A Threshold Equation for Action Potential Initiation

In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation function near spike initiation (about −55 mV), while its parameters are adjusted near half-activation voltage (about −30 mV), which might explain why many models exhibit little threshold variability, contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.     

Statistical Analysis of Membrane Potential Fluctuations: Relation with Presynaptic Spike Train

In a study of integration at the single neuron level, the relationships between the postsynaptic membrane potential and the presynaptic spike train were analyzed. Fluctuations in membrane potential of neurons in the visceral ganglion of Aplysia were measured and described by histograms. The histogram estimates the probability density function of the membrane potential. Comparisons were made among histograms when there was no synaptic input, and when there was a single input in which variations were made in the PSP (postsynaptic potential) sign, i.e. excitatory or inhibitory, and arrival statistics, e.g. slow or fast, regular, Poisson-like, or patterned. This was examined in cells where the membrane potential was constant and in cells in which there was spontaneous pacemaker activity. The form of the histogram depended on whether the neuron was spontaneously quiescent or a pacemaker, or whether it received presynaptic input and, if it did, on the sign and temporal characteristics of such input. From such histograms the mean firing rate of output spike trains can be predicted; additional information of a temporal nature is required, however, to predict features of the interval structure of the output train. Suggestions are made concerning the way the nervous system might utilize the information summarized in the membrane potential histogram.     

神经元放电阈值的可变性及其意义

神经元能够将不同时空模式的突触输入转化为时序精确的动作电位输出，这种灵活、可靠的信息编码方式是神经集群在动态环境或特定任务下产生所需活动模式的重要基础。动作电位的产生遵循全或无规律，只有当细胞膜电压达到放电阈值时，神经元才产生动作电位。放电阈值在细胞内和细胞间具有高度可变性，具体动态依赖于刺激输入和放电历史。特别是，放电阈值对动作电位起始前的膜电压变化十分敏感，这种状态依赖性产生的生物物理根源包括Na⁺失活和K⁺激活。在绝大多数神经元中，动作电位的触发位置是轴突起始端，这个位置处的阈值可变性是决定神经元对时空输入转化规律的关键因素。但是，电生理实验中动作电位的记录位置却通常是胞体或近端树突，此处的阈值可变性高于轴突起始端，而其产生的重要根源是轴突动作电位的反向传播。基于胞体测量的相关研究显示，放电阈值动态能够增强神经元的时间编码、特征选择、增益调控和同时侦测能力。本文首先介绍放电阈值的概念及量化方法，然后详细梳理近年来国内外关于放电阈值可变性及产生根源的研究进展，在此基础上归纳总结放电阈值可变性对神经元编码的重要性，最后对未来放电阈值的研究方向进行展望。     

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.     

Changes in potentiation and timing of sensorimotor transmission after adaptation of a postsynaptic transient outward current in a simulated cortical neuron

How adaptation of a postsynaptic transient outward current might affect the efficacy of sensorimotor transmission was investigated. The transmission signals that were studied were a 5 ms conditioned stimulus (CS) and a 60 ms US drawn from intracellularly recorded, depolarizing postsynaptic potentials (PSPs) elicited in pyramidal neurons of the cat motor cortex by a click CS and a glabella tap US, respectively. SPICE, a program used to analyze electrical circuits, was used to simulate the cortical neuron containing the adaptive outward current. Changes in the magnitude and latency of rise to firing threshold of the PSPs were compared i) after presynaptic augmentation of a CS input in the absence of an adaptive postsynaptic current and ii) after decreasing the magnitude of an adaptive postsynaptic current that was rapidly activated by depolarization. Effects of short (6 ms) and long (24 ms) inactivation time constants of the postsynaptic current were also studied. In both presynaptic adaptation and postsynaptic adaptation, the potentiation of the magnitude of the CS-induced PSP was similar, with the latency to threshold being reduced by " 1 ms in both cases. The effects on the US PSP differed. Presynaptic adaptation affecting the CS had no effect on the US. Adaptation of the CS by a postsynaptic outward current with a 6 ms inactivation time constant, reduced the latency to threshold of an EPSP from a nearby US synapse by up to 6 ms by augmenting the initial portion of the slowly rising US-induced PSP. Adaptation of a postsynaptic current with a 24 ms inactivation time constant reduced the latency of response to the US PSP by up to 16 ms. When the US synapse was relocated to the soma, the reduction in US latency caused by adaptation of the outward current at the CS synapse was reduced by up to one half. The latency of slowly rising components of integrated synaptic responses to compound CSs of > 5 ms duration from multiple synaptic inputs would be expected to show reductions corresponding to those of the US. We conclude that potentiation of synaptic transmission by adaptation of a postsynaptic outward current can result in reductions of latency of sensorimotor transmission that can significantly affect the timing and accuracy of controlled motor tasks. These effects depend significantly on the locations of the synaptic inputs within the cell.     

