Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.     

Modeling the Electric Potential across Neuronal Membranes: The Effect of Fixed Charges on Spinal Ganglion Neurons and Neuroblastoma Cells

We present a model for the electric potential profile across the membranes of neuronal cells. We considered the resting and action potential states, and analyzed the influence of fixed charges of the membrane on its electric potential, based on experimental values of membrane properties of the spinal ganglion neuron and the neuroblastoma cell. The spinal ganglion neuron represents a healthy neuron, and the neuroblastoma cell, which is tumorous, represents a pathological neuron. We numerically solved the non-linear Poisson-Boltzmann equation for the regions of the membrane model we have adopted, by considering the densities of charges dissolved in an electrolytic solution and fixed on both glycocalyx and cytoplasmic proteins. Our model predicts that there is a difference in the behavior of the electric potential profiles of the two types of cells, in response to changes in charge concentrations in the membrane. Our results also describe an insensitivity of the neuroblastoma cell membrane, as observed in some biological experiments. This electrical property may be responsible for the low pharmacological response of the neuroblastoma to certain chemotherapeutic treatments.     

On the dynamics of a pair of coupled neurons subject to alternating input rates

We present a statistical analysis of the firing activity of two coupled neuronal units that interact according to a 'sending-receiving' model. The membrane potential's behavior of both units is described by the Stein equations under the additional assumption that the spikes released by the sending neuron constitute an extra excitation for the receiving one. We also assume the presence of an alternating behavior for the rates of inputs to the sending neuron. By means of ad hoc simulations, we obtain, and then discuss, some statistical results concerning the spike production times of the units within the subintervals of the alternating inputs, as well as the reaction times of the receiving neuron.     

Two-terminal electronic circuit neuron model with excitable membrane V-I-t characteristics

Two-terminal electronic circuit neuron model is described. The model has time-variant voltagecurrent characteristics of an excitable membrane. Corresponding equivalent circuit is shown by the use of time-invariant elements as a voltage-controlled oscillator. The design principle of the model is outlined. Two application examples are demonstrated: (1) Simulator of the excitable membrane for the investigation of a voltage-clamp instrument, which works in the low current and low voltage region. (2) Circuit model for an electric organ of a weakly electric fish, Apteronotus.     

Medullary neurons mediating the inhibition of inspiration by intercostal muscle tendon organs?

Studies were conducted to test the hypothesis that nonrespiratory-modulated units are last-order interneurons mediating the effects of intercostal muscle tendon organs on medullary inspiratory neuron activity. Vagotomized, anesthetized, or decerebrate cats were used. Results show the following. 1) Afferents from different receptor types (i.e., intercostal tendon organs and chest wall cutaneous receptors) that inhibit medullary inspiratory neuron activities evoke the same units. 2) Gastrocnemius muscle group I afferent fibers evoke some of the same units as intercostal afferents but do not alter respiratory activity. 3) The "pneumotaxic center" and laryngeal nerve afferents, which inhibit medullary inspiratory activity, evoke different medullary units than intercostal afferents. 4) Evoked units are not active in spontaneously breathing cats. Additional results suggest that a few respiratory neurons near the retrofacial nucleus may be involved in the mediation of the inspiratory inhibitory effects of intercostal tendon organs. These results do not establish the mechanism by which intercostal muscle tendon organs reduces medullary inspiratory activity.     

Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junctions in the nervous system, a model system consisting of two nearly identical neurons electrotonically coupled is studied. We assume that each neuron discharges a train of impulses or bursts either spontaneously or under constant stimulus via chemical synapses. It is known that not only an electric current but also chemical substances whose molecular weight is about 1000 can pass through the junction of an electrical synapse (gap junction). So, our model system is regarded as a set of non-linear oscillators coupled by diffusion, and it may be described by a system of ordinary differential equations. Neurons are excited constantly when they are stimulated by an electric current above the threshold level. Therefore, we expect Hopf bifurcation to occur at the critical magnitude of a stimulating electric current in the system of differential equations which describes the dynamics of a single neuron. Studying our model system according to the theory of Hopf bifurcation, we found regions of diffusion constants of the electrical junction which give two kinds of periodic solutions. One is the solution where two neurons oscillate in phase synchrony. The other is where two neurons oscillate 180° out of phase. In the case where one neuron is described by the BVP model, the following was found by computer simulation. When the initial difference between the phase of two neurons is small, the two neurons come to oscillate synchronously. If the initial difference is large, however, the two come to be excited alternately. The physiological implications of these results are discussed.Department of Behaviorology, Faculty of Human Sciences     

