Classical Hodgkin-Huxley Model Research Articles

Closed-loop baroreflex model with biophysically detailed afferent pathway.

In this work, we couple a lumped-parameter closed-loop model of the cardiovascular system with a physiologically-detailed mathematical description of the baroreflex afferent pathway. The model features a classical Hodgkin-Huxley current-type model for the baroreflex afferent limb (primary neuron) and for the second-order neuron in the central nervous system. The pulsatile arterial wall distension triggers a frequency-modulated sequence of action potentials at the afferent neuron. This signal is then integrated at the brainstem neuron model. The efferent limb, representing the sympathetic and parasympathetic nervous system, is described as a transfer function acting on heart and blood vessel model parameters in order to control arterial pressure. Three in silico experiments are shown here: a step increase in the aortic pressure to evaluate the functionality of the reflex arch, a hemorrhagic episode and an infusion simulation. Through this model, it is possible to study the biophysical dynamics of the ionic currents proposed for the afferent limb components of the baroreflex during the cardiac cycle, and the way in which currents dynamics affect the cardiovascular function. Moreover, this system can be further developed to study in detail each baroreflex loop component, helping to unveil the mechanisms involved in the cardiovascular afferent information processing.

Mathematical Modeling of Neuron Using Hodgkin-Huxley Model

The growing rate of mental illnesses has brought the idea to cure anxiety disorders through altering heart rate variability using sympathetic and parasympathetic stimulations. The parasympathetic activity in the sinoatrial node being responsible for arrythmia has been considered in the already existing cardiovascular model. This study presents a mathematical model of a single neuron and simulated its activity for a single burst to be amalgamated in a closed-loop system of the heart and uses heart rate variability as a control variable to treat anxiety disorders and decrease the rate of increasing mental disorders in the world population. The proposed study shows the results that rises from 7mV and goes up to 107mV and then again falls back to 7mV depicting a single burst. These results is being validated by comparing the model of neuron presented in classical Hodgkin-Huxley model using MATLAB.

Dynamics and stability of neural systems with indirect interactions involved energy levels

Dynamical modelling for neural systems with direct and indirect connections is to understand how these connections contribute to neural dynamics. Despite recent findings suggesting the existence of indirect connections in neural systems, their dynamical characteristics remain poorly understood. In this paper, we propose a simplified circuit model with indirect interactions inspired by the indirect connection between two neurons, from an energy perspective. Through bifurcation and dynamics analysis, we find that the presented model has a striking resemblance with the classical Hodgkin-Huxley neuronal model. Moreover, stability in a neural network coupled with energy is demonstrated by combining stability analysis and numerical simulation. Our analysis sheds light on the excitability dynamics and multi-stability that can emerge in biophysical systems with nonlinear interactions inspired by the neural systems and highlights the role of energy in propagating electrical activities.

Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling.

Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.

Phase synchronization in a sparse network of randomly connected neurons under the effect of Poissonian spike inputs

This article investigates the emergence of phase synchronization in a network of randomly connected neurons by chemical synapses. The study uses the classic Hodgkin–Huxley model to simulate the neuronal dynamics under the action of a train of Poissonian spikes. In such a scenario, we observed the emergence of irregular spikes for a specific range of conductances and also that the phase synchronization of the neurons is reached when the external current is strong enough to induce spiking activity but without overcoming the coupling current. Conversely, if the external current assumes very high values, then an opposite effect is observed, i.e., the prevention of the network synchronization. We explain such behaviors considering different mechanisms involved in the system, such as incoherence, minimization of currents, and stochastic effects from the Poissonian spikes. Furthermore, we present some numerical simulations where the stimulation of only a fraction of neurons, for instance, can induce phase synchronization in the non-stimulated fraction of the network, besides cases in which for larger coupling values, it is possible to propagate the spiking activity in the network when considering stimulation over only one neuron.

