Generalized cable theory for neurons in complex and heterogeneous media

Loss of Ephaptic Contacts in the Murine Thalamus during Osmotic Demyelination Syndrome

BACKGROUND AND AIM

A murine model mimicking osmotic demyelination syndrome (ODS) revealed with histology in the relay posterolateral (VPL) and ventral posteromedial (VPM) thalamic nuclei adjoined nerve cell bodies in chronic hyponatremia, amongst the damaged 12 h and 48 h after reinstatement of osmolality. This report aims to verify and complement with ultrastructure other neurophysiology, immunohistochemistry, and molecular biochemistry data to assess the connexin-36 protein, as part of those hinted close contacts.This ODS investigation included four groups of mice: Sham (NN; n = 13), hyponatremic (HN; n = 11), those sacrificed 12 h after a fast restoration of normal natremia (ODS12h; n = 6) and mice sacrificed 48 h afterward, or ODS48 h (n = 9). Out of these, thalamic zones samples included NN (n = 2), HN (n = 2), ODS12h (n = 3) and ODS48h (n = 3).

RESULTS

Ultrastructure illustrated junctions between nerve cell bodies that were immunolabeled with connexin36 (Cx36) with light microscopy and Western blots. These cell's junctions were reminiscent of low resistance junctions characterized in other regions of the CNS with electrophysiology. Contiguous neurons showed neurolemma contacts in intact and damaged tissues according to their location in the ODS zones, at 12 h and 48 h post correction along with other demyelinating alterations. Neurons and ephaptic contact measurements indicated the highest alterations, including nerve cell necrosis in the ODS epicenter and damages decreased toward the outskirts of the demyelinated zone.

CONCLUSION

Ephapses contained C × 36between intact or ODS injured neurons in the thalamus appeared to be resilient beyond the core degraded tissue injuries. These could maintain intercellular ionic and metabolite exchanges between these lesser injured regions and, thus, would partake to some brain plasticity repairs.

Extracellular and intracellular components of the impedance of neural tissue

Electric phenomena in brain tissue can be measured using extracellular potentials, such as the local field potential, or the electro-encephalogram. The interpretation of these signals depends on the electric structure and properties of extracellular media, but the measurements of these electric properties are still debated. Some measurements point to a model in which the extracellular medium is purely resistive, and thus parameters such as electric conductivity and permittivity should be independent of frequency. Other measurements point to a pronounced frequency dependence of these parameters, with scaling laws that are consistent with capacitive or diffusive effects. However, these experiments correspond to different preparations, and it is unclear how to correctly compare them. Here, we provide for the first time, impedance measurements (in the 1-10 kHz frequency range) using the same setup in various preparations, from primary cell cultures to acute brain slices, and a comparison with similar measurements performed in artificial cerebrospinal fluid with no biological material. The measurements show that when the current flows across a cell membrane, the frequency dependence of the macroscopic impedance between intracellular and extracellular electrodes is significant, and cannot be captured by a model with resistive media. Fitting a mean-field model to the data shows that this frequency dependence could be explained by the ionic diffusion mainly associated with Debye layers surrounding the membranes. We conclude that neuronal membranes and their ionic environment induce strong deviations to resistivity that should be taken into account to correctly interpret extracellular potentials generated by neurons.

Is the Extracellular Impedance High and Non-resistive in Cerebral Cortex?

A recent commentary to Biophysical Journal criticized a previous study published in the same journal by Gomes et al. in 2016, and an alternative interpretation of the measurements was proposed. We reply here to these criticisms and provide some additional clarification, in particular, about a possible misinterpretation of the electrical circuit corresponding to these experiments. We suggest that, indeed, the extracellular impedance in cerebral cortex could be high and non-resistive, and we propose further experiments to settle this issue.

