Cable theory in neurons with active, linearized membranes

New Theoretical Model of Nerve Conduction in Unmyelinated Nerves

Nerve conduction in unmyelinated fibers has long been described based on the equivalent circuit model and cable theory. However, without the change in ionic concentration gradient across the membrane, there would be no generation or propagation of the action potential. Based on this concept, we employ a new conductive model focusing on the distribution of voltage-gated sodium ion channels and Coulomb force between electrolytes. Based on this new model, the propagation of the nerve conduction was suggested to take place far before the generation of action potential at each channel. We theoretically showed that propagation of action potential, which is enabled by the increasing Coulomb force produced by inflowing sodium ions, from one sodium ion channel to the next sodium channel would be inversely proportionate to the density of sodium channels on the axon membrane. Because the longitudinal number of sodium ion channel would be proportionate to the square root of channel density, the conduction velocity of unmyelinated nerves is theoretically shown to be proportionate to the square root of channel density. Also, from a viewpoint of equilibrium state of channel importation and degeneration, channel density was suggested to be proportionate to axonal diameter. Based on these simple basis, conduction velocity in unmyelinated nerves was theoretically shown to be proportionate to the square root of axonal diameter. This new model would also enable us to acquire more accurate and understandable vision on the phenomena in unmyelinated nerves in addition to the conventional electric circuit model and cable theory.

Incorporating inductances in tissue-scale models of cardiac electrophysiology

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and we examine the effect of intracellular and extracellular inductances on the virtual electrode phenomenon.

Branching morphology determines signal propagation dynamics in neurons

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

The role of negative conductances in neuronal subthreshold properties and synaptic integration

Based on passive cable theory, an increase in membrane conductance produces a decrease in the membrane time constant and input resistance. Unlike the classical leak currents, voltage-dependent currents have a nonlinear behavior which can create regions of negative conductance, despite the increase in membrane conductance (permeability). This negative conductance opposes the effects of the passive membrane conductance on the membrane input resistance and time constant, increasing their values and thereby substantially affecting the amplitude and time course of postsynaptic potentials at the voltage range of the negative conductance. This paradoxical effect has been described for three types of voltage-dependent inward currents: persistent sodium currents, L- and T-type calcium currents and ligand-gated glutamatergic N-methyl-D-aspartate currents. In this review, we describe the impact of the creation of a negative conductance region by these currents on neuronal membrane properties and synaptic integration. We also discuss recent contributions of the quasi-active cable approximation, an extension of the passive cable theory that includes voltage-dependent currents, and its effects on neuronal subthreshold properties.

Subthreshold linear modeling of dendritic trees: a computational approach

The design of communication systems based on the transmission of information through neurons is envisioned as a key technology for the pervasive interconnection of future wearable and implantable devices. While previous literature has mainly focused on modeling propagation of electrochemical spikes carrying natural information through the nervous system, in recent work the authors of this paper proposed the so-called subthreshold electrical stimulation as a viable technique to propagate artificial information through neurons. This technique promises to limit the interference with natural communication processes, and it can be successfully approximated with linear models. In this paper, a novel model is proposed to account for the subthreshold stimuli propagation from the dendritic tree to the soma of a neuron. A computational approach is detailed to obtain this model for a given realistic 3D dendritic tree with an arbitrary morphology. Numerical results from the model are obtained over a stimulation signal bandwidth of 1KHz, and compared with the results of a simulation through the NEURON software.

Competition between Persistent Na+ and Muscarine-Sensitive K+ Currents Shapes Perithreshold Resonance and Spike Tuning in CA1 Pyramidal Neurons

Neurons from many brain regions display intrinsic subthreshold theta-resonance, responding preferentially to theta-frequency oscillatory stimuli. Resonance may contribute to selective communication among neurons and to orchestrate brain rhythms. CA1 pyramidal neurons receive theta activity, generating place fields. In these neurons the expression of perithreshold frequency preference is controversial, particularly in the spiking regime, with evidence favoring either non-resonant (integrator-like) or resonant behavior. Perithreshold dynamics depends on the persistent Na⁺ current I_NaP developing above −70 mV and the muscarine-sensitive K⁺ current I_M activating above −60 mV. We conducted current and voltage clamp experiments in slices to investigate perithreshold excitability of CA1 neurons under oscillatory stimulation. Around 20% of neurons displayed perithreshold resonance that is expressed in spiking. The remaining neurons (~80%) acted as low-pass filters lacking frequency preference. Paired voltage clamp measurement of I_NaP and I_M showed that perithreshold activation of I_M is in general low while I_NaP is high enough to depolarize neurons toward threshold before resonance expression, explaining the most abundant non-resonant perithreshold behavior. Partial blockade of I_NaP by pharmacological tools or dynamic clamp changed non-resonant to resonant behavior. Furthermore, shifting I_M activation toward hyperpolarized potentials by dynamic clamp also transformed non-resonant neurons into resonant ones. We propose that the relative levels of I_NaP and I_M control perithreshold behavior of CA1 neurons constituting a gating mechanism for theta resonance in the spiking regime. Both currents are regulated by intracellular signaling and neuromodulators which may allow dynamic switching of perithreshold behavior between resonant and non-resonant.

