Cable properties of cat spinal motoneurones measured by combining voltage clamp, current clamp and intracellular staining

Integrating single-cell transcriptomics and microcircuit computer modeling

Biophysically realistic computer modeling of neuronal microcircuitry has served as a testing ground for hypotheses related to the structure and function of different brain microcircuits. Recent advances in single-cell transcriptomics provide snapshots of a neuron's molecular state and have demonstrated that cell-specific genetic markers engineer the electrophysiological properties of a neuron. Integrating these molecular details with biophysical modeling can allow unprecedented mechanistic insights. In this opinion review, we consider systems biology-based strategies involving statistical deconvolution and gene ontology to integrate the two approaches. We foresee that this integration will infer the nonlinear interactions between the transcriptomically detailed neurons in different brain states. For an initial assessment of these integrative strategies, we recommend testing them on a penetrant phenotype such as epilepsy or a basic organism model such as Caenorhabditis elegans.

Dendritic distributions of L-type Ca2+ and SKL channels in spinal motoneurons: a simulation study

Persistent inward currents are important to motoneuron excitability and firing behaviors and also have been implicated in excitotoxicity. In particular, L-type Ca2+ channels, usually located on motoneuron dendrites, play a primary role in amplification of synaptic inputs. However, recent experimental findings on L-type Ca2+ channel behaviors challenge some fundamental assumptions that have been used in interpreting experimental and computational modeling data. Thus, the objectives of this study were to incorporate recent experimental data into an updated, high-fidelity computational model in order to explain apparent inconsistencies and to better elucidate the spatial distributions, expression patterns, and functional roles of L-type Ca2+ and SKL channels. Specifically, the updated model incorporated asymmetric channel activation/deactivation kinetics, depolarization-dependent facilitation, randomness in channel gating, and coactivation of SKL channels. Our simulation results suggest that L-type Ca2+ and SKL channels colocalize primarily on distal dendrites of motoneurons in a punctate expression. Also, punctate expression, as opposed to a homogeneous expression, provides high synaptic current amplification, limits bistability and firing rates, and robustly regulates the Ca2+ persistent inward current, thereby reducing risk of excitotoxicity. The hysteresis and bistability observed experimentally in current-voltage and frequency-current relationships result from the L-type Ca2+ channels' distal location and intrinsic warm-up. Accordingly, our results indicate that punctate expression of L-type Ca2+ and SKL channels is a potent mechanism for regulating excitability, which would provide a strong neuroprotective effect. Our results could provide broader insights into the functional significance of warm-up and punctate expression of ion channels to regulation of cell excitability.NEW & NOTEWORTHY Recent experimental findings on L-type Ca2+ channels challenge fundamental assumptions used in interpreting experimental and computational modeling data. Here, we incorporated recent experimental data into an updated, high-fidelity computational model to explain apparent inconsistencies and better elucidate the distributions, expression patterns, and functional roles of L-type Ca2+ and SKL channels. Our results indicate that punctate expression of L-type Ca2+ and SKL channels is a potent mechanism for regulating motoneuron excitability, providing a strong neuroprotective effect.

Exact solutions to cable equations in branching neurons with tapering dendrites

Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which provides a general mathematical modelling framework of spatio-temporal voltage dynamics. Typically such models need to be solved numerically unless the cell membrane is modelled either by passive or quasi-active dynamics, in which cases analytical solutions can be reduced to calculation of the Green's function describing the fundamental input-output relationship in a given morphology. Such analytically tractable models often assume individual dendritic segments to be cylinders. However, it is known that dendritic segments in many types of neurons taper, i.e. their radii decline from proximal to distal ends. Here we consider a generalised form of cable theory which takes into account both branching and tapering structures of dendritic trees. We demonstrate that analytical solutions can be found in compact algebraic forms in an arbitrary branching neuron with a class of tapering dendrites studied earlier in the context of single neuronal cables by Poznanski (Bull. Math. Biol. 53(3):457-467, 1991). We apply this extended framework to a number of simplified neuronal models and contrast their output dynamics in the presence of tapering versus cylindrical segments.

Determination of effective synaptic conductances using somatic voltage clamp

The interplay between excitatory and inhibitory neurons imparts rich functions of the brain. To understand the synaptic mechanisms underlying neuronal computations, a fundamental approach is to study the dynamics of excitatory and inhibitory synaptic inputs of each neuron. The traditional method of determining input conductance, which has been applied for decades, employs the synaptic current-voltage (I-V) relation obtained via voltage clamp. Due to the space clamp effect, the measured conductance is different from the local conductance on the dendrites. Therefore, the interpretation of the measured conductance remains to be clarified. Using theoretical analysis, electrophysiological experiments, and realistic neuron simulations, here we demonstrate that there does not exist a transform between the local conductance and the conductance measured by the traditional method, due to the neglect of a nonlinear interaction between the clamp current and the synaptic current in the traditional method. Consequently, the conductance determined by the traditional method may not correlate with the local conductance on the dendrites, and its value could be unphysically negative as observed in experiment. To circumvent the challenge of the space clamp effect and elucidate synaptic impact on neuronal information processing, we propose the concept of effective conductance which is proportional to the local conductance on the dendrite and reflects directly the functional influence of synaptic inputs on somatic membrane potential dynamics, and we further develop a framework to determine the effective conductance accurately. Our work suggests re-examination of previous studies involving conductance measurement and provides a reliable approach to assess synaptic influence on neuronal computation.

To understand synaptic mechanisms underlying neuronal computations, a fundamental approach is to use voltage clamp to measure the dynamics of excitatory and inhibitory input conductances. Due to the space clamp effect, the measured conductance in general deviates from the local input conductance on the dendrites, hence its biological interpretation is questionable, as we demonstrate in this work. We further propose the concept of effective conductance that is proportional to the local input conductance on the dendrites and reflects directly the synaptic impact on spike generation, and develop a framework to determine the effective conductance reliably. Our work provides a biologically plausible metric for elucidating synaptic influence on neuronal computation under the constraint of the space clamp effect.

Electrotonic signal processing in AII amacrine cells: compartmental models and passive membrane properties for a gap junction-coupled retinal neuron

Amacrine cells are critical for processing of visual signals, but little is known about their electrotonic structure and passive membrane properties. AII amacrine cells are multifunctional interneurons in the mammalian retina and essential for both rod- and cone-mediated vision. Their dendrites are the site of both input and output chemical synapses and gap junctions that form electrically coupled networks. This electrical coupling is a challenge for developing realistic computer models of single neurons. Here, we combined multiphoton microscopy and electrophysiological recording from dye-filled AII amacrine cells in rat retinal slices to develop morphologically accurate compartmental models. Passive cable properties were estimated by directly fitting the current responses of the models evoked by voltage pulses to the physiologically recorded responses, obtained after blocking electrical coupling. The average best-fit parameters (obtained at - 60 mV and ~ 25 °C) were 0.91 µF cm^-2 for specific membrane capacitance, 198 Ω cm for cytoplasmic resistivity, and 30 kΩ cm² for specific membrane resistance. We examined the passive signal transmission between the cell body and the dendrites by the electrotonic transform and quantified the frequency-dependent voltage attenuation in response to sinusoidal current stimuli. There was significant frequency-dependent attenuation, most pronounced for signals generated at the arboreal dendrites and propagating towards the soma and lobular dendrites. In addition, we explored the consequences of the electrotonic structure for interpreting currents in somatic, whole-cell voltage-clamp recordings. The results indicate that AII amacrines cannot be characterized as electrotonically compact and suggest that their morphology and passive properties can contribute significantly to signal integration and processing.

The effects of model composition design choices on high-fidelity simulations of motoneuron recruitment and firing behaviors

OBJECTIVE

Computational models often require tradeoffs, such as balancing detail with efficiency; yet optimal balance should incorporate sound design features that do not bias the results of the specific scientific question under investigation. The present study examines how model design choices impact simulation results.

APPROACH

We developed a rigorously-validated high-fidelity computational model of the spinal motoneuron pool to study three long-standing model design practices which have yet to be examined for their impact on motoneuron recruitment, firing rate, and force simulations. The practices examined were the use of: (1) generic cell models to simulate different motoneuron types, (2) discrete property ranges for different motoneuron types, and (3) biological homogeneity of cell properties within motoneuron types.

MAIN RESULTS

Our results show that each of these practices accentuates conditions of motoneuron recruitment based on the size principle, and minimizes conditions of mixed and reversed recruitment orders, which have been observed in animal and human recordings. Specifically, strict motoneuron orderly size recruitment occurs, but in a compressed range, after which mixed and reverse motoneuron recruitment occurs due to the overlap in electrical properties of different motoneuron types. Additionally, these practices underestimate the motoneuron firing rates and force data simulated by existing models.

SIGNIFICANCE

Our results indicate that current modeling practices increase conditions of motoneuron recruitment based on the size principle, and decrease conditions of mixed and reversed recruitment order, which, in turn, impacts the predictions made by existing models on motoneuron recruitment, firing rate, and force. Additionally, mixed and reverse motoneuron recruitment generated higher muscle force than orderly size motoneuron recruitment in these simulations and represents one potential scheme to increase muscle efficiency. The examined model design practices, as well as the present results, are applicable to neuronal modeling throughout the nervous system.

Impact of the localization of dendritic calcium persistent inward current on the input-output properties of spinal motoneuron pool: a computational study

The goal of this study is to investigate how the dendritic Ca-PIC location influences nonlinear input-output properties and depends on the type of motoneurons across the motoneuron pool. A model motoneuron pool consisting of 10 motoneurons was constructed using a recently developed two-compartment modeling approach that reflected key cell type-associated properties experimentally identified. The dendritic excitability and firing output depended systematically on both the PIC location and the motoneuron type. The PIC onset and offset in the current-voltage (I-V) relationship tended to occur at more hyperpolarized voltages as the path length to the PIC channels from the soma increased and as the cell type shifted from high- to low-threshold motoneurons. At the same time, the firing acceleration and frequency hysteresis in the frequency-current (F-I) relationship became faster and larger, respectively. However, the PIC onset-offset hysteresis increased as the path length and the recruitment threshold increased. Furthermore, the gain of frequency-current function before full PIC activation was larger for PIC channels located over distal dendritic regions in low- compared with high-threshold motoneurons. When compared with previously published experimental observations, the modeling concurred when Ca-PIC channels were placed closer to the soma in high- than low-threshold motoneurons in the model motoneuron pool. All of these results suggest that the negative relationship of Ca-PIC location and cell recruitment threshold may underlie the systematic variation in I-V and F-I transformation across the motoneuron pool.NEW & NOTEWORTHY How does the dendritic location of calcium persistent inward current (Ca-PIC) influence dendritic excitability and firing behavior across the spinal motoneuron pool? This issue was investigated developing a model motoneuron pool that reflected key motoneuron type-specific properties experimentally identified. The simulation results point out the negative relationship between the distance of Ca-PIC source from the soma and cell recruitment threshold as a basis underlying the systematic variation in input-output properties of motoneurons over the motoneuron pool.

Is realistic neuronal modeling realistic?

Scientific models are abstractions that aim to explain natural phenomena. A successful model shows how a complex phenomenon arises from relatively simple principles while preserving major physical or biological rules and predicting novel experiments. A model should not be a facsimile of reality; it is an aid for understanding it. Contrary to this basic premise, with the 21st century has come a surge in computational efforts to model biological processes in great detail. Here we discuss the oxymoronic, realistic modeling of single neurons. This rapidly advancing field is driven by the discovery that some neurons don't merely sum their inputs and fire if the sum exceeds some threshold. Thus researchers have asked what are the computational abilities of single neurons and attempted to give answers using realistic models. We briefly review the state of the art of compartmental modeling highlighting recent progress and intrinsic flaws. We then attempt to address two fundamental questions. Practically, can we realistically model single neurons? Philosophically, should we realistically model single neurons? We use layer 5 neocortical pyramidal neurons as a test case to examine these issues. We subject three publically available models of layer 5 pyramidal neurons to three simple computational challenges. Based on their performance and a partial survey of published models, we conclude that current compartmental models are ad hoc, unrealistic models functioning poorly once they are stretched beyond the specific problems for which they were designed. We then attempt to plot possible paths for generating realistic single neuron models.

Contribution of sublinear and supralinear dendritic integration to neuronal computations

Nonlinear dendritic integration is thought to increase the computational ability of neurons. Most studies focus on how supralinear summation of excitatory synaptic responses arising from clustered inputs within single dendrites result in the enhancement of neuronal firing, enabling simple computations such as feature detection. Recent reports have shown that sublinear summation is also a prominent dendritic operation, extending the range of subthreshold input-output (sI/O) transformations conferred by dendrites. Like supralinear operations, sublinear dendritic operations also increase the repertoire of neuronal computations, but feature extraction requires different synaptic connectivity strategies for each of these operations. In this article we will review the experimental and theoretical findings describing the biophysical determinants of the three primary classes of dendritic operations: linear, sublinear, and supralinear. We then review a Boolean algebra-based analysis of simplified neuron models, which provides insight into how dendritic operations influence neuronal computations. We highlight how neuronal computations are critically dependent on the interplay of dendritic properties (morphology and voltage-gated channel expression), spiking threshold and distribution of synaptic inputs carrying particular sensory features. Finally, we describe how global (scattered) and local (clustered) integration strategies permit the implementation of similar classes of computations, one example being the object feature binding problem.

Axon-somatic back-propagation in detailed models of spinal alpha motoneurons

Antidromic action potentials following distal stimulation of motor axons occasionally fail to invade the soma of alpha motoneurons in spinal cord, due to their passing through regions of high non-uniformity. Morphologically detailed conductance-based models of cat spinal alpha motoneurons have been developed, with the aim to reproduce and clarify some aspects of the electrophysiological behavior of the antidromic axon-somatic spike propagation. Fourteen 3D morphologically detailed somata and dendrites of cat spinal alpha motoneurons have been imported from an open-access web-based database of neuronal morphologies, NeuroMorpho.org, and instantiated in neurocomputational models. An axon hillock, an axonal initial segment and a myelinated axon are added to each model. By sweeping the diameter of the axonal initial segment (AIS) and the axon hillock, as well as the maximal conductances of sodium channels at the AIS and at the soma, the developed models are able to show the relationships between different geometric and electrophysiological configurations and the voltage attenuation of the antidromically traveling wave. In particular, a greater than usually admitted sodium conductance at AIS is necessary and sufficient to overcome the dramatic voltage attenuation occurring during antidromic spike propagation both at the myelinated axon-AIS and at the AIS-soma transitions.

Development of modified cable models to simulate accurate neuronal active behaviors

In large network and single three-dimensional (3-D) neuron simulations, high computing speed dictates using reduced cable models to simulate neuronal firing behaviors. However, these models are unwarranted under active conditions and lack accurate representation of dendritic active conductances that greatly shape neuronal firing. Here, realistic 3-D (R3D) models (which contain full anatomical details of dendrites) of spinal motoneurons were systematically compared with their reduced single unbranched cable (SUC, which reduces the dendrites to a single electrically equivalent cable) counterpart under passive and active conditions. The SUC models matched the R3D model's passive properties but failed to match key active properties, especially active behaviors originating from dendrites. For instance, persistent inward currents (PIC) hysteresis, frequency-current (FI) relationship secondary range slope, firing hysteresis, plateau potential partial deactivation, staircase currents, synaptic current transfer ratio, and regional FI relationships were not accurately reproduced by the SUC models. The dendritic morphology oversimplification and lack of dendritic active conductances spatial segregation in the SUC models caused significant underestimation of those behaviors. Next, SUC models were modified by adding key branching features in an attempt to restore their active behaviors. The addition of primary dendritic branching only partially restored some active behaviors, whereas the addition of secondary dendritic branching restored most behaviors. Importantly, the proposed modified models successfully replicated the active properties without sacrificing model simplicity, making them attractive candidates for running R3D single neuron and network simulations with accurate firing behaviors. The present results indicate that using reduced models to examine PIC behaviors in spinal motoneurons is unwarranted.

