Biophysical basis for three distinct dynamical mechanisms of action potential initiation

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.     

Extension of the neuronal membrane model to account for suppression of the action potential by a constant magnetic field

A biophysical explanation of the reduced excitability in neurons exposed to a constant magnetic field is based on an extended neuronal membrane model. In the presence of a constant magnetic field, reduced excitability is manifested as an increase in the excitation threshold and a decrease in the frequency of action potentials. The proposed explanation for the reduced excitability rests on the well-known Hall effect. The separation of charges resulting from the Lorentz force exerted on moving intracellular ions leads to the formation of a Hall electric field in a direction perpendicular to that of action-potential transmission. Consequently, the ion current for discharging the membrane capacitance is reduced in the presence of a magnetic field, thereby limiting initiation of the action potential. The validity of the proposed biophysical explanation is justified analytically and verified by simulations based on the Hodgkin and Huxley model for the electrical excitability of a neuron. Based on derivation of the current segregation ratio α characterizing the reduction in the stimulating current from first principles, the equivalent circuit model of the neuronal membrane is extended to account for the reduced excitability of neurons exposed to a constant magnetic field.     

Interaction of vasomotor and exocrine neurons in bullfrog paravertebral sympathetic ganglia

A 2 min sample of an intracellular recording of in vivo synaptic activity from a vasomotor C-neuron in a bullfrog sympathetic ganglion was converted to a series of stimulus pulses. This physiologically derived activity was used to stimulate preganglionic C-fibres of similar ganglia studied in vitro. Intracellular recordings were made from exocrine B-cells within the ganglia. Although they do not receive fast, nicotinic synaptic input from preganglionic C-fibres, B-cell excitability was profoundly increased by stimulation of C-fibres with physiologically derived activity. Also, subthreshold depolarizing current pulses that failed to generate action potentials in B-cells under control conditions almost always generated action potentials whilst C-fibres were activated. These effects were attenuated or prevented by the luteinizing hormone releasing hormone antagonist, [D-pyro-Glu1,D-Phe2,D-Trp3,6]-LHRH (70 microM). The physiological release of luteinizing hormone releasing hormone from C-fibres therefore causes an interaction between vasomotor and exocrine outflow within a paravertebral sympathetic ganglion.     

A gradient model of cardiac pacemaker myocytes

We have formulated a spatial-gradient model of action potential heterogeneity within the rabbit sinoatrial node (SAN), based on cell-specific ionic models of electrical activity from its central and peripheral regions. The ionic models are derived from a generic cell model, incorporating five background and exchange currents, and seven time-dependent currents based on three- or four-state Markov schemes. State transition rates are given by non-linear sigmoid functions of membrane potential.

By appropriate selection of parameters, the generic model is able to accurately reproduce a wide range of action potential waveforms observed experimentally. Specifically, the model can fit recordings from central and peripheral regions of the SAN with RMS errors of 0.3987 and 0.7628 mV, respectively. Using a custom least squares parameter optimisation routine, we have constructed a spatially-varying gradient model that exhibits a smooth transition in action potential characteristics from the central to the peripheral region, whilst ensuring individual membrane currents remain physiologically accurate. Smooth transition action potential characteristics include maximum diastolic potential, overshoot potential, upstroke velocity, action potential duration and cycle length. The gradient model is suitable for developing higher dimensional models of the right atrium, in which action potential heterogeneity within nodal tissue may be readily incorporated.     

The retrograde frequency response of passive dendritic trees constrains the nonlinear firing behaviour of a reduced neuron model

Our goal was to investigate how the propagation of alternating signals (i.e. AC), like action potentials, into the dendrites influenced nonlinear firing behaviour of motor neurons using a systematically reduced neuron model. A recently developed reduced modeling approach using only steady-current (i.e. DC) signaling was analytically expanded to retain features of the frequency-response analysis carried out in multicompartment anatomically reconstructed models. Bifurcation analysis of the extended model showed that the typically overlooked parameter of AC amplitude attenuation was positively correlated with the current threshold for the activation of a plateau potential in the dendrite. Within the multiparameter space map of the reduced model the region demonstrating "fully-bistable" firing was bounded by directional DC attenuation values that were negatively correlated to AC attenuation. Based on these results we conclude that analytically derived reduced models of dendritic trees should be fit on DC and AC signaling, as both are important biophysical parameters governing the nonlinear firing behaviour of motor neurons.     

