Unequal diameters and their effects on time-varying voltages in branched neurons.

A theoretical method, developed in a previous paper, enables one to calculate analytical expressions for time-varying voltages at specific locations in branching dendritic systems in response to synaptic current inputs at other sites. Exact results were obtained for a number of dendritic trees that possessed certain symmetries: all branch lengths had to be integral multiples of one another, and all branch diameters had to be equal. Because the second of these conditions is unrealistic, the method has been generalized to treat dendritic trees whose branches differ in diameter. The method entails adding onto the symmetric results a sum of correction terms. It is found that the correction terms, as well as the symmetric results, can be expressed as combinations of two families of functions. These functions, generalizations of those found in our earlier paper, provide a precise formalism for analyzing how voltage transients depend on the geometrical structure of the dendritic tree. Examples are given that show how the correction terms affect the value of the voltage, and how variations in branch diameters alter the behavior of the propagated postsynaptic potential. The implications of these results for our understanding of neuronal functioning are discussed.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.     

Fast spatiotemporal smoothing of calcium measurements in dendritic trees

We discuss methods for fast spatiotemporal smoothing of calcium signals in dendritic trees, given single-trial, spatially localized imaging data obtained via multi-photon microscopy. By analyzing the dynamics of calcium binding to probe molecules and the effects of the imaging procedure, we show that calcium concentration can be estimated up to an affine transformation, i.e., an additive and multiplicative constant. To obtain a full spatiotemporal estimate, we model calcium dynamics within the cell using a functional approach. The evolution of calcium concentration is represented through a smaller set of hidden variables that incorporate fast transients due to backpropagating action potentials (bAPs), or other forms of stimulation. Because of the resulting state space structure, inference can be done in linear time using forward-backward maximum-a-posteriori methods. Non-negativity constraints on the calcium concentration can also be incorporated using a log-barrier method that does not affect the computational scaling. Moreover, by exploiting the neuronal tree structure we show that the cost of the algorithm is also linear in the size of the dendritic tree, making the approach applicable to arbitrarily large trees. We apply this algorithm to data obtained from hippocampal CA1 pyramidal cells with experimentally evoked bAPs, some of which were paired with excitatory postsynaptic potentials (EPSPs). The algorithm recovers the timing of the bAPs and provides an estimate of the induced calcium transient throughout the tree. The proposed methods could be used to further understand the interplay between bAPs and EPSPs in synaptic strength modification. More generally, this approach allows us to infer the concentration on intracellular calcium across the dendritic tree from noisy observations at a discrete set of points in space.     

Active dendrites,potassium channels and synaptic plasticity

The dendrites of CA1 pyramidal neurons in the hippocampus express numerous types of voltage-gated ion channel, but the distributions or densities of many of these channels are very non-uniform. Sodium channels in the dendrites are responsible for action potential (AP) propagation from the axon into the dendrites (back-propagation); calcium channels are responsible for local changes in dendritic calcium concentrations following back-propagating APs and synaptic potentials; and potassium channels help regulate overall dendritic excitability. Several lines of evidence are presented here to suggest that back-propagating APs, when coincident with excitatory synaptic input, can lead to the induction of either long-term depression (LTD) or long-term potentiation (LTP). The induction of LTD or LTP is correlated with the magnitude of the rise in intracellular calcium. When brief bursts of synaptic potentials are paired with postsynaptic APs in a theta-burst pairing paradigm, the induction of LTP is dependent on the invasion of the AP into the dendritic tree. The amplitude of the AP in the dendrites is dependent, in part, on the activity of a transient, A-type potassium channel that is expressed at high density in the dendrites and correlates with the induction of the LTP. Furthermore, during the expression phase of the LTP, there are local changes in dendritic excitability that may result from modulation of the functioning of this transient potassium channel. The results support the view that the active properties of dendrites play important roles in synaptic integration and synaptic plasticity of these neurons.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.     

The path integral for dendritic trees

We construct the path integral for determining the potential on any dendritic tree described by a linear cable equation. This is done by generalizing Brownian motion from a line to a tree. We also construct the path integral for dendritic structures with spatially-varying and/or time-dependent membrane conductivities due, for example, to synaptic inputs. The path integral allows novel computational techniques to be applied to cable problems. Our anlaysis leads ultimately to an exact expression for the Green's function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules providing a fast and efficient method for solving complex cable problems.Research supported by Department of Energy Contracts DE-AC02-76ER03230 and DE-AC02-76ER03069 and by National Institute of Mental Health grant MH46742     

