A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

Excitability and Threshold Mechanism for Enhanced Neuronal Response Induced by Inhibition Preceding Excitation

Postinhibitory facilitation (PIF) of neural firing presents a paradoxical phenomenon that the inhibitory effect induces enhancement instead of reduction of the firing activity, which plays important roles in sound location of the auditory nervous system, awaited theoretical explanations. In the present paper, excitability and threshold mechanism for the PIF phenomenon is presented in the Morris-Lecar model with type I, II, and III excitabilities. Firstly, compared with the purely excitatory stimulations applied to the steady state, the inhibitory preceding excitatory stimulation to form pairs induces the firing rate increased for type II and III excitabilities instead of type I excitability, when the interval between the inhibitory and excitatory stimulation within each pair is suitable. Secondly, the threshold mechanism for the PIF phenomenon is acquired. For type II and III excitabilities, the inhibitory stimulation induces subthreshold oscillations around the steady state. During the middle and ending phase of the ascending part and the beginning phase of the descending part within a period of the subthreshold oscillations, the threshold to evoke an action potential by an excitatory stimulation becomes weaker, which is the cause for the PIF phenomenon. Last, a theoretical estimation for the range of the interval between the inhibitory and excitatory stimulation for the PIF phenomenon is acquired, which approximates half of the intrinsic period of the subthreshold oscillations for the relatively strong stimulations and becomes narrower for the relatively weak stimulations. The interval for the PIF phenomenon is much shorter for type III excitability, which is closer to the experiment observation, due to the shorter period of the subthreshold oscillations. The results present the excitability and threshold mechanism for the PIF phenomenon, which provide comprehensive and deep explanations to the PIF phenomenon.

Protonic conductor: better understanding neural resting and action potential

With the employment of the transmembrane electrostatic proton localization theory with a new membrane potential equation, neural resting and action potential is now much better understood as the voltage contributed by the localized protons/cations at a neural liquid- membrane interface. Accordingly, the neural resting/action potential is essentially a protonic/cationic membrane capacitor behavior. It is now understood with a newly formulated action potential equation: when action potential is <0 (negative number), the localized protons/cations charge density at the liquid-membrane interface along the periplasmic side is >0 (positive number); when the action potential is >0, the concentration of the localized protons and localized nonproton cations is <0, indicating a "depolarization" state. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. With the use of the action potential equation, the biological significance of axon myelination is now also elucidated as to provide protonic insulation and prevent any ions both inside and outside of the neuron from interfering with the action potential signal, so that the action potential can quickly propagate along the axon with minimal (e.g., 40 times less) energy requirement.NEW & NOTEWORTHY The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The biological significance of axon myelination is now elucidated as to provide protonic insulation and prevent any ions from interfering with action potential signal.

A thermodynamic description for physiological transmembrane transport

A general formulation for both passive and active transmembrane transport is derived from basic thermodynamical principles. The derivation takes into account the energy required for the motion of molecules across membranes and includes the possibility of modeling asymmetric flow. Transmembrane currents can then be described by the general model in the case of electrogenic flow. As it is desirable in new models, it is possible to derive other well-known expressions for transmembrane currents as particular cases of the general formulation. For instance, the conductance-based formulation for current turns out to be a linear approximation of the general formula for current. Also, under suitable assumptions, other formulas for current based on electrodiffusion, like the constant field approximation by Goldman, can be recovered from the general formulation. The applicability of the general formulations is illustrated first with fits to existing data, and after, with models of transmembrane potential dynamics for pacemaking cardiocytes and neurons. The general formulations presented here provide a common ground for the biophysical study of physiological phenomena that depend on transmembrane transport.

