The low-threshold spike (LTS), generated by the transient Ca(2+) current I(T), plays a pivotal role in thalamic relay cell responsiveness and thus in the nature of the thalamic relay. By injecting depolarizing current ramps at various rates to manipulate the slope of membrane depolarization (dV/dt), we found that an LTS occurred only if dV/dt exceeded a minimum value of approximately 5-12 mV/sec. We injected current ramps of variable dV/dt into relay cells that were sufficiently hyperpolarized to de-inactivate I(T) completely. Higher values of dV/dt activated an LTS. However, lower values of dV/dt eventually led to tonic firing without ever activating an LTS; apparently, the inactivation of I(T) proceeded before I(T) could be recruited. Because the maximum rate of rise of the LTS decreased with slower activating ramps of injected current, we conclude that slower ramps allow increasing inactivation of I(T) before the threshold for its activation gating is reached, and when the injected ramps have a sufficiently low dV/dt, the inactivation is severe enough to prevent activation of an LTS. In the presence of Cs(+), we found that even the lowest dV/dt that we applied led to LTS activation, apparently because Cs(+) reduced the K(+) "leak" conductance and increased neuronal input resistance. Nonetheless, under normal conditions, our data suggest that there is neither significant window current (related to the overlap of the inactivation and activation curves for I(T)), rhythmogenic properties, nor bistability properties for these neurons. Our theoretical results using a minimal model of LTS excitability in these neurons are consistent with the experimental observations and support our conclusions. We suggest that inputs activating very slow EPSPs (i.e., via metabotropic receptors) may be able to inactivate I(T) without generating sizable I(T) and a spurious burst of action potentials to cortex.

Spontaneous episodic activity occurs throughout the developing nervous system because immature circuits are hyperexcitable. It is not fully understood how the temporal pattern of this activity is regulated. Here, we study the role of activity-dependent depression of network excitability in the generation and regulation of spontaneous activity in the embryonic chick spinal cord. We demonstrate that the duration of an episode of activity depends on the network excitability at the beginning of the episode. We found a positive correlation between episode duration and the preceding inter-episode interval, but not with the following interval, suggesting that episode onset is stochastic whereas episode termination occurs deterministically, when network excitability falls to a fixed level. This is true over a wide range of developmental stages and under blockade of glutamatergic or GABAergic/glycinergic synapses.We also demonstrate that during glutamatergic blockade the remaining part of the network becomes more excitable, compensating for the loss of glutamatergic synapses and allowing spontaneous activity to recover. This compensatory increase in the excitability of the remaining network reflects the progressive increase in synaptic efficacy that occurs in the absence of activity. Therefore, the mechanism responsible for the episodic nature of the activity automatically renders this activity robust to network disruptions. The results are presented using the framework of our computational model of spontaneous activity in the developing cord. Specifically, we show how they follow logically from a bistable network with a slow activity-dependent depression switching periodically between the active and inactive states.

Spontaneous episodic activity is a general feature of developing neural networks. In the chick spinal cord, the activity comprises episodes of rhythmic discharge (duration 5-90 sec; cycle rate 0.1-2 Hz) that recur every 2-30 min. The activity does not depend on specialized connectivity or intrinsic bursting neurons and is generated by a network of functionally excitatory connections. Here, we develop an idealized, qualitative model of a homogeneous, excitatory recurrent network that could account for the multiple time-scale spontaneous activity in the embryonic chick spinal cord. We show that cycling can arise from the interplay between excitatory connectivity and fast synaptic depression. The slow episodic behavior is attributable to a slow activity-dependent network depression that is modeled either as a modulation of cellular excitability or as synaptic depression. Although the two descriptions share many features, the model with a slow synaptic depression accounts better for the experimental observations during blockade of excitatory synapses.

Dynamic activation of ion channels formed by colicin Ia incorporated into a planar bilayer lipid membrane (BLM) was investigated by the voltage clamp technique using different step voltage stimuli. We have demonstrated a critical resting interval, Δtc, between two identical successive voltage pulses. If the second pulse is applied within Δtc, it produces a predictable current response. On the contrary, if the second pulse is applied after Δtc, the current response cannot be reliably predicted. Computer simulations based on an idealized mathematical model, developed in this paper, qualitatively reproduce the system's dynamic responses to stimulus trains. The behavior of the ion channels, when the resting period exceeds Δtc, may be interpreted as a transient gain or loss or resetting of memory, as revealed by a specific sequence of electrical pulses used for stimulation.