Spatio-temporal modeling of saltatory conduction in neurons using Poisson–Nernst–Planck treatment and estimation of conduction velocity

Action potential propagation along the axons and across the dendrites is the foundation of the electrical activity observed in the brain and the rest of the nervous system. Theoretical and numerical modeling of this action potential activity has long been a key focus area of electro-chemical neuronal modeling, and over the years, electrical network models of varying complexity have been proposed. Specifically, considering the presence of nodes of Ranvier along the myelinated axon, single-cable models of the propagation of action potential have been popular. Building on these models, and considering a secondary electrical conduction pathway below the myelin sheath, the double-cable model has been proposed. Such cable theory based treatments, including the classical Hodgkin–Huxley model, single-cable model, and double-cable model have been extensively studied in the literature. But these have inherent limitations in their lack of a representation of the spatio-temporal evolution of the neuronal electro-chemistry. In contrast, a Poisson–Nernst–Planck (PNP) based electro-diffusive framework accounts for the underlying spatio-temporal ionic concentration dynamics and is a more general and comprehensive treatment. In this work, a high-fidelity implementation of the PNP model is demonstrated. This electro-diffusive model is shown to produce results similar to the cable theory based electrical network models, and in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple nodes of Ranvier, its correlation with cable theory based models, and multiple variants of the electro-diffusive model — PNP without myelin, PNP with myelin, and PNP with the myelin sheath and peri-axonal space. Further, we apply this spatio-temporal model to numerically estimate conduction velocity in a rat axon using the three model variants. Specifically, spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.Statement of Significance: In this work, we present a comprehensive PDE based treatment for modeling neuronal action potential generation and propagation and provide a first-of-its-kind framework for computationally estimating action potential conduction velocities. This electro-diffusive model, based on a Poisson-Nernst-Planck (PNP) formulation, is shown to produce results similar to the cable theory based electrical network models, and in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple nodes of Ranvier, its correlation with cable theory based models, and multiple variants of the electro-diffusive model - PNP without myelin, PNP with myelin, and PNP with the myelin sheath and periaxonal space. Further, we apply this spatio-temporal model to numerically estimate conduction velocity in a rat axon using the three model variants. Specifically, spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.

Model Analysis of Post-Stimulation Effect on Axonal Conduction and Block.

To reveal the possible contribution of changes in membrane ion concentration gradients and ion pump activity to axonal conduction/block induced by long-duration electrical stimulation. A new model for conduction and block of unmyelinated axons based on the classical Hodgkin-Huxley (HH) equations is developed to include changes in Na+ and K+ concentrations and ion pumps. The effect of long-duration stimulation on axonal conduction/block is analyzed by computer simulation using this new model. The new model successfully simulates initiation, propagation, and block of action potentials induced by short-duration (multiple milliseconds) stimulations that do not significantly change the ion concentrations in the classical HH model. In addition, the activity-dependent effects such as action potential attenuation and broadening observed in animal studies are also successfully simulated by the new model. Finally, the model successfully simulates axonal block occurring after terminating a long-duration (multiple seconds) direct current (DC) stimulation as observed in recent animal studies and reveals 3 different mechanisms for the post-DC block of axonal conduction. Ion concentrations and pumps play an important role in post-stimulation effects and activity-dependent effects on axonal conduction/block. The duration of stimulation is a determinant factor because it influences the total charges applied to the axon, which in turn determines the ion concentrations inside and outside the axon. Despite recent clinical success of many neurostimulation therapies, the effects of long-duration stimulation on axonal conduction/block are poorly understood. This new model could significantly impact our understanding of the mechanisms underlying different neurostimulation therapies.

Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model

The classic Hodgkin-Huxley model is widely used for understanding the electrophysiological dynamics of a single neuron. While applying a low-amplitude constant current to the system results in a single voltage spike, it is possible to produce multiple voltage spikes by applying time-varying currents, which may not be experimentally measurable. The aim of this work is to estimate time-varying applied currents of different deterministic forms given noisy voltage data. In particular, we utilize an augmented ensemble Kalman filter with parameter tracking to estimate four different time-varying applied current parameters and associated Hodgkin-Huxley model states, along with uncertainty bounds in each case. We test the efficiency of the parameter tracking algorithm in this setting by analyzing the effects of changing the standard deviation of the parameter drift and the frequency of data available on the resulting time-varying applied current estimates and related uncertainty.

Neuronal synchronization enhanced by neuron–astrocyte interaction

In this study, we consider the problem of signal processing in neuron–astrocyte networks where the intercellular communication is described on the basis of a tripartite synapse concept. This type of communication involves combined contributions of pre- and postsynaptic neuronal compartments and components of the surrounding astrocyte to the processes of information transmission. Astrocyte-mediated regulation of neuronal activity is considered through the analysis of the response changes in the classical Hodgkin–Huxley model driven by excitatory synaptic current. It is shown that the complicated astrocyte-dependent dynamics of this current can lead to non-trivial changes in individual postsynaptic neuronal activity and, hence, in the cooperative activation of neuronal groups linked by the “astrocyte-mediated bridge”.

