the random connectivity (determined by the synaptic probability distributions; see Fig. 2A) are the only sources of noise in the network.

Model neurons

Especially in vivo, intracellular voltage records show clear transitions between two well-defined stable membrane potentials (Cowan and Wilson 1994; Stern et al. 1997). It has been argued that intrinsic channels may shape the neuronal membrane potential via an inward rectifier K⁺ channel (I_AR) and a slowly inactivating K⁺ channel activated by depolarization (I_KS) (Nisenbaum et al. 1994; Wilson 1992; Wilson and Kawaguchi 1996). In our model pyramidal neurons, we include these and other channels found in cortical pyramidal cells.

Our model pyramidal cells have a somatic and a dendritic compartment (Pinsky and Rinzel 1994). The spiking currents, I_Na and I_K, are located in the soma, together with a leak current I_L, a fast A-type K⁺ current I_A, a non-inactivating slow K⁺ current I_KS, and a Na⁺-dependent K⁺ current I_KNa. The dendrite contains a high-threshold Ca²⁺ current I_Ca, a Ca²⁺-dependent K⁺-current I_KCa, a non-inactivating (persistent) Na⁺ current I_NaP, and an inward rectifier (activated by hyperpolarization) non-inactivating K⁺ current I_AR. The dynamical equations for the somatic voltage (V_s) and the dendritic voltage (V_d) are

with the membrane capacitance C_m = 1 µF/cm² and the areas being A_s = 0.015 mm² and A_d = 0.035 mm² for the soma and dendrite, respectively. The coupling between soma and dendrite is determined by g_sd = 1.75 ± 0.1 µS (axial resistance 0.57 M, standard deviation indicates the degree to which this parameter is randomly varied from cell to cell). I_syn,s and I_syn,d are the synaptic currents impinging on the soma and dendrites, respectively. In our simulations, all excitatory synapses target the dendritic compartment and all inhibitory synapses are localized on the somatic compartment of postsynaptic pyramidal neurons.

Interneurons are modeled with just Hodgkin-Huxley spiking currents, I_Na and I_K, and a leak current I_L in their single compartment
with the total neuronal surface area being A_i = 0.02 mm².

Ion channel kinetics and conductances

All ion channels are modeled following the Hodgkin-Huxley formalism, with gating variables x governed by the first-order kinetics equation dx/dt =  [_x(V)(1  x)  _x(V)x] = [x(V)  x]/_x(V).  being the temperature factor ( = 1 unless otherwise indicated).

For pyramidal cells, the sodium and potassium spiking currents are modeled following (Wang 1998) with slight variations. The sodium current I_Na = g_Namh(V  V_Na) has a maximum conductance of g_Na = 50 mS/cm², its rapid activation variable is replaced by its steady-state m = _m/(_m + _m) with _m = 0.1(V + 33)/[1  exp((V + 33)/10)] and _m = 4 exp((V + 53.7)/12) and the inactivation variable has _h = 0.07 exp((V + 50)/10) and _h = 1/[1 + exp((V + 20)/10)]. The temperature factor is  = 4. The delayed rectifier I_K = g_Kn⁴(V  V_K) has a maximal conductance g_K = 10.5 mS/cm² and its inactivation kinetics are set by _n = 0.01(V + 34)/[1  exp((V + 34)/10)] and _n = 0.125 exp[(V + 44)/25], with  = 4. The leakage current I_L = g_L(V  V_L) is a passive channel with conductance g_L = 0.0667 ± 0.0067 mS/cm² (Gaussian-distributed in the population, mean ± SD given). The fast A-type K⁺-channel is as in Golomb and Amitai (1997); I_A = g_Amh(V  V_K) has its fast activation variable replaced by its steady-state m = 1/[1 + exp((V + 50)/20)] and the inactivation variable is governed by h = 1/[1 + exp((V + 80)/6)] and _h = 15 ms. Its maximal conductance is g_A = 1 mS/cm². The non-inactivating K⁺-channel is modeled as in (Wang 1999a) but with no inactivation variable: I_KS = g_KSm(V  V_K). It has a maximal conductance g_KS = 0.576 mS/cm² and an activation controlled by m = 1/[1 + exp((V + 34)/6.5)] and _m = 8/[exp((V + 55)/30) + exp((V + 55)/30)]. The remaining currents are modeled with instantaneous activation because their activation is sufficiently fast and removing these additional variables significantly reduces the time required to perform our network simulations.