Changes in potentiation and timing of sensorimotor transmission after adaptation of a postsynaptic transient outward current in a simulated cortical neuron

How adaptation of a postsynaptic transient outward current might affect the efficacy of sensorimotor transmission was investigated. The transmission signals that were studied were a 5 ms conditioned stimulus (CS) and a 60 ms US drawn from intracellularly recorded, depolarizing postsynaptic potentials (PSPs) elicited in pyramidal neurons of the cat motor cortex by a click CS and a glabella tap US, respectively. SPICE, a program used to analyze electrical circuits, was used to simulate the cortical neuron containing the adaptive outward current. Changes in the magnitude and latency of rise to firing threshold of the PSPs were compared i) after presynaptic augmentation of a CS input in the absence of an adaptive postsynaptic current and ii) after decreasing the magnitude of an adaptive postsynaptic current that was rapidly activated by depolarization. Effects of short (6 ms) and long (24 ms) inactivation time constants of the postsynaptic current were also studied. In both presynaptic adaptation and postsynaptic adaptation, the potentiation of the magnitude of the CS-induced PSP was similar, with the latency to threshold being reduced by < or = 1 ms in both cases. The effects on the US PSP differed. Presynaptic adaptation affecting the CS had no effect on the US. Adaptation of the CS by a postsynaptic outward current with a 6 ms inactivation time constant, reduced the latency to threshold of an EPSP from a nearby US synapse by up to 6 ms by augmenting the initial portion of the slowly rising US-induced PSP. Adaptation of a postsynaptic current with a 24 ms inactivation time constant reduced the latency of response to the US PSP by up to 16 ms. When the US synapse was relocated to the soma, the reduction in US latency caused by adaptation of the outward current at the CS synapse was reduced by up to one half. The latency of slowly rising components of integrated synaptic responses to compound CSs of > 5 ms duration from multiple synaptic inputs would be expected to show reductions corresponding to those of the US. We conclude that potentiation of synaptic transmission by adaptation of a postsynaptic outward current can result in reductions of latency of sensorimotor transmission that can significantly affect the timing and accuracy of controlled motor tasks. These effects depend significantly on the locations of the synaptic inputs within the cell.     

Adaptive Spike Threshold Enables Robust and Temporally Precise Neuronal Encoding

Neural processing rests on the intracellular transformation of information as synaptic inputs are translated into action potentials. This transformation is governed by the spike threshold, which depends on the history of the membrane potential on many temporal scales. While the adaptation of the threshold after spiking activity has been addressed before both theoretically and experimentally, it has only recently been demonstrated that the subthreshold membrane state also influences the effective spike threshold. The consequences for neural computation are not well understood yet. We address this question here using neural simulations and whole cell intracellular recordings in combination with information theoretic analysis. We show that an adaptive spike threshold leads to better stimulus discrimination for tight input correlations than would be achieved otherwise, independent from whether the stimulus is encoded in the rate or pattern of action potentials. The time scales of input selectivity are jointly governed by membrane and threshold dynamics. Encoding information using adaptive thresholds further ensures robust information transmission across cortical states i.e. decoding from different states is less state dependent in the adaptive threshold case, if the decoding is performed in reference to the timing of the population response. Results from in vitro neural recordings were consistent with simulations from adaptive threshold neurons. In summary, the adaptive spike threshold reduces information loss during intracellular information transfer, improves stimulus discriminability and ensures robust decoding across membrane states in a regime of highly correlated inputs, similar to those seen in sensory nuclei during the encoding of sensory information.     