Current clamp and modeling studies of low-threshold calcium spikes in cells of the cat's lateral geniculate nucleus. All thalamic relay cells display a voltage-dependent low-threshold Ca2+ spike that plays an important role in relay of information to cortex. We investigated activation properties of this spike in relay cells of the cat's lateral geniculate nucleus using the combined approach of current-clamp intracellular recording from thalamic slices and simulations with a reduced model based on voltage-clamp data. Our experimental data from 42 relay cells showed that the actual Ca2+ spike activates in a nearly all-or-none manner and in this regard is similar to the conventional Na+/K+ action potential except that its voltage dependency is more hyperpolarized and its kinetics are slower. When the cell's membrane potential was hyperpolarized sufficiently to deinactivate much of the low-threshold Ca2+ current (IT) underlying the Ca2+ spike, depolarizing current injections typically produced a purely ohmic response when subthreshold and a full-blown Ca2+ spike of nearly invariant amplitude when suprathreshold. The transition between the ohmic response and activated Ca2+ spikes was abrupt and reflected a difference in depolarizing inputs of <1 mV. However, activation of a full-blown Ca2+ spike was preceded by a slower period of depolarization that was graded with the amplitude of current injection, and the full-blown Ca2+ spike activated when this slower depolarization reached a sufficient membrane potential, a quasithreshold. As a result, the latency of the evoked Ca2+ spike became less with stronger activating inputs because a stronger input produced a stronger depolarization that reached the critical membrane potential earlier. Although Ca2+ spikes were activated in a nearly all-or-none manner from a given holding potential, their actual amplitudes were related to these holding potentials, which, in turn, determined the level of IT deinactivation. Our simulations could reproduce all of the main experimental observations. They further suggest that the voltage-dependent K+ conductance underlying IA, which is known to delay firing in many cells, does not seem to contribute to the variable latency seen in activation of Ca2+ spikes. Instead the simulations indicate that the activation of IT starts initially with a slow and graded depolarization until enough of the underling transient (or T) Ca2+ channels are recruited to produce a fast, "autocatalytic" depolarization seen as the Ca2+ spike. This can produce variable latency dependent on the strength of the initial activation of T channels. The nearly all-or-none nature of Ca2+ spike activation suggests that when a burst of action potentials normally is evoked as a result of a Ca2+ spike and transmitted to cortex, this signal is largely invariant with the amplitude of the input activating the relay cell.

The isolated lumbosacral cord of the chick embryo generates spontaneous episodes of rhythmic activity. Muscle nerve recordings show that the discharge of sartorius (flexor) and femorotibialis (extensor) motoneurons alternates even though the motoneurons are depolarized simultaneously during each cycle. The alternation occurs because sartorius motoneuron firing is shunted or voltage-clamped by its synaptic drive at the time of peak femorotibialis discharge. Ablation experiments have identified a region dorsomedial to the lateral motor column that may be required for the alternation of sartorius and femorotibialis motoneurons. This region overlaps the location of interneurons activated by ventral root stimulation. Wholecell recordings from interneurons receiving short latency ventral root input indicate that they fire at an appropriate time to contribute to the cyclical pause in firing of sartorius motoneurons. Spontaneous activity was modeled by the interaction of three variables: network activity and two activity-dependent forms of network depression. A "slow" depression which regulates the occurrence of episodes and a "fast" depression that controls cycling during an episode. The model successfully predicts several aspects of spinal network behavior including spontaneous rhythmic activity and the recovery of network activity following blockade of excitatory synaptic transmission.

