Changing excitation and inhibition in simulated neural networks: effects on induced bursting behavior

 The development of synchronous bursting in neuronal ensembles represents an important change in network behavior. To determine the influences on development of such synchronous bursting behavior we study the dynamics of small networks of sparsely connected excitatory and inhibitory neurons using numerical simulations. The synchronized bursting activities in networks evoked by background spikes are investigated. Specifically, patterns of bursting activity are examined when the balance between excitation and inhibition on neuronal inputs is varied and the fraction of inhibitory neurons in the network is changed. For quantitative comparison of bursting activities in networks, measures of the degree of synchrony are used. We demonstrate how changes in the strength of excitation on inputs of neurons can be compensated by changes in the strength of inhibition without changing the degree of synchrony in the network. The effects of changing several network parameters on the network activity are analyzed and discussed. These changes may underlie the transition of network activity from normal to potentially pathologic (e.g., epileptic) states. Received: 21 May 2002 / Accepted in revised form: 3 December 2002 / Published online: 7 March 2003 Correspondence to: P. Kudela (e-mail: pkudela@jhmi.edu) Acknowledgements. This research was supported by NIH grant NS 38958.     

Spike-Frequency Adaptation of a Generalized Leaky Integrate-and-Fire Model Neuron

Although spike-frequency adaptation is a commonly observed property of neurons, its functional implications are still poorly understood. In this work, using a leaky integrate-and-fire neural model that includes a Ca²⁺-activated K⁺ current (I _AHP), we develop a quantitative theory of adaptation temporal dynamics and compare our results with recent in vivo intracellular recordings from pyramidal cells in the cat visual cortex. Experimentally testable relations between the degree and the time constant of spike-frequency adaptation are predicted. We also contrast the I _AHP model with an alternative adaptation model based on a dynamical firing threshold. Possible roles of adaptation in temporal computation are explored, as a a time-delayed neuronal self-inhibition mechanism. Our results include the following: (1) given the same firing rate, the variability of interspike intervals (ISIs) is either reduced or enhanced by adaptation, depending on whether the I _AHP dynamics is fast or slow compared with the mean ISI in the output spike train; (2) when the inputs are Poisson-distributed (uncorrelated), adaptation generates temporal anticorrelation between ISIs, we suggest that measurement of this negative correlation provides a probe to assess the strength of I _AHP in vivo; (3) the forward masking effect produced by the slow dynamics of I _AHP is nonlinear and effective at selecting the strongest input among competing sources of input signals.     

Response of Human Thalamic Neurons to High-Frequency Stimulation

Thalamic deep brain stimulation (DBS) is an effective treatment for tremor, but the mechanisms of action remain unclear. Previous studies of human thalamic neurons to noted transient rebound bursting activity followed by prolonged inhibition after cessation of high frequency extracellular stimulation, and the present study sought to identify the mechanisms underlying this response. Recordings from 13 thalamic neurons exhibiting low threshold spike (LTS) bursting to brief periods of extracellular stimulation were made during surgeries to implant DBS leads in 6 subjects with Parkinson''s disease. The response immediately after cessation of stimulation included a short epoch of burst activity, followed by a prolonged period of silence before a return to LTS bursting. A computational model of a population of thalamocortical relay neurons and presynaptic axons terminating on the neurons was used to identify cellular mechanisms of the observed responses. The model included the actions of neuromodulators through inhibition of a non-pertussis toxin sensitive K⁺ current (I_KL), activation of a pertussis toxin sensitive K⁺ current (I_KG), and a shift in the activation curve of the hyperpolarization-activated cation current (I_h). The model replicated well the measured responses, and the prolonged inhibition was associated most strongly with changes in I_KG while modulation of I_KL or I_h had minimal effects on post-stimulus inhibition suggesting that neuromodulators released in response to high frequency stimulation are responsible for mediating the post-stimulation bursting and subsequent long duration silence of thalamic neurons. The modeling also indicated that the axons of the model neurons responded robustly to suprathreshold stimulation despite the inhibitory effects on the soma. The findings suggest that during DBS the axons of thalamocortical neurons are activated while the cell bodies are inhibited thus blocking the transmission of pathological signals through the network and replacing them with high frequency regular firing.     

