Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects.

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction-diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction-diffusion system is Baer and Rinzel's continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon "complete accommodation."

A model of activity-dependent changes in dendritic spine density and spine structure.

Recent evidence indicates that the morphology and density of dendritic spines are regulated during synaptic plasticity. See, for instance, a review by Hayashi and Majewska [9]. In this work, we extend previous modeling studies [27] by combining a model for activity-dependent spine density with one for calcium-mediated spine stem restructuring. The model is based on the standard dimensionless cable equation, which represents the change in the membrane potential in a passive dendrite. Additional equations characterize the change in spine density along the dendrite, the current balance equation for an individual spine head, the change in calcium concentration in the spine head, and the dynamics of spine stem resistance. We use computational studies to investigate the changes in spine density and structure for differing synaptic inputs and demonstrate the effects of these changes on the input-output properties of the dendritic branch. Moderate amounts of high-frequency synaptic activation to dendritic spines result in an increase in spine stem resistance that is correlated with spine stem elongation. In addition, the spine density increases both inside and outside the input region. The model is formulated so that this long-term potentiation-inducing stimulus eventually leads to structural stability. In contrast, a prolonged low-frequency stimulation paradigm that would typically induce long-term depression results in a decrease in stem resistance (correlated with stem shortening) and an eventual decrease in spine density.

Impact of time-dependent changes in spine density and spine shape on the input-output properties of a dendritic branch: a computational study.

Populations of dendritic spines can change in number and shape quite rapidly as a result of synaptic activity. Here, we explore the consequences of such changes on the input-output properties of a dendritic branch. We consider two models: one for activity-dependent spine densities and the other for calcium-mediated spine-stem restructuring. In the activity-dependent density model we find that for repetitive synaptic input to passive spines, changes in spine density remain local to the input site. For excitable spines, the spine density increases both inside and outside the input region. When the spine stem resistances are relatively high, the transition to higher dendritic output is abrupt; when low, the rate of increase is gradual and resembles long-term potentiation. In the second model, spine density is held constant, but the stem dimensions are allowed to change as a result of stimulation-induced calcium influxes. The model is formulated so that a moderate amount of synaptic activation results in spine stem elongation, whereas high levels of activation result in stem shortening. Under these conditions, passive spines receiving modest stimulation progressively increase their spine stem resistance and head potentials, but little change occurs in the dendritic output. For excitable spines, modest stimulation frequencies cause a lengthening of both stimulated and neighboring spines and the stimulus eventually propagates. High-frequency stimulation that causes spines to shorten in the stimulated region decreases the amplitude of the dendritic output slightly or drastically, depending on initial spine densities and stem resistances.

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)

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.

An analysis of a dendritic neuron model with an active membrane site.

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.

Analysis of an excitable dendritic spine with an activity-dependent stem conductance.

Dendritic spines are the major target for excitatory synaptic inputs in the vertebrate brain. They are tiny evaginations of the dendritic surface consisting of a bulbous head and a tenuous stem. Spines are considered to be an important locus for plastic changes underlying memory and learning processes. The findings that synaptic morphology may be activity-dependent and that spine head membrane may be endowed with voltage-dependent (excitable) channels is the motivation for this study. We first explore the dynamics, when an excitable, yet morphologically fixed spine receives a constant current input. Two parameter Andronov-Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and steady-states. We show how these boundaries can change as a function of both the spine stem conductance and the conductance load of the attached dendrite. Building on this reference case an idealized model for an activity-dependent spine is formulated and analyzed. Specifically we examine the possibility that the spine stem resistance, the tunable "synaptic weight" parameter identified by Rall and Rinzel, is activity-dependent. In the model the spine stem conductance depends (slowly) on the local electrical interactions between the spine head and the dendritic cable; parameter regimes are found for bursting, steady states, continuous spiking, and more complex oscillatory behavior. We find that conductance load of the dendrite strongly influences the burst pattern as well as other dynamics. When the spine head membrane potential exhibits relaxation oscillations a simple model is formulated that captures the dynamical features of the full model.

Asymptotic analysis of noise sensitivity in a neuronal burster.

A combination of asymptotic approaches provides a new analysis of the effect of small noise on the bursting cycle of a neuronal burster of elliptic type (type III). The analysis is applied to a stochastic model of an excitable spine, with an activity-dependent stem conductance, that exhibits conditional burst dynamics. First, we give an asymptotic approximation to the probability density for the state of the system. This density is used to compute several quantities which describe the influence of the noise on the transition from the silent to the active phase. Second, we also use a multiscale method to provide a reduced system for analysing the effect of noise on the transition out of the active phase. The combination of these two approaches results in a new framework for a quantitative description of how noise shortens the burst cycle, which measures the significant influence of small noise. For the stochastic spine model, this study suggests that small amplitude noise can significantly influence the activity-dependent morphological plasticity of dendritic spines. The techniques used in this paper combine probabilistic and asymptotic methods, and have been generalized for other noisy nonlinear systems.

Analysis of an autonomous phase model for neuronal parabolic bursting.

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol 25, 653-675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.

Background-induced flicker enhancement in cat retinal horizontal cells. II. Spatial properties.

1. Intracellular recordings have been made from cat retinal horizontal cells stimulated with flickering test spots. Dim backgrounds increase flicker amplitudes in response to small but not large test stimuli. 2. This background-induced flicker enhancement has been measured for different slit- and square-test stimulus widths and the results compared with two spatial models for the enhancement effect. 3. In the "dark test-region" model it is argued that rods within the test region are unresponsive to background stimuli because of prior saturation by the test stimulus. Background-evoked rod signals decay passively from regions outside the test stimulus through a syncytial network into the recording site, where they act on the cone-to-horizontal-cell synapse, increasing its gain. 4. In the "changing length-constant" model rod signals reduce the length constant of a syncytial network by uncoupling the cells within it. This causes an increased response to small but not large test stimuli. 5. Both models are analytically evaluated with the use of a conductive-sheet approximation to the syncytial network. Expressions are derived for network polarization [(V(0, 0)] as a function of stimulus size. The specific stimulus shapes considered are disks, rectangles, slits, and squares in both bright and dark varieties. From these expressions predictions of response enhancement as a function of stimulus size are made for both models. 6. The dark test-region model provides for an exponential decay of flicker enhancement as a function of slit width but a steeper-than-exponential decay with the width of squares, in close agreement with experimental data. 7. The changing length-constant model makes qualitatively similar predictions. Flicker enhancement declines nearly exponentially with slit width. For square-shaped test stimuli the predicted decline of flicker enhancement with size is somewhat shallower than either the dark test-region-model curve or the experimentally determined curve. 8. As recorded in the same set of cells and under the same set of stimulus conditions (with the use of both slit- and square-test stimuli), the mean length constant of the peak-to-peak flicker component in the horizontal-cell response is 168 +/- 18 (SE) microns with the background and 232 +/- 45 microns in the dark. The mean length constant for the background-induced flicker enhancement, as fit by dark test-region-model curves, is 186 +/- 22 microns (n = 9).(ABSTRACT TRUNCATED AT 250 WORDS)