Dendrites as electrical systems with distributed parameters differ from electrical systems with lumped parameters in an important aspect: any dendritic site can be considered as either input or output or both. In that sense, we deal with an electrical system such that the inputs and outputs are distributed in space, over the whole dendritic space.

The voltage is a standard and direct indicator of electric states. Similarity or dissimilarity of voltages reflects similarity or dissimilarity of electric states at different locations in space. The sign and magnitude of the voltage, that is the difference in the transmembrane potential between the sites, determines what electrically communicates with what and the intensity of the sent/received signals. The sites communicate by sending/receiving charges, i.e. by currents. The current flows in the direction of the voltage drop. Hence, considering the path profiles of the transmembrane voltage, one can see from where and to where the current flows in the given domain. Given the core resistance of the dendritic cable, the current between neighbouring sites is proportional to the voltage difference. Since one cannot ‘observe’ the path map of resistances, the path map of the voltages is informative, however not exhaustively. For an exhaustive characterization of the electric states and of the electric communication between sites over the dendritic space, a complementary map of the membrane currents is required (see Chapter 6).

Dendritic cables in the steady state

In the steady state, there are no temporal changes in voltage, voltage-sensitive conductance and current. The time derivatives of all values are zero. Therefore the capacitive current is zero.

After the single dendritic path analyzed in the preceding chapter, an elementary bifurcation introduces the simplest case of a second discrete dimension for navigating over the dendrites. A set of elementary bifurcations forms the so-called binary tree, which is most typical for dendritic arborizations of neurons. Obviously, the bifurcation, also named binary branching or dichotomic branching is topologically symmetrical but most often metrically asymmetrical, due to differences in lengths and/or diameters of the sister branches.

In this chapter, we study the proximal-to-distal electrical relationship in two sister paths by comparing sites situated at the same path distance from their common origin but on different paths. The path length of the shorter branch determines the path distance extent of the domain in which we can compare equidistant sites, since it is obvious that the most distal sites on the longer branch do not have equidistant counterparts on the shorter one! Remaining at the same path distance from the origin in the first continuous dimension and ‘jumping’ from one branch to the other in the second discrete dimension, we can compare electrical states of equidistant sites (Figure 8.1). Such a structure is most convenient for studying the impact of the metrical asymmetry of branching on the electrical structure of the paths. The difference in the electrical states reveals the critical impact of metrical asymmetry.

Theory for different configurations

Consider a simple example of metrical asymmetry: two branches of the same diameter d and different lengths, shorter l and longer l′ = l + Δl (Δl > 0) arising from the common origin.

Brief historical background

The lack of methods of fixation and the lack of staining techniques seriously handicapped the earlier workers in their observations of nervous tissues during the nineteenth century. The story changed enormously during the last decades of the twentieth century when Golgi found that osmic dichromate fixation followed by silver impregnation gave pictures that could not be achieved by any other method. The enthusiastic description by Ramón y Cajal of the beauty of the successful Golgi preparations depicts for the first time the richness of these histological images:

The gifted hand of Ramón y Cajal produced the first seminal book on nervous systems proposing prophetic views on the functions of dendritic arborizations (Ramón y Cajal, 1911). But the lengthy and elaborate descriptions of Golgi preparations discouraged the students who were frustrated with this type of investigation and, influenced by the new results of local electrical stimulation, tended to devote themselves to the new promising approach of electrophysiology.

A relevant mathematical tool to describe the bioelectricity in the neuronal dendrites is cable theory, which is based on application of the so-called cable equation to the core conductor model (Kernleitermodel) of the dendritic structure conducting currents and voltages. Exhaustive description of the cable theory and its application to analysis of electrical phenomena in the dendrites is provided in several excellent works (Taylor, 1963; Jack et al., 1975; Rall, 1977; Rall and Agmon-Snir, 1999; Koch, 1999). In this chapter, the basics of this theory are given with accentuation of the issues important for understanding the material in the following chapters.

Cable theory requires space in which electrical parameters are distributed. The dendritic space is shaped by the membrane into a tube-like branching structure. The tube diameter ranges from several micrometres (or even a fraction of a micrometre) to several tens of micrometers in diameter and the tube length can read hundreds or even thousands of micrometres. These dimensions of the dendritic space are much greater than the thickness of the membrane across which the charges are spatially separated to create the electric field. What happens in the space along and over the membrane tubes? This depends on spatial properties of the electrical field in this larger domain. If the charge separation and the electric field produced by molecular machines is different at different locations in the cable, then electrical voltage occurs and the corresponding current flows between these locations. The routes in the space along which the current flows are shaped by the dendritic structure. We focus on the electrical events in the dendritic cables.