The influences of somatic and dendritic inhibition on bursting patterns in a neuronal circuit model

The balance between inhibition and excitation plays a crucial role in the generation of synchronous bursting activity in neuronal circuits. In human and animal models of epilepsy, changes in both excitatory and inhibitory synaptic inputs are known to occur. Locations and distribution of these excitatory and inhibitory synaptic inputs on pyramidal cells play a role in the integrative properties of neuronal activity, e.g., epileptiform activity. Thus the location and distribution of the inputs onto pyramidal cells are important parameters that influence neuronal activity in epilepsy. However, the location and distribution of inhibitory synapses converging onto pyramidal cells have not been fully studied. The objectives of this study are to investigate the roles of the relative location of inhibitory synapses on the dendritic tree and soma in the generation of bursting activity. We investigate influences of somatic and dendritic inhibition on bursting activity patterns in several paradigms of potential connections using a simplified multicompartmental model. We also investigate the effects of distribution of fast and slow components of GABAergic inhibition in pyramidal cells. Interspike interval (ISI) analysis is used for examination of bursting patterns. Simulations show that the inhibitory interneuron regulates neuronal bursting activity. Bursting behavior patterns depend on the synaptic weight and delay of the inhibitory connection as well as the location of the synapse. When the inhibitory interneuron synapses on the pyramidal neuron, inhibitory action is stronger if the inhibitory synapse is close to the soma. Alterations of synaptic weight of the interneuron can be compensatory for changes in the location of synaptic input. The relative changes in these parameters exert a considerable influence on whether synchronous bursting activity is facilitated or reduced. Additional simulations show that the slow GABAergic inhibitory component is more effective than the fast component in distal dendrites. Taken together, these findings illustrate the potential for GABAergic inhibition in the soma and dendritic tree to play an important modulatory role in bursting activity patterns.     

Understanding effects on excitability of simulated <Emphasis Type="Italic">I</Emphasis><Subscript>h</Subscript> modulation in simple neuronal models

The hyperpolarization-activated, inward, mixed cation current, I _h, appears in a wide variety of cells in the nervous system, contributes to diverse neuronal properties, and is up-regulated by a number of important neurotransmitters. Up-regulation of I _h is usually associated with an excitability-enhancing depolarization of resting membrane potential and an excitability-depressing shunting effect caused by a decrease in input resistance. In order to gain a better understanding of the interaction of these effects and their influence on excitability with I _h modulation, we systematically analyze changes in neuronal properties associated with excitability during I _h modulation in simplified, yet, biophysical neuron models based on a hippocampal pyramidal neuron. We simulate I _h modulation by varying both its maximal conductance and its half-activation voltage, mimicking the effects of cAMP-linked neurotransmitters, through ranges of physiologically realistic parameter regimes. Of particular interest is the contribution of the different effects of I _h up-regulation when membrane potentials are held at common levels and neuronal excitability is probed. Our modeling results suggest that, although holding potentials at common levels may compensate for changes in resting membrane potentials, this protocol may exaggerate the excitability-depressing influences of changes in input resistances with I _h up-regulation.     

Targeted Deletion of Kcne2 Impairs HCN Channel Function in Mouse Thalamocortical Circuits

Background

Hyperpolarization-activated, cyclic nucleotide-gated (HCN) channels generate the pacemaking current, I_h, which regulates neuronal excitability, burst firing activity, rhythmogenesis, and synaptic integration. The physiological consequence of HCN activation depends on regulation of channel gating by endogenous modulators and stabilization of the channel complex formed by principal and ancillary subunits. KCNE2 is a voltage-gated potassium channel ancillary subunit that also regulates heterologously expressed HCN channels; whether KCNE2 regulates neuronal HCN channel function is unknown.

