Stochastic simulations on the reliability of action potential propagation in thin axons

It is generally assumed that axons use action potentials (APs) to transmit information fast and reliably to synapses. Yet, the reliability of transmission along fibers below 0.5 μm diameter, such as cortical and cerebellar axons, is unknown. Using detailed models of rodent cortical and squid axons and stochastic simulations, we show how conduction along such thin axons is affected by the probabilistic nature of voltage-gated ion channels (channel noise). We identify four distinct effects that corrupt propagating spike trains in thin axons: spikes were added, deleted, jittered, or split into groups depending upon the temporal pattern of spikes. Additional APs may appear spontaneously; however, APs in general seldom fail (<1%). Spike timing is jittered on the order of milliseconds over distances of millimeters, as conduction velocity fluctuates in two ways. First, variability in the number of Na channels opening in the early rising phase of the AP cause propagation speed to fluctuate gradually. Second, a novel mode of AP propagation (stochastic microsaltatory conduction), where the AP leaps ahead toward spontaneously formed clusters of open Na channels, produces random discrete jumps in spike time reliability. The combined effect of these two mechanisms depends on the pattern of spikes. Our results show that axonal variability is a general problem and should be taken into account when considering both neural coding and the reliability of synaptic transmission in densely connected cortical networks, where small synapses are typically innervated by thin axons. In contrast we find that thicker axons above 0.5 μm diameter are reliable.     

Identifying critical regions for spike propagation in axon segments

Morphological reconstructions of axon segments reveal the abundance of geometrical ultrastructures that can dramatically affect the propagation of Action Potentials (AP). Moreover, deformations and swellings in axons resulting from brain traumas are associated to many neural dysfunctions and disorders. Our aim is to develop a computational framework to distinguish between geometrical enlargements that lead to minor changes in propagation from those that result in critical phenomenon such as reflection or blockage of the original traveling spike. We use a few geometrical parameters to model a prototypical shaft enlargement and explore the parameter space characterizing all possible propagation regimes and dynamics in an unmylienated AP model. Contrary to earlier notions that large diameter increases mostly lead to blocking, we demonstrate transmission is stable provided the geometrical changes occur in a slow manner. Our method also identifies a narrow range of parameters leading to a reflection regime. The distinction between these three regimes can be evaluated by a simple function of the geometrical parameters inferred through numerical simulations. We suggest that evaluating this function along axon segments can detect regions most susceptible to (i) transmission failure due to perturbations, (ii) structural plasticity, (iii) critical swellings caused by brain traumas and/or (iv) neurological disorders associated with the break down of spike train propagation.     

Axons as computing devices: basic insights gained from models.

Detailed models of single neurons are typically focused on the dendritic tree and ignore the axonal tree, assuming that the axon is a simple transmission line. In the last 40 years, however, several theoretical and experimental studies have suggested that axons could implement information processing tasks by exploiting: 1) the time delay in action potential (AP) propagation along the axon; 2) the differential filtering of APs into the axonal subtrees; and 3) their activity-dependent excitability. Models for axonal trees have attempted to examine the feasibility of these ideas. However, because the physiological and anatomical data on axons are seriously limited, realistic models of axons have not been developed. The present paper summarizes the main insights that were gained from simplified models of axons; it also highlights the stochastic nature of axons, a topic that was largely neglected in classical models of axons. The advance of new experimental techniques makes it now possible to pay a very close experimental visit to axons. Theoretical tools and fast computers enable to go beyond the simplified models and to construct realistic models of axons. When tightly linked, experiments and theory will help to unravel how axons share the information processing tasks that single neurons implement.     

