So far, everything we’ve done with recursion we could have also done with for loops. The recursive functions that we’ve seen were conceptually somewhat different from the for loop versions, but recursion hasn’t yet given us powers to compute things that we couldn’t compute before. But now it will!

Peptide Fragments

Imagine that we have a set of fragments of proteins, where each fragment has a given mass. For example, the masses of five protein fragments might be [2, 3, 8, 10, 12] in some unit of mass. In addition, we know the mass of the original protein, say 25 units for the sake of example. The question is whether or not there is a subset of our list of fragment masses that add up to the mass of the protein. In this example, the answer is “yes”: fragments with masses 3, 10, and 12 add up to 25. On the other hand, if the fragments had masses [2, 15, 17, 20], we could not have found a subset that adds up to 25. For now, we’re assuming that each mass in the list can be used at most once. This problem arises in the study of protein structure and has been the focus of a recent study.

The computer is the most powerful general-purpose tool available to biologists.

In part, this is due to the continuing rapid growth of biological data. For example, at the time of writing, the GenBank database had over 100 million genetic sequences with over 100 billion DNA characters. Among the contents of that database are genes from many organisms, annotated with what’s known about their function.

Imagine that you’re studying a bacterium and wish to understand what causes it to be infectious. One promising approach is to identify genes in the bacterium and compare these to known genes in GenBank. If you’re able to find similar genes whose function is known, it will tell you a great deal about the role of the genes in your bacterium. This approach represents a computational challenge, and is, in fact, the topic of Part I of this book.

But searching enormous databases is not the only reason that computers are so useful to biologists. Many biological problems have a large number of different possible solutions and only a computer – programmed with carefully designed computational recipes or “algorithms” – has any chance of finding the right one. For example, biological molecules such as proteins and RNA fold into complex shapes that strongly impact their function. Computational techniques have been developed to predict how these molecules fold. Such techniques help us understand how proteins and RNA work and can even help us design new molecules to treat disease.

Acquired Immune Deficiency Syndrome or AIDS was first recognized in the early 1980s. Within a few years of its discovery, scientists had identified the virus that causes it, called the Human Immunodeficiency Virus or HIV. Through the 1980s and early 1990s, the HIV/AIDS epidemic grew steadily, with the number of affected people and the number of deaths increasing every year. By the late 1990s it was causing millions of deaths per year worldwide, and had a prevalence of more than 20% in the adult population of some countries.

An epidemic of this magnitude requires multiple types of response. One approach has been to limit the spread of the disease through education, for example, by encouraging the use of condoms and discouraging drug users from sharing needles. A second approach has been to carry out research into the basic biology of HIV in an effort to develop better treatments or even a vaccine.

The programming problems for this section are connected with basic research into HIV. They involve predicting the RNA secondary structure in a single gene from the HIV genome. We’ll talk about secondary structure and its prediction shortly. But first, let’s briefly discuss HIV’s genome and life cycle.

The neuron models discussed in the previous chapters are deterministic and generate, for most choices of parameters, spike trains that look regular when driven by a constant stimulus. In vivo recordings of neuronal activity, however, are characterized by a high degree of irregularity. The spike train of an individual neuron is far from being periodic, and correlations between the spike timings of neighboring neurons are weak. If the electrical activity picked up by an extracellular electrode is made audible by a loudspeaker then what we basically hear is noise. The question whether this is indeed just noise or rather a highly efficient way of coding information cannot easily be answered. Indeed, listening to a computer modem 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 flicker is one of the most burning problems in neuroscience.

Several experiments have been undertaken to tackle this problem. It seems that a neuron in vitro, once it is isolated from the network, can react in a very reliable and reproducible manner to a fluctuating input current, and so can neurons in the sensory cortex in vivo when driven by a strong time-dependent signal. On the other hand, neurons produce irregular spike trains in the absence of any temporally structured stimuli. Irregular spontaneous activity, i.e., activity that is not related in any obvious way to external stimulation, and trial-to-trial variations in neuronal responses are often considered as noise.

In the network models discussed in Parts III and IV, each synapse has so far been characterized by a single constant parameter wij, called the synaptic weight, synaptic strength, or synaptic efficacy. If wij is constant, the amplitude of the response of a postsynaptic neuron i to the arrival of action potentials from a presynaptic neuron j should always be the same. Electrophysiological experiments, however, show that the response amplitude is not fixed but can change over time. In experimental neuroscience, changes of the synaptic strength are called synaptic plasticity.

Appropriate stimulation paradigms can induce changes of the postsynaptic response that last for hours or days. If the stimulation paradigm leads to a persistent increase of the synaptic 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. LTP and LTD are different from short-term synaptic plasticity such as synaptic facilitation or depression that we have encountered in Section 3.1. Facilitated or depressed synapses decay back to their normal strength within less than a few seconds, whereas, after an LTP or LTD protocol, synapses keep their new values for hours. The long-term storage of the new values is thought to be the basis of long-lasting memories.

The firing of action potentials has been successfully described by the Hodgkin–Huxley model, originally for the spikes in the giant axon of the squid but also, with appropriate modifications of the model, for other neuron types. The Hodgkin–Huxley model is defined by four nonlinear differential equations. The behavior of high-dimensional systems of nonlinear differential equations is difficult to visualize – and even more difficult to analyze. For an understanding of the firing behavior of the Hodgkin–Huxley model, we therefore need to turn to numerical simulations of the model. In Section 4.1 we show, as an example, some simulation results in search of the firing threshold of the Hodgkin–Huxley model. However, it remains to show whether we can get some deeper insights into the observed behavior of the model.

Four equations are in fact just two more than two: in Section 4.2 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. Two-dimensional differential equations can be studied in a transparent manner by means of a technique known as “phase plane analysis.” Section 4.3 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 for an auxiliary variable.

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 described their dynamics in terms of differential equations. Their paper published in 1952, which presents beautiful experiments combined with an elegant mathematical theory (Hodgkin and Huxley, 1952), was rapidly recognized as groundbreaking work and eventually led to the Nobel Prize for Hodgkin and Huxley in 1963. In this chapter, the Hodgkin–Huxley model is reviewed and its behavior illustrated by several examples.

The Hodