Propagating activity patterns in large-scale inhibitory neuronal networks.

The propagation of activity is studied in a spatially structured network model of gamma-aminobutyric acid-containing (GABAergic) neurons exhibiting postinhibitory rebound. In contrast to excitatory-coupled networks, recruitment spreads very slowly because cells fire only after the postsynaptic conductance decays, and with two possible propagation modes. If the connection strength decreases monotonically with distance (on-center), then propagation occurs in a discontinuous manner. If the self- and nearby connections are absent (off-center), propagation can proceed smoothly. Modest changes in the synaptic reversal potential can result in depolarization-mediated waves that are 25 times faster. Functional and developmental roles for these behaviors and implications for thalamic circuitry are suggested.

InsP3-induced Ca2+ excitability of the endoplasmic reticulum.

Oscillations in intracellular Ca2+ can be induced by a variety of cellular signalling processes (Woods et al., 1986; Berridge 1988; Jacob et al., 1988) and appear to play a role in secretion (Stojilkovic et al., 1994), fertilization (Miyazaki et al., 1993), and smooth muscle contraction (Iino and Tsukioka, 1994). Recently, great progress has been made in understanding the mechanisms involved in a particular class of Ca2+ oscillation, associated with the second messenger inositol 1,4,5-trisphosphate (InsP3) (Berridge, 1993). Working in concert with intracellular Ca2+, InsP3 controls Ca2+ release via the InsP3 receptor in the endoplasmic reticulum (ER) (Berridge and Irvine, 1989). The IP3 receptor is regulated by its coagonists InsP3 and Ca2+, which both activate and inhibit Ca2+ release (Finch et al., 1991; Bezprozvanny et al., 1991; De Young and Keizer, 1992). These processes, together with the periodic activation of Ca2+ uptake into the ER, have been identified as key features in the mechanism of InsP3-induced Ca2+ oscillations in pituitary gonadotrophs (Li et al., 1994), Xenopus laevis oocytes (Lechleiter and Clapham, 1992; Atri et al., 1993), and other cell types (Keizer and De Young, 1993). Earlier discussions and models of InsP3-induced Ca2+ oscillations focused on the nature and number of internal releasable pools of Ca2+ (Goldbeter et al., 1990; Swillens and Mercan, 1990; Somogyi and Stucki, 1991), the importance of oscillations in InsP3 (Meyer and Stryer, 1988), and other issues not based on detailed experimental findings in specific cells types.

The role of activity-dependent network depression in the expression and self-regulation of spontaneous activity in the developing spinal cord.

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.

Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network.

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.

Dynamics of low-threshold spike activation in relay neurons of the cat lateral geniculate nucleus.

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.

Modeling plasma membrane and endoplasmic reticulum excitability in pituitary cells.

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.

Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons.

The sleep spindle rhythm of thalamic origin (7-14 Hz) displays widespread synchronization among thalamic nuclei and over most of the neocortex. The mechanisms which mediate such global synchrony are not yet well understood. Here, we theoretically address the hypothesis of Steriade and colleagues that the reticularis thalami nucleus may be considered as a genuine pacemaker for thalamocortical spindles. Interestingly, the reticularis consists of a population of neurons which are GABAergic and synaptically coupled. These cells, as do thalamic relay cells, exhibit a transient depolarization following release from sustained hyperpolarization. This postinhibitory rebound property is due to a T-type calcium ionic current which is inactivated at rest but de-inactivated by hyperpolarization. Theoretically, rebound-capable cells coupled by inhibition can generate rhythmic activity, although such oscillations are usually alternating (out-of-phase), rather than synchronous (in-phase). Here, we develop and apply to Steriade's pacemaker hypothesis our earlier finding that mutual inhibition can in fact synchronize cells, provided that the postsynaptic conductance decays sufficiently slowly. Indeed, postsynaptic receptors of the GABAB subtype mediate inhibition with a large decay time-constant (approximately 200 ms). In contrast, chloride-dependent, GABAA-mediated inhibitory postsynaptic potentials are fast and brief. Both GABAA and GABAB receptor binding sites are present in most thalamic regions, including the reticularis. We suggest that if GABAB receptors exist postsynaptically in the reticularis, they may play a critical role in the rhythmic synchronization among reticular neurons, hence in the thalamocortical system.

Modeling N-methyl-D-aspartate-induced bursting in dopamine neurons.

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.

Mechanisms for nonuniform propagation along excitable cables.

