Stochastic differential equation model for cerebellar granule cell excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.     

An analysis of the mammalian ventricular action potential

Summary A Hodgkin-Huxley model for ventricular excitation is abstracted from electrophysiological data. A singular perturbation analysis of the 8-dimensional phase portrait of the model characterizes the role of calcium during the plateau phase of the ventricular action potential and demonstrates how the calcium refractory period prevents tetanization. Supported in part by the Undergraduate Research Opportunities Program, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA     

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.     

Native Gating Behavior of Ion Channels in Neurons with Null-Deviation Modeling

Computational modeling has emerged as an indispensable approach to resolve and predict the intricate interplay among the many ion channels underlying neuronal excitability. However, simulation results using the classic formula-based Hodgkin-Huxley (H-H) model or the superior Markov kinetic model of ion channels often deviate significantly from native cellular signals despite using carefully measured parameters. Here we found that the filters of patch-clamp amplifier not only delayed the signals, but also introduced ringing, and that the residual series resistance in experiments altered the command voltages, which had never been fully eliminated by improving the amplifier itself. To remove all the above errors, a virtual device with the parameters exactly same to that of amplifier was introduced into Markov kinetic modeling so as to establish a null-deviation model. We demonstrate that our novel null-deviation approach fully restores the native gating-kinetics of ion-channels with the data recorded at any condition, and predicts spike waveform and firing patterns clearly distinctive from those without correction.     

A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity

Many neurons possess dendrites enriched with sodium channels and are capable of generating action potentials. However, the role of dendritic sodium spikes remain unclear. Here, we study computational models of neurons to investigate the functional effects of dendritic spikes. In agreement with previous studies, we found that point neurons or neurons with passive dendrites increase their somatic firing rate in response to the correlation of synaptic bombardment for a wide range of input conditions, i.e. input firing rates, synaptic conductances, or refractory periods. However, neurons with active dendrites show the opposite behavior: for a wide range of conditions the firing rate decreases as a function of correlation. We found this property in three types of models of dendritic excitability: a Hodgkin-Huxley model of dendritic spikes, a model with integrate and fire dendrites, and a discrete-state dendritic model. We conclude that fast dendritic spikes confer much broader computational properties to neurons, sometimes opposite to that of point neurons.     

Stabilizing role of calcium store-dependent plasma membrane calcium channels in action-potential firing and intracellular calcium oscillations

In many biological systems, cells display spontaneous calcium oscillations (CaOs) and repetitive action-potential firing. These phenomena have been described separately by models for intracellular inositol trisphosphate (IP3)-mediated CaOs and for plasma membrane excitability. In this study, we present an integrated model that combines an excitable membrane with an IP3-mediated intracellular calcium oscillator. The IP3 receptor is described as an endoplasmic reticulum (ER) calcium channel with open and close probabilities that depend on the cytoplasmic concentration of IP3 and Ca2+. We show that simply combining this ER model for intracellular CaOs with a model for membrane excitability of normal rat kidney (NRK) fibroblasts leads to instability of intracellular calcium dynamics. To ensure stable long-term periodic firing of action potentials and CaOs, it is essential to incorporate calcium transporters controlled by feedback of the ER store filling, for example, store-operated calcium channels in the plasma membrane. For low IP3 concentrations, our integrated NRK cell model is at rest at -70 mV. For higher IP3 concentrations, the CaOs become activated and trigger repetitive firing of action potentials. At high IP3 concentrations, the basal intracellular calcium concentration becomes elevated and the cell is depolarized near -20 mV. These predictions are in agreement with the different proliferative states of cultures of NRK fibroblasts. We postulate that the stabilizing role of calcium channels and/or other calcium transporters controlled by feedback from the ER store is essential for any cell in which calcium signaling by intracellular CaOs involves both ER and plasma membrane calcium fluxes.     