The persistent sodium channel I_NaP = g_NaPm(V  V_Na) has maximal conductance g_NaP = 0.0686 mS/cm², it activates instantaneously according to m = 1/[1 + exp((V + 55.7)/7.7)] and it does not inactivate. It is borrowed with parameter modification from (Fleidervish et al. 1996). The inwardly rectifying K⁺ channel was modeled as in (Stern et al. 1997; Spain et al. 1987) and adjusting the parameters: I_AR = g_ARh(V  V_K) activates instantaneously below a low-lying threshold following h = 1/[1 + exp((V + 75)/4)] and it has a maximal conductance g_AR = 0.0257 mS/cm². The high-threshold Ca²⁺-channel I_Ca = g_Cam(V  V_Ca) has g_Ca = 0.43 mS/cm² and is instantaneously activated at very depolarized voltages, thus making it effectively a very transient current. The voltage dependency is given by m = 1/[1 + exp((V + 20)/9)]. The concentration of intracellular calcium, [Ca²⁺], follows first-order kinetics as d[Ca²⁺]/dt = _CaA_dI_Ca  [Ca²⁺]/_Ca with _Ca = 0.005 µM/(nA · ms) and _Ca = 150 ms. The Ca²⁺-dependent K⁺ channel I_KCa = g_KCa[Ca²⁺]/([Ca²⁺]+K_D)(V  V_K) (with K_D = 30 µM) activates instantaneously in the presence of intracellular calcium [Ca²⁺], and it has a maximal conductance g_KCa = 0.57 mS/cm². All the mechanisms involving intracellular calcium are taken from Wang (1998). As for the intracellular sodium concentration [Na⁺], its dynamics are somewhat more involved because they incorporate a Na-K pump (Li et al. 1996): d[Na⁺]/dt = _Na(A_sI_Na + A_dI_NaP)R_pump{[Na⁺]³/([Na⁺]³ + 15³)  [Na⁺]/([Na⁺] + 15³)}, with _Na = 0.01 mM/(nA · ms), R_pump = 0.018 mM/ms, and [Na⁺]_eq = 9.5 mM. The Na²⁺-dependent K⁺ channel I_KNa = g_KNaw([Na⁺])(V  V_K) has a conductance g_KNa = 1.33 mS/cm² and w([Na⁺]) = 0.37/[1 + (38.7/[Na⁺])^3.5] (Bischoff et al. 1998). The kinetics for [Na⁺] and I_KNa are taken from (Liu 1999; Wang et al. 2002). For all these channels, the reversal potentials used are V_L = 60.95 ± 0.3 mV, V_Na = 55 mV, V_K = 100 mV, and V_Ca = 120 mV.

For the last three figures, a slight modification in the implementation of the I_KNa current was introduced because we realized that I_KNa was continuously contributing a sizable [Na⁺]-independent, voltage-independent hyperpolarizing current that was confounded with the leakage current. Because for those figures we were interested in changing independently g_KNa and g_L, we opted to subtract the constant part from I_KNa: I_KNa = g_KNa (w([Na⁺])  w([Na⁺]_eq))(V  V_K) and change the leakage properties to compensate for this: g_L = 0.07 mS/cm² and V_L = 62.8 mV.

For interneurons, the model was taken from Wang and Buzsáki (1996). The sodium current I_Na = g_Namh(V  V_Na) has a maximal conductance g_Na = 35 mS/cm², and its rapid activation is replaced by its steady-state value m = _m/(_m + _m) with _m = 0.5(V + 35)/[1  exp((V + 35)/10)] and _m = 20 exp((V + 60)/18). The inactivation gating variable is controlled by _h = 0.35 exp((V + 58)/20) and _h = 5/[1 + exp((V + 28)/10)]. The delayed rectifier I_K = g_Kn⁴(V  V_K) has g_K = 9 mS/cm² and it activates with kinetics given by _n = 0.05(V + 34)/[1  exp((V + 34)/10)] and _n = 0.625 exp((V + 44)/80). The leakage current I_L = g_L(V  V_L) is a passive channel with conductance g_L = 0.1025 ± 0.0025 mS/cm². The reversal potentials are V_L = 63.8 ± 0.15 mV, V_Na = 55 mV, and V_K = 90 mV.

Model pyramidal neurons set according to these parameters fire at an average of 22 Hz when they are injected a depolarizing current of 0.25 nA for 0.5 s. The firing pattern corresponds to a regular spiking neuron with some adaptation, no bursting pattern was ever observed. In contrast, a model interneuron fires at ~75 Hz when equally stimulated and has the firing pattern of a fast spiking neuron.

Model synapses

Kinetics of synaptic currents is modeled as in (Wang 1999b): a postsynaptic current I_syn = g_syns(V  V_syn) enters the postsynaptic neuron when the presynaptic neuron's action potential activates the gating variable s(t) following ds/dt = f(V_pre)  s/, with f(V_pre) = 1/[1 + exp((V_pre  20)/2)]. For AMPAR-mediated synaptic transmission,  = 3.48,  = 2 ms, and V_syn = 0; while for inhibitory synaptic transmission  = 1,  = 10 ms, and V_syn = 70 mV. In the case of NMDAR-mediated synaptic transmission, the gating variable follows a second-order kinetic scheme: ds/dt = (1  s)  s/, dx/dt = _xf(V_pre)  x/_x ( = 0.5,  = 100 ms, _x = 3.48, _x = 2 ms, V_syn = 0) so that the ensuing excitatory postsynaptic current (EPSC) has a slower rise phase and saturates at high presynaptic firing rates.

Unless specified otherwise, the synaptic conductances' maximal strengths are set to the following values: pyramidal to pyramidal: g = 5.4 nS, g = 0.9 nS; pyramidal to interneuron: g = 2.25 nS, g = 0.5 nS; interneuron to pyramidal: g_IE = 4.15 nS; and interneuron to interneuron: g_II = 0.165 nS. In Fig. 11 (except for A, left) the synaptic conductances are changed to g = 5.75 nS, g = 0.75 nS, g = 2.75 nS, g = 0.6 nS, g_IE = 4.25 nS, g_II = 0.135 nS. These values were chosen so that the network would show stable periodic propagating discharges with characteristics compatible with experimental observations. The precise network activity pattern is sensitive to these parameters but the qualitative presence of traveling waves and oscillations is robust to synaptic parameter changes (see examples in Fig. 6).

