The response of a nerve cylinder to spatially distributed white noise inputs

The response of a passive nerve cylinder (or dendritic tree in the equivalent cylinder representation) to random white noise input currents is determined. Results for the mean, variance and covariance of the depolarization are obtained for an arbitrary number of independent spatially distributed inputs. The case of a cylinder with sealed ends is considered in detail. The differences that arise when the input currents are distributed over a small but finite region of space instead of concentrated at a point are investigated. In the case of distributed inputs, the expectation is smoother near the stimulus and the variance becomes finite over the entire cable length including the region of the applied stimulus. Away from the stimulus, there are no appreciable differences between the responses for the two cases. The interaction between an excitatory input and an inhibitory input at various locations is examined and one case of more than two inputs is also analysed to study effects which could not have been discerned from point models for a neuron with random inputs.     

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.     

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f ^3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.     

Branch Input Resistance and Steady Attenuation for Input to One Branch of a Dendritic Neuron Model

Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.     

Extrapolation methods for climate time series revisited – Spatial correlations in climatic fluctuations influence simulated tree species abundance and migration

Simulations of tree population dynamics under past and future climatic changes with time- and space-discrete models often suffer from a lack of detailed long-term climate time series that are required to drive these models. Inter- and extrapolation methods which are applied to generate long-term series differ in terms of whether they do or do not account for spatial correlation of climatic fluctuations. In this study we compared tree species abundance and migration outcomes from simulations using extrapolation methods generating spatially correlated (SC) and spatially independent (SI) climatic fluctuations. We used the spatially explicit and linked forest-landscape model TreeMig and a simple cellular automaton to demonstrate that spatial correlation of climatic fluctuations affects simulation outcomes. We conclude that methods to generate long-term climate time series should account for the spatial correlation of climatic fluctuations found in available climate records when simulating tree species abundance and migration.     

Stochastic analogues of deterministic single-species population models

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models--additive normally distributed noise and multiplicative lognormally distributed noise--generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.     

The spatial pattern of light determines the kinetics and modulates backpropagation of optogenetic action potentials

Optogenetics offers an unprecedented ability to spatially target neuronal stimulations. This study investigated via simulation, for the first time, how the spatial pattern of excitation affects the response of channelrhodopsin-2 (ChR2) expressing neurons. First we described a methodology for modeling ChR2 in the NEURON simulation platform. Then, we compared four most commonly considered illumination strategies (somatic, dendritic, axonal and whole cell) in a paradigmatic model of a cortical layer V pyramidal cell. We show that the spatial pattern of illumination has an important impact on the efficiency of stimulation and the kinetics of the spiking output. Whole cell illumination synchronizes the depolarization of the dendritic tree and the soma and evokes spiking characteristics with a distinct pattern including an increased bursting rate and enhanced back propagation of action potentials (bAPs). This type of illumination is the most efficient as a given irradiance threshold was achievable with only 6 % of ChR2 density needed in the case of somatic illumination. Targeting only the axon initial segment requires a high ChR2 density to achieve a given threshold irradiance and a prolonged illumination does not yield sustained spiking. We also show that patterned illumination can be used to modulate the bAPs and hence spatially modulate the direction and amplitude of spike time dependent plasticity protocols. We further found the irradiance threshold to increase in proportion to the demyelination level of an axon, suggesting that measurements of the irradiance threshold (for example relative to the soma) could be used to remotely probe a loss of neural myelin sheath, which is a hallmark of several neurodegenerative diseases.     

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.     

An analytical method for investigating transient potentials in neurons with branching dendritic trees.

An analytical method is developed that allows one to explore the way in which the geometrical structure of a neuron's dendritic tree affects the time-course and amplitude of transient potentials generated at different locations on dendritic branches. The method requires that, for a given dendritic arborization, one associates a symmetric geometry for which exact mathematical expressions for time-varying dendritic potentials can be calculated. The value of the dendritic potential for the asymmetric geometry is evaluated by adding correction terms to the results for the symmetric geometry. Several model trees are examined, and in each case the analytical results are expressed in terms of two closely related families of functions. These functions provide a precise formulation for systematically analyzing the way in which the voltage transient at a given point depends upon the geometrical structure of the dentritic tree. Several numerical examples are presented. A discussion of how to generalize the method and of some potential applications are given.     