Hybrid Scheme for Modeling Local Field Potentials from Point-Neuron Networks

With rapidly advancing multi-electrode recording technology, the local field potential (LFP) has again become a popular measure of neuronal activity in both research and clinical applications. Proper understanding of the LFP requires detailed mathematical modeling incorporating the anatomical and electrophysiological features of neurons near the recording electrode, as well as synaptic inputs from the entire network. Here we propose a hybrid modeling scheme combining efficient point-neuron network models with biophysical principles underlying LFP generation by real neurons. The LFP predictions rely on populations of network-equivalent multicompartment neuron models with layer-specific synaptic connectivity, can be used with an arbitrary number of point-neuron network populations, and allows for a full separation of simulated network dynamics and LFPs. We apply the scheme to a full-scale cortical network model for a ∼1 mm² patch of primary visual cortex, predict laminar LFPs for different network states, assess the relative LFP contribution from different laminar populations, and investigate effects of input correlations and neuron density on the LFP. The generic nature of the hybrid scheme and its public implementation in hybridLFPy form the basis for LFP predictions from other and larger point-neuron network models, as well as extensions of the current application with additional biological detail.

Response functions for electrically coupled neuronal network: a method of local point matching and its applications

Neuronal networks connected by electrical synapses, also referred to as gap junctions, are present throughout the entire central nervous system. Many instances of gap-junctional coupling are formed between dendritic arbours of individual cells, and these dendro-dendritic gap junctions are known to play an important role in mediating various brain rhythms in both normal and pathological states. The dynamics of such neuronal networks modelled by passive or quasi-active (resonant) membranes can be described by the Green's function which provides the fundamental input-output relationships of the entire network. One of the methods for calculating this response function is the so-called 'sum-over-trips' framework which enables the construction of the Green's function for an arbitrary network as a convergent infinite series solution. Here we propose an alternative and computationally efficient approach for constructing the Green's functions on dendro-dendritic gap junction-coupled neuronal networks which avoids any infinite terms in the solutions. Instead, the Green's function is constructed from the solution of a system of linear algebraic equations. We apply this new method to a number of systems including a simple single cell model and two-cell neuronal networks. We also demonstrate that the application of this novel approach allows one to reduce a model with complex dendritic formations to an equivalent model with a much simpler morphological structure.

Active subthreshold dendritic conductances shape the local field potential

Key points

The local field potential (LFP), the low‐frequency part of extracellular potentials recorded in neural tissue, is often used for probing neural circuit activity. Interpreting the LFP signal is difficult, however.

While the cortical LFP is thought mainly to reflect synaptic inputs onto pyramidal neurons, little is known about the role of the various subthreshold active conductances in shaping the LFP.

By means of biophysical modelling we obtain a comprehensive qualitative understanding of how the LFP generated by a single pyramidal neuron depends on the type and spatial distribution of active subthreshold currents.

For pyramidal neurons, the h‐type channels probably play a key role and can cause a distinct resonance in the LFP power spectrum.

Our results show that the LFP signal can give information about the active properties of neurons and imply that preferred frequencies in the LFP can result from those cellular properties instead of, for example, network dynamics.

Abstract

The main contribution to the local field potential (LFP) is thought to stem from synaptic input to neurons and the ensuing subthreshold dendritic processing. The role of active dendritic conductances in shaping the LFP has received little attention, even though such ion channels are known to affect the subthreshold neuron dynamics. Here we used a modelling approach to investigate the effects of subthreshold dendritic conductances on the LFP. Using a biophysically detailed, experimentally constrained model of a cortical pyramidal neuron, we identified conditions under which subthreshold active conductances are a major factor in shaping the LFP. We found that, in particular, the hyperpolarization‐activated inward current, I_h, can have a sizable effect and cause a resonance in the LFP power spectral density. To get a general, qualitative understanding of how any subthreshold active dendritic conductance and its cellular distribution can affect the LFP, we next performed a systematic study with a simplified model. We found that the effect on the LFP is most pronounced when (1) the synaptic drive to the cell is asymmetrically distributed (i.e. either basal or apical), (2) the active conductances are distributed non‐uniformly with the highest channel densities near the synaptic input and (3) when the LFP is measured at the opposite pole of the cell relative to the synaptic input. In summary, we show that subthreshold active conductances can be strongly reflected in LFP signals, opening up the possibility that the LFP can be used to characterize the properties and cellular distributions of active conductances.

The local field potential (LFP), the low‐frequency part of extracellular potentials recorded in neural tissue, is often used for probing neural circuit activity. Interpreting the LFP signal is difficult, however.

While the cortical LFP is thought mainly to reflect synaptic inputs onto pyramidal neurons, little is known about the role of the various subthreshold active conductances in shaping the LFP.

By means of biophysical modelling we obtain a comprehensive qualitative understanding of how the LFP generated by a single pyramidal neuron depends on the type and spatial distribution of active subthreshold currents.

For pyramidal neurons, the h‐type channels probably play a key role and can cause a distinct resonance in the LFP power spectrum.

Our results show that the LFP signal can give information about the active properties of neurons and imply that preferred frequencies in the LFP can result from those cellular properties instead of, for example, network dynamics.