A cross-interval spike train analysis: the correlation between spike generation and temporal integration of doublets

A stochastic spike train analysis technique is introduced to reveal the correlation between the firing of the next spike and the temporal integration period of two consecutive spikes (i.e., a doublet). Statistics of spike firing times between neurons are established to obtain the conditional probability of spike firing in relation to the integration period. The existence of a temporal integration period is deduced from the time interval between two consecutive spikes fired in a reference neuron as a precondition to the generation of the next spike in a compared neuron. This analysis can show whether the coupled spike firing in the compared neuron is correlated with the last or the second-to-last spike in the reference neuron. Analysis of simulated and experimentally recorded biological spike trains shows that the effects of excitatory and inhibitory temporal integration are extracted by this method without relying on any subthreshold potential recordings. The analysis also shows that, with temporal integration, a neuron driven by random firing patterns can produce fairly regular firing patterns under appropriate conditions. This regularity in firing can be enhanced by temporal integration of spikes in a chain of polysynaptically connected neurons. The bandpass filtering of spike firings by temporal integration is discussed. The results also reveal that signal transmission delays may be attributed not just to conduction and synaptic delays, but also to the delay time needed for temporal integration. Received: 3 March 1997 / Accepted in revised form: 6 November 1997     

Membrane Potential Fluctuations Determine the Precision of Spike Timing and Synchronous Activity: A Model Study

It is much debated on what time scale information is encoded by neuronal spike activity. With a phenomenological model that transforms time-dependent membrane potential fluctuations into spike trains, we investigate constraints for the timing of spikes and for synchronous activity of neurons with common input. The model of spike generation has a variable threshold that depends on the time elapsed since the previous action potential and on the preceding membrane potential changes. To ensure that the model operates in a biologically meaningful range, the model was adjusted to fit the responses of a fly visual interneuron to motion stimuli. The dependence of spike timing on the membrane potential dynamics was analyzed. Fast membrane potential fluctuations are needed to trigger spikes with a high temporal precision. Slow fluctuations lead to spike activity with a rate about proportional to the membrane potential. Thus, for a given level of stochastic input, the frequency range of membrane potential fluctuations induced by a stimulus determines whether a neuron can use a rate code or a temporal code. The relationship between the steepness of membrane potential fluctuations and the timing of spikes has also implications for synchronous activity in neurons with common input. Fast membrane potential changes must be shared by the neurons to produce synchronous activity.     

Effects of phase on homeostatic spike rates

Recent experimental results by Talathi et al. (Neurosci Lett 455:145–149, 2009) showed a divergence in the spike rates of two types of population spike events, representing the putative activity of the excitatory and inhibitory neurons in the CA1 area of an animal model for temporal lobe epilepsy. The divergence in the spike rate was accompanied by a shift in the phase of oscillations between these spike rates leading to a spontaneous epileptic seizure. In this study, we propose a model of homeostatic synaptic plasticity which assumes that the target spike rate of populations of excitatory and inhibitory neurons in the brain is a function of the phase difference between the excitatory and inhibitory spike rates. With this model of homeostatic synaptic plasticity, we are able to simulate the spike rate dynamics seen experimentally by Talathi et al. in a large network of interacting excitatory and inhibitory neurons using two different spiking neuron models. A drift analysis of the spike rates resulting from the homeostatic synaptic plasticity update rule allowed us to determine the type of synapse that may be primarily involved in the spike rate imbalance in the experimental observation by Talathi et al. We find excitatory neurons, particularly those in which the excitatory neuron is presynaptic, have the most influence in producing the diverging spike rates and causing the spike rates to be anti-phase. Our analysis suggests that the excitatory neuronal population, more specifically the excitatory to excitatory synaptic connections, could be implicated in a methodology designed to control epileptic seizures.     