Another interpretation of the Goldman-Hodgkin-Katz equation based on Ling's adsorption theory

According to standard membrane theory, the generation of membrane potential is attributed to transmembrane ion transport. However, there have been a number of reports of membrane behavior in conflict with the membrane theory of cellular potential. Putting aside the membrane theory, we scrutinized the generation mechanism of membrane potential from the view of the long-dismissed adsorption theory of Ling. Ling's adsorption theory attributes the membrane potential generation to mobile ion adsorption. Although Ling's adsorption theory conflicts with the broadly accepted membrane theory, we found that it well reproduces experimentally observed membrane potential behavior. Our theoretical analysis finds that the potential formula based on the GHK eq., which is a fundamental concept of membrane theory, coincides with the potential formula based on Ling's adsorption theory. Reinterpreting the permeability coefficient in the GHK eq. as the association constant between the mobile ion and adsorption site, the GHK eq. turns into the potential formula from Ling's adsorption theory. We conclude that the membrane potential is generated by ion adsorption as Ling's adsorption theory states and that the membrane theory of cellular potential should be amended even if not discarded.

Detection of nonstationary transition to synchronized states of a neural network using recurrence analyses

We study the stability of asymptotic states displayed by a complex neural network. We focus on the loss of stability of a stationary state of networks using recurrence quantifiers as tools to diagnose local and global stabilities as well as the multistability of a coupled neural network. Numerical simulations of a neural network composed of 1024 neurons in a small-world connection scheme are performed using the model of Braun et al. [Int. J. Bifurcation Chaos 08, 881 (1998)IJBEE40218-127410.1142/S0218127498000681], which is a modified model from the Hodgkin-Huxley model [J. Phys. 117, 500 (1952)]. To validate the analyses, the results are compared with those produced by Kuramoto's order parameter [Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin Heidelberg, 1984)]. We show that recurrence tools making use of just integrated signals provided by the networks, such as local field potential (LFP) (LFP signals) or mean field values bring new results on the understanding of neural behavior occurring before the synchronization states. In particular we show the occurrence of different stationary and nonstationarity asymptotic states.

Phasic Firing and Coincidence Detection by Subthreshold Negative Feedback: Divisive or Subtractive or, Better, Both

Phasic neurons typically fire only for a fast-rising input, say at the onset of a step current, but not for steady or slow inputs, a property associated with type III excitability. Phasic neurons can show extraordinary temporal precision for phase locking and coincidence detection. Exemplars are found in the auditory brain stem where precise timing is used in sound localization. Phasicness at the cellular level arises from a dynamic, voltage-gated, negative feedback that can be recruited subthreshold, preventing the neuron from reaching spike threshold if the voltage does not rise fast enough. We consider two mechanisms for phasicness: a low threshold potassium current (subtractive mechanism) and a sodium current with subthreshold inactivation (divisive mechanism). We develop and analyze three reduced models with either divisive or subtractive mechanisms or both to gain insight into the dynamical mechanisms for the potentially high temporal precision of type III-excitable neurons. We compare their firing properties and performance for a range of stimuli. The models have characteristic non-monotonic input-output relations, firing rate vs. input intensity, for either stochastic current injection or Poisson-timed excitatory synaptic conductance trains. We assess performance according to precision of phase-locking and coincidence detection by the models' responses to repetitive packets of unitary excitatory synaptic inputs with more or less temporal coherence. We find that each mechanism contributes features but best performance is attained if both are present. The subtractive mechanism confers extraordinary precision for phase locking and coincidence detection but only within a restricted parameter range when the divisive mechanism of sodium inactivation is inoperative. The divisive mechanism guarantees robustness of phasic properties, without compromising excitability, although with somewhat less precision. Finally, we demonstrate that brief transient inhibition if properly timed can enhance the reliability of firing.