Linking dendritic network structures to population demogenetics: The downside of connectivity

Spatial structures strongly influence ecological processes. Connectivity is known to positively influence metapopulation demography and genetics by increasing the rescue effect and thus favoring individual and gene flow between populations. This result has not been tested in the special case of dendritic networks, which encompass stream and cave ecosystem for instance. We propose a first approach using an individual based model to explore the population demography and genetics in various dendritic networks. To do so, we first generate a large number of different networks, and we analyze the relationship between their hydrographical characteristics and connectivity. We show that connectivity mean and variance of connectivity are strongly correlated in dendritic networks. Connectivity segregates two types of networks: Hortonian and non‐Hortonian networks. We then simulate the population dynamics for a simple life cycle in each of the generated networks, and we analyze persistence time as well as populations structure at quasi‐stationary state. Our main results show that connectivity in dendritic networks can promote local extinction and genetic isolation by distance at low dispersal and diminish the size of the metapopulation at high dispersal. We discuss these unexpected findings in the light of connectivity spatial distribution in dendritic networks in the case of our model.     

Dependence of Transformation of Intrinsic Rhythmic Impulse Activity of Neurons on Spatio-Temporal Organization of Synaptic Actions on Dendrites: A Simulation Study

The aim of the study to elucidate the biophysical mechanisms able to determine specific transformations of the patterns of output signals of neurons (neuronal impulse codes) depending on the spatio-temporal organization of synaptic actions coming to the dendrites. We studied mathematical models of the neocortical layer 5 pyramidal neurons built according to the results of computer reconstruction of their dendritic arborizations and experimental data on the voltage-dependent conductivities of their dendritic membrane. This work is a continuation of our previous studies that showed the existence of certain relations between the complexity of neural impulse codes, on the one hand, and the complexity, size, metrical asymmetry of branching, and nonlinear membrane properties of the dendrites, on the other hand. This relation determines synchronous (with some phase shifts) or asynchronous transitions of asymmetrical dendritic subtrees between high and low depolarization states during the generation of output impulse patterns in response to distributed tonic activation of dendritic inputs. In this work we demonstrate the first time that the appearance and pattern of transformations of complex periodical impulse trains at the neuron’s output associated with receiving a short series of presynaptic action potentials are determined not only by the time of arrival of such a series, but also by their spatial addressing to asymmetric dendritic subtrees; the latter, in this case, may be in the same (synchronous transitions) or different (asynchronous transitions) electrical states. Biophysically, this phenomenon is based on a significant excess of the driving potential for a synaptic excitatory current in low-depolarization regions, as compared with that in high-depolarization dendritic regions receiving phasic synaptic stimuli. These findings open a novel aspect of the functioning of neurons and neuronal networks.     

Exploiting the potential of three dimensional spatial wavelet analysis to explore nesting of temporal oscillations and spatial variance in simultaneous EEG-fMRI data

Synchronization of the activity in neural networks is a fundamental mechanism of brain function, putatively serving the integration of computations on multiple spatial and temporal scales. Time scales are thought to be nested within distinct spatial scales, so that whereas fast oscillations may integrate local networks, slow oscillations might integrate computations across distributed brain areas. We here describe a newly developed approach that provides potential for the further substantiation of this hypothesis in future studies. We demonstrate the feasibility and important caveats of a novel wavelet-based means of relating time series of three-dimensional spatial variance (energy) of fMRI data to time series of temporal variance of EEG. The spatial variance of fMRI data was determined by employing the three-dimensional dual-tree complex wavelet transform. The temporal variance of EEG data was estimated by using traditional continuous complex wavelets. We tested our algorithm on artificial signals with known signal-to-noise ratios and on empirical resting state EEG-fMRI data obtained from four healthy human subjects. By employing the human posterior alpha rhythm as an exemplar, we demonstrated face validity of the approach. We believe that the proposed method can serve as a suitable tool for future research on the spatiotemporal properties of brain dynamics, hence moving beyond analyses based exclusively in one domain or the other.     

Remote Modulation of Current Transfer from Dendritic Glutamatergic Synapses by GABA-ergic Synapses of the Somatic Zone of Motoneurons: a Simulation Study