Asymmetry in signal propagation between the soma and dendrites plays a key role in determining dendritic excitability in motoneurons

It is widely recognized that propagation of electrophysiological signals between the soma and dendrites of neurons differs depending on direction, i.e. it is asymmetric. How this asymmetry influences the activation of voltage-gated dendritic channels, and consequent neuronal behavior, remains unclear. Based on the analysis of asymmetry in several types of motoneurons, we extended our previous methodology for reducing a fully reconstructed motoneuron model to a two-compartment representation that preserved asymmetric signal propagation. The reduced models accurately replicated the dendritic excitability and the dynamics of the anatomical model involving a persistent inward current (PIC) dispersed over the dendrites. The relationship between asymmetric signal propagation and dendritic excitability was investigated using the reduced models while varying the asymmetry in signal propagation between the soma and the dendrite with PIC density constant. We found that increases in signal attenuation from soma to dendrites increased the activation threshold of a PIC (hypo-excitability), whereas increases in signal attenuation from dendrites to soma decreased the activation threshold of a PIC (hyper-excitability). These effects were so strong that reversing the asymmetry in the soma-to-dendrite vs. dendrite-to-soma attenuation, reversed the correlation between PIC threshold and distance of this current source from the soma. We propose the tight relation of the asymmetric signal propagation to the input resistance in the dendrites as a mechanism underlying the influence of the asymmetric signal propagation on the dendritic excitability. All these results emphasize the importance of maintaining the physiological asymmetry in dendritic signaling not only for normal function of the cells but also for biophysically realistic simulations of dendritic excitability.

Burst firing versus synchrony in a gap junction connected olfactory bulb mitral cell network model

A key player in olfactory processing is the olfactory bulb (OB) mitral cell (MC). We have used dual whole-cell patch-clamp recordings from the apical dendrite and cell soma of MCs to develop a passive compartmental model based on detailed morphological reconstructions of the same cells. Matching the model to traces recorded in experiments we find: C_m = 1.91 ± 0.20 μF cm⁻², R_m = 3547 ± 1934 Ω cm² and R_i = 173 ± 99 Ω cm. We have constructed a six MC gap-junction (GJ) network model of morphologically accurate MCs. These passive parameters (PPs) were then incorporated into the model with Na⁺, Kdr, and KA conductances and GJs from Migliore et al. (2005). The GJs were placed in the apical dendrite tuft (ADT) and their conductance adjusted to give a coupling ratio between MCs consistent with experimental findings (~0.04). Firing at ~50 Hz was induced in all six MCs with continuous current injections (0.05–0.07 nA) at 20 locations to the ADT of two of the MCs. It was found that MCs in the network synchronized better when they shared identical PPs rather than using their own PPs for the fit suggesting that the OB may have populations of MCs tuned for synchrony. The addition of calcium-activated potassium channels (iKCa) and L-type calcium channels (iCa(L)) (Bhalla and Bower, 1993) to the model enabled MCs to generate burst firing. However, the GJ coupling was no longer sufficient to synchronize firing. When cells were stimulated by a continuous current injection there was an initial period of asynchronous burst firing followed after ~120 ms by synchronous repetitive firing. This occurred as intracellular calcium fell due to reduced iCa(L) activity. The kinetics of one of the iCa(L) gate variables, which had a long activation time constant (τ ~ range 18–150 ms), was responsible for this fall in iCa(L). The model makes predictions about the nature of the kinetics of the calcium current that will need experimental verification.

Generating synchrony from the asynchronous: compensation for cochlear traveling wave delays by the dendrites of individual brainstem neurons

Broadband transient sounds, such as clicks and consonants, activate a traveling wave in the cochlea. This wave evokes firing in auditory nerve fibers that are tuned to high frequencies several milliseconds earlier than in fibers tuned to low frequencies. Despite this substantial traveling wave delay, octopus cells in the brainstem receive broadband input and respond to clicks with submillisecond temporal precision. The dendrites of octopus cells lie perpendicular to the tonotopically organized array of auditory nerve fibers, placing the earliest arriving inputs most distally and the latest arriving closest to the soma. Here, we test the hypothesis that the topographic arrangement of synaptic inputs on dendrites of octopus cells allows octopus cells to compensate the traveling wave delay. We show that in mice the full cochlear traveling wave delay is 1.6 ms. Because the dendrites of each octopus cell spread across approximately one-third of the tonotopic axis, a click evokes a soma-directed sweep of synaptic input lasting 0.5 ms in individual octopus cells. Morphologically and biophysically realistic, computational models of octopus cells show that soma-directed sweeps with durations matching in vivo measurements result in the largest and sharpest somatic EPSPs. A low input resistance and activation of a low-voltage-activated potassium conductance that are characteristic of octopus cells are important determinants of sweep sensitivity. We conclude that octopus cells have dendritic morphologies and biophysics tailored to accomplish the precise encoding of broadband transient sounds.

Somato-dendritic morphology and dendritic signal transfer properties differentiate between fore- and hindlimb innervating motoneurons in the frog Rana esculenta

Background

The location specific motor pattern generation properties of the spinal cord along its rostro-caudal axis have been demonstrated. However, it is still unclear that these differences are due to the different spinal interneuronal networks underlying locomotions or there are also segmental differences in motoneurons innervating different limbs. Frogs use their fore- and hindlimbs differently during jumping and swimming. Therefore we hypothesized that limb innervating motoneurons, located in the cervical and lumbar spinal cord, are different in their morphology and dendritic signal transfer properties. The test of this hypothesis what we report here.

Results

Discriminant analysis classified segmental origin of the intracellularly labeled and three-dimensionally reconstructed motoneurons 100% correctly based on twelve morphological variables. Somata of lumbar motoneurons were rounder; the dendrites had bigger total length, more branches with higher branching orders and different spatial distributions of branch points. The ventro-medial extent of cervical dendrites was bigger than in lumbar motoneurons. Computational models of the motoneurons showed that dendritic signal transfer properties were also different in the two groups of motoneurons. Whether log attenuations were higher or lower in cervical than in lumbar motoneurons depended on the proximity of dendritic input to the soma. To investigate dendritic voltage and current transfer properties imposed by dendritic architecture rather than by neuronal size we used standardized distributions of transfer variables. We introduced a novel combination of cluster analysis and homogeneity indexes to quantify segmental segregation tendencies of motoneurons based on their dendritic transfer properties. A segregation tendency of cervical and lumbar motoneurons was detected by the rates of steady-state and transient voltage-amplitude transfers from dendrites to soma at all levels of synaptic background activities, modeled by varying the specific dendritic membrane resistance. On the other hand no segregation was observed by the steady-state current transfer except under high background activity.

Conclusions

We found size-dependent and size-independent differences in morphology and electrical structure of the limb moving motoneurons based on their spinal segmental location in frogs. Location specificity of locomotor networks is therefore partly due to segmental differences in motoneurons driving fore-, and hindlimbs.

A simple method for characterizing passive and active neuronal properties: application to striatal neurons

The study of active and passive neuronal dynamics usually relies on a sophisticated array of electrophysiological, staining and pharmacological techniques. We describe here a simple complementary method that recovers many findings of these more complex methods but relies only on a basic patch-clamp recording approach. Somatic short and long current pulses were applied in vitro to striatal medium spiny (MS) and fast spiking (FS) neurons from juvenile rats. The passive dynamics were quantified by fitting two-compartment models to the short current pulse data. Lumped conductances for the active dynamics were then found by compensating this fitted passive dynamics within the current-voltage relationship from the long current pulse data. These estimated passive and active properties were consistent with previous more complex estimations of the neuron properties, supporting the approach. Relationships within the MS and FS neuron types were also evident, including a graduation of MS neuron properties consistent with recent findings about D1 and D2 dopamine receptor expression. Application of the method to simulated neuron data supported the hypothesis that it gives reasonable estimates of membrane properties and gross morphology. Therefore detailed information about the biophysics can be gained from this simple approach, which is useful for both classification of neuron type and biophysical modelling. Furthermore, because these methods rely upon no manipulations to the cell other than patch clamping, they are ideally suited to in vivo electrophysiology.