Cortical microcircuit connectivity

The neurons in the network are sparsely connected to each other through a fixed number of connections that are set at the beginning of the simulation. Neurons make 20 ± 5 (SD) contacts to their postsynaptic partners (multiple contacts onto the same target, but no autapses, are allowed). For each pair of neurons separated by a distance x in the network (see Fig. 2A), the probability that they are connected in each direction is decided by a Gaussian probability distribution centered at 0 and with a prescribed standard deviation : P(x) = exp(x²/2²)/. In our simulations, the total length of the model network is assumed to be 5 mm, and we let  = 250 µm for excitatory connections so that the typical size of a patch of connections coming from a single pyramidal neuron is 500 µm. This number is consistent with anatomical (Rockland 1985) data for local connections within ferret visual cortex. Anatomical studies have also shown that excitatory horizontal connections in cortex extend further away creating a periodic patchy pattern (Gilbert and Wiesel 1983; Rockland 1985). In some simulations, we include this by using a probability of connection given by P(x) = [exp(x²/2²) + s exp((x  d)²/2²) + s exp((x + d)²/2²)]/(1 + 2s)/ so that additional probability of connection (whose strength is controlled by the parameter s) is added at a distance d of the soma (see Fig. 11A, top right). For inhibitory connections, a Gaussian probability distribution is also used but with a smaller standard deviation  = 125 µm, except for simulations in Fig. 12, where  = 500 µm is used. Anatomical and physiological data indicate that axonal arbors from inhibitory (basket) cells vary considerably, ranging from narrow to widespread (Crook et al. 1998). Here we mostly work with the narrow inhibition architecture but we also briefly explore the case of broader inhibition (Fig. 12).

Robustness of the model

The model network that we present here has proven its robustness to parameter modification in a variety of tests. The model is robust to connectivity sparsity and randomness and to neuronal inhomogeneity. Furthermore, a certain amount of randomness and heterogeneity seems to confer more stability to smooth wave propagation. Also, intrinsic neuronal properties can be varied substantially without changing the essential propagation and oscillation properties. Using a network of integrate-and-fire neurons, instead of Hodgkin-Huxley neurons, shows similar network dynamics (see van Vreeswijk and Hansel 2001; Wang 1999b for non-traveling slow oscillations), but the model cannot reproduce experimental intracellular data quantitatively. However, as indicated by Goldman et al. (2001) in a different context, there are some intrinsic parameters that do affect the collective network dynamics importantly. The reversal leak potential, for instance, controls very finely the excitability of the neurons, and its mean value across the population has very marked effects on the frequency of the slow oscillations. However, the standard deviation with which the reversal leak potential is distributed across the neuronal population does not have such an important effect on the activity characteristics. In particular, increasing threefold the reversal leak potential standard deviation augments the slow oscillation frequency by just 50% (from 0.27 to 0.4 Hz), the wave velocity 40% (from 5 to 7 mm/s), and the overall firing frequency 60% (from 1.1 to 1.8 Hz). Instead, with the same manipulation in a network with blocked excitatory synaptic transmission, the average spontaneous rate of firing of the network increases more than fourfold (from 0.06 to 0.26 Hz) and the number of spontaneously active neurons more than doubles (from 12 to 30%). This implies that the model is robust with respect to the exact fraction of neurons spontaneously active in the absence of synaptic excitation, but it is more sensitive to the overall excitability level of the network. As for neuron number, our simulations show that doubling or halving the number of neurons in our model network does not change either the oscillation frequency, or the wave propagation velocity, or the average firing rate. This is so because the connectivity of the model is unaffected by the number of neurons. Any given neuron connects to an average of 20 postsynaptic cells independently of the size of the network.

Numerical methods

The model was implemented in a C++ code and simulated using a forth-order Runge-Kutta method with a time step of 0.06 ms.

Experimental methods

Experimental data depicted in Fig. 1 was collected extracellularly in prefrontal cortex slices of the ferret. Details of the methods can be found in Sanchez-Vives and McCormick, 2000.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION REFERENCES

Excitatory synaptic block reveals spontaneous neuronal firing

Extracellular multiple unit recordings in layer V revealed the basic characteristics of the slow oscillation in vitro, including the recurrence of synchronized bursts of activity in neighboring neurons, and the presence of spontaneous activity between up states (Fig. 1, B-D). The rate of firing of this multiple unit activity typically decreased following an up state but increased prior to the onset of the next up state (Fig. 1, B-D). At least some of this spontaneous activity was not dependent upon fast glutamatergic excitation because it survived block of AMPA and NMDA receptors with bath application of 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX; 20 µM) and DL-2-amino-5-phosphonovaleric acid (DL-APV; 50 µM; n = 6 slices). Block of glutamatergic excitatory postsynaptic potentials (EPSPs) resulted in both a block of the up state (see Sanchez-Vives and McCormick 2000) and a significant reduction in spontaneous activity during the down state. However, in 4 of 10 recorded layer V sites in CNQX/APV, significant spontaneous activity remained (Fig. 1, B-D), suggesting that at least some layer V pyramidal cells discharge spontaneously through intrinsic membrane mechanisms (e.g. see Wang and McCormick 1992). We used these and other (Sanchez-Vives and McCormick 2000) features of the slow oscillation as guide lines in the generation of our model of this activity.