Methodology/Principal Findings

We investigated the effects of Kcne2 gene deletion on I_h properties and excitability in ventrobasal (VB) and cortical layer 6 pyramidal neurons using brain slices prepared from Kcne2 ^+/+ and Kcne2 ^−/− mice. Kcne2 deletion shifted the voltage-dependence of I_h activation to more hyperpolarized potentials, slowed gating kinetics, and decreased I_h density. Kcne2 deletion was associated with a reduction in whole-brain expression of both HCN1 and HCN2 (but not HCN4), although co-immunoprecipitation from whole-brain lysates failed to detect interaction of KCNE2 with HCN1 or 2. Kcne2 deletion also increased input resistance and temporal summation of subthreshold voltage responses; this increased intrinsic excitability enhanced burst firing in response to 4-aminopyridine. Burst duration increased in corticothalamic, but not thalamocortical, neurons, suggesting enhanced cortical excitatory input to the thalamus; such augmented excitability did not result from changes in glutamate release machinery since miniature EPSC frequency was unaltered in Kcne2 ^−/− neurons.

Conclusions/Significance

Loss of KCNE2 leads to downregulation of HCN channel function associated with increased excitability in neurons in the cortico-thalamo-cortical loop. Such findings further our understanding of the normal physiology of brain circuitry critically involved in cognition and have implications for our understanding of various disorders of consciousness.     

Mathematical Model of Bursting in Dissociated Purkinje Neurons

In vitro, Purkinje cell behaviour is sometimes studied in a dissociated soma preparation in which the dendritic projection has been cleaved. A fraction of these dissociated somas spontaneously burst. The mechanism of this bursting is incompletely understood. We have constructed a biophysical Purkinje soma model, guided and constrained by experimental reports in the literature, that can replicate the somatically driven bursting pattern and which hypothesises Persistent Na⁺ current (I_NaP) to be its burst initiator and SK K⁺ current (I_SK) to be its burst terminator.     

Monosynaptic pathway responsible for generation of bursting activity in theHelix pomatia RPal neuron

The connection between a visceral ganglia interneuron initiating bursting pacemaker activity in the RPal neuron and the RPal neuron itself was investigated inHelix pomatia. Stimulating the interneuron either initiated or intensified bursting activity in the RPal neuron, depending on initial electrical activity in this cell. Replacing calcium with magnesium ions in the extracellular fluid and adding CdCl₂ to this fluid reversibly inhibited the effect of interneuronal stimulation on the RPal neuron. The latter effect was unaffected by increasing the concentration of extracellular Ca²⁺ 10 to 70 mM. Intracellular injection of both Cs⁺ and TEA into the interneuron produced an increase in the duration of its action potentials and rendered the link connecting the neurons more effective. It is deduced that a monosynaptic chemical connection exists between the interneuron and the RPal neuron for which a peptide compound serves as transmitter.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 19, No. 1, pp. 20–28, January–February, 1987.     

A unified model for two modes of bursting in GnRH neurons

Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca²⁺ current I _s, and the Ca²⁺ -activated K⁺ current I _KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.     

Mechanisms of membrane potential oscillation in bursting neurons on the snail,Helix pomatia