中枢神经系统内神经元信号处理(英文)

一种新颖的轴突断端(axon bleb)膜片钳记录方法大力促进了中枢神经系统轴突功能的研究。我们的工作应用这一方法揭示了大脑皮层锥体神经元的数码信号(具全或无特性的动作电位)的爆发和传播机制。在轴突始段(axon initial segment,AIS)远端高密度聚集的低阈值Na+通道亚型Nav1.6决定动作电位的爆发;而在AIS近端高密度聚集的高阈值Na+通道亚型Nav1.2促进动作电位向胞体和树突的反向传播。应用胞体和轴突的同时记录,我们发现胞体阈下膜电位的变化可以在轴突上传播较长的距离并可到达那些离胞体较近的突触前终末。进一步的研究证明了胞体膜电位的变化调控动作电位触发的突触传递,该膜电位依赖的突触传递是一种模拟式的信号传递。轴突上一类特殊K+通道(Kv1)的活动调制动作电位的波形,特别是其波宽,从而调控各种突触前膜电位水平下突触强度的变化。突触前终末的背景Ca2+浓度也可能参与模拟信号的传递。这些发现深化了我们对中枢神经系统内神经信号处理基本原理的认识,进而帮助我们理解脑如何工作。     

The corpus callosum in primates: processing speed of axons and the evolution of hemispheric asymmetry

Interhemispheric communication may be constrained as brain size increases because of transmission delays in action potentials over the length of axons. Although one might expect larger brains to have progressively thicker axons to compensate, spatial packing is a limiting factor. Axon size distributions within the primate corpus callosum (CC) may provide insights into how these demands affect conduction velocity. We used electron microscopy to explore phylogenetic variation in myelinated axon density and diameter of the CC from 14 different anthropoid primate species, including humans. The majority of axons were less than 1 µm in diameter across all species, indicating that conduction velocity for most interhemispheric communication is relatively constant regardless of brain size. The largest axons within the upper 95th percentile scaled with a progressively higher exponent than the median axons towards the posterior region of the CC. While brain mass among the primates in our analysis varied by 97-fold, estimates of the fastest cross-brain conduction times, as conveyed by axons at the 95th percentile, varied within a relatively narrow range between 3 and 9 ms across species, whereas cross-brain conduction times for the median axon diameters differed more substantially between 11 and 38 ms. Nonetheless, for both size classes of axons, an increase in diameter does not entirely compensate for the delay in interhemispheric transmission time that accompanies larger brain size. Such biophysical constraints on the processing speed of axons conveyed by the CC may play an important role in the evolution of hemispheric asymmetry.     

Action Potential Energy Efficiency Varies Among Neuron Types in Vertebrates and Invertebrates

The initiation and propagation of action potentials (APs) places high demands on the energetic resources of neural tissue. Each AP forces ATP-driven ion pumps to work harder to restore the ionic concentration gradients, thus consuming more energy. Here, we ask whether the ionic currents underlying the AP can be predicted theoretically from the principle of minimum energy consumption. A long-held supposition that APs are energetically wasteful, based on theoretical analysis of the squid giant axon AP, has recently been overturned by studies that measured the currents contributing to the AP in several mammalian neurons. In the single compartment models studied here, AP energy consumption varies greatly among vertebrate and invertebrate neurons, with several mammalian neuron models using close to the capacitive minimum of energy needed. Strikingly, energy consumption can increase by more than ten-fold simply by changing the overlap of the Na⁺ and K⁺ currents during the AP without changing the APs shape. As a consequence, the height and width of the AP are poor predictors of energy consumption. In the Hodgkin–Huxley model of the squid axon, optimizing the kinetics or number of Na⁺ and K⁺ channels can whittle down the number of ATP molecules needed for each AP by a factor of four. In contrast to the squid AP, the temporal profile of the currents underlying APs of some mammalian neurons are nearly perfectly matched to the optimized properties of ionic conductances so as to minimize the ATP cost.     