ON and OFF inhibition as mechanisms for forward masking in the inferior colliculus: a modeling study

Masking effects of a preceding stimulus on the detection or perception of a signal have been found in several sensory systems in mammals, including humans and rodents. In the auditory system, it has been hypothesized that a central "OFF-inhibitory" mechanism, which is generated by neurons that respond after a sound is terminated, may contribute to the observed psychophysics. The present study constructed a systems model for the inferior colliculus that includes major ascending monaural and binaural auditory pathways. The fundamental characteristics of several neuron types along the pathways were captured by Hodgkin-Huxley models with specific membrane and synaptic properties. OFF responses were reproduced with a model of the superior paraolivary nucleus containing a hyperpolarization-activated h current and a T-type calcium current. When the gap between the end of the masker and the onset of the signal was large, e.g., >5 ms, OFF inhibition generated strong suppressive effects on the signal response. For smaller gaps, an additional inhibitory source, which was modeled as ON inhibition from the contralateral dorsal nucleus of the lateral lemniscus, showed the potential of explaining the psychophysics. Meanwhile, the effect of a forward masker on the binaural sensitivity to a low-frequency signal was examined, which was consistent with previous psychophysical findings related to sound localization.

A Novel Method for the Description of Voltage-Gated Ionic Currents Based on Action Potential Clamp Results-Application to Hippocampal Mossy Fiber Boutons

Action potential clamp (AP-clamp) recordings of the delayed rectifier K(+) current I K and the fast-activated Na(+) current I Na in rat hippocampal mossy fiber boutons (MFBs) are analyzed using a computational technique recently reported. The method is implemented using a digitized AP from an MFB and computationally applying that data set to published models of I K and I Na. These numerical results are compared with experimental AP-clamp recordings. The I Na result is consistent with experiment; the I K result is not. The difficulty with the I K model concerns the fully activated current-voltage relation, which is described here by the Goldman-Hodgkin-Katz dependence on the driving force (V-E K) rather than (V-E K) itself, the standard model for this aspect of ion permeation. That revision leads to the second-a much steeper voltage dependent activation curve for I K than the one obtained from normalization of a family of I K records by (V-E K). The revised model provides an improved description of the AP-clamp measurement of I K in MFBs compared with the standard approach. The method described here is general. It can be used to test models of ionic currents in any excitable cell. In this way it provides a novel approach to the relationship between ionic current and membrane excitability in neurons.

Optogenetic versus Electrical Stimulation of Human Cardiomyocytes: Modeling Insights

Optogenetics provides an alternative to electrical stimulation to manipulate membrane voltage, and trigger or modify action potentials (APs) in excitable cells. We compare biophysically and energetically the cellular responses to direct electrical current injection versus optical stimulation mediated by genetically expressed light-sensitive ion channels, e.g., Channelrhodopsin-2 (ChR2). Using a computational model of ChR2(H134R mutant), we show that both stimulation modalities produce similar-in-morphology APs in human cardiomyocytes, and that electrical and optical excitability vary with cell type in a similar fashion. However, whereas the strength-duration curves for electrical excitation in ventricular and atrial cardiomyocytes closely follow the theoretical exponential relationship for an equivalent RC circuit, the respective optical strength-duration curves significantly deviate, exhibiting higher nonlinearity. We trace the origin of this deviation to the waveform of the excitatory current-a nonrectangular self-terminating inward current produced in optical stimulation due to ChR2 kinetics and voltage-dependent rectification. Using a unifying charge measure to compare energy needed for electrical and optical stimulation, we reveal that direct electrical current injection (rectangular pulse) is more efficient at short pulses, whereas voltage-mediated negative feedback leads to self-termination of ChR2 current and renders optical stimulation more efficient for long low-intensity pulses. This applies to cardiomyocytes but not to neuronal cells (with much shorter APs). Furthermore, we demonstrate the cell-specific use of ChR2 current as a unique modulator of intrinsic activity, allowing for optical control of AP duration in atrial and, to a lesser degree, in ventricular myocytes. For self-oscillatory cells, such as Purkinje, constant light at extremely low irradiance can be used for fine control of oscillatory frequency, whereas constant electrical stimulation is not feasible due to electrochemical limitations. Our analysis offers insights for designing future new energy-efficient stimulation strategies in heart or brain.