• 1.1. The mechanism of generation of membrane potential (MP) oscillations was studied in identified bursting neurons from the snail Helix pomatia.
• 2.2. Long-lasting stimulation of an identified peptidergic interneuron produced a persistent bursting activity in a non-active burster.
• 3.3. External application of calcium channel blockers (1 mM Cd²⁺ or 5 mM La²⁺) resulted in a transient increase in the slow-wave amplitude and subsequent prevention of pacemaker activity generation in bursting neurons. Application of these blockers together with endogenous neuropeptide initiating bursting activity generation, increased MP wave amplitude without prevention of bursting activity generation.
• 4.4. Replacement of all NaCl in normal Ringer's solution with isoosmotic CaCl₂, glucose or Tris-HCl produced a reversible block of bursting activity generation. Stationary current-voltage relation (CVR) of bursting neuron membrane has a region of negative resistance (NRR) and does not intersect the potential axis in threshold region for action potential (AP) generation in normal Ringer's solution. In Na-free solution stationary CVR is linear and intersects the potential axis near — 52 mV.
• 5.5. Novel potential- and time-dependent outward (E_rev = − 58 mV) current, I_B, activated by hyperpolarization was found in the bursting neuron membrane. Having achieved a maximal value, this current decayed with a time constant of about 1 sec. Hyperpolarization inactivated maximal conductance, g_B, responsible for I_B, and depolarization abolished inactivation of g_B.
• 6.6. Short-lasting (0.01 sec) hyperpolarization of the bursting neuron membrane by inward current pulse induced the development of prolonged hyperpolarization wave lasting up to 10 sec.
• 7.7. These results suggest that: (a) persistent bursting activity of RPal neuron in the snail Helix pomatia is not endogenous but is due to a constant activation of peptidergic synaptic inputs of these neurons; (b) Ca²⁺ ions do not play a pivotal role in the ionic mechanism of MP oscillations but play a determining role in the process of secretion of a peptide initiating bursting activity by the interneuron presynaptic terminal; (c) depolarizing phase of the MP wave is due to specific properties of stationary CVR and hyperpolarization phase is due to regenerative properties of hyperpolarization-activated outward current I_B. The minimal mathematical version of MP oscillations based on the experimental data is presented.

     

The role of a transient potassium current in a bursting neuron model

Presented here is a biophysical cell model which can exhibit low-frequency repetitive activity and bursting behavior. The model is developed from previous models (Av-Ron et al. 1991, 1993) for excitability, oscillations and bursting. A stepwise development of the present model shows the contribution of a transient potassium current (I _A ) to the overall dynamics. By changing a limited set of model parameters one can describe different firing patterns; oscillations with frequencies ranging from 2–200 Hz and a wide range of bursting behaviors in terms of the durations of bursting and quiescence, peak firing frequency and rate of change of the firing frequency.     

Computational study of enhanced excitability in Hermissenda: membrane conductances modulated by 5-HT

Serotonin (5-HT) applied to the exposed but otherwise intact nervous system results in enhanced excitability of Hermissenda type-B photoreceptors. Several ion currents in the type-B photoreceptors are modulated by 5-HT, including the A-type K⁺ current (I_K,A), sustained Ca²⁺ current (I_Ca,S), Ca-dependent K⁺ current (I_K,Ca), and a hyperpolarization-activated inward rectifier current (I_h). In this study, we developed a computational model that reproduces physiological characteristics of type B photoreceptors, e.g. resting membrane potential, dark-adapted spike activity, spike width, and the amplitude difference between somatic and axonal spikes. We then used the model to investigate the contribution of different ion currents modulated by 5-HT to the magnitudes of enhanced excitability produced by 5-HT. Ion currents were systematically varied within limits observed experimentally, both individually and in combinations. A reduction of I_K,A or I_K,Ca, or an increase in I_h enhanced excitability by 20–50%. Decreasing I_Ca,S produced a dramatic decrease in excitability. Reductions of I_K,V produced only minimal increases in excitability, suggesting that I_K,V probably plays a minor role in 5-HT induced enhanced excitability. Combinations of changes in I_K,A, I_K,Ca, I_h and I_Ca,S produced increases in excitability comparable to experimental observations. After 5-HT application, the cell's depolarization force is shifted from the I_h–I_Ca,S combination to predominantly I_h.     

Electrophysiological and Theoretical Analysis of Depolarization-Dependent Outward Currents in the Dendritic Membrane of an Identified Nonspiking Interneuron in Crayfish