The Nodes of Ranvier: Molecular Assembly and Maintenance

Action potential (AP) propagation in myelinated nerves requires clustered voltage gated sodium and potassium channels. These channels must be specifically localized to nodes of Ranvier where the AP is regenerated. Several mechanisms have evolved to facilitate and ensure the correct assembly and stabilization of these essential axonal domains. This review highlights the current understanding of the axon intrinsic and glial extrinsic mechanisms that control the formation and maintenance of the nodes of Ranvier in both the peripheral nervous system (PNS) and central nervous system (CNS).Axons conduct electrical signals, called action potentials (APs), among neurons in a circuit in response to sensory input, and between motor neurons and muscles. In mammals and other vertebrates, many axons are myelinated. Myelin, made by Schwann cells and oligodendrocytes in the peripheral nervous system (PNS) and central nervous system (CNS), respectively, is a multilamellar sheet of glial membrane that wraps around axons to increase transmembrane resistance and decrease membrane capacitance. Although myelin is traditionally viewed as a passive contributor to nervous system function, it is now recognized that myelinating glia also play many active roles including regulation of axon diameter, axonal energy metabolism, and the clustering of ion channels at gaps in the myelin sheath called nodes of Ranvier. Together, the active and passive properties conferred on axons by myelin, result in axons with high AP conduction velocities, low metabolic demands, and reduced space requirements as compared with unmyelinated axons. Thus, myelin and the clustering of ion channels in axons permitted the evolution of the complex nervous systems found in vertebrates. This review highlights the current understanding of the axonal intrinsic and glial extrinsic mechanisms that control the formation and maintenance of the nodes of Ranvier in both the PNS and CNS.     

First node of Ranvier facilitates high-frequency burst encoding

In central neurons the first node of Ranvier is located at the first axonal branchpoint, ～ 100 μm from the axon initial segment where synaptic inputs are integrated and converted into action potentials (APs). Whether the first node contributes to this signal transformation is not well understood. Here it was found that in neocortical layer 5 axons, the first branchpoint is required for intrinsic high-frequency (≥ 100 Hz) AP bursts. Furthermore, block of nodal Na(+) channels or axotomy of the first node in intrinsically bursting neurons depolarized the somatic AP voltage threshold (～ 5 mV) and eliminated APs selectively within a high-frequency cluster in response to steady currents or simulated synaptic inputs. These results indicate that nodal persistent Na(+) current exerts an anterograde influence on AP initiation in the axon initial segment, revealing a computational role of the first node of Ranvier beyond conduction of the propagating AP.     

An analysis of the relationships between subthreshold electrical properties and excitability in skeletal muscle

Skeletal muscle activation requires action potential (AP) initiation followed by its sarcolemmal propagation and tubular excitation to trigger Ca(2+) release and contraction. Recent studies demonstrate that ion channels underlying the resting membrane conductance (G(M)) of fast-twitch mammalian muscle fibers are highly regulated during muscle activity. Thus, onset of activity reduces G(M), whereas prolonged activity can markedly elevate G(M). Although these observations implicate G(M) regulation in control of muscle excitability, classical theoretical studies in un-myelinated axons predict little influence of G(M) on membrane excitability. However, surface membrane morphologies differ markedly between un-myelinated axons and muscle fibers, predominantly because of the tubular (t)-system of muscle fibers. This study develops a linear circuit model of mammalian muscle fiber and uses this to assess the role of subthreshold electrical properties, including G(M) changes during muscle activity, for AP initiation, AP propagation, and t-system excitation. Experimental observations of frequency-dependent length constant and membrane-phase properties in fast-twitch rat fibers could only be replicated by models that included t-system luminal resistances. Having quantified these resistances, the resulting models showed enhanced conduction velocity of passive current flow also implicating elevated AP propagation velocity. Furthermore, the resistances filter passive currents such that higher frequency current components would determine sarcolemma AP conduction velocity, whereas lower frequency components excite t-system APs. Because G(M) modulation affects only the low-frequency membrane impedance, the G(M) changes in active muscle would predominantly affect neuromuscular transmission and low-frequency t-system excitation while exerting little influence on the high-frequency process of sarcolemmal AP propagation. This physiological role of G(M) regulation was increased by high Cl(-) permeability, as in muscle endplate regions, and by increased extracellular [K(+)], as observed in working muscle. Thus, reduced G(M) at the onset of exercise would enhance t-system excitation and neuromuscular transmission, whereas elevated G(M) after sustained activity would inhibit these processes and thereby accentuate muscle fatigue.     