Until now, information concerning spatial interaction of postsynaptic excitation and inhibition in neuronal dendrites remains rather limited. In model experiments, we studied spatial effects of tonic co-activation of GABA-ergic synapses situated on the soma and axon hillock of a motoneuron and dendritic glutamatergic synapses with receptors sensitive or insensitive to N-methyl-D-aspartate. We analyzed distribution maps of the transmembrane potentials and excitatory currents transferred toward the soma over the reconstructed dendritic arborization of a rat abducens motoneuron (three-dimensional reconstruction). In the motoneuron, isolated tonic excitation of glutamatergic synapses induced two stable states of low (downstate) or high (upstate) spatially heterogeneous dendritic depolarization, which decayed with unequal rates along different dendritic paths. In this case, the local steady-state current-voltage relation of the dendritic membrane became N-shaped due to a limb of the negative slope within a certain voltage range. The upstate corresponding to plateau potentials associated with stereotyped motor activity patterns was analyzed in detail. In this state, most proximal dendritic sites were the main sources of the excitatory current reaching the soma, while the contribution from distal sites was negligible. Co-activation of GABA-synapses located at the soma and axon hillock reduced this depolarization and shifted the main excitatory current source from a perisomatic location to the middle, structurally more complex, region of the dendritic arborization. The more remote dendritic region having a greater membrane area and receiving a greater number of synaptic contacts became directly involved in the supply of the trigger zone by the excitatory current. We suggest that a special, not described earlier, operational mechanism of postsynaptic inhibition is manifested in the above spatial effects of activation of strategically located inhibitory synapses, and that the list of known crucial inhibitory mechanisms (namely hyperpolarization and shunting of the postsynaptic membrane) must be expanded.     

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.     

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.     

Cryptic variation in butterfly eyespot development: the importance of sample size in gene expression studies

Previous studies have shown that development can be robust to variation in parameters such as the timing or level of gene expression. This leads to the prediction that natural populations should be able to host developmental variation that has little phenotypic effect. Cryptic variation is of particular interest because it can result in selectable phenotypes when "released" by environmental or genetic factors. Currently, however, we have little idea of how variation is distributed between genes or over time in pattern formation processes. Here we survey expression of Notch (N), Spalt (Sal), and Engrailed (En) during butterfly eyespot determination to better understand how pattern formation may vary within a population. We observed substantial heterochronic variance in the progress of spatial expression patterns for all three proteins, suggesting some degree of developmental buffering in eyespot development. Peak variance for different proteins was found at both early and late stages of development, contrasting with previous models suggesting that the distribution of variance should be more temporally focused during pattern formation. We speculate that our observations are representative of a standing reservoir of cryptic variation that may contribute to phenotypic evolution under certain circumstances. Our results also provide a strong cautionary message that gene expression studies with limited sample sizes can be positively misleading in terms of inferring expression pattern time series, as well as for making cross-species phylogenetic comparisons.     

Distribution of survival times in the Fix and Neyman model

In this paper we consider a general illness-death stochastic model in which the transition intensities all vary proportionally to a time function ϕ(t). We extend Chiang's earlier work to include processes which are both reversible and have parameters which are time varying, and obtain survival time distributions and the expectation and variance of survival times.     

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.     

Solutions for transients in arbitrarily branching cables: II. Voltage clamp theory

Analytical solutions are derived for arbitrarily branching passive neurone models with a soma and somatic shunt, for synaptic inputs and somatic voltage commands, for both perfect and imperfect somatic voltage clamp. The solutions are infinite exponential series. Perfect clamp decouples different dendritic trees at the soma: each exponential component exists only in one tree; its time constant is independent of stimulating and recording position within the tree; its amplitude is the product of a factor constant over that entire tree and factors dependent on stimulating and recording positions. Imperfect clamp to zero is mathematically equivalent to voltage recording with a shunt. As the series resistance increases, different dendritic trees become more strongly coupled. A number of interesting response symmetries are evident. The solutions reveal parameter dependencies, including an insensitivity of the early parts of the responses to specific membrane resistivity and somatic shunt, and an approximately linear dependence of the slower time constants on series resistance, for small series resistances. The solutions are illustrated using a “cartoon” representation of a CA1 pyramidal cell and a two-cylinder + soma model.     

An integrated approach to identify spatiotemporal and individual-level determinants of animal home range size

Animal home range use is a central focus of ecological research. However, how and why home range size varies between individuals is not well studied or understood for most species. We develop a hierarchical analytical approach--using generalized linear mixed-effects modeling of time series of home range sizes--that allows variance in home range size to be decomposed into components due to variation in temporal, spatial, and individual-level processes, also facilitating intra- and interspecific comparative analyses. We applied the approach to data from a roe deer population radiotracked in central Italy. Over multiple timescales, temporal variation is explained by photoperiod and climate and spatial variation by the distribution of habitat types and spatial variance in radiotracking error. Differences between individuals explained a substantial amount of variance in home range size, but only a relatively minor part was explained by the individual attributes of sex and age. We conclude that the choice of temporal scale at which data are collected and the definition of home range can significantly influence biological inference. We suggest that the appropriate choice of scale and definition requires a good understanding of the ecology and life history of the study species. Our findings contrast with several common assumptions about roe deer behavior.