Depolarization-dependent outward currents were analyzed using the single-electrode voltage clamp technique in the dendritic membrane of an identified nonspiking interneuron (LDS interneuron) in situ in the terminal abdominal ganglion of crayfish. When the membrane was depolarized by more than 20 mV from the resting potential (65.0 ± 5.7 mV), a transient outward current was observed to be followed by a sustained outward current. Pharmacological experiments revealed that these outward currents were composed of 3 distinct components. A sustained component (I _s) was activated slowly (half rise time > 5 msec) and blocked by 20 mM TEA. A transient component (I _t1) that was activated and inactivated very rapidly (peak time < 2.5 msec, half decay time < 1.2 msec) was also blocked by 20 mM TEA. Another transient component (I _t2) was blocked by 100 M 4AP, activated rapidly (peak time < 10.0 msec) and inactivated slowly (half decay time > 131.8 msec). Two-step pulse experiments have revealed that both sustained and transient components are not inactivated at the resting potential: the half-maximal inactivation was attained at –21.0 mV in I _t1, and –38.0 mV in I _t2. I _s showed no noticeable inactivation. When the membrane was initially held at the resting potential level and clamped to varying potential levels, the half-maximal activation was attained at –36.0 mV in I _s, –31.0 mV in I _t1 and –40.0 mV in I _t2. The activation and inactivation time constants were both voltage dependent. A mathematical model of the LDS interneuron was constructed based on the present electrophysiological records to simulate the dynamic interaction of outward currents during membrane depolarization. The results suggest that those membrane conductances found in this study underlie the outward rectification of the interneuron membrane as well as depolarization-dependent shaping of the excitatory synaptic potential observed in current-clamp experiments.     

The role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: A computational study

Experimental evidences point out the participation of nonsynaptic mechanisms (e.g., fluctuations in extracellular ions) in epileptiform bursting and spreading depression (SD). During these abnormal oscillatory patterns, it is observed an increase of extracellular potassium concentration [K⁺]_o and a decrease of extracellular calcium concentration [Ca²⁺]_o which raises the neuronal excitability. However, whether the high [K⁺]_o triggers and propagates these abnormal neuronal activities or plays a secondary role into this process is unclear. To better understand the influence of extracellular potassium dynamics in these oscillatory patterns, the experimental conditions of high [K⁺]_o and zero [Ca²⁺]_o were replicated in an extended Golomb model where we added important regulatory mechanisms of ion concentration as Na⁺-K⁺ pump, ion diffusion and glial buffering. Within these conditions, simulations of the cell model exhibit seizure-like discharges (ictal bursting). The SD was elicited by the interruption of the Na⁺−K⁺ pump activity, mimicking the effect of cellular hypoxia (an experimental protocol to elicit SD, the hypoxia-induced SD). We used the bifurcation theory and the fast-slow method to analyze the interference of K⁺ dynamics in the cellular excitability. This analysis indicates that the system loses its stability at a high [K⁺]_o, transiting to an elevated state of neuronal excitability. Effects of high [K⁺]_o are observed in different stages of ictal bursting and SD. In the initial stage, the increase of [K⁺]_o creates favorable conditions to trigger both oscillatory patterns. During the neuronal activity, a continuous growth of [K⁺]_o by outward K⁺ flow depresses K⁺ currents in a positive feedback way. At the last stage, due to the depression of K⁺ currents, the Na⁺-K⁺ pump is the main mechanism in the end of neuronal activity. Thus, this work suggests that [K⁺]_o dynamics may play a fundamental role in these abnormal oscillatory patterns.     

Reduction of a model for an Onchidium pacemaker neuron

The eight-variable model for the giant neuron localized in the esophageal ganglia of the marine pulmonate mollusk Onchidium verruculatum is reduced to four-and-three-dimensional systems by regrouping variables with similar time scales. These reduced models replicate the complex behavior including beating, periodic bursting and aperiodic bursting displayed by the original full model when the parameter I _ext representing the intensity of the constant DC current stimulation is varied across a wide range. The complex behavior of the full model arises from the interaction of fast and slow dynamics, and depends on the time scale C _s of the slow dynamics. The four-variable reduced model is constructed independently from the parameter C _s so that it reproduces the two-dimensional bifurcation structure of the full model for the two parameters I _ext and C _s . The three-variable reduced model is derived for a specific value of C _s . The parameters of this model are tuned so that its one-parameter bifurcation diagram for I _ext closely matches that of the full model. Correspondence between bifurcation structures ensures that both reduced models reproduce the various discharge patterns of the full model. Similarity between the full and reduced models is also confirmed by comparing mean firing frequencies and membrane potential waveforms in various regimes. The reduction exposes the factors essential for reproducing the dynamics of the full model; indeed, it shows that the eight variables representing the membrane potential and seven gating variables of six ionic currents in the full model account, in fact, for three basic processes responsible for excitability, post-discharge refractoriness and slow membrane modulation. Received: 7 May 1996 / Accepted 4 December 1997     

