So far the discussion of network behavior in Chapters 6–8 has been restricted to homogeneous populations of neurons. In this chapter we turn to networks that have a spatial structure. In doing so we emphasize two characteristic features of the cerebral cortex, namely the high density of neurons and its virtually two-dimensional architecture.

Each cubic millimeter of cortical tissue contains about 105 neurons. This impressive number suggests that a description of neuronal dynamics in terms of an averaged population activity is more appropriate than a description at the single-neuron level. Furthermore, the cerebral cortex is huge. More precisely, the unfolded cerebral cortex of humans covers a surface of 2200–2400 cm2, but its thickness amounts on average to only 2.5–3.0 mm2. If we do not look too closely, the cerebral cortex can hence be treated as a continuous two-dimensional sheet of neurons. Neurons will no longer be labeled by discrete indices but by continuous variables that give their spatial position on the sheet. The coupling of two neurons i and j is replaced by the average coupling strength between neurons at position x and those at position y, or, even more radically simplified, by the average coupling strength of two neurons being separated by the distance |x − y|. Similar to the notion of an average coupling strength, we will also introduce the average activity of neurons located at position x and describe the dynamics of the network in terms of these averaged quantities only.

In Chapters 10 and 11 we have explored the principle of Hebbian synaptic plasticity. In this final chapter we would like to close the chain of arguments that we have followed throughout the book and establish a link between synaptic plasticity and the problems of neuronal coding and signal transmission. We will start the chapter with the question of rapid and reliable signal transmission, a question that we have encountered on several occasions in this book. In Section 12.1 we will see that an asymmetric spike-time-dependent learning rule is capable of detecting early events that may serve as predictors for others. Such a mechanism can speed up signal processing and, hence, the reaction time. In Section 12.2 we show that spike-time-dependent plasticity can enhance signal transmission by selectively strengthening synaptic connections that transmit precisely timed spikes at the expense of those synapses that transmit poorly timed spikes. In Section 12.3 we turn to sequence learning and explore whether spike-time-dependent plasticity can support coding schemes that are based on spatio-temporal spike patterns with a millisecond resolution. The last two sections study the coding properties of specific neuronal systems. In Section 12.4 we will illustrate the role of an inverted (or anti-)Hebb rule for the subtraction of expectations – which has been hypothesized as an important component of signal processing in electric fish. Finally, in Section 12.5 we will see that a spike-time-dependent Hebbian rule can play an important role in the developmental tuning of signal transmission in the auditory system of barn owls.

In the neuron models discussed so far each synapse is characterized by a single constant parameter wij that determines the amplitude of the postsynaptic response to an incoming action potential. Electrophysiological experiments, however, show that the response amplitude is not fixed but can change over time. Appropriate stimulation paradigms can systematically induce changes of the postsynaptic response that last for hours or days. If the stimulation paradigm leads to a persistent increase of the synaptic transmission efficacy, the effect is called long-term potentiation of synapses, or LTP for short. If the result is a decrease of the synaptic efficacy, it is called long-term depression (LTD). These persistent changes are thought to be the neuronal correlate of “learning” and “memory”.

In the formal theory of neural networks the weight wij of a connection from neuron j to i is considered as a parameter that can be adjusted so as to optimize the performance of a network for a given task. The process of parameter adaptation is called learning and the procedure for adjusting the weights is referred to as a learning rule. Here learning is meant in its widest sense. It may refer to synaptic changes during development just as well as to the specific changes necessary to memorize a visual pattern or to learn a motor task. There are so many different learning rules that we cannot cover them all in this book.

The behavior of high-dimensional nonlinear differential equations is difficult to visualize – and even more difficult to analyze. Two-dimensional differential equations, however, can be studied in a transparent manner by means of a phase plane analysis. A reduction of the four-dimensional equation of Hodgkin and Huxley to a two-variable neuron model is thus highly desirable. In the first section of this chapter we exploit the temporal properties of the gating variables of the Hodgkin–Huxley model so as to approximate the four-dimensional differential equation by a two-dimensional one. Section 3.2 is devoted to the phase plane analysis of generic neuron models consisting of two coupled differential equations, one for the membrane potential and the other one for the so-called relaxation variable. One of the questions to which we will return repeatedly throughout this chapter is the problem of the firing threshold. Section 3.3 summarizes some results on threshold and excitability in two-dimensional models. As a first step, however, we have to go through the approximations that are necessary for a reduction of the Hodgkin–Huxley model to two dimensions.

Reduction to two dimensions

In this section we perform a systematic reduction of the four-dimensional Hodgkin–Huxley model to two dimensions. To do so, we have to eliminate two of the four variables. The essential ideas of the reduction can also be applied to detailed neuron models that may contain many different ion channels. In this case, more than two variables would have to be eliminated, but the procedure would be completely analogous (Kepler et al., 1992).