Increased excitability of acidified skeletal muscle: role of chloride conductance

Generation of the action potentials (AP) necessary to activate skeletal muscle fibers requires that inward membrane currents exceed outward currents and thereby depolarize the fibers to the voltage threshold for AP generation. Excitability therefore depends on both excitatory Na+ currents and inhibitory K+ and Cl- currents. During intensive exercise, active muscle loses K+ and extracellular K+ ([K+]o) increases. Since high [K+]o leads to depolarization and ensuing inactivation of voltage-gated Na+ channels and loss of excitability in isolated muscles, exercise-induced loss of K+ is likely to reduce muscle excitability and thereby contribute to muscle fatigue in vivo. Intensive exercise, however, also leads to muscle acidification, which recently was shown to recover excitability in isolated K(+)-depressed muscles of the rat. Here we show that in rat soleus muscles at 11 mM K+, the almost complete recovery of compound action potentials and force with muscle acidification (CO2 changed from 5 to 24%) was associated with reduced chloride conductance (1731 +/- 151 to 938 +/- 64 microS/cm2, P < 0.01) but not with changes in potassium conductance (405 +/- 20 to 455 +/- 30 microS/cm2, P < 0.16). Furthermore, acidification reduced the rheobase current by 26% at 4 mM K+ and increased the number of excitable fibers at elevated [K+]o. At 11 mM K+ and normal pH, a recovery of excitability and force similar to the observations with muscle acidification could be induced by reducing extracellular Cl- or by blocking the major muscle Cl- channel, ClC-1, with 30 microM 9-AC. It is concluded that recovery of excitability in K(+)-depressed muscles induced by muscle acidification is related to reduction in the inhibitory Cl- currents, possibly through inhibition of ClC-1 channels, and acidosis thereby reduces the Na+ current needed to generate and propagate an AP. Thus short term regulation of Cl- channels is important for maintenance of excitability in working muscle.     

Mathematical analysis of the changes in the parameters of the action potentials,membrane and ionic currents of frog muscle fibre during the recovery cycle

The method of mathematical modelling was used to study the excitability changes of the membrane of a frog skeletal muscle fibre and the parameters of the action potentials, membrane and ionic currents during the first 30 ms of the recovery cycle.The threshold current for a fibre at rest was found to be 0.32 A and the durations of the absolute and relative refractory periods were respectively 4 ms and 5.2 ms. With increasing interpulse interval, the subnormality of the membrane excitability is followed by supernomality. Under the same condition the supernormality in the velocity recovery cycle is not obtained.In the recovery cycle, the shape (polarity, sequence and number of phases) of the action potentials, of the membrane and ionic currents and their conductances, are unchanged. Only the time and amplitude parameters of the quantities listed above are known to vary. With increasing the interpulse interval, the amplitudes of the quantities increase and their durations are shortened attaining the values of the corresponding quantities of the initial action potential.The membrane properties are recovered 30 ms after application of the initial pulse, but the supernormality of the excitability is still preserved.     