Intrinsic bursting activity in the pre-Bötzinger Complex: Role of persistent sodium and potassium currents

Computational models of single pacemaker neuron and neural population in the pre-Bötzinger Complex (pBC) were developed based on the previous models by Butera et al. (1999a,b). Our modeling study focused on the conditions that could define endogenous bursting vs. tonic activity in single pacemaker neurons and population bursting vs. asynchronous firing in populations of pacemaker neurons. We show that both bursting activity in single pacemaker neurons and population bursting activity may be released or suppressed depending on the expression of persistent sodium (I_NaP) and delayed-rectifier potassium (I_K) currents. Specifically, a transition from asynchronous firing to population bursting could be induced by a reduction of I_K via a direct suppression of the potassium conductance or through an elevation of extracellular potassium concentration. Similar population bursting activity could be triggered by an augmentation of I_NaP. These findings are discussed in the context of the possible role of population bursting activity in the pBC in the respiratory rhythm generation in vivo vs. in vitro and during normal breathing in vivo vs. gasping.     

Dynamics of epileptiform activity in mouse hippocampal slices

Increase of the extracellular K⁺ concentration mediates seizure-like synchronized activities in vitro and was proposed to be one of the main factors underlying epileptogenesis in some types of seizures in vivo. While underlying biophysical mechanisms clearly involve cell depolarization and overall increase in excitability, it remains unknown what qualitative changes of the spatio-temporal network dynamics occur after extracellular K⁺ increase. In this study, we used multi-electrode recordings from mouse hippocampal slices to explore changes of the network activity during progressive increase of the extracellular K⁺ concentration. Our analysis revealed complex spatio-temporal evolution of epileptiform activity and demonstrated a sequence of state transitions from relatively simple network bursts into complex bursting, with multiple synchronized events within each burst. We describe these transitions as qualitative changes of the state attractors, constructed from experimental data, mediated by elevation of extracellular K⁺ concentration.     

Magnesium induces neuronal apoptosis by suppressing excitability

In clinical obstetrics, magnesium sulfate (MgSO₄) use is widespread, but effects on brain development are unknown. Many agents that depress neuronal excitability increase developmental neuroapoptosis. In this study, we used dissociated cultures of rodent hippocampus to examine the effects of Mg⁺⁺ on excitability and survival. Mg⁺⁺-induced caspase-3-associated cell loss at clinically relevant concentrations. Whole-cell patch-clamp techniques measured Mg⁺⁺ effects on action potential threshold, action potential peak amplitude, spike number and changes in resting membrane potential. Mg⁺⁺ depolarized action potential threshold, presumably from surface charge screening effects on voltage-gated sodium channels. Mg⁺⁺ also decreased the number of action potentials in response to fixed current injection without affecting action potential peak amplitude. Surprisingly, Mg⁺⁺ also depolarized neuronal resting potential in a concentration-dependent manner with a +5.2 mV shift at 10 mM. Voltage ramps suggested that Mg⁺⁺ blocked a potassium conductance contributing to the resting potential. In spite of this depolarizing effect of Mg⁺⁺, the net inhibitory effect of Mg⁺⁺ nearly completely silenced neuronal network activity measured with multielectrode array recordings. We conclude that although Mg⁺⁺ has complex effects on cellular excitability, the overall inhibitory influence of Mg⁺⁺ decreases neuronal survival. Taken together with recent in vivo evidence, our results suggest that caution may be warranted in the use of Mg⁺⁺ in clinical obstetrics and neonatology.     

The potassium A-current,low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I _A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI _A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI _A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _A parameters given by Connor and Stevens for the crab axon. When theI _A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _A decreases and as inactivation rate decreases. We also study howI _A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.