Dynamics of globally coupled inhibitory neurons with heterogeneity.
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Journal Article↗
Biomedical subjects
J Rinzel
Publications and source records attributed to J Rinzel.
Rhythmogenic effects of weak electrotonic coupling in neuronal models.
Strong gap-junctional coupling can synchronize the electrical oscillations of cells, but we show, in a theoretical model, that weak coupling can phase lock two cells 180 degrees out-of-phase. Antiphase oscillations can exist in parameter regimens where in-phase oscillations break down. Some consequences are (i) coupling two excitable cells leads to pacemaking, (ii) coupling two pacemaker cells leads to bursting, and (iii) coupling two bursters increases burst period. The latter shows that details of the fast spikes can affect macroscopic properties of the slow bursts. These effects hold in other models for bursting and may play a role in the collective behavior of cellular ensembles.
Animals↗
Model for synchronization of pancreatic beta-cells by gap junction coupling.
Pancreatic beta-cells coupled by gap junctions in sufficiently large clusters exhibit regular electrical bursting activity, which is described by the Chay-Keizer model and its variants. According to most reports, however, isolated cells exhibit disorganized spiking. We have previously (Sherman, A. J. Rinzel, and J. Keizer, 1988. Biophys. J. 54:411-425) modeled these behaviors by hypothesizing that stochastic channel fluctuations disrupt the bursts. We showed that when cells are coupled by infinite conductance gap junctions, so that the cluster is isopotential and may be viewed as a single "supercell," the fluctuations are shared over a larger membrane area and hence dampened. Bursting emerges when there are more than approximately 50 cells in the cluster. In the model the temporal organization of spikes into bursts increases the amplitude of intracellular calcium oscillations, which may be relevant for insulin secretion. We now extend the previous work by considering the case of a true "multicell" model with finite gap junctional conductance. Whereas the previous study assumed that the cells were synchronized, we can now study the process of synchronization itself. We show that, for sufficiently large clusters, the cells both synchronize and begin to burst with moderate, physiologically reasonable gap junctional conductance. An unexpected finding is that the burst period is longer, and calcium amplitude greater, than when coupling is infinitely strong, with an optimum in the range of 150-250 pS. Our model is in good agreement with recent experimental data of Perez-Armendariz, M., D. C. Spray, and M. V. L. Bennett. (1991. Biophys. J. 59:76-92) showing extensive gap junctions in beta-cell pairs with mean interfacial conductance of 213 +/- 113 pS. The optimality property of our model is noteworthy because simple slow-wave models without spikes do not show the same behavior.
Animals↗
Propagation of dendritic spikes mediated by excitable spines: a continuum theory.
1. Neuroscientists are currently hypothesizing on how voltage-dependent channels, in dendrites with spines, may be spatially distributed or how their numbers may divide between spine heads and the dendritic base. A new cable theory is formulated to investigate electrical interactions between many excitable and/or passive dendritic spines. The theory involves a continuum formulation in which the spine density, the membrane potential in spine heads, and the spine stem current vary continuously in space and time. The spines, however, interact only indirectly by voltage spread along the dendritic shaft. Active membrane in the spine heads is modeled with Hodgkin-Huxley (HH) kinetics. Synaptic currents are generated by transient conductance increases. For most simulations the membrane of spine stems and dendritic shaft is assumed passive. 2. Action-potential generation and propagation occur as localized excitatory synaptic input into spine heads causes a few excitable spines to fire, which then initiates a chain reaction of spine firings along a branch. This sustained wavelike response is possible for a certain range of spine densities and electrical parameters. Propagation is precluded for spine stem resistance (Rss) either too large or too small. Moreover, even if Rss lies in a suitable range for the local generation of an action potential (resulting from local synaptic excitatory input), this range may not be suitable to initiate a chain reaction of spine firings along the dendrite; success or failure of impulse propagation depends on an even narrower range of Rss values. 3. The success or failure of local excitation to spread as a chain reaction depends on the spatial distribution of spines. Impulse propagation is unlikely if the excitable spines are spaced too far apart. However, propagation may be recovered by redistributing the same number of equally spaced spines into clusters. 4. The spread of excitation in a distal dendritic arbor is also influenced by the branching geometry. Input to one branch can initiate a chain reaction that accelerates into the sister branch but rapidly attenuates as it enters the parent branch. In branched dendrites with many excitable and passive spines, regions of decreased conductance load (e.g., near sealed ends) can facilitate attenuating waves and enhance waves that are successfully propagating. Regions of increased conductance load (e.g., near common branch points) promote attenuation and tend to block propagation. Non-uniform loading and/or nonuniform spine densities can lead to complex propagation characteristics. 5. Some analytic results of classical cable theory are generalized for the case of a passive spiny dendritic cable.(ABSTRACT TRUNCATED AT 400 WORDS)
Action Potentials↗
A model of the T-type calcium current and the low-threshold spike in thalamic neurons.
