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

Frequency-domain analysis of membrane polarization in two-compartment model neurons with weak alternating electric fields

Transcranial alternating current stimulation (tACS) is widely used in studying brain functions and the treatment of neuropsychiatric diseases in a frequency-specific manner. However, how tACS works on neuronal activity has been poorly understood. In this paper, we use linear system analysis to investigate how weak alternating electric fields (EFs) affect the membrane polarization of neurons in the frequency domain. Two biophysically realistic conductance-based two-compartment models of cortical pyramidal neurons are developed to simulate subthreshold membrane polarization with weak alternating EFs. We linearize the original nonlinear models at the stable equilibrium points and further simplify them to the two- or three-dimensional linear systems. Thus, we calculate the transfer functions of the low-dimensional linear models to model neuronal polarization patterns. Based on the transfer functions, we compute the amplitude- and phase-frequency characteristics to describe the relationship between weak EFs and membrane polarization. We also computed the parameters (gain, zeros, and poles) and structures (the number of zeros and poles) of transfer functions to reveal how neuronal intrinsic properties affect the parameters and structure of transfer functions and thus the frequency-dependent membrane polarization with alternating EFs. We find that the amplitude and phase of membrane polarization both strongly depended on EF frequency, and these frequency responses are modulated by the intrinsic properties of neurons. The compartment geometry, internal coupling conductance, and ionic currents (except Ih) affect the frequency-dependent polarization by mainly changing the gain and pole of transfer functions. Larger gain contributes to larger amplitude-frequency characteristics. The closer the pole is to the imaginary axis, the lower phase-frequency characteristics. However, Ih changes the structure of transfer function in the dendrite by introducing a new pair of zero-pole points, which decrease the amplitude at low frequencies and thus lead to a visible resonance. These results highlight the effects of passive properties and active ion currents on subthreshold membrane polarization with alternating EFs in the frequency domain, which provide an explainable connection of how intrinsic properties of neurons modulate the neuronal input-output functions with weak EF stimulation.

Weak electric fields promote resonance in neuronal spiking activity: Analytical results from two-compartment cell and network models

Transcranial brain stimulation and evidence of ephaptic coupling have sparked strong interests in understanding the effects of weak electric fields on the dynamics of neuronal populations. While their influence on the subthreshold membrane voltage can be biophysically well explained using spatially extended neuron models, mechanistic analyses of neuronal spiking and network activity have remained a methodological challenge. More generally, this challenge applies to phenomena for which single-compartment (point) neuron models are oversimplified. Here we employ a pyramidal neuron model that comprises two compartments, allowing to distinguish basal-somatic from apical dendritic inputs and accounting for an extracellular field in a biophysically minimalistic way. Using an analytical approach we fit its parameters to reproduce the response properties of a canonical, spatial model neuron and dissect the stochastic spiking dynamics of single cells and large networks. We show that oscillatory weak fields effectively mimic anti-correlated inputs at the soma and dendrite and strongly modulate neuronal spiking activity in a rather narrow frequency band. This effect carries over to coupled populations of pyramidal cells and inhibitory interneurons, boosting network-induced resonance in the beta and gamma frequency bands. Our work contributes a useful theoretical framework for mechanistic analyses of population dynamics going beyond point neuron models, and provides insights on modulation effects of extracellular fields due to the morphology of pyramidal cells.

The elongated spatial structure of pyramidal neurons, which possess large apical dendrites, plays an important role for the integration of synaptic inputs and mediates sensitivity to weak extracellular electric fields. Modeling studies at the population level greatly contribute to our mechanistic understanding but face a methodological challenge because morphologically detailed neuron models are too complex for use in noisy, in-vivo like conditions and large networks in particular. Here we present an analytical approach based on a two-compartment spiking neuron model that can distinguish synaptic inputs at the apical dendrite from those at the somatic region and accounts for an extracellular field in a biophysically minimalistic way. We devised efficient methods to approximate the responses of a spatially more detailed pyramidal neuron model, and to study the spiking dynamics of single neurons and sparsely coupled large networks in the presence of fluctuating inputs. Using these methods we focused on how responses are affected by oscillatory weak fields. Our results suggest that ephaptic coupling may play a mechanistic role for oscillations of population activity and indicate the potential to entrain networks by weak electric stimulation.

