Passive Dendritic Tree Research Articles

Effects of passive dendritic tree properties on the firing dynamics of a leaky-integrate-and-fire neuron

We study the effects of dendritic tree topology and biophysical properties on the firing dynamics of a leaky-integrate-and-fire (LIF) neuron that explicitly includes spiking dynamics. We model the dendrites as a multi-compartment tree with passive dynamics. Owing to the simplicity of the system, we obtain the full analytical solution for the model which we use to derive a lower dimensional return map that captures the complete dynamics of the system. Using the map, we explore how biophysical properties and dendritic tree architecture affect firing dynamics. As was first reported in earlier work by one of the authors, we also find that the addition of the dendritic tree can induce bistability between periodic firing and quiescence. However, we go beyond their results by systematically examining how dendritic tree topology affects the appearance of this bistable behavior. We find that the structure of the dendritic tree can have significant quantitative effects on the bifurcation structure of the system, with branchier topologies tending to promote bistable behavior over unbranched chain topologies. We also show that this effect occurs even when the input conductance at the soma is held fixed, indicating that the topology of the dendritic tree is mainly responsible for this quantitative change in the bifurcation structure. Lastly, we demonstrate how our framework can be used to explore the effect of biophysical properties on the firing dynamics of a neuron with a more complex dendritic tree topology.

The retrograde frequency response of passive dendritic trees constrains the nonlinear firing behaviour of a reduced neuron model.

Our goal was to investigate how the propagation of alternating signals (i.e. AC), like action potentials, into the dendrites influenced nonlinear firing behaviour of motor neurons using a systematically reduced neuron model. A recently developed reduced modeling approach using only steady-current (i.e. DC) signaling was analytically expanded to retain features of the frequency-response analysis carried out in multicompartment anatomically reconstructed models. Bifurcation analysis of the extended model showed that the typically overlooked parameter of AC amplitude attenuation was positively correlated with the current threshold for the activation of a plateau potential in the dendrite. Within the multiparameter space map of the reduced model the region demonstrating “fully-bistable” firing was bounded by directional DC attenuation values that were negatively correlated to AC attenuation. Based on these results we conclude that analytically derived reduced models of dendritic trees should be fit on DC and AC signaling, as both are important biophysical parameters governing the nonlinear firing behaviour of motor neurons.

Democratization in a passive dendritic tree: an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democracy". Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green's representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.

A software for simulating steady-state properties of passive dendrites based on the cable theory

In this study, a computer software, CableTeo, is introduced for simulating the steady-state electrical properties of passive dendrite based on the cable theory. The cable theory for dendritic neurons addresses to current–voltage relations in a continuous passive dendritic tree. It is briefly summarized that the cable theory related to passive cables and dendrites, which is a useful approximation and an important reference for excitable cases. The proposed software can be used to construct user-defined dendritic tree model. The user can define the model in detail, display the constructed dendritic tree, and examine the basic electrical properties of the dendritic tree using transfer impedance approach. The software addresses to ones who want to run simple simulations of the cable theory without need to any programming language skills or expensive software. It can also be used for educational purposes.

Temporal sub units in dendritic trees

This simulation study examines the possibility that dendritic sub units can be defined according to temporal aspects in the timing of populations of synaptic inputs. A two cell model with passive dendritic trees is used, which is subject to both common and independent synaptic inputs, the presence of common synaptic input results in a tendency for correlated firing in the two cell model. The strength of this correlation is used to measure the efficacy of the common synaptic inputs in modulating the output discharge of each neurone. Our results suggest that a small fraction of the total synaptic input can effectively modulate the timing of output spikes, this phenomenon is not dependent on the physical location of the inputs on the dendritic tree. This phenomenon depends on the presence of temporal correlation between the pre-synaptic spike trains that provide the common input. We propose to refer to these as temporal sub units.

Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In this paper we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. In the first instance this system is shown to support saltatory waves with the same qualitative features as those observed in a model with Hodgkin-Huxley kinetics in the spine-head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time-varying external input we find that the distribution of spines can play a crucial role in determining spatio-temporal filtering properties. In particular, the SDS model in response to periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune [1999, 'Mechanisms for generating temporal filters in the electrosensory system,' The Journal of Experimental Biology 202: 1281-1289]. Further, we demonstrate the robustness of observed wave properties to natural sources of noise that arise both in the cable and the spine-head, and highlight the possibility of purely noise induced waves and coherent oscillations.

Dendritic cable with active spines: A modelling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Here we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded in a simple one-dimensional cable. This system is shown to support saltatory waves as a result of the discrete distribution of spines. Moreover, we demonstrate one of the ways to incorporate noise into the spine-head whilst retaining computational tractability of the model. The SDS model sustains a variety of propagating patterns.

Influence of active dendritic currents on input-output processing in spinal motoneurons in vivo.

The extensive dendritic tree of the adult spinal motoneuron generates a powerful persistent inward current (PIC). We investigated how this dendritic PIC influenced conversion of synaptic input to rhythmic firing. A linearly increasing, predominantly excitatory synaptic input was generated in triceps ankle extensor motoneurons by slow stretch (duration: 2-10 s) of the Achilles tendon in the decerebrate cat preparation. The firing pattern evoked by stretch was measured by injecting a steady current to depolarize the cell to threshold for firing. The effective synaptic current (I(N), the net synaptic current reaching the soma of the cell) evoked by stretch was measured during voltage clamp. Hyperpolarized holding potentials were used to minimize the activation of the dendritic PIC and thus estimate stretch-evoked I(N) for a passive dendritic tree (I(N,PASS)). Depolarized holding potentials that approximated the average membrane potential during rhythmic firing allowed strong activation of the dendritic PIC and thus resulted in marked enhancement of the total stretch-evoked I(N) (I(N,TOT)). The net effect of the dendritic PIC on the generation of rhythmic firing was assessed by plotting stretch-evoked firing (strong PIC activation) versus stretch-evoked I(N,PASS) (minimal PIC activation). The gain of this input-output function for the neuron (I-O(N)) was found to be ~2.7 times as high as for the standard injected frequency current (F-I) function in low-input conductance neurons. However, about halfway through the stretch, firing rate tended to become constant, resulting in a sharp saturation in I-O(N) that was not present in F-I. In addition, the gain of I-O(N) decreased sharply with increasing input conductance, resulting in much lower stretch-evoked firing rates in high-input conductance cells. All three of these phenomena (high initial gain, saturation, and differences in low- and high-input conductance cells) were also readily apparent in the differences between stretch-evoked I(N,TOT) and I(N, PASS) and thus could be accounted for by the activation of the dendritic PIC. As a result, stretch-evoked I(N,TOT) and F-I provided an accurate prediction of the overall change in stretch-evoked firing. However, in about half of the low-input conductance cells, the rate of rise of firing in response to stretch was not smoothly graded but instead consisted of a rapid surge. Stretch-evoked I(N,TOT) was always smoothly graded. This suggests that although stretch-evoked I(N,TOT) can be used to predict the overall change in firing, prediction of the dynamics of firing may be less accurate.

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.

On the analytic solution of electrotonic spread in branched passive dendritic trees.

From the classical work of Rall it is known that the spread of electric potential in a passive dendritic tree may be obtained by expressing the initial conditions as a linear combination of a set of trigonometric eigenfunctions, each decaying with the associated time constant. It is shown here that in order to evaluate the permissible parameters in these eigenfunctions one may formulate the boundary conditions at all the junctions and endings of the dendritic tree as a set of homogeneous linear equations in which the parameters in the eigenfunctions are the unknowns. These equations have a nontrivial solution if the relevant determinant vanishes, a condition that permits the evaluation of the various parameters, thus providing an analytic approach to the expression of the eigenfunctions as well as the decay time constants. The above approach is illustrated by application to a dendritic tree that has a parent segments and two generations of offspring segments, without any restrictions as to the relative diameters or lengths of the various segments in the tree. General properties of the tree may be readily derived, like the variation of the eigenvalues on scaling of the lengths or diameters of all the segments. A few special cases with specified dimensions of the various segments are derived from the general case. In the case of a dendritic tree that fulfills the "equivalent cylinder" conditions, all of the eigenvalues and eigenfunctions of the tree may be determined by the proposed method, including those that do not apply to the equivalent cylinder. The orthogonality properties of the eigenfunctions are discussed.