Non-linear leak currents affect mammalian neuron physiology

In their seminal works on squid giant axons, Hodgkin, and Huxley approximated the membrane leak current as Ohmic, i.e., linear, since in their preparation, sub-threshold current rectification due to the influence of ionic concentration is negligible. Most studies on mammalian neurons have made the same, largely untested, assumption. Here we show that the membrane time constant and input resistance of mammalian neurons (when other major voltage-sensitive and ligand-gated ionic currents are discounted) varies non-linearly with membrane voltage, following the prediction of a Goldman-Hodgkin-Katz-based passive membrane model. The model predicts that under such conditions, the time constant/input resistance-voltage relationship will linearize if the concentration differences across the cell membrane are reduced. These properties were observed in patch-clamp recordings of cerebellar Purkinje neurons (in the presence of pharmacological blockers of other background ionic currents) and were more prominent in the sub-threshold region of the membrane potential. Model simulations showed that the non-linear leak affects voltage-clamp recordings and reduces temporal summation of excitatory synaptic input. Together, our results demonstrate the importance of trans-membrane ionic concentration in defining the functional properties of the passive membrane in mammalian neurons as well as other excitable cells.

Regulatory evolution and voltage-gated ion channel expression in squid axon: selection-mutation balance and fitness cliffs

It has been suggested that optimization of either axonal conduction velocity or the energy efficiency of action potential conduction predominates in the selection of voltage-gated sodium conductance levels in the squid axon. A population genetics model of channel gene regulatory function was used to examine the role of these and other evolutionary forces on the selection of both sodium and potassium channel expression levels. In this model, the accumulating effects of mutations result in degradation of gene regulatory function, causing channel gene expression to fall to near-zero in the absence of positive selection. In the presence of positive selection, channel expression levels fall to the lowest values consistent with the selection criteria, thereby establishing a selection-mutation balance. Within the parameter space of sodium and potassium conductance values, the physiological performance of the squid axon model showed marked discontinuities associated with conduction failure and excitability. These discontinuities in physiological function may produce fitness cliffs. A fitness cliff associated with conduction failure, combined with the effects of phenotypic noise, can account for the selection of sodium conductance levels, without considering either conduction velocity or metabolic cost. A fitness cliff associated with a transition in axonal excitability, combined with phenotypic noise, can explain the selection of potassium channel expression levels. The results suggest that voltage-gated ion channel expression will fall to low levels, consistent with key functional constraints, even in the absence of positive selection for energy efficiency. Channel expression levels and individual variation in channel expression within the population can be explained by regulatory evolution in combination with genetic variation in regulatory function and phenotypic noise, without resorting to more complex mechanisms, such as activity-dependent homeostasis. Only a relatively small region of the large, nominally isofunctional parameter space for channel expression will normally be occupied, because of the effects of mutation.

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

Inducing a switch in neuronal state using energy optimal stimuli is relevant to a variety of problems in neuroscience. Analytical techniques from optimal control theory can identify such stimuli; however, solutions to the optimization problem using indirect variational approaches can be elusive in models that describe neuronal behavior. Here we develop and apply a direct gradient-based optimization algorithm to find stimulus waveforms that elicit a change in neuronal state while minimizing energy usage. We analyze standard models of neuronal behavior, the Hodgkin-Huxley and FitzHugh-Nagumo models, to show that the gradient-based algorithm: (1) enables automated exploration of a wide solution space, using stochastically generated initial waveforms that converge to multiple locally optimal solutions; and (2) finds optimal stimulus waveforms that achieve a physiological outcome condition, without a priori knowledge of the optimal terminal condition of all state variables. Analysis of biological systems using stochastically-seeded gradient methods can reveal salient dynamical mechanisms underlying the optimal control of system behavior. The gradient algorithm may also have practical applications in future work, for example, finding energy optimal waveforms for therapeutic neural stimulation that minimizes power usage and diminishes off-target effects and damage to neighboring tissue.

