A modified cable formalism for modeling neuronal membranes at high frequencies

A General, Noise-Driven Mechanism for the 1/f-Like Behavior of Neural Field Spectra

Consistent observations across recording modalities, experiments, and neural systems find neural field spectra with 1/f-like scaling, eliciting many alternative theories to explain this universal phenomenon. We show that a general dynamical system with stochastic drive and minimal assumptions generates 1/f-like spectra consistent with the range of values observed in vivo without requiring a specific biological mechanism or collective critical behavior.

Modeling the effect of magnetoelectric nanoparticles on neuronal electrical activity: An analog circuit approach

This paper introduces a physical neuron model that incorporates magnetoelectric nanoparticles (MENPs) as an essential electrical circuit component to wirelessly control local neural activity. Availability of such a model is important as MENPs, due to their magnetoelectric effect, can wirelessly and noninvasively modulate neural activity, which, in turn, has implications for both finding cures for neurological diseases and creating a wireless noninvasive high-resolution brain-machine interface. When placed on a neuronal membrane, MENPs act as magnetic-field-controlled finite-size electric dipoles that generate local electric fields across the membrane in response to magnetic fields, thus allowing to controllably activate local ion channels and locally initiate an action potential. Herein, the neuronal electrical characteristic description is based on ion channel activation and inhibition mechanisms. A MENP-based memristive Hodgkin-Huxley circuit model is extracted by combining the Hodgkin-Huxley model and an equivalent circuit model for a single MENP. In this model, each MENP becomes an integral part of the neuron, thus enabling wireless local control of the neuron's electric circuit itself. Furthermore, the model is expanded to include multiple MENPs to describe collective effects in neural systems.

The 1/f-like behavior of neural field spectra are a natural consequence of noise driven brain dynamics

Consistent observations across recording modalities, experiments, and neural systems find neural field spectra with 1/f-like scaling, eliciting many alternative theories to explain this universal phenomenon. We show that a general dynamical system with stochastic drive and minimal assumptions generates 1/f-like spectra consistent with the range of values observed in vivo, without requiring a specific biological mechanism or collective critical behavior.

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.

A Single Differential Equation Description of Membrane Properties Underlying the Action Potential and the Axon Electric Field

In a succession of articles published over 65 years ago, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley established what now forms our physical understanding of excitation in nerve, and how the axon conducts the action potential. They uniquely quantified the movement of ions in the nerve cell during the action potential, and demonstrated that the action potential is the result of a depolarizing event across the cell membrane. They confirmed that a complete depolarization event is followed by an abrupt increase in voltage that propagates longitudinally along the axon, accompanied by considerable increases in membrane conductance. In an elegant theoretical framework, they rigorously described fundamental properties of the Na⁺ and K⁺ conductances intrinsic to the action potential. Notwithstanding the elegance of Hodgkin and Huxley's incisive and explicative series of discoveries, their model is mathematically complex, relies on no small number of stochastic factors, and has no analytical solution. Solving for the membrane action potential and the ionic currents requires integrations approximated using numerical methods. In this article I present an analytical formalism of the nerve action potential, V_m and that of the accompanying cell membrane electric field, E_m . To conclude, I present a novel description of V_m in terms of a single, nonlinear differential equation. This is an original stand-alone article: the major contribution is the latter, and how this description coincides with the cell membrane electric field. This work has necessitated unifying information from two preceding papers [1,2], each being concerned with the development of closed-form descriptions of the nerve action potential, V_m .

Electrothermal Equivalent Three-Dimensional Finite-Element Model of a Single Neuron

OBJECTIVE

We propose a novel approach for modelling the interdependence of electrical and mechanical phenomena in nervous cells, by using electrothermal equivalences in finite element (FE) analysis so that existing thermomechanical tools can be applied.

METHODS

First, the equivalence between electrical and thermal properties of the nerve materials is established, and results of a pure heat conduction analysis performed in Abaqus CAE Software 6.13-3 are validated with analytical solutions for a range of steady and transient conditions. This validation includes the definition of equivalent active membrane properties that enable prediction of the action potential. Then, as a step toward fully coupled models, electromechanical coupling is implemented through the definition of equivalent piezoelectric properties of the nerve membrane using the thermal expansion coefficient, enabling prediction of the mechanical response of the nerve to the action potential.