VC Dimension of an Integrate-and-Fire Neuron Model

We compute the VC dimension of a leaky integrate-and-fire neuron model. The VC dimension quantifies the ability of a function class to partition an input pattern space, and can be considered a measure of computational capacity. In this case, the function class is the class of integrate-and-fire models generated by varying the integration time constant T and the threshold θ, the input space they partition is the space of continuous-time signals, and the binary partition is specified by whether or not the model reaches threshold at some specified time. We show that the VC dimension diverges only logarithmically with the input signal bandwidth N. We also extend this approach to arbitrary passive dendritic trees. The main contributions of this work are (1) it offers a novel treatment of computational capacity of this class of dynamic system; and (2) it provides a framework for analyzing the computational capabilities of the dynamic systems defined by networks of spiking neurons.

A brief history of time (constants).

That the cerebral cortex processes information at prodigious speeds cannot be doubted. Yet the passive time constant, tau(m), of neurons, often thought of as a measure of the neuron's "response time' to synaptic input, is relatively long. In the 1950s, tau(m) was estimated to be only a few milliseconds for mammalian central neurons; with improvement in recording techniques, its estimated value grew over the years and it now stands near 20-100 msec. However, as we will argue here, the functional meaning of tau(m) is ambiguous. On the basis of a newly introduced definition of local delay, we show that the time window for synaptic integration in passive dendritic trees can be much smaller than the time constant. We argue that the voltage response to very brief synaptic inputs is essentially independent of tau(m). We discuss how tau(m) can change dynamically with the global activity of the network, as well as the difficulties of defining a time constant in structures with voltage-dependent elements. We conclude that the classically defined tau(m) only provides a very rough estimate, typically an overestimate, of the response time of neurons and that alternative measures are required to capture the dependency of the time course of the membrane potential on ligand-gated and/or voltage-dependent membrane conductances.

A novel theoretical approach to the analysis of dendritic transients

A novel theoretical framework for analyzing dendritic transients is introduced. This approach, called the method of moments, is an extension of Rall's cable theory for dendrites. It provides analytic investigation of voltage attenuation, signal delay, and synchronization problems in passive dendritic trees. In this method, the various moments of a transient signal are used to characterize the properties of the transient. The strength of the signal is measured by the time integral of the signal, its characteristic time is determined by its centroid ("center of gravity"), and the width of the signal is determined by a measure similar to the standard deviation in probability theory. Using these signal properties, the method of moments provides theorems, expressions, and efficient algorithms for analyzing the voltage response in arbitrary passive trees. The method yields new insights into spatiotemporal integration, coincidence detection mechanisms, and the properties of local interactions between synaptic inputs in dendritic trees. The method can also be used for matching dendritic neuron models to experimental data and for the analysis of synaptic inputs recorded experimentally.

Compartmental modelling with artificial neural networks

In this paper, we introduce an abbreviated compartmental modelling scheme which may be of interest to those in neuron- based adaptive systems because of the additional scope it provides for studying biologically-inspired learning mechanisms. The scheme, although not as flexible and precise as the general compartmental approach, allows one to design Hodgkin-Huxley style cells, and passive dendritic trees with an arbitrary number of synaptic connections. The trade-offs made for computational performance, may make the modelling scheme more appropriate for practical applications. The modelling scheme is based upon artificial neural networks, which we have used to represent cylindrical compartments (both passive and active) of different lengths, two types of voltage-dependent channels, and basic chemical synapses with variable time constants.