Accommodation to hyperpolarization of human axons assessed in the frequency domain

Human axons in vivo were subjected to subthreshold currents with a threshold impedance amplitude profile to allow the use of frequency domain techniques to determine the propensity for resonant behavior and to clarify the relative contributions of different ion channels to their low-frequency responsiveness. Twenty-four studies were performed on the motor and sensory axons of the median nerve in six subjects. The response to oscillatory currents was tested between direct current (DC) and 16 Hz. A resonant peak at ∼2-2.5 Hz was found in the response of hyperpolarized axons, but there was only a small broad response in axons at resting membrane potential (RMP). A mathematical model of axonal excitability developed using DC pulses provided a good fit to the frequency response for human axons and indicated that the hyperpolarization-activated current Ih and the slow potassium current IKs are principally responsible for the resonance. However, the results indicate that if axons are hyperpolarized by more than -60% of resting threshold, the only conductances that are appreciably active are Ih and the leak conductance, i.e., that the activity of these conductances can be studied in vivo virtually in isolation at hyperpolarized membrane potentials. Given that the leak conductance dampens resonance, it is suggested that the -60% hyperpolarization used here is optimal for Ih As expected, differences between the frequency responses of motor and sensory axons were present and best explained by reduced slow potassium conductance GKs, up-modulation of Ih, and increased persistent Na(+) current INaP (due to depolarization of RMP) in sensory axons.

The Global Spike: Conserved Dendritic Properties Enable Unique Ca2+ Spike Generation in Low-Threshold Spiking Neurons

Low-threshold Ca²⁺ spikes (LTS) are an indispensible signaling mechanism for neurons in areas including the cortex, cerebellum, basal ganglia, and thalamus. They have critical physiological roles and have been strongly associated with disorders including epilepsy, Parkinson's disease, and schizophrenia. However, although dendritic T-type Ca²⁺ channels have been implicated in LTS generation, because the properties of low-threshold spiking neuron dendrites are unknown, the precise mechanism has remained elusive. Here, combining data from fluorescence-targeted dendritic recordings and Ca²⁺ imaging from low-threshold spiking cells in rat brain slices with computational modeling, the cellular mechanism responsible for LTS generation is established. Our data demonstrate that key somatodendritic electrical conduction properties are highly conserved between glutamatergic thalamocortical neurons and GABAergic thalamic reticular nucleus neurons and that these properties are critical for LTS generation. In particular, the efficiency of soma to dendrite voltage transfer is highly asymmetric in low-threshold spiking cells, and in the somatofugal direction, these neurons are particularly electrotonically compact. Our data demonstrate that LTS have remarkably similar amplitudes and occur synchronously throughout the dendritic tree. In fact, these Ca²⁺ spikes cannot occur locally in any part of the cell, and hence we reveal that LTS are generated by a unique whole-cell mechanism that means they always occur as spatially global spikes. This all-or-none, global electrical and biochemical signaling mechanism clearly distinguishes LTS from other signals, including backpropagating action potentials and dendritic Ca²⁺/NMDA spikes, and has important consequences for dendritic function in low-threshold spiking neurons.

SIGNIFICANCE STATEMENT Low-threshold Ca²⁺ spikes (LTS) are critical for important physiological processes, including generation of sleep-related oscillations, and are implicated in disorders including epilepsy, Parkinson's disease, and schizophrenia. However, the mechanism underlying LTS generation in neurons, which is thought to involve dendritic T-type Ca²⁺ channels, has remained elusive due to a lack of knowledge of the dendritic properties of low-threshold spiking cells. Combining dendritic recordings, two-photon Ca²⁺ imaging, and computational modeling, this study reveals that dendritic properties are highly conserved between two prominent low-threshold spiking neurons and that these properties underpin a whole-cell somatodendritic spike generation mechanism that makes the LTS a unique global electrical and biochemical signal in neurons.

A Sparse Reformulation of the Green's Function Formalism Allows Efficient Simulations of Morphological Neuron Models

We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale as O(n) with the number n of inputs locations, contrary to the previously reported O(n(2)) scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.

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.

Firing-rate resonances in the peripheral auditory system of the cricket, Gryllus bimaculatus

In many communication systems, information is encoded in the temporal pattern of signals. For rhythmic signals that carry information in specific frequency bands, a neuronal system may profit from tuning its inherent filtering properties towards a peak sensitivity in the respective frequency range. The cricket Gryllus bimaculatus evaluates acoustic communication signals of both conspecifics and predators. The song signals of conspecifics exhibit a characteristic pulse pattern that contains only a narrow range of modulation frequencies. We examined individual neurons (AN1, AN2, ON1) in the peripheral auditory system of the cricket for tuning towards specific modulation frequencies by assessing their firing-rate resonance. Acoustic stimuli with a swept-frequency envelope allowed an efficient characterization of the cells' modulation transfer functions. Some of the examined cells exhibited tuned band-pass properties. Using simple computational models, we demonstrate how different, cell-intrinsic or network-based mechanisms such as subthreshold resonances, spike-triggered adaptation, as well as an interplay of excitation and inhibition can account for the experimentally observed firing-rate resonances. Therefore, basic neuronal mechanisms that share negative feedback as a common theme may contribute to selectivity in the peripheral auditory pathway of crickets that is designed towards mate recognition and predator avoidance.

Mathematical modeling of subthreshold resonant properties in pyloric dilator neurons

Various types of neurons exhibit subthreshold resonance oscillation (preferred frequency response) to fluctuating sinusoidal input currents. This phenomenon is well known to influence the synaptic plasticity and frequency of neural network oscillation. This study evaluates the resonant properties of pacemaker pyloric dilator (PD) neurons in the central pattern generator network through mathematical modeling. From the pharmacological point of view, calcium currents cannot be blocked in PD neurons without removing the calcium-dependent potassium current. Thus, the effects of calcium (I_Ca) and calcium-dependent potassium (I_KCa) currents on resonant properties remain unclear. By taking advantage of Hodgkin-Huxley-type model of neuron and its equivalent RLC circuit, we examine the effects of changing resting membrane potential and those ionic currents on the resonance. Results show that changing the resting membrane potential influences the amplitude and frequency of resonance so that the strength of resonance (Q-value) increases by both depolarization and hyperpolarization of the resting membrane potential. Moreover, hyperpolarization-activated inward current (I_h) and I_Ca (in association with I_KCa) are dominant factors on resonant properties at hyperpolarized and depolarized potentials, respectively. Through mathematical analysis, results indicate that I_h and I_KCa affect the resonant properties of PD neurons. However, I_Ca only has an amplifying effect on the resonance amplitude of these neurons.

Dorsoventral differences in Kv7/M-current and its impact on resonance, temporal summation and excitability in rat hippocampal pyramidal cells

In rodent hippocampi, the connections, gene expression and functions differ along the dorsoventral (D-V) axis. CA1 pyramidal cells show increasing excitability along the D-V axis, although the underlying mechanism is not known. In the present study, we investigated how the M-current (IM ), caused by Kv7/M (KCNQ) potassium channels, and known to often control neuronal excitability, contributes to D-V differences in intrinsic properties of CA1 pyramidal cells. Using whole-cell patch clamp recordings and the selective Kv7/M blocker 10,10-bis(4-pyridinylmethyl)-9(10H)-anthracenone dihydrochloride (XE991) in hippocampal slices from 3- to 4-week-old rats, we found that: (i) IM had a stronger impact on subthreshold electrical properties in dorsal than ventral CA1 pyramidal cells, including input resistance, temporal summation of artificial synaptic potentials, and M-resonance; (ii) IM activated at more negative potentials (left-shifted) and had larger peak amplitude in the dorsal than ventral CA1; and (iii) the initial spike threshold (during ramp depolarizations) was elevated, and the medium after-hyperpolarization and spike frequency adaptation were increased (i.e. excitability was lower) in the dorsal rather than ventral CA1. These differences were abolished or reduced by application of XE991, indicating that they were caused by IM . Thus, it appears that IM has stronger effects in dorsal than in ventral rat CA1 pyramidal cells because of a larger maximal M-conductance and left-shifted activation curve in the dorsal cells. These mechanisms may contribute to D-V differences in the rate and phase coding of position by CA1 place cells, and may also enhance epileptiform activity in ventral CA1.

Neural population tuning links visual cortical anatomy to human visual perception

The anatomy of cerebral cortex is characterized by two genetically independent variables, cortical thickness and cortical surface area, that jointly determine cortical volume. It remains unclear how cortical anatomy might influence neural response properties and whether such influences would have behavioral consequences. Here, we report that thickness and surface area of human early visual cortices exert opposite influences on neural population tuning with behavioral consequences for perceptual acuity. We found that visual cortical thickness correlated negatively with the sharpness of neural population tuning and the accuracy of perceptual discrimination at different visual field positions. In contrast, visual cortical surface area correlated positively with neural population tuning sharpness and perceptual discrimination accuracy. Our findings reveal a central role for neural population tuning in linking visual cortical anatomy to visual perception and suggest that a perceptually advantageous visual cortex is a thinned one with an enlarged surface area.

Cell-type-specific resonances shape the responses of striatal neurons to synaptic input

Neurons respond to synaptic inputs in cell-type-specific ways. Each neuron type may thus respond uniquely to shared patterns of synaptic input. We applied statistically identical barrages of artificial synaptic inputs to four striatal cell types to assess differences in their responses to a realistic input pattern. Each interneuron type fired in phase with a specific input-frequency component. The fast-spiking interneuron fired in relation to the gamma-band (and higher) frequencies, the low-threshold spike interneuron to the beta-band frequencies, and the cholinergic neurons to the delta-band frequencies. Low-threshold spiking and cholinergic interneurons showed input impedance resonances at frequencies matching their spiking resonances. Fast-spiking interneurons showed resonance of input impedance but at lower than gamma frequencies. The spiny projection neuron's frequency preference did not have a fixed frequency but instead tracked its own firing rate. Spiny cells showed no input impedance resonance. Striatal interneurons are each tuned to a specific frequency band corresponding to the major frequency components of local field potentials. Their influence in the circuit may fluctuate along with the contribution of that frequency band to the input. In contrast, spiny neurons may tune to any of the frequency bands by a change in firing rate.

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.

Systematic analysis of the contributions of stochastic voltage gated channels to neuronal noise

Electrical signaling in neurons is mediated by the opening and closing of large numbers of individual ion channels. The ion channels' state transitions are stochastic and introduce fluctuations in the macroscopic current through ion channel populations. This creates an unavoidable source of intrinsic electrical noise for the neuron, leading to fluctuations in the membrane potential and spontaneous spikes. While this effect is well known, the impact of channel noise on single neuron dynamics remains poorly understood. Most results are based on numerical simulations. There is no agreement, even in theoretical studies, on which ion channel type is the dominant noise source, nor how inclusion of additional ion channel types affects voltage noise. Here we describe a framework to calculate voltage noise directly from an arbitrary set of ion channel models, and discuss how this can be use to estimate spontaneous spike rates.