Low dimensional model of bursting neurons

A computationally efficient, biophysically-based model of neuronal behavior is presented; it incorporates ion channel dynamics in its two fast ion channels while preserving simplicity by representing only one slow ion current. The model equations are shown to provide a wide array of physiological dynamics in terms of spiking patterns, bursting, subthreshold oscillations, and chaotic firing. Despite its simplicity, the model is capable of simulating an extensive range of spiking patterns. Several common neuronal behaviors observed in vivo are demonstrated by varying model parameters. These behaviors are classified into dynamical classes using phase diagrams whose boundaries in parameter space prove to be accurately delineated by linear stability analysis. This simple model is suitable for use in large scale simulations involving neural field theory or neuronal networks.     

Contributions of T-type calcium channel isoforms to neuronal firing

Low voltage-activated (LVA) T-type calcium channels play critical roles in the excitability of many cell types and are a focus of research aimed both at understanding the physiological basis of calcium channel-dependent signaling and the underlying pathophysiology associated with hyperexcitability disorders such as epilepsy. These channels play a critical role towards neuronal firing in both conducting calcium ions during action potentials and also in switching neurons between distinct modes of firing. In this review the properties of the CaV3.1, CaV3.2 and CaV3.3 T-type channel isoforms is discussed in relation to their individual contributions to action potentials during burst and tonic firing states as well their roles in switching between firing states.     

The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model

We have previously discussed qualitative models for bursting and thalamic neurons that were obtained by modifying a simple two-dimensional model for repetitive firing. In this paper we report the results of making a similar sequence of modifications to a more elaborate six-dimensional model of repetitive firing which is based on the Hodgkin-Huxley equations. To do this we first reduce the six-dimensional model to a two-dimensional model that resembles our original two-dimensional qualitative model. This is achieved by defining a new variable, which we call q. We then add a subthreshold inward current and a subthreshold outward current having a variable, z, that changes slowly. This gives a three-dimensional (v,q,z) model of the Hodgkin-Huxley type, which we refer to as the z-model. Depending on the choice of parameter values this model resembles our previous models of bursting and thalamic neurons. At each stage in the development of these models we return to the corresponding seven-dimensional model to confirm that we can obtain similar solutions by using the complete system of equations. The analysis of the three-dimensional model involves a state diagram and a stability diagram. The state diagram shows the projection of the phase path from v,q,z space into the v,z plane, together with the projections of the curves z = 0 and v = q = 0. The stability of the points on the curve v = q = 0, which we call the v, q nullcurve, is determined by the stability diagram. Taken together the state and stability diagrams show how to assemble the ionic currents to produce a given firing pattern.     

Numerical simulation of motility patterns of the small bowel. 1. formulation of a mathematical model

A complete mathematical model of the periodic myoelectrical activity of a functional unit of the small intestine is presented. Based on real morphological and electrophysiological data, the model assumes that: the functional unit is an electromyogenic syncytium; the kinetics of L-type Ca2+, T-type Ca2+, Ca2+-activated K+, voltage dependent K+and Cl-channels determine the electrical activity of the functional unit; the enteric nervous system is satisfactorily represented by an efferent cholinergic neuron that provides an excitatory input to the functional unit through receptor-linked L-type Ca2+channels and by an afferent pathway composed of the primary and secondary sensory neurons; the dynamics of propagation of the wave of depolarization along the unmyelinated nerve axons satisfy the Hodgkin-Huxley model; the electrical activity of the neural soma reflects the interaction of N-type Ca2+channels, Ca2+-activated K+and voltage dependent Na+, K+and Cl-channels; the smooth muscle syncytium of the locus is a null-dimensional contractile system. With the proposed model the dynamics of active force generation are determined entirely by the concentration of cytosolic calcium. The model describes: the mechanical excitation of the free nerve endings of the mechanoreceptor of the receptive field of the pathway; the electrical processes of the propagation of excitation along the afferent and efferent neural circuits; the chemical mechanisms of nerve-pulse transmission at the synaptic zones; the slow wave and bursting type electrical activity; cytosolic calcium concentration; the dynamics of active force generation. Numerical simulations have shown that the model can display different electrical patterns and mechanical responses of the locus. The results show good qualitative and quantitative agreement with the results of experiments conducted on the small intestine.     

