Temporal disparity of action potentials triggered in axon initial segments and distal axons in the neocortex.

Neural population activity determines the timing of synaptic inputs, which arrive to dendrites, cell bodies, and axon initial segments (AISs) of cortical neurons. Action potential initiation in the AIS (AIS-APs) is driven by input integration, and the phase preference of AIS-APs during network oscillations is characteristic to cell classes. Distal regions of cortical axons do not receive synaptic inputs, yet experimental induction protocols can trigger retroaxonal action potentials (RA-APs) in axons distal from the soma. We report spontaneously occurring RA-APs in human and rodent cortical interneurons that appear uncorrelated to inputs and population activity. Network-linked triggering of AIS-APs versus input-independent timing of RA-APs of the same interneurons results in disparate temporal contribution of a single cell to in vivo network operation through perisomatic and distal axonal firing.

In vivo Assessment of Microtubule Dynamics and Orientation in Caenorhabditis elegans Neurons.

In neurons, microtubule orientation has been a key assessor to identify axons that have plus-end out microtubules and dendrites that generally have mixed orientation. Here we describe methods to label, image, and analyze the microtubule dynamics and growth during the development and regeneration of touch neurons in C. elegans. Using genetically encoded fluorescent reporters of microtubule tips, we imaged the axonal microtubules. The local changes in microtubule behavior that initiates axon regeneration following axotomy can be quantified using this protocol. This assay is adaptable to other neurons and genetic backgrounds to investigate the regulation of microtubule dynamics in various cellular processes.

Computationally going where experiments cannot: a dynamical assessment of dendritic ion channel currents during in vivo-like states.

Background: Despite technological advances, how specific cell types are involved in brain function remains shrouded in mystery. Further, little is known about the contribution of different ion channel currents to cell excitability across different neuronal subtypes and their dendritic compartments in vivo. The picture that we do have is largely based on somatic recordings performed in vitro. Uncovering dendritic ion channel current contributions in neuron subtypes that represent a minority of the neuronal population is not currently a feasible task using purely experimental means. Methods: We employ two morphologically-detailed multi-compartment models of a specific type of inhibitory interneuron, the oriens lacunosum moleculare (OLM) cell. The OLM cell is a well-studied cell type in CA1 hippocampus that is important in gating sensory and contextual information. We create in vivo-like states for these cellular models by including levels of synaptic bombardment that would occur in vivo. Using visualization tools and analyses we assess the ion channel current contribution profile across the different somatic and dendritic compartments of the models. Results: We identify changes in dendritic excitability, ion channel current contributions and co-activation patterns between in vitro and in vivo-like states. Primarily, we find that the relative timing between ion channel currents are mostly invariant between states, but exhibit changes in magnitudes and decreased propagation across dendritic compartments. We also find enhanced dendritic hyperpolarization-activated cyclic nucleotide-gated channel (h-channel) activation during in vivo-like states, which suggests that dendritically located h-channels are functionally important in altering signal propagation in the behaving animal. Conclusions: Overall, we have demonstrated, using computational modelling, the dynamical changes that can occur to ion channel mechanisms governing neuronal spiking in vitro and in vivo. In particular, we have shown that the magnitudes of some ion channel current contributions are differentially altered during in vivo-like states relative to in vitro.

The Inverse First Passage time method for a two dimensional Ornstein Uhlenbeck process with neuronal application.

The Inverse First Passage time problem seeks to determine the boundary corresponding to a given stochastic process and a fixed first passage time distribution. Here, we determine the numerical solution of this problem in the case of a two dimensional Gauss-Markov diffusion process. We investigate the boundary shape corresponding to Inverse Gaussian or Gamma first passage time distributions for different choices of the parameters, including heavy and light tails instances. Applications in neuroscience framework are illustrated.

Wiring Economy of Pyramidal Cells in the Juvenile Rat Somatosensory Cortex.