Virtual NEURON: a strategy for merged biochemical and electrophysiological modeling

Because of its highly branched dendrite, the Purkinje neuron requires significant computational resources if coupled electrical and biochemical activity are to be simulated. To address this challenge, we developed a scheme for reducing the geometric complexity; while preserving the essential features of activity in both the soma and a remote dendritic spine. We merged our previously published biochemical model of calcium dynamics and lipid signaling in the Purkinje neuron, developed in the Virtual Cell modeling and simulation environment, with an electrophysiological model based on a Purkinje neuron model available in NEURON. A novel reduction method was applied to the Purkinje neuron geometry to obtain a model with fewer compartments that is tractable in Virtual Cell. Most of the dendritic tree was subject to reduction, but we retained the neuron's explicit electrical and geometric features along a specified path from spine to soma. Further, unlike previous simplification methods, the dendrites that branch off along the preserved explicit path are retained as reduced branches. We conserved axial resistivity and adjusted passive properties and active channel conductances for the reduction in surface area, and cytosolic calcium for the reduction in volume. Rallpacks are used to validate the reduction algorithm and show that it can be generalized to other complex neuronal geometries. For the Purkinje cell, we found that current injections at the soma were able to produce similar trains of action potentials and membrane potential propagation in the full and reduced models in NEURON; the reduced model produces identical spiking patterns in NEURON and Virtual Cell. Importantly, our reduced model can simulate communication between the soma and a distal spine; an alpha function applied at the spine to represent synaptic stimulation gave similar results in the full and reduced models for potential changes associated with both the spine and the soma. Finally, we combined phosphoinositol signaling and electrophysiology in the reduced model in Virtual Cell. Thus, a strategy has been developed to combine electrophysiology and biochemistry as a step toward merging neuronal and systems biology modeling.

Evidence from computer simulations for alterations in the membrane biophysical properties and dendritic processing of synaptic inputs in mutant superoxide dismutase-1 motoneurons

A critical step in improving our understanding of the development of amyotrophic lateral sclerosis (ALS) is to identify the factors contributing to the alterations in the excitability of motoneurons and assess their individual contributions. Here we investigated the early alterations in the passive electrical and morphological properties of neonatal spinal motoneurons that occur by 10 d after birth, long before disease onset. We identified some of the factors contributing to these alterations, and estimated their individual contributions. To achieve this goal, we undertook a computer simulation analysis using realistic morphologies of reconstructed wild-type (WT) and mutant superoxide dismutase-1 (mSOD1) motoneurons. Ion channel parameters of these models were then tuned to match the experimental data on electrical properties obtained from these same motoneurons. We found that the reduced excitability of mSOD1 models was accompanied with decreased specific membrane resistance by approximately 25% and efficacy of synaptic inputs (slow and fast) by 12-22%. Linearity of summation of synaptic currents was similar to WT. We also assessed the contribution of the alteration in dendritic morphology alone to this decreased excitability and found that it reduced the input resistance by 10% and the efficacy of synaptic inputs by 7-15%. Our results were also confirmed in models with dendritic active conductances. Our simulations indicated that the alteration in passive electrical properties of mSOD1 models resulted from concurrent alterations in their morphology and membrane biophysical properties, and consequently altered the motoneuronal dendritic processing of synaptic inputs. These results clarify new aspects of spinal motoneurons malfunction in ALS.

Dendritic vulnerability in neurodegenerative disease: insights from analyses of cortical pyramidal neurons in transgenic mouse models

In neurodegenerative disorders, such as Alzheimer's disease, neuronal dendrites and dendritic spines undergo significant pathological changes. Because of the determinant role of these highly dynamic structures in signaling by individual neurons and ultimately in the functionality of neuronal networks that mediate cognitive functions, a detailed understanding of these changes is of paramount importance. Mutant murine models, such as the Tg2576 APP mutant mouse and the rTg4510 tau mutant mouse have been developed to provide insight into pathogenesis involving the abnormal production and aggregation of amyloid and tau proteins, because of the key role that these proteins play in neurodegenerative disease. This review showcases the multidimensional approach taken by our collaborative group to increase understanding of pathological mechanisms in neurodegenerative disease using these mouse models. This approach includes analyses of empirical 3D morphological and electrophysiological data acquired from frontal cortical pyramidal neurons using confocal laser scanning microscopy and whole-cell patch-clamp recording techniques, combined with computational modeling methodologies. These collaborative studies are designed to shed insight on the repercussions of dystrophic changes in neocortical neurons, define the cellular phenotype of differential neuronal vulnerability in relevant models of neurodegenerative disease, and provide a basis upon which to develop meaningful therapeutic strategies aimed at preventing, reversing, or compensating for neurodegenerative changes in dementia.

Experimentally guided modelling of dendritic excitability in rat neocortical pyramidal neurones

Constructing physiologically relevant compartmental models of neurones is critical for understanding neuronal activity and function. We recently suggested that measurements from multiple locations along the soma, dendrites and axon are necessary as a data set when using a genetic optimization algorithm to constrain the parameters of a compartmental model of an entire neurone. However, recordings from L5 pyramidal neurones can routinely be performed simultaneously from only two locations. Now we show that a data set recorded from the soma and apical dendrite combined with a parameter peeling procedure is sufficient to constrain a compartmental model for the apical dendrite of L5 pyramidal neurones. The peeling procedure was tested on several compartmental models showing that it avoids local minima in parameter space. Based on the requirements of this analysis procedure, we designed and performed simultaneous whole-cell recordings from the soma and apical dendrite of rat L5 pyramidal neurones. The data set obtained from these recordings allowed constraining a simplified compartmental model for the apical dendrite of L5 pyramidal neurones containing four voltage-gated conductances. In agreement with experimental findings, the optimized model predicts that the conductance density gradients of voltage-gated K(+) conductances taper rapidly proximal to the soma, while the density gradient of the voltage-gated Na(+) conductance tapers slowly along the apical dendrite. The model reproduced the back-propagation of the action potential and the modulation of the resting membrane potential along the apical dendrite. Furthermore, the optimized model provided a mechanistic explanation for the back-propagation of the action potential into the apical dendrite and the generation of dendritic Na(+) spikes.

Modeling a layer 4-to-layer 2/3 module of a single column in rat neocortex: interweaving in vitro and in vivo experimental observations

We report a step in constructing an in silico model of a neocortical column, focusing on the synaptic connection between layer 4 (L4) spiny neurons and L2/3 pyramidal cells in rat barrel cortex. It is based first on a detailed morphological and functional characterization of synaptically connected pairs of L4-L2/3 neurons from in vitro recordings and second, on in vivo recordings of voltage responses of L2/3 pyramidal cells to current pulses and to whisker deflection. In vitro data and a detailed compartmental model of L2/3 pyramidal cells enabled us to extract their specific membrane resistivity ( approximately 16,000 ohms x cm(2)) and capacitance ( approximately 0.8 microF/cm(2)) and the spatial distribution of L4-L2/3 synaptic contacts. The average peak conductance per L4 synaptic contact is 0.26 nS for the alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid and 0.2 nS for NMDA receptors. The in vivo voltage response for current steps was then used to calibrate the model for in vivo conditions in the Down state. Consequently, the effect of a single whisker deflection was modeled by converging, on average, 350 +/- 20 L4 axons onto the modeled L2/3 pyramidal cell. Based on values of synaptic conductance, the spatial distribution of L4 synapses on L2/3 dendrites, and the average in vivo spiking probability of L4 spiny neurons, the model predicts that the feed-forward L4-L2/3 connection on its own does not fire the L2/3 neuron. With a broader distribution in the number of L4 neurons or with slight synchrony among them, the L2/3 model does spike with low probability.