In contrast to the limited response properties observed under normal experimental conditions, spinal motoneurons generate complex firing patterns, such as Ca2+-dependent regenerative spiking and plateaus, in the presence of certain neurotransmitters and ion-channel blockers. We have developed a quantitative motoneuron model, based on turtle motoneuron data, toinvestigate the roles of specific ionic currents and the effects of their soma and dendritic distribution in generating these complex firing patterns. In addition, the model is used to explore the effects of multiple ion channel blockers and neurotransmitters that are known to modulate motoneuron firing patterns. To represent the distribution of ionic currents across the soma and dendrites, the model contains two compartments. The soma compartment, representing the soma and proximal dendrites, contains Hodgkin-Huxley-like sodium (INa) and delayed rectifier K+ (IK-dr) currents, an N-like Ca2+ current (ICa-N), and a calcium-dependent K+ current [IK(Ca)]. The dendritic compartment, representing the lumped distal dendrites, contains, in addition to ICa-N and IK(Ca) as in the soma, a persistent L-like calcium current (ICa-L). We determined kinetic parameters for INa, IK-dr, ICa-N, and IK(Ca) in order to reproduce normal action-potential firing observed in turtle spinal motoneurons, including fast and slow afterhyperpolarizations (AHPs) and a linear steady-state frequency-current relation. With this parameter set as default, a sequence of pharmacological manipulations were systematically simulated. A small reduction of IK-dr [mimicking the experimental effect of tetraethylammonium (TEA) in low concentration] enhanced the slow AHP and caused calcium spiking (mediated by ICa-N) when INa was blocked. Firing patterns observed experimentally in high TEA [and tetrodotoxin (TTX)], namely calcium spikes riding on a calcium plateau, were reproduced only when both IK-dr and IK(Ca) were reduced. Dendritic plateau potentials, mediated by ICa-L, were reliably unmasked when IK(Ca) was reduced, mimicking the experimental effect of the bee venom apamin. The effect of 5-HT, which experimentally induces the ability to generate calcium-dependent plateau potentials but not calcium spiking, was reproduced in the model by reducing IK(Ca) alone. The plateau threshold current level, however, was reduced substantially if a simultaneous increase in ICa-L was simulated, suggesting that serotonin (5-HT) induces plateau potentials by regulating more than one conductance. The onset of the plateau potential showed significant delays in response to near-threshold, depolarizing current steps. In addition, the delay times were sensitive to the current step amplitude. The delay and its sensitivity were explained by examining the model's behavior near the threshold for plateau onset. This modeling study thus accurately accounts for the basic firing behavior of vertebrate motoneurons as well as a range of complex firing patterns invoked by ion-channel blockers and 5-HT. In addition, our computational results support the hypothesis that the electroresponsiveness of motoneurons depends on a nonuniform distribution of ionic conductances, and they predict modulatory effects of 5-HT and properties of plateau activation that have yet to be tested experimentally.

The mechanism underlying subthreshold oscillations in inferior olivary cells is not known. To study this question, we developed a single-compartment, two-variable, Hodgkin-Huxley-like model for inferior olive neurons. The model consists of a leakage current and a low-threshold calcium current, whose kinetics were experimentally measured in slices. Depending on the maximal calcium and leak conductances, we found that a neuron model's response to current injection could be of four qualitatively different types: always stable, spontaneously oscillating, oscillating with injection of current, and bistable with injection of current. By the use of phase plane techniques, numerical integration, and bifurcation analysis, we subdivided the two-parameter space of channel densities into four regions corresponding to these behavioral types. We further developed, with the use of such techniques, an empirical rule of thumb that characterizes whether two cells when coupled electrically can generate sustained, synchronized oscillations like those observed in inferior olivary cells in slices, of low amplitude (0.1-10 mV) in the frequency range 4-10 Hz. We found that it is not necessary for either cell to be a spontaneous oscillator to obtain a sustained oscillation. On the other hand, two spontaneous oscillators always form an oscillating network when electrically coupled with any arbitrary coupling conductance. In the case of an oscillating pair of electrically coupled nonidentical cells, the coupling current varies periodically and is nonzero even for very large coupling values. The coupling current acts as an equalizing current to reconcile the differences between the two cells' ionic currents. It transiently depolarizes one cell and/or hyperpolarizes the other cell to obtain the regenerative response(s) required for the synchronized oscillation. We suggest that the subthreshold oscillations observed in the inferior olive can emerge from the electrical coupling between neurons with different channel densities, even if the inferior olive nucleus contains no or just a small proportion of spontaneously oscillating neurons.

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ mobilization leads to depletion of the endoplasmic reticulum (ER) and an increase in Ca2+ entry. We show here for the gonadotroph, an excitable endocrine cell, that sensing of ER Ca2+ content can occur without the Ca2+ release-activated Ca2+ current (Icrac), but rather through the coupling of IP3-induced Ca2+ oscillations to plasma membrane voltage spikes that gate Ca2+ entry. Thus we demonstrate that capacitative Ca2+ entry is accomplished through Ca(2+)-controlled Ca2+ entry. We develop a comprehensive model, with parameter values constrained by available experimental data, to simulate the spatiotemporal behavior of agonist-induced Ca2+ signals in both the cytosol and ER lumen of gonadotrophs. The model combines two previously developed models, one for ER-mediated Ca2+ oscillations and another for plasma membrane potential-driven Ca2+ oscillations. Simulations show agreement with existing experimental records of store content, cytosolic Ca2+ concentration ([Ca2+]i), and electrical activity, and make a variety of new, experimentally testable predictions. In particular, computations with the model suggest that [Ca2+]i in the vicinity of the plasma membrane acts as a messenger for ER content via Ca(2+)-activated K+ channels and Ca2+ pumps in the plasma membrane. We conclude that, in excitable cells that do not express Icrac, [Ca2+]i profiles provide a sensitive mechanism for regulating net calcium flux through the plasma membrane during both store depletion and refilling.