Spatially distributed dendritic resonance selectively filters synaptic input

An important task performed by a neuron is the selection of relevant inputs from among thousands of synapses impinging on the dendritic tree. Synaptic plasticity enables this by strenghtening a subset of synapses that are, presumably, functionally relevant to the neuron. A different selection mechanism exploits the resonance of the dendritic membranes to preferentially filter synaptic inputs based on their temporal rates. A widely held view is that a neuron has one resonant frequency and thus can pass through one rate. Here we demonstrate through mathematical analyses and numerical simulations that dendritic resonance is inevitably a spatially distributed property; and therefore the resonance frequency varies along the dendrites, and thus endows neurons with a powerful spatiotemporal selection mechanism that is sensitive both to the dendritic location and the temporal structure of the incoming synaptic inputs.

Electrical resonance in the θ frequency range in olfactory amygdala neurons

The cortical amygdala receives direct olfactory inputs and is thought to participate in processing and learning of biologically relevant olfactory cues. As for other brain structures implicated in learning, the principal neurons of the anterior cortical nucleus (ACo) exhibit intrinsic subthreshold membrane potential oscillations in the θ-frequency range. Here we show that nearly 50% of ACo layer II neurons also display electrical resonance, consisting of selective responsiveness to stimuli of a preferential frequency (2–6 Hz). Their impedance profile resembles an electrical band-pass filter with a peak at the preferred frequency, in contrast to the low-pass filter properties of other neurons. Most ACo resonant neurons displayed frequency preference along the whole subthreshold voltage range. We used pharmacological tools to identify the voltage-dependent conductances implicated in resonance. A hyperpolarization-activated cationic current depending on HCN channels underlies resonance at resting and hyperpolarized potentials; notably, this current also participates in resonance at depolarized subthreshold voltages. KV7/KCNQ K⁺ channels also contribute to resonant behavior at depolarized potentials, but not in all resonant cells. Moreover, resonance was strongly attenuated after blockade of voltage-dependent persistent Na⁺ channels, suggesting an amplifying role. Remarkably, resonant neurons presented a higher firing probability for stimuli of the preferred frequency. To fully understand the mechanisms underlying resonance in these neurons, we developed a comprehensive conductance-based model including the aforementioned and leak conductances, as well as Hodgkin and Huxley-type channels. The model reproduces the resonant impedance profile and our pharmacological results, allowing a quantitative evaluation of the contribution of each conductance to resonance. It also replicates selective spiking at the resonant frequency and allows a prediction of the temperature-dependent shift in resonance frequency. Our results provide a complete characterization of the resonant behavior of olfactory amygdala neurons and shed light on a putative mechanism for network activity coordination in the intact brain.

Somatic versus dendritic resonance: differential filtering of inputs through non-uniform distributions of active conductances

Synaptic inputs to neurons are processed in a frequency-dependent manner, with either low-pass or resonant response characteristics. These types of filtering play a key role in the frequency-specific information flow in neuronal networks. While the generation of resonance by specific ionic conductances is well investigated, less attention has been paid to the spatial distribution of the resonance-generating conductances across a neuron. In pyramidal neurons – one of the major excitatory cell-types in the mammalian brain – a steep gradient of resonance-generating h-conductances with a 60-fold increase towards distal dendrites has been demonstrated experimentally. Because the dendritic trees of these cells are large, spatial compartmentalization of resonant properties can be expected. Here, we use mathematical descriptions of spatially extended neurons to investigate the consequences of such a distal, dendritic localization of h-conductances for signal processing. While neurons with short dendrites do not exhibit a pronounced compartmentalization of resonance, i.e. the filter properties of dendrites and soma are similar, we find that neurons with longer dendrites ( space constant) can show distinct filtering of dendritic and somatic inputs due to electrotonic segregation. Moreover, we show that for such neurons, experimental classification as resonant versus nonresonant can be misleading when based on somatic recordings, because for these morphologies a dendritic resonance could easily be undetectable when using somatic input. Nevertheless, noise-driven membrane-potential oscillations caused by dendritic resonance can propagate to the soma where they can be recorded, hence contrasting with the low-pass filtering at the soma. We conclude that non-uniform distributions of active conductances can underlie differential filtering of synaptic input in neurons with spatially extended dendrites, like pyramidal neurons, bearing relevance for the localization-dependent targeting of synaptic input pathways to these cells.

Resonance in neocortical neurons and networks

Neocortical networks produce oscillations that often correspond to characteristic physiological or pathological patterns. However, the mechanisms underlying the generation of and the transitions between such oscillatory states remain poorly understood. In this study, we examined resonance in mouse layer V neocortical pyramidal neurons. To accomplish this, we employed standard electrophysiology to describe cellular resonance parameters. Bode plot analysis revealed a range of resonance magnitude values in layer V neurons and demonstrated that both magnitude and phase response characteristics of layer V neocortical pyramidal neurons are modulated by changes in the extracellular environment. Specifically, increased resonant frequencies and total inductive areas were observed at higher extracellular potassium concentrations and more hyperpolarised membrane potentials. Experiments using pharmacological agents suggested that current through hyperpolarization-activated cyclic nucleotide-gated channels (I(h) ) acts as the primary driver of resonance in these neurons, with other potassium currents, such as A-type potassium current and delayed-rectifier potassium current (Kv1.4 and Kv1.1, respectively), contributing auxiliary roles. The persistent sodium current was also shown to play a role in amplifying the magnitude of resonance without contributing significantly to the phase response. Although resonance effects in individual neurons are small, their properties embedded in large networks may significantly affect network behavior and may have potential implications for pathological processes.