Subthreshold dendritic signal processing and coincidence detection in dentate gyrus granule cells

Although dendritic signal processing has been extensively investigated in hippocampal pyramidal cells, only little is known about dendritic integration of synaptic potentials in dentate gyrus granule cells, the first stage in the hippocampal trisynaptic circuit. Here we combined dual whole-cell patch-clamp recordings with high-resolution two-photon microscopy to obtain detailed passive cable models of hippocampal granule cells from adult mice. Passive cable properties were determined by direct fitting of the compartmental model to the experimentally measured voltage responses to short and long current pulses. The data are best fit by a cable model with homogenously distributed parameters, including an average specific membrane resistance (R(m)) of 38.0 kohms cm2, a membrane capacitance (C(m)) of 1.0 microF cm(-2), and an intracellular resistivity (R(i)) of 194 ohms cm. Computational analysis shows that signal propagation from somata into dendrites is more efficient in granule cells compared with CA1 pyramidal cells for both steady-state and sinusoidal voltage waveforms up to the gamma frequency range (f50% of 74 Hz). Similarly, distal synaptic inputs from entorhinal fibers can efficiently depolarize the somatic membrane of granule cells. Furthermore, the time course of distal dendritic synaptic potentials is remarkably fast, and temporal summation is restricted to a narrow time window in the range of approximately 10 ms attributable to the rapid dendritic charge redistribution during transient voltage signals. Therefore, the structure of the granule cell dendritic tree may be critically important for precise dendritic signal processing and coincidence detection during hippocampus-dependent memory formation and retrieval.

Synaptic Amplification Versus Bistability in Motoneuron Dendritic Processing: A Top-Down Modeling Approach

Motoneurons have been shown to exhibit both bistable firing and synaptic amplification. Both of these behaviors have generally been attributed to a single mechanism—dendritic plateau potentials based on L-type Ca2+ conductances. However, our recent discovery of a fast-amplification mode calls this into question. Here we examine the possibility that two mechanisms underlie these behaviors, one being a slow-mode bistability mechanism (i.e., the L-type Ca2+-conductance–based dendritic plateaus) and the other being a theoretical fast-mode amplification mechanism. A “top-down” motoneuron model that encapsulated these and other hypotheses was developed in which these mechanisms could be explored. The resulting final model simultaneously exhibits synaptic amplification, plateau potential formation, bistable firing patterns, and current–voltage ( I– V) and frequency–current ( F– I) hystereses. This model suggests that amplification and plateaus are mutually exclusive in the same dendrite/dendritic branch. Thus we predict that plateau generation does not occur in all dendritic branches. This could be readily accomplished by a large degree of variation in the density of L-type Ca2+ channels believed to underlie plateau formation in these cells with the added benefit of spreading plateau onset over a wider voltage range, as is observed experimentally.

Changes in the structural complexity of the aged brain

Structural changes of neurons in the brain during aging are complex and not well understood. Neurons have significant homeostatic control of essential brain functions, including synaptic excitability, gene expression, and metabolic regulation. Any deviations from the norm can have severe consequences as seen in aging and injury. In this review, we present some of the structural adaptations that neurons undergo throughout normal and pathological aging and discuss their effects on electrophysiological properties and cognition. During aging, it is evident that neurons undergo morphological changes such as a reduction in the complexity of dendrite arborization and dendritic length. Spine numbers are also decreased, and because spines are the major sites for excitatory synapses, changes in their numbers could reflect a change in synaptic densities. This idea has been supported by studies that demonstrate a decrease in the overall frequency of spontaneous glutamate receptor-mediated excitatory responses, as well as a decrease in the levels of alpha-amino-3-hydroxy-5-methylisoxazole-4-propionic acid and N-methyl-d-aspartate receptor expression. Other properties such as gamma-aminobutyric acid A receptor-mediated inhibitory responses and action potential firing rates are both significantly increased with age. These findings suggest that age-related neuronal dysfunction, which must underlie observed decline in cognitive function, probably involves a host of other subtle changes within the cortex that could include alterations in receptors, loss of dendrites, and spines and myelin dystrophy, as well as the alterations in synaptic transmission. Together these multiple alterations in the brain may constitute the substrate for age-related loss of cognitive function.

Estimates of the location of L-type Ca2+ channels in motoneurons of different sizes: a computational study

In the presence of monoamines, L-type Ca(2+) channels on the dendrites of motoneurons contribute to persistent inward currents (PICs) that can amplify synaptic inputs two- to sixfold. However, the exact location of the L-type Ca(2+) channels is controversial, and the importance of the location as a means of regulating the input-output properties of motoneurons is unknown. In this study, we used a computational strategy developed previously to estimate the dendritic location of the L-type Ca(2+) channels and test the hypothesis that the location of L-type Ca(2+) channels varies as a function of motoneuron size. Compartmental models were constructed based on dendritic trees of five motoneurons that ranged in size from small to large. These models were constrained by known differences in PIC activation reported for low- and high-conductance motoneurons and the relationship between somatic PIC threshold and the presence or absence of tonic excitatory or inhibitory synaptic activity. Our simulations suggest that L-type Ca(2+) channels are concentrated in hotspots whose distance from the soma increases with the size of the dendritic tree. Moving the hotspots away from these sites (e.g., using the hotspot locations from large motoneurons on intermediate-sized motoneurons) fails to replicate the shifts in PIC threshold that occur experimentally during tonic excitatory or inhibitory synaptic activity. In models equipped with a size-dependent distribution of L-type Ca(2+) channels, the amplification of synaptic current by PICs depends on motoneuron size and the location of the synaptic input on the dendritic tree.

A new stochastic tridimensional model of neonatal rat spinal motoneuron for investigating compartmentalization of neuronal conductances and their influence on firing

During postnatal development spinal motoneurons play a major role in expressing basic behaviours like reflex reactions and in allowing the onset of the locomotor programme. For this purpose it is useful to clarify how various inputs are integrated at the level of the motoneuron soma to generate phasic or rhythmic firing. Although existing models of motoneurons have indicated the distributed role of certain conductances in regulating firing, it is unclear how the spatial distribution of certain currents is ultimately shaping motoneuron output. Thus, it would be helpful to build a bridge between histological and electrophysiological data. The present report is based on the construction of a 3D motoneuron model based on available parameters applicable to the neonatal spinal cord. The presented algorithm allows building up a complex, variable dendrogram which, together with the somatic and axonic compartments, enables strategic location of certain voltage or ligand gated conductances and simulation of resulting electrical behaviour. One application of the present model has been exploring the functional location of the recently reported cystic fibrosis transmembrane conductance regulator (CFTR) which controls Cl(-) homeostasis of postnatal motoneurons. The 3D model is made available for free, user friendly use via the dedicated web site http://www.mn-morphology.org.

Dendritic voltage-gated K+ conductance gradient in pyramidal neurones of neocortical layer 5B from rats

Voltage-gated potassium channels effectively regulate dendritic excitability in neurones. It has been suggested that in the distal apical dendrite of layer 5B (L5B) neocortical pyramidal neurones, K+ conductances participate in active dendritic synaptic integration and control regenerative dendritic potentials. The ionic mechanism for triggering these regenerative potentials has yet to be elucidated. Here we used two-electrode voltage clamp (TEVC) to quantitatively record K+ conductance densities of a sustained K+ conductance in the soma and apical dendrite of L5B neurones of adult rats. We report that the somatic and proximal dendritic sustained voltage-gated K+ conductance density is more than 10-fold larger than previous estimates. The results obtained using TEVC were corroborated using current-clamp experiments in combination with compartmental modelling. Possible error sources, including inaccurate measurement of the passive membrane parameters and unknown axonal and basal dendritic conductance distributions, were shown not to distort the density estimation considerably. The sustained voltage-gated K+ conductance density was found to decrease steeply along the apical dendrite. The steep negative K+ conductance density gradient along the apical dendrite may help to define a distal, low-threshold region for amplification of distal synaptic input in L5B pyramidal neurones.

Contribution of morphology and membrane resistance to integration of fast synaptic signals in two thalamic cell types

Thalamocortical cells (TCs) and interneurons (INs) in the lateral geniculate nucleus process visual information from the retina. The TCs have many short dendrites, whereas the INs have fewer and longer dendrites. Because of these morphological differences, it has been suggested that transmission of synaptic signals from dendritic synapses to soma is more efficient in TCs than in INs. However, a higher membrane resistance (R(m)) for the INs could, in theory, compensate for the attenuating effect of their long dendrites and allow distal synaptic inputs to significantly depolarize the soma. Compartmental models were made from biocytin filled TCs (n = 15) and INs (n = 3) and adjusted to fit the current- and voltage-clamp recordings from the individual cells. The confidence limits for the passive electrical parameters were explored by simulating the influence of noise, morphometric errors and non-uniform and active conductances. One of the useful findings was that R(m) was accurately estimated despite realistic levels of active conductance. Simulations to explore the somatic influence of dendritic synapses showed that a small (0.5 nS) excitatory synapse placed at different dendritic positions gave similar somatic potentials in the individual TCs, within the TC population and also between TCs and INs. A linear increase in the conductance of the synapse gave increases in somatic potentials that were more sublinear in INs than TCs. However, when the total synaptic conductance was increased by simultaneously activating many small, spatially distributed synapses, the INs converted the synaptic signals to soma potentials almost as efficiently as the TCs. Thus, INs can transfer fast synaptic signals to soma as efficiently as TCs except when the focal conductance is large.