1. A model of the transient, low-threshold voltage-dependent (T-type) Ca2+ current is constructed using recent whole-cell voltage-clamp data from enzymatically isolated rat thalamocortical relay neurons. The T-type Ca2+ current is described according to the Hodgkin-Huxley scheme, using the m3h format, with rate constants determined from the experimental data (22-24 degrees C; extracellular Ca2+ concentration [Ca2+]o = 3 mM). 2. The T-type Ca2+ current inactivates rapidly during maintained depolarization (time constant, Tau h approximately 20 ms at -20 mV), yet recovery from inactivation is slow (time constant, Tau r approximately 270 ms at -80 mV). To reconcile these observations, a two-step kinetic scheme is proposed for the inactivation gate. Each of the time constants in this scheme is voltage dependent, with a maximum at about -85 mV (45 ms for one and 275 ms for the other). 3. Numerical simulations of recovery in a two-pulse, voltage-clamp protocol compare favorably with experimental results obtained by Coulter et al. as well as those obtained in an independent series of experiments with guinea pig thalamic neurons ([Ca2+]o = 10 mM). 4. For current-clamp simulations, a leakage current gL (V-VL) is included; with VL = -65 mV, the calculated resting membrane potential is -63 mV. 5. It is shown that the T-type Ca2+ current together with the leakage current suffices to describe the low-threshold spike (LTS), a slow, triangular-shaped depolarizing event that can be evoked only from relatively hyperpolarized membrane potentials and that underlies the burst firing of Na(+)-dependent action potentials in thalamic neurons. Outward currents are not required to reproduce the basic shape of the LTS. 6. The LTS can be activated with either a depolarizing current step from a sufficiently hyperpolarized level or on termination of a hyperpolarizing current step. In either case, the amplitude of the LTS is a monotonically increasing, sigmoid-shape function of the hyperpolarizing current step intensity. 7. Because of the slower kinetic step of the channel's inactivation gate, our model predicts that recovery of the LTS to greater than one-half amplitude would require a prolonged hyperpolarization of greater than 100 ms (at body temperature). This imposes an upper limit (approximately 10 Hz) on the frequency of repetitive hyperpolarization that can elicit a train of LTSs and hence on the frequency of any rhythm that requires LTS-mediated bursting of thalamic neurons.(ABSTRACT TRUNCATED AT 400 WORDS)
Action Potentials↗
Impulse patterning and relaxational propagation in excitable media.
Wavetrains of impulses in homogeneous excitable media relax during propagation toward constant-speed patterns. Here we present a study of this relaxation process. Starting with the basic reaction-diffusion or cable equations, we derive kinematics for the trajectories of widely spaced impulses in the form of ordinary differential equations for the set of times at which impulses arrive at a given point in space. Stability criteria derived from these equations allow us to determine the possible asymptotic forms of propagating trains. When the recovery after excitation is monotonic, only one stable train exists for a given propagation speed. In the case of an oscillatory recovery, however, many stable trains are possible. This essential difference between monotonic and oscillatory recoveries manifests itself in qualitatively distinct relaxational behaviors.
Biophysical Phenomena↗
Modulation of the frequency of glucose-dependent bursts of electrical activity by HCO3/CO2 in rodent pancreatic B-cells: experimental and theoretical results.