RESULTS

Results of the coupled electromechanical model are validated with previously published experimental results of deformation for squid giant axon, crab nerve fibre, and garfish olfactory nerve fibre.

CONCLUSION

A simplified coupled electromechanical modelling approach is established through an electrothermal equivalent FE model of a nervous cell for biomedical applications.

SIGNIFICANCE

One of the key findings is the mechanical characterization of the neural activity in a coupled electromechanical domain, which provides insights into the electromechanical behaviour of nervous cells, such as thinning of the membrane. This is a first step toward modelling three-dimensional electromechanical alteration induced by trauma at nerve bundle, tissue, and organ levels.

Differential fiber-specific block of nerve conduction in mammalian peripheral nerves using kilohertz electrical stimulation

Kilohertz electrical stimulation (KES) has been shown to induce repeatable and reversible nerve conduction block in animal models. In this study, we characterized the ability of KES stimuli to selectively block specific components of stimulated nerve activity using in vivo preparations of the rat sciatic and vagus nerves. KES stimuli in the frequency range of 5-70 kHz and amplitudes of 0.1-3.0 mA were applied. Compound action potentials were evoked using either electrical or sensory stimulation, and block of components was assessed through direct nerve recordings and muscle force measurements. Distinct observable components of the compound action potential had unique conduction block thresholds as a function of frequency of KES. The fast component, which includes motor activity, had a monotonically increasing block threshold as a function of the KES frequency. The slow component, which includes sensory activity, showed a nonmonotonic block threshold relationship with increasing KES frequency. The distinct trends with frequency of the two components enabled selective block of one component with an appropriate choice of frequency and amplitude. These trends in threshold of the two components were similar when studying electrical stimulation and responses of the sciatic nerve, electrical stimulation and responses of the vagus nerve, and sensorimotor stimulation and responses of the sciatic nerve. This differential blocking effect of KES on specific fibers can extend the applications of KES conduction block to selective block and stimulation of neural signals for neuromodulation as well as selective control of neural circuits underlying sensorimotor function.

Power laws from linear neuronal cable theory: power spectral densities of the soma potential, soma membrane current and single-neuron contribution to the EEG

Power laws, that is, power spectral densities (PSDs) exhibiting behavior for large frequencies f, have been observed both in microscopic (neural membrane potentials and currents) and macroscopic (electroencephalography; EEG) recordings. While complex network behavior has been suggested to be at the root of this phenomenon, we here demonstrate a possible origin of such power laws in the biophysical properties of single neurons described by the standard cable equation. Taking advantage of the analytical tractability of the so called ball and stick neuron model, we derive general expressions for the PSD transfer functions for a set of measures of neuronal activity: the soma membrane current, the current-dipole moment (corresponding to the single-neuron EEG contribution), and the soma membrane potential. These PSD transfer functions relate the PSDs of the respective measurements to the PSDs of the noisy input currents. With homogeneously distributed input currents across the neuronal membrane we find that all PSD transfer functions express asymptotic high-frequency power laws with power-law exponents analytically identified as for the soma membrane current, for the current-dipole moment, and for the soma membrane potential. Comparison with available data suggests that the apparent power laws observed in the high-frequency end of the PSD spectra may stem from uncorrelated current sources which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. While the PSD noise spectra at low frequencies may be dominated by synaptic noise, our findings suggest that the high-frequency power laws may originate in noise from intrinsic ion channels. The significance of this finding goes beyond neuroscience as it demonstrates how power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation.

The common observation of power laws in nature and society, that is, quantities or probabilities that follow distributions, has for long intrigued scientists. In the brain, power laws in the power spectral density (PSD) have been reported in electrophysiological recordings, both at the microscopic (single-neuron recordings) and macroscopic (EEG) levels. We here demonstrate a possible origin of such power laws in the basic biophysical properties of neurons, that is, in the standard cable-equation description of neuronal membranes. Taking advantage of the mathematical tractability of the so called ball and stick neuron model, we demonstrate analytically that high-frequency power laws in key experimental neural measures will arise naturally when the noise sources are evenly distributed across the neuronal membrane. Comparison with available data further suggests that the apparent high-frequency power laws observed in experiments may stem from uncorrelated current sources, presumably intrinsic ion channels, which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. The significance of this finding goes beyond neuroscience as it demonstrates how power laws power-law exponents α may arise from a simple, linear physics equation.