View larger version (44K): [in this window] [in a new window]   Fig. 1. Block of excitatory synaptic transmission eliminates the slow oscillation and reveals spontaneous action potential activity in a subset of layer V cells. A: extracellular multiple unit recording from layer V in prefrontal cortex during the bath application of 6-cyano-7-nitroquinoxalene-2,3-dione (CNQX, 20 µM) and DL-2-amino-5-phosphonovaleric acid (dl-APV, 50 µM). The slow oscillation (see expansion in B) is completely blocked and spontaneous discharge is greatly reduced. However, occasional action potential discharges still occur. B: expansion of recording in A prior (left) and following (right) block of excitatory synaptic transmission. C: recording at another site in layer V before and after block of excitatory transmission. D: another example of the slow oscillation recorded in control solution, and spontaneous activity recorded in a nearby site following the block of synaptic transmission.

Slow oscillation and wave propagation of network activity

In accordance with the experimental observations of Sanchez-Vives and McCormick (2000), neurons in our model show spontaneous activity as repetitive episodes of low-rate neuronal firing, separated by long-lasting silences of ~2.5 s (Fig. 2). The oscillation frequency is thus ~0.4 Hz. Activity patterns are organized spatially as synchronous waves that propagate from one site to neighboring sites, thus recruiting the whole network at each firing episode of the slow oscillation (Fig. 2B). Typically, the most active part of the network initiates the discharge at each cycle (in Fig. 2B, 3rd row from bottom), but the initiation site is not unique and varies from simulation trial to trial.

View larger version (48K): [in this window] [in a new window]   Fig. 2. The model reproduces spontaneous slow oscillations observed in the in vitro slice preparation. A: schematic representation of the spatial connectivity in the network model. A particular realization of the probabilistic connectivity for a pyramidal neuron and an interneuron is illustrated. - - -, probability distribution of synaptic connections from one neuron at the center to the rest of the network. Neurons are not drawn to scale with probability distributions. B: the spontaneous network activity can be visualized as multiunit recordings (5 neighboring cells per site, sites are spatially separated by 500 µm) to compare with the experiments of Sanchez-Vives and McCormick (2000). C: intracellular somatic voltage Vm and intracellular sodium concentration [Na+]i of 3 representative pyramidal neurons. Calibration bars for central traces apply to all corresponding traces in C and D. Time scale as in B. D: blow-up of an individual depolarized episode of the cells in C. In C and D, s point at 75 mV for voltage traces and at 10 mM for internal sodium concentration.

The membrane potential of modeled pyramidal neurons, as also seen in experiments, undergoes transitions between more depolarized states with higher spiking activity (up states) and more hyperpolarized states with virtually no spike discharges (down states; Fig. 2C). Slow oscillation of the membrane potential occurs in parallel with waxing and waning of the intracellular sodium concentration: [Na⁺]_i accumulates slowly due to spike-triggered sodium influx during the up state and decays by an extrusion process to the extracellular medium during the down state (Fig. 2, C and D). We will show that the slow [Na⁺]_i dynamics is critical to the generation of slow oscillations (see following text).

Because neurons are not identical in the heterogeneous network model, the membrane potential of pyramidal cells shows quantitatively different firing patterns. Some (highly excitable) cells show spike firing prior to the onset of an up state, and a relatively small after-hyperpolarization in the down state (Fig. 2, C and D, top). Other less excitable cells do not show spiking during the down state and have a pronounced and slowly recovering afterhyperpolarization following the firing discharge in the up state (Fig. 2, C and D, middle). And finally, a subpopulation of (the least excitable) pyramidal neurons exhibit clearly defined voltage up states and down states separated by ~10 mV (Fig. 2, C and D, bottom). These differences arise from the random initialization of intrinsic properties for each cell, and, therefore, other neurons show intermediate behaviors between these three characteristic examples.

Propagating discharges can also be evoked by local external stimulation, for example by a brief depolarizing current injection to a subpopulation of 50 neurons (indicated by arrows in Fig. 3). Colliding waves usually merge and extinguish (3rd and 4th waves in Fig. 3A). The wave nature can be more clearly revealed when spontaneous activity is absent if all pyramidal cells are slightly hyperpolarized so that they are unable to trigger any network event by themselves. In this case, briefly stimulating by current injection a restricted area of the network triggers a discharge episode that travels across the network as a wave front (Fig. 3B, left) and merges with waves traveling in the opposite direction (Fig. 3B, right). Furthermore, stimulation immediately after an evoked discharge is unable to recruit the network for a renewed propagation (Fig. 3B, left, 2nd stimulation arrow). Experiments on the slice have yielded very similar results: external stimulation during the hyperpolarized phase of the oscillation generated a wave that propagated across the slice (Sanchez-Vives and McCormick 2000), and the slice was refractory immediately following one of these network events.