Dendritic Properties Control Energy Efficiency of Action Potentials in Cortical Pyramidal Cells

Neural computation is performed by transforming input signals into sequences of action potentials (APs), which is metabolically expensive and limited by the energy available to the brain. The metabolic efficiency of single AP has important consequences for the computational power of the cell, which is determined by its biophysical properties and morphologies. Here we adopt biophysically-based two-compartment models to investigate how dendrites affect energy efficiency of APs in cortical pyramidal neurons. We measure the Na⁺ entry during the spike and examine how it is efficiently used for generating AP depolarization. We show that increasing the proportion of dendritic area or coupling conductance between two chambers decreases Na⁺ entry efficiency of somatic AP. Activating inward Ca²⁺ current in dendrites results in dendritic spike, which increases AP efficiency. Activating Ca²⁺-activated outward K⁺ current in dendrites, however, decreases Na⁺ entry efficiency. We demonstrate that the active and passive dendrites take effects by altering the overlap between Na⁺ influx and internal current flowing from soma to dendrite. We explain a fundamental link between dendritic properties and AP efficiency, which is essential to interpret how neural computation consumes metabolic energy and how biophysics and morphologies contribute to such consumption.

Morphology controls how hippocampal CA1 pyramidal neuron responds to uniform electric fields: a biophysical modeling study

Responses of different neurons to electric field (EF) are highly variable, which depends on intrinsic properties of cell type. Here we use multi-compartmental biophysical models to investigate how morphologic features affect EF-induced responses in hippocampal CA1 pyramidal neurons. We find that the basic morphologies of neuronal elements, including diameter, length, bend, branch, and axon terminals, are all correlated with somatic depolarization through altering the current sources or sinks created by applied field. Varying them alters the EF threshold for triggering action potentials (APs), and then determines cell sensitivity to suprathreshold field. Introducing excitatory postsynaptic potential increases cell excitability and reduces morphology-dependent EF firing threshold. It is also shown that applying identical subthreshold EF results in distinct polarizations on cell membrane with different realistic morphologies. These findings shed light on the crucial role of morphologies in determining field-induced neural response from the point of view of biophysical models. The predictions are conducive to better understanding the variability in modulatory effects of EF stimulation at the cellular level, which could also aid the interpretations of how applied fields activate central nervous system neurons and affect relevant circuits.

Action potential initiation in a two-compartment model of pyramidal neuron mediated by dendritic Ca2+ spike

Dendritic Ca²⁺ spike endows cortical pyramidal cell with powerful ability of synaptic integration, which is critical for neuronal computation. Here we propose a two-compartment conductance-based model to investigate how the Ca²⁺ activity of apical dendrite participates in the action potential (AP) initiation to affect the firing properties of pyramidal neurons. We have shown that the apical input with sufficient intensity triggers a dendritic Ca²⁺ spike, which significantly boosts dendritic inputs as it propagates to soma. Such event instantaneously shifts the limit cycle attractor of the neuron and results in a burst of APs, which makes its firing rate reach a plateau steady-state level. Delivering current to two chambers simultaneously increases the level of neuronal excitability and decreases the threshold of input-output relation. Here the back-propagating APs facilitate the initiation of dendritic Ca²⁺ spike and evoke BAC firing. These findings indicate that the proposed model is capable of reproducing in vitro experimental observations. By determining spike initiating dynamics, we have provided a fundamental link between dendritic Ca²⁺ spike and output APs, which could contribute to mechanically interpreting how dendritic Ca²⁺ activity participates in the simple computations of pyramidal neuron.

Direct Current Stimulation Alters Neuronal Input/Output Function

BACKGROUND

Direct current stimulation (DCS) affects both neuronal firing rate and synaptic efficacy. The neuronal input/output (I/O) function determines the likelihood that a neuron elicits an action potential in response to synaptic input of a given strength. Changes of the neuronal I/O function by DCS may underlie previous observations in animal models and human testing, yet have not been directly assessed.

OBJECTIVE

Test if the neuronal input/output function is affected by DCS METHODS: Using rat hippocampal brain slices and computational modeling, we provide evidence for how DCS modulates the neuronal I/O function.

RESULTS

We show for the first time that DCS modulates the likelihood of neuronal firing for a given and fixed synaptic input. Opposing polarization of soma and dendrite may have a synergistic effect for anodal stimulation, increasing the driving force of synaptic activity while simultaneously increasing spiking probability at the soma. For cathodal stimulation, however, the opposing effects tend to cancel. This results in an asymmetry in the strength of the effects of stimulation for opposite polarities.

CONCLUSIONS

Our results may explain the asymmetries observed in acute and long term effects of transcranial direct current stimulation.

Effects of polarization induced by non-weak electric fields on the excitability of elongated neurons with active dendrites

An externally-applied electric field can polarize a neuron, especially a neuron with elongated dendrites, and thus modify its excitability. Here we use a computational model to examine, predict, and explain these effects. We use a two-compartment Pinsky-Rinzel model neuron polarized by an electric potential difference imposed between its compartments, and we apply an injected ramp current. We vary three model parameters: the magnitude of the applied potential difference, the extracellular potassium concentration, and the rate of current injection. A study of the Time-To-First-Spike (TTFS) as a function of polarization leads to the identification of three regions of polarization strength that have different effects. In the weak region, the TTFS increases linearly with polarization. In the intermediate region, the TTFS increases either sub- or super-linearly, depending on the current injection rate and the extracellular potassium concentration. In the strong region, the TTFS decreases. Our results in the weak and strong region are consistent with experimental observations, and in the intermediate region, we predict novel effects that depend on experimentally-accessible parameters. We find that active channels in the dendrite play a key role in these effects. Our qualitative results were found to be robust over a wide range of inter-compartment conductances and the ratio of somatic to dendritic membrane areas. In addition, we discuss preliminary results where synaptic inputs replace the ramp injection protocol. The insights and conclusions were found to extend from our polarized PR model to a polarized PR model with I h dendritic currents. Finally, we discuss the degree to which our results may be generalized.

Cortical neuron activation induced by electromagnetic stimulation: a quantitative analysis via modelling and simulation

Previous simulation works concerned with the mechanism of non-invasive neuromodulation has isolated many of the factors that can influence stimulation potency, but an inclusive account of the interplay between these factors on realistic neurons is still lacking. To give a comprehensive investigation on the stimulation-evoked neuronal activation, we developed a simulation scheme which incorporates highly detailed physiological and morphological properties of pyramidal cells. The model was implemented on a multitude of neurons; their thresholds and corresponding activation points with respect to various field directions and pulse waveforms were recorded. The results showed that the simulated thresholds had a minor anisotropy and reached minimum when the field direction was parallel to the dendritic-somatic axis; the layer 5 pyramidal cells always had lower thresholds but substantial variances were also observed within layers; reducing pulse length could magnify the threshold values as well as the variance; tortuosity and arborization of axonal segments could obstruct action potential initiation. The dependence of the initiation sites on both the orientation and the duration of the stimulus implies that the cellular excitability might represent the result of the competition between various firing-capable axonal components, each with a unique susceptibility determined by the local geometry. Moreover, the measurements obtained in simulation intimately resemble recordings in physiological and clinical studies, which seems to suggest that, with minimum simplification of the neuron model, the cable theory-based simulation approach can have sufficient verisimilitude to give quantitatively accurate evaluation of cell activities in response to the externally applied field.

Multilevel computational models for predicting the cellular effects of noninvasive brain stimulation

Since 2000, there has been rapid acceleration in the use of tDCS in both clinical and cognitive neuroscience research, encouraged by the simplicity of the technique (two electrodes and a battery powered stimulator) and the perception that tDCS protocols can be simply designed by placing the anode over the cortex to "excite," and the cathode over cortex to "inhibit." A specific and predictive understanding of tDCS needs experimental data to be placed into a quantitative framework. Biologically constrained computational models provide a useful framework within which to interpret results from empirical studies and generate novel, testable hypotheses. Although not without caveats, computational models provide a tool for exploring cognitive and brain processes, are amenable to quantitative analysis, and can inspire novel empirical work that might be difficult to intuit simply by examining experimental results. We approach modeling the effects of tDCS on neurons from multiple levels: modeling the electric field distribution, modeling single-compartment effects, and finally with multicompartment neuron models.

Biophysical Insights into How Spike Threshold Depends on the Rate of Membrane Potential Depolarization in Type I and Type II Neurons

Dynamic spike threshold plays a critical role in neuronal input-output relations. In many neurons, the threshold potential depends on the rate of membrane potential depolarization (dV/dt) preceding a spike. There are two basic classes of neural excitability, i.e., Type I and Type II, according to input-output properties. Although the dynamical and biophysical basis of their spike initiation has been established, the spike threshold dynamic for each cell type has not been well described. Here, we use a biophysical model to investigate how spike threshold depends on dV/dt in two types of neuron. It is observed that Type II spike threshold is more depolarized and more sensitive to dV/dt than Type I. With phase plane analysis, we show that each threshold dynamic arises from the different separatrix and K+ current kinetics. By analyzing subthreshold properties of membrane currents, we find the activation of hyperpolarizing current prior to spike initiation is a major factor that regulates the threshold dynamics. The outward K+ current in Type I neuron does not activate at the perithresholds, which makes its spike threshold insensitive to dV/dt. The Type II K+ current activates prior to spike initiation and there is a large net hyperpolarizing current at the perithresholds, which results in a depolarized threshold as well as a pronounced threshold dynamic. These predictions are further attested in several other functionally equivalent cases of neural excitability. Our study provides a fundamental description about how intrinsic biophysical properties contribute to the threshold dynamics in Type I and Type II neurons, which could decipher their significant functions in neural coding.

