This chapter examines techniques for evaluating the probability of finding a synaptic contact between neurons. A simple case for which one might want to estimate that probability is shown in Figure 2.0.1, where neurons from region A send their axons into region B and establish synaptic contacts there.

One of the principal reasons to evaluate the probability of such contact is to compare the evaluated probability and the experimental findings. Such comparisons can reveal the existence of rules that govern the connectivity between neurons. To illustrate this point, let us suppose that we are able to obtain an estimate of the probability of contact between a neuron from A and a neuron from B, assuming complete randomness (i.e., every neuron from A has the same probability of establishing contact with every neuron in B). We can then conduct experiments in which electrodes are thrust into A and B. With one electrode we stimulate a neuron from A, and with the other we record the response of a neuron from B. Such an experiment can help us decide if the cell we stimulate indeed affects the recorded cell synaptically. By repeating the experiment many times, we can experimentally evaluate the probability that a neuron from A will make a functioning synaptic contact on a neuron from B.

If the probability of contact that we determined experimentally agrees with the probability evaluated theoretically, then the idea of completely random connectivity can be accepted.

In Chapter 3 we discussed the input-output relations of synaptic connections, as measured by spike-train analysis. In this chapter we attempt to relate these phenomenological transmission curves to intracellular synaptic mechanisms. There appears to be no uniquely accurate way of translating the membrane potential changes into firing rates, though two extreme cases can be understood fairly well. The first case is that of a neuron whose membrane potential hyperpolarizes strongly after each action potential, and then begins to depolarize gradually until it hits the threshold and fires again. Firing times of such neurons are quasi-periodic. The second case is that of a neuron whose membrane potential fluctuates strongly around a constant mean level and whose excitatory postsynaptic potentials (EPSPs) and inhibitory postsynaptic potentials (IPSPs) are small.

Both types of neurons can be found in the mammalian nervous system. We refer to them as the periodically firing neurons and the randomly firing neurons. Sections 4.1 and 4.2 deal with the analysis of firing times in these two types of neurons. Section 4.3 describes the autocorrelation function and shows how it can be used to distinguish between the two types of neurons. It also shows that cortical neurons behave like randomly firing neurons. Sections 4.4 and 4.5 describe the expected modulations of firing rates generated by a postsynaptic potential in these two types of neurons.

Introduction

The methods and data presented in Chapters 2,3, and 4 are essential for assessing the feasibility of neuronal circuits composed of a small number of elements. However, the neural network models presented in recent years have demonstrated that computations can also be carried out by massive interactions among a multitude of neurons. This chapter offers an introduction to current trends in neural network modeling.

Modeling of neural networks has been carried out extensively in the past five years. Among the first attempt to build circuits that would compute were McCullouch and Pitts [1943], who showed how to compute logic predicates with neurons. They later constructed a neuronal circuit that recognized shapes, regardless of their position in the visual field [Pitts and McCulloch, 1947]. Subsequently there were many other attempts to construct computing circuits from neural-like elements [e.g., Wooldridge, 1979], but most of those attempts did not leave a lasting impression in the neurosciences. The original McCulloch and Pitts paper was difficult to follow, even for mathematicians [Palm, 1986b]; thus, adoption and development of their ideas by neurophysiologists did not follow. The image-recognizing circuit they developed had connectivities that, so far as we know, did not exist in the visual areas. That seems to have been the fate of most “neural computers” suggested in the past.

Our discussion of the transmission of activity through synapses concentrated on asynchronous transmission. We defined the ASG as the number of spikes added to the postsynaptic stream after a presynaptic spike, but we were not concerned with the exact time at which these extra spikes were produced. We have shown that if several spikes converge on one postsynaptic target, their effects add up linearly as long as their EPSPs do not overlap in time.

In this chapter we deal with the properties of synchronous transmission of activity. We have already shown that if a cell receives convergent excitations from several sources, their combined effect will be greater than the sum of their isolated effects. Here we examine this property quantitatively. An examination of the properties of synchronous activation of neurons will lead us to the conclusion that diverging/converging chains are bound to transmit activity synchronously.

The last section deals with the question how one can observe synchronous activity experimentally. In fact, this entire book came about because in studies in which we recorded the activity of several neurons simultaneously, we observed phenomena that could be explained only by assuming synchronous transmission through diverging/converging chains.

This chapter presents a brief overview of the macroscopic structure, the microscopic structure, and the embryological genesis of the cerebral cortex. In the last section of this chapter, the quantitative aspects of the cortical morphology are considered. It should be noted that a comprehensive survey of all the known facts about the cortex would be an impossible task. Any selection of facts necessarily reflects the personal opinions of the author regarding the relevance of particular facts and issues. The selection of topics and facts presented here is certainly biased by the general aims of the later chapters and by my view that the brain is essentially deterministic on a large scale, but probabilistic on a small scale.