Characterization of Myenteric Sensory Neurons in the Mouse Small Intestine

We recorded from myenteric AH/Dogiel type II cells, demonstrated mechanosensitive responses, and characterized their basic properties. Recordings were obtained using the mouse longitudinal muscle myenteric plexus preparation with patch-clamp and sharp intracellular electrodes. The neurons had an action potential hump and a slow afterhyperpolarization (AHP) current. The slow AHP was carried by intermediate conductance Ca2+-dependent K+-channel currents sensitive to charybdotoxin and clotrimazole. All possessed a hyperpolarization-activated current that was blocked by extracellular cesium. They also expressed a TTX-resistant Na+ current with an onset near the resting potential. Pressing on the ganglion containing the patched neuron evoked depolarizing potentials in 17/18 cells. The potentials persisted after synaptic transmission was blocked. Volleys of presynaptic electrical stimuli evoked slow excitatory postsynaptic potentials (EPSPs) in 9/11 sensory neurons, but 0/29 cells received fast EPSP input. The slow EPSP was generated by removal of a voltage-insensitive K+ current. Patch-clamp recording with a KMeSO4-containing, but not a conventional KCl-rich, intracellular solution reproduced the single-spike slow AHPs and low input resistances seen with sharp intracellular recording. Cell-attached recording of intermediate conductance potassium channels supported the conclusion that the single-spike slow AHP is an intrinsic property of intestinal AH/sensory neurons. Unitary current recordings also suggested that the slow AHP current probably does not contribute significantly to the high resting background conductance seen in these cells. The characterization of mouse myenteric sensory neurons opens the way for the study of their roles in normal and pathological physiology.

Creation and reduction of a morphologically detailed model of a leech heart interneuron

Conductance-based neuron models aid in understanding the role intrinsic and synaptic currents play in producing neuronal activity. Incorporating morphological detail into a model allows for additional analysis of nonhomogeneous distributions of active and synaptic conductances, as well as spatial segregation of electrical events. We developed a morphologically detailed "Full Model" of a leech heart interneuron that replicates reasonably well intracellular recordings from these interneurons. However, it constitutes hundreds of compartments, each increasing parameter space and simulation time. To reduce the number of compartments of the Full Model, while preserving conductance densities and distributions, its compartments were grouped into functional groups that each share identical conductance densities. Each functional group was sequentially reduced to one or two compartments, preserving surface area, conductance densities, and its contribution to input resistance. As a result, the input resistance and membrane time constant were preserved. The axial resistances of several compartments were rescaled to match the amplitude of synaptic currents and low-threshold calcium currents and the shape of action potentials to those in the Full Model. This reduced model, with intrinsic conductances, matched the activity of the Full Model for a variety of simulated current-clamp and voltage-clamp data. Because surface area and conductance distribution of the functional groups of the Full Model were maintained, parameter changes introduced into the reduced model can be directly translated to the Full Model. Thus our computationally efficient reduced morphology model can be used as a tool for exploring the parameter space of the Full Model and in network simulations.

Computational Estimation of the Distribution of L-type Ca2+Channels in Motoneurons Based on Variable Threshold of Activation of Persistent Inward Currents

In the presence of neuromodulators such as serotonin and noradrenaline, motoneurons exhibit persistent inward currents (PICs) that serve to amplify synaptic inputs. A major component of these PICs is mediated by L-type Ca(2+) channels. Estimates based on electrophysiological studies indicate that these channels are located on the dendrites, but immunohistochemical studies of their precise distribution have yielded different results. Our goal was to determine the distribution of these channels using computational methods. A theoretical analysis of the activation of PICs by a somatic current injection in the absence or presence of synaptic activity suggests that L-type Ca(2+) channels may be segregated to discrete hot spots 25-200 microm long and centered 100-400 microm from the soma in the dendritic tree. Compartmental models based on detailed anatomical measurements of the structure of feline neck motoneurons with L-type Ca(2+) channels incorporated in these regions produced plateau potentials resulting from PIC activation. Furthermore, we replicated the experimental observation that the somatic threshold at which PICs were activated was depolarized by tonic activation of inhibitory synapses and hyperpolarized by tonic activation of excitatory synapses. Models with L-type Ca(2+) channels distributed uniformly were unable to replicate the change in somatic threshold of PIC activation. Therefore we conclude that the set of L-type Ca(2+) channels mediating plateau potentials is restricted to discrete regions in the dendritic tree. Furthermore, this distribution leads to the compartmentalization of the dendritic tree of motoneurons into subunits whose sequential activation lead to the graded amplification of synaptic inputs.

Structure and function relationship in the abdominal stretch receptor organs of the crayfish

The structure/function relationship in the rapidly and slowly adapting stretch receptor organs of the crayfish (Astacus leptodactylus) was investigated using confocal microscopy and neuronal modeling methods. Both receptor muscles were single muscle fibers with structural properties closely related to the function of the receptors. Dendrites of the rapidly adapting neuron terminated in a common pile of nerve endings going in all directions. Dendrites of the slowly adapting neuron terminated in a characteristic T shape in multiple regions of the receptor muscle. The slowly adapting main dendrite, which was on average 2.1 times longer and 21% thinner than the rapidly adapting main dendrite, induced larger voltage attenuation. The somal surface area of the slowly adapting neuron was on average 51% larger than that of the rapidly adapting neuron. Variation in the neuronal geometry was greatest among the slowly adapting neurons. A computational model of a neuron pair demonstrated that the rapidly and the slowly adapting neurons attenuated the dendritic receptor potential like low-pass filters with cut-off frequencies at 100 and 20 Hz, respectively. Recurrent dendrites were observed mostly in the slowly adapting neurons. Voltage signals were calculated to be propagated 23% faster in the rapidly adapting axon, which is 51% thicker than the slowly adapting axon. The present findings support the idea that the morphology of the rapidly and the slowly adapting neurons evolved to optimally sense the dynamic and the static features of the mechanical stimulus, respectively.

Simulation of dendritic CaV1.3 channels in cat lumbar motoneurons: spatial distribution

We used computer simulations to study the dendritic spatial distribution of low voltage-activated L-type calcium (Ca(V)1.3 type) channels, which mediate hysteretic persistent inward current (PIC) in spinal motoneurons. This study was prompted by the growing experimental evidence of the functional interactions between synaptic inputs and active conductances over the motoneuron dendritic tree. A compartmental cable model of an adult cat alpha-motoneuron was developed in NEURON simulation environment constituting the detailed morphology of type-identified triceps surae alpha-motoneuron and realistic distribution of group Ia afferent-to-motoneuron contacts. Simulations of different distributions of Ca(V)1.3 channels were conducted and the resultant behavior was compared to experimental data. Our results suggest that Ca(V)1.3 channels do not uniformly cover the whole motoneuron dendritic tree. Instead, their distribution is similar to that of synaptic contacts. We found that Ca(V)1.3 channels are primarily localized to a wide intermediate band overlapping with the dendritic Ia-synaptic territory at dendritic distances of 300 to 850 microm (0.62 +/- 0.21lambda) from the soma in triceps surae alpha-motoneurons. These findings explain the functional interaction between synaptic inputs and the Ca(V)1.3 channels over the motoneuron dendritic tree.