The superposition of the electrical profiles of all individual dendrites of a given neuron represents the electrical structure of the whole arborization. The electrical path profiles form a complex tree-like structure that is topologically equivalent (homeomorphous) to the morphological dendritic arborization as both of them are composed of the same number of identically connected branches and paths. Due to this one-to-one correspondence between the dendritic paths and their electrical profiles, the tree-like electrical structure acquires several important features of the morphological tree such as branching pattern, path distance extent and complexity function.

Which new aspects are brought in when several individual dendrites are united in the whole arborization? Although one can expect occurrence or absence of new properties induced by the whole system, we focus on the new emerging properties in this chapter.

To find out what is the case, we explore the whole arborization electrical structure by applying the same protocols in the same sequence (passive membrane and singlesite inputs, passive or active membrane and distributed inputs) as those applied to individual dendrites in the preceding section. Namely, we start with the passive electrical structure of the whole arborization represented by the superposition of the current transfer profiles T (x) of individual dendrites, which remain the same as those computed for each extracted individual dendrite.

Organization of the spatial electrical profiles

We start by computing passive electrical structures of the whole arborization transferring single-site inputs for different neurons. The careful observation of the patterns of spatial electrical profiles reveals two main organizations.

Biological neurons have complex and diverse shape and size, which are mainly defined by their dendritic arborization (see Chapter 2). Considering the complex arborizations given by nature one can recognize the elementary structures considered in Chapters 7 and 8. The uniform segments, symmetrical or, more often, asymmetrical bifurcations as structural components are present in biological arborizations en masse and in various, unpredictable combinations. The geometrical information required for building the electrical structure of biological dendrites is the same as for the elementary artificial dendritic structures: the branching pattern, lengths and diameters of the branches, whereas the 3D organization does not matter. In the 3D biological arborization, because of the complexity all these structural details are seen hardly if at all, and so retrieving and relating the structural and electrical features are hampered. To deal with this problem we have to separate different aspects of geometry of the dendritic space.

One aspect could be considered as intrinsic, irrelevant of the 3D arrangement of a neuron in the space of the brain or spinal cord. The components of the dendritic structure are characterized only in terms of their lengths and diameters. The multiplicity of the structural components (paths, branches and bifurcations) imparts the complexity to the biological dendrites. In a given arborization, one meets unpredictably connected branches with unpredictably varying lengths and heterogeneous diameters and, in that sense, the dendritic geometry is stochastic both topologically and metrically.

The biologist

The shapes of the dendritic arborization of vertebrate neurons is a unique property which differentiates the nervous tissue from all the other tissues of the organism. The neuron doctrine, which we owe to Santiago Ramón y Cajal (Ramón y Cajal, 1904, 1911),was established 50 years after the cellular theory proposed by Schwann in 1839. This long period of trial and error and of vigourous opposition by the adherents of the reticularism is simply explained by the great difficulty of recognizing a nerve cell on histological preparations (Figure 1.1).

It was only after the discovery of the Golgi method, which is a selective technique for visualizing nerve cells and their prolongations that Ramón y Cajal established the first fundamental concept of neuroscience:

He called the nervous tissue the most intricate structure known in the living world. He observed a great number of neurons stained with the Golgi method in a variety of species. The comparison of dendritic morphologies of neurons located in homologous regions of the brains of different animals led him to formulate what we call the ‘shape hypothesis’. It was in the darwinism context of the time and tuned with the comparative phylogenetic approach.

Currents flowing between dendritic sites redistribute charges over the dendritic space. The spatial maps of the net current are complementary to the those of the membrane voltage. The current density maps show contributions, positive or negative, of different dendritic sites to the core current flowing in the dendrites. In neurons, the currents are transferred by ions, which are not only elementary charges but also elementary amounts of substance. The current flow into or out of a unitary volume of the dendritic space changes the amount of substance per unit volume, that is the concentration. Both electrical and chemical signalling in neurons is concentration dependent. The well-known examples include the Nernst equilibrium potentials for the transmembrane ion currents, the concentration-dependent currents such as calcium-dependent potassium current, concentration-dependent ion pumps in the plasma membrane and in the membrane of intracellular organelles, and finally ion concentration-dependent intracellular biochemical reactions of many vitally important substances. Hence, the current density maps are necessary for understanding the contribution of the current flow and substance fluxes across the membrane to the dynamics of ion concentration over the dendritic space.