A hydraulic-photosynthetic model based on extended HLH and its application to Coast redwood (Sequoia sempervirens)

Although recent investigations [Ryan, M.G., Yoder, B.J., 1997. Hydraulic limits to tree height and tree growth. Bioscience 47, 235-242; Koch, G.W., Sillett, S.C.,Jennings, G.M.,Davis, S.D., 2004. The limits to tree height. Nature 428, 851-854; Niklas, K.J., Spatz, H., 2004. Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass. Proc. Natl Acad. Sci. 101, 15661-15663; Ryan, M.G., Phillips, N., Bond, B.J., 2006. Hydraulic limitation hypothesis revisited. Plant Cell Environ. 29, 367-381; Niklas, K.J., 2007. Maximum plant height and the biophysical factors that limit it. Tree Physiol. 27, 433-440; Burgess, S.S.O., Dawson, T.E., 2007. Predicting the limits to tree height using statistical regressions of leaf traits. New Phytol. 174, 626-636] suggested that the hydraulic limitation hypothesis (HLH) is the most plausible theory to explain the biophysical limits to maximum tree height and the decline in tree growth rate with age, the analysis is largely qualitative or based on statistical regression. Here we present an integrated biophysical model based on the principle that trees develop physiological compensations (e.g. the declined leaf water potential and the tapering of conduits with heights [West, G.B., Brown, J.H., Enquist, B.J., 1999. A general model for the structure and allometry of plant vascular systems. Nature 400, 664-667]) to resist the increasing water stress with height, the classical HLH and the biochemical limitations on photosynthesis [von Caemmerer, S., 2000. Biochemical Models of Leaf Photosynthesis. CSIRO Publishing, Australia]. The model has been applied to the tallest trees in the world (viz. Coast redwood (Sequoia sempervirens)). Xylem water potential, leaf carbon isotope composition, leaf mass to area ratio at different heights derived from the model show good agreements with the experimental measurements of Koch et al. [2004. The limits to tree height. Nature 428, 851-854]. The model also well explains the universal trend of declining growth rate with age.     

Modulation of M-current by intracellular Ca2+

IM is a voltage- and time-dependent K+ current that is suppressed by muscarinic receptor activation. IM augmentation following agonist washout was blocked by heavily buffering [Ca2+]i using BAPTA. Although IM is not primarily Ca2+ dependent, small increases in [Ca2+]i by photolysis of the "caged" Ca2+ chelator nitr-5 or by evoking action potentials augmented, while larger increases inhibited, IM. Raising [Ca2+]i for prolonged periods, by nitr-5 photolysis, reduced its sensitivity to agonist, leaving a poorly reversible response. These results suggest that IM can be regulated by physiologically relevant changes in [Ca2+]i, placing IM in a unique position to modulate cell excitability.     

A mathematical model of action potentials of mouse sinoatrial node cells with molecular bases

Genetically modified mice are popular experimental models for studying the molecular bases and mechanisms of cardiac arrhythmia. A postgenome challenge is to classify the functional roles of genes in cardiac function. To unveil the functional role of various genetic isoforms of ion channels in generating cardiac pacemaking action potentials (APs), a mathematical model for spontaneous APs of mouse sinoatrial node (SAN) cells was developed. The model takes into account the biophysical properties of membrane ionic currents and intracellular mechanisms contributing to spontaneous mouse SAN APs. The model was validated by its ability to reproduce the physiological exceptionally short APs and high pacing rates of mouse SAN cells. The functional roles of individual membrane currents were evaluated by blocking their coding channels. The roles of intracellular Ca(2+)-handling mechanisms on cardiac pacemaking were also investigated in the model. The robustness of model pacemaking behavior was evaluated by means of one- and two-parameter analyses in wide parameter value ranges. This model provides a predictive tool for cellular level outcomes of electrophysiological experiments. It forms the basis for future model development and further studies into complex pacemaking mechanisms as more quantitative experimental data become available.     

