Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

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.     

A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon.

Sodium channel gating behavior was modeled with Markovian models fitted to currents from the cut-open squid giant axon in the absence of divalent cations. Optimum models were selected with maximum likelihood criteria using single-channel data, then models were refined and extended by simultaneous fitting of macroscopic ionic currents, ON and OFF gating currents, and single-channel first latency densities over a wide voltage range. Best models have five closed states before channel opening, with inactivation from at least one closed state as well as the open state. Forward activation rate constants increase with depolarization, and deactivation rate constants increase with hyperpolarization. Rates of inactivation from the open or closed states are generally slower than activation or deactivation rates and show little or no voltage dependence. Channels tend to reopen several times before inactivating. Macroscopic rates of activation and inactivation result from a combination of closed, open and inactivated state transitions. At negative potentials the time to first opening dominates the macroscopic current due to slow activation rates compared with deactivation rates: channels tend to reopen rarely, and often inactivate from closed states before they reopen. At more positive potentials, the time to first opening and burst duration together produce the macroscopic current.     

Consistency and Diversity of Spike Dynamics in the Neurons of Bed Nucleus of Stria Terminalis of the Rat: A Dynamic Clamp Study

Neurons display a high degree of variability and diversity in the expression and regulation of their voltage-dependent ionic channels. Under low level of synaptic background a number of physiologically distinct cell types can be identified in most brain areas that display different responses to standard forms of intracellular current stimulation. Nevertheless, it is not well understood how biophysically different neurons process synaptic inputs in natural conditions, i.e., when experiencing intense synaptic bombardment in vivo. While distinct cell types might process synaptic inputs into different patterns of action potentials representing specific “motifs” of network activity, standard methods of electrophysiology are not well suited to resolve such questions. In the current paper we performed dynamic clamp experiments with simulated synaptic inputs that were presented to three types of neurons in the juxtacapsular bed nucleus of stria terminalis (jcBNST) of the rat. Our analysis on the temporal structure of firing showed that the three types of jcBNST neurons did not produce qualitatively different spike responses under identical patterns of input. However, we observed consistent, cell type dependent variations in the fine structure of firing, at the level of single spikes. At the millisecond resolution structure of firing we found high degree of diversity across the entire spectrum of neurons irrespective of their type. Additionally, we identified a new cell type with intrinsic oscillatory properties that produced a rhythmic and regular firing under synaptic stimulation that distinguishes it from the previously described jcBNST cell types. Our findings suggest a sophisticated, cell type dependent regulation of spike dynamics of neurons when experiencing a complex synaptic background. The high degree of their dynamical diversity has implications to their cooperative dynamics and synchronization.     

On the mechanisms underlying the depolarization block in the spiking dynamics of CA1 pyramidal neurons

Under sustained input current of increasing strength neurons eventually stop firing, entering a depolarization block. This is a robust effect that is not usually explored in experiments or explicitly implemented or tested in models. However, the range of current strength needed for a depolarization block could be easily reached with a random background activity of only a few hundred excitatory synapses. Depolarization block may thus be an important property of neurons that should be better characterized in experiments and explicitly taken into account in models at all implementation scales. Here we analyze the spiking dynamics of CA1 pyramidal neuron models using the same set of ionic currents on both an accurate morphological reconstruction and on its reduction to a single-compartment. The results show the specific ion channel properties and kinetics that are needed to reproduce the experimental findings, and how their interplay can drastically modulate the neuronal dynamics and the input current range leading to a depolarization block. We suggest that this can be one of the rate-limiting mechanisms protecting a CA1 neuron from excessive spiking activity.     

