Propagation of action potentials along complex axonal trees. Model and implementation

Electrophysiology using coaxial atom probe array: live imaging reveals hidden circuits of a hippocampal neural network

Since the 1960s, it is held that when a neuron fires, a nerve spike passes only through the selective branches, the calculated choice is a key to learning by rewiring. It is argued by chemically estimating the membrane's ion channel density that different axonal branches get active to pass the spike-branches blink at firing at different time domains. Here, using a new time-lapse dielectric imaging, we visualize the classic branch selection process; thenceforth, hidden circuits operating at different time domains become visible. The fractal grid of coaxial probes captures wireless snapshots of material's vibration at various depths below the membrane by setting a suitable frequency. Thus far, branch selection observed emitted energy or particle but never the emitters, what they do. As each dielectric material transmits and reflects signals of different frequencies, we image live how filaments search for many branch-made circuits, choose a unique pathway 10³ times faster than a single nerve spike. It reveals that neural branches and circuit visible in a microscope are not absolute, there coexist many circuits each operating in different dime domains, operating at a time.NEW & NOTEWORTHY Using dielectric resonance scanner, we show electromagnetic field connections between physically separated neurons. Electromagnetic field creates field lines that pass through gap junctions, connect Axon initial segment with the dendrites through Soma, and connect axonal or dendritic branches even if there is no synaptic junction. Consequently, many distinct loops connecting various branches form coexisting circuits. Our discovery suggests that physically appearing neural circuit is a fractional view of many simultaneously operating circuits in different time domains in a neural network.

Branching morphology determines signal propagation dynamics in neurons

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

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.

Versatile morphometric analysis and visualization of the three-dimensional structure of neurons

The computational properties of a neuron are intimately related to its morphology. However, unlike electrophysiological properties, it is not straightforward to collapse the complexity of the three-dimensional (3D) structure into a small set of measurements accurately describing the structural properties. This strong limitation leads to the fact that many studies involving morphology related questions often rely solely on empirical analysis and qualitative description. It is possible however to acquire hierarchical lists of positions and diameters of points describing the spatial structure of the neuron. While there is a number of both commercially and freely available solutions to import and analyze this data, few are extendable in the sense of providing the possibility to define novel morphometric measurements in an easy to use programming environment. Fewer are capable of performing morphometric analysis where the output is defined over the topology of the neuron, which naturally requires powerful visualization tools. The computer application presented here, Py3DN, is an open-source solution providing novel tools to analyze and visualize 3D data collected with the widely used Neurolucida (MBF) system. It allows the construction of mathematical representations of neuronal topology, detailed visualization and the possibility to define non-standard morphometric analysis on the neuronal structures. Above all, it provides a flexible and extendable environment where new types of analyses can be easily set up allowing a high degree of freedom to formulate and test new hypotheses. The application was developed in Python and uses Blender (open-source software) to produce detailed 3D data representations.

Identifying critical regions for spike propagation in axon segments

Morphological reconstructions of axon segments reveal the abundance of geometrical ultrastructures that can dramatically affect the propagation of Action Potentials (AP). Moreover, deformations and swellings in axons resulting from brain traumas are associated to many neural dysfunctions and disorders. Our aim is to develop a computational framework to distinguish between geometrical enlargements that lead to minor changes in propagation from those that result in critical phenomenon such as reflection or blockage of the original traveling spike. We use a few geometrical parameters to model a prototypical shaft enlargement and explore the parameter space characterizing all possible propagation regimes and dynamics in an unmylienated AP model. Contrary to earlier notions that large diameter increases mostly lead to blocking, we demonstrate transmission is stable provided the geometrical changes occur in a slow manner. Our method also identifies a narrow range of parameters leading to a reflection regime. The distinction between these three regimes can be evaluated by a simple function of the geometrical parameters inferred through numerical simulations. We suggest that evaluating this function along axon segments can detect regions most susceptible to (i) transmission failure due to perturbations, (ii) structural plasticity, (iii) critical swellings caused by brain traumas and/or (iv) neurological disorders associated with the break down of spike train propagation.