View larger version (49K): [in this window] [in a new window]   Fig. 3. Excitatory and inhibitory neurons fire in phase during the slow oscillation with interneurons typically firing before pyramidal neurons at the onset of the up state. A: rastergram of network activity during 20 s of simulation. Time is on the horizontal axis and the network is represented along the y axis. Each red dot corresponds to 1 spike at time x in the pyramidal cell located at position y. Blue dots are spikes fired by the inhibitory population and may be overlapped by red dots. Note spontaneously occurring discharges and evoked episodes (triggered by brief external stimulation indicated by tilted arrows). B, left: rastergram for an evoked discharge (brief external stimulation indicated by arrow) when pyramidal neurons are slightly hyperpolarized to prevent spontaneous oscillations. Where spikes from excitatory and inhibitory cells coincide in space and time, red and blue dots overlap and are illustrated as green. Note how the same external stimulation applied right after an evoked propagating wave fails to elicit a new discharge because of the refractoriness of the network. Right: example of a collision of 2 evoked waves. Note the faint blue rim (interneuron spikes) surrounding the wave in both panels, indicating that interneurons typically fire first and for a longer time during the up state. C: histogram of the intervals between the 1st spike of each pyramidal and its immediately adjacent interneuron in the time window shown in B (directly stimulated neurons are not shown). The histogram is biased toward positive time lags, indicating that on average interneuron firing leads pyramidal cell firing (by ~50 ms) at the onset of the slow oscillation discharge episode. D: firing rate averaged across neurons for the time window shown in B. Within the considered window, we substract from the spike times for each excitatory (inhibitory) neuron the time of the 1st spike of the closest lying inhibitory (excitatory) neuron, and then we construct the time histogram of those intervals in the red (blue) curve. Notice that interneurons firing leads pyramidal cell firing by ~50 ms, as also shown in C.

Model pyramidal neurons and interneurons fire in phase during the slow oscillation (Fig. 3, A and B). When pyramidal cells and interneurons at a given spatial location are closely examined, we found that interneurons typically discharge first in response to the arrival of the wave front (blue rim around activity in B, rightward shift in peak of histogram in C, and shift in average activity of pyramidal cells and interneurons in D). Pyramidal neurons have an average maximal rate of ~10 Hz during the network discharge while interneurons fire at ~20 Hz (Fig. 3D).

Change of input resistance during the slow oscillation

In Fig. 5 are plotted the time courses of synaptic and intrinsic ionic conductances in a pyramidal neuron during the slow oscillation. In particular, the input resistance of pyramidal neurons (here computed as the inverse of all open conductances across the membrane, see Fig. 4) is the highest during the down state, reaching its maximal value right before the onset of the discharge episode. The input resistance is at its minimum during the initial phase of the up state and increases gradually in the course of the up state (Fig. 5B). This is in agreement with experimental recordings during the occurrence of the slow oscillation in cortical cells of the anesthetized cat in vivo (Contreras et al. 1996). On the basis of this observation, Contreras et al. (1996) suggested that a K⁺ conductance contributing to spike frequency adaptation cannot be responsible for the termination of the up state because this would lead to a gradual reduction, not increase, of the input resistance as the spike discharge progressed. This argument assumes that the K⁺ conductance dominates the input resistance. In our model, the mechanism terminating the up state is the activation of a Na⁺-activated K⁺ conductance g_KNa (see following text). However, the input resistance is determined by the sum of all conductances and is dominated by the synaptic conductances rather than by g_KNa. During an up state, the slow activation of g_KNa produces spike-frequency adaptation; reduced neural firing leads to a decrease of recurrent synaptic conductances and other intrinsic ion conductances, hence an overall increase of the input resistance (Fig. 5, C and D).

View larger version (20K): [in this window] [in a new window]   Fig. 4. Comparison between the input resistance calculated as the inverse of open membrane conductances in the model (gray plot in bottom) and that by the usual method employed in intracellular recordings through brief hyperpolarizing pulses (voltage trace in top, current pulses 0.3 nA, 100 ms). , the input resistances calculated as the ratio between the voltage deflection caused by each current pulse and the magnitude of the current (0.3 nA). Notice the quantitative agreement between the 2 estimates. The neuron was kept hyperpolarized (0.25 nA) in the course of the network simulation to prevent spikes riding on the up state, which confounds Rin calculations with the pulse method.

View larger version (30K): [in this window] [in a new window]   Fig. 5. Membrane input resistance and various ionic conductance contributions in the course of the slow oscillation. A: membrane voltage trace of 1 neuron while the network undergoes slow oscillations. B: total input resistance. Maximal resistance occurs right before the discharge onset and the resistance is minimal at the beginning of the up state. There is a gradual increase of the input resistance during the up state. C: excitatory and inhibitory synaptic conductances decrease as the discharge progresses. D: intrinsic conductances during the slow oscillation. gAR is closed, while gKS and gNaP are activated, during the up state. Contributions from the other intrinsic channels, mostly from the passive leakage, are not shown but are included for input resistance calculation. Activity-dependent K+ conductances (gKCa and gKNa) are weaker in magnitude (notice the scale on the y axis). For the sake of clarity, the data in this graph correspond to simulations with reduced gKNa (gKNa = 0.33 mS/cm2), that have much longer up states. We have confirmed by performing averages over many neurons that the trends illustrated here hold true for the reference parameter set, such as that used in Figs. 2 and 3.