Factors mediating powerful voltage attenuation along CA1 pyramidal neuron dendrites

We performed simultaneous patch-electrode recordings from the soma and apical dendrite of CA1 pyramidal neurons in hippocampal slices, in order to determine the degree of voltage attenuation along CA1 dendrites. Fifty per cent attenuation of steady-state somatic voltage changes occurred at a distance of 238 microm from the soma in control and 409 microm after blocking the hyperpolarization-activated (H) conductance. The morphology of three neurons was reconstructed and used to generate computer models, which were adjusted to fit the somatic and dendritic voltage responses. These models identify several factors contributing to the voltage attenuation along CA1 dendrites, including high axial cytoplasmic resistivity, low membrane resistivity, and large H conductance. In most cells the resting membrane conductances, including the H conductances, were larger in the dendrites than the soma. Simulations suggest that synaptic potentials attenuate enormously as they propagate from the dendrite to the soma, with greater than 100-fold attenuation for synapses on many small, distal dendrites. A prediction of this powerful EPSP attenuation is that distal synaptic inputs are likely only to be effective in the presence of conductance scaling, dendritic excitability, or both.

Rallian "equivalent" cylinders reconsidered: comparisons with literal compartments

In Rall's "equivalent" cylinder morphological-to-electrical transformation, neuronal arborizations are reduced to single unbranched core-conductors. The conventional assumption that such an "equivalent" reconstructs the electrical properties of the fibers it represents was tested directly; electrical properties and responses of "equivalent" cylinders were compared with those of their literal branch constituents for fibers with a single symmetrical bifurcation. The numerical solution methods were validated independently by their accurate reconstruction of the responses of an analog circuit configured with compartmental architecture to solve the cable equation for passive fibers with a symmetrical bifurcation. In passive fibers, "equivalent" cylinders misestimated the spatial distribution of voltage amplitudes and steady-state input resistance, partly due to the lack of axial current bifurcation. In active fibers with a single propagating action potential, the spatial distributions of point-to-point conduction velocity values (measured in meters/second) for a literal branch point differed significantly from those of their "equivalent" cylinders. "Equivalent" cylinders also underestimated the diameter-dependent delay in propagation through the branch point and branches, due to the larger "equivalent" diameter. Corrections to the "equivalent" cylinder did not reconcile differences between "equivalent" and literal models. However, "equivalent" and literal branch fibers had the same (a) steady-state resistance "looking into" an isolated symmetrical branch point and (b) geometry-independent point-to-point propagation velocity when measured in space constants per millisecond except within +/-1 space constant from the geometrical inhomogeneity. In summary, Rall's "equivalent" cylinders did not accurately reconstruct all passive or active electrophysiological properties and responses of their literal compartments. For the modeling of individual neurons, the requirement of single-branch resolution is discussed.

Developmental changes in spinal motoneuron dendrites in neonatal mice

We examined the age-dependent morphological changes of lumbar spinal motoneurons (MNs) in neonatal Swiss-Webster mice during the first 2 weeks of postnatal life. Neurons labeled by intracellular injection of biocytin in hemisected lumbosacral spinal cords in vitro were reconstructed from serial sections. Digitized data were compared for young (P3; postnatal days 2-4; n = 9) and older animals (P11; postnatal days 10-13; n = 8). As expected, measures of dendritic size (e.g., stem branch diameter, total surface area, maximum distance to tips, and lateral tree spread) were all significantly greater for P11 than for P3 mice. In contrast, the number of dendrites per MN and parameters related to tree topology (e.g., terminations per tree and maximum branch order), although slightly greater for P11 animals, were not significantly different between the two ages. Dendrite growth appeared to be proportional throughout the tree because the ratios between average terminal and internal branch lengths were similar for the two groups. Furthermore, this elongation was proportional to enlargement of overall spinal cord dimensions. A variety of other morphometric measures showed no significant difference between age groups. The relative constancy of MN dendritic topology up to P13 was surprising, given the striking maturation in motor function during this time period.

Depressed responses of facilitatory synapses

We show that when temporal summation takes place, depression of postsynaptic responses may ensue when the underlying synaptic conductance change is constant or even facilitatory. We term this phenomenon "apparent depression." Such apparent depression is most notable for slow synaptic conductance changes, for high frequency, and when the synapse is located at distal dendritic sites. We show that, when temporal summation ensues, the erroneous estimation of short-term synaptic plasticity arises partially from the conventional measurement of synaptic dynamics at postsynaptic potential peak time. This can be corrected when measuring overlapping synaptic responses at fixed intervals after stimulus time. Somatic voltage clamp also helps to partially correct for the apparent depression, but a good model of the neuron can do even better in providing a more accurate view of the underlying synaptic conductances.

Determination of the Location and Magnitude of Synaptic Conductance Changes in Spinal Motoneurons by Impedance Measurements

The relation between impedance change and the location and magnitude of a tonic synaptic conductance was examined in compartmental motoneuron models based on previously published data. The dependency of motoneuron impedance on system time constant (τ), electrotonic length (L), and dendritic-to-somatic conductance ratio (ρ) was examined, showing that the relation between impedance phase and ρ differed markedly between models with uniform and nonuniform membrane resistivity. Dendritic synaptic conductances decreased impedance magnitude at low frequencies; at higher frequencies, impedance magnitude increased. The frequency at which the change in impedance magnitude reversed from a decrease to an increase—the reversal frequency, Fr—was a good estimator of electrotonic synaptic location. A measure of the average normalized impedance change at frequencies less than Fr, cuΔZ, estimated relative synaptic conductance. Fr and cuΔZ provided useful estimates of synaptic location and conductance in models with nonuniform (step, sigmoidal) and uniform membrane resistivity. Fr also provided good estimates of spatial synaptic location on the equivalent cable in both step and sigmoidal models. Variability in relations between Fr, cuΔZ, and conductance location and magnitude between neurons was reduced by normalization with ρ and τ. The effects on Fr and cuΔZ of noise in experimental recordings, different synaptic distributions, and voltage-dependent conductances were also assessed. This study indicates that location and conductance of tonic dendritic conductances can be estimated from Fr, cuΔZ, and basic electrotonic motoneuron parameters with the exercise of suitable precautions.

Location and Magnitude of Conductance Changes Produced by Renshaw Recurrent Inhibition in Spinal Motoneurons

The mean location of Renshaw synapses on spinal motoneurons and their synaptic conductance were estimated from changes in impedance magnitude produced by sustained recurrent inhibition. Motoneuron impedance was determined by injecting quasi-white noise current into lumbosacral motoneurons of pentobarbital-anesthetized cats. Synaptic location and conductance were estimated by comparing observed impedance changes to simulation results obtained using standard motoneuron models and compartmental models fit to each impedance function. Estimated synaptic locations ranged from 0.10 to 0.41λ, with a mean of 0.19 or 0.24λ, depending on the estimation method. Average dendritic path length was 262 μm. Average synaptic conductance was 23 to 27 nS (range: 6.7 to 57.9 nS), corresponding to conductance changes of 78 to 88% of resting membrane conductance. Estimated accuracy was supported by consistency using different estimation methods, agreement with Fyffe's 1991 morphological data, and comparisons of observed and simulated recurrent IPSP amplitudes. Synaptic location, but not synaptic conductance, was correlated with rheobase, a measure of motoneuron excitability. Synaptic conductance did not depend on synaptic location. A regression analysis demonstrated that synaptic conductance and cell impedance were the principal factors determining recurrent IPSP amplitude. Simulations using the observed values and locations of Renshaw conductance demonstrate that recurrent inhibition can require as much as an additional 14 to 18% sustained excitatory synaptic conductance to depolarize motoneurons sufficiently to activate somatic or dendritic inward currents and recruit motoneurons or amplify excitatory synaptic currents.

Estimation of the Electrical Parameters of Spinal Motoneurons Using Impedance Measurements

Electrical parameters of spinal motoneurons were estimated by optimizing the parameters of motoneuron models to match experimentally determined impedance functions with those of the models. The model was described by soma area, somatic and dendritic membrane resistivities, and the diameter of an equivalent dendritic cable having a standard profile. The impedance functions of motoneurons and optimized models usually differed (rms error) by <2% of input resistance. Consistent estimates for most parameters were obtained from repeated impedance determinations in individual motoneurons; estimates of dendritic resistivity were most variable. The few cells that could not be fit well had reduced impedance phase lag consistent with dendritic penetrations. Most fits were improved by inclusion of a voltage-dependent conductance GV with time constant τV. A uniformly distributed GV with τV >5 ms provided a better fit for most cells. The magnitude of this conductance decreased with depolarization. Impedance functions of other cells were adequately fit by a passive model or by a model with a somatic GV and τV <5 ms. Most of these neurons (7/8) had resting potentials positive to −60 mV. The electrotonic parameters ρ, τ, and L, estimated from model parameters, were consistent with published distributions. Most motoneuron parameters obtained in somatic shunt and sigmoidal models were well correlated, and parameters were moderately affected by changes in dendritic profile. These results demonstrate the utility and limitations of impedance measurements for estimating motoneuron parameters and suggest that voltage-dependent conductances are a substantial component of resting electrical properties.