Ever since Cajal hypothesized that the structure of neurons is designed in such a way as to save space, time and matter, numerous researchers have analyzed wiring properties at different scales of brain organization. Here we test the hypothesis that individual pyramidal cells, the most abundant type of neuron in the cerebral cortex, optimize brain connectivity in terms of wiring length. In this study, we analyze the neuronal wiring of complete basal arborizations of pyramidal neurons in layer II, III, IV, Va, Vb and VI of the hindlimb somatosensory cortical region of postnatal day 14 rats. For each cell, we search for the optimal basal arborization and compare its length with the length of the real dendritic structure. Here the optimal arborization is defined as the arborization that has the shortest total wiring length provided that all neuron bifurcations are respected and the extent of the dendritic arborizations remain unchanged. We use graph theory and evolutionary computation techniques to search for the minimal wiring arborizations. Despite morphological differences between pyramidal neurons located in different cortical layers, we found that the neuronal wiring is near-optimal in all cases (the biggest difference between the shortest synthetic wiring found for a dendritic arborization and the length of its real wiring was less than 5%). We found, however, that the real neuronal wiring was significantly closer to the best solution found in layers II, III and IV. Our studies show that the wiring economy of cortical neurons is related not to the type of neurons or their morphological complexities but to general wiring economy principles.

Fluctuations in the open time of synaptic channels: an application to noise analysis based on charge.

Synaptic channels are stochastic devices. Even recording from large ensembles of channels, the fluctuations, described by Markov transition matrices, can be used to extract single channel properties. Here we study fluctuations in the open time of channels, which is proportional to the charge flowing through the channel. We use the results to implement a novel type of noise analysis that uses the charge rather than the current to extract fundamental channel parameters. We show in simulations that this charge based noise analysis is more robust if the synapse is located on the dendrites and thus subject to cable filtering. However, we also demonstrate that when multiple synapses are distributed on the dendrites, noise analysis breaks down. We finally discuss applications of our results to other biological processes.

Simulation of signal flow in 3D reconstructions of an anatomically realistic neural network in rat vibrissal cortex.

The three-dimensional (3D) structure of neural circuits represents an essential constraint for information flow in the brain. Methods to directly monitor streams of excitation, at subcellular and millisecond resolution, are at present lacking. Here, we describe a pipeline of tools that allow investigating information flow by simulating electrical signals that propagate through anatomically realistic models of average neural networks. The pipeline comprises three blocks. First, we review tools that allow fast and automated acquisition of 3D anatomical data, such as neuron soma distributions or reconstructions of dendrites and axons from in vivo labeled cells. Second, we introduce NeuroNet, a tool for assembling the 3D structure and wiring of average neural networks. Finally, we introduce a simulation framework, NeuroDUNE, to investigate structure-function relationships within networks of full-compartmental neuron models at subcellular, cellular and network levels. We illustrate the pipeline by simulations of a reconstructed excitatory network formed between the thalamus and spiny stellate neurons in layer 4 (L4ss) of a cortical barrel column in rat vibrissal cortex. Exciting the ensemble of L4ss neurons with realistic input from an ensemble of thalamic neurons revealed that the location-specific thalamocortical connectivity may result in location-specific spiking of cortical cells. Specifically, a radial decay in spiking probability toward the column borders could be a general feature of signal flow in a barrel column. Our simulations provide insights of how anatomical parameters, such as the subcellular organization of synapses, may constrain spiking responses at the cellular and network levels.

A model of NMDA receptor control of F-actin treadmilling in synaptic spines and their growth.

Synaptic spines grow as a consequence of the formation of F-actin filaments at the spine head. The dynamics of F-actin in the spine head upon excitation of N-methy-D-aspartate (NMDA) receptors has recently been investigated experimentally, but there is no quantitative account of how these dynamic changes occur upon activation of these receptors; this we now supply. Dynamics of F-actin at the apex of lamellipodia have been investigated in detail, giving rise to the treadmilling theory of F-actin dynamics, involving catalysis by profilin, for which quantitative models are now available. Here, we adapt such a model to describe the dynamics of F-actin in the synaptic-spine head and show that it gives quantitative descriptions of this treadmilling phenomena which are well fitted by Monte Carlo simulations. Next, the means by which excitation of NMDA receptors enhances the activity of profilin through activity of the Rho small GTPase RhoA and the specific kinase ROCK is discussed. This is then used to model the NMDA receptor excitatory enhancement of profilin and so the treadmilling process of F-actin dynamics in spine growth. Such modelling provides a quantitative description of the synaptic-spine dynamics of the filamentous to globular actin ratio that is observed experimentally.