Accelerating compartmental modeling on a graphical processing unit

Compartmental modeling is a widely used tool in neurophysiology but the detail and scope of such models is frequently limited by lack of computational resources. Here we implement compartmental modeling on low cost Graphical Processing Units (GPUs), which significantly increases simulation speed compared to NEURON. Testing two methods for solving the current diffusion equation system revealed which method is more useful for specific neuron morphologies. Regions of applicability were investigated using a range of simulations from a single membrane potential trace simulated in a simple fork morphology to multiple traces on multiple realistic cells. A runtime peak 150-fold faster than the CPU was achieved. This application can be used for statistical analysis and data fitting optimizations of compartmental models and may be used for simultaneously simulating large populations of neurons. Since GPUs are forging ahead and proving to be more cost-effective than CPUs, this may significantly decrease the cost of computation power and open new computational possibilities for laboratories with limited budgets.

Axon Physiology

Axons are generally considered as reliable transmission cables in which stable propagation occurs once an action potential is generated. Axon dysfunction occupies a central position in many inherited and acquired neurological disorders that affect both peripheral and central neurons. Recent findings suggest that the functional and computational repertoire of the axon is much richer than traditionally thought. Beyond classical axonal propagation, intrinsic voltage-gated ionic currents together with the geometrical properties of the axon determine several complex operations that not only control signal processing in brain circuits but also neuronal timing and synaptic efficacy. Recent evidence for the implication of these forms of axonal computation in the short-term dynamics of neuronal communication is discussed. Finally, we review how neuronal activity regulates both axon morphology and axonal function on a long-term time scale during development and adulthood.

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.

Deep brain stimulation mechanisms: beyond the concept of local functional inhibition

Deep brain electrical stimulation has become a recognized therapy in the treatment of a variety of motor disorders and has potentially promising applications in a wide range of neurological diseases including neuropsychiatry. Behavioural observation that electrical high-frequency stimulation of a given brain area induces an effect similar to a lesion suggested a mechanism of functional inhibition. In vitro and in vivo experiments as well as per operative recordings in patients have revealed a variety of effects involving local changes of neuronal excitability as well as widespread effects throughout the connected network resulting from activation of axons, including antidromic activation. Here we review current data regarding the local and network activity changes induced by high-frequency stimulation of the subthalamic nucleus and discuss this in the context of motor restoration in Parkinson's disease. Stressing the important functional consequences of axonal activation in deep brain stimulation mechanisms, we highlight the importance of developing anatomical knowledge concerning the fibre connections of the putative therapeutic targets.

Trends in programming languages for neuroscience simulations

Neuroscience simulators allow scientists to express models in terms of biological concepts, without having to concern themselves with low-level computational details of their implementation. The expressiveness, power and ease-of-use of the simulator interface is critical in efficiently and accurately translating ideas into a working simulation. We review long-term trends in the development of programmable simulator interfaces, and examine the benefits of moving from proprietary, domain-specific languages to modern dynamic general-purpose languages, in particular Python, which provide neuroscientists with an interactive and expressive simulation development environment and easy access to state-of-the-art general-purpose tools for scientific computing.

Extending Cable Theory to Heterogeneous Dendrites

Dendrites may exhibit many types of electrical and morphological heterogeneities at the scale of a few micrometers. Models of neurons, even so-called detailed models, rarely consider such heterogeneities. Small-scale fluctuations in the membrane conductances and the diameter of dendrites are generally disregarded and spines merely incorporated into the dendritic shaft. Using the two-scales method known as homogenization, we establish explicit expressions for the small-scale fluctuations of the membrane voltage, and we derive the cable equation satisfied by the voltage when these fluctuations are averaged out. This allows us to rigorously establish under what conditions a heterogeneous dendrite can be approximated by a homogeneous cable. We consider different distributions of synapses, orderly or random, on a passive dendrite, and we investigate when replacing excitatory and inhibitory synaptic conductances by their local averages leads to a small error in the voltage. This indicates in which regimes the approximations made in compartmental models are justified. We extend these results to active membranes endowed with voltage-dependent conductances or NMDA receptors. Then we examine under which conditions a spiny dendrite behaves as a smooth dendrite. We discover a new regime where this holds true, namely, when the conductance of the spine neck is small compared to the conductance of the synapses impinging on the spine head. Spines can then be taken into account by an effective excitatory current, the capacitance of the dendrite remaining unchanged. In this regime, the synaptic current transmitted from a spine to the dendritic shaft is strongly attenuated by the weak coupling conductance, but the total current they deliver can be quite substantial. These results suggest that pedunculated spines and stubby spines might play complementary roles in synaptic integration. Finally, we analyze how varicosities affect voltage diffusion in dendrites and discuss their impact on the spatiotemporal integration of synaptic input.

