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.”     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

Spine-neck geometry determines NMDA receptor-dependent Ca2+ signaling in dendrites

Increases in cytosolic Ca2+ concentration ([Ca2+]i) mediated by NMDA-sensitive glutamate receptors (NMDARs) are important for synaptic plasticity. We studied a wide variety of dendritic spines on rat CA1 pyramidal neurons in acute hippocampal slices. Two-photon uncaging and Ca2+ imaging revealed that NMDAR-mediated currents increased with spine-head volume and that even the smallest spines contained a significant number of NMDARs. The fate of Ca2+ that entered spine heads through NMDARs was governed by the shape (length and radius) of the spine neck. Larger spines had necks that permitted greater efflux of Ca2+ into the dendritic shaft, whereas smaller spines manifested a larger increase in [Ca2+]i within the spine compartment as a result of a smaller Ca2+ flux through the neck. Spine-neck geometry is thus an important determinant of spine Ca2+ signaling, allowing small spines to be the preferential sites for isolated induction of long-term potentiation.     

Theoretical study on electrical properties of dendritic spines

In most parts of mammalian central nervous system the majority of synapses are located on dendritic spines. Several suggestions have been made about the functional significance of the dendritic spines. We investigate electrical properties of dendritic spines in the neurons with arbitrary dendritic geometry. Following Butz & Cowan (1974), all dendritic branches, including spines, are treated as cylinders of uniform passive membrane. We show that the postsynaptic potential due to the synapse on the spine is represented as a convolution integral of the following two functions. The first is the postsynaptic potential caused by the same synapse on the branching point where the spine stalk is attached to the main dendritic trunk. The second function is determined mainly by the morphological and electrical properties of the spine and it represents the attenuation effect of the spine. On the assumption that the diameter of the spine stalk is sufficiently small compared to that of the parent dendrite to which the spine stem is attached, we obtain an approximation of the second function and conclude that morphological change of the spine does not produce an effective change of the postsynaptic potential, hence does not provide the neural basis for learning or memory simply by changing cable properties of dendrites. Moreover, we show that synapses on the dendritic spine are not effectively isolated from other synapses on the same assumption.     

The biophysical properties of spines as a basis for their electrical function: a comment on Kawato & Tsukahara (1983)

In a theoretical study of the passive cable properties of dendritic spines Kawato & Tsukahara (1983) claim to have proved that "the dendritic spine has no significant electrical function" (from their discussion). However, Kawato & Tsukahara restrict their analysis to current inputs to spines. Since the dimensions of spines are very small, their input resistance is expected to be very large and the synaptic input to spines has to be modeled as conductance change. Under this assumption, spines show interesting (non-linear) electrical properties: i) the somatic potential induced by an excitatory synapse on a spine may depend strongly on the shape of the spine and ii) the effect of inhibition might be confined to the spine.     

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. Received: 10 January 1997/Revised version: 25 March 1997     

Single-Voxel Recording of Voltage Transients in Dendritic Spines

We report sensitive recording of membrane potential in single dendritic spines in cortical neurons within a brain slice using two-photon excitation and a new, fluorinated, intracellularly loaded organic dye, di-2-AN(F)EPPTEA. With a two-photon excitation wavelength of 1060 nm, we achieve voltage sensitivity of >16% change in fluorescence per 100 mV. By targeting single spines in single-voxel recordings, we attain excellent single/noise quality, with back-propagating action potentials (bAPs) visible in single sweeps while recording at 10 kHz. This recording rate allows us to reliably assess fast bAP dynamics on single sweeps including bAP rise times of 0.5 ms. The amplitude and propagation delays of the bAPs are similar among different spines located within the same dendritic region, and this is true despite large differences in spine size. The interregion differences in bAP waveforms in spines vary in relation to their distance from the soma and the caliber of their parent dendrites.     