Channelrhodopsin2 current during the action potential: "optical AP clamp" and approximation

The most widely used optogenetic tool, Channelrhodopsin2 (ChR2), is both light- and voltage-sensitive. A light-triggered action potential or light-driven perturbations of ongoing electrical activity provide instant voltage feedback, shaping ChR2 current. Therefore, depending on the cell type and the light pulse duration, the typically reported voltage-clamp-measured ChR2 current traces are often not a good surrogate for the ChR2 current during optically-triggered action potentials. We discuss two experimental methods to reveal ChR2 current during an action potential: an “optical AP clamp” and its approximation employing measured current-voltage curve for ChR2. The methods are applicable to voltage- and light-sensitive ion currents operating in excitable cells, e.g. cardiomyocytes or neurons.

Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection

We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single-neuron models and their relation with Hodgkin's classification in 1948 of repetitive firing properties. We discuss the mathematical dissection of bursting oscillations using fast/slow analysis and demonstrate the approach using single-cell and mean-field network models. Finally, we illustrate the properties of Type III excitability in which case repetitive firing for constant or slow inputs is absent. Rather, firing is in response only to rapid enough changes in the stimulus. Our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution.

Ionic mechanism underlying optimal stimuli for neuronal excitation: role of Na+ channel inactivation

The ionic mechanism underlying optimal stimulus shapes that induce a neuron to fire an action potential, or spike, is relevant to understanding optimal information transmission and therapeutic stimulation in the nervous system. Here we analyze for the first time the ionic basis for stimulus optimality in the Hodgkin and Huxley model and for eliciting a spike in squid giant axons, the preparation for which the model was devised. The experimentally determined stimulus is a smoothly varying biphasic current waveform having a relatively long and shallow hyperpolarizing phase followed by a depolarizing phase of briefer duration. The hyperpolarizing phase removes a small degree of the resting level of Na⁺ channel inactivation. This result together with the subsequent depolarizing phase provides a signal that is energetically more efficient for eliciting spikes than rectangular current pulses. Sodium channel inactivation is the only variable that is changed during the stimulus waveform, other than the membrane potential, V. The activation variables for Na⁺ and K⁺ channels are unchanged throughout the stimulus. This result demonstrates how an optimal stimulus waveform relates to ionic dynamics and may have implications for energy efficiency of neural excitation in many systems including the mammalian brain.

A comparative analysis of models of Na+ channel gating for mammalian and invertebrate nonmyelinated axons: relationship to energy efficient action potentials

The rapidly activating, voltage gated Na(+) current, INa, has recently been measured in mammalian nonmyelinated axons. Those results have been incorporated in simulations of the action potential, results that demonstrate a significant separation in time during the spike between INa and the repolarizing K(+) current, IK. The original Hodgkin and Huxley (1952) model of Na(+) channel gating, m(3)h, where m and h are channel activation and inactivation, respectively, has been used in this analysis. This model was originally developed for invertebrate nonmyelinated axons, squid giant axons in particular. The model has not survived challenges based on results from invertebrate preparations using a double-step voltage clamp protocol and measurements of gating currents, results that demonstrate a kinetic link between activation and inactivation leading to a delayed onset of inactivation following a voltage step. These processes are independent of each other in the Hodgkin and Huxley (1952) model. Application of the double-step protocol to the m(3)h model for mammalian INa results reveals a surprising prediction, an apparent delay in onset of inactivation even though activation and inactivation are uncoupled in the model. Other results, most notably gating currents, will be required to demonstrate such a link, if indeed it exists for mammalian Na(+) channels. The information obtained will be significant in determining the way in which the Na(+) channel is sequestered away from its open state during repolarization, thereby allowing for a separation in time between INa and IK during a spike, an energetically efficient mechanism of neuronal signaling in the mammalian brain.

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.