Mathematical Models of Electrical Activity in Plants

Electrical activity plays an important role in plant life; in particular, electrical responses can participate in the reception of the action of stressors (local electrical responses and oscillations) and signal transduction into unstimulated parts of the plant (action potential, variation potential and system potential). Understanding the mechanisms of electrical responses and subsequent changes in physiological processes and the prediction of plant responses to stressors requires the elaboration of mathematical models of electrical activity in plant organisms. Our review describes approaches to the simulation of plant electrogenesis and summarizes current models of electrical activity in these organisms. It is shown that there are numerous models of the generation of electrical responses, which are based on various descriptions (from modifications of the classical Hodgkin-Huxley model to detailed models, which consider ion transporters, regulatory processes, buffers, etc.). A moderate number of works simulate the propagation of electrical signals using equivalent electrical circuits, systems of excitable elements with local electrical coupling and descriptions of chemical signal propagation. The transmission of signals from a plasma membrane to intracellular compartments (endoplasmic reticulum, vacuole) during the generation of electrical responses is much less modelled. Finally, only a few works simulate plant physiological changes that are connected with electrical responses or investigate the inverse problem: reconstruction of the type and parameters of stimuli through the analysis of electrical responses. In the conclusion of the review, we discuss future perspectives on the simulation of electrical activity in plants.

Spike-Threshold Variability Originated from Separatrix-Crossing in Neuronal Dynamics.

The threshold voltage for action potential generation is a key regulator of neuronal signal processing, yet the mechanism of its dynamic variation is still not well described. In this paper, we propose that threshold phenomena can be classified as parameter thresholds and state thresholds. Voltage thresholds which belong to the state threshold are determined by the ‘general separatrix’ in state space. We demonstrate that the separatrix generally exists in the state space of neuron models. The general form of separatrix was assumed as the function of both states and stimuli and the previously assumed threshold evolving equation versus time is naturally deduced from the separatrix. In terms of neuronal dynamics, the threshold voltage variation, which is affected by different stimuli, is determined by crossing the separatrix at different points in state space. We suggest that the separatrix-crossing mechanism in state space is the intrinsic dynamic mechanism for threshold voltages and post-stimulus threshold phenomena. These proposals are also systematically verified in example models, three of which have analytic separatrices and one is the classic Hodgkin-Huxley model. The separatrix-crossing framework provides an overview of the neuronal threshold and will facilitate understanding of the nature of threshold variability.

Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input

Nous considerons un modele stochastique decrivant un neurone stimule par un signal qu’il recoit a travers son systeme dendritique. Ce signal aleatoire, de type Ornstein–Uhlenbeck, resulte de la perturbation d’un signal periodique deterministe. Pour decrire l’evolution du potentiel de membrane du neurone, nous faisons appel au modele classique de Hodgkin et Huxley. Le systeme stochastique obtenu est fortement degenere. Nous demontrons que la condition de Hormander faible est satisfaite en tout point de l’espace d’etats. L’existence d’une fonction de Lyapunov permet de conclure qu’il existe au plus un nombre fini de mesures ergodiques pour ce modele. Par consequent, la complexite du systeme est reduite de maniere significative par rapport a la situation deterministe.