The burst pattern of electrical activity recorded from pancreatic B-cells in response to 11 mM glucose shows a large islet to islet variability. The relationship between burst frequency and glucose sensing (the threshold for electrical activity and the graded increase in electrical response to glucose, i.e. active phase %) has not been investigated within the same islet. In this work, we show that low HCO3 (5 mM) Hepes buffered solutions reversibly reduce the frequency of bursts compared to control (25 mM) HCO3 buffered solutions in the same islet. There was no change in the threshold or active phase (%). Using the mathematical model of Sherman et al. 1988, we explored mechanisms for a change in frequency independent of a change in active phase (%). Increased exchangeable calcium pool size and increased cell to cell coupling were the two theoretical treatments which could reproduce the experimental data. We conclude that burst frequency can be modulated independent of the active phase and that alteration of a calcium pool size best fits the experimental data.
Animals↗
Domain model for Ca2(+)-inactivation of Ca2+ channels at low channel density.
The "shell" model for Ca2(+)-inactivation of Ca2+ channels is based on the accumulation of Ca2+ in a macroscopic shell beneath the plasma membrane. The shell is filled by Ca2+ entering through open channels, with the elevated Ca2+ concentration inactivating both open and closed channels at a rate determined by how fast the shell is filled. In cells with low channel density, the high concentration Ca2+ "shell" degenerates into a collection of nonoverlapping "domains" localized near open channels. These domains form rapidly when channels open and disappear rapidly when channels close. We use this idea to develop a "domain" model for Ca2(+)-inactivation of Ca2+ channels. In this model the kinetics of formation of an inactivated state resulting from Ca2+ binding to open channels determines the inactivation rate, a mechanism identical with that which explains single-channel recordings on rabbit-mesenteric artery Ca2+ channels (Huang Y., J. M. Quayle, J. F. Worley, N. B. Standen, and M. T. Nelson. 1989. Biophys. J. 56:1023-1028). We show that the model correctly predicts five important features of the whole-cell Ca2(+)-inactivation for mouse pancreatic beta-cells (Plants, T. D. 1988. J. Physiol. 404:731-747) and that Ca2(+)-inactivation has only minor effects on the bursting electrical activity of these cells.
Animals↗
Mechanisms for nonuniform propagation along excitable cables.
We have discussed two classes of mechanisms that can lead to propagation with nonconstant velocity, and to disruption of temporal patterning of action potentials. Inhomogenieties along the cable due to geometrical change or to altered cell coupling can result in conduction delays, with the possibility of block or reflection. Such conduction irregularities have been considered relevant to cardiac reentry phenomena. Our simulations with a discrete number of excitable cells, coupled by gap junctions, showed that the underlying mathematical structure of a saddle point threshold in an ionic model also contributes in an important way to creating a long delay. Such threshold behavior, although not yet demonstrated for some of the most well-studied models of excitability, should not be viewed as extraordinary; we have seen it in models other than those of references 6 and 7, and have produced it in the Hodgkin-Huxley model with reasonable parameter variations (but have not yet checked for reflections with these modifications). We are unaware of any computations with theoretical models of cardiac membrane that yield robust reflection behavior. Perhaps modifications of these models will be necessary in order to obtain adequate delays for reflection. The mechanism we have described here may serve as a guideline for additional features to seek in such parametric tuning. A different class of factors that contribute to interferring with action potential timing include the effects of previous activity on propagation speed. These influences may be described in terms of the dispersion relation, c(T), the dependence of speed on time between action potentials. The form of this function, for large T, reflects the exponential behavior of the action potential's return to rest. Supernormal conduction reveals itself in the dispersion relation when there is an overshoot of excitability in the return to rest, either a single overshoot or an alternating sequence of over- and undershoots (as seen in some nerve membrane models). A simple kinematic recipe was described for quantitatively predicting timing changes during propagation without having to solve the full cable equations. To apply these concepts to cardiac models it will be necessary to compute the dispersion relation for these models. By studying the dependence of c(T) and the waveform trajectory (including conductances as well as membrane potential) on various parameters one may gain insight into the ionic basis for experimentally observed supernormal conduction.
Action Potentials↗
Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing.