The effect of dendritic voltage-gated conductances on the neuronal impedance: a quantitative model

Neuronal impedance characterizes the magnitude and timing of the subthreshold response of a neuron to oscillatory input at a given frequency. It is known to be influenced by both the morphology of the neuron and the presence of voltage-gated conductances in the cell membrane. Most existing theoretical accounts of neuronal impedance considered the effects of voltage-gated conductances but neglected the spatial extent of the cell, while others examined spatially extended dendrites with a passive or spatially uniform quasi-active membrane. We derived an explicit mathematical expression for the somatic input impedance of a model neuron consisting of a somatic compartment coupled to an infinite dendritic cable which contained voltage-gated conductances, in the more general case of non-uniform dendritic membrane potential. The validity and generality of this model was verified through computer simulations of various model neurons. The analytical model was then applied to the analysis of experimental data from real CA1 pyramidal neurons. The model confirmed that the biophysical properties and predominantly dendritic localization of the hyperpolarization-activated cation current I (h) were important determinants of the impedance profile, but also predicted a significant contribution from a depolarization-activated fast inward current. Our calculations also implicated the interaction of I (h) with amplifying currents as the main factor governing the shape of the impedance-frequency profile in two types of hippocampal interneuron. Our results provide not only a theoretical advance in our understanding of the frequency-dependent behavior of nerve cells, but also a practical tool for the identification of candidate mechanisms that determine neuronal response properties.

The stochastic properties of input spike trains control neuronal arithmetic

In the nervous system, the representation of signals is based predominantly on the rate and timing of neuronal discharges. In most everyday tasks, the brain has to carry out a variety of mathematical operations on the discharge patterns. Recent findings show that even single neurons are capable of performing basic arithmetic on the sequences of spikes. However, the interaction of the two spike trains, and thus the resulting arithmetic operation may be influenced by the stochastic properties of the interacting spike trains. If we represent the individual discharges as events of a random point process, then an arithmetical operation is given by the interaction of two point processes. Employing a probabilistic model based on detection of coincidence of random events and complementary computer simulations, we show that the point process statistics control the arithmetical operation being performed and, particularly, that it is possible to switch from subtraction to division solely by changing the distribution of the inter-event intervals of the processes. Consequences of the model for evaluation of binaural information in the auditory brainstem are demonstrated. The results accentuate the importance of the stochastic properties of neuronal discharge patterns for information processing in the brain; further studies related to neuronal arithmetic should therefore consider the statistics of the interacting spike trains.

Weak resonant double Hopf bifurcations in an inertial four-neuron model with time delay

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms

Cortical information processing originates from the exchange of action potentials between many cell types. To capture the essence of these interactions, it is of critical importance to build mathematical models that reflect the characteristic features of spike generation in individual neurons. We propose a framework to automatically extract such features from current-clamp experiments, in particular the passive properties of a neuron (i.e., membrane time constant, reversal potential, and capacitance), the spike-triggered adaptation currents, as well as the dynamics of the action potential threshold. The stochastic model that results from our maximum likelihood approach accurately predicts the spike times, the subthreshold voltage, the firing patterns, and the type of frequency-current curve. Extracting the model parameters for three cortical cell types revealed that cell types show highly significant differences in the time course of the spike-triggered currents and moving threshold, that is, in their adaptation and refractory properties but not in their passive properties. In particular, GABAergic fast-spiking neurons mediate weak adaptation through spike-triggered currents only, whereas regular spiking excitatory neurons mediate adaptation with both moving threshold and spike-triggered currents. GABAergic nonfast-spiking neurons combine the two distinct adaptation mechanisms with reduced strength. Differences between cell types are large enough to enable automatic classification of neurons into three different classes. Parameter extraction is performed for individual neurons so that we find not only the mean parameter values for each neuron type but also the spread of parameters within a group of neurons, which will be useful for future large-scale computer simulations.

Properties and functional implications of I (h) in hippocampal area CA3 interneurons

The present study examines the biophysical properties and functional implications of I (h) in hippocampal area CA3 interneurons with somata in strata radiatum and lacunosum-moleculare. Characterization studies showed a small maximum h-conductance (2.6 ± 0.3 nS, n = 11), shallow voltage dependence with a hyperpolarized half-maximal activation (V (1/2) = -91 mV), and kinetics characterized by double-exponential functions. The functional consequences of I (h) were examined with regard to temporal summation and impedance measurements. For temporal summation experiments, 5-pulse mossy fiber input trains were activated. Blocking I (h) with 50 μM ZD7288 resulted in an increase in temporal summation, suggesting that I (h) supports sensitivity of response amplitude to relative input timing. Impedance was assessed by applying sinusoidal current commands. From impedance measurements, we found that I (h) did not confer theta-band resonance, but flattened the impedance-frequency relations instead. Double immunolabeling for hyperpolarization-activated cyclic nucleotide-gated proteins and glutamate decarboxylase 67 suggests that all four subunits are present in GABAergic interneurons from the strata considered for electrophysiological studies. Finally, a model of I (h) was employed in computational analyses to confirm and elaborate upon the contributions of I (h) to impedance and temporal summation.