Intraretinal axon diameter: a single cell analysis in the marmoset (Callithrix jacchus)

We have labelled individual retinal ganglion cells of a New World primate, the common marmoset (Callithrix jacchus) with neurobiotin and then measured axon, soma and dendritic field diameter. A total of 111 cells were analysed (62 parasol cells, 22 midget cells, 16 hedge cells and 11 small bistratified cells). When all retinal ganglion cells were grouped together axon diameter was positively correlated to soma diameter. When analysed according to cell class only midget cells showed a positive correlation between soma size and mean axon diameter. Dendritic field diameter and mean axon diameter of both parasol and midget cells showed significant correlations. Axon diameter is not constant along the intraretinal length of the axon and the rate of change in diameter appears to be related to the cell class and the initial size of the axon. Midget cell axons showed a rapid increase of up to 20% over the first 200 μm in contrast to parasol cell axons which increased more slowly over this distance but then showed a marked increase in diameter of up to 40% over the next 450 μm. However, axon diameter did not remain at these increased diameters but decreased at greater distances from the soma. The degree to which an axon changes its diameter is related to retinal ganglion cell class and the initial size of the axon. We postulate that these variations in intraretinal axon diameter may have a direct influence on conduction velocity and reflect a compensatory mechanism to minimise spatiotemporal dispersion along the visual pathway.     

The time course of the evoked secretion of the mediator quanta in various regions of the frog motor nerve ending

Apart from the fact that the gradient of the velocity of the AP propagation along the nerve terminal and the intensity of secretion do exist, the kinetics of a quanta transmitter release may also be revealed in different parts of the terminal. The velocity of the propagation and the minimum sympatric delay tend to diminish along with moving away from the myelinated part of axon, whereas the synchronicity of the quanta release rises. The distinctions in the time course of secretion in different parts of the terminal were amplified when the calcium ion concentration in the medium was enhanced. The observed peculiarities of the secretion kinetics in different regions of nerve ending seem to compensate for diminishing of the amplitude of multiquantal endplate current.     

Theoretical study of transmembrane and extracellular potentials under propagation block due to geometrical inhomogeneity

A mathematical model was used to study transmembrane and extracellular potentials produced by active geometrically inhomogeneous excitable structures under conditions of propagation block. The structures were electrical analogues of intact or damaged unmyelinated nerve fibres, of the soma to axon transition, or of branching axons or dendrites. It was shown that: (1) damage to a cell is equivalent to the presence of a geometrical inhomogeneity, namely of a region of increased diameter; (2) propagation block caused by a geometrical inhomogeneity, results in; (a) a sharp decrease in the calculated transmembrane potential amplitude not only for the blocked region but also before it; (b) a considerable increase in the amplitude of both the negative phase of extracellular potentials at the points of the volume conductor preceding the blocked region and the first positive phase at points in the proximity of the region; (c) a more pronounced increase in the first positive phase amplitude at small radial distances, if the geometrical inhomogeneity is short compared with the length constant (gamma); (3) the membrane damage results in recording of potentials resembling "giant" ones.     