Neural Avalanches at the Critical Point between Replay and Non-Replay of Spatiotemporal Patterns

We model spontaneous cortical activity with a network of coupled spiking units, in which multiple spatio-temporal patterns are stored as dynamical attractors. We introduce an order parameter, which measures the overlap (similarity) between the activity of the network and the stored patterns. We find that, depending on the excitability of the network, different working regimes are possible. For high excitability, the dynamical attractors are stable, and a collective activity that replays one of the stored patterns emerges spontaneously, while for low excitability, no replay is induced. Between these two regimes, there is a critical region in which the dynamical attractors are unstable, and intermittent short replays are induced by noise. At the critical spiking threshold, the order parameter goes from zero to one, and its fluctuations are maximized, as expected for a phase transition (and as observed in recent experimental results in the brain). Notably, in this critical region, the avalanche size and duration distributions follow power laws. Critical exponents are consistent with a scaling relationship observed recently in neural avalanches measurements. In conclusion, our simple model suggests that avalanche power laws in cortical spontaneous activity may be the effect of a network at the critical point between the replay and non-replay of spatio-temporal patterns.     

Modeling action potential generation and propagation in NRK fibroblasts

Normal rat kidney (NRK) fibroblasts change their excitability properties through the various stages of cell proliferation. The present mathematical model has been developed to explain excitability of quiescent (serum deprived) NRK cells. It includes as cell membrane components, on the basis of patch-clamp experiments, an inwardly rectifying potassium conductance (G_Kir), an L-type calcium conductance (G_CaL), a leak conductance (G_leak), an intracellular calcium-activated chloride conductance [G_Cl(Ca)], and a gap junctional conductance (G_gj), coupling neighboring cells in a hexagonal pattern. This membrane model has been extended with simple intracellular calcium dynamics resulting from calcium entry via G_CaL channels, intracellular buffering, and calcium extrusion. It reproduces excitability of single NRK cells and cell clusters and intercellular action potential (AP) propagation in NRK cell monolayers. Excitation can be evoked by electrical stimulation, external potassium-induced depolarization, or hormone-induced intracellular calcium release. Analysis shows the roles of the various ion channels in the ultralong (30 s) NRK cell AP and reveals the particular role of intracellular calcium dynamics in this AP. We support our earlier conclusion (De Roos A, Willems PH, van Zoelen EJ, and Theuvenet AP. Am J Physiol Cell Physiol 273: C1900–C1907, 1997) that AP generation and propagation may act as a rapid mechanism for the propagation of intracellular calcium waves, thus contributing to fast intercellular calcium signaling. The present model serves as a starting point to further analyze excitability changes during contact inhibition and cell transformation. Hodgkin-Huxley model; intracellular calcium dynamics; L-type calcium conductance; inward rectifier; calcium-activated chloride conductance; gap junctional coupling     

Long-lasting increases in intrinsic excitability triggered by inhibition

Although experience-dependent changes in neural circuits are commonly assumed to be mediated by synaptic plasticity, modifications of intrinsic excitability may serve as a complementary mechanism. In whole-cell recordings from spontaneously firing vestibular nucleus neurons, brief periods of inhibitory synaptic stimulation or direct membrane hyperpolarization triggered long-lasting increases in spontaneous firing rates and firing responses to intracellular depolarization. These increases in excitability, termed firing rate potentiation, were induced by decreases in intracellular calcium and expressed as reductions in the sensitivity to the BK-type calcium-activated potassium channel blocker iberiotoxin. Firing rate potentiation is a novel form of cellular plasticity that could contribute to motor learning in the vestibulo-ocular reflex.     

A novel disease connection for TRPM2 channels

Low voltage-activated (LVA) T-type calcium channels play critical roles in the excitability of many cell types and are a focus of research aimed both at understanding the physiological basis of calcium channel-dependent signaling and the underlying pathophysiology associated with hyperexcitability disorders such as epilepsy. These channels play a critical role towards neuronal firing in both conducting calcium ions during action potentials and also in switching neurons between distinct modes of firing. In this review the properties of the Ca_V3.1, Ca_V3.2 and Ca_V3.3 T-type channel isoforms is discussed in relation to their individual contributions to action potentials during burst and tonic firing states as well their roles in switching between firing states.     