Distribution of vestibulospinal contacts on the dendrites of ipsilateral splenius motoneurons: an anatomical substrate for push-pull interactions during vestibulocollic reflexes.

Excitatory and inhibitory synapses may control neuronal output through a push-pull mechanism--that is, increases in excitation are coupled to simultaneous decreases in inhibition or vice versa. This pattern of activity is characteristic of excitatory and inhibitory vestibulospinal axons that mediate vestibulocollic reflexes. Previously, we showed that medial vestibulospinal tract (MVST) neurons in the rostral descending vestibular nucleus (DVN), an excitatory pathway, primarily innervate the medial dendrites of contralateral splenius motoneurons. In the present study, we tested the hypothesis that the counterparts of the push-pull mechanism, the ipsilateral inhibitory MVST synapses, are distributed on the dendritic tree such that the interactions with excitatory MVST synapses are enhanced. We combined anterograde tracing and intracellular staining in adult felines and show that most contacts (approximately 70%) between inhibitory MVST neurons in the rostral DVN and ipsilateral splenius motoneurons are also located on medial dendrites. There was a weak bias towards proximal dendrites. Using computational methods, we further show that the organization of excitatory and inhibitory MVST synapses on splenius motoneurons increases their likelihood for interaction. We found that if either excitatory or inhibitory MVST synapses were uniformly distributed throughout the dendritic tree, the proportion of inhibitory contacts in close proximity to excitatory contacts decreased. Thus, the compartmentalized distribution of excitatory and inhibitory MVST synapses on splenius motoneurons may be specifically designed to enhance their interactions during vestibulocollic reflexes. This suggests that the push-pull modulation of motoneuron output is based, in part, on the spatial arrangement of synapses on the dendritic tree.

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 = 10(4) 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.

Quantitative assessment of the distributions of membrane conductances involved in action potential backpropagation along basal dendrites.

Basal dendrites of prefrontal cortical neurons receive strong synaptic drive from recurrent excitatory synaptic inputs. Synaptic integration within basal dendrites is therefore likely to play an important role in cortical information processing. Both synaptic integration and synaptic plasticity depend crucially on dendritic membrane excitability and the backpropagation of action potentials. We carried out multisite voltage-sensitive dye imaging of membrane potential transients from thin basal branches of prefrontal cortical pyramidal neurons before and after application of channel blockers. We found that backpropagating action potentials (bAPs) are predominantly controlled by voltage-gated sodium and A-type potassium channels. In contrast, pharmacologically blocking the delayed rectifier potassium, voltage-gated calcium, or I(h) conductance had little effect on dendritic AP propagation. Optically recorded bAP waveforms were quantified and multicompartmental modeling was used to link the observed behavior with the underlying biophysical properties. The best-fit model included a nonuniform sodium channel distribution with decreasing conductance with distance from the soma, together with a nonuniform (increasing) A-type potassium conductance. AP amplitudes decline with distance in this model, but to a lesser extent than previously thought. We used this model to explore the mechanisms underlying two sets of published data involving high-frequency trains of APs and the local generation of sodium spikelets. We also explored the conditions under which I(A) down-regulation would produce branch strength potentiation in the proposed model. Finally, we discuss the hypothesis that a fraction of basal branches may have different membrane properties compared with sister branches in the same dendritic tree.

Electric fields due to synaptic currents sharpen excitatory transmission.

The synaptic response waveform, which determines signal integration properties in the brain, depends on the spatiotemporal profile of neurotransmitter in the synaptic cleft. Here, we show that electrophoretic interactions between AMPA receptor-mediated excitatory currents and negatively charged glutamate molecules accelerate the clearance of glutamate from the synaptic cleft, speeding up synaptic responses. This phenomenon is reversed upon depolarization and diminished when intracleft electric fields are weakened through a decrease in the AMPA receptor density. In contrast, the kinetics of receptor-mediated currents evoked by direct application of glutamate are voltage-independent, as are synaptic currents mediated by the electrically neutral neurotransmitter GABA. Voltage-dependent temporal tuning of excitatory synaptic responses may thus contribute to signal integration in neural circuits.

A cost-benefit analysis of neuronal morphology.