A Compartmental Model for Activity-Dependent Dendritic Spine Branching

Dendritic spines are small, mushroom-like protrusions from the arbor of a neuron in the central nervous system. Interdependent changes in the morphology, biochemistry, and activity of spines have been associated with learning and memory. Moreover, post-mortem cortices from patients with Alzheimer’s or Parkinson’s disease exhibit biochemical and physical alterations within their dendritic arbors and a reduction in the number of dendritic spines. For over a decade, experimentalists have observed perforations in postsynaptic densities on dendritic spines after induction of long-term potentiation, a sustained enhancement of response to a brief electrical or chemical stimulus, associated with learning and memory. In more recent work, some suggest that activity-dependent intraspine calcium may regulate the surface area of the spine head, and reorganization of postsynaptic densities on the surface. In this paper, we develop a model of a dendritic spine with the ability to partition its transmission and receptor zones, as well as the entire spine head. Simulations are initially performed with fixed parameters for morphology to study electrical properties and identify parameters that increase efficacy of the synaptic connection. Equations are then introduced to incorporate calcium as a second messenger in regulating continuous changes in morphology. In the model, activity affects compartmental calcium, which regulates spine head morphology. Conversely, spine head morphology affects the level of local activity, whether the spines are modeled with passive membrane properties, or excitable membrane using Hodgkin–Huxley kinetics. Results indicate that merely separating the postsynaptic receptors on the surface of the spine may add to the diversity of circuitry, but does not change the efficacy of the synapse. However, when the surface area of the spine is a dynamic variable, efficacy of the synapse may vary continuously over time.     

An equivalent cable model for neuronal trees with active membrane

A non-uniform equivalent cable model of membrane voltage changes in branching neuronal trees with active ion channels has been developed. A general branching condition is formulated, extending Rall's 3/2 power rule for passive dendritic trees so that non-uniform cable segments can be treated. The theoretical results support the use of the dendritic profile model of Clements and Redman. The theory is then applied to dendrites of different morphological type yielding qualitative different response behaviour. Received: 25 September 1997 / Accepted: 13 November 1997     

Fast Kalman filtering on quasilinear dendritic trees

Optimal filtering of noisy voltage signals on dendritic trees is a key problem in computational cellular neuroscience. However, the state variable in this problem—the vector of voltages at every compartment—is very high-dimensional: realistic multicompartmental models often have on the order of N = 10⁴ compartments. Standard implementations of the Kalman filter require O(N ³) time and O(N ²) space, and are therefore impractical. Here we take advantage of three special features of the dendritic filtering problem to construct an efficient filter: (1) dendritic dynamics are governed by a cable equation on a tree, which may be solved using sparse matrix methods in O(N) time; and current methods for observing dendritic voltage (2) provide low SNR observations and (3) only image a relatively small number of compartments at a time. The idea is to approximate the Kalman equations in terms of a low-rank perturbation of the steady-state (zero-SNR) solution, which may be obtained in O(N) time using methods that exploit the sparse tree structure of dendritic dynamics. The resulting methods give a very good approximation to the exact Kalman solution, but only require O(N) time and space. We illustrate the method with applications to real and simulated dendritic branching structures, and describe how to extend the techniques to incorporate spatially subsampled, temporally filtered, and nonlinearly transformed observations.     

Critical scale of propagation influences dynamics of waves in a model of excitable medium

Background  

Duration and speed of propagation of the pulse are essential factors for stability of excitation waves. We explore the propagation of excitation waves resulting from periodic stimulation of an excitable cable to determine the minimal stable pulse duration in a rate-dependent modification of a Chernyak-Starobin-Cohen reaction-diffusion model.     

Structural and functional plasticity of dendritic spines – root or result of behavior?

Dendritic spines are multifunctional integrative units of the nervous system and are highly diverse and dynamic in nature. Both internal and external stimuli influence dendritic spine density and morphology on the order of minutes. It is clear that the structural plasticity of dendritic spines is related to changes in synaptic efficacy, learning and memory and other cognitive processes. However, it is currently unclear whether structural changes in dendritic spines are primary instigators of changes in specific behaviors, a consequence of behavioral changes, or both. In this review, we first examine the basic structure and function of dendritic spines in the brain, as well as laboratory methods to characterize and quantify morphological changes in dendritic spines. We then discuss the existing literature on the temporal and functional relationship between changes in dendritic spines in specific brain regions and changes in specific behaviors mediated by those regions. Although technological advancements have allowed us to better understand the functional relevance of structural changes in dendritic spines that are influenced by environmental stimuli, the role of spine dynamics as an underlying driver or consequence of behavior still remains elusive. We conclude that while it is likely that structural changes in dendritic spines are both instigators and results of behavioral changes, improved research tools and methods are needed to experimentally and directly manipulate spine dynamics in order to more empirically delineate the relationship between spine structure and behavior.     