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.

The interaction between membrane kinetics and membrane geometry in the transmission of action potentials in non-uniform excitable fibres: a finite element approach

By solving the partial differential equations for an axonal segment using a finite element method, the interaction between membrane kinetics and axonal inhomogeneities, measured by their influence on propagated action potentials and stochastic spike trains, is investigated for Morris-Lecar and Hodgkin-Huxley membrane models. To facilitate comparisons of both kinetic models, parameter values are matched to give approximately the same speed for propagated action potentials. In all cases examined, the Morris-Lecar membrane model is more sensitive to geometric inhomogeneities than the comparable Hodgkin-Huxley membrane model. This difference in sensitivity can, in part, be attributed to significant differences in the membrane current supplied by each kinetic model ahead of the action potential. Also, the Morris-Lecar membrane model did not generate reflected action potentials whereas these were observed over a narrow range of geometric parameters for the comparable Hodgkin-Huxley membrane model. Simulations using stochastic spike train input showed that the presence of a sharp flare could significantly modify the statistical characteristics of the spike train output. The behaviour of action potentials governed by Morris-Lecar kinetics were more sensitive to changes in axonal geometry than those generated by comparable Hodgkin-Huxley kinetics. As a consequence of the fine balance between membrane kinetics and axon geometry, local changes in membrane properties, such as those caused by synaptic activity, can be expected to have a strong influence on the behaviour of stochastic spike trains at regions of changing axonal geometry.

Impact of active dendrites and structural plasticity on the memory capacity of neural tissue

We consider the combined effects of active dendrites and structural plasticity on the storage capacity of neural tissue. We compare capacity for two different modes of dendritic integration: (1) linear, where synaptic inputs are summed across the entire dendritic arbor, and (2) nonlinear, where each dendritic compartment functions as a separately thresholded neuron-like summing unit. We calculate much larger storage capacities for cells with nonlinear subunits and show that this capacity is accessible to a structural learning rule that combines random synapse formation with activity-dependent stabilization/elimination. In a departure from the common view that memories are encoded in the overall connection strengths between neurons, our results suggest that long-term information storage in neural tissue could reside primarily in the selective addressing of synaptic contacts onto dendritic subunits.

Axons as computing devices: basic insights gained from models

Detailed models of single neurons are typically focused on the dendritic tree and ignore the axonal tree, assuming that the axon is a simple transmission line. In the last 40 years, however, several theoretical and experimental studies have suggested that axons could implement information processing tasks by exploiting: 1) the time delay in action potential (AP) propagation along the axon; 2) the differential filtering of APs into the axonal subtrees; and 3) their activity-dependent excitability. Models for axonal trees have attempted to examine the feasibility of these ideas. However, because the physiological and anatomical data on axons are seriously limited, realistic models of axons have not been developed. The present paper summarizes the main insights that were gained from simplified models of axons; it also highlights the stochastic nature of axons, a topic that was largely neglected in classical models of axons. The advance of new experimental techniques makes it now possible to pay a very close experimental visit to axons. Theoretical tools and fast computers enable to go beyond the simplified models and to construct realistic models of axons. When tightly linked, experiments and theory will help to unravel how axons share the information processing tasks that single neurons implement.

Visual inter-hemispheric processing: constraints and potentialities set by axonal morphology

The largest bundle of axonal fibers in the entire mammalian brain, namely the corpus callosum, is the pathway through which almost half a billion neurons scattered over all neocortical areas can exert an influence on their contralateral targets. These fibers are thus crucial participants in the numerous cortical functions requiring collaborative processing of information across the hemispheres. One of such operations is to combine the two partial cortical maps of the visual field into a single, coherent representation. This paper reviews recent anatomical, computational and electrophysiological studies on callosal connectivity in the cat visual system. We analyzed the morphology of individual callosal axons linking primary visual cortices using three-dimensional light-microscopic techniques. While only a minority of callosal axons seem to perform a strict 'point-to-point' mapping between retinotopically corresponding sites in both hemispheres, many others have widespread arbors and terminate into a handful of distant, radially oriented tufts. Therefore, the firing of a single callosal neuron might influence several cortical columns within the opposite hemisphere. Computer simulation was then applied to investigate how the intricate geometry of these axons might shape the spatio-temporal distribution of trans-callosal inputs. Based on the linear relation between diameter and conduction velocity of myelinated fibers, the theoretical delays required for a single action potential to reach all presynaptic boutons of a given arbor were derived from the caliber, g-ratio and length of successive axonal segments. This analysis suggests that the architecture of callosal axons is, in principle, suitable to promote the synchronous activation of multiple targets located across distant columns in the opposite hemisphere. Finally, electrophysiological recordings performed in several laboratories have shown the existence of stimulus-dependent synchronization of visual responses across the two hemispheres. Possible implications of these findings are discussed in the context of temporal tagging of neuronal assemblies.