Over hundreds of millions of years, evolution has optimized brain design to maximize its functionality while minimizing costs associated with building and maintenance. This observation suggests that one can use optimization theory to rationalize various features of brain design. Here, we attempt to explain the dimensions and branching structure of dendritic arbors by minimizing dendritic cost for given potential synaptic connectivity. Assuming only that dendritic cost increases with total dendritic length and path length from synapses to soma, we find that branching, planar, and compact dendritic arbors, such as those belonging to Purkinje cells in the cerebellum, are optimal. The theory predicts that adjacent Purkinje dendritic arbors should spatially segregate. In addition, we propose two explicit cost function expressions, falsifiable by measuring dendritic caliber near bifurcations.

Modeling synaptic dynamics driven by receptor lateral diffusion.

The synaptic weight between a pre- and a postsynaptic neuron depends in part on the number of postsynaptic receptors. On the surface of neurons, receptors traffic by random motion in and out from a microstructure called the postsynaptic density (PSD). In the PSD, receptors can be stabilized at the membrane when they bind to scaffolding proteins. We propose a mathematical model to compute the postsynaptic counterpart of the synaptic weight based on receptor trafficking. We take into account the receptor fluxes at the PSD, which can be regulated by neuronal activity, and the interactions of receptors with the scaffolding molecules. Using a Markovian approach, we estimate the mean and the fluctuations of the number of bound receptors. When the number of receptors is large, a deterministic system is also derived. Moreover, these equations can be used, for example, to fit fluorescence-recovery-after-photobleaching experiments to determine, in living neurons, the chemical binding constants for the receptors/scaffolding molecules interaction at synapses.

Efficient estimation of detailed single-neuron models.

Biophysically accurate multicompartmental models of individual neurons have significantly advanced our understanding of the input-output function of single cells. These models depend on a large number of parameters that are difficult to estimate. In practice, they are often hand-tuned to match measured physiological behaviors, thus raising questions of identifiability and interpretability. We propose a statistical approach to the automatic estimation of various biologically relevant parameters, including 1) the distribution of channel densities, 2) the spatiotemporal pattern of synaptic input, and 3) axial resistances across extended dendrites. Recent experimental advances, notably in voltage-sensitive imaging, motivate us to assume access to: i) the spatiotemporal voltage signal in the dendrite and ii) an approximate description of the channel kinetics of interest. We show here that, given i and ii, parameters 1-3 can be inferred simultaneously by nonnegative linear regression; that this optimization problem possesses a unique solution and is guaranteed to converge despite the large number of parameters and their complex nonlinear interaction; and that standard optimization algorithms efficiently reach this optimum with modest computational and data requirements. We demonstrate that the method leads to accurate estimations on a wide variety of challenging model data sets that include up to about 10(4) parameters (roughly two orders of magnitude more than previously feasible) and describe how the method gives insights into the functional interaction of groups of channels.

Statistical determinants of dendritic morphology in hippocampal pyramidal neurons: A hidden Markov model.

Dendritic structure is traditionally characterized by distributions and interrelations of morphometric parameters, such as Sholl-like plots of the number of branches versus dendritic path distance. However, how much of a given morphology is effectively captured by any statistical description is generally unknown. In this work, we assemble a small number of standard geometrical parameters measured from experimental data in a simple stochastic algorithm to describe the dendrograms of hippocampal pyramidal cells. The model, consistent with the hidden Markov framework, is feedforward, local, and causal. It relies on two "hidden" local variables: the expected number of terminal tips in a given subtree, and the current path distance from the soma. The algorithm generates dendrograms that statistically reproduce all morphological essentials of dendrites observed in real neurons, including the distributions of branching and termination points, branch lengths, membrane area, topological asymmetry, and (assuming passive membrane parameters within physiological range) electrotonic characteristics. Thus, this algorithm and the small number of its morphometric parameters constitute a remarkably complete description of the dendrograms of hippocampal pyramidal cells. Specifically, it is found that CA3 and CA1 basal dendrites and CA3 apical dendrites can each be described as homogeneous morphological classes. In contrast, the accurate generation of CA1 apical dendrites necessitates the separate sampling of two types of branches, main and oblique, suggesting their derivations from different developmental mechanisms (terminal and interstitial growth, respectively). We further offer a plausible biophysical interpretation of the model hidden variables, relating them to microtubules and other intracellular resources.

