Joël Tabak

Mechanisms of Spontaneous Activity in the Developing Spinal Cord and Their Relevance to Locomotion

Abstract: 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. Whole‐cell 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.

Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents

Dopamine (DA) released from the hypothalamus tonically inhibits pituitary lactotrophs. DA (at micromolar concentration) opens potassium channels, hyperpolarizing the lactotrophs and thus preventing the calcium influx that triggers prolactin hormone release. Surprisingly, at concentrations ∼1000 lower, DA can stimulate prolactin secretion. Here, we investigated whether an increase in a K+ current could mediate this stimulatory effect. We considered the fast K+ currents flowing through large-conductance BK channels and through A-type channels. We developed a minimal lactotroph model to investigate the effects of these two currents. Both IBK and IA could transform the electrical pattern of activity from spiking to bursting, but through distinct mechanisms. IBK always increased the intracellular Ca2+ concentration, while IA could either increase or decrease it. Thus, the stimulatory effects of DA could be mediated by a fast K+ conductance which converts tonically spiking cells to bursters. In addition, the study illustrates that a heterogeneous distribution of fast K+ conductances could cause heterogeneous lactotroph firing patterns.

Modeling Spontaneous Activity in the Developing Spinal Cord Using Activity-Dependent Variations of Intracellular Chloride

We investigated how spontaneous activity is generated in developing, hyperexcitable networks. We focused our study on the embryonic chick spinal cord, a preparation that exhibits rhythmic discharge on multiple timescales: slow episodes (lasting minutes) and faster intraepisode cycling (∼1 Hz frequency). For this purpose, we developed a mean field model of a recurrent network with slow chloride dynamics and a fast depression variable. We showed that the model, in addition to providing a biophysical mechanism for the slow dynamics, was able to account for the experimentally observed activity. The model made predictions on how interval and duration of episodes are affected when changing chloride-mediated synaptic transmission or chloride flux across cell membrane. These predictions guided experiments, and the model results were compared with experimental data obtained with electrophysiological recordings. We found agreement when transmission was affected through changes in synaptic conductance and good qualitative agreement when chloride flux was varied through changes in external chloride concentration or in the rate of the Na+-K+-2Cl- cotransporter. Furthermore, the model made predictions about the time course of intracellular chloride concentration and chloride reversal potential and how these are affected by changes in synaptic conductance. Based on the comparison between modeling and experimental results, we propose that chloride dynamics could be an important mechanism in rhythm generation in the developing chick spinal cord.

Mixed Mode Oscillations as a Mechanism for Pseudo-Plateau Bursting

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called “pseudo-plateau bursting”. Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic β-cells.

Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field

Many developing neural systems exhibit spontaneous activity (O’Donovan, Curr Opin Neurobiol 9:94–104, 1999; Feller, Neuron 22:653–656, 1999) characterized by episodes of discharge (active phases) when many cells are firing, separated by silent phases during which few cells fire. Various models exhibit features of episodic behavior by means of recurrent excitation for supporting an episode and slow activity-dependent depression for terminating one. The basic mechanism has been analyzed using mean-field, firing-rate models. Firing-rate models are typically formulated ad hoc, not derived from a spiking network description, and the effects of substantial heterogeneity amongst the units are not usually considered. Here we develop an excitatory network of spiking neurons (N-cell model) with slow synaptic depression to model episodic rhythmogenesis. This N-cell model displays episodic behavior over a range of heterogeneity in bias currents. Important features of the episodic behavior include orderly recruitment of individual cells during silent phases and existence of a dynamical process whereby a small critical subpopulation of intermediate excitability conveys synaptic drive from active to silent cells. We also derive a general self-consistency equation for synaptic drive that includes cell heterogeneity explicitly. We use this mean-field description to expose the dynamical bistability that underlies episodic behavior in the heterogeneous network. In a systematic numerical study we find that the robustness of the episodic behavior improves with increasing heterogeneity. Furthermore, the heterogeneity of depression variables (imparted by the heterogeneity in cellular firing thresholds) plays an important role in this improvement: it renders the network episodic behavior more robust to variations in excitability than if depression is uniformized. We also investigate the effects of noise vs. heterogeneity on the robustness of episodic behavior, especially important for the developing nervous system. We demonstrate that noise-induced episodes are very fragile, whereas heterogeneity-produced episodic rhythm is robust.

Fast-activating voltage- and calcium-dependent potassium (BK) conductance promotes bursting in pituitary cells: a dynamic clamp study.