Statistical model of the hippocampal CA3 region

A model with intermediate complexity is introduced to reproduce the basic firing modes of the CA3 pyramidal cell. Our model consists of a single compartment, has two variables (membrane potential and internal calcium concentration), and involves two separate stages for interspike mechanisms and firing. Interspike dynamics is governed by voltage- and calcium-dependent ionic channels but no channel kinetics are provided. This model is suitable to be included in our statistical population model (Part II, following paper). Bifurcation analysis reveals that interspike dynamics rather than sodium firing has the dominant role in the control of bursting/nonbursting behavior. Received: 29 August 1997 / Accepted in revised form: 17 July 1998     

A homeostatic model of neuronal firing governed by feedback signals from the extracellular matrix

Molecules of the extracellular matrix (ECM) can modulate the efficacy of synaptic transmission and neuronal excitability. These mechanisms are crucial for the homeostatic regulation of neuronal firing over extended timescales. In this study, we introduce a simple mathematical model of neuronal spiking balanced by the influence of the ECM. We consider a neuron receiving random synaptic input in the form of Poisson spike trains and the ECM, which is modeled by a phenomenological variable involved in two feedback mechanisms. One feedback mechanism scales the values of the input synaptic conductance to compensate for changes in firing rate. The second feedback accounts for slow fluctuations of the excitation threshold and depends on the ECM concentration. We show that the ECM-mediated feedback acts as a robust mechanism to provide a homeostatic adjustment of the average firing rate. Interestingly, the activation of feedback mechanisms may lead to a bistability in which two different stable levels of average firing rates can coexist in a spiking network. We discuss the mechanisms of the bistability and how they may be related to memory function.     

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.      

Period doubling of calcium spike firing in a model of a Purkinje cell dendrite

Recordings from cerebellar Purkinje cell dendrites have revealed that in response to sustained current injection, the cell firing pattern can move from tonic firing of Ca²⁺ spikes to doublet firing and even to quadruplet firing or more complex firing. These firing patterns are not modified substantially if Na⁺ currents are blocked. We show that the experimental results can be viewed as a slow transition of the neuronal dynamics through a period-doubling bifurcation. To further support this conclusion and to understand the underlying mechanism that leads to doublet firing, we develop and study a simple, one-compartment model of Purkinje cell dendrite. The neuron can also exhibit quadruplet and chaotic firing patterns that are similar to the firing patterns that some of the Purkinje cells exhibit experimentally. The effects of parameters such as temperature, applied current, and potassium reversal potential in the model resemble their effects in experiments. The model dynamics involve three time scales. Ca²⁺- dependent K⁺ currents, with intermediate time scales, are responsible for the appearance of doublet firing, whereas a very slow hyperpolarizing current transfers the neuron from tonic to doublet firing. We use the fast-slow analysis to separate the effects of the three time scales. Fast-slow analysis of the neuronal dynamics, with the activation variable of the very slow, hyperpolarizing current considered as a parameter, reveals that the transitions occurs via a cascade of period-doubling bifurcations of the fast and intermediate subsystem as this slow variable increases. We carry out another analysis, with the Ca²⁺ concentration considered as a parameter, to investigate the conditions for the generation of doublet firing in systems with one effective variable with intermediate time scale, in which the rest state of the fast subsystem is terminated by a saddle-node bifurcation. We find that the scenario of period doubling in these systems can occur only if (1) the time scale of the intermediate variable (here, the decay rate of the calcium concentration) is slow enough in comparison with the interspike interval of the tonic firing at the transition but is not too slow and (2) there is a bistability of the fast subsystem of the spike-generating variables.     

Dynamical description of sinoatrial node pacemaking: improved mathematical model for primary pacemaker cell

We developed an improved mathematical model for a single primary pacemaker cell of the rabbit sinoatrial node. Original features of our model include 1) incorporation of the sustained inward current (I(st)) recently identified in primary pacemaker cells, 2) reformulation of voltage- and Ca(2+)-dependent inactivation of the L-type Ca(2+) channel current (I(Ca,L)), 3) new expressions for activation kinetics of the rapidly activating delayed rectifier K(+) channel current (I(Kr)), and 4) incorporation of the subsarcolemmal space as a diffusion barrier for Ca(2+). We compared the simulated dynamics of our model with those of previous models, as well as with experimental data, and examined whether the models could accurately simulate the effects of modulating sarcolemmal ionic currents or intracellular Ca(2+) dynamics on pacemaker activity. Our model represents significant improvements over the previous models, because it can 1) simulate whole cell voltage-clamp data for I(Ca,L), I(Kr), and I(st); 2) reproduce the waveshapes of spontaneous action potentials and ionic currents during action potential clamp recordings; and 3) mimic the effects of channel blockers or Ca(2+) buffers on pacemaker activity more accurately than the previous models.     