Weak electric fields detectability in a noisy neural network

We investigate the detectability of weak electric field in a noisy neural network based on Izhikevich neuron model systematically. The neural network is composed of excitatory and inhibitory neurons with similar ratio as that in the mammalian neocortex, and the axonal conduction delays between neurons are also considered. It is found that the noise intensity can modulate the detectability of weak electric field. Stochastic resonance (SR) phenomenon induced by white noise is observed when the weak electric field is added to the network. It is interesting that SR almost disappeared when the connections between neurons are cancelled, suggesting the amplification effects of the neural coupling on the synchronization of neuronal spiking. Furthermore, the network parameters, such as the connection probability, the synaptic coupling strength, the scale of neuron population and the neuron heterogeneity, can also affect the detectability of the weak electric field. Finally, the model sensitivity is studied in detail, and results show that the neural network model has an optimal region for the detectability of weak electric field signal.     

An ephaptic transmission model of CA3 pyramidal cells: an investigation into electric field effects

Extracellular electric fields existing throughout the living brain affect the neural coding and information processing via ephaptic transmission, independent of synapses. A two-compartment whole field effect model (WFEM) of pyramidal neurons embedded within a resistive array which simulates the extracellular medium i.e. ephapse is developed to study the effects of electric field on neuronal behaviors. We derive the two linearized filed effect models (LFEM-1 and LFEM-2) from WFEM at the stable resting state. Through matching these simplified models to the subthreshold membrane response in experiments of the resting pyramidal cells exposed to applied electric fields, we not only verify our proposed model’s validity but also found the key parameters which dominate subthreshold frequency response characteristic. Moreover, we find and give its underlying biophysical mechanism that the unsymmetrical properties of active ion channels results in the very different low-frequency response of somatic and dendritic compartments. Following, WFEM is used to investigate both direct-current (DC) and alternating-current field effect on the neural firing patterns by bifurcation analyses. We present that DC electric field could modulate neuronal excitability, with the positive field improving the excitability, the modest negative field suppressing the excitability, but interestingly, the larger negative field re-exciting the neuron back into spiking behavior. The neuron exposed to the sinusoidal electric field exhibits abundant firing patterns sensitive to the input frequency and intensity. In addition, the electrical properties of ephapse can modulate the efficacy of field effect. Our simulated results are qualitatively in line with the relevant experimental results and can explain some experimental phenomena. Furthermore, they are helpful to provide the predictions which can be tested in future experiments.     

Ultrastructure of neurosecretory cells in the pars intercerebralis of Rhodnius prolixus (Hemiptera).

Six neuron types are distinguished in the pars intercerebralis of the starved fifth instar of Rhodnius prolixus. All neuron types contain electron dense secretory granules derived from Golgi complexes which are of characteristic size and morphology in each type. The neuron types are not thought to represent stages in a secretory cycle. The variety of neuron types described is related to that revealed by staining sections of the same cells with paraldehyde fuchsin. Active synthesis of neurosecretory granules continues throughout starvation and the lysosomal system appears to be involved in the continual degradation of secretory granules. Some of the variations in granule morphology observed may be a consequence of granule fusion and the importance of cytoplasmic events in the development of neurosecretory granules is discussed.     

龙虾胃肠神经系统的数值分析

利用抑制神经系统的WinnerLess Competition(WLC)模型,通过数值方法分析Mulloney型龙虾胃肠神经系统神经元的电位发放,得到胃研磨囊和幽门神经系统中各个神经元的电位发放和系统的节律变化。结果表明,胃研磨系统内神经元的发放规律显示两侧牙齿和中间牙齿出现切断、挤压和研磨食物等状态,幽门系统内神经元的发放规律显示幽门节律出现依次发放的三个部分。两个神经系统的数值结果,不仅解释了龙虾胃肠神经系统中神经元电位发放与肌肉运动的关系,而且理论再现了龙虾胃肠神经系统的节律变化和实验结果。     

Determination of the fraction of active inputs required by a neuron to fire

What fraction of the inputs to a neuron in the primary visual cortex (V1) need to be active for that neuron to reach its firing threshold? The paper describes a numerical method for estimating the selectivity of visual neurons, in terms of the required fraction of active excitatory inputs, from standard data produced by intracellular electro-physiological recordings. The method also provides an estimate of the relative strength of the feedforward inhibition in a push-pull model of the inputs to V1 simple cells. The method is tested on two V1 cells described in Carandini and Ferster [Carandini, M., Ferster, D., 2000. Membrane potential and firing rate in cat primary visual cortex. J. Neurosci. 20, 470-484]. The results indicate that the maximum strength of feedforward inhibition is around 30% of the maximum strength of feedforward excitation. The two V1 neurons investigated fire if more than around 40% of their excitatory LGN inputs are active.     