Balance between synaptic excitation and inhibition

It is apparent in Fig. 5C that the excitatory and inhibitory synaptic conductances onto pyramidal neurons closely follow each other during the up state. This is shown more clearly in Fig. 6A, where the two synaptic conductances are not plotted against time, but against each other for seven different neurons (middle), or averaged across 32 neurons equally spaced in the network (right). This plot yields a linear relationship, which means that the excitatory and inhibitory synaptic conductances increase and decrease in time in such a way that the ratio of the two g_exc:g_inh remains fixed. This ratio varies from neuron to neuron, ranging from 1:1 to 1:5. On average, inhibitory synaptic conductances are typically four times larger than excitatory synaptic conductances, g_exc:g_inh  4. Assuming that the average potential in the up state is about V = 55 mV, the driving force for the excitatory current (with a reversal potential of V_exc = 0 mV) is (V_exc  V)  55 mV, whereas that of the inhibitory current (with a reversal potential V_inh = 70 mV) is (V  V_inh)  15 mV. Hence, the driving force for excitation is approximately four times larger than for inhibition. Therefore, an excitatory conductance four times smaller than the inhibitory conductance will yield comparable excitatory and inhibitory postsynaptic currents. In other words, there is an approximate balance between synaptic excitation and inhibition that is preserved over time throughout the up state, when pyramidal neural firing is sustained at relatively low rates (10 Hz).

View larger version (54K): [in this window] [in a new window]   Fig. 6. Excitatory and inhibitory synaptic conductances preserve an approximate proportional relation to each other in the course of the slow oscillation. A: control network as in Fig. 2. B: same network with 10% enhanced inhibition to pyramidal neurons. C: network with 10% reduced excitation to interneurons. Left: network activity shown in a multiunit array record, distance between adjacent recording channels 0.5 mm. Middle: excitatory synaptic conductance plotted vs. inhibitory synaptic conductance for 7 different cells. Note the linear relationship between the 2 conductances, with the ratio (the slope of the linear plot) varying from neuron to neuron. Right: the excitatory and inhibitory conductances averaged across 32 neurons equally spaced in the network (calculated at each time step) and plotted against each other. The approximately linear relationship indicates that the balance between synaptic excitation and inhibition is maintained during the up state. Strictly, conductances do not keep an exact linear relationship but tend to depart systematically from it tracing an elongated closed cycle. This is especially evident in C, where arrows indicate the sense in which the cycle is traced in the course of the up state.

To establish whether this excitation-inhibition balance is the result of a particularly well-tuned parameter choice or is an emergent property of the network, we drew the same graphs for network simulations where one synaptic conductance had been modified (Fig. 6, B and C). We observe that in all cases, the approximately linear relationship between excitatory and inhibitory conductances remains, but the proportionality ratio varies. It also becomes clear that the relationship is not exactly linear but follows a closed cycle, with excitatory conductances above (below) the average linear relationship during the down-up (up-down) transition (see Fig. 6C, right).

The exact ratio of excitatory to inhibitory conductances is likely to depend on other parameters like V_L. Indeed, in a much simplified model including only leak and synaptic currents, the membrane equation is C_mdV/dt = g_L(V  V_L)  g_E(V  V_E)  g_I(V  V_I). Assume that V_E = 0, V_L = V_I = 75 mV, and g_L = 20 nS. The steady state of the voltage is given by V_ss = (g_L + g_I)V_L/(g_L + g_I + g_E). Then, the membrane potential could stabilize around its threshold even with an excitation-inhibition ratio of g_E:g_I = 1(V_ss ~ 56 mV, if g_E = g_I = 10 nS). However, the issue of whether such a model could maintain its stability throughout a recurrently generated up state does not seem trivial because synaptic conductance increases proportional to the network activity could lead to reverberatory instability.

Mechanisms of the slow oscillation in the model

The basic mechanism for the emergence of the oscillatory activity in the network model is the interplay between neuronal spontaneous firing amplified by recurrent excitation, and a negative feedback due to slow activity-dependent K⁺ currents. These positive and negative feedback processes operate along the lines illustrated in Fig 7: because the reversal potential of the leakage current is distributed randomly (see METHODS), some neurons are spontaneously active at very low rates (0.6 ± 0.2 Hz). Occasionally, a sufficient number of closely adjacent spontaneously active neurons fire together, thus triggering a cascade of recurrent excitation that locally brings the network into the firing regime of the oscillation (up state). At that point, activity-dependent K⁺ currents (notably the Na⁺-dependent K⁺ current) start accumulating in active pyramidal neurons, reducing their excitability. The decremental excitability of pyramidal neurons eventually makes the network recurrence unable to sustain the firing state, and the local network reverts to the silent state via a slow afterhyperpolarization. The decay time of the currents responsible for this afterhyperpolarization sets the time scale for the reappearance of spontaneous firing and determines the periodicity of the oscillatory cycle.

View larger version (18K): [in this window] [in a new window]   Fig. 7. Mechanism of the slow oscillation: some neurons have slightly lower spiking threshold and fire spontaneously (bottom left). This spontaneous firing will occasionally trigger the recruitment of all the cells in a subregion of the network through recurrent excitation and bring those cells up into the firing state (top left). While neurons fire, their activity-dependent K+ currents (especially IKNa) accumulate slowly. A point is reached in which the neurons are not excitable enough to maintain this self-sustained spiking state and they revert to the silent mode (middle right). Only after the Na+-dependent K+ current decays can the spontaneous firing resume and eventually trigger a new discharge episode (bottom left).

