The assembly of ionic currents in a thalamic neuron. III. The seven-dimensional model

We replace our earlier three-dimensional ha-model of a thalamic neuron (Rose & Hindmarsh, Proc. R. Soc. Lond. B 237, 289-312 (1989b)) by a seven-dimensional ha-model. The stability and state diagrams for this seven-dimensional model are shown to be similar to those of the three-dimensional system with which it is compared. Two examples that illustrate how the seven-dimensional model can be related to experimental recordings are then discussed in detail. In each case we show the state and stability diagrams and the time courses of the different ionic currents during the burst response. We discuss how the various components of the state diagram could be determined experimentally. These experiments are illustrated by using our model to simulate the results of actual experiments. Finally the state and stability diagrams are transferred to a current-voltage diagram. This shows the advantage of the state diagram compared with the current-voltage diagram for representing the dynamics of the firing of a thalamic neuron.     

A model of a thalamic neuron

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.     

The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model

We have previously discussed qualitative models for bursting and thalamic neurons that were obtained by modifying a simple two-dimensional model for repetitive firing. In this paper we report the results of making a similar sequence of modifications to a more elaborate six-dimensional model of repetitive firing which is based on the Hodgkin-Huxley equations. To do this we first reduce the six-dimensional model to a two-dimensional model that resembles our original two-dimensional qualitative model. This is achieved by defining a new variable, which we call q. We then add a subthreshold inward current and a subthreshold outward current having a variable, z, that changes slowly. This gives a three-dimensional (v,q,z) model of the Hodgkin-Huxley type, which we refer to as the z-model. Depending on the choice of parameter values this model resembles our previous models of bursting and thalamic neurons. At each stage in the development of these models we return to the corresponding seven-dimensional model to confirm that we can obtain similar solutions by using the complete system of equations. The analysis of the three-dimensional model involves a state diagram and a stability diagram. The state diagram shows the projection of the phase path from v,q,z space into the v,z plane, together with the projections of the curves z = 0 and v = q = 0. The stability of the points on the curve v = q = 0, which we call the v, q nullcurve, is determined by the stability diagram. Taken together the state and stability diagrams show how to assemble the ionic currents to produce a given firing pattern.     

Analysis of bursting in a thalamic neuron model

We extend a quantitative model for low-voltage, slow-wave excitability based on the T-type calcium current (Wang et al. 1991) by juxtaposing it with a Hodgkin-Huxley-like model for fast sodium spiking in the high voltage regime to account for the distinct firing modes of thalamic neurons. We employ bifurcation analysis to illustrate the stimulus-response behavior of the full model under both voltage regimes. The model neuron shows continuous sodium spiking when depolarized sufficiently from rest. Depending on the parameters of calcium current inactivation, there are two types of low-voltage responses to a hyperpolarizing current step: a single rebound low threshold spike (LTS) upon release of the step and periodic LTSs. Bursting is seen as sodium spikes ride the LTS crest. In both cases, we analyze the LTS burst response by projecting its trajectory into a fast/slow phase plane. We also use phase plane methods to show that a potassium A-current shifts the threshold for sodium spikes, reducing the number of fast sodium spikes in an LTS burst. It can also annihilate periodic bursting. We extend the previous work of Rose and Hindmarsh (1989a–c) for a thalamic neuron and propose a simpler model for thalamic activity. We consider burst modulation by using a neuromodulator-dependent potassium leakage conductance as a control parameter. These results correspond with experiments showing that the application of certain neurotransmitters can switch firing modes. Received: 18 July 1993/Accepted in revised form: 22 January 1994     