The Soliton and the Action Potential - Primary Elements Underlying Sentience

At present the neurological basis of sentience is poorly understood and this problem is exacerbated by only a partial knowledge of how one of the primary elements of sentience, the action potential, actually works. This has consequences for our understanding of how communication within the brain and in artificial brain neural networks (BNNs). Reverse engineering models of brain activity assume processing works like a conventional binary computer and neglects speed of cognition, latencies, error in nerve conduction and the true dynamic structure of neural networks in the brain. Any model of nerve conduction that claims inspiration from nature must include these prerequisite parameters, but current western computer modeling of artificial BNNs assumes that the action potential is binary and binary mathematics has been assumed by force of popular acceptance to mediate computation in the brain. Here we present evidence that the action potential is a temporal compound ternary structure, described as the computational action potential (CAP). The CAP contains the refractory period, an analog third phase capable of phase-ternary computation via colliding action potentials. This would best fit a realistic BNN and provides a plausible mechanism to explain transmission, in preference to Cable Theory. The action potential pulse (APPulse), is made up of the action potential combined with a coupled synchronized soliton pressure pulse in the cell membrane. We describe a model of an ion channel in a membrane where a soliton deforms the channel sufficiently to destroy the electrostatic insulation thereby instigating a mechanical contraction across the membrane by electrostatic forces. Such a contraction has the effect of redistributing the force lengthways thereby increasing the volume of the ion channel in the membrane. Na ions, once attracted to the interior, balance the forces and the channel reforms to its original shape. A refractory period then occurs until the Na ions diffuse from the adjacent interior space. Finally, a computational model of the action potential (the CAP) is proposed with single action potentials significantly including the refractory period as a computational element capable of computation between colliding action potentials.

Coding of self-motion-induced and self-independent visual motion in the rat dorsomedial striatum

Evolutionary development of vision has provided us with the capacity to detect moving objects. Concordant shifts of visual features suggest movements of the observer, whereas discordant changes are more likely to be indicating independently moving objects, such as predators or prey. Such distinction helps us to focus attention, adapt our behavior, and adjust our motor patterns to meet behavioral challenges. However, the neural basis of distinguishing self-induced and self-independent visual motions is not clarified in unrestrained animals yet. In this study, we investigated the presence and origin of motion-related visual information in the striatum of rats, a hub of action selection and procedural memory. We found that while almost half of the neurons in the dorsomedial striatum are sensitive to visual motion congruent with locomotion (and that many of them also code for spatial location), only a small subset of them are composed of fast-firing interneurons that could also perceive self-independent visual stimuli. These latter cells receive their visual input at least partially from the secondary visual cortex (V2). This differential visual sensitivity may be an important support in adjusting behavior to salient environmental events. It emphasizes the importance of investigating visual motion perception in unrestrained animals.

Locally induced neuronal synchrony precisely propagates to specific cortical areas without rhythm distortion

Propagation of oscillatory spike firing activity at specific frequencies plays an important role in distributed cortical networks. However, there is limited evidence for how such frequency-specific signals are induced or how the signal spectra of the propagating signals are modulated during across-layer (radial) and inter-areal (tangential) neuronal interactions. To directly evaluate the direction specificity of spectral changes in a spiking cortical network, we selectively photostimulated infragranular excitatory neurons in the rat primary visual cortex (V1) at a supra-threshold level with various frequencies, and recorded local field potentials (LFPs) at the infragranular stimulation site, the cortical surface site immediately above the stimulation site in V1, and cortical surface sites outside V1. We found a significant reduction of LFP powers during radial propagation, especially at high-frequency stimulation conditions. Moreover, low-gamma-band dominant rhythms were transiently induced during radial propagation. Contrastingly, inter-areal LFP propagation, directed to specific cortical sites, accompanied no significant signal reduction nor gamma-band power induction. We propose an anisotropic mechanism for signal processing in the spiking cortical network, in which the neuronal rhythms are locally induced/modulated along the radial direction, and then propagate without distortion via intrinsic horizontal connections for spatiotemporally precise, inter-areal communication.

Solitonic conduction of electrotonic signals in neuronal branchlets with polarized microstructure

A model of solitonic conduction in neuronal branchlets with microstructure is presented. The application of cable theory to neurons with microstructure results in a nonlinear cable equation that is solved using a direct method to obtain analytical approximations of traveling wave solutions. It is shown that a linear superposition of two oppositely directed traveling waves demonstrate solitonic interaction: colliding waves can penetrate through each other, and continue fully intact as the exact pulses that entered the collision. These findings indicate that microstructure when polarized can sustain solitary waves that propagate at a constant velocity without attenuation or distortion in the absence of synaptic transmission. Solitonic conduction in a neuronal branchlet arising from polarizability of its microstructure is a novel signaling mode of electrotonic signals in thin processes (<0.5 μm diameter).