According to this scheme, recurrent excitation should be responsible for quickly bringing neurons into the firing state in a collective manner, and the activity-dependent slow I_KNa eventually leads to a transition back to the down state. More specifically, we hypothesized that recurrent synaptic excitation produces a network bistability with an active up state and an inactive down state and that the slow kinetics of I_KNa drives the network to switch back and forth between these two states (see Fig. 8). We tested this prediction by substituting the time-varying I_KNa current with a constant hyperpolarizing current of varying intensities simultaneously in all model neurons. We found that the network indeed exhibits two stable dynamic states: the silent state with large hyperpolarization and the persistent firing state with low hyperpolarization. There is a range of current intensity over which these two states coexist (bistability; Fig. 8, A and B). In the control network simulation, the role of the injected hyperpolarizing current of Fig. 8 is fulfilled by the time-varying I_KNa, whose slow dynamics causes the neurons to cyclically trace this hysteresis loop (perimeter of shaded area in Fig. 8B): as I_KNa progressively builds up in the firing state (upper solid line in Fig. 8B), pyramidal neurons experience an increasing hyperpolarizing current. Eventually, point A in Fig. 8B is reached and neurons in the network collectively and sharply fall into the silent state because recurrent excitation is no longer sufficient to sustain their firing. Then neurons remain silent while I_KNa recovers slowly and the network regains the level of excitability (point B in Fig. 8B) where the silent state becomes unstable again and a collective sharp transition to the up state is generated. Thus the cycle goes on indefinitely, driven by the slow kinetics of I_KNa. Transitions between up and down states are sharp because of the sudden loss of network stability at a given degree of neuronal excitability.

View larger version (24K): [in this window] [in a new window]   Fig. 8. Recurrent excitation produces network bistability for a range of values of the slowly decaying current IKNa. The Na+-dependent K+ channel was blocked in all pyramidal cells, and a constant hyperpolarizing current Iext was injected as a replacement in all of them. Simulations were repeated for a range of intensities of Iext with the stimulation protocol illustrated in A (bottom): 2.5 s after the initiation of a simulation trial, a brief depolarizing pulse is injected into all pyramidal neurons, the network settles into a stable state for another 2.5 s, and then a brief hyperpolarizing pulse is injected. The average network firing rate is calculated before the depolarizing and hyperpolarizing pulses (triangles). The rastergram shows the network activity, where spikes are represented by dots at the firing time (along the x axis) and the position of the neuron on the network (along y axis). Below is the instantaneous population firing rate and a sample "intracellular" trace of one of the cells in the network. B: for a range of constant injected current (shaded area), the network can stably remain in the silent state or in a persistent firing state sustained by excitatory reverberation. Triangles correspond to the network firing rates in the example simulation of A, at the time points indicated by the same triangles. In the full model, the time-varying IKNa plays the role of hyperpolarization current. When the neurons are in the sustained firing state, [Na+]i slowly accumulates and the activation of IKNa makes neurons move leftward along the upper branch of the graph (left-pointing arrow), up to the left end-point (`left-knee', point A) when the network abruptly drops down to the silent state. When neurons cease firing, [Na+]i decays back to the baseline and IKNa decreases slowly, so that neurons move rightward along the lower branch of the graph (right-pointing arrow), until they reach the right end-point (`right-knee', point B), when the network jumps back to the persistently firing state again. C: reducing excitatory feedback (E-to-E conductances) by 20% abolishes the network hysteresis, demonstrating that bistability requires strongly recurrent synaptic excitation.

To further test the importance of feedback excitation to the bistability phenomenon, we reduced the maximum conductance of the recurrent excitatory synapses to pyramidal neurons by 20%. The result is a loss of the bistable range (Fig. 8C). Now the network is either silent or, upon progressive depolarization of pyramidal neurons, it becomes spontaneously active. There is, however, no range of input currents for which both a quiescent state and a persistent firing state are simultaneously stable.

The mechanism for adaptation-induced network oscillations has been mathematically analyzed by van Vreeswijk and Hansel (2001) (see also Fuhrmann et al. 2002; Wang 1999b).

Pharmacological manipulations

To compare with the experimental results of Sanchez-Vives and McCormick (2000), we explored the effect of various synaptic receptor blockers on our slowly oscillating cortical network (Fig. 9). Blocking AMPAR-mediated transmission abolishes this rhythmic activity completely, and a subpopulation of excitable neurons fire spontaneously without any apparent network amplification. Similarly, the slow oscillation disappears under NMDARs blockade but this disruption is not as dramatic as when AMPARs are blocked, and occasional bursts of activity are able to recruit neighboring neurons into a network event via AMPAR-mediated transmission. On the other hand, blocking GABA_ARs results in a more exuberant discharge that propagates at a much faster velocity across the cortical network. Also, the time intervals between burst episodes are significantly prolonged, i.e. the oscillation period is longer. All these behaviors are very similar to the results of Sanchez-Vives and McCormick (2000) after bath application of various synaptic receptor blockers in the slice preparation (compare the Fig. 9, left and right).

View larger version (63K): [in this window] [in a new window]   Fig. 9. Simulations with synaptic transmission blockade reproduce the experimental results. Left: simulation results plotted as multiunit records with blockade of the various synaptic receptors. Right: experimental results when CNQX, APV, or picrotoxin were applied in the bath solution of a slowly oscillating cortical slice. The experimental results shown as control correspond to the same slice as the results for N-methyl-D-aspartate (NMDA) blockade. Control traces are very similar (not shown) for the AMPA and GABAA blockade.