Bursting as an Effective Relay Mode in a Minimal Thalamic Model

In recent years, accumulating evidence indicates that thalamic bursts are present during wakefulness and participate in information transmission as an effective relay mode with distinctive properties from the tonic activity. Thalamic bursts originate from activation of the low threshold calcium cannels via a local feedback inhibition, exerted by the thalamic reticular neurons upon the relay neurons. This article, examines if this simple mechanism is sufficient to explain the distinctive properties of thalamic bursting as an effective relay mode. A minimal model of thalamic circuit composed of a retinal spike train, a relay neuron and a reticular neuron is simulated to generate the tonic and burst firing modes. The integrate-and-fire-or-burst model is used to simulate the neurons. After discriminating the burst events with criteria based on inter-spike-intervals, statistical indices show that the bursts of the minimal model are stereotypic events. The relation between the rate of bursts and the parameters of the input spike train demonstrates marked nonlinearities. Burst response is shown to be selective to spike-silence-spike sequences in the input spike train. Moreover, burst events represent the input more reliably than the tonic spike in a considerable range of the parameters of the model. In conclusion, many of the distinctive properties of thalamic bursts such as stereotypy, nonlinear dependence on the sensory stimulus, feature selectivity and reliability are reproducible in the minimal model. Furthermore, the minimal model predicts that while the bursts are more frequent in the spike train of the off-center X relay neurons (corresponding to off-center X retinal ganglion cells), they are more reliable when generated by the on-center ones (corresponding to on-center X ganglion cells).     

A unified model of CA1/3 pyramidal cells: an investigation into excitability

After-depolarisation is a hallmark of excitability in hippocampal pyramidal cells of CA1 and CA3 regions, because it constitutes the subthreshold relation between inward and outward ionic currents. This relationship determines the nominal response to stimuli and provides the necessary conditions for firing a spike or a burst of action potentials. Nevertheless, after-depolarisation is an inherently transient phenomenon that is not very well understood. We study after-depolarisation using a single-compartment pyramidal-cell model based on recent voltage- and current-clamp experimental data. We systematically investigate CA1 and CA3 behaviour and show that changes to maximal conductances of T-type Ca(2+)-current and muscarinic-sensitive and delayed rectifier K(+)-currents are sufficient to switch the behaviour of the model from a CA3 to a CA1 neuron. We use model analysis to define after-depolarisation and bursting threshold. We also explain the influence of particular ionic currents on this phenomenon. This study ends with a sensitivity analysis that demonstrates the influence of specific currents on excitability. Counter-intuitively, we find that a decrease of Na(+)-current could cause an increase in excitability. Our analysis suggests that a change of high-voltage activated Ca(2+)-current can have a similar effect.     

Analysis of long cortical evoked potentials

Evoked potentials arising in the motor cortex in response to its direct stimulation (dendritic and slow negative potentials), to stimulation of the ventrolateral (primary response) and intralaminar (nonspecific response) thalamic nuclei, and to stimulation of the pyramidal tracts (antidromic response), and also postsynaptic responses of neurons corresponding to them were studied in acute experiments on curarized cats. Evoked potentials arising in response to direct cortical stimulation and also to stimulation of the specific and nonspecific thalamic nuclei and pyramidal tracts were recorded from the same point of the motor cortex, and the corresponding intracellular responses were recorded from the same neuron. Slow negative potentials arising under these conditions of stimulation and the IPSPs corresponding to them were shown to have an identical time course. The results show that slow negative potentials are a reflection of hyperpolarization of pyramidal neurons. It is suggested that the individual components of responses evoked by direct stimulation of the cortex and thalamic nuclei have a common genesis.I. S. Beritashvili Institute of Physiology, Academy of Sciences of the Georgian SSR, Tbilisi. Translated from Neirofiziologiya, Vol. 14, No. 2, pp. 115–121, March–April, 1982.     

Decreased Response to Acetylcholine during Aging of Aplysia Neuron R15

How aging affects the communication between neurons is poorly understood. To address this question, we have studied the electrophysiological properties of identified neuron R15 of the marine mollusk Aplysia californica. R15 is a bursting neuron in the abdominal ganglia of the central nervous system and is implicated in reproduction, water balance, and heart function. Exposure to acetylcholine (ACh) causes an increase in R15 burst firing. Whole-cell recordings of R15 in the intact ganglia dissected from mature and old Aplysia showed specific changes in burst firing and properties of action potentials induced by ACh. We found that while there were no significant changes in resting membrane potential and latency in response to ACh, the burst number and burst duration is altered during aging. The action potential waveform analysis showed that unlike mature neurons, the duration of depolarization and the repolarization amplitude and duration did not change in old neurons in response to ACh. Furthermore, single neuron quantitative analysis of acetylcholine receptors (AChRs) suggested alteration of expression of specific AChRs in R15 neurons during aging. These results suggest a defect in cholinergic transmission during aging of the R15 neuron.     