Responses of neurons in the electrosensory lateral line lobe of the weakly electric fish <Emphasis Type="Italic">Gnathonemus petersii</Emphasis> to simple and complex electrosensory stimuli

Mormyrid fish use active electrolocation to detect and analyze objects. The electrosensory lateral line lobe in the brain receives input from electroreceptors and an efference copy of the command to discharge the electric organ. In curarized fish, we recorded extracellularly from neurons of the electrosensory lateral line lobe while stimulating in the periphery with either a local point stimulus or with a more natural whole-body stimulus. Two classes of neurons were found: (1) three types of E-cells, which were excited by a point stimulus; and (2) two types of I-cells, which were inhibited by point stimulus and responded with excitation to the electric organ corollary discharge. While all neurons responded to a point stimulus, only one out of two types of I-units and two of the three types of E-units changed their firing behavior to a whole-body stimulus or when an object was present. In most units, the responses to whole-body stimuli and to point stimuli differed substantially. Many electrosensory lateral line lobe units showed neural plasticity after prolonged sensory stimulation. However, plastic effects during whole body stimulation were often unlike those occurring during point stimuli, suggesting that under natural conditions electrosensory lateral line lobe network effects play an important role in shaping neural plasticity.     

Manifestation of a monosynaptic excitatory connection in the structure of the cross-correlation histogram

Manifestation of a monosynaptic excitatory connection in the structure of the cross-correlation histogram was studied on a mathematical model of interneuronal interaction. Specific features of the shape and position of the primary peak on the cross-correlation histogram distinguishing monosynaptic excitation of one neuron by another from monosynaptic excitation of these units by a third neuron are described. The height of the primary peak is shown to change with a change in certain parameters of the model describing mechanisms of action potential generation and synaptic transmission of excitation. The structure of the autocorrelation histogram is shown to be reflected to some extent in the structure of the cross-correlation histogram on either side of the primary peak and to take part in the formation of secondary peaks and troughs.Institute of Higher Nervous Activity and Neurophysiology, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 11, No. 4, pp. 348–354, July–August, 1979.     

Functional multimodality of axonal tree in invertebrate neurons.

This review, based on invertebrate neuron examples, aims at highlighting the functional consequences of axonal tree organization. The axonal organization of invertebrate neurons is very complex both morphologically and physiologically. The first part shows how the transfer of information along sensory axons is modified by presynaptic inhibition mechanisms. In primary afferents, presynaptic inhibition is involved in: 1) increasing the dynamic range of the sensory response; 2) processing the sensory information such as increasing spatial and/or temporal selectivity; 3) discriminating environmental information from sensory activities generated by the animal's own movement; and 4) modulating the gain of negative feedback (resistance reflex) during active rhythmic movements such as locomotion. In a second part, the whole organization of other types of neurons is considered, and evidence is given that a neuron may not work as a unit, but rather as a mosaic of disconnected 'integrate-and-fire' units. Examples of invertebrate neurons are presented in which several spike initiating zones exist, such as in some stomatogastric neurons. The separation of a neuron into two functionally distinct entities may be almost total with distinct arborizations existing in different ganglia. However, this functional separation is not definitive and depends on the state of the neuron. In conclusion, the classical integrate-and-fire representation of the neuron, with its dendritic arborization, its spike initiating zone, its axon and axonal tree seems to be no more applicable to invertebrate neurons. A better knowledge of the function of vertebrate neurons would probably demonstrate that it is the case for a large number of them, as suggested by the complex architecture of some reticular interneurons in vertebrates.     