Signal delay and input synchronization in passive dendritic structures

1. A novel approach for analyzing transients in passive structures called "the method of moments" is introduced. It provides, as a special case, an analytic method for calculating the time delay and speed of propagation of electrical signals in any passive dendritic tree without the need for numerical simulations. 2. Total dendritic delay (TD) between two points (y, x) is defined as the difference between the centroid (the center of gravity) of the transient current input, I, at point y[tI(y)] and the centroid of the transient voltage response, V, at point x [tV(x)]. The TD measured at the input points is nonzero and is called the local delay (LD). Propagation delay, PD(y, x), is then defined as TD(y, x)--LD(y) whereas the net dendritic delay, NDD(y, 0), of an input point, y, is defined as TD(y, 0) - LD(0), where 0 is the target point, typically the soma. The signal velocity at a point x0 in the tree, theta(x0), is defined as [1/(dtv(x)/dx)[x = x0. 3. With the use of these definitions, several properties of dendritic delay exist. First, the delay between any two points in a given tree is independent of the properties (shape and duration) of the transient current input. Second, the velocity of the signal at any given point (y) in a given direction from (y) does not depend on the morphology of the tree "behind" the signal, and of the input location. Third, TD(y, x) = TD(x, y), for any two points, x, y. 4. Two additional properties are useful for efficiently calculating delays in arbitrary passive trees. 1) The subtrees connected at the ends of any dendritic segment can each be functionally lumped into an equivalent isopotential R-C compartment. 2) The local delay at any given point (y) in a tree is the mean of the local delays of the separate structures (subtrees) connected at y, weighted by the relative input conductance of the corresponding subtrees. 5. Because the definitions for delays utilize difference between centroids, the local delay and the total delay can be interpreted as measures for the time window in which synaptic inputs affect the voltage response at a target/decision point. Large LD or TD is closely associated with a relatively wide time window, whereas small LD or TD imply that inputs have to be well synchronized to affect the decision point. The net dendritic delay may be interpreted as the cost (in terms of delay) of moving a synapse away from the target point.(ABSTRACT TRUNCATED AT 400 WORDS)

On trees as equivalent cables.

By means of a suitable transformation, any passive dendritic tree may be reduced to an equivalent, possibly non-uniform cable. Under certain conditions the equivalent cable has disjoint sections of which only one communicates with the soma. Inputs that map on to the disconnected sections cannot be seen by the soma. Ralls's equivalent cylinder and its generalizations emerge naturally as the simplest cases of this behaviour. Even where, as is more usual, decomposition does not occur exactly the equivalent cable together with the input mapping from the tree to the cable provides a readily visualisable and intuitively appealing description of quite subtle relationships on the tree. The structure of the equivalent cable is dominated by approximate geometric symmetries of the tree. These symmetries cause well-defined subspaces of the total space of synaptic inputs to arrive at the soma at different times, thus allowing them, in principle, to be reflected, for example in the temporal statistics of the neurons' spike output.

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 x 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.(ABSTRACT TRUNCATED AT 250 WORDS)

Regularity analysis in a compartmental model of chopper units in the anteroventral cochlear nucleus.