The response of gonadotrophs to secretagogues involves dose-dependent, complex dynamic patterns of electrical activity and inositol 1,4,5-trisphosphate (InsP(3))-induced Ca(2+) mobilization, including pulsatility and oscillations on multiple time scales from milliseconds to minutes. Detailed in vitro experiments have enabled the identification of key mechanisms that underlie the plasma membrane (PM) electrical excitability and endoplasmic reticulum (ER) calcium excitability. We summarize these findings and review computer simulations of a biophysical model that resynthesizes and couples these components and that reproduces quantitatively the observed time courses and dose-response characteristics, as well as effects of various pharamacological manipulations. The theory suggests that cytosolic calcium is the primary messenger in coordinating the PM and ER regenerative behaviors during ER depletion and refilling.

1. We study the propagation and dynamics of spindle waves in thalamic slices by developing and analyzing a model of reciprocally coupled populations of excitatory thalamocortical (TC) neurons and inhibitory thalamic reticular (RE) neurons. 2. Each TC neuron has three intrinsic ionic currents: a low-threshold T-type Ca+2 current (ICa-T), a hyperpolarization-activated cation ("sag") current (Ih) and a leak current. Each RE cell also has three currents: ICa-T, a leak current, and a calcium-activated potassium current (IAHP). Isolated TC cells are at rest, can burst when released or depolarized from a hyperpolarized level, and burst rhythmically under moderate constant hyperpolarizing current. Isolated RE cells are at a hyperpolarized resting membrane potential and can burst when depolarized. 3. TC cells excite RE cells with fast alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) synapses, and RE cells inhibit TC cells with fast gamma-aminobutyric acid-A (GABAA) and slow GABAB synapses and inhibit each other with GABAA synapses only. GABAB postsynaptic conductances operate far from saturation, and the slow inhibitory postsynaptic potentials (IPSPs) increase with the width of the presynaptic burst. The model network is a one-dimensional cellular array with localized coupling. The synaptic coupling strength decays with the distance between the pre- and postsynaptic cells, either exponentially or as a step function. 4. The "intact" network can oscillate with partial synchrony and a population frequency of approximately 10 Hz. RE cells emit bursts almost at every oscillation cycle, whereas TC cells do so almost at every other cycle. Block of GABAB receptors hardly changes the network behavior. Block of GABAA receptors leads the network to a slowed oscillatory state, where the population frequency is approximately 4 Hz and both RE and TC cells fire unusually long bursts at every cycle and in full synchrony. These results are consistent with the experimental observations of von Krosigk, Bal, and McCormick. We obtain such consistency only when the above assumptions regarding the synaptic dynamics, particularly nonsaturating GABAB synapses, are fulfilled. 5. The slice model has a stable rest state with no neural activity. By initially depolarizing a few neurons at one end of the slice while all the other cells are at rest, a recruitment process may be initiated, and a wavefront of oscillatory activity propagates across the slice. Ahead of the wavefront, neurons are quiescent; neurons behind it oscillate. We find that the wave progresses forward in a lurching manner. TC cells that have just become inhibited must be hyperpolarized for a long enough time before they can fire rebound bursts and recruit RE cells. This step limits the wavefront velocity and may involve a substantial part of the cycle when no cells at the front are depolarized. 6. The wavefront velocity increases linearly with the characteristic spatial length of the connectivity (the footprint length). It increases only gradually with the synaptic strength, logarithmically in the case of an exponential connection function and only slightly for a step connection function. It also decreases gradually with a potassium leak conductance that hyperpolarizes RE cells. 7. To reproduce the experimentally measured wavefront velocity of approximately 1 mm/s, together with other in vitro observations, both the RE-to-TC and the TC-to-RE projections in the model should be spatially localized. The sum of the RE-to-TC and the TC-to-RE synaptic footprint lengths should be on the order of 100 microns. (ABSTRACT TRUNCATED AT 250 WORDS)