Role of active dendritic conductances in subthreshold input integration

Dendrites of many types of neurons contain voltage-dependent conductances that are active at subthreshold membrane potentials. To understand the computations neurons perform it is key to understand the role of active dendrites in the subthreshold processing of synaptic inputs. We examine systematically how active dendritic conductances affect the time course of postsynaptic potentials propagating along dendrites, and how they affect the interaction between such signals. Voltage-dependent currents can be classified into two types that have qualitatively different effects on subthreshold input responses: regenerative dendritic currents boost and broaden EPSPs, while restorative currents attenuate and narrow EPSPs. Importantly, the effects of active dendritic currents on EPSP shape increase as the EPSP travels along the dendrite. The effectiveness of active currents in modulating the EPSP shape is determined by their activation time constant: the faster it is, the stronger the effect on EPSP amplitude, while the largest effects on EPSP width occur when it is comparable to the membrane time constant. We finally demonstrate that the two current types can differentially improve precision and robustness of neural computations: restorative currents enhance coincidence detection of dendritic inputs, whereas direction selectivity to sequences of dendritic inputs is enhanced by regenerative dendritic currents.

Persistent currents and discharge patterns in rat hindlimb motoneurons

We report here the first direct measurements of persistent inward currents (PICs) in rat hindlimb motoneurons, obtained from ketamine-xylazine anesthetized rats during slow voltage ramps performed by single-electrode somatic voltage clamp. Most motoneurons expressed PICs and current-voltage (I-V) relations often contained a negative-slope region (NSR; 13/19 cells). PICs activated at -52.7 ± 3.89 mV, 9 mV negative to spike threshold. NSR onset was -44.2 ± 4.1 mV. PIC amplitudes were assessed by maximum inward currents measured relative to extrapolated leak current and to NSR-onset current. PIC conductance at potentials just positive to activation was assessed by the relative change in slope conductance (g(in)/g(leak)). PIC amplitudes varied widely; some exceeded 5 and 10 nA relative to current at NSR onset or leak current, respectively. PIC amplitudes did not vary significantly with input conductance, but PIC amplitudes normalized by recruitment current decreased with increasing input conductance. Similarly, g(in)/g(leak) decreased with increasing input conductance. Currents near resting potential on descending limbs of I-V relations were often outward, relative to ascending-limb currents. This residual outward current was correlated with increases in leak conductance on the descending limb and with input conductance. Excluding responses with accommodation, residual outward currents matched differences between recruitment and derecruitment currents, suggesting a role for residual outward current in frequency adaptation. Comparison of potentials for PIC activation and NSR onset with interspike trajectories during discharge demonstrated correspondence between PIC activation and frequency-current (f-I) range boundaries. Contributions of persistent inward and outward currents to motoneuron discharge characteristics are discussed.

Mode locking in a periodically forced resonate-and-fire neuron model

The resonate-and-fire (RF) model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system (impact oscillator). We analyze the response of the RF neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one-dimensional time map. Based on such a firing map, we describe mode-locked solutions and their stability, leading to the so-called Arnol'd tongues. The boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow. Despite the fact that the periodically driven RF system shows periodic firing, its behavior may become chaotic when the forcing frequency is near the resonant frequency. We compare these results to numerical simulations of the model undergoing sinusoidal forcing. Furthermore, upon varying a system parameter, the RF system can be reduced to the integrate-and-fire system and in this case we show the consistency of the results on mode-locked solutions.

The role of ongoing dendritic oscillations in single-neuron dynamics

The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as both temporally and spatially localized. Under this localist account, neurons compute near-instantaneous mappings from their current input to their current output, brought about by somatic summation of dendritic contributions that are generated in functionally segregated compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought; notably that local dendritic activity may be a mechanism for generating on-going whole-cell voltage oscillations.

A central issue in biology is how local processes yield global consequences. This is especially relevant for neurons since these spatially extended cells process local synaptic inputs to generate global action potential output. The dendritic tree of a neuron, which receives most of the inputs, expresses ion channels that can generate nonlinear dynamics. A prominent phenomenon resulting from such ion channels are voltage oscillations. The distribution of the active membrane channels throughout the cell is often highly non-uniform. This can turn the dendritic tree into a network of sparsely spaced local oscillators. Here we analyze whether local dendritic oscillators can produce cell-wide voltage oscillations. Our mathematical theory shows that indeed even when the dendritic oscillators are weakly coupled, they lock their phases and give global oscillations. We show how the biophysical properties of the dendrites determine the global locking and how it can be controlled by synaptic inputs. As a consequence of global locking, even individual synaptic inputs can affect the timing of action potentials. In fact, dendrites locking in synchrony can lead to sustained firing of the cell. We show that dendritic trees can be bistable, with dendrites locking in either synchrony or asynchrony, which may provide a novel mechanism for single cell-based memory.

An oscillatory circuit underlying the detection of disruptions in temporally-periodic patterns

Neurons in diverse brain areas can respond to the interruption of a regular stimulus pattern by firing bursts of spikes. Here we describe a simple model, which permits such responses to periodic stimuli over a substantial frequency range. Focusing on the omitted stimulus response (OSR) in isolated retinas subjected to periodic patterns of dark flashes, we develop a biophysically-realistic model which accounts for resonances in ON bipolar cells. The bipolar cell terminal contains an LRC oscillator whose inductance is modulated by a transient calcium concentration, thus adjusting its resonant frequency to approximately match that of the stimulus. The model reproduces ganglion cell outputs, which sum the ON and OFF bipolar pathways, and it responds to omitted flashes with approximately constant latencies, as observed experimentally.