Axonal Noise as a Source of Synaptic Variability

Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside synapses, yet recent advances in experimental methods and biophysical understanding have led us to reconsider the role of axons as highly reliable transmission channels. We show that in many thin axons of our brain, the action potential (AP) waveform and thus the Ca⁺⁺ signal controlling vesicle release at synapses will be significantly affected by the inherent variability of ion channel gating. We investigate how and to what extent fluctuations in the AP waveform explain observed PSP variability. Using both biophysical theory and stochastic simulations of central and peripheral nervous system axons from vertebrates and invertebrates, we show that channel noise in thin axons (<1 µm diameter) causes random fluctuations in AP waveforms. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by up to 20 mV and 0.5 ms while a single AP propagates in C-fibre axons. We show how AP height and width variabilities increase with a ¾ power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find for example that for mammalian unmyelinated axons with 0.2 µm diameter (matching cerebellar parallel fibres) axonal noise alone can explain half of the PSP variability in cerebellar synapses. We conclude that axonal variability may have considerable impact on synaptic response variability. Thus, in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of presynaptic neurons, causes of variability may have been confounded. We thereby show how bottom-up aggregation of molecular noise sources contributes to our understanding of variability observed at higher levels of biological organisation.     

Study using the Noble model of excitation propagation through a branching node]

Excitation propagation through a branching node with 3 thin branches fused into a thick one has been studied on Noble model. When the ratio of the diameters of thin and thick branches equals 1 : 10 the propagation is blocked. When the diameter ratio is 1 : 8 and excitation comes asynchroneously (along one of input branches 10 msec later than along two others) a 30 msec delay of propagation is observed. The importance of the results obtained for explaining atrio-ventricular delay is discussed.     

Optical Coherence Tomography Phase Measurement of Transient Changes in Squid Giant Axons During Activity

Noncontact optical measurements reveal that transient changes in squid giant axons are associated with action potential propagation and altered under different environmental (i.e., temperature) and physiological (i.e., ionic concentrations) conditions. Using a spectral-domain optical coherence tomography system, which produces real-time cross-sectional images of the axon in a nerve chamber, axonal surfaces along a depth profile are monitored. Differential phase analyses show transient changes around the membrane on a millisecond timescale, and the response is coincident with the arrival of the action potential at the optical measurement area. Cooling the axon slows the electrical and optical responses and increases the magnitude of the transient signals. Increasing the NaCl concentration bathing the axon, whose diameter is decreased in the hypertonic solution, results in significantly larger transient signals during action potential propagation. While monophasic and biphasic behaviors are observed, biphasic behavior dominates the results. The initial phase detected was constant for a single location but alternated for different locations; therefore, these transient signals acquired around the membrane appear to have local characteristics.     

Spontaneous firing and myelination of very small axons

In this article the question of what evolutionary factors guided acquisition of myelin in the nervous system is addressed. The conclusion that conduction velocity of action potentials along the axon has been the only motive force needs reformulation, as other factors may have played a central role as well. In particular, protection against firing of spontaneous action potentials which may result from the simultaneous opening of only few (less than 10) sodium channels at the nodes of small (less than 1 micron diameter) myelinated axons, may have greatly contributed to discouraging myelination of axons smaller than 1 micron.     

Active dendrites,potassium channels and synaptic plasticity

The dendrites of CA1 pyramidal neurons in the hippocampus express numerous types of voltage-gated ion channel, but the distributions or densities of many of these channels are very non-uniform. Sodium channels in the dendrites are responsible for action potential (AP) propagation from the axon into the dendrites (back-propagation); calcium channels are responsible for local changes in dendritic calcium concentrations following back-propagating APs and synaptic potentials; and potassium channels help regulate overall dendritic excitability. Several lines of evidence are presented here to suggest that back-propagating APs, when coincident with excitatory synaptic input, can lead to the induction of either long-term depression (LTD) or long-term potentiation (LTP). The induction of LTD or LTP is correlated with the magnitude of the rise in intracellular calcium. When brief bursts of synaptic potentials are paired with postsynaptic APs in a theta-burst pairing paradigm, the induction of LTP is dependent on the invasion of the AP into the dendritic tree. The amplitude of the AP in the dendrites is dependent, in part, on the activity of a transient, A-type potassium channel that is expressed at high density in the dendrites and correlates with the induction of the LTP. Furthermore, during the expression phase of the LTP, there are local changes in dendritic excitability that may result from modulation of the functioning of this transient potassium channel. The results support the view that the active properties of dendrites play important roles in synaptic integration and synaptic plasticity of these neurons.     