TYPE III EXCITABILITY, SLOPE SENSITIVITY AND COINCIDENCE DETECTION

Some neurons in the nervous system do not show repetitive firing for steady currents. For time-varying inputs, they fire once if the input rise is fast enough. This property of phasic firing is known as Type III excitability. Type III excitability has been observed in neurons in the auditory brainstem (MSO), which show strong phase-locking and accurate coincidence detection. In this paper, we consider a Hodgkin-Huxley type model (RM03) that is widely-used for phasic MSO neurons and we compare it with a modification of it, showing tonic behavior. We provide insight into the temporal processing of these neuron models by means of developing and analyzing two reduced models that reproduce qualitatively the properties of the exemplar ones. The geometric and mathematical analysis of the reduced models allows us to detect and quantify relevant features for the temporal computation such as nearness to threshold and a temporal integration window. Our results underscore the importance of Type III excitability for precise coincidence detection.

Optimal stimulus shapes for neuronal excitation

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.

Computational neuroscience seeks to understand the mechanisms by which signals excite a neuron or a neuronal network. An important consideration in these studies is optimality, i.e., what signal most effectively causes excitation. Optimization of neuronal signaling is important for networks that need to minimize energy costs, for sensory neurons to selectively respond to specific stimulus features, and for therapeutic deep brain stimulators to maximize battery life. Here we show in a classic mathematical model of the action potential and in experiments on a single cell preparation that: 1) a single neuron can be highly discriminatory for the shape of low amplitude stimuli that elicit an action potential and 2) the shape of the optimal stimulus depends upon the overall state of inputs to the neuron. Our findings reveal a surprising complexity of computation at the single cell level that may be important for understanding physiological function of the nervous system.

Sound localization: Jeffress and beyond

Many animals use the interaural time differences (ITDs) to locate the source of low frequency sounds. The place coding theory proposed by Jeffress has long been a dominant model to account for the neural mechanisms of ITD detection. Recent research, however, suggests a wider range of strategies for ITD coding in the binaural auditory brainstem. We discuss how ITD is coded in avian, mammalian, and reptilian nervous systems, and review underlying synaptic and cellular properties that enable precise temporal computation. The latest advances in recording and analysis techniques provide powerful tools for both overcoming and utilizing the large field potentials in these nuclei.

Response of Morris-Lecar neurons to various stimuli

We studied the responses of three classes of Morris-Lecar neurons to sinusoidal inputs and synaptic pulselike stimuli with deterministic and random interspike intervals (ISIs). It was found that the responses of the output frequency of class 1 and 2 neurons showed similar evolution properties by varying input amplitudes and frequencies, whereas class 3 neuron exhibited substantially different properties. Specifically, class 1 and 2 neurons display complicated phase locking (p : q, p > q, denoting output action potentials per input spikes) in low-frequency sinusoidal input area when the input amplitude is above their threshold, but a class 3 neuron does not fire action potentials in this area even if the amplitude is much higher. In the case of the deterministic ISI synaptic injection, all the three classes of neurons oscillate spikes with an arbitrary small frequency. When increasing the input frequency (both sinusoidal and deterministic ISI synaptic inputs), all neurons display 1 : 1 phase locking, whereas the response frequency decreases even fall to zero in the high-frequency input area. When the random ISI synaptic pulselike stimuli are injected into the neurons, one can clearly see the low-pass filter behaviors from the return map. The output ISI distribution depends on the mean ISI of input train as well as the ISI variation. Such different responses of three classes of neurons result from their distinct dynamical mechanisms of action potential initiation. It was suggested that the intrinsic dynamical cellular properties are very important to neuron information processing.