In studying this chapter, the reader is expected to understand the “classical” views of the cortical anatomy and microscopy, to understand methods and concepts that are dealt with in current neuromorphological research, and to be able to interpret microscopic images in a manner suitable for calculating the probabilities of contact, as described in Chapter 2. Most of this chapter is intended for the reader with little background in neurobiology. The reader who is familiar with the cortical structure may skim this chapter, paying detailed attention only to the tables in Section 1.5.

Introduction

Given a record of the spiking activity of a neuron, the question arises as to what information can be obtained from it. There are many aspects to this question. A primary distinction must be made, of course, between spontaneous and driven activity.

It is a fact that the majority of CNS cells sampled thus far exhibit spontaneous activity. For example, in the cat cochlear nucleus, Pfeiffer and Kiang (1965) found that 80% of the 269 cells studied spontaneously emitted spikes. It is tacitly assumed in most studies that the spontaneous activity is not significantly affected by small doses of anaesthetic. This has been demonstrated for cells of Clarke's column (Pyatigorskii 1966).

The spontaneous, or background, activity is an important object to study for several reasons. First, it is a property of the cell and is of interest in itself. Furthermore, in conjunction with physiologically realistic mathematical models of nerve cells, it may be possible to infer actual nerve-cell properties from the spontaneous activity. Attempts of this kind have been made by Bishop et al. (1964), Braitenberg et al. (1965), Smith and Smith (1965), Correia and Landolt (1977), Tuckwell and Richter (1978), and Tuckwell (1979a). Statistical estimation procedures for parameters in some neuronal random processes have been outlined by Lansky (1983) and will be elaborated upon in Section 10.11.

In Volume 1, Linear Cable Theory and Dendritic Structure, the classical theory of membrane potentials, the Lapicque model neuron, and linear cable theory were introduced. The subthreshold responses described were always linear and the mathematical theory relatively simple.

In this volume, Nonlinear and Stochastic Theories, more complicated models are considered with concomitant increases in mathematical complexity. The principal new elements are nonlinear systems of reaction—diffusion systems and probability and stochastic processes. However, the required mathematics is developed in the text. The material also involves numerical methods of solution of partial differential equations and the statistical analysis of point processes. It is expected that this volume would be suitable for a second one-term course in quantitative neurophysiology with Volume 1 as a prerequisite.

Chapter 7 introduces reversal potentials for ionic currents into the cable equation with emphasis on those that arise during synaptic transmission. A derivation of the cable equation with synaptic reversal potentials is given as there did not appear to be one in the literature. The space-clamped problems are dealt with first, for the sake of completeness of presentation, and are used to explore the effects of relative timing of excitation and inhibition. Using the Green's function technique, exact solutions are then obtained for the full cable equation with periodic synaptic excitation.

Introduction

In Chapter 1 we saw that sufficient depolarization of a motoneuron, due either to current injection or excitatory synaptic input, could cause the occurrence of an action potential. During the action potential the polarity of the cell membrane was briefly reversed, the inside of the cell becoming temporarily at a positive electrical potential relative to the outside. Furthermore, this depolarizing pulse propagated along the axon of the cell to eventually reach the target muscle cells that were excited at the neuromuscular synapses. The reader should also review the first experimental recording of the action potential, that for the squid giant axon shown in Figure 2.1.

In this chapter a theory of the action potential will be presented. This theory relies almost solely on the works of the great trio of physiologists, Hodgkin, Huxley, and Katz. Not only did they perform the many key experiments but they also developed cogent mathematical models in the form of systems of differential equations — the Hodgkin—Huxley equations. These equations have given considerable impetus to the mathematical study of nonlinear partial differential equations. The equations cannot be solved in the sense that solutions of the linear (Chapters 5 and 6) and bilinear (Chapter 7) cable equations could be found.

Introduction

So far we have described a number of mathematical models of electrically excitable cells and, at least for some of the models, we have indicated the kind of mathematical questions and analysis that arise in work on these models. The next major step is to describe in detail how the mathematical analysis can be carried out. In order to obtain this description, we will first give some theory of differential equations. By doing this, we will introduce the mathematical language that is appropriate for discussing the problems that are of concern to us. We also describe some mathematical techniques and results that will be useful in the study of the models.

Basic theory

In this section we describe some basic properties of solutions of differential equations. These are used very frequently in the analysis of all the physiological models.

Existence theorems and extension theorems

It is reasonable that our first concern should be for the existence of solutions. To a reader whose experience with differential equations is limited to an introductory course following calculus, such a concern may seem unnecessarily fussy. In a first course in differential equations various techniques for computing solutions are described and it might be expected that our chief concern would be to summarize such computational techniques; but all the physiological models in which we are interested are nonlinear systems, and, consequently, it is usually impossible to get explicit expressions for the solutions (i.e., closed solutions).