A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees

Transient potentials in dendritic trees can be calculated by approximating the dendrite by a set of connected cylinders. The profiles for the currents and potentials in the whole system can then be obtained by imposing the proper boundary conditions and calculating these profiles along each individual cylinder. An elegant implementation of this method has been described by Holmes (1986), and is based on the Laplace transform of the cable equation. By calculating the currents and potentials only at the ends of the cylinders, the whole system of connected cylinders can be described by a set of n equations, where n denotes the number of internal and external nodes (points of connection and endpoints of the cylinders). The present study shows that the set of equations can be formulated by a simple vector equation which is essentially a generalization of Ohm's law for the whole system. The current and potential n-vectors are coupled by a n × n conductance matrix whose structure immediately reflects the connectivity pattern of the connected cylinders. The vector equation accounts for conductances, associated with driving potentials, which may be local or distributed over the membrane. It is shown that the vector equation can easily be adapted for the calculation of transients over a period in which stepwise changes in system parameters have occurred. In this adaptation it is assumed that the initial conditions for the potential profiles at the start of a new period after a stepwise change can be approximated by steady-state solutions. The vector representation of the Laplace-transformed equations is attractive because of its simplicity and because the structure of the conductance matrix directly corresponds to the connectivity pattern of the dendritic tree. Therefore it will facilitate the automatic generation of the equations once the geometry of the branching structure is known.     

Liapunov functions: Geometry and stability

Summary Several geometrical interpretations of Liapunov functions for non-linear ecological models are examined and their limitations pointed out. In particular the geometrical nonuniqueness of Liapunov functions is illustrated by displaying explicitly and comparing four different Liapunov functions for the symmetric competition model for two species. The main point is that considerable care must be taken in using the geometrical properties of an arbitrary Liapunov function as a guide to stability under perturbations.     

Transient Response in a Dendritic Neuron Model for Current Injected at One Branch

Mathematical expressions are obtained for the response function corresponding to an instantaneous pulse of current injected to a single dendritic branch in a branched dendritic neuron model. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. The response function when used in a convolution formula enables one to compute the voltage transient at any specified point in the dendritic tree for an arbitrary current injection at a given input location. A particular numerical example, for a brief current injection at a branch terminal, illustrates the attenuation and delay characteristics of the depolarization peak as it spreads throughout the neuron model. In contrast to the severe attenuation of voltage transients from branch input sites to the soma, the fraction of total input charge actually delivered to the soma and other trees is calculated to be about one-half. This fraction is independent of the input time course. Other numerical examples, which compare a branch terminal input site with a soma input site, demonstrate that, for a given transient current injection, the peak depolarization is not proportional to the input resistance at the injection site and, for a given synaptic conductance transient, the effective synaptic driving potential can be significantly reduced, resulting in less synaptic current flow and charge, for a branch input site. Also, for the synaptic case, the two inputs are compared on the basis of the excitatory post-synaptic potential (EPSP) seen at the soma and the total charge delivered to the soma.     

Syntheses of spinal cord field potentials in the terrapin

A compartmental model of a terrapin motoneuron has been set up to compute membrane potential variations associated with synaptic input at different locations or with antidromic invasion. Membrane potential distributions obtained in that way were used to compute field potentials by means of a volume conduction formalism. The model was used to simulated field potentials measured in the spinal cord in response to stimulation of a muscle nerve with the intention to discriminate between different activation hypothesis for the generation of the spinal cord potential. Extracellular potentials calculated with an excitatory input distributed over the whole dorsal dendritic tree were found to give better reconstruction when compared with excitation restricted to the distal part of the dorsal dendrites, or with somatic inhibition.     

Modeling activity-dependent synapse restructuring

The spread of electrical activity in a dendritic tree is shaped, in part, by its morphology. Conversely, experimental evidence is growing that electrical and chemical activity can slowly shape the morphology of the dendrite. In this theoretical study, the dendritic spines are dynamic elements, with biophysical properties that change in response to patterns of electrical activity. Recent experiments and diagrammatic models suggest that activity-dependent processes can regulate structural modifications in dendritic spines as well as their distribution along the dendrite. This study considers how local changes in spine structure (minutes to hours) can influence patterns of electrical activity along the dendrite; and how electrical activity due to synaptic events and excitable membrane dynamics can, over time, influence the morphology of the dendrite. The model presents a slow subsystem for structural synaptic plasticity associated with long-term potentiation. A perturbation problem evolves naturally when the spine stem shortens, since the ratio of spine stem resistance to input resistance is small. Hence, the difference between the spine head and dendritic potentials become negligible. This paper presents an asymptotic expansion of head potential in terms of dendritic potential. This leads to a reduced model for post-synaptic restructuring that captures the dynamics of the full model in a briefer computation period when the spines are well connected to the dendrite.     