The cyclic GMP-induced inward current in neuron R14 of Aplysia californica: similarity to a FMRFamide-induced inward current

Injecting cGMP into Aplysia neuron R14 induced an inward current similar to one elicited by application of FMRFamide to the outside of that cell. In contrast, injection of cAMP into R14 caused a long-lasting outward current and conductance increase. Phosphodiesterase inhibitors increased the cGMP and FMRFamide-induced inward currents in R14. The cGMP-induced inward current is voltage dependent and is largely carried by Na+. It is also strongly and inversely dependent on both external [Ca2+] and [Cl-], although these ions are not significant current carriers. Changing external [K+] had no effect. Voltage and ion dependencies of the cGMP-induced inward current are similar to those of an inward current induced by FMRFamide. Thus cGMP may be a second messenger to FMRFamide in producing a slow inward current in R14. cGMP does not appear to be a second messenger to FMRFamide in most Aplysia neurons.     

A study of the connection between the interneuron initiating pacemaker activity in a bursting neuron and the bursting neuron of the snailHelix pomatia

The connection between an interneuron initiating pacemaker activity in the bursting RPa1 neuron and the bursting neuron itself (Pin and Gola, 1983) has been analyzed in the snail Helix pomatia. Prolonged depolarization of the interneuronal membrane produced in it a series of action potentials as well as a parallel initiation or enhancement of bursting activity in the RPa1 neuron. If the discharge in the interneuron was evoked by short current pulses of threshold amplitude, no bursting activity was seen in the RPa1 neuron. However, short stimuli delivered on the background of subthreshold depolarization of the interneuronal membrane produced bursting activity in the RPa1 neuron. Under voltage-clamp conditions a slow inward current could be recorded in the RPa1 neuronal membrane after stimulation of the interneuron with a latency of about 2 sec. Short shifts of the holding potential in the hyperpolarizing direction at the maximum of this current produced a transient outward current. Replacement of extracellular Ca2+ by Mg2+ ions, as well as addition of 1 mM CdCl2 to the external solution, prevented the response to the interneuronal stimulation in the RPa1 neuron. Electron microscopic investigation of the interneuron has shown the abundance of Golgi complexes in its cytoplasm with electron-dense granules in their vicinity. It is concluded that the connection between the interneuron and the bursting neuron is of chemical origin, based on secretion by the former of some substances which activate at least two types of ionic channels in the membrane of the RPa1 neuron.     

Interspike interval densities of resonate and fire neurons

The subthreshold dynamics of a neuron can follow one of the two patterns: resonant neurons generate intrinsic subthreshold membrane potential oscillations, whereas in nonresonant neurons these oscillations are not observed. Here, we investigate how these subthreshold behaviors affect the suprathreshold response. Both types of neurons are described by a resonate and fire model, with the stable fixpoint being either a focus or a node. Using analytic expression for a linear oscillator model with threshold and reset, we calculate the multimodal interspike interval densities. We show that a change in model parameters induces qualitative changes in the interspike interval densities.     

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.     

Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons. PACS 05.45.-a, 05.40.Ca, 87.18.Sn, 87.19     

A Relaxation Oscillator Description of the Burst-Generating Mechanism in the Cardiac Ganglion of the Lobster,Homarus americanus

Properties of the neural mechanism responsible for generating the periodic burst of spike potentials in the nine ganglion neurons were investigated by applying brief, single shocks to the four small cells with extracellular electrodes placed near the trigger zones of the small cells. The shock elicited a burst if presented during the latter portion of the silent period, terminated a burst during the latter portion of the burst period, and was followed by a newly initiated burst during the early portion of the burst period. The resultant changes in burst and silent period durations were quantitatively described by a second-order non-linear differential equation similar to the van der Pol equation for a relaxation oscillator. The equation also qualitatively described changes in firing threshold of the small cells during the burst cycle. The first derivative of the solution to the equation is similar to slow transmembrane potentials in neurons that are involved in generation of burst activity in other crustacean cardiac ganglia.     