The electrical activity pattern of endocrine pituitary cells regulates their basal secretion level. Rat somatotrophs and lactotrophs exhibit spontaneous bursting and have high basal levels of hormone secretion, while gonadotrophs exhibit spontaneous spiking and have low basal hormone secretion. It has been proposed that the difference in electrical activity between bursting somatotrophs and spiking gonadotrophs is due to the presence of large conductance potassium (BK) channels on somatotrophs but not on gonadotrophs. This is one example where the role of an ion channel type may be clearly established. We demonstrate here that BK channels indeed promote bursting activity in pituitary cells. Blocking BK channels in bursting lacto-somatotroph GH4C1 cells changes their firing activity to spiking, while further adding an artificial BK conductance via dynamic clamp restores bursting. Importantly, this burst-promoting effect requires a relatively fast BK activation/deactivation, as predicted by computational models. We also show that adding a fast-activating BK conductance to spiking gonadotrophs converts the activity of these cells to bursting. Together, our results suggest that differences in BK channel expression may underlie the differences in electrical activity and basal hormone secretion levels among pituitary cell types and that the rapid rate of BK channel activation is key to its role in burst promotion.

Parameter Estimation Methods for Single Neuron Models

With the advancement in computer technology, it has become possible to fit complex models to neuronal data. In this work, we test how two methods can estimate parameters of simple neuron models (passive soma) to more complex ones (neuron with one dendritic cylinder and two active conductances). The first method uses classical voltage traces resulting from current pulses injection (time domain), while the second uses measures of the neurons response to sinusoidal stimuli (frequency domain). Both methods estimate correctly the parameters in all cases studied. However, the time-domain method is slower and more prone to estimation errors in the cable parameters than the frequency-domain method. Because with noisy data the goodness of fit does not distinguish between different solutions, we suggest that running the estimation procedure a large number of times might help find a good solution and can provide information about the interactions between parameters. Also, because the formulation used for the models response in the frequency domain is analytical, one can derive a local sensitivity analysis for each parameter. This analysis indicates how well a parameter is likely to be estimated and helps choose an optimal stimulation protocol. Finally, the tests suggest a strategy for fitting single-cell models using the two methods examined.

Differential control of active and silent phases in relaxation models of neuronal rhythms

Rhythmic bursting activity, found in many biological systems, serves a variety of important functions. Such activity is composed of episodes, or bursts (the active phase, AP) that are separated by quiescent periods (the silent phase, SP). Here, we use mean field, firing rate models of excitatory neural network activity to study how AP and SP durations depend on two critical network parameters that control network connectivity and cellular excitability. In these models, the AP and SP correspond to the networks underlying bistability on a fast time scale due to rapid recurrent excitatory connectivity. Activity switches between the AP and SP because of two types of slow negative feedback: synaptic depression—which has a divisive effect on the network input/output function, or cellular adaptation—a subtractive effect on the input/output function. We show that if a model incorporates the divisive process (regardless of the presence of the subtractive process), then increasing cellular excitability will speed up the activity, mostly by decreasing the silent phase. Reciprocally, if the subtractive process is present, increasing the excitatory connectivity will slow down the activity, mostly by lengthening the active phase. We also show that the model incorporating both slow processes is less sensitive to parameter variations than the models with only one process. Finally, we note that these network models are formally analogous to a type of cellular pacemaker and thus similar results apply to these cellular pacemakers.

The dynamics underlying pseudo-plateau bursting in a pituitary cell model

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.

A-type k+ current can act as a trigger for bursting in the absence of a slow variable

Models of bursting in single cells typically include two subsystems with different timescales. Variations in one or more slow variables switch the system between a silent and a spiking state. We have developed a model for bursting in the pituitary lactotroph that does not include any slow variable. The model incorporates fast, noninactivating calcium and potassium currents (the spike-generating mechanism), as well as the fast, inactivating A-type potassium current (IA). IA is active only briefly at the beginning of a burst, but this brief impulse of IA acts as a burst trigger, injecting the spike trajectory close to an unstable steady state. The spiraling of the trajectory away from the steady state produces a period of low-amplitude spiking typical of lactotrophs. Increasing the conductance of A-type potassium current brings the trajectory closer to the unstable steady state, increasing burst duration. However, this also increases interburst interval, and for larger conductance values, all activity stops. To our knowledge, this is the first example of a physiologically based, single-compartmental model of bursting with no slow subsystem.