Charge transfer ratio

The charge transfer ratio also called the relative effectiveness of the charge transfer, was first introduced by Barrett and Crill (1974) to characterize the contributions from different individual dendritic sites to the total somatopetal current transferred to the soma.

There is no bioelectricity without space. To produce an electric field, electrical charges must be separated in space. The distance separating the charge carriers – the ions – on the neuronal membrane ranges between 6 and 10 nanometres. The charges are separated by the action of non-electrical forces that must be organized in space. Pump molecules embedded in the membrane operate by chemical binding and unbinding of the ions on the opposite sites of the membrane. The shape of the cell membrane together with the membrane material determine the membrane capacitance.

Ions as carriers of current

In neurons the currents are carried by ions flowing in the conductive intra- and extracellular media, the cytoplasm and cerebro-spinal fluid. The intra- and extra-cellular media are conductors of the second class, the electrolytes. The ions are elementary species of both charge and substance. Therefore, two driving forces move them: electrical and non-electrical, diffusive or chemical. Both types of forces occur due to special properties of the neuronal membrane separating the intracellular solution from the extracellular one. Electrical forces originate from the voltage difference (gradient). Diffusive forces are due to gradient of ion concentration. Ions change their spatial location also as a result of chemical reactions, e.g. with intracellular or membrane molecules. Main ion species carrying currents are sodium (Na+), potassium (K+), calcium (Ca2+) and chloride (Cl−). Each species has different concentrations inside and outside the cell.

The dendritic job to process synaptic inputs ends by generating patterns of output discharges. If the site of initiation of action potentials has long been known, the mechanisms by which the axo-somatic trigger zone is finally put into action is an open question. A 60-year-old large consensus admits the simple explanation: the current shifts the voltage at the initial segment and when a threshold is reached, the neuron fires. The reasons for the numerous different types of output patterns observed from a single neuron are skipped and remain unknown. How the output patterns are formed by the electrical dendritic arborization with non-linear, active membrane is explained in this chapter.

We select two types of neurons with clearly different geometry and cocktails of voltage-dependent channels in their dendrites, and simulate generation of output discharge patterns in response to tonic activation of synaptic inputs distributed over the dendritic membrane to find out the rules that govern the neuronal code.

Terminology to describe the repertoire of neuronal discharges

We propose the following terminology to describe the types of electrical activity of neurons that we observe in our models.

Elementary electrical event at the neuron output (axon) is a single action potential (spike) or a burst of action potentials. Other examples of elementary events recorded from the soma or dendrites are slow depolarization waves or postsynaptic potentials.

Burst of action potentials is a group of sequential action potentials separated by the same or different time intervals, the duration of which is compatible with the refractoriness period. Examples are groups of two (doublet), three (triplet) or four (quadruplet) action potentials.

For the biologist, the gap between results of model computation and live neurons is filled when an electrical structure is mapped on the static anatomy of the dendritic field of neurons. The speaking likeness of these 3D images opens a new way of thinking by providing a functional image of 3D dendritic space. Indeed it is new because the question of the electrical state of the whole dendritic space is rarely addressed, although critical for understanding how the neuron processes its inputs.

The three types of neurons investigated here have their own idiosyncratic 3D dendritic pattern so well described in morphological words. In Chapters 10 and 11, we explore the arborizations as determinants of their spatial electrical properties without referring to the dendritic geometry as an object inserted in physical 3D space. The restriction of this view can be explained with a simple analogy. This is a view of the dendrites ‘from the inside’.

We can imagine the dendritic arborization as a ‘cave maze’ in the brain's depths. When we are inside the maze, we do not perceive its 3D shape. We can wander inside, uncoiling ‘Ariadne's thread’ on our path from the entry (the soma) to the deadlocks (the distal tips) and then measure the length of the threads between those points to find the path lengths. Another spatial information available from the interior view is the diameter of the ‘cave’ at each site along the path. Knowing the lengths and diameters of all the paths in such a labyrinth is sufficient for computing the spatial electrical structures.

Any live dendrite contains constitutive parts, such as the artificial elements used in Chapters 7 and 8 to demonstrate the biophysical laws that rule the proximal-todistal and path-to-path electrical relationships. In fact, the live dendrite is made of similar elements and is ruled by the same laws. However, any piece of dendrite observed under a microscope in an histological preparation displays a much more complex shape than artificial elements. It appears tortuous, irregular, often nodular with branching points and daughter branches of different lengths and diameters. These idiosyncratic attributes are totally unpredictable, making dendrites unique.