Velocity of wave propagation

Figure 9 shows that the network can sustain two different types of propagating waves depending on whether intracortical inhibition is functional (control case) or the network is disinhibited (GABA_AR block). Notably, these two waves have very different propagation speeds, as in slice experiments (Golomb and Amitai 1997; Sanchez-Vives and McCormick 2000; Wu et al. 2001). Both in the model and in the experiment, the propagation of the disinhibited wave is almost an order of magnitude faster than that of the wave in control conditions. The velocity of the disinhibited wave is largely determined by the efficacy of AMPAR-mediated excitatory synaptic transmission (Fig. 10A), consistent with previous analytical and simulation results (Ermentrout 1998a; Golomb and Amitai 1997). The dependence of the propagation speed on the strength of feedback inhibition is plotted in Fig. 10B. Either an increase of the inhibitory conductance onto excitatory cells or of the excitatory conductances onto inhibitory cells gradually decreases the wave propagation speed, according to a smooth sigmoid function (fitted curve).

View larger version (34K): [in this window] [in a new window]   Fig. 10. Control of the propagation speed by recurrent excitation and by feedback inhibition. A: with inhibition blockade, the wave velocity increases linearly with the strength of fast recurrent excitation (AMPA E-to-E conductances on x axis, relative to the reference value). B: parametrical dependence of the wave velocity on the strength of feedback inhibition. Left: inhibitory conductance onto pyramidal neurons on x axis, relative to the reference value. Right: excitatory conductances (AMPA and NMDA) onto inhibitory neurons on x axis, relative to the reference value. C: setting AMPA-mediated E-to-E conductances 4-fold stronger yields correct propagation velocity in disinhibited networks (see A), but inhibition fails to control the wave to generate smooth slower propagation. Above a critical level (shaded area), the wave phenomenon is abolished and the network activity becomes spatially uniform. Below this critical value, the wave dynamics is complex: the network activity propagates at 2 distinct velocities (, faster velocity; , slower velocity). D: example to illustrate the coexistence of 2 velocities of propagation in the network dynamics. The 2 distinct propagation velocities are indicated (- - -).

Although the model reproduces roughly a 10-fold increase in the wave speed with inhibition blockade, a closer examination revealed that the absolute wave speeds in the model are significantly off the experimentally measured values. In control conditions, the velocity is <3 mm/s, whereas in the experiments it was ~10 mm/s; with inhibition blockade, the wave propagation is <20 mm/s, when the experiment yielded a value ~80 mm/s. We have explored possible solutions to this discrepancy. Figure 10A shows that with sufficiently strong E-to-E coupling, a wave velocity of ~100 mm/s could be achieved in the disinhibited network. This suggests that one could first increase the strength of recurrent AMPA-mediated excitatory synapses to achieve the desired velocity in the disinhibited network; then, by gradually increasing the inhibitory feedback (as in Fig. 10B), slower propagation at a desired velocity could be obtained. However, this is not the case (Fig. 10C): with much stronger excitatory feedback (4-fold E-to-E AMPA conductances), increasing inhibition in a network does not lead to the slower propagation mode observed in the experiment. Instead, when the feedback inhibition is above a threshold value, the wave phenomenon disappears and the network shows a spatially uniform (unstructured) tonic firing state (shaded areas in Fig. 10C). Below this threshold but near it (inhibitory synapses are still very strong) a complex activity pattern emerges (see illustrative rastergram in Fig. 10D), in which activity propagates at two different velocities. The rastergram in a small time window reveals that activity spreads quickly, as in the disinhibited case ("fast wave" indicated in Fig. 10D). Examination at a longer timescale reveals also a slower propagation, which is nevertheless not a smooth wavefront ("slow propagation mode" indicated in Fig. 10D). The global network pattern is clearly not comparable to the one observed in the slice experiment of Sanchez-Vives and McCormick (2000).

Another possibility is to increase the spatial extent (the "footprint" _EE) of excitatory connectivity because the wave velocity increases proportionally with _EE (Ermentrout 1998b). However, to achieve the experimentally comparable wave velocities, one would have to increase _EE of the E-to-E coupling by three- to fourfold, from 250 µm (see METHODS) to ~2 mm, which would be inconsistent with the anatomical and physiological data. We examined the more plausible scenario that weak but long-range patchy horizontal connections could increase the propagation speeds significantly. This pattern of connection is prominent in the mammalian cortex (Gilbert and Wiesel 1983; Rockland 1985), and it has been observed in ferret visual cortex in both supragranular and infragranular layers (Rockland 1985). Typically, horseradish peroxidase injection in a restricted area of cortex results in orthograde striped staining of intracortical connections with stripe width of 250 µm and center-to-center distance 0.5-0.7 mm (Rockland 1985). We model this kind of connectivity as depicted in Fig. 11A (right). When we use this type of connectivity in our model, discharges propagate much faster in both conditions (Fig. 11A, right compare with left for non-patchy connectivity). For the wave in the disinhibited network, the velocity increased to >50 mm/s; whereas for the network with functional inhibition, the propagating wave has a speed close to 10 mm/s. On the other hand, the inclusion of long-range excitatory connections does not lead to a significant increase of the pyramidal neural activity, the firing rates are comparable to the situation without patchy excitatory connections. Both velocities depend markedly on the spatial extent and strength of the patchy horizontal connections (Fig. 11,C and D). Figure 11C shows how the center-to-center distance for the patchy horizontal connections influences the propagation speeds. The more separated the patc