Maxsim, software for the analysis of multiple axonal arbors and their simulated activation

In order to analyze the structural organization of complex axonal arbors reconstructed from histological serial sections, and to investigate the functional implications of their geometrical properties, we developed software providing the following facilities: (1) direct importation of data files generated by a commercially available 3-D light-microscopic reconstruction system, including routine procedures for identification and correction of data acquisition errors; (2) real-time 3-D rotations of the arbors in the stack of serial sections; (3) multiple interactive display modes; (4) possibility of modifying diameter and/or connectivity of different branches; (5) simulation of the invasion of the arbor by a single action potential initiated at any chosen point, and visualization of spatio-temporal profiles of activation; (6) extraction of quantitative data converted to standard file formats compatible with available mathematical software. All these tools can be applied to single or multiple axons, individually or simultaneously. The software, called Maxsim, is a highly flexible C-written program running on graphical workstations using the UNIX operating system and X-Window environment.

Backpropagation of action potentials generated at ectopic axonal loci: hypothesis that axon terminals integrate local environmental signals

This review deals with the fascinating complexity of presynaptic axon terminals that are characterized by a high degree of functional distinctiveness. In vertebrate and invertebrate neurons, all-or-none APs can take off not only from the axon hillock, but also from ectopic axonal loci including terminals. Invertebrate neurons display EAPs, for instance alternating with somatic APs, during survival functions. In vertebrate, EAPs have been recorded in the peripheral and central nervous systems in time relationship with physiological or pathological neuronal activities. In motor or sensory axon, EAP generation may be the cause of motor dysfunctioning or sensory perceptions and pain respectively. Locomotion is associated with rhythmic depolarizations of the presynaptic axonal membrane of primary afferents, which are ridden by robust EAP bursts. In central axons lying within an epileptic tissue EAP discharges, coinciding with paroxysmal ECoG waves, get longer as somatic discharges get shorter during seizure progression. Once invaded by an orthodromic burst, an ectopic axonal locus can display an EAP after discharge. Such loci can also fire during hyperpolarization or the postinhibitory excitatory period of the parent somata, but not during their tonic excitation. Neurons are thus endowed with electrophysiological intrinsic properties making possible the alternate discharges of somatic APs and EAPs. In invertebrate and vertebrate neurons, ectopic axonal loci fire while the parent somata stop firing, further suggesting that axon terminal networks are unique and individual functional entities. The functional importance of EAPs in the nervous systems is, however, not yet well understood. Ectopically generated axonal APs propagate backwards and forwards along the axon, thus acting as a retrograde and anterograde signal. In invertebrate neurons, somatically and ectopically generated APs cannot have the same effect on the postsynaptic membrane. As suggested by studies related to the dorsal root reflex, EAPs may not only be implied in the presynaptic modulation of transmitter release but also contribute significantly during their backpropagation to a powerful control (collision process) of incoming volleys. From experimental data related to epileptiform activities it is proposed that EAPs, once orthodromically conducted, might potentiate synapses, initiate, spread or maintain epileptic cellular processes. For instance, paroxysmal discharges of EAPs would exert, like a booster-driver, a powerful synchronizing synaptic drive upon a large number of excitatory and inhibitory postsynaptic neurons. We have proposed that, once backpropagated, EAPs are likewise capable of initiating (and anticipating) threshold and low-threshold somatodendritic depolarizations. Interestingly, an antidromic EAP can modulate the excitability of the parent soma.(ABSTRACT TRUNCATED AT 400 WORDS)

Silicon neuron simulation with SPICE: tool for neurobiology and neural networks

The paper deals with computer simulations of 'silicon neurons', which are assemblies of CMOS circuits that generate the equivalents of the ionic currents and of the action potentials of real (biological) neurons. The circuit simulation program SPICE is used to simulate the generation of action potentials by a silicon neuron. Moreover, the equivalent circuits of silicon synapses are described and the behaviours of simple two- and three-neuron networks are analysed. Implications for the areas of neurobiology and formal neural networks are briefly considered.