Pancreatic beta-cells in an intact Islet of Langerhans exhibit bursting electrical behavior. The Chay-Keizer model describes this using a calcium-activated potassium (K-Ca) channel, but cannot account for the irregular spiking of isolated beta-cells. Atwater I., L. Rosario, and E. Rojas, Cell Calcium. 4:451-461, proposed that the K-Ca channels, which are rarely open, are shared by several cells. This suggests that the chaotic behavior of isolated cells is stochastic. We have revised the Chay-Keizer model to incorporate voltage clamp data of Rorsman and Trube and extended it to include stochastic K-Ca channels. This model can describe the behavior of single cells, as well as that of clusters of cells tightly coupled by gap junctions. As the size of the clusters is increased, the electrical activity shows a transition from chaotic spiking to regular bursting. Although the model of coupling is over-simplified, the simulations lend support to the hypothesis that bursting is the result of channel sharing.
Action Potentials↗
Threshold for repetitive activity for a slow stimulus ramp: a memory effect and its dependence on fluctuations.
We have obtained new insights into the behavior of a class of excitable systems when a stimulus, or parameter, is slowly tuned through a threshold value. Such systems do not accommodate no matter how slowly a stimulus ramp is applied, and the stimulus value at onset of repetitive activity shows a curious, nonmonotonic dependence on ramp speed. (Jakobsson, E. and R. Guttman. Biophys. J. 1980. 31:293-298.) demonstrated this for squid axon and for the Hodgkin-Huxley (HH) model. Furthermore, they showed theoretically that for moderately slow ramps the threshold increases as the ramp speed decreases, but for much slower ramp speeds threshold decreases as the ramp speed decreases. This latter feature was found surprising and it was suggested that the HH model, and squid axon in low calcium, exhibits reverse accommodation. We have found that reverse accommodation reflects the influence of persistent random fluctuations, and is a feature of all such excitable systems. We have derived an analytic condition which yields an approximation for threshold in the case of a slow ramp when the effect of fluctuations are negligible. This condition predicts, and numerical calculations confirm, that the onset of oscillations occurs beyond the critical stimulus value which is predicted by treating the stimulus intensity as a static parameter, i.e., the dynamic aspect of a ramp leads to a delay in the onset. The condition further demonstrates a memory effect, i.e., firing threshold is dependent on the initial state of the system. For very slow ramps then, fluctuations diminish both the delay and memory effects. We characterize the class of excitable systems for which these behaviors are expected, and we illustrate the phenomena for the HH model and for a model of cAMP-receptor dynamics in Dictyostelium discoideum.
Animals↗
Dissection of a model for neuronal parabolic bursting.
We have obtained new insight into the mechanisms for bursting in a class of theoretical models. We study Plant's model for Aplysia R-15 to illustrate our view of these so-called "parabolic" bursters, which are characterized by low spike frequency at the beginning and end of a burst. By identifying and analyzing the fast and slow processes we show how they interact mutually to generate spike activity and the slow wave which underlies the burst pattern. Our treatment is essentially the first step of a singular perturbation approach presented from a geometrical viewpoint and carried out numerically with AUTO. We determine the solution sets (steady state and oscillatory) of the fast subsystem with the slow variables treated as parameters. These solutions form the slow manifold over which the slow dynamics then define a burst trajectory. During the silent phase of a burst, the solution trajectory lies approximately on the steady state branch of the slow manifold and during the active phase of spiking, the trajectory sweeps through the oscillation branch. The parabolic nature of bursting arises from the (degenerate) homoclinic transition between the oscillatory branch and the steady state branch. We show that, for some parameter values, the trajectory remains strictly on the steady state branch (to produce a resting steady state or a pure slow wave without spike activity) or strictly in the oscillatory branch (continuous spike activity without silent phases). Plant's model has two slow variables: a calcium conductance and the intracellular free calcium concentration, which activates a potassium conductance. We also show how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium. These and other biophysical interpretations are discussed.
Animals↗
Prediction of the glucose-induced changes in membrane ionic permeability and cytosolic Ca2+ by mathematical modeling.
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Animals↗
A biophysical model of cochlear processing: intensity dependence of pure tone responses.