Propagation of CaMKII translocation waves in heterogeneous spiny dendrites

CaMKII (Ca²⁺-calmodulin-dependent protein kinase II) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed experimentally that stimulating a local region of dendrite not only induces the local translocation of CaMKII from the dendritic shaft to synaptic targets within spines, but also initiates a wave of CaMKII translocation that spreads distally through the dendrite with an average speed of order 1μm/s. We have previously developed a simple reaction–diffusion model of CaMKII translocation waves that can account for the observed wavespeed and predicts wave propagation failure if the density of spines is too high. A major simplification of our previous model was to treat the distribution of spines as spatially uniform. However, there are at least two sources of heterogeneity in the spine distribution that occur on two different spatial scales. First, spines are discrete entities that are joined to a dendritic branch via a thin spine neck of submicron radius, resulting in spatial variations in spine density at the micron level. The second source of heterogeneity occurs on a much longer length scale and reflects the experimental observation that there is a slow proximal to distal variation in the density of spines. In this paper, we analyze how both sources of heterogeneity modulate the speed of CaMKII translocation waves along a spiny dendrite. We adapt methods from the study of the spread of biological invasions in heterogeneous environments, including homogenization theory of pulsating fronts and Hamilton–Jacobi dynamics of sharp interfaces.     

Bifurcations of lurching waves in a thalamic neuronal network

We consider a two-layer, one-dimensional lattice of neurons; one layer consists of excitatory thalamocortical neurons, while the other is comprised of inhibitory reticular thalamic neurons. Such networks are known to support “lurching” waves, for which propagation does not appear smooth, but rather progresses in a saltatory fashion; these waves can be characterized by different spatial widths (different numbers of neurons active at the same time). We show that these lurching waves are fixed points of appropriately defined Poincaré maps, and follow these fixed points as parameters are varied. In this way, we are able to explain observed transitions in behavior, and, in particular, to show how branches with different spatial widths are linked with each other. Our computer-assisted analysis is quite general and could be applied to other spatially extended systems which exhibit this non-trivial form of wave propagation.     

Effects of noise on models of spiny dendrites

We study the effects of noise in two models of spiny dendrites. Through the introduction of different types of noise to both the Spike-diffuse-spike (SDS) and Baer–Rinzel (BR) models we investigate the change in behaviour of the travelling wave solution present in both deterministic systems, as noise intensity increases. We show that the speed of wave propagation in both the SDS and BR models respectively differs as the noise intensity in the spine heads increases. In contrast the cable is very robust to noise and as such the speed shows very little variation from the deterministic system. We introduce a space-dependent spine density, ρ(x), to the original Baer–Rinzel model and show how this modified model can mimic behaviour (under influence of noise) of both original systems, through variation of one parameter. We also show that the correlation time and length scales of the noise can enhance propagation of travelling wave solutions where the white noise dominates the underlying signal and produces noise induced phenomena.     

Investigating Sub-Spine Actin Dynamics in Rat Hippocampal Neurons with Super-Resolution Optical Imaging

Morphological changes in dendritic spines represent an important mechanism for synaptic plasticity which is postulated to underlie the vital cognitive phenomena of learning and memory. These morphological changes are driven by the dynamic actin cytoskeleton that is present in dendritic spines. The study of actin dynamics in these spines traditionally has been hindered by the small size of the spine. In this study, we utilize a photo-activation localization microscopy (PALM)–based single-molecule tracking technique to analyze F-actin movements with ∼30-nm resolution in cultured hippocampal neurons. We were able to observe the kinematic (physical motion of actin filaments, i.e., retrograde flow) and kinetic (F-actin turn-over) dynamics of F-actin at the single-filament level in dendritic spines. We found that F-actin in dendritic spines exhibits highly heterogeneous kinematic dynamics at the individual filament level, with simultaneous actin flows in both retrograde and anterograde directions. At the ensemble level, movements of filaments integrate into a net retrograde flow of ∼138 nm/min. These results suggest a weakly polarized F-actin network that consists of mostly short filaments in dendritic spines.     