Median-based robust algorithms for tracing neurons from noisy confocal microscope images

This paper presents a method to exploit rank statistics to improve fully automatic tracing of neurons from noisy digital confocal microscope images. Previously proposed exploratory tracing (vectorization) algorithms work by recursively following the neuronal topology, guided by responses of multiple directional correlation kernels. These algorithms were found to fail when the data was of lower quality (noisier, less contrast, weak signal, or more discontinuous structures). This type of data is commonly encountered in the study of neuronal growth on microfabricated surfaces. We show that by partitioning the correlation kernels in the tracing algorithm into multiple subkernels, and using the median of their responses as the guiding criterion improves the tracing precision from 41% to 89% for low-quality data, with a 5% improvement in recall. Improved handling was observed for artifacts such as discontinuities and/or hollowness of structures. The new algorithms require slightly higher amounts of computation, but are still acceptably fast, typically consuming less than 2 seconds on a personal computer (Pentium III, 500 MHz, 128 MB). They produce labeling for all somas present in the field, and a graph-theoretic representation of all dendritic/axonal structures that can be edited. Topological and size measurements such as area, length, and tortuosity are derived readily. The efficiency, accuracy, and fully-automated nature of the proposed method makes it attractive for large-scale applications such as high-throughput assays in the pharmaceutical industry, and study of neuron growth on nano/micro-fabricated structures. A careful quantitative validation of the proposed algorithms is provided against manually derived tracing, using a performance measure that combines the precision and recall metrics.

Relationship between morphoelectrotonic properties of motoneuron dendrites and their trajectory

The distribution and geometry of the dendritic trees of spinal motoneurons obey several well-established rules. Some of these rules are based on systematic relationships between quantitative geometrical features (e.g., total dendritic length) and the three-dimensional trajectory followed by dendrites from their origin to their termination. Because dendritic geometry partially determines the transmission of current and voltage signals generated by synapses on the dendritic tree, our goal was to compare the efficacy of signal transmission by dendritic trajectories that followed different directions. To achieve this goal, we constructed detailed compartmental models of the dendritic trees of three intracellularly stained biventer cervicis/complexus (BCCM) motoneurons and calculated the electronic properties of 361 dendritic paths. Each trajectory was classified according to its orientation, e.g., rostral, rostral-dorsal-lateral. The attenuation of current and voltage signals en route to the soma was strongly related to trajectory orientation. Trajectories with similar attenuation factors formed functional subunits that were arranged in distinct domains within the ventral horn. Changes in R(m) or R(i) had little effect on which trajectories belonged to each functional subunit. However, differences in the efficacy of signal transmission between subunits increased during high network activity (mimicked by decreases in R(m)). The most efficient subunit delivered two times more current and four times more voltage to the soma than the least efficient subunit. These results indicate that the input-output properties of motoneurons depend on the direction of the path taken by dendrites from their origin at the cell body to their terminals.

Comparison of the morphological and electrotonic properties of Renshaw cells, Ia inhibitory interneurons, and motoneurons in the cat

The morphological and electrotonic properties of 4 motoneurons, 8 Ia inhibitory interneurons, and 4 Renshaw cells were compared. The morphological analysis, based on 3-D reconstructions of the cells, revealed that dendrites of motoneurons are longer and more extensively branched. Renshaw cells have dendrites that are shorter and simpler in structure. Dendrites of Ia inhibitory interneurons could be as long as those of motoneurons but the branching structure resembled that of Renshaw cells. Compartmental models were used to determine the electrotonic properties of the paths from each dendritic terminal to the soma. The attenuations of steady-state voltage changes in motoneurons were 3 and 7 times larger than in Ia inhibitory interneurons and Renshaw cells, respectively. The same relative order was observed for current attenuation and electrotonic length. The dendritic input resistances in Renshaw cells were 2 and 4 times larger than in Ia inhibitory interneurons and motoneurons, respectively. The difference in these electrotonic properties increased during higher synaptic activity as modeled by a decrease of Rm. The peak amplitudes of voltage transients at sites of brief, synaptic-like changes in conductance were highly dependent on cell class and were largest in Renshaw cells and smallest in motoneurons. In combination with class-specific differences in the attenuation of transient voltage signals, this led to large differences in the peak amplitudes of somatic voltage transients. Differences in the rise times and half-widths of the voltage transients were observed as well. Thus, based on passive properties, each cell class has a unique set of input/output properties.

Analytical and numerical construction of equivalent cables

The mathematical complexity experienced when applying cable theory to arbitrarily branched dendrites has lead to the development of a simple representation of any branched dendrite called the equivalent cable. The equivalent cable is an unbranched model of a dendrite and a one-to-one mapping of potentials and currents on the branched model to those on the unbranched model, and vice versa. The piecewise uniform cable, with a symmetrised tri-diagonal system matrix, is shown to represent the canonical form for an equivalent cable. Through a novel application of the Laplace transform it is demonstrated that an arbitrary branched model of a dendrite can be transformed to the canonical form of an equivalent cable. The characteristic properties of the equivalent cable are extracted from the matrix for the transformed branched model. The one-to-one mapping follows automatically from the construction of the equivalent cable. The equivalent cable is used to provide a new procedure for characterising the location of synaptic contacts on spinal interneurons.

A large pool of releasable vesicles in a cortical glutamatergic synapse

To probe exocytosis at a cortical glutamatergic synapse, we made capacitance measurements in whole-cell recorded hippocampal mossy fiber terminals. Evaluation of different methods by using a morphology-based equivalent electrical model revealed that quantitative capacitance measurements are possible in this presynaptic structure. Voltage pulses leading to presynaptic Ca2+ inflow evoked large capacitance signals that showed saturation with increasing pulse duration. The mean peak capacitance increase was 100 fF, corresponding to a pool of approximately 1,400 releasable vesicles. Thus hippocampal mossy fiber synapses have a vesicular "maxipool." Large pool size and rapid vesicle recycling may underlie the uniquely large extent of activity-dependent plasticity in this synapse.

Action potentials in basal and oblique dendrites of rat neocortical pyramidal neurons

Basal and oblique dendrites comprise ~2/3 of the total excitable membrane in the mammalian cerebral cortex, yet they have never been probed with glass electrodes, and therefore their electrical properties and overall impact on synaptic processing are unknown. In the present study, fast multi-site voltage-sensitive dye imaging combined with somatic recording was used to provide a detailed description of the membrane potential transients in basal and oblique dendrites of pyramidal neurons during single and trains of action potentials (APs). The optical method allowed simultaneous measurements from several dendrites in the visual field up to 200 microm from the soma, thus providing a unique report on how an AP invades the entire dendritic tree. In contrast to apical dendrites, basal and oblique branches: (1) impose very little amplitude and time course modulation on backpropagating APs; (2) are strongly invaded by the somatic spike even when somatic firing rates reach 40 Hz (activity-independent backpropagation); and (3) do not exhibit signs of a 'calcium shoulder' on the falling phase of the AP. A compartmental model incorporating AP peak latencies and half-widths obtained from the apical, oblique and basal dendrites indicates that the specific intracellular resistance (Ri) is less than 100 omicron cm. The combined experimental and modelling results also provide evidence that all synaptic locations along basal and oblique dendrites, situated within 200 microm from the soma, experience strong and near-simultaneous (latency < 1 ms) voltage transients during somatic firing. The cell body, axon hillock and basal dendritic compartments achieve unique synchronization during each AP. Therefore, with respect to a retrograde signal (AP), basal and proximal oblique dendrites should be considered as an integral part of the axo-somatic compartment.