Modeling of inter-neuronal coupling medium and its impact on neuronal synchronization

In this paper, modeling of the coupling medium between two neurons, the effects of the model parameters on the synchronization of those neurons, and compensation of coupling strength deficiency in synchronization are studied. Our study exploits the inter-neuronal coupling medium and investigates its intrinsic properties in order to get insight into neuronal-information transmittance and, there from, brain-information processing. A novel electrical model of the coupling medium that represents a well-known RLC circuit attributable to the coupling medium’s intrinsic resistive, inductive, and capacitive properties is derived. Surprisingly, the integration of such properties reveals the existence of a natural three-term control strategy, referred to in the literature as the proportional integral derivative (PID) controller, which can be responsible for synchronization between two neurons. Consequently, brain-information processing can rely on a large number of PID controllers based on the coupling medium properties responsible for the coherent behavior of neurons in a neural network. Herein, the effects of the coupling model (or natural PID controller) parameters are studied and, further, a supervisory mechanism is proposed that follows a learning and adaptation policy based on the particle swarm optimization algorithm for compensation of the coupling strength deficiency.

An Evaluation of the Accuracy of Classical Models for Computing the Membrane Potential and Extracellular Potential for Neurons

Two mathematical models are part of the foundation of Computational neurophysiology; (a) the Cable equation is used to compute the membrane potential of neurons, and, (b) volume-conductor theory describes the extracellular potential around neurons. In the standard procedure for computing extracellular potentials, the transmembrane currents are computed by means of (a) and the extracellular potentials are computed using an explicit sum over analytical point-current source solutions as prescribed by volume conductor theory. Both models are extremely useful as they allow huge simplifications of the computational efforts involved in computing extracellular potentials. However, there are more accurate, though computationally very expensive, models available where the potentials inside and outside the neurons are computed simultaneously in a self-consistent scheme. In the present work we explore the accuracy of the classical models (a) and (b) by comparing them to these more accurate schemes. The main assumption of (a) is that the ephaptic current can be ignored in the derivation of the Cable equation. We find, however, for our examples with stylized neurons, that the ephaptic current is comparable in magnitude to other currents involved in the computations, suggesting that it may be significant-at least in parts of the simulation. The magnitude of the error introduced in the membrane potential is several millivolts, and this error also translates into errors in the predicted extracellular potentials. While the error becomes negligible if we assume the extracellular conductivity to be very large, this assumption is, unfortunately, not easy to justify a priori for all situations of interest.

Cable equation for general geometry

The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.

Ion diffusion may introduce spurious current sources in current-source density (CSD) analysis

Standard CSD analysis does not account for ionic diffusion. Using biophysically realistic computer simulations, we show that unaccounted-for diffusive currents can lead to the prediction of spurious current sources. This finding may be of strong interest for in vivo electrophysiologists doing extracellular recordings in general, and CSD analysis in particular.

Current-source density (CSD) analysis is a well-established method for analyzing recorded local field potentials (LFPs), that is, the low-frequency part of extracellular potentials. Standard CSD theory is based on the assumption that all extracellular currents are purely ohmic, and thus neglects the possible impact from ionic diffusion on recorded potentials. However, it has previously been shown that in physiological conditions with large ion-concentration gradients, diffusive currents can evoke slow shifts in extracellular potentials. Using computer simulations, we here show that diffusion-evoked potential shifts can introduce errors in standard CSD analysis, and can lead to prediction of spurious current sources. Further, we here show that the diffusion-evoked prediction errors can be removed by using an improved CSD estimator which accounts for concentration-dependent effects.

NEW & NOTEWORTHY Standard CSD analysis does not account for ionic diffusion. Using biophysically realistic computer simulations, we show that unaccounted-for diffusive currents can lead to the prediction of spurious current sources. This finding may be of strong interest for in vivo electrophysiologists doing extracellular recordings in general, and CSD analysis in particular.

Analytic Modeling of Neural Tissue: I. A Spherical Bidomain

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

Extending Integrate-and-Fire Model Neurons to Account for the Effects of Weak Electric Fields and Input Filtering Mediated by the Dendrite