Cortical area and species differences in dendritic spine morphology

Dendritic spines receive most excitatory inputs in the neocortex and are morphologically very diverse. Recent evidence has demonstrated linear relationships between the size and length of dendritic spines and important features of its synaptic junction and time constants for calcium compartmentalisation. Therefore, the morphologies of dendritic spines can be directly interpreted functionally. We sought to explore whether there were potential differences in spine morphologies between areas and species that could reflect potential functional differences. For this purpose, we reconstructed and measured thousands of dendritic spines from basal dendrites of layer III pyramidal neurons from mouse temporal and occipital cortex and from human temporal cortex. We find systematic differences in spine densities, spine head size and spine neck length among areas and species. Human spines are systematically larger and longer and exist at higher densities than those in mouse cortex. Also, mouse temporal spines are larger than mouse occipital spines. We do not encounter any correlations between the size of the spine head and its neck length. Our data suggests that the average synaptic input is modulated according to cortical area and differs among species. We discuss the implications of these findings for common algorithms of cortical processing.     

Subtle re-location of dendritic spine branches containing membrane with fast spiking mechanisms can alter synaptic efficacy

The potential physiological impact of morphological changes in the active dendritic spines, which are believed to be associated with altered synaptic efficacy, was investigated in a computer simulation study using the NEURON package [1]. A compartmental model of a simplified neuron was built, which included 30 complex spines (neck, head, and active zone) and accommodating AMPA-type synaptic inputs with alpha-function conductances. Hodgkin-Huxley type excitable membranes were inserted into the spine heads. It was shown that arranging spines in dense clusters, as opposed to a uniformly random spine distribution, has a negligible effect on the synaptic signal transfer (other model conditions, including synaptic input and spine density, remained unchanged). However, if a proportion (e.g., 3–20%) of the spines partly fuse with their neighbors forming branched spines, this could increase dramatically the cell response to the unchanged synaptic input. Results of this pilot study provide the basis for a more detailed investigation of the relationship between the spine arrangement and synaptic function, considering dual-component synaptic currents and mechanisms controlling ion fluxes in the dendritic compartments.     

Fast extraction of neuron morphologies from large-scale SBFSEM image stacks

Neuron morphology is frequently used to classify cell-types in the mammalian cortex. Apart from the shape of the soma and the axonal projections, morphological classification is largely defined by the dendrites of a neuron and their subcellular compartments, referred to as dendritic spines. The dimensions of a neuron’s dendritic compartment, including its spines, is also a major determinant of the passive and active electrical excitability of dendrites. Furthermore, the dimensions of dendritic branches and spines change during postnatal development and, possibly, following some types of neuronal activity patterns, changes depending on the activity of a neuron. Due to their small size, accurate quantitation of spine number and structure is difficult to achieve (Larkman, J Comp Neurol 306:332, 1991). Here we follow an analysis approach using high-resolution EM techniques. Serial block-face scanning electron microscopy (SBFSEM) enables automated imaging of large specimen volumes at high resolution. The large data sets generated by this technique make manual reconstruction of neuronal structure laborious. Here we present NeuroStruct, a reconstruction environment developed for fast and automated analysis of large SBFSEM data sets containing individual stained neurons using optimized algorithms for CPU and GPU hardware. NeuroStruct is based on 3D operators and integrates image information from image stacks of individual neurons filled with biocytin and stained with osmium tetroxide. The focus of the presented work is the reconstruction of dendritic branches with detailed representation of spines. NeuroStruct delivers both a 3D surface model of the reconstructed structures and a 1D geometrical model corresponding to the skeleton of the reconstructed structures. Both representations are a prerequisite for analysis of morphological characteristics and simulation signalling within a neuron that capture the influence of spines.