A neural mass model based on single cell dynamics to model pathophysiology

Neural mass models are successful in modeling brain rhythms as observed in macroscopic measurements such as the electroencephalogram (EEG). While the synaptic current is explicitly modeled in current models, the single cell electrophysiology is not taken into account. To allow for investigations of the effects of channel pathologies, channel blockers and ion concentrations on macroscopic activity, we formulate neural mass equations explicitly incorporating the single cell dynamics by using a bottom-up approach. The mean and variance of the firing rate and synaptic input distributions are modeled. The firing rate curve (F(I)-curve) is used as link between the single cell and macroscopic dynamics. We show that this model accurately reproduces the behavior of two populations of synaptically connected Hodgkin-Huxley neurons, also in non-steady state.     

Effect of firing rate on the calcium permeability in adult neurons during spontaneous action potentials.

Calcium channels in neurons mediate a wide variety of essential functions. In addition to contributing to action potential shape, they furnish a substrate that acts as an intracellular second messenger. This study shows that the shape of the neuronal action potential has characteristics that promote long openings of L-type (high threshold) calcium channels. We also present evidence that a change in the firing rate of isolated neurons modulates gating of single calcium channels. This mechanism could be important in modulating neuron excitability and providing a rise in intracellular Ca, when needed.     

Formation and Dynamics of Waves in a Cortical Model of Cholinergic Modulation

Acetylcholine (ACh) is a regulator of neural excitability and one of the neurochemical substrates of sleep. Amongst the cellular effects induced by cholinergic modulation are a reduction in spike-frequency adaptation (SFA) and a shift in the phase response curve (PRC). We demonstrate in a biophysical model how changes in neural excitability and network structure interact to create three distinct functional regimes: localized asynchronous, traveling asynchronous, and traveling synchronous. Our results qualitatively match those observed experimentally. Cortical activity during slow wave sleep (SWS) differs from that during REM sleep or waking states. During SWS there are traveling patterns of activity in the cortex; in other states stationary patterns occur. Our model is a network composed of Hodgkin-Huxley type neurons with a M-current regulated by ACh. Regulation of ACh level can account for dynamical changes between functional regimes. Reduction of the magnitude of this current recreates the reduction in SFA the shift from a type 2 to a type 1 PRC observed in the presence of ACh. When SFA is minimal (in waking or REM sleep state, high ACh) patterns of activity are localized and easily pinned by network inhomogeneities. When SFA is present (decreasing ACh), traveling waves of activity naturally arise. A further decrease in ACh leads to a high degree of synchrony within traveling waves. We also show that the level of ACh determines how sensitive network activity is to synaptic heterogeneity. These regimes may have a profound functional significance as stationary patterns may play a role in the proper encoding of external input as memory and traveling waves could lead to synaptic regularization, giving unique insights into the role and significance of ACh in determining patterns of cortical activity and functional differences arising from the patterns.     

Modulation of neuronal excitability by intracellular calcium buffering: from spiking to bursting

We have investigated the detailed regulation of neuronal firing pattern by the cytosolic calcium buffering capacity using a combination of mathematical modeling and patch-clamp recording in acute slice. Theoretical results show that a high calcium buffer concentration alters the characteristic regular firing of cerebellar granule cells and that a transition to various modes of oscillations occurs, including bursting. Using bifurcation analysis, we show that this transition from spiking to bursting is a consequence of the major slowdown of calcium dynamics. Patch-clamp recordings on cerebellar granule cells loaded with a high concentration of the fast calcium buffer BAPTA (15 mM) reveal dramatic alterations in their excitability as compared to cells loaded with 0.15 mM BAPTA. In high calcium buffering conditions, granule cells exhibit all bursting behaviors predicted by the model whereas bursting is never observed in low buffering. These results suggest that cytosolic calcium buffering capacity can tightly modulate neuronal firing patterns leading to generation of complex patterns and therefore that calcium-binding proteins may play a critical role in the non-synaptic plasticity and information processing in the central nervous system.