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1.  Linking Macroscopic with Microscopic Neuroanatomy Using Synthetic Neuronal Populations 
PLoS Computational Biology  2014;10(10):e1003921.
Dendritic morphology has been shown to have a dramatic impact on neuronal function. However, population features such as the inherent variability in dendritic morphology between cells belonging to the same neuronal type are often overlooked when studying computation in neural networks. While detailed models for morphology and electrophysiology exist for many types of single neurons, the role of detailed single cell morphology in the population has not been studied quantitatively or computationally. Here we use the structural context of the neural tissue in which dendritic trees exist to drive their generation in silico. We synthesize the entire population of dentate gyrus granule cells, the most numerous cell type in the hippocampus, by growing their dendritic trees within their characteristic dendritic fields bounded by the realistic structural context of (1) the granule cell layer that contains all somata and (2) the molecular layer that contains the dendritic forest. This process enables branching statistics to be linked to larger scale neuroanatomical features. We find large differences in dendritic total length and individual path length measures as a function of location in the dentate gyrus and of somatic depth in the granule cell layer. We also predict the number of unique granule cell dendrites invading a given volume in the molecular layer. This work enables the complete population-level study of morphological properties and provides a framework to develop complex and realistic neural network models.
Author Summary
Computational models of neurons and neural networks provide a valuable avenue to test our understanding of brain regions and to make predictions to guide future experimentation. Each neuron has a unique dendritic tree, features of which can vary depending on the location of the neuron within the particular brain region. In this study, we generated a complete population of dendritic trees for the most numerous type of neuron in the hippocampus, the dentate gyrus granule cell, using a realistic three-dimensional structural context to drive the generation process. Morphological properties can now be studied at the level of complete neuronal populations, and this work provides a foundation to build upon in the construction of large-scale, data-driven neuroanatomical and network models.
doi:10.1371/journal.pcbi.1003921
PMCID: PMC4207466  PMID: 25340814
2.  Preserving Neural Function under Extreme Scaling 
PLoS ONE  2013;8(8):e71540.
Important brain functions need to be conserved throughout organisms of extremely varying sizes. Here we study the scaling properties of an essential component of computation in the brain: the single neuron. We compare morphology and signal propagation of a uniquely identifiable interneuron, the HS cell, in the blowfly (Calliphora) with its exact counterpart in the fruit fly (Drosophila) which is about four times smaller in each dimension. Anatomical features of the HS cell scale isometrically and minimise wiring costs but, by themselves, do not scale to preserve the electrotonic behaviour. However, the membrane properties are set to conserve dendritic as well as axonal delays and attenuation as well as dendritic integration of visual information. In conclusion, the electrotonic structure of a neuron, the HS cell in this case, is surprisingly stable over a wide range of morphological scales.
doi:10.1371/journal.pone.0071540
PMCID: PMC3747245  PMID: 23977069
4.  The Dendritic Density Field of a Cortical Pyramidal Cell 
Much is known about the computation in individual neurons in the cortical column. Also, the selective connectivity between many cortical neuron types has been studied in great detail. However, due to the complexity of this microcircuitry its functional role within the cortical column remains a mystery. Some of the wiring behavior between neurons can be interpreted directly from their particular dendritic and axonal shapes. Here, I describe the dendritic density field (DDF) as one key element that remains to be better understood. I sketch an approach to relate DDFs in general to their underlying potential connectivity schemes. As an example, I show how the characteristic shape of a cortical pyramidal cell appears as a direct consequence of connecting inputs arranged in two separate parallel layers.
doi:10.3389/fnana.2012.00002
PMCID: PMC3269636  PMID: 22347169
modeling; dendrite; axon; pyramidal cell; Ramón y Cajal
5.  One Rule to Grow Them All: A General Theory of Neuronal Branching and Its Practical Application 
PLoS Computational Biology  2010;6(8):e1000877.
Understanding the principles governing axonal and dendritic branching is essential for unravelling the functionality of single neurons and the way in which they connect. Nevertheless, no formalism has yet been described which can capture the general features of neuronal branching. Here we propose such a formalism, which is derived from the expression of dendritic arborizations as locally optimized graphs. Inspired by Ramón y Cajal's laws of conservation of cytoplasm and conduction time in neural circuitry, we show that this graphical representation can be used to optimize these variables. This approach allows us to generate synthetic branching geometries which replicate morphological features of any tested neuron. The essential structure of a neuronal tree is thereby captured by the density profile of its spanning field and by a single parameter, a balancing factor weighing the costs for material and conduction time. This balancing factor determines a neuron's electrotonic compartmentalization. Additions to this rule, when required in the construction process, can be directly attributed to developmental processes or a neuron's computational role within its neural circuit. The simulations presented here are implemented in an open-source software package, the “TREES toolbox,” which provides a general set of tools for analyzing, manipulating, and generating dendritic structure, including a tool to create synthetic members of any particular cell group and an approach for a model-based supervised automatic morphological reconstruction from fluorescent image stacks. These approaches provide new insights into the constraints governing dendritic architectures. They also provide a novel framework for modelling and analyzing neuronal branching structures and for constructing realistic synthetic neural networks.