Dendritic signal transmission induced by intracellular charge inhomogeneities

Signal propagation in neuronal dendrites represents the basis for interneuron communication and information processing in the brain. Here we take into account charge inhomogeneities arising in the vicinity of ion channels in cytoplasm and obtain a modified cable equation. We show that charge inhomogeneities acting on a millisecond time scale can lead to the appearance of propagating waves with wavelengths of hundreds of micrometers. They correspond to a certain frequency band predicting the appearance of resonant properties in brain neuron signaling. We also show that membrane potential in spiny dendrites obeys the modified cable equation suggesting a crucial role of the spines in dendritic subthreshold resonance.

Generalized cable theory for neurons in complex and heterogeneous media

Cable theory has been developed over the last decade, usually assuming that the extracellular space around membranes is a perfect resistor. However, extracellular media may display more complex electrical properties due to various phenomena, such as polarization, ionic diffusion, or capacitive effects, but their impact on cable properties is not known. In this paper, we generalize cable theory for membranes embedded in arbitrarily complex extracellular media. We outline the generalized cable equations, then consider specific cases. The simplest case is a resistive medium, in which case the equations recover the traditional cable equations. We show that for more complex media, for example, in the presence of ionic diffusion, the impact on cable properties such as voltage attenuation can be significant. We illustrate this numerically, always by comparing the generalized cable to the traditional cable. We conclude that the nature of intracellular and extracellular media may have a strong influence on cable filtering as well as on the passive integrative properties of neurons.

Generalized theory for current-source-density analysis in brain tissue

The current-source density (CSD) analysis is a widely used method in brain electrophysiology, but this method rests on a series of assumptions, namely that the surrounding extracellular medium is resistive and uniform, and in some versions of the theory, that the current sources are exclusively made by dipoles. Because of these assumptions, this standard model does not correctly describe the contributions of monopolar sources or of nonresistive aspects of the extracellular medium. We propose here a general framework to model electric fields and potentials resulting from current source densities, without relying on the above assumptions. We develop a mean-field formalism that is a generalization of the standard model and that can directly incorporate nonresistive (nonohmic) properties of the extracellular medium, such as ionic diffusion effects. This formalism recovers the classic results of the standard model such as the CSD analysis, but in addition, we provide expressions to generalize the CSD approach to situations with nonresistive media and arbitrarily complex multipolar configurations of current sources. We found that the power spectrum of the signal contains the signature of the nature of current sources and extracellular medium, which provides a direct way to estimate those properties from experimental data and, in particular, estimate the possible contribution of electric monopoles.

Evidence for frequency-dependent extracellular impedance from the transfer function between extracellular and intracellular potentials: intracellular-LFP transfer function

We examine the properties of the transfer function F(T)=V(m)/V(LFP) between the intracellular membrane potential (V(m)) and the local field potential (V(LFP)) in cerebral cortex. We first show theoretically that, in the subthreshold regime, the frequency dependence of the extracellular medium and that of the membrane potential have a clear incidence on F(T). The calculation of F(T) from experiments and the matching with theoretical expressions is possible for desynchronized states where individual current sources can be considered as independent. Using a mean-field approximation, we obtain a method to estimate the impedance of the extracellular medium without injecting currents. We examine the transfer function for bipolar (differential) LFPs and compare to simultaneous recordings of V(m) and V(LFP) during desynchronized states in rat barrel cortex in vivo. The experimentally derived F(T) matches the one derived theoretically, only if one assumes that the impedance of the extracellular medium is frequency-dependent, and varies as 1/√ω (Warburg impedance) for frequencies between 3 and 500 Hz. This constitutes indirect evidence that the extracellular medium is non-resistive, which has many possible consequences for modeling LFPs.