The task of understanding the principles of information processing in the brain poses, apart from numerous experimental questions, challenging theoretical problems on all levels from molecules to behavior. This books concentrates on modeling approaches at the level of neurons and small populations of neurons, since we think that this is an appropriate level to address fundamental questions of neuronal coding, signal transmission, or synaptic plasticity. In this text we concentrate on theoretical concepts and phenomenological models derived from them. We think of a neuron primarily as a dynamic element that emits output pulses whenever the excitation exceeds some threshold. The resulting sequence of pulses or “spikes” contains all the information that is transmitted from one neuron to the next. In order to understand signal transmission and signal processing in neuronal systems, we need an understanding of their basic elements, i.e., the neurons, which is the topic of Part I. New phenomena emerge when several neurons are coupled. Part II introduces network concepts, in particular pattern formation, collective excitations, and rapid signal transmission between neuronal populations. Learning concepts presented in Part III are based on spike-time-dependent synaptic plasticity.

We wrote this book as an introduction to spiking neuron models for advanced undergraduate or graduate students. It can be used either as the main text for a course that focuses on neuronal dynamics, or as part of a larger course in computational neuroscience, theoretical biology, neuronal modeling, biophysics, or neural networks.

From a biophysical point of view, action potentials are the result of currents that pass through ion channels in the cell membrane. In an extensive series of experiments on the giant axon of the squid, Hodgkin and Huxley succeeded in measuring these currents and describing their dynamics in terms of differential equations. In Section 2.2, the Hodgkin–Huxley model is reviewed and its behavior illustrated by several examples.

The Hodgkin–Huxley equations are the starting point for detailed neuron models which account for numerous ion channels, different types of synapse, and the specific spatial geometry of an individual neuron. Ion channels, synaptic dynamics, and the spatial structure of dendrites are the topics of Sections 2.3–2.5. The Hodgkin–Huxley model is also an important reference model for the derivation of simplified neuron models in Chapters 3 and 4. Before we can turn to the Hodgkin–Huxley equations, we need to give some additional information on the equilibrium potential of ion channels.

Equilibrium potential

Neurons are, just as other cells, enclosed by a membrane which separates the interior of the cell from the extracellular space. Inside the cell the concentration of ions is different from that in the surrounding liquid. The difference in concentration generates an electrical potential which plays an important role in neuronal dynamics. In this section, we want to provide some background information and give an intuitive explanation of the equilibrium potential.

In vivo recordings of neuronal activity are characterized by a high degree of irregularity. The spike train of individual neurons is far from being periodic and relations between the firing patterns of several neurons seem to be random. If the electrical activity picked up by an extracellular electrode is made audible by a loudspeaker then we basically hear – noise. The question of whether this is indeed just noise or rather a highly efficient way of coding information cannot easily be answered. Listening to a computer modern or a fax machine might also leave the impression that this is just noise. Being able to decide whether we are witnessing the neuronal activity that is underlying the composition of a poem (or the electronic transmission of a love letter) and not just meaningless noise is one of the most burning problems in neuroscience.

Several experiments have been undertaken to tackle this problem. It seems as if neurons can react in a very reliable and reproducible manner to fluctuating currents that are injected via intracellular electrodes. As long as the same time course of the injected current is used the action potentials occur with precisely the same timing relative to the stimulation (Bryant and Segundo, 1976; Mainen and Sejnowski, 1995). A related phenomenon can be observed by using nonstationary sensory stimulation. Spatially uniform random flicker, for example, elicits more or less the same spike train in retinal ganglion cells if the same flicker sequence is presented again (Berry et al., 1997).

Basic concepts

We present in this chapter algorithms to search for regular expressions in texts or biological sequences. Regular expressions are often used in text retrieval or computational biology applications to represent search patterns that are more complex than a string, a set of strings, or an extended string. We begin with a formal definition of a regular expression and the language (set of strings) it represents.

DefinitionA regular expression RE is a string on the set of symbols Σ ∪ { ε, |, ·, ⋆, (,) }, which is recursively defined as the empty character ε a character α ∈ Σ and (RE1), (RE1 · RE2), (RE1 | RE2), and (RE1⋆), where RE1and RE2are regular expressions.

For instance, in this chapter we consider the regular expression (((A·T) | (G·A))·(((A·G) | ((A·A)·A))⋆)). When there is no ambiguity, we simplify our expressions by writing RE1RE2 instead of (RE1 · RE2). This way, we obtain a more readable expression, in our case (AT|GA) ((AG|AAA)⋆). It is usual to use also the precedence order “*”, “·”, “|” to remove more parentheses, but we do not do this here. The symbols “·”, “|”, “⋆” are called operators. It is customary to add an extra postfix operator “+” to mean RE+ = RE · RE⋆. We define now the language represented by a regular expression.

DefinitionThe language represented by a regular expression RE is a set of strings over Σ which is defined recursively on the structure of RE as follows:

1. • If RE is ε, then L(RE) = {ε}, the empty string.

2. • If RE is α ∈ Σ, then L(RE) = {α}, a single string of one character.

3. • If RE is of the form (RE1), then L(RE) = L(RE1).

4. • If RE is of the form (RE1 · RE2), then L(RE) = L(RE1) · L(RE2), where W1 · W2is the set of strings w such that w = w1w2, with w1 ∈ W1and w2 ∈ W2. The operator “·” represents the classical concatenation of strings.

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