We have discussed two classes of mechanisms that can lead to propagation with nonconstant velocity, and to disruption of temporal patterning of action potentials. Inhomogenieties along the cable due to geometrical change or to altered cell coupling can result in conduction delays, with the possibility of block or reflection. Such conduction irregularities have been considered relevant to cardiac reentry phenomena. Our simulations with a discrete number of excitable cells, coupled by gap junctions, showed that the underlying mathematical structure of a saddle point threshold in an ionic model also contributes in an important way to creating a long delay. Such threshold behavior, although not yet demonstrated for some of the most well-studied models of excitability, should not be viewed as extraordinary; we have seen it in models other than those of references 6 and 7, and have produced it in the Hodgkin-Huxley model with reasonable parameter variations (but have not yet checked for reflections with these modifications). We are unaware of any computations with theoretical models of cardiac membrane that yield robust reflection behavior. Perhaps modifications of these models will be necessary in order to obtain adequate delays for reflection. The mechanism we have described here may serve as a guideline for additional features to seek in such parametric tuning. A different class of factors that contribute to interferring with action potential timing include the effects of previous activity on propagation speed. These influences may be described in terms of the dispersion relation, c(T), the dependence of speed on time between action potentials. The form of this function, for large T, reflects the exponential behavior of the action potential's return to rest. Supernormal conduction reveals itself in the dispersion relation when there is an overshoot of excitability in the return to rest, either a single overshoot or an alternating sequence of over- and undershoots (as seen in some nerve membrane models). A simple kinematic recipe was described for quantitatively predicting timing changes during propagation without having to solve the full cable equations. To apply these concepts to cardiac models it will be necessary to compute the dispersion relation for these models. By studying the dependence of c(T) and the waveform trajectory (including conductances as well as membrane potential) on various parameters one may gain insight into the ionic basis for experimentally observed supernormal conduction.

Mechanisms of spontaneous activity in the developing spinal cord and their relevance to locomotion.

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.

Synaptic amplification by active membrane in dendritic spines.

The suspected functional role of dendritic spines as loci of neuronal plasticity (possibly memory and learning) is greatly enriched when active membrane properties are assumed at the spine head. Computations with reasonable electrical and structural parameter values (corresponding to an optimal range for spine stem resistance) show that an active spine head membrane can provide very significant synaptic amplification and also strongly non-linear properties that could modulate the integration of input from many afferent sources.

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.

Voltage transients in neuronal dendritic trees.

An analytical method is outlined for calculating the passive voltage transient at each point in an extensively branched neuron model for arbitrary current injection at a single branch. The method is based on a convolution formula that employs the transient response function, the voltage response to an instantaneous pulse of current. For branching that satisfies Rall's equivalent cylinder constraint, the response function is determined explicitly. Voltage transients, for a brief current injected at a branch terminal, are evaluated at several locations to illustrate the attenuation and delay characteristics of passive spread. A comparison with the same transient input terminal input, the fraction of input charge dissipated by various branches in the neuron model is illustrated. These fractions are independent of the input time course. For transient synaptic conductance change at a single branch terminal, a numerical example demonstrates the nonlinear effect of reduced synaptic driving potential. The branch terminal synaptic input is compared with the same synaptic conductance input applied to the soma on the basis of excitatory postsynaptic potential amplitude at the soma and charge delivered to the soma.

Excitation dynamics: insights from simplified membrane models.

Excitable nerve membranes and models for their electrical activity exhibit a broad repertoire of dynamic behavior. To reveal these behaviors the theoretician seeks a model that is simple enough to analyze yet one that retains adequate biophysical realism. Here we strike such a balance by describing a two-variable simplification of the Hodgkin-Huxley (HH) model, which exhibits many membrane phenomena and reproduces, with good agreement, many HH responses. Comparisons and illustrations are presented for the single spike response, repetitive firing (and its cessation by a brief current pulse), and bistable behavior for increased extracellular K+ concentrations.

Repetitive activity and Hopf bifurcation under point-stimulation for a simple FitzHugh-Naguma nerve conduction model.

In response to point-stimulation with a constant current, a neuron may propagate a repetitive train of action potentials along its axon. For maintained repetitive activity, the current strength I must be, typically, neither too small nor too large. For I outside some range, time independent steady behavior is observed following a transient phase just after the current is applied. We present analytical results for a piecewise linear FitzHugh-Nagumo model for a point-stimulated (non-space-clamped) nerve which are consistent with this qualitative experimental picture. For each value of I there is a unique, spatially nonuniform, steady state solution. We show that this solution is stable except for an interval (I*, I(*)) of I values. Stability for I too small or too large corresponds to experiments with sub-threshold I or the excessive I which leads to 'nerve block'. For I = I*, I(*) we find Hopf bifurcation of spatially nonuniform, time periodic solutions. We conclude that (I*, I(*)) lies interior to the range of I values for repetitive activity. The values of I* and I(*) and their dependence on the model parameters are determined. Qualitative differences between results for the point-stimulated configuration and the space-clamped case are discussed.

Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons.

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory's predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.

Rhythmogenic effects of weak electrotonic coupling in neuronal models.

Strong gap-junctional coupling can synchronize the electrical oscillations of cells, but we show, in a theoretical model, that weak coupling can phase lock two cells 180 degrees out-of-phase. Antiphase oscillations can exist in parameter regimens where in-phase oscillations break down. Some consequences are (i) coupling two excitable cells leads to pacemaking, (ii) coupling two pacemaker cells leads to bursting, and (iii) coupling two bursters increases burst period. The latter shows that details of the fast spikes can affect macroscopic properties of the slow bursts. These effects hold in other models for bursting and may play a role in the collective behavior of cellular ensembles.