Effect of power-law ionic conductances in the Hodgkin and Huxley model

An increasing number of results show that the voltage and spiking activity of a neuron follows scale free adaptation. Under such conditions, the voltage, or firing rate, of a neuron cannot be characterized by a unique time constant, instead these processes are characterized by power-laws. Power-law behaviors suggest that the components responsible for the voltage are strongly interacting across temporal scales, slow processes affect the rates of fast processes and vice versa. We recently introduced the fractional leaky integrate-and-fire model, which we have used to replicate the firing rate activity of adapting cortical neurons [1]. Our results have shown that spike rate adaptation can be modeled by a fractional derivative of low order. Thus suggesting, a strong interaction across conductances in cortical neurons. this work we decided to study the effects of power-law behavior in a biophysical model of spiking activity, the Hodgkin-Huxley model. The classical Hodgkin-Huxley model is described by CdVdt=−(gm(V−Erest)+∑i=1Ngi(V−Ei))+I. where C is the capacitance; V, voltage; gm, the membrane conductance; Erest, the resting potential; gi={gNa, gK } the sodium or potassium conductances with corresponding reversal potentials (Ei={ ENa, EK }); and I, the input curret. The conductances are gK=gK¯n4 and gNa=gNa¯m3h, with gK¯ and gNa¯ the maximum conductances. The gating variables n, m, and h are defined by the general equation dxdt=αxV,t1-x-βx(V,t)x where x={n, m, h}. To implement power-law behavior in either gating variable we substitute the classical derivative by the fractional derivative of order η using the Caputo definition dηfdtη=1Γn-η∫0tf(n)tt-uη+1-ndu where is the Gamma function, and n is the nearest integer larger than η. In our case, n = 1 because this models how the interaction of previous activity slows down the activation of the gating variables. The fractional derivative value is the result of integrating the activity of the function over all past activities weighted by a function that follows a power-law. The weighted values are called the memory trace. Thus, the fractional derivative provides information over all past activity, in contrast with the classical derivative that only takes into account the value of the function in the immediate previous time point. We have recently developed efficient ways to computationally solve these equations [2]. Our results show the emergence of a wide range of spiking behaviors (bursting, spiking and sub-threshold oscillations) in response to constant stimulation as a function of the fractional order in the different activation/inactivation variables. In the case of n, the neuron shows reduction if spiking response and emergence of sub-threshold oscillations. While fractional h results in bursting activity. This emergent richness in spiking activity, while only modeling two conductances, allows study of the overall effects of power-law behavior in neuronal activity. At the biophysical level, our results suggest that voltage dependent ion channels that deviate from a classical Markov process could increase the amount of spiking patterns in single cells, thus increasing their information content capacity.

A study of quantum mechanical probabilities in the classical Hodgkin-Huxley model.

The Hodgkin-Huxley (HH) model is a powerful model to explain different aspects of spike generation in excitable cells. However, the HH model was proposed in 1952 when the real structure of the ion channel was unknown. It is now common knowledge that in many ion-channel proteins the flow of ions through the pore is governed by a gate, comprising a so-called "selectivity filter" inside the ion channel, which can be controlled by electrical interactions. The selectivity filter (SF) is believed to be responsible for the selection and fast conduction of particular ions across the membrane of an excitable cell. Other (generally larger) parts of the molecule such as the pore-domain gate control the access of ions to the channel protein. In fact, two types of gates are considered here for ion channels: the "external gate", which is the voltage sensitive gate, and the "internal gate" which is the selectivity filter gate (SFG). Some quantum effects are expected in the SFG due to its small dimensions, which may play an important role in the operation of an ion channel. Here, we examine parameters in a generalized model of HH to see whether any parameter affects the spike generation. Our results indicate that the previously suggested semi-quantum-classical equation proposed by Bernroider and Summhammer (BS) agrees strongly with the HH equation under different conditions and may even provide a better explanation in some cases. We conclude that the BS model can refine the classical HH model substantially.

Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs.

The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

Characterizing spiking in noisy type II neurons.

Understanding the dynamics of noisy neurons remains an important challenge in neuroscience. Here, we describe a simple probabilistic model that accurately describes the firing behavior in a large class (type II) of neurons. To demonstrate the usefulness of this model, we show how it accurately predicts the interspike interval (ISI) distributions, bursting patterns and mean firing rates found by: (1) simulations of the classic Hodgkin-Huxley model with channel noise, (2) experimental data from squid giant axon with a noisy input current and (3) experimental data on noisy firing from a neuron within the suprachiasmatic nucleus (SCN). This simple model has 6 parameters, however, in some cases, two of these parameters are coupled and only 5 parameters account for much of the known behavior. From these parameters, many properties of spiking can be found through simple calculation. Thus, we show how the complex effects of noise can be understood through a simple and general probabilistic model.