Slope-based stochastic resonance: how noise enables phasic neurons to encode slow signals

Fundamental properties of phasic firing neurons are usually characterized in a noise-free condition. In the absence of noise, phasic neurons exhibit Class 3 excitability, which is a lack of repetitive firing to steady current injections. For time-varying inputs, phasic neurons are band-pass filters or slope detectors, because they do not respond to inputs containing exclusively low frequencies or shallow slopes. However, we show that in noisy conditions, response properties of phasic neuron models are distinctly altered. Noise enables a phasic model to encode low-frequency inputs that are outside of the response range of the associated deterministic model. Interestingly, this seemingly stochastic-resonance (SR) like effect differs significantly from the classical SR behavior of spiking systems in both the signal-to-noise ratio and the temporal response pattern. Instead of being most sensitive to the peak of a subthreshold signal, as is typical in a classical SR system, phasic models are most sensitive to the signal's rising and falling phases where the slopes are steep. This finding is consistent with the fact that there is not an absolute input threshold in terms of amplitude; rather, a response threshold is more properly defined as a stimulus slope/frequency. We call the encoding of low-frequency signals with noise by phasic models a slope-based SR, because noise can lower or diminish the slope threshold for ramp stimuli. We demonstrate here similar behaviors in three mechanistic models with Class 3 excitability in the presence of slow-varying noise and we suggest that the slope-based SR is a fundamental behavior associated with general phasic properties rather than with a particular biological mechanism.

Principal brain cells, called neurons, show a tremendous amount of diversity in their responses to driving stimuli. A widely present but understudied class of neurons prefers to respond to high-frequency inputs and neglect slow variations; these cells are called phasic neurons. Although phasic neurons do not normally respond to slow signals, we show that noise, a ubiquitous neural input, can enable them to respond to distinct features of slow signals. We emphasize the fact that, in the presence of noise, they are still sensitive to the change in stimulus, rather than to the constant part of the slow inputs, just as they are for fast inputs without noise. This feature distinguishes the response of phasic neurons from those of other neurons, which show more sensitivity to the amplitude of their inputs. We believe that our study has significantly broadened the understanding about the information-processing ability and functional roles of phasic neurons.

Determining k channel activation curves from k channel currents often requires the goldman-hodgkin-katz equation

Potassium ion current in nerve membrane, I(K), has traditionally been described by I(K) = g(K)(V - E(K)), where g(K) is the K ion conductance, V is membrane potential and E(K) is the K(+) Nernst potential. This description has been unchallenged by most investigators in neuroscience since its introduction almost 60 years ago. The problem with the I(K) approximately (V - E(K)) proportionality is that it is inconsistent with the unequal distribution of K ions in the intra- and extracellular bathing media. Under physiological conditions the intracellular K(+) concentration is significantly higher than the extracellular concentration. Consequently, the slope conductance at potentials positive to E(K) cannot be the same as that for potentials negative to E(K), as the linear proportionality between I(K) and (V - E(K)) requires. Instead I(K) has a non-linear dependence on (V - E(K)) which is well described by the Goldman-Hodgkin-Katz equation. The implications of this result for K(+) channel gating and membrane excitability are reviewed in this report.

Probability distributions of Markovian sodium channel states during propagating axonal impulses with or without recovery supernormality

This study addressed a macroscopic neurophysiological phenomenon - supernormality during the recovery phase of propagating axonal impulses - in explicit chemical terms. Excitation was reconstructed numerically using the kinetic scheme of multiple-state probabilistic transitions within a population of voltage-dependent sodium channels (NaCh) derived by Vandenberg and Bezanilla ("PC" scheme). Each NaCh transition was characterized as a reversible Markov process with voltage-dependent rate constants associated with each respective directional transition. While recovery reconstructed with the Hodgkin-Huxley formalism included a supernormal period, the PC scheme did not. The present analysis showed that the occurrence and degree of supernormality with the PC scheme was determined by the relative speed of the transitions within the closed loop of the kinetic scheme; supernormality was promoted by speeding these kinetics. The analysis also showed that concurrent with supernormality, the faster loop kinetics caused (1) an elevation in the C(1) --> C(2) transitions, and (2) a reduction in the I(4) --> I(5) transitions. Thus, macroscopic functionality in information processing could be expressed in terms of probabilistic interstate transitions among a population of NaCh molecules.