Burst firing of dopaminergic neurons of the substantia nigra pars compacta can be induced in vitro by the glutamate agonist N-methyl-D-aspartate. It has been suggested that the interburst hyperpolarization is due to Na+ extrusion by a ouabain-sensitive pump [Johnson et al. (1992) Science 258, 665-667]. We formulate and explore a theoretical model, with a minimal number of currents, for this novel mechanism of burst generation. This minimal model is further developed into a more elaborate model based on observations of additional currents and hypotheses about their spatial distribution in dopaminergic neurons [Hounsgaard (1992) Neuroscience 50, 513-518; Llinás et al. (1984) Brain Res. 294, 127-132]. Using the minimal model, we confirm that interaction between the regenerative, inward N-methyl-D-aspartate-mediated current and the outward Na(+)-pump current is sufficient to generate the slow oscillation (approximately 0.5 Hz) underlying the burst. The negative-slope region of the N-methyl-D-aspartate channel's current-voltage relation is indispensable for this slow rhythm generation. The time-scale of Na(+)-handling determines the burst's slow frequency. Moreover, we show that, given the constraints of sodium handling, such bursting is best explained mechanistically by using at least two spatial, cable-like compartments: a soma where action potentials are produced and a dendritic compartment where the slow rhythm is generated. Our result is consistent with recent experimental evidence that burst generation originates in distal dendrites [Seutin et al. (1994) Neuroscience 58, 201-206]. Responses of the model to a number of electrophysiological and pharmacological stimuli are consistent with known responses observed under similar conditions. These include the persistence of the slow rhythm when the tetrodotoxin-sensitive Na+ channel is blocked and when the soma is voltage-clamped at -60 mV. Using our more elaborate model, we account for details of the observed frequency adaptation in N-methyl-D-aspartate-induced bursting, the origin of multiple spiking and bursting mechanisms, and the interaction between two different bursting mechanisms. Besides reproducing several well established firing patterns, this model also suggests that new firing modes, not yet recorded, might also occur in dopaminergic neurons. This model provides mechanistic insights and explanations into the origin of a variety of experimentally observed membrane potential firing patterns in dopaminergic neurons, including N-methyl-D-aspartate-induced bursting and its dendritic origin. Such a model, capable of reproducing a number of realistic behaviors of dopaminergic neurons, could be useful in further studies of the basal ganglia-thalamocortical motor circuit. It may also shed light on bursting that involves N-methyl-D-aspartate channel activity in other neuron types.

The rhythmogenesis of 10-Hz sleep spindles is studied in a large-scale thalamic network model with two cell populations: the excitatory thalamocortical (TC) relay neurons and the inhibitory nucleus reticularis thalami (RE) neurons. Spindle-like bursting oscillations emerge naturally from reciprocal interactions between TC and RE neurons. We find that the network oscillations can be synchronized coherently, even though the RE-TC connections are random and sparse, and even though individual neurons fire rebound bursts intermittently in time. When the fast gamma-aminobutyrate type A synaptic inhibition is blocked, synchronous slow oscillations resembling absence seizures are observed. Near-maximal network synchrony is established with even modest convergence in the RE-to-TC projection (as few as 5-10 RE inputs per TC cell suffice). The hyperpolarization-activated cation current (Ih) is found to provide a cellular basis for the intermittency of rebound bursting that is commonly observed in TC neurons during spindles. Such synchronous oscillations with intermittency can be maintained only with a significant degree of convergence for the TC-to-RE projection.

Single pituitary cells often fire spontaneous action potentials (APs), which are believed to underlie spiking fluctuations in cytosolic calcium concentration ([Ca2+](i)). To address how these basal [Ca2+](i) fluctuations depend on changes in plasma membrane voltage (V), simultaneous measurements of V and [Ca2+](i) were performed in rat pituitary gonadotrophs. The data show that each [Ca2+](i) spike is produced by the Ca2+ entry during a single AP. Using these and previously obtained patch-clamp data, we develop a quantitative mathematical model of this plasma membrane oscillator and the accompanying spatiotemporal [Ca2+](i) oscillations. The model demonstrates that AP-induced [Ca2+](i) spiking is prominent only in a thin shell layer neighboring the cell surface. This localized [Ca2+](i) spike transiently activates the Ca2+-dependent K+ current resulting in a sharp afterhyperpolarization following each voltage spike. In accord with experimental observations, the model shows that the frequency and amplitude of the voltage spikes are highly sensitive to current injection and to the blocking of the Ca2+-sensitive current. Computations also predict that, leaving the membrane channels intact, the firing rate can be modified by changing the Ca2+ handling parameters: the Ca2+ diffusion rate, the Ca2+ buffering capacity, and the plasma membrane Ca2+ pump rate. Finally, the model suggests reasons that spontaneous APs were seen in some gonadotrophs but not in others. This model provides a basis for further exploring how plasma membrane electrical activity is involved in the control of cytosolic calcium level in unstimulated as well as agonist-stimulated gonadotrophs.

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol 25, 653-675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.