Firing properties of the soma and axon of the abdominal stretch receptor neurons in the crayfish (Astacus leptodactylus)

Action potentials (APs) and impulse responses in the soma and axon of the rapidly and slowly adapting (SA) abdominal stretch receptor neurons of the crayfish (Astacus leptodactylus) were recorded with single microelectrode current-clamp technique. Impulse frequency response to constant current injection was almost constant in the SA neuron while the response decayed completely in the rapidly adapting (RA) neuron. Mean impulse frequency responses to current stimulations were similar in the receptor neuron pairs. In the RA neuron additional current steps evoked additional impulses while a sudden drop in the current amplitude caused adaptation. Impulse duration was dependent on the rate of rise when current ramps were used. Adaptation was facilitated when calculated receptor current was used. Exposing the neuron to 3 mmol/l TEA or scorpion venom resulted in partly elongated impulse responses. SA neuron could continuously convert the current input into impulse frequency irrespective of previous stimulation conditions. Exposing the SA neuron to 3 mmol/l TEA or 1 mmol/l Lidocaine reduced impulse duration to large current stimulations. The SA neuron fired spontaneously if it was exposed to 5-10 mmol/l Lidocaine or 10(-2) mg/ml Leiurus quinquestriatus venom. The action potential (AP) amplitudes in the RA soma, RA axon, SA soma, and SA axon were significantly different between components of all pairs. Duration of the AP in the axon of the RA neuron was significantly shorter than those in the RA soma, SA soma, and SA axon. Diameter of the RA axon was larger than that of the SA axon. Non-adapting impulse responses were promptly observed only in the SA axons. The results indicate that the RA neuron is a sort of rate receptor transducing the rapid length changes in the receptor muscle while the SA neuron is capable of transducing the maintained length changes in the receptor muscle. The differences in firing properties mainly originate from the differences in the active and passive properties of the receptor neurons.     

Modelling the Effects of Electrical Coupling between Unmyelinated Axons of Brainstem Neurons Controlling Rhythmic Activity

Gap junctions between fine unmyelinated axons can electrically couple groups of brain neurons to synchronise firing and contribute to rhythmic activity. To explore the distribution and significance of electrical coupling, we modelled a well analysed, small population of brainstem neurons which drive swimming in young frog tadpoles. A passive network of 30 multicompartmental neurons with unmyelinated axons was used to infer that: axon-axon gap junctions close to the soma gave the best match to experimentally measured coupling coefficients; axon diameter had a strong influence on coupling; most neurons were coupled indirectly via the axons of other neurons. When active channels were added, gap junctions could make action potential propagation along the thin axons unreliable. Increased sodium and decreased potassium channel densities in the initial axon segment improved action potential propagation. Modelling suggested that the single spike firing to step current injection observed in whole-cell recordings is not a cellular property but a dynamic consequence of shunting resulting from electrical coupling. Without electrical coupling, firing of the population during depolarising current was unsynchronised; with coupling, the population showed synchronous recruitment and rhythmic firing. When activated instead by increasing levels of modelled sensory pathway input, the population without electrical coupling was recruited incrementally to unpatterned activity. However, when coupled, the population was recruited all-or-none at threshold into a rhythmic swimming pattern: the tadpole “decided” to swim. Modelling emphasises uncertainties about fine unmyelinated axon physiology but, when informed by biological data, makes general predictions about gap junctions: locations close to the soma; relatively small numbers; many indirect connections between neurons; cause of action potential propagation failure in fine axons; misleading alteration of intrinsic firing properties. Modelling also indicates that electrical coupling within a population can synchronize recruitment of neurons and their pacemaker firing during rhythmic activity.