1. We investigate the discharge patterns of chopper units in the anteroventral cochlear nucleus (AVCN) by developing an equivalent cylinder compartmental model of AVCN stellate cells, which are the sources of the chopper response pattern. The model consists of a passive dendritic tree connected to somatic and axonal compartments with voltage-sensitive channels. Synaptic inputs to the model are simulated auditory nerve fiber responses to best-frequency tones. 2. We adjust the anatomic and electrical parameters of the model to agree with available intracellular data from stellate cells in the AVCN of the mouse and the cat and compare the response of the model to injected current with responses recorded in vitro. The model shows approximately linear current-voltage characteristics for small hyperpolarizing currents. The model's input resistance and the time course of its response to hyperpolarizing current applied at the soma are comparable with those measured from stellate cells in vitro. In response to sustained depolarizing current, the model fires repetitively with nearly perfect regularity, a property also observed in vitro. 3. Auditory nerve inputs to the cell are modeled as deadtime-modified Poisson processes with a multiexponential adaptation in the Poisson rate. We are able to adjust the number, rate, and location of excitatory and inhibitory inputs to the model and succeed in simulating chopper response patterns seen in vivo. 4. Chopper units exhibit a variety of regularity and adaptation patterns in response to tone stimuli. Physiological data from brain slice experiments and experiments in vivo imply that this heterogeneity is primarily due to differences in input configurations. By systematically varying the number and position of excitatory and inhibitory inputs, we can simulate a range of chopper response patterns. 5. We quantify the regularity of the model's response using the coefficient of variation (CV) of the interspike interval. We find that the CV decreases, i.e., the regularity increases, as the number of converging inputs or their distance from the soma increases. The regularity of the output is more sensitive to the number of converging inputs than to their location on the dendritic tree. The statistics of the first spike latency (FSL) are also sensitive to the configuration of excitatory inputs. The mean and minimum FSL are more sensitive to the electrotonic distance of the inputs from the soma than to the number of inputs, whereas the standard deviation of the FSL is highly dependent on the number of converging inputs and is nearly independent of their location.(ABSTRACT TRUNCATED AT 400 WORDS)

Visibility of synaptically induced conductance changes: theory and simulations of anatomically characterized cortical pyramidal cells

A recent report has provided evidence that there are no significant increases in the neuronal input conductance during the response of cortical cells in cat visual cortex to non-preferred visual stimuli (Douglas et al., 1988). A criticism of experiments of this kind is that changes in the membrane conductance occurring in the dendritic tree may not be visible from electrodes that impale the soma. Our paper describes theoretical and numerical results concerning the visibility of synaptically induced conductance changes from intracellular electrodes, in both ideal and anatomically well-characterized cortical neurons. Based on earlier work by Rall (1967), we here derive theoretical expressions for the change in input conductance at any location in a passive dendritic tree resulting from activation of a single synapse and obtain bounds for the effects of multiple synapses. We find that the conductance change measured at the cell body is always less than the sum of the synaptic conductance changes and that this observed conductance change does not depend on the synaptic reversal potential. For the case of an infinite dendritic cylinder, the change in input resistance due to a single synaptic input decays exponentially with distance of the synapse from the recording site. Numerical simulations of synaptic inputs that change approximately as fast as the membrane time-constant produce an increase in input conductance that is only slightly less visible than that of a constant input. We also compute the changes in somatic input conductance of 2 morphologically identified pyramidal cells from cat visual cortex during activity of a single inhibitory basket cell with known synaptic input locations. We find that the increase in conductance due to the activity of the inhibitory basket cells is clearly visible from the cell body of the pyramidal cells and that a 70% reduction in the amplitude of excitation is associated with at least a 30% increase in somatic input conductance, which would be visible in intracellular recordings. Taken together with the negative experimental evidence of Douglas et al. (1988), our results cast doubt on a large class of models of direction selectivity that rely on synaptically mediated inhibitory conductance increases to veto or block excitatory conductances increases.

Analysis of a Postsynaptic Scheme based on a Tapering Equivalent Cable Model

An analytic solution of a modified cable equation with reversal potentials is used to explore nonlinear synaptic effects in passive dendritic trees of arbitrary geometry. It is shown that shunting inhibition can be effective when located off the direct path between the excitation and the soma. It is also shown that a peripherally placed excitatory input juxtaposed with a shunting inhibitory input may produce a voltage-peak minimum at the soma in order to suppress the initiation of an action potential at the axon hillock. The applicability of this postsynaptic scheme as a basis for a directionally selective signal generation in the retina is discussed.