A unified model of CA1/3 pyramidal cells: an investigation into excitability

After-depolarisation is a hallmark of excitability in hippocampal pyramidal cells of CA1 and CA3 regions, because it constitutes the subthreshold relation between inward and outward ionic currents. This relationship determines the nominal response to stimuli and provides the necessary conditions for firing a spike or a burst of action potentials. Nevertheless, after-depolarisation is an inherently transient phenomenon that is not very well understood. We study after-depolarisation using a single-compartment pyramidal-cell model based on recent voltage- and current-clamp experimental data. We systematically investigate CA1 and CA3 behaviour and show that changes to maximal conductances of T-type Ca(2+)-current and muscarinic-sensitive and delayed rectifier K(+)-currents are sufficient to switch the behaviour of the model from a CA3 to a CA1 neuron. We use model analysis to define after-depolarisation and bursting threshold. We also explain the influence of particular ionic currents on this phenomenon. This study ends with a sensitivity analysis that demonstrates the influence of specific currents on excitability. Counter-intuitively, we find that a decrease of Na(+)-current could cause an increase in excitability. Our analysis suggests that a change of high-voltage activated Ca(2+)-current can have a similar effect.     

Pre & postsynaptic tuning of action potential timing by spontaneous GABAergic activity

Frequency and timing of action potential discharge are key elements for coding and transfer of information between neurons. The nature and location of the synaptic contacts, the biophysical parameters of the receptor-operated channels and their kinetics of activation are major determinants of the firing behaviour of each individual neuron. Ultimately the intrinsic excitability of each neuron determines the input-output function. Here we evaluate the influence of spontaneous GABAergic synaptic activity on the timing of action potentials in Layer 2/3 pyramidal neurones in acute brain slices from the somatosensory cortex of young rats. Somatic dynamic current injection to mimic synaptic input events was employed, together with a simple computational model that reproduce subthreshold membrane properties. Besides the well-documented control of neuronal excitability, spontaneous background GABAergic activity has a major detrimental effect on spike timing. In fact, GABA(A) receptors tune the relationship between the excitability and fidelity of pyramidal neurons via a postsynaptic (the reversal potential for GABA(A) activity) and a presynaptic (the frequency of spontaneous activity) mechanism. GABAergic activity can decrease or increase the excitability of pyramidal neurones, depending on the difference between the reversal potential for GABA(A) receptors and the threshold for action potential. In contrast, spike time jitter can only be increased proportionally to the difference between these two membrane potentials. Changes in excitability by background GABAergic activity can therefore only be associated with deterioration of the reliability of spike timing.     

A thermodynamic model for the cooperative functional properties of the tetraheme cytochrome c3 from Desulfovibrio gigas.

A thermodynamic model is presented to describe the redox behaviour of the tetraheme cytochrome c3 from Desulfovibrio gigas. This molecule displays different intrinsic redox potentials for the four hemes and during the redox titration process, interactions among different hemes occur, thus altering the values of redox potentials according to which of the hemes are oxidized [Santos, H., Moura, J.J.G., Moura, I., LeGall, J. & Xavier, A.V. (1984) Eur. J. Biochem. 141, 283-296]. This complex cooperative behaviour [Xavier, A.V. (1986) J. Inorg. Biochem. 28, 239-243] has been analyzed here using an I2H4-interaction network [Cornish-Bowden, A. & Koshland, D.E. Jr (1970) J. Biol. Chem. 245, 6241-6250] coupled to a proton-linked equilibrium between two tertiary structures. Such a formalism, which requires a reduced number of parameters, is able to fully account quantitatively for the pH dependence of the NMR redox-titration curves. The 'redox-Bohr' effect is discussed in terms of the available structure and thermodynamic data and a functional mechanism is proposed.     

Realization of biophysical models of cardiac electrical activity

Considered are the principles of realization of biophysical models of heart ventricle electrical activity in the form of a double electric layer on the surface of the electrically active myocardium (epicardium and endocardium) and the boundary surfaces dividing the model compartments with different electrophysiological characteristics. The model parameters are the electrophysiological and anatomical characteristics of the heart such as the geometry of the ventricles and the specialized His-Purkinje conduction system, the velocity of depolarization spread over myocardium, the ratio of the velocities of excitation transmission through the Myocardium / His / Purkinje elements of the model, the shape of transmembrane action potentials on the boundary surfaces, the orientation of the intrinsic anatomical axes of the heart relative to the initial set of coordinates, and some other biophysical characteristics of the myocardium. This model is the main unit of a computer simulation system, which includes databases of real and simulated electrocardiosignals.     