A mathematical model of cochlear processing is developed to account for the nonlinear dependence of frequency selectivity on intensity in inner hair cell and auditory nerve fiber responses. The model describes the transformation from acoustic stimulus to intracellular hair cell potentials in the cochlea. It incorporates a linear formulation of basilar membrane mechanics and subtectorial fluid-cilia displacement coupling, and a simplified description of the inner hair cell nonlinear transduction process. The analysis at this stage is restricted to low-frequency single tones. The computed responses to single tone inputs exhibit the experimentally observed nonlinear effects of increasing intensity such as the increase in the bandwidth of frequency selectivity and the downward shift of the best frequency. In the model, the first effect is primarily due to the saturating effect of the hair cell nonlinearity. The second results from the combined effects of both the nonlinearity and of the inner hair cell low-pass transfer function. In contrast to these shifts along the frequency axis, the model does not exhibit intensity dependent shifts of the spatial location along the cochlea of the peak response for a given single tone. The observed shifts therefore do not contradict an intensity invariant tonotopic code.
Acoustic Stimulation↗
Synaptic amplification by active membrane in dendritic spines.
The suspected functional role of dendritic spines as loci of neuronal plasticity (possibly memory and learning) is greatly enriched when active membrane properties are assumed at the spine head. Computations with reasonable electrical and structural parameter values (corresponding to an optimal range for spine stem resistance) show that an active spine head membrane can provide very significant synaptic amplification and also strongly non-linear properties that could modulate the integration of input from many afferent sources.
Cell Membrane↗
Bursting, beating, and chaos in an excitable membrane model.
We have studied periodic as well as aperiodic behavior in the self-sustained oscillations exhibited by the Hodgkin-Huxley type model of Chay, T. R., and J. Keizer (Biophys. J., 1983, 42:181-190) for the pancreatic beta-cell. Numerical solutions reveal a variety of patterns as the glucose-dependent parameter kCa is varied. These include regimes of periodic beating (continuous spiking) and bursting modes and, in the transition between these modes, aperiodic responses. Such aperiodic behavior for a nonrandom system has been called deterministic chaos and is characterized by distinguishing features found in previous studies of chaos in nonbiophysical systems and here identified for an (endogenously active) excitable membrane model. To parallel the successful analysis of chaos in other physical/chemical contexts we introduce a simplified, but quantitative, one-variable, discrete-time representation of the dynamics. It describes the evolution of intracellular calcium (which activates a potassium conductance) from one spike upstroke to the next and exhibits the various modes of behavior.
Animals↗
Signal enhancement in distal cortical dendrites by means of interactions between active dendritic spines.
Pyramidal neurons in the cerebral cortex characteristically give rise to an apical dendrite, whose distal dendritic branches in layer I are covered with spines. These spines are known to be sites of synaptic connections, but the physiological properties of the spines and the functional significance of their responses are still largely unknown. The main function attributed thus far to these synaptic responses, situated at a great distance from the neuronal cell body, is slow background modulation of impulse output in the axon. In pursuing computer simulation analysis of electrical properties of dendrites, we have obtained results suggesting interactions between distal dendritic spines. If the heads of dendritic spines have excitable membrane properties, the spread of current from one or several spines could bring adjacent spines to their thresholds for impulse generation. This could give rise to a sequence of spine head action potentials, representing a saltatory propagation, from one or more excitable spine heads to nearby excitable spine heads, in the distal dendritic branches. Both the amplification due to several spine action potentials and the possibility of propagation into more proximal branches would increase the efficacy of distal synaptic inputs. Because of nonlinear dependence upon several modifiable parameters (such as spine stem resistance and membrane excitability) and upon the spatio-temporal pattern of synaptic input, such contingent synaptic enhancement would be particularly relevant to cortical functions underlying information processing and to plasticity underlying learning and memory.
Dendrites↗
Excitation dynamics: insights from simplified membrane models.
Excitable nerve membranes and models for their electrical activity exhibit a broad repertoire of dynamic behavior. To reveal these behaviors the theoretician seeks a model that is simple enough to analyze yet one that retains adequate biophysical realism. Here we strike such a balance by describing a two-variable simplification of the Hodgkin-Huxley (HH) model, which exhibits many membrane phenomena and reproduces, with good agreement, many HH responses. Comparisons and illustrations are presented for the single spike response, repetitive firing (and its cessation by a brief current pulse), and bistable behavior for increased extracellular K+ concentrations.
Animals↗