A model of slow wave propagation and entrainment along the stomach

Interstitial cells of Cajal (ICC) isolated from different regions of the stomach generate spontaneous electrical slow wave activity at different frequencies, with cells from the proximal stomach pacing faster than their distal counterparts. However, in vivo there exists a uniform pacing frequency; slow waves propagate aborally from the proximal stomach and subsequently entrain distal tissues. Significant resting membrane potential (RMP) gradients also exist within the stomach whereby membrane polarization generally increases from the fundus to the antrum. Both of these factors play a major role in the macroscopic electrical behavior of the stomach and as such, any tissue or organ level model of gastric electrophysiology should ensure that these phenomena are properly described. This study details a dual-cable model of gastric electrical activity that incorporates biophysically detailed single-cell models of the two predominant cell types, the ICC and smooth muscle cells. Mechanisms for the entrainment of the intrinsic pacing frequency gradient and for the establishment of the RMP gradient are presented. The resulting construct is able to reproduce experimentally recorded slow wave activity and provides a platform on which our understanding of gastric electrical activity can advance.

Network-state modulation of power-law frequency-scaling in visual cortical neurons

Various types of neural-based signals, such as EEG, local field potentials and intracellular synaptic potentials, integrate multiple sources of activity distributed across large assemblies. They have in common a power-law frequency-scaling structure at high frequencies, but it is still unclear whether this scaling property is dominated by intrinsic neuronal properties or by network activity. The latter case is particularly interesting because if frequency-scaling reflects the network state it could be used to characterize the functional impact of the connectivity. In intracellularly recorded neurons of cat primary visual cortex in vivo, the power spectral density of V_m activity displays a power-law structure at high frequencies with a fractional scaling exponent. We show that this exponent is not constant, but depends on the visual statistics used to drive the network. To investigate the determinants of this frequency-scaling, we considered a generic recurrent model of cortex receiving a retinotopically organized external input. Similarly to the in vivo case, our in computo simulations show that the scaling exponent reflects the correlation level imposed in the input. This systematic dependence was also replicated at the single cell level, by controlling independently, in a parametric way, the strength and the temporal decay of the pairwise correlation between presynaptic inputs. This last model was implemented in vitro by imposing the correlation control in artificial presynaptic spike trains through dynamic-clamp techniques. These in vitro manipulations induced a modulation of the scaling exponent, similar to that observed in vivo and predicted in computo. We conclude that the frequency-scaling exponent of the V_m reflects stimulus-driven correlations in the cortical network activity. Therefore, we propose that the scaling exponent could be used to read-out the “effective” connectivity responsible for the dynamical signature of the population signals measured at different integration levels, from Vm to LFP, EEG and fMRI.

Intracellular recording of neocortical neurons provides an opportunity of characterizing the statistical signature of the synaptic bombardment to which it is submitted. Indeed the membrane potential displays intense fluctuations which reflect the cumulative activity of thousands of input neurons. In sensory cortical areas, this measure could be used to estimate the correlational structure of the external drive. We show that changes in the statistical properties of network activity, namely the local correlation between neurons, can be detected by analyzing the power spectrum density (PSD) of the subthreshold membrane potential. These PSD can be fitted by a power-law function 1/f^α in the upper temporal frequency range. In vivo recordings in primary visual cortex show that the α exponent varies with the statistics of the sensory input. Most remarkably, the exponent observed in the ongoing activity is indistinguishable from that evoked by natural visual statistics. These results are emulated by models which demonstrate that the exponent α is determined by the local level of correlation imposed in the recurrent network activity. Similar relationships are also reproduced in cortical neurons recorded in vitro with artificial synaptic inputs by controlling in computo the level of correlation in real time.

Characterizing neuronal activity by describing the membrane potential as a stochastic process

Cortical neurons behave similarly to stochastic processes, as a consequence of their irregularity and dense connectivity. Their firing pattern is close to a Poisson process, and their membrane potential (V(m)) is analogous to colored noise. One way to characterize this activity is to identify V(m) to a multidimensional stochastic process. We review here this approach and how it can be used to extract important statistical signatures of neuronal activity. The "VmD method" consists of fitting the V(m) distribution obtained intracellularly to analytic expressions derived from stochastic processes, and thereby deduce synaptic conductance parameters. However, this method requires at least two levels of V(m), which prevents applications to single-trial measurements. We also discuss methods that can be applied to single V(m) traces, such as power spectral analysis and the "STA method" to calculate spike-triggered average conductances based on a maximum likelihood procedure. A recently proposed method, the "VmT method", is based on the fusion of these two concepts. This method is analogous to the VmD method and estimates the mean excitatory and inhibitory conductances and their variances. However, it does so by using a maximum-likelihood estimation, and can thus be applied to single V(m) traces. All methods were tested using controlled conductance injection in dynamic-clamp experiments.