Spike Inference from Calcium Imaging Using Sequential Monte Carlo Methods

As recent advances in calcium sensing technologies facilitate simultaneously imaging action potentials in neuronal populations, complementary analytical tools must also be developed to maximize the utility of this experimental paradigm. Although the observations here are fluorescence movies, the signals of interest—spike trains and/or time varying intracellular calcium concentrations—are hidden. Inferring these hidden signals is often problematic due to noise, nonlinearities, slow imaging rate, and unknown biophysical parameters. We overcome these difficulties by developing sequential Monte Carlo methods (particle filters) based on biophysical models of spiking, calcium dynamics, and fluorescence. We show that even in simple cases, the particle filters outperform the optimal linear (i.e., Wiener) filter, both by obtaining better estimates and by providing error bars. We then relax a number of our model assumptions to incorporate nonlinear saturation of the fluorescence signal, as well external stimulus and spike history dependence (e.g., refractoriness) of the spike trains. Using both simulations and in vitro fluorescence observations, we demonstrate temporal superresolution by inferring when within a frame each spike occurs. Furthermore, the model parameters may be estimated using expectation maximization with only a very limited amount of data (e.g., ∼5-10 s or 5-40 spikes), without the requirement of any simultaneous electrophysiology or imaging experiments.     

A self-organizing state-space-model approach for parameter estimation in hodgkin-huxley-type models of single neurons

Traditional approaches to the problem of parameter estimation in biophysical models of neurons and neural networks usually adopt a global search algorithm (for example, an evolutionary algorithm), often in combination with a local search method (such as gradient descent) in order to minimize the value of a cost function, which measures the discrepancy between various features of the available experimental data and model output. In this study, we approach the problem of parameter estimation in conductance-based models of single neurons from a different perspective. By adopting a hidden-dynamical-systems formalism, we expressed parameter estimation as an inference problem in these systems, which can then be tackled using a range of well-established statistical inference methods. The particular method we used was Kitagawa's self-organizing state-space model, which was applied on a number of Hodgkin-Huxley-type models using simulated or actual electrophysiological data. We showed that the algorithm can be used to estimate a large number of parameters, including maximal conductances, reversal potentials, kinetics of ionic currents, measurement and intrinsic noise, based on low-dimensional experimental data and sufficiently informative priors in the form of pre-defined constraints imposed on model parameters. The algorithm remained operational even when very noisy experimental data were used. Importantly, by combining the self-organizing state-space model with an adaptive sampling algorithm akin to the Covariance Matrix Adaptation Evolution Strategy, we achieved a significant reduction in the variance of parameter estimates. The algorithm did not require the explicit formulation of a cost function and it was straightforward to apply on compartmental models and multiple data sets. Overall, the proposed methodology is particularly suitable for resolving high-dimensional inference problems based on noisy electrophysiological data and, therefore, a potentially useful tool in the construction of biophysical neuron models.     

Canards and mixed-mode oscillations in a forest pest model

We consider a three-variable forest pest model, proposed by Rinaldi & Muratori (1992) [Rinaldi, S., Muratori, S., 1992. Limit cycles in slow–fast forest–pest models. Theor. Popul. Biol. 41, 26–43]. The model allows relaxation oscillations where long pest-free periods are interspersed with outbreaks of high pest concentration. For small values of the timescale of the young trees, the model can be reduced to a two-dimensional model. By a geometrical analysis we identify a canard explosion in the reduced model, that is, a change over a narrow parameter interval from outbreak dynamics to small oscillations around an endemic state. For larger values of the timescale of the young trees the two-dimensional approximation breaks down, and a broader parameter interval with mixed-mode oscillations appear, replacing the simple canard explosion. The analysis only relies on simple and generic properties of the model, and is expected to be applicable in a larger class of multiple timescale dynamical models.     

Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g _L and g ₁. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g _L ???g ₁ parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g ₁ increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I _h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.