Vasopressin induces depolarization and state-dependent firing patterns in rat thalamic paraventricular nucleus neurons in vitro

The thalamic midline paraventricular nucleus (PVT) is prominently innervated by vasopressin-immunoreactive neurons from the suprachiasmatic nucleus (SCN), site of the brain's biological clock. Using patch-clamp recordings in slice preparations taken from Wistar rats during the subjective day, we examined 90 PVT neurons for responses to bath-applied AVP (0.5-2 microM; 1-3 min). In current clamp at resting membrane potentials (-65 +/- 1 mV), PVT neurons displayed low-threshold spikes (LTSs) and burst firing patterns. In 50% of cells tested, AVP induced a slowly rising, prolonged membrane depolarization and tonic firing, returning to burst firing upon recovery. AVP modulated hyperpolarization-activated LTSs by decreasing the time to the initial sodium spike at the onset of LTS, also increasing the duration of the afterdepolarization. Responses were blockable with a V(1a) receptor antagonist (Manning compound). Under voltage clamp, AVP induced a TTX-resistant, slowly rising, and prolonged (approximately 15 min) inward current (<40 pA). Current-voltage relationship (I-V) analyses of the AVP responses revealed a decrease in membrane conductance to 73.1 +/- 6.2% of control, with net AVP current reversing at -106 +/- 4 mV, and decreased inward rectification at negative potentials. These observations are consistent with an AVP-induced closure of an inwardly rectifying potassium conductance. On the basis of these in vitro observations, we suggest that the SCN vasopressinergic innervation of PVT is excitatory in nature, possibly releasing AVP with circadian rhythmicity and contributing to state-dependent firing patterns in PVT neurons over the sleep-wake cycle.     

Interval analysis of bursting discharges in aplysia neurons

Time intervals of 12 records of bursting discharges in Aplysia neurons were analysed by digital computer to determine the interrelations between the burst period, the interburst interval and the burst duration. The effects of membrane potential changes on the parameters of bursting discharges were examined also. A low correlation was found between burst duration and burst period in the majority of cases, and this was interpreted as an indication of probable independence between the mechanisms governing these parameters. Also, a specific temporal organization of interspike intervals seems to be present in each type of neuron. The results suggest that the mechanism governing the burst period is characterized by a slow membrane potential oscillation resembling that observed in bursting neurons when actions potentials are blocked by tetrodotoxin. The burst duration would be determined by the response of the neuron to suprathreshold depolarization.     

Mechanisms of firing patterns in fast-spiking cortical interneurons

Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na(+), delayed-rectifier K(+), and slowly inactivating d-type K(+) conductances. The model is analyzed using nonlinear dynamical system theory. For small Na(+) window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, gd, and it is delayed for larger gd. As gd further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na(+) window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.     

Modulation of Activity of One Neuron by Subthreshold Slow Potentials in Another in Lobster Cardiac Ganglion

A direct demonstration is given of interaction between specific neurons without impulses, via graded slow potentials electrotonically spread from one cell to another. Repetitive polarizing or depolarizing current pulses of 50 to 200 msec. and subthreshold intensity were passed through an intracellular electrode in the soma of a follower cell of the isolated ganglion. When the frequency is near the natural rhythm of impulse bursts corresponding to heart beats and arising in a pacemaker cell 5 to 10 mm. posteriorly, the bursts rapidly become synchronized with the pulses. The effect disappears upon withdrawing the intracellular electrode. Brief pulses or full spikes in the follower are not effective. Hyperpolarizing long pulses attract the burst to a fixed period after the end of the pulse, depolarizations after the beginning of the pulse. The natural rhythm promptly reappears when the pulses are stopped and occasionally breaks through during weak repetitive pulses. Current pulses in postsynaptic cells also alter the threshold of a presynaptic neuron to externally applied stimuli. Some kind of direct, low resistance pathway for electrotonic spread, discriminating against spikes because of their brevity, is inferred, providing a basis for subthreshold interaction which is specific and not by way of a field effect. Due to the sensitivity of modulation of ongoing rhythms, electrotonic currents can be effective even after decrementing over several millimeters.     

Ionic mechanisms underlying excitation-to-frequency transduction: studies by voltage clamp methods

The transduction of synaptic activity to impulse generation is controlled by the active and passive properties of neurons. The voltage dependent conductances of cat motoneurons, as we understand them, are presented and related to repetitive firing behavior. Both outward potassium and inward calcium currents are activated in the subthreshold region. Accomodation of the initial segment allows tonic activation of these currents during repetitive firing and the response properties of the neuron depend upon the balance of inward and outward currents. The effects of putative neurotransmitters and changes in ionic concentration upon the active ionic currents and upon the response properties of neurons are also discussed.