Transcranial brain stimulation and evidence of ephaptic coupling have recently sparked strong interests in understanding the effects of weak electric fields on the dynamics of brain networks and of coupled populations of neurons. The collective dynamics of large neuronal populations can be efficiently studied using single-compartment (point) model neurons of the integrate-and-fire (IF) type as their elements. These models, however, lack the dendritic morphology required to biophysically describe the effect of an extracellular electric field on the neuronal membrane voltage. Here, we extend the IF point neuron models to accurately reflect morphology dependent electric field effects extracted from a canonical spatial “ball-and-stick” (BS) neuron model. Even in the absence of an extracellular field, neuronal morphology by itself strongly affects the cellular response properties. We, therefore, derive additional components for leaky and nonlinear IF neuron models to reproduce the subthreshold voltage and spiking dynamics of the BS model exposed to both fluctuating somatic and dendritic inputs and an extracellular electric field. We show that an oscillatory electric field causes spike rate resonance, or equivalently, pronounced spike to field coherence. Its resonance frequency depends on the location of the synaptic background inputs. For somatic inputs the resonance appears in the beta and gamma frequency range, whereas for distal dendritic inputs it is shifted to even higher frequencies. Irrespective of an external electric field, the presence of a dendritic cable attenuates the subthreshold response at the soma to slowly-varying somatic inputs while implementing a low-pass filter for distal dendritic inputs. Our point neuron model extension is straightforward to implement and is computationally much more efficient compared to the original BS model. It is well suited for studying the dynamics of large populations of neurons with heterogeneous dendritic morphology with (and without) the influence of weak external electric fields.

How extracellular electric fields—as generated endogenously or through transcranial brain stimulation—affect the dynamics of neuronal populations is of great interest but not well understood. To study neuronal activity at the network level single-compartment neuron models have been proven very successful, because of their computational efficiency and analytical tractability. Unfortunately, these models lack the dendritic morphology to biophysically account for the effects of electric fields, and for changes in synaptic integration due to morphology alone. Here, we consider a canonical, spatially extended model neuron and characterize its responses to fluctuating synaptic input as well as an oscillatory, weak electric field. In order to accurately reproduce these responses we analytically derive an extension for the popular integrate-and-fire point neuron models. We show that the dendritic cable acts as a filter for the synaptic input current, which depends on the input location, and that an electric field modulates the neuronal spike rate strongest at a certain (preferred) field frequency. These phenomena can be successfully reproduced using integrate-and-fire models, extended by a small number of components that are straightforward to implement. The extended point models are thus well suited for studying populations of coupled neurons with different morphology, exposed to extracellular electric fields.

Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue

Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. The framework uses (1) the NEURON simulator to compute the activity and ionic output currents from multicompartmental neuron models, and (2) the electrodiffusive Kirchhoff-Nernst-Planck framework to simulate the resulting dynamics of the potential and ion concentrations in the ECS, accounting for the effect of electrical migration as well as diffusion. Using this framework, we explore the effect that ECS diffusion has on the electrical potential surrounding a small population of 10 pyramidal neurons. The neural model was tuned so that simulations over ∼100 seconds of biological time led to shifts in ECS concentrations by a few millimolars, similar to what has been seen in experiments. By comparing simulations where ECS diffusion was absent with simulations where ECS diffusion was included, we made the following key findings: (i) ECS diffusion shifted the local potential by up to ∼0.2 mV. (ii) The power spectral density (PSD) of the diffusion-evoked potential shifts followed a 1/f2 power law. (iii) Diffusion effects dominated the PSD of the ECS potential for frequencies up to several hertz. In scenarios with large, but physiologically realistic ECS concentration gradients, diffusion was thus found to affect the ECS potential well within the frequency range picked up in experimental recordings.

Intracellular Impedance Measurements Reveal Non-ohmic Properties of the Extracellular Medium around Neurons

Determining the electrical properties of the extracellular space around neurons is important for understanding the genesis of extracellular potentials, as well as for localizing neuronal activity from extracellular recordings. However, the exact nature of these extracellular properties is still uncertain. Here, we introduce a method to measure the impedance of the tissue, one that preserves the intact cell-medium interface using whole-cell patch-clamp recordings in vivo and in vitro. We find that neural tissue has marked non-ohmic and frequency-filtering properties, which are not consistent with a resistive (ohmic) medium, as often assumed. The amplitude and phase profiles of the measured impedance are consistent with the contribution of ionic diffusion. We also show that the impact of such frequency-filtering properties is possibly important on the genesis of local field potentials, as well as on the cable properties of neurons. These results show non-ohmic properties of the extracellular medium around neurons, and suggest that source estimation methods, as well as the cable properties of neurons, which all assume ohmic extracellular medium, may need to be reevaluated.