Spike-frequency adaptation of a two-compartment neuron modulated by extracellular electric fields

Spike-frequency adaptation has been shown to play an important role in neural coding. Based on a reduced two-compartment model, here we investigate how two common adaptation currents, i.e., voltage-sensitive potassium current (I(M)) and calcium-sensitive potassium current (I(AHP)), modulate neuronal responses to extracellular electric fields. It is shown that two adaptation mechanisms lead to distinct effects on the dynamical behavior of the neuron to electric fields. These effects depend on a neuronal morphological parameter that characterizes the ratio of soma area to total membrane area and internal coupling conductance. In the case of I(AHP) current, changing the morphological parameter switches spike initiation dynamics between saddle-node on invariant cycle bifurcation and supercritical Hopf bifurcation, whereas it only switches between subcritical and supercritical Hopf bifurcations for I(M) current. Unlike the morphological parameter, internal coupling conductance is unable to alter the bifurcation scenario for both adaptation currents. We also find that the electric field threshold for triggering neuronal steady-state firing is determined by two parameters, especially by the morphological parameter. Furthermore, the neuron with I(AHP) current generates mixed-mode oscillations through the canard phenomenon for some small values of the morphological parameter. All these results suggest that morphological properties play a critical role in field-induced effects on neuronal dynamics, which could qualitatively alter the outcome of adaptation by modulating internal current between soma and dendrite. The findings are readily testable in experiments, which could help to reveal the mechanisms underlying how the neuron responds to electric field stimulus.

Input-output relation and energy efficiency in the neuron with different spike threshold dynamics

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na(+) and K(+) currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism.

Endogenous field feedback promotes the detectability for exogenous electric signal in the hybrid coupled population

This paper presents the endogenous electric field in chemical or electrical synaptic coupled networks, aiming to study the role of endogenous field feedback in the signal propagation in neural systems. It shows that the feedback of endogenous fields to network activities can reduce the required energy of the noise and enhance the transmission of input signals in hybrid coupled populations. As a common and important nonsynaptic interactive method among neurons, particularly, the endogenous filed feedback can not only promote the detectability of exogenous weak signal in hybrid coupled neural population but also enhance the robustness of the detectability against noise. Furthermore, with the increasing of field coupling strengths, the endogenous field feedback is conductive to the stochastic resonance by facilitating the transition of cluster activities from the no spiking to spiking regions. Distinct from synaptic coupling, the endogenous field feedback can play a role as internal driving force to boost the population activities, which is similar to the noise. Thus, it can help to transmit exogenous weak signals within the network in the absence of noise drive via the stochastic-like resonance.

Neuronal spike initiation modulated by extracellular electric fields

Based on a reduced two-compartment model, the dynamical and biophysical mechanism underlying the spike initiation of the neuron to extracellular electric fields is investigated in this paper. With stability and phase plane analysis, we first investigate in detail the dynamical properties of neuronal spike initiation induced by geometric parameter and internal coupling conductance. The geometric parameter is the ratio between soma area and total membrane area, which describes the proportion of area occupied by somatic chamber. It is found that varying it could qualitatively alter the bifurcation structures of equilibrium as well as neuronal phase portraits, which remain unchanged when varying internal coupling conductance. By analyzing the activating properties of somatic membrane currents at subthreshold potentials, we explore the relevant biophysical basis of spike initiation dynamics induced by these two parameters. It is observed that increasing geometric parameter could greatly decrease the intensity of the internal current flowing from soma to dendrite, which switches spike initiation dynamics from Hopf bifurcation to SNIC bifurcation; increasing internal coupling conductance could lead to the increase of this outward internal current, whereas the increasing range is so small that it could not qualitatively alter the spike initiation dynamics. These results highlight that neuronal geometric parameter is a crucial factor in determining the spike initiation dynamics to electric fields. The finding is useful to interpret the functional significance of neuronal biophysical properties in their encoding dynamics, which could contribute to uncovering how neuron encodes electric field signals.