Pre & postsynaptic tuning of action potential timing by spontaneous GABAergic activity

Frequency and timing of action potential discharge are key elements for coding and transfer of information between neurons. The nature and location of the synaptic contacts, the biophysical parameters of the receptor-operated channels and their kinetics of activation are major determinants of the firing behaviour of each individual neuron. Ultimately the intrinsic excitability of each neuron determines the input-output function. Here we evaluate the influence of spontaneous GABAergic synaptic activity on the timing of action potentials in Layer 2/3 pyramidal neurones in acute brain slices from the somatosensory cortex of young rats. Somatic dynamic current injection to mimic synaptic input events was employed, together with a simple computational model that reproduce subthreshold membrane properties. Besides the well-documented control of neuronal excitability, spontaneous background GABAergic activity has a major detrimental effect on spike timing. In fact, GABA(A) receptors tune the relationship between the excitability and fidelity of pyramidal neurons via a postsynaptic (the reversal potential for GABA(A) activity) and a presynaptic (the frequency of spontaneous activity) mechanism. GABAergic activity can decrease or increase the excitability of pyramidal neurones, depending on the difference between the reversal potential for GABA(A) receptors and the threshold for action potential. In contrast, spike time jitter can only be increased proportionally to the difference between these two membrane potentials. Changes in excitability by background GABAergic activity can therefore only be associated with deterioration of the reliability of spike timing.     

Correlation of Transmitter Release with Membrane Properties of the Presynaptic Fiber of the Squid Giant Synapse

Depolarization of the presynaptic terminal by current produced a postsynaptic potential (PSP) which increased with increasing presynaptic polarization and then reached a plateau. Iontophoretic injection of tetraethylammonium ions (TEA) into the presynaptic axon near the terminal produced a prolonged presynaptic spike. The resulting PSP is increased in size and its time course closely followed that of the presynaptic spike. The presynaptic fiber no longer exhibited rectification and strong depolarizations revealed that the PSP reached a maximum with about 110 mv depolarization. Further depolarization produced a decrease in PSP amplitude and finally transmission was blocked. However, a PSP then always appeared on withdrawal of the depolarizing current. Under the conditions of these experiments, the PSP could be considered a direct measure of transmitter release. Bathing the TEA-injected synapse with concentrations of tetrodotoxin (TTX) sufficient to block spike activity in both pre- and postsynaptic axons did not greatly modify postsynaptic electrogenesis. However, doubling TTX concentration reversibly blocked PSP. Thus the permeability changes to Na and K accompanying the spike do not appear necessary for transmitter release. Some other processes related to the level of presynaptic polarization must be involved to explain the data. The inhibition of transmitter release by strong depolarizations appears to be related to Ca action. A membrane Ca current may also be necessary for normal transmitter release.     

Dynamics of action potential initiation in the GABAergic thalamic reticular nucleus in vivo

Understanding the neural mechanisms of action potential generation is critical to establish the way neural circuits generate and coordinate activity. Accordingly, we investigated the dynamics of action potential initiation in the GABAergic thalamic reticular nucleus (TRN) using in vivo intracellular recordings in cats in order to preserve anatomically-intact axo-dendritic distributions and naturally-occurring spatiotemporal patterns of synaptic activity in this structure that regulates the thalamic relay to neocortex. We found a wide operational range of voltage thresholds for action potentials, mostly due to intrinsic voltage-gated conductances and not synaptic activity driven by network oscillations. Varying levels of synchronous synaptic inputs produced fast rates of membrane potential depolarization preceding the action potential onset that were associated with lower thresholds and increased excitability, consistent with TRN neurons performing as coincidence detectors. On the other hand the presence of action potentials preceding any given spike was associated with more depolarized thresholds. The phase-plane trajectory of the action potential showed somato-dendritic propagation, but no obvious axon initial segment component, prominent in other neuronal classes and allegedly responsible for the high onset speed. Overall, our results suggest that TRN neurons could flexibly integrate synaptic inputs to discharge action potentials over wide voltage ranges, and perform as coincidence detectors and temporal integrators, supported by a dynamic action potential threshold.     

Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

Biophysical mechanism of spike threshold dependence on the rate of rise of the membrane potential by sodium channel inactivation or subthreshold axonal potassium current

Spike threshold filters incoming inputs and thus gates activity flow through neuronal networks. Threshold is variable, and in many types of neurons there is a relationship between the threshold voltage and the rate of rise of the membrane potential (dVm/dt) leading to the spike. In primary sensory cortex this relationship enhances the sensitivity of neurons to a particular stimulus feature. While Na⁺ channel inactivation may contribute to this relationship, recent evidence indicates that K⁺ currents located in the spike initiation zone are crucial. Here we used a simple Hodgkin-Huxley biophysical model to systematically investigate the role of K⁺ and Na⁺ current parameters (activation voltages and kinetics) in regulating spike threshold as a function of dVm/dt. Threshold was determined empirically and not estimated from the shape of the Vm prior to a spike. This allowed us to investigate intrinsic currents and values of gating variables at the precise voltage threshold. We found that Na⁺ inactivation is sufficient to produce the relationship provided it occurs at hyperpolarized voltages combined with slow kinetics. Alternatively, hyperpolarization of the K⁺ current activation voltage, even in the absence of Na⁺ inactivation, is also sufficient to produce the relationship. This hyperpolarized shift of K⁺ activation allows an outward current prior to spike initiation to antagonize the Na⁺ inward current such that it becomes self-sustaining at a more depolarized voltage. Our simulations demonstrate parameter constraints on Na⁺ inactivation and the biophysical mechanism by which an outward current regulates spike threshold as a function of dVm/dt.     

Presumed mechanisms of a long-term increase in the intrinsic excitability of cerebellar granule cells: A model study

Persistent use-dependent changes in the intrinsic neuronal excitability determine the long-term dynamics of the activity of these neurons. In synergy with the long-lasting modification of synaptic transmission, such changes in the excitability presumably contribute to the formation of a memory trace in the brain. Nevertheless, neither particular transmembrane ion conductances implicated in the intrinsic plasticity nor the mechanisms of regulation of such conductances have been identified in most neurons where this plasticity was observed. In our model study, we tried to determine those membrane conductances in cerebellar granule cells (GrCs) whose changes can result in a persistent increase in the input resistance and a decrease in the spike threshold observed after high-frequency stimulation of presynaptic neurons. For this purpose, published experimental results were simulated with the use of a slightly modified model of the electroresponsiveness of rat cerebellar GrCs. It was concluded that experimentally observed changes in the input resistance of the neuron, in the minimum current step needed to fire action potentials (APs), in the spike threshold, in the average spike frequency, and in the delay of the first spike may be caused only by changes in the background voltage-independent potassium conductance and persistent sodium conductance. Hyperpolarization-directed shifts in the activation and inactivation curves of fast sodium channels are also possible. The observed changes in the intrinsic excitability evoke the shift in the peak of the frequency-response curve in such a manner that it becomes close to the frequency of oscillations recorded in the cerebellar granular layer during realization of voluntary movements. Neirofiziologiya/Neurophysiology, Vol. 38, No. 2, pp. 119–130, March–April, 2006.     

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker–Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).     

Action potential shape change in an electrically coupled network during propagation: a computer simulation

We applied compartmental computer modeling to test a model of spike shape change in the jellyfish, Polyorchis penicillatus, to determine whether adaptive spike shortening can be attributed to the inactivation properties of a potassium channel. We modeled the jellyfish outer nerve-ring as a continuous linear segment, using ion channel and membrane properties derived in earlier studies. The model supported action potentials that shortened as they propagated away from the site of initiation and this was found to be largely independent of potassium channel inactivation. Spike broadening near the site of initiation was found to be due to a depolarization plateau that collapsed as two spikes spread from the point of initiation. The lifetime of this plateau was found to depend critically on the inward current flux and the space constant of the membrane. These data suggest that the spike shape changes may be due not only to potassium channel inactivation, but also to the passive properties of the membrane.