Poincare return mapping for models of elliptic neurons

Event Abstract Back to Event Poincare return mapping for models of elliptic neurons Jeremy Wojcik1* and Andrey Shilnikov1 1 Georgia State University, Department of Mathematics and Statistics and the Neuroscience Institute, United States We propose a novel computer assisted method for the effective construction and accurate examination of families of Poincaré return mappings for voltage maxima in models of elliptic bursting neurons. Such busters are adequately described by dynamical systems with two characteristic time scales: slow and fast. A feature of a slow-fast dynamical model is that its solutions stay close to the so-called slow motion manifolds, comprised of equilibria and limit cycles of its fast subsystem. In the context of neurodynamics they are respectively called quiescent and tonic spiking equilibria. We reveal the topology of the manifolds in the phase space of an elliptic neuron model to define the Poincaré mappings for oscillatory dynamics of the fast membrane potential. The algorithms discussed allow us to create a full family of Poincare mappings to get insight into driving forces on the dynamics of a model in question over an entire parameter range. Of special interests are these mappings for nonlocal bifurcations at transitions between states of the model. This includes mechanisms of transitions between tonic spiking and bursting, between quiescence and tonic spiking, bifurcations of bursting, the emergence of various mixed mode oscillations, and their interactions with bursting and quiescence etc. The examination of nonlocal bifurcation at these transitions is accomplished through reduction of the multidimensional model to a one-dimensional voltage next amplitude mapping: T: Vn ?Vn+1. Such mappings allow us to identify and study bifurcations and complex dynamics of the model at such transitions. We note that the reduction to next amplitude mappings for maximal values of oscillatory voltage alone is highly advantageous as the behavior of any neuron model are manifest through the evolution of accumulative voltage dynamics which can only be accessed in experimental studies in most cases. We examine the mathematical Fitzhugh-Nagumo-Rinzel model, a bursting modification of the classical Hodgkin-Huxley model, as well as the Terman-Rubin model for the external segment of globus pallidus. We begin by creation of the respective slow motion manifolds through parameter continuation techniques. Next we are then able to extrapolate a set of maxima's, for each limit cycle of the manifold, to be used as initial conditions for the solution set for a given parameter. We then find the next maxima for the set of initial conditions creating a next amplitude map. Varying the parameter yields a family of next amplitude maps that allow for the identification of periodic and chaotic attractors, as well system repellers and homoclinic solutions that globally organize the dynamics of the model. We are then able to predict and/or identify the mechanisms for state transitions in the models, before the actual transition. Conference: 2010 South East Nerve Net (SENN) and Georgia/South Carolina Neuroscience Consortium (GASCNC) conferences, Atlanta , United States, 5 Mar - 7 Mar, 2010. Presentation Type: Poster Presentation Topic: Posters Citation: Wojcik J and Shilnikov A (2010). Poincare return mapping for models of elliptic neurons. Front. Neurosci. Conference Abstract: 2010 South East Nerve Net (SENN) and Georgia/South Carolina Neuroscience Consortium (GASCNC) conferences. doi: 10.3389/conf.fnins.2010.04.00088 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 18 Mar 2010; Published Online: 18 Mar 2010. * Correspondence: Jeremy Wojcik, Georgia State University, Department of Mathematics and Statistics and the Neuroscience Institute, Atlanta, United States, jwojcik1@gsu.edu Login Required This action requires you to be registered with Frontiers and logged in. 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Abstract Action Potential Models for Toxin Recognition

In this paper, we present a robust methodology using mathematical pattern recognition schemes to detect and classify events in action potentials for recognizing toxins in biological cells. We focus on event detection in action potential via abstraction of information content into a low dimensional feature vector within the constrained computational environment of a biosensor. We use generated families of action potentials from a classic Hodgkin–Huxley model to verify our methodology and build toxin recognition engines. We demonstrate that good recognition rates are achievable with our methodology.

The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971, J. Physiol. Lond. 214, 31-53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (IA). Using a reduction of the classical Hodgkin-Huxley model, we study the effects of IA on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based generic IA parameters, the model does not fire at arbitrarily low rates, although it can for the more atypical IA parameters given by Connor and Stevens for the crab axon. When the IA inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases as gA decreases and as inactivation rate decreases. We also study how IA affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.

Possible dual effect of synapses that are putatively purely excitatory or purely inhibitory: bases in stability theory and implications for neural network behavior

Depolarization of an excitable membrane has a dual effect; excitatory in that it causes rapid opening of calcium and/or sodium channels but inhibitory in that it also causes those channels to inactivate. We considered whether apparently paradoxical or dual behavior might be exhibited by excitatory and inhibitory synaptic inputs. We used the classic Hodgkin-Huxley model for voltage-gated channels plus leakage channels of appropriate selectivity for ligand-gated postsynaptic channels. We summarize a model cell's behavior by calculating elicited firing frequency as a function of reversal potential and conductance of summed synaptic inputs, using stability theory and direct simulations. Dual behavior is elicited in the model with reasonable densities of ligand-gated channels. Thus a particular synaptic input to a neuron may be either excitatory or inhibitory depending on simultaneous activity of other synaptic inputs to the cell. This input-output map may give rise to biologically realistic and rich behaviors as an element of computed neural networks, and still be computationally tractable.