Action potential propagation and propagation block by GABA in rat posterior pituitary nerve terminals

1. A theoretical model was developed to investigate action potential propagation in posterior pituitary nerve terminals. This model was then used to evaluate the efficacy of depolarizing and shunting GABA responses on action potential propagation. 2. Experimental data obtained from the posterior pituitary with patch clamp techniques were used to derive empirical expressions for the voltage and time dependence of the nerve terminal Na+ and K+ channels. The essential structure employed here was based on anatomical and cable data from the posterior pituitary, and consisted of a long cylindrical axon (diameter, 0.5 mm) with a large spherical swelling (diameter, 4-21 mm) in the middle. 3. In the absence of an inhibitory conductance, simulated action potentials propagated with high fidelity through the nerve terminal. Swellings could block propagation, but only when sizes exceeded those observed in the posterior pituitary. Adding axonal branches reduced the critical size only slightly. These results suggested that action potentials invade the entire posterior pituitary nerve terminal in the absence of inhibition or depression. 4. The addition of inhibitory conductance to a swelling caused simulated action potentials to fail at the swelling. Depolarizing inhibitory conductances were 1.6 times more effective than shunting inhibitory conductances in blocking propagation. 5. Inhibitory conductances within the range of experimentally observed magnitudes and localized to swellings in the observed range of sizes were too weak to block simulated action potentials. However, twofold enhancement of GABA responses by neurosteroid resulted in currents strong enough to block propagation in realistic swelling sizes. 6. GABA could block simulated propagation without neurosteroid enhancement provided that GABA was present throughout a region in the order of a few hundred micrometres. For this widespread inhibition depolarizing conductance was 2.2 times more effective than shunting conductance. 7. These results imply two modes of propagation block, one resulting from highly localized release of inhibitory transmitter under conditions potentiating GABA responses, and the other resulting from widespread release of GABA in the absence of receptor potentiation. 8. The Na+ channels of the posterior pituitary nerve terminal have a unique voltage dependence that allows small depolarizations to inactivate without causing activation. The voltage dependence of this Na+ channel may serve as a specialized adaptation that facilitates in allowing small depolarizing conductances to block action potential propagation.

Defined neuronal arborizations by guided outgrowth of leech neurons in culture

Identified neurons of Hirudo medicinalis were cultivated on a protein extract of the extracellular matrix (ECM) of the leech. Microscopic patterns of active ECM protein were prepared by UV photolithography using aluminium masks. The shape of the patterns was visualized by a colour pattern formed in a dye-polymer substrate. The neurons were explanted on the root of branched ECM patterns. The patterns guided the outgrowth of neurites along linear lanes and they induced a bifurcation of the neurites under certain conditions. Neurons with a reproducible, regular shape of arborization were obtained within 1-2 days.

On trees as equivalent cables

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

Exclusive-OR function of single arborized neuron

We designed four arborized neurons which are able to evaluate the exclusive-or (XOR) function from two inputs. The input neurons form exclusively excitatory synapses on a dendritic tree which is a patchwork of "passive" (ohmic) and "active" cable segments. The active segments are described by the Hodgkin-Huxley model. The dynamics of the neurons and their output are obtained by numerical integration of the cable equation. In neurons 1 and 2 the XOR function is based on the annihilation of colliding action potentials. In neuron No. 3 the design takes advantage of the refractory period of action potentials. In neuron No. 4 voltage inversion is used as it occurs for inactivated sodium conductance in the Hodgkin-Huxley model. In all cases the XOR function depends critically on an appropriate timing of the input signals and on delays of the voltage transients in different branches of the dendrite.

Single neurone models: oversimple, complex and reduced

The single neurone stands in the midst of a controversy among modelers. Some believe that its details are functionally superfluous when the neurone operates in a large network, and very simple models can be used to represent the input/output characteristics of neurones. Others claim that the unique morphology and electrical properties of neurones do play an important role. Complicated models of neurones are developed to reveal how the various kinds of 'neurone-ware' (dendrites, spines, axons, membrane channels and synapses) create a computationally powerful unit. Various models are discussed, including new carefully reduced models that retain essential features of more complex models. Such intermediate models will play a central role in our efforts to understand information processing in large neuronal networks.