A general diffusion model for analyzing the efficacy of synaptic input to threshold neurons

We describe a general diffusion model for analyzing the efficacy of individual synaptic inputs to threshold neurons. A formal expression is obtained for the system propagator which, when given an arbitrary initial state for the cell, yields the conditional probability distribution for the state at all later times. The propagator for a cell with a finite threshold is written as a series expansion, such that each term in the series depends only on the infinite threshold propagator, which in the diffusion limit reduces to a Gaussian form. This procedure admits a graphical representation in terms of an infinite sequence of diagrams. To connect the theory to experiment, we construct an analytical expression for the primary correlation kernel (PCK) which profiles the change in the instantaneous firing rate produced by a single postsynaptic potential (PSP). Explicit solutions are obtained in the diffusion limit to first order in perturbation theory. Our approximate expression resembles the PCK obtained by computer simulation, with the accuracy depending strongly on the mode of firing. The theory is most accurate when the synaptic input drives the membrane potential to a mean level more than one standard deviation below the firing threshold, making such cells highly sensitive to synchronous synaptic input.     

The role of a transient potassium current in a bursting neuron model

Presented here is a biophysical cell model which can exhibit low-frequency repetitive activity and bursting behavior. The model is developed from previous models (Av-Ron et al. 1991, 1993) for excitability, oscillations and bursting. A stepwise development of the present model shows the contribution of a transient potassium current (I _A ) to the overall dynamics. By changing a limited set of model parameters one can describe different firing patterns; oscillations with frequencies ranging from 2–200 Hz and a wide range of bursting behaviors in terms of the durations of bursting and quiescence, peak firing frequency and rate of change of the firing frequency.     

Spatiotemporal BOLD dynamics from a poroelastic hemodynamic model

A quantitative theory is developed for the relationship between stimulus and the resulting blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal, including both spatial and temporal dynamics for the first time. The brain tissue is modeled as a porous elastic medium, whose interconnected pores represent the vasculature. The model explicitly incorporates conservation of blood mass, interconversion of oxygenated and deoxygenated hemoglobin, force balance within the blood and of blood pressure with vessel walls, and blood flow modulation due to neuronal activity. In appropriate limits it is shown to reproduce prior Balloon models of hemodynamic response, which do not include spatial variations. The regime of validity of such models is thereby clarified by elucidating their assumptions, and when these break down, for example when voxel sizes become small.     

Neuronal models for sleep-wake regulation and synaptic reorganization in the sleeping hippocampus

In this article, we discuss mathematical models that address the control of sleep-wake behavior in the infant and adult rodent and a model that addresses changes in single-cell firing patterns in the hippocampus across wake and rapid eye movement (REM) sleep states. Each of the models describes the dynamics of experimentally identified neuronal components--either the firing activity of wake-and sleep-promoting neuronal populations or the spiking activity of hippocampal pyramidal neurons. Our discussion of each model illustrates how a mathematical model that describes the temporal dynamics of the modeled neuronal components can reveal specifics about proposed neuronal mechanisms that underlie sleep-wake regulation or sleep-specific firing patterns. For example, the dynamics of the models developed for sleep-wake regulation in the infant rodent lend insight into the involved brain-stem neuronal populations and the evolution of the network during maturation. The results of the model for sleep-wake regulation in the adult rodent suggest distinct properties of the involved neuronal populations and their interactions that account for long-lasting and brief waking bouts. The dynamics of the model for sleep-specific hippocampal neural activity proposes neural mechanisms to account for observed activity changes that can invoke synaptic reorganization associated with learning and memory consolidation.     

Exact neural mass model for synaptic-based working memory

A synaptic theory of Working Memory (WM) has been developed in the last decade as a possible alternative to the persistent spiking paradigm. In this context, we have developed a neural mass model able to reproduce exactly the dynamics of heterogeneous spiking neural networks encompassing realistic cellular mechanisms for short-term synaptic plasticity. This population model reproduces the macroscopic dynamics of the network in terms of the firing rate and the mean membrane potential. The latter quantity allows us to gain insight of the Local Field Potential and electroencephalographic signals measured during WM tasks to characterize the brain activity. More specifically synaptic facilitation and depression integrate each other to efficiently mimic WM operations via either synaptic reactivation or persistent activity. Memory access and loading are related to stimulus-locked transient oscillations followed by a steady-state activity in the β-γ band, thus resembling what is observed in the cortex during vibrotactile stimuli in humans and object recognition in monkeys. Memory juggling and competition emerge already by loading only two items. However more items can be stored in WM by considering neural architectures composed of multiple excitatory populations and a common inhibitory pool. Memory capacity depends strongly on the presentation rate of the items and it maximizes for an optimal frequency range. In particular we provide an analytic expression for the maximal memory capacity. Furthermore, the mean membrane potential turns out to be a suitable proxy to measure the memory load, analogously to event driven potentials in experiments on humans. Finally we show that the γ power increases with the number of loaded items, as reported in many experiments, while θ and β power reveal non monotonic behaviours. In particular, β and γ rhythms are crucially sustained by the inhibitory activity, while the θ rhythm is controlled by excitatory synapses.     

Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory's predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.     

Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.     

A computational grid-to-place-cell transformation model indicates a synaptic driver of place cell impairment in early-stage Alzheimer’s Disease

Alzheimer’s Disease (AD) is characterized by progressive neurodegeneration and cognitive impairment. Synaptic dysfunction is an established early symptom, which correlates strongly with cognitive decline, and is hypothesised to mediate the diverse neuronal network abnormalities observed in AD. However, how synaptic dysfunction contributes to network pathology and cognitive impairment in AD remains elusive. Here, we present a grid-cell-to-place-cell transformation model of long-term CA1 place cell dynamics to interrogate the effect of synaptic loss on network function and environmental representation. Synapse loss modelled after experimental observations in the APP/PS1 mouse model was found to induce firing rate alterations and place cell abnormalities that have previously been observed in AD mouse models, including enlarged place fields and lower across-session stability of place fields. Our results support the hypothesis that synaptic dysfunction underlies cognitive deficits, and demonstrate how impaired environmental representation may arise in the early stages of AD. We further propose that dysfunction of excitatory and inhibitory inputs to CA1 pyramidal cells may cause distinct impairments in place cell function, namely reduced stability and place map resolution.     

Cerebellar supervised learning revisited: biophysical modeling and degrees-of-freedom control

The biophysical models of spike-timing-dependent plasticity have explored dynamics with molecular basis for such computational concepts as coincidence detection, synaptic eligibility trace, and Hebbian learning. They overall support different learning algorithms in different brain areas, especially supervised learning in the cerebellum. Because a single spine is physically very small, chemical reactions at it are essentially stochastic, and thus sensitivity-longevity dilemma exists in the synaptic memory. Here, the cascade of excitable and bistable dynamics is proposed to overcome this difficulty. All kinds of learning algorithms in different brain regions confront with difficult generalization problems. For resolution of this issue, the control of the degrees-of-freedom can be realized by changing synchronicity of neural firing. Especially, for cerebellar supervised learning, the triangle closed-loop circuit consisting of Purkinje cells, the inferior olive nucleus, and the cerebellar nucleus is proposed as a circuit to optimally control synchronous firing and degrees-of-freedom in learning.