Simplest relationship between local field potential and intracellular signals in layered neural tissue

The relationship between the extracellularly measured electric field potential resulting from synaptic activity in an ensemble of neurons and intracellular signals in these neurons is an important but still open question. Based on a model neuron with a cylindrical dendrite and lumped soma, we derive a formula that substantiates a proportionality between the local field potential and the total somatic transmembrane current that emerges from the difference between the somatic and dendritic membrane potentials. The formula is tested by intra- and extracellular recordings of evoked synaptic responses in hippocampal slices. Additionally, the contribution of different membrane currents to the field potential is demonstrated in a two-population mean-field model. Our formalism, which allows for a simple estimation of unknown dendritic currents directly from somatic measurements, provides an interpretation of the local field potential in terms of intracellularly measurable synaptic signals. It is also applicable to the study of cortical activity using two-compartment neuronal population models.

Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations

Neurons generate magnetic fields which can be recorded with macroscopic techniques such as magnetoencephalography. The theory that accounts for the genesis of neuronal magnetic fields involves dendritic cable structures in homogeneous resistive extracellular media. Here we generalize this model by considering dendritic cables in extracellular media with arbitrarily complex electric properties. This method is based on a multiscale mean-field theory where the neuron is considered in interaction with a "mean" extracellular medium (characterized by a specific impedance). We first show that, as expected, the generalized cable equation and the standard cable generate magnetic fields that mostly depend on the axial current in the cable, with a moderate contribution of extracellular currents. Less expected, we also show that the nature of the extracellular and intracellular media influence the axial current, and thus also influence neuronal magnetic fields. We illustrate these properties by numerical simulations and suggest experiments to test these findings.

Generation of field potentials and modulation of their dynamics through volume integration of cortical activity

Field potentials (FPs) recorded within the brain, often called "local field potentials" (LFPs), are useful measures of net synaptic activity in a neuronal ensemble. However, due to volume conduction, FPs spread beyond regions of underlying synaptic activity, and thus an "LFP" signal may not accurately reflect the temporal patterns of synaptic activity in the immediately surrounding neuron population. To better understand the physiological processes reflected in FPs, we explored the relationship between the FP and its membrane current generators using current source density (CSD) analysis in conjunction with a volume conductor model. The model provides a quantitative description of the spatiotemporal summation of immediate local and more distant membrane currents to produce the FP. By applying the model to FPs in the macaque auditory cortex, we have investigated a critical issue that has broad implications for FP research. We have shown that FP responses in particular cortical layers are differentially susceptible to activity in other layers. Activity in the supragranular layers has the strongest contribution to FPs in other cortical layers, and infragranular FPs are most susceptible to contributions from other layers. To define the physiological processes generating FPs recorded in loci of relatively weak synaptic activity, strong effects produced by synaptic events in the vicinity have to be taken into account. While outlining limitations and caveats inherent to FP measurements, our results also suggest specific peak and frequency band components of FPs can be related to activity in specific cortical layers. These results may help improving the interpretability of FPs.

Transcranial extracellular impedance control (tEIC) modulates behavioral performances

Electric brain stimulations such as transcranial direct current stimulation (tDCS), transcranial random noise stimulation (tRNS), and transcranial alternating current stimulation (tACS) electrophysiologically modulate brain activity and as a result sometimes modulate behavioral performances. These stimulations can be viewed from an engineering standpoint as involving an artificial electric source (DC, noise, or AC) attached to an impedance branch of a distributed parameter circuit. The distributed parameter circuit is an approximation of the brain and includes electric sources (neurons) and impedances (volume conductors). Such a brain model is linear, as is often the case with the electroencephalogram (EEG) forward model. Thus, the above-mentioned current stimulations change the current distribution in the brain depending on the locations of the electric sources in the brain. Now, if the attached artificial electric source were to be replaced with a resistor, or even a negative resistor, the resistor would also change the current distribution in the brain. In light of the superposition theorem, which holds for any linear electric circuit, attaching an electric source is different from attaching a resistor; the resistor affects each active electric source in the brain so as to increase (or decrease in some cases of a negative resistor) the current flowing out from each source. From an electrophysiological standpoint, the attached resistor can only control the extracellular impedance and never causes forced stimulation; we call this technique transcranial extracellular impedance control (tEIC). We conducted a behavioral experiment to evaluate tEIC and found evidence that it had real-time enhancement and depression effects on EEGs and a real-time facilitation effect on reaction times. Thus, tEIC could be another technique to modulate behavioral performance.