In this chapter, as we tackle much more complex live objects, we consider first natural dendritic structures of moderate complexity: individual dendrites extracted from a whole arborization. We study electrical structures of natural dendrites in the same way as we did in the previous chapters dealing with simplified artificially built structures. We look for geometry-related features in the electrical structures in relation to structural heterogeneities and branchings in their natural occurrence. As the recognizable geometry-related features of electric structures are found for individual dendrites, they will be used as navigation tools in electrical structures of complex arborizations of different neuron types described in Chapter 11.

Geometry of an example dendrite

Here we show an example of a systematic study performed on one individual dendrite extracted from the reconstructed arborization of an abducens motoneuron of the rat (cell M5 in Figure 9.5). The location of the selected dendrite in the 3D space surrounding the soma is shown in Figure 10.1, A (Korogod et al., 1998).

Coming back from our journey into dendritic space, we bring back with us the conviction that we have discovered some new critical notions which are the messages that we must pass on.

What did we learn about the features pictured in many handbooks under such key words as branching dendrites, complex trees, tapering branches, branching point, excitable dendrites, etc. These words give a feeling of déjávu. Do they really take on a special new significance in the description of operating neuronal dendrites as dynamical electrical devices? We strongly believe that the answer is yes indeed, as we attribute a specific role to every one of these structural features in a harmonious working ensemble!

The scope of our book is restricted to the dynamical electrical picture of dendritic space. This picture is composed of spatial profiles of electrical values along dendritic branches. We refer to what is observed as electrical states of parts of the whole dendritic arborization. Mapping these values on the reconstructed images of the dendrites provides a specific mosaic of electrical states of its parts, characterizing the electrical state of the whole arborization. The dendritic structure is both the bearer and determinant of its electrical properties as a whole.

To make these geometric features and the mechanisms underlying their function clearer, we select special artificial conditions which help to unveil some hidden neuronal operations. We use models imperfect and simplified, but efficient to analyze reality. They replace impossible observations by simulations.

Dear Reader,

We invite you to travel in space with us! This will be a very peculiar space: the dendritic space of neurons that is the cosmos for neuroscientists. It is mysterious and practically unexplored like the outer space we glimpse at in the sky. Curiously, we can further extend this analogy: the tools of astronomy can be turned from the sky to the microscope stage to explore shining brain stars, the neurons radiating their dendrites into the surrounding space. This was performed in the pioneering work by Paul Gogan and co-workers using a modified astronomical camera to image the microstructure of the dendritic membrane during the excitation of single live neurons in culture (see references in Chapter 14). The explorers of the dendritic space still have to invent the appropriate spacecrafts and technologies. As in cosmology, experimentation is limited, and mathematical and computer models are the only way of gaining insight into the nature of the dendritic space. The itinerary of our travel relies on these tools.

We start with a brief historical background to the dendritic problem and describe the origin of the structural data used for further morphometric and computer simulation studies of the dendritic arborizations (Chapters 1 and 2). Chapter 3 describes basic bioelectricity with emphasis on space.We show how charge carriers are separated in space and thus electric fields and currents are created across the neuronal membrane. An important generalization is that, despite multiplicity and diversity of channel types, the number of different types of current-voltage relations is restricted to three.

The mathematical tools described in the preceding chapters can now be applied first to simple artificial structures for the sake of demonstration of the electrical relations between proximal and distal dendritic sites. Studying these relationships means analyzing the electrical states of the sites. In a dendritic cable, the local electrical state, that is the state of a site, is characterized by the transmembrane voltage, current and/or conductance. A set of values of voltage (current, conductance) defined at consecutive sites along a path forms the so-called path profile of the corresponding values. It is graphically represented by a plot of these values as a function of the path distance from the soma.

A single dendritic path has a unique dimension measured in units of distance along the dendrite. Electrical relationships between all the sites situated in this continuous one-dimension space at shorter or longer distances from the reference point, usually the soma, provide a one-dimension representation of the electrical structure of a path. The electrical relation between proximal and distal sites is the only type of spatial relationship that can be assessed by the electrical picture of a single path. As a single dendritic path (Figure 7.1) is the most simple building block of an arborization, its study provides basic insights into the complexity of the dendritic structure.

In this chapter, the impact of a variation in diameter on the electrical structure of a single dendritic path is analyzed in detail.