Complexity of calcium signaling in synaptic spines.

Long-term potentiation and long-term depression are thought to be cellular mechanisms contributing to learning and memory. Although the physiological phenomena have been well characterized, little consensus of their underlying molecular mechanisms has emerged. One reason for this may be the under-appreciated complexity of the signaling pathways that can arise if key signaling molecules are discretely localized within the synapse. Recent findings suggest an unanticipated degree of structural organization at the synapse, and improved methods in cellular imaging of living tissue have provided much-needed information about the intracellular dynamics of Ca(2+), thought to be critical for both LTP and LTD. In this review, we briefly summarize some of these developments, and show that a more complete understanding of cellular signaling depends on the successful integration of traditional biochemistry and molecular biology with the spatial and temporal details of synaptic ultrastructure. Biophysically realistic computer simulations can have an important role in bridging these disciplines.

Subthreshold voltage noise due to channel fluctuations in active neuronal membranes.

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na+ and K+ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.

Electrophysiological assessment of the cutaneous arborization of Adelta-fiber nociceptors.

Little is known about the relationship between the branching structure and function of physiologically identified cutaneous nociceptor terminals. The axonal arborization itself, however, has an impact on the afferent signal that is conveyed along the parent axon to the CNS. We therefore developed electrophysiological techniques to investigate the branching structure of cutaneous nociceptors. Single-fiber recordings were obtained from physiologically identified nociceptors that innervated the hairy skin of the monkey. Electrodes for transcutaneous stimulation were fixed at two separate locations inside the receptive field. For 32 Adelta-fiber nociceptors, distinct steps in latency of the recorded action potential were observed as the intensity of the transcutaneous electrical stimulus increased, indicating discrete sites for action potential initiation. The number of discrete latencies at each stimulation location ranged from 1 to 9 (3.7 +/- 0. 2; mean +/- SE) and the mean size of the latency step was 9.9 +/- 1. 0 ms (range: 0.4-89.1 ms). For seven Adelta fibers, collision techniques were used to locate the position of the branch point where the daughter fibers that innervated the two locations within the receptive field join the parent axon. To correct for changes in electrical excitability at the peripheral terminals, collision experiments between the two skin locations and between each skin location and a nerve trunk electrode were necessary. Nine branch points were studied in the seven Adelta fibers; the mean propagation time from the action potential initiation site to the branch point was 31 +/- 5 ms corresponding to a distance of 54 +/- 10 mm. Almost half of the daughter branches were unmyelinated. These results demonstrate that collision techniques can be used to study the functional anatomy of physiologically identified nociceptive afferent terminals. Furthermore these results indicate that some nociceptive afferents branch quite proximal to their peripheral receptive field. Occlusion of action potential activity can occur in these long branches such that the shorter branches dominate in the response to natural stimuli.

Critical assessment of the involvement of perforations, spinules, and spine branching in hippocampal synapse formation.

Several studies propose that long-term enhancement of synaptic transmission between neurons results from the enlargement, perforation, and splitting of synapses and dendritic spines. Unbiased analyses through serial electron microscopy were used to assess the morphological basis for synapse spilitting in hippocampal area CA1. Few perforated synapses and almost no split (i.e., branched) spines occurred at postnatal day 15, an age of high synaptogenesis; thus, synapse splitting is unlikely to be important during development. The synapse splitting hypothesis predicts an intermediate stage of branched spines with both heads sharing the same presynaptic bouton. Ninety-one branched dendritic spines were traced through serial sections, and the different branches never synapsed with the same presynaptic bouton. Projections from spines, called "spinules," have been thought to extend from perforations in the postsynaptic density (PSD), thereby dividing the presynaptic bouton. Forty-six spinules were traced, and only 13% emerged from perforations in the PSD. Most spinules emerged from the edges of nonperforated PSDs, or from spine necks, where they extended into boutons that were not presynaptic to the spine. In summary, these morphological characteristics are inconsistent with synapse and spine splitting. An alternative is discussed whereby perforated synapses and spinules are transient components of synaptic activation, and branched spines appear from synapses forming in close proximity to one another.