Spike-triggered averages for passive and resonant neurons receiving filtered excitatory and inhibitory synaptic drive

A path-integral approach is developed for the analysis of spike-triggered average quantities in neurons with voltage-gated subthreshold currents. Using a linearization procedure to reduce the models to the generalized integrate-and-fire form, analytical expressions are obtained in an experimentally relevant limit of fluctuation-driven firing. The influences of voltage-gated channels as well as excitatory and inhibitory synaptic filtering are shown to affect significantly the neuronal dynamics prior to the spike. Analytical forms are given for all relevant physiological quantities, such as the mean voltage triggered to the spike, mean current flowing through voltage-gated channels, and the mean excitatory and inhibitory conductance waveforms prior to a spike. The mathematical results are shown to be in good agreement with numerical simulations of the underlying nonlinear conductance-based models. The method promises to provide a useful analytical tool for the prediction and interpretation of the temporal structure of spike-triggered averages measured experimentally.

The h channel mediates location dependence and plasticity of intrinsic phase response in rat hippocampal neurons

The presence of phenomenological inductances in neuronal membrane has been known for more than one-half a century. Despite this, the dramatic contributions of such inductive elements to the amplitude and, especially, phase of neuronal impedance, and their roles in modulating temporal dynamics of neuronal responses have surprisingly remained unexplored. In this study, we demonstrate that the h channel contributes a location-dependent and plastic phenomenological inductive component to the input impedance of CA1 pyramidal neurons. Specifically, we show that the h channels introduce an apparent negative delay in the local voltage response of these neurons with respect to the injected current within the theta frequency range. The frequency range and the extent of this lead expand with increases in h current either through hyperpolarization, or with increasing distance of dendritic location from the soma. We also demonstrate that a spatially widespread increase in this inductive phase component accompanies long-term potentiation. Finally, using impedance analysis, we show that both location and activity dependence of intrinsic phase response are attributable not to changes in a capacitive or a leak component, but to changes in h-channel properties. Our results suggest that certain voltage-gated ion channels can differentially regulate internal time delays within neurons, thus providing them with an independent control mechanism in temporal coding of neuronal information. Our analyses and results also establish impedance as a powerful measure of intrinsic dynamics and excitability, given that it quantifies temporal relationships among signals and excitability as functions of input frequency.

Active dendrites: colorful wings of the mysterious butterflies

Santiago Ramón y Cajal had referred to neurons as the 'mysterious butterflies of the soul.' Wings of these butterflies--their dendrites--were traditionally considered as passive integrators of synaptic information. Owing to a growing body of experimental evidence, it is now widely accepted that these wings are colorful, endowed with a plethora of active conductances, with each family of these butterflies made of distinct hues and shades. Furthermore, rapidly evolving recent literature also provides direct and indirect demonstrations for activity-dependent plasticity of these active conductances, pointing toward chameleonic adaptability in these hues. These experimental findings firmly establish the immense computational power of a single neuron, and thus constitute a turning point toward the understanding of various aspects of neuronal information processing. In this brief historical perspective, we track important milestones in the chameleonic transmogrification of these mysterious butterflies.

Resonant or not, two amplification modes of proprioceptive inputs by persistent inward currents in spinal motoneurons

Why do motoneurons possess two persistent inward currents (PICs), a fast sodium current and a slow calcium current? To answer this question, we replaced the natural PICs with dynamic clamp-imposed artificial PICs at the soma of spinal motoneurons of anesthetized cats. We investigated how PICs with different kinetics (1-100 ms) amplify proprioceptive inputs. We showed that their action depends on the presence or absence of a resonance created by the I(h) current. In resonant motoneurons, a fast PIC enhances the resonance and amplifies the dynamic component of Ia inputs elicited by ramp-and-hold muscle stretches. This facilitates the recruitment of these motoneurons, which likely innervate fast contracting motor units developing large forces, e.g., to restore balance or produce ballistic movements. In nonresonant motoneurons, in contrast, a fast PIC easily triggers plateau potentials, which leads to a dramatic amplification of the static component of Ia inputs. This likely facilitates the recruitment of these motoneurons, innervating mostly slowly contracting and fatigue-resistant motor units, during postural activities. Finally, a slow PIC may switch a resonant motoneuron to nonresonant by counterbalancing I(h), thus changing the action of the fast PIC. A modeling study shows that I(h) needs to be located on the dendrites to create the resonance, and it predicts that dendritic PICs amplify synaptic input in the same manner as somatic PICs.

Measurement and analysis of postsynaptic potentials using a novel voltage-deconvolution method

Accurate measurement of postsynaptic potential amplitudes is a central requirement for the quantification of synaptic strength, dynamics of short-term and long-term plasticity, and vesicle-release statistics. However, the intracellular voltage is a filtered version of the underlying synaptic signal and so a method of accounting for the distortion caused by overlapping postsynaptic potentials must be used. Here a voltage-deconvolution technique is demonstrated that defilters the entire voltage trace to reveal an underlying signal of well-separated synaptic events. These isolated events can be cropped out and reconvolved to yield a set of isolated postsynaptic potentials from which voltage amplitudes may be measured directly-greatly simplifying this common task. The method also has the significant advantage of providing a higher temporal resolution of the dynamics of the underlying synaptic signal. The versatility of the method is demonstrated by a variety of experimental examples, including excitatory and inhibitory connections to neurons with passive membranes and those with activated voltage-gated currents. The deconvolved current-clamp voltage has many features in common with voltage-clamp current measurements. These similarities are analyzed using cable theory and a multicompartment cell reconstruction, as well as direct comparison to voltage-clamp experiments.