Noise analysis of an ion channel in the cardiac myocyte--study based on a Markov model

Mechanical contraction of a cardiac muscle cell is related to the electric activation of the plasma membrane. As in the neuron cell, inflow of the Na(+) ions across the cell membrane causes electric activation with amplitude of about 100 mV. However, differently from the nerve cell, the action potential lasts a few hundred milliseconds before repolarization. Moreover, several types of K(+) channel such as the classical inward rectifier K(+) channel, the voltage dependent channel and others are responsible for the formation of the action potential. The mechanism of opening and closing the K(+) channels is not thoroughly elucidated. In the present paper, a four state Markov model with one open and three closed states is studied to obtain open and close probabilities of the gates constituting a specific ionic channel. The probability density functions of durations of opening and closing of the channel are also discussed.     

Dendritic Integration in Olfactory Sensory Neurons: A Steady-State Analysis of How the Neuron Structure and Neuron Environment Influence the Coding of Odor Intensity

Response properties of the receptor potential at steady state were analyzed in a biophysical model of an olfactory sensory neuron embedded in a multicell environment. The neuron structure was described as a set of several identical dendrites (or cilia) bearing the transduction mechanisms, joined to a nonsensory part—dendritic knob, soma, and axon. The different ionic compositions of the media surrounding the neuron sensory and nonsensory parts and the extraneuronal voltage sources, which both result from the presence of auxiliary cells, were also taken into account. Analytical solutions were found to describe how the receptor potential at the nonsensory part responds to a uniform change in the odorant-dependent conductance resulting from odorant stimulation of the sensory dendrites. We investigated the influence of various geometrical and electrical parameters on the receptor-potential response in the classical model neuron within a homogeneous environment and in the model neuron surrounded with auxiliary cells. First, it was found that the maximum amplitude of the receptor potential is independent of the neuron structure in the absence of auxiliary cells but not in their presence. In the latter case, the amplitude decreases with the length and number of sensory dendrites and with the input resistance of the nonsensory part. Second, the sensitivity (as measured by the increase in membrane conductance at half-maximum response) of the neuron model in the absence of auxiliary cells is higher, but its dynamic range is narrower than in their presence. The dynamic range is wide and the sensitivity low when the input resistance of the nonsensory part is small and the sensory dendrite is unbranched. Both sensitivity and dynamic range are higher for a longer dendrite. These results help understand the morphology of insect olfactory sensilla and can be generalized to other neuron types.     

The Green’s function formalism as a bridge between single- and multi-compartmental modeling

Neurons are spatially extended structures that receive and process inputs on their dendrites. It is generally accepted that neuronal computations arise from the active integration of synaptic inputs along a dendrite between the input location and the location of spike generation in the axon initial segment. However, many application such as simulations of brain networks use point-neurons—neurons without a morphological component—as computational units to keep the conceptual complexity and computational costs low. Inevitably, these applications thus omit a fundamental property of neuronal computation. In this work, we present an approach to model an artificial synapse that mimics dendritic processing without the need to explicitly simulate dendritic dynamics. The model synapse employs an analytic solution for the cable equation to compute the neuron’s membrane potential following dendritic inputs. Green’s function formalism is used to derive the closed version of the cable equation. We show that by using this synapse model, point-neurons can achieve results that were previously limited to the realms of multi-compartmental models. Moreover, a computational advantage is achieved when only a small number of simulated synapses impinge on a morphologically elaborate neuron. Opportunities and limitations are discussed.     

Atmospheric-pressure electric discharge as an instrument of chemical activation of water solutions

Results of experimental studies and numerical simulations of physicochemical characteristics of plasmas generated in different types of atmospheric-pressure discharges (pulsed streamer corona, gliding electric arc, dielectric barrier discharge, glow-discharge electrolysis, diaphragmatic discharge, and dc glow discharge) used to initiate various chemical processes in water solutions are analyzed. Typical reactor designs are considered. Data on the power supply characteristics, plasma electron parameters, gas temperatures, and densities of active particles in different types of discharges excited in different gases and their dependences on the external parameters of discharges are presented. The chemical composition of active particles formed in water is described. Possible mechanisms of production and loss of plasma particles are discussed.     

Fast temporal adaptation of on-off units in the first optic chiasm of the blowfly

1. We recorded from spiking units in the first optic chiasm between lamina and medulla in the brain of the blowfly (Calliphora vicina). Both previously characterized neuron types, on-off units and sustaining units, were encountered. On-off units had a temporal frequency response with a lower cut-off frequency than blowfly photoreceptors. This low cut-off frequency is related to a fast temporal adaptation of the on-off units to trains of short light pulses. Temporal adaptation occurred independently for short on- and off-pulses. 2. On-off units only responded to stimuli of relatively large contrast. Contrasts of less than 10% gave little or no response.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.