Electrical advantages of dendritic spines

Many neurons receive excitatory glutamatergic input almost exclusively onto dendritic spines. In the absence of spines, the amplitudes and kinetics of excitatory postsynaptic potentials (EPSPs) at the site of synaptic input are highly variable and depend on dendritic location. We hypothesized that dendritic spines standardize the local geometry at the site of synaptic input, thereby reducing location-dependent variability of local EPSP properties. We tested this hypothesis using computational models of simplified and morphologically realistic spiny neurons that allow direct comparison of EPSPs generated on spine heads with EPSPs generated on dendritic shafts at the same dendritic locations. In all morphologies tested, spines greatly reduced location-dependent variability of local EPSP amplitude and kinetics, while having minimal impact on EPSPs measured at the soma. Spine-dependent standardization of local EPSP properties persisted across a range of physiologically relevant spine neck resistances, and in models with variable neck resistances. By reducing the variability of local EPSPs, spines standardized synaptic activation of NMDA receptors and voltage-gated calcium channels. Furthermore, spines enhanced activation of NMDA receptors and facilitated the generation of NMDA spikes and axonal action potentials in response to synaptic input. Finally, we show that dynamic regulation of spine neck geometry can preserve local EPSP properties following plasticity-driven changes in synaptic strength, but is inefficient in modifying the amplitude of EPSPs in other cellular compartments. These observations suggest that one function of dendritic spines is to standardize local EPSP properties throughout the dendritic tree, thereby allowing neurons to use similar voltage-sensitive postsynaptic mechanisms at all dendritic locations.     

海马CA1区锥体细胞树突分级峰电位的模型分析

脑神经元复杂的树突树结构和电压激活离子通道赋予其复杂的信息整合功能。通过神经元形态的分枝结构模型和电压激活离子通道动力学模型 ,包括远端树突高密度A型K 通道 ,对海马CA1区锥体细胞树突的信息整合特性进行了仿真研究。结果表明 ,远端树突峰电位在很大范围内具有分级放大作用 ,在一定的刺激条件下峰电位的发起部位可在树突轴上移动 ,并且轴突反传动作电位幅度在远端树突会发生突然减小。     

The electro-dynamics of the dendritic space in Purkinje cells of the cerebellum

The functional geometry of the reconstructed dendritic arborization of Purkinje neurons is the object of this work. The combined effects of the local geometry of the dendritic branches and of the membrane mechanisms are computed in passive configuration to obtain the electrotonic structure of the arborization. Steady-currents applied to the soma and expressed as a function of the path distance from the soma form different clusters of profiles in which dendritic branches are similar in voltages and current transfer effectiveness. The locations of the different clusters are mapped on the dendrograms and 3D representations of the arborization. It reveals the presence of different spatial dendritic sectors clearly separated in 3D space that shape the arborization in ordered electrical domains, each with similar passive charge transfer efficiencies. Further simulations are performed in active configuration with a realistic cocktail of conductances to find out whether similar spatial domains found in the passive model also characterize the active dendritic arborization. During tonic activation of excitatory synaptic inputs homogeneously distributed over the whole arborization, the Purkinje cell generates regular oscillatory potentials. The temporal patterns of the electrical oscillations induce similar spatial sectors in the arborization as those observed in the passive electrotonic structure. By taking a video of the dendritic maps of the membrane potentials during a single oscillation, we demonstrate that the functional dendritic field of a Purkinje neuron displays dynamic changes which occur in the spatial distribution of membrane potentials in the course of the oscillation. We conclude that the branching pattern of the arborization explains such continuous reconfiguration and discuss its functional implications.     

Effects of rectification on synaptic efficacy.

We have investigated the effects of postsynaptic membrane properties on the shape of synaptic potentials generated by time-varying synaptic conductances. We used numerical simulation techniques to model cells of several different geometrical forms, from an isopotential sphere to a neuron with a soma and a dendritic tree. A variety of postsynaptic membrane properties were tested: (a) a passive resistance-capacitance membrane, (b) a membrane represented by the Hodgkin and Huxley (HH) equations, and (c) a membrane that was passive except for a delayed rectification represented by a voltage- and time-dependent increase in GK. In all cases we investigated the effects of these postsynaptic membrane properties on synaptic potentials produced by synaptic conductances that were fast or slow compared with the membrane time constant. In all cases the effects of postsynaptic rectification occurred on postsynaptic potentials of amplitudes as low as 1 mV. The HH model (compared with the passive model) produced an increased peak amplitude (from the increase in GNa) but a decreased half-width and a decreased time integral (from the increase in GK). These effects of the HH GK change were duplicated by a simple analytical rectifier model.     

Synaptic integration in dendritic trees

Most neurons have elaborate dendritic trees that receive tens of thousands of synaptic inputs. Because postsynaptic responses to individual synaptic events are usually small and transient, the integration of many synaptic responses is needed to depolarize most neurons to action potential threshold. Over the past decade, advances in electrical and optical recording techniques have led to new insights into how synaptic responses propagate and interact within dendritic trees. In addition to their passive electrical and morphological properties, dendrites express active conductances that shape individual synaptic responses and influence synaptic integration locally within dendrites. Dendritic voltage-gated Na(+) and Ca(2+) channels support action potential backpropagation into the dendritic tree and local initiation of dendritic spikes, whereas K(+) conductances act to dampen dendritic excitability. While all dendrites investigated to date express active conductances, different neuronal types show specific patterns of dendritic channel expression leading to cell-specific differences in the way synaptic responses are integrated within dendritic trees. This review explores the way active and passive dendritic properties shape synaptic responses in the dendrites of central neurons, and emphasizes their role in synaptic integration.     

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.