Author Summary
More than a century has passed since Ramón y Cajal presented a set of fundamental biological laws of neuronal branching. He described how the shape of the core elements of the neural circuitry – axons and dendrites – are constrained by physical parameters such as space, cytoplasmic volume, and conduction time. The existence of these laws enabled him to organize his histological observations, to formulate the neuron doctrine, and to infer directionality in signal flow in the nervous system. We show that Cajal's principles can be used computationally to generate synthetic neural circuits. These principles rigorously constrain the shape of real neuronal structures, providing direct validation of his theories. At the same time, this strategy provides us with a powerful set of tools for generating synthetic neurons, as well as a model-based approach for automated reconstructions of neuronal trees from confocal image stacks.
doi:10.1371/journal.pcbi.1000877
PMCID: PMC2916857  PMID: 20700495
6.  Traveling waves in developing cerebellar cortex mediated by asymmetrical Purkinje cell connectivity 
Nature neuroscience  2009;12(4):463-473.
Correlated network activity plays a critical role in the development of many neural circuits. Purkinje cells are among the first neurons to populate the cerebellar cortex, where they sprout exuberant axon collaterals. Here we use multiple patch-clamp recordings targeted with two-photon microscopy to characterize monosynaptic connections between Purkinje cells of juvenile mice. We show that Purkinje cell axon collaterals project asymmetrically in the sagittal plane, directed away from the lobule apex. Based on our anatomical and physiological characterization of this connection, we construct a network model that robustly generates waves of activity traveling along chains of connected Purkinje cells. Consistent with the model, we observe traveling waves of activity in Purkinje cells in sagittal slices from young mice that require GABAA receptor-mediated transmission and intact Purkinje cell axon collaterals. These traveling waves are absent in adult animals, suggesting they play a developmental role in wiring the cerebellar cortical microcircuit.
doi:10.1038/nn.2285
PMCID: PMC2912499  PMID: 19287389
7.  A New Approach for Determining Phase Response Curves Reveals that Purkinje Cells Can Act as Perfect Integrators 
PLoS Computational Biology  2010;6(4):e1000768.
Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.
Author Summary
By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron (and thus obtaining the so-called phase response curve), it should be possible to predict a full spike train in response to more complex stimulation patterns. When we applied this traditional protocol to obtain phase response curves in cerebellar Purkinje cells in the presence of noise, we observed a triangular region devoid of data points near the end of the spiking cycle. This “Bermuda Triangle” revealed a flaw in the classical method for constructing phase response curves. We developed a new approach to eliminate this flaw and used it to construct phase response curves of Purkinje cells over a range of spiking rates. Surprisingly, at low firing rates, phase changes were independent of the phase of the injected current pulses, implying that the Purkinje cell is a perfect integrator under these conditions. This mechanism has not yet been described in other cell types and may be crucial for the information processing capabilities of these neurons.
doi:10.1371/journal.pcbi.1000768
PMCID: PMC2861707  PMID: 20442875
8.  The Morphological Identity of Insect Dendrites 
PLoS Computational Biology  2008;4(12):e1000251.
Dendrite morphology, a neuron's anatomical fingerprint, is a neuroscientist's asset in unveiling organizational principles in the brain. However, the genetic program encoding the morphological identity of a single dendrite remains a mystery. In order to obtain a formal understanding of dendritic branching, we studied distributions of morphological parameters in a group of four individually identifiable neurons of the fly visual system. We found that parameters relating to the branching topology were similar throughout all cells. Only parameters relating to the area covered by the dendrite were cell type specific. With these areas, artificial dendrites were grown based on optimization principles minimizing the amount of wiring and maximizing synaptic democracy. Although the same branching rule was used for all cells, this yielded dendritic structures virtually indistinguishable from their real counterparts. From these principles we derived a fully-automated model-based neuron reconstruction procedure validating the artificial branching rule. In conclusion, we suggest that the genetic program implementing neuronal branching could be constant in all cells whereas the one responsible for the dendrite spanning field should be cell specific.
Author Summary
Neural computation has been shown to be heavily dependent not only on the connectivity of single neurons but also on their specific dendritic shape—often used as a key feature for their classification. Still, very little is known about the constraints determining a neuron's morphological identity. In particular, one would like to understand what cells with the same or similar function share anatomically, what renders them different from others, and whether one can formalize this difference objectively. A large number of approaches have been proposed, trying to put dendritic morphology in a parametric frame. A central problem lies in the wide variety and variability of dendritic branching and function even within one narrow cell class. We addressed this problem by investigating functionally and anatomically highly conserved neurons in the fly brain, where each neuron can easily be individually identified in different animals. Our analysis shows that the pattern of dendritic branching is not unique in any particular cell, only the features of the area that the dendrites cover allow a clear classification. This leads to the conclusion that all fly dendrites share the same growth program but a neuron's dendritic field shape, its “anatomical receptive field”, is key to its specific identity.
doi:10.1371/journal.pcbi.1000251
PMCID: PMC2588660  PMID: 19112481
9.  Optimization principles of dendritic structure 
Background
Dendrites are the most conspicuous feature of neurons. However, the principles determining their structure are poorly understood. By employing cable theory and, for the first time, graph theory, we describe dendritic anatomy solely on the basis of optimizing synaptic efficacy with minimal resources.
Results
We show that dendritic branching topology can be well described by minimizing the path length from the neuron's dendritic root to each of its synaptic inputs while constraining the total length of wiring. Tapering of diameter toward the dendrite tip – a feature of many neurons – optimizes charge transfer from all dendritic synapses to the dendritic root while housekeeping the amount of dendrite volume. As an example, we show how dendrites of fly neurons can be closely reconstructed based on these two principles alone.
doi:10.1186/1742-4682-4-21
PMCID: PMC1924501  PMID: 17559645

Results 1-9 (9)