Gene circuit designs for noisy excitable dynamics

Certain cellular processes take the form of activity pulses that can be interpreted in terms of noise-driven excitable dynamics. Here we present an overview of different gene circuit architectures that exhibit excitable pulses of protein expression, when subject to molecular noise. Different types of excitable dynamics can occur depending on the bifurcation structure leading to the specific excitable phase-space topology. The bifurcation structure is not, however, linked to a particular circuit architecture. Thus a given gene circuit design can sustain different classes of excitable dynamics depending on the system parameters.     

Organization of excitable dynamics in hierarchical biological networks

This study investigates the contributions of network topology features to the dynamic behavior of hierarchically organized excitable networks. Representatives of different types of hierarchical networks as well as two biological neural networks are explored with a three-state model of node activation for systematically varying levels of random background network stimulation. The results demonstrate that two principal topological aspects of hierarchical networks, node centrality and network modularity, correlate with the network activity patterns at different levels of spontaneous network activation. The approach also shows that the dynamic behavior of the cerebral cortical systems network in the cat is dominated by the network's modular organization, while the activation behavior of the cellular neuronal network of Caenorhabditis elegans is strongly influenced by hub nodes. These findings indicate the interaction of multiple topological features and dynamic states in the function of complex biological networks.     

Spatio-temporal protein dynamics in single living cells

The development and application of single cell optical imaging has identified dynamic and oscillatory signalling processes in individual cells. This requires single cell analyses since the processes may otherwise be masked by the population average. These oscillations range in timing from seconds/minutes (e.g. calcium) to minutes/hours (e.g. NF-kappaB, Notch/Wnt and p53) and hours/days (e.g. circadian clock and cell cycle). Quantitative live cell measurement of the protein processes underlying these complex networks will allow characterisation of the core mechanisms that drive these signalling pathways and control cell function. Ultimately, such studies can be applied to develop predictive models of whole tissues and organisms.     

Bursting excitable cell models by a slow Ca2+ current

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.     

Gap junctions in excitable cells

Gap junction channels are an integral part of the conduction or propagation of an action potential from cell to cell. Gap junctions have rather unique gating and permeability properties which permit the movement of molecules from cell to cell. These molecules may not be directly linked to action potentials but can alter nonjunctional processes within cells, which in turn can affect conduction velocity. The data described in this review reveal that, for the majority of excitable cells, there are two limiting factors, with respect to gap junctions, that affect the conduction/propagation of action potentials. These are (1) the total number of channels and (2) the selective permeability of the channels. Interestingly, voltage dependence and the time course of voltage inactivation (kinetics) are not rate limiting steps under normal physiological conditions for any of the connexins studied so far. Only specialized rectifying electrical synapses utilize strong voltage dependence and rapid kinetics to permit or deny the continued propagation of an action potential.     

High frequency electrostimulation of excitable cells

Reactions of nerve fibers to high frequency electrical stimulation are examined with three nerve models. Switching on the signal produces a single AP at the threshold current. Stronger currents lead into a region of repetitive firing. The firing rate depends on the current and the fibers more distant from the electrode will have a lower rate. The AP's are not synchronized. In the "House-Urban" cochlear implant a 16 kHz carrier is used for stimulation. It is modulated by electrical signals derived from sound pressure. An analysis of the modulation shows which signals can produce APs synchronized with the source signal.     

Computer simulations of the effect of non-inactivating sodium channels on the electric behavior of excitable cells

Non-inactivating sodium channels have been discovered in various cell types. Additionally, normal voltage-gated sodium channels can be induced to lose their ability to inactivate by treatment with proteolytic enzymes, with certain chemical reagents, or with toxins. The presence of non-inactivating sodium channels in the outer membrane of a cell is expected to profoundly modify the electrical properties of the cell, because the electrical depolarization of the cell and the opening of these channels reciprocally reinforce each other without intrinsic control. The normal resting state may thus be destabilized and a new resting state at depolarized resting potentials may become possible. In this study, computer simulations were carried out to systematically explore the patterns of behavior of excitable cells which have non-inactivating sodium channels in their plasma membrane. The cells were assumed to be space clamped and the relevant Hodgkin and Huxley equations were assumed to describe the electrical behavior of the cells, except that some or all of the sodium channels could not inactivate. The sodium currents were thus represented by the sum of two terms: FI.gNa.m3.h.(V-ENa) + (1-FI).gNa.m3(V-ENa), where FI(0 less than or equal to FI less than or equal to 1) is the fraction of sodium channels which inactivate normally, and the other symbols have their usual significance. The behavior of non-inactivating sodium channels created by pronase treatment or reaction with chemical reagents was found to conform with that predicted by the second term in this expression. The simulations thus quantitatively apply to excitable cells thus treated, but may serve additionally to qualitatively illustrate patterns of electrical activity induced by non-inactivating sodium channels also in other cases. A variety of possible types of electrical behavior was obtained: Normal behavior, including capability of firing action potentials, requires values of FI which are not far from unity, the permissible range depending on the fully activated potassium ion conductance, gK. Bistability, at which the cell may exist in one of two stable states of different resting potential, occurs when the value of FI is lowered. Transitions from the polarized to the depolarized resting states, and vice versa, may be brought about by depolarizing and hyperpolarizing triggers, respectively. Such behavior is like that of memory storage devices. Monostability at depolarized potentials is favored by low FI values and can occur if gK is less than the Hodgkin and Huxley value.(ABSTRACT TRUNCATED AT 400 WORDS)     

Spontaneous secondary spiking in excitable cells

Kepler & Marder (1993, Biol. Cybern.68, 209-214) proposed a model describing the electrical activity of a crab neuron in which a train of directly induced action potentials is sometimes followed by one or more spontaneous action potentials, referred to as spontaneous secondary spikes. We reduce their five-dimensional model to three dimensions in two different ways in order to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a qualitative explanation of the phenomenon using phase-plane and bifurcation analysis. We demonstrate that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model. The exact dynamics of the excitable system are not important, but the relative time constant and amplitude of the slow inward current are crucial. Using the numerical bifurcation analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential for all values of the bifurcation parameter. At a critical value of the bifurcation parameter, a stable tonic firing mode arises via a saddle-node of periodics bifurcation. Whether or not the models can exhibit transient or continuous spontaneous spiking depends on their position in parameter space relative to this saddle-node of periodics.     

Collision of two action potentials in a single excitable cell

BackgroundIt is a common incident in nature, that two waves or pulses run into each other head-on. The outcome of such an event is of special interest, because it allows conclusions about the underlying physical nature of the pulses. The present experimental study dealt with the head-on meeting of two action potentials (AP) in a single excitable plant cell (Chara braunii internode).MethodsThe membrane potential was monitored with multiple sensors along a single excitable cell. In control experiments, an AP was excited electrically at either end of the cell cylinder. Subsequently, stimuli were applied simultaneously at both ends of the cell in order to generate two APs that met each other head-on.ResultsWhen two action potentials propagated into each other, the pulses did not penetrate but annihilated (N = 26 experiments in n = 10 cells).ConclusionsAPs in excitable plant cells did not penetrate upon meeting head-on. In the classical electrical model, this behavior is specifically attributed to relaxation of ion channel proteins. From an acoustic point of view, annihilation can be viewed as a result of nonlinear material properties (e.g. a phase change).General significanceThe present results suggest that APs in excitable animal and plant cells belong to a similar class of nonlinear phenomena. Intriguingly, other excitation waves in biology (intracellular waves, cortical spreading depression, etc.) also annihilate upon collision and are thus expected to follow the same underlying principles as the observed action potentials.     

Controlling the active properties of excitable cells

Molecular perturbations of neurons, including genetic knockout and transgenic approaches, have provided insight into the cellular processes underlying neuronal function and plasticity. A detailed understanding of how individual neurons participate in the circuitry that controls behavior, however, will require the ability to experimentally manipulate the active properties of neurons in vivo. Recent technologies have greatly advanced our experimental ability to modulate the active properties of neurons with spatial and temporal precision; technical advances have been applied to the investigation of a diverse array of neurobiological questions.     

A general approach to modeling conduction and concentration dynamics in excitable cells of concentric cylindrical geometry.

This paper discusses mathematical approaches for modeling the propagation of the action potential and ion concentration dynamics in a general class of excitable cells and cell assemblies of concentric cylindrical geometry. Examples include myelinated and unmyelinated axons, single strands of interconnected cardiac cells and outer hair cells. A key feature in some of the cells is the presence of a small working volume such as the periaxonal space between the myelin sheath and the axon in the myelinated axon and the extracisternal space between the plasma membrane and the subsurface cisterna of the outer hair cell. Proper treatment of these cell types requires a modeling approach which can readily address these anatomical properties and the non-uniform biophysical properties of the concentric membranes and the ionic composition of the volumes between the membranes. An electrodiffusion approach is first developed in which the Nernst-Planck equation is used to characterize axial ion fluxes. It is then demonstrated that this "full" model can be stepwise reduced, eventually becoming equivalent to the standard cable equation formulation. This is done in a manner that permits direct comparisons between the full and simplified models by running simulations using a single parameter set. An intermediate approach where the contributions of the axial currents to ion concentration changes and the effect of varying ion concentrations on solution conductivities are ignored is derived and is found adequate in many cases. Two application examples are given: a "cardiac strand" model, for which the intermediate formulation is shown sufficient and a model of the myelinated axon, for which the full electrodiffusion formulation is clearly necessary. The latter finding is due to spatial inhomogeneities in the anatomy and distribution of ion channels and transporters in the myelinated axon and the restricted periaxonal space between the myelin sheath and the axon.     

Exploring dynamics in living cells by tracking single particles

In the last years, significant advances in microscopy techniques and the introduction of a novel technology to label living cells with genetically encoded fluorescent proteins revolutionized the field of Cell Biology. Our understanding on cell dynamics built from snapshots on fixed specimens has evolved thanks to our actual capability to monitor in real time the evolution of processes in living cells. Among these new tools, single particle tracking techniques were developed to observe and follow individual particles. Hence, we are starting to unravel the mechanisms driving the motion of a wide variety of cellular components ranging from organelles to protein molecules by following their way through the cell. In this review, we introduce the single particle tracking technology to new users. We briefly describe the instrumentation and explain some of the algorithms commonly used to locate and track particles. Also, we present some common tools used to analyze trajectories and illustrate with some examples the applications of single particle tracking to study dynamics in living cells.     

Investigation of the effects of continuous-wave,pulse- and amplitude-modulated microwaves on single excitable cells of chara corallina

Single internodal excitable cells of Chara corallina were exposed to CW, pulse-modulated and sinusoidally modulated S-band microwave fields in a temperature-controled waveguide exposure chamber. All electrical measurements were made external to the waveguide (ie, under no impressed microwave field). The dependent variables measured before, during, and after exposure to the S-band microwave fields included: resting potential, amplitude of the action potential, rise and decay time of the action potential, conduction velocity, and excitability. Cells maintained at 22 ± 0.1 °C during exposure showed no consistent or statistically significant microwave-dependent alterations in any of the dependent variables.     

Computational model for autophagic vesicle dynamics in single cells

Macroautophagy (autophagy) is a cellular recycling program essential for homeostasis and survival during cytotoxic stress. This process, which has an emerging role in disease etiology and treatment, is executed in four stages through the coordinated action of more than 30 proteins. An effective strategy for studying complicated cellular processes, such as autophagy, involves the construction and analysis of mathematical or computational models. When developed and refined from experimental knowledge, these models can be used to interrogate signaling pathways, formulate novel hypotheses about systems, and make predictions about cell signaling changes induced by specific interventions. Here, we present the development of a computational model describing autophagic vesicle dynamics in a mammalian system. We used time-resolved, live-cell microscopy to measure the synthesis and turnover of autophagic vesicles in single cells. The stochastically simulated model was consistent with data acquired during conditions of both basal and chemically-induced autophagy. The model was tested by genetic modulation of autophagic machinery and found to accurately predict vesicle dynamics observed experimentally. Furthermore, the model generated an unforeseen prediction about vesicle size that is consistent with both published findings and our experimental observations. Taken together, this model is accurate and useful and can serve as the foundation for future efforts aimed at quantitative characterization of autophagy.     

A numerical method to model excitable cells

We have extended a fast, stable, and accurate method for the numerical solution of cable equations to include changes in geometry and membrane properties in order to model a single excitable cell realistically. In addition, by including the provision that the radius may be a function of distance along an axis, we have achieved a general and powerful method for simulating a cell with any number of branched processes, any or all of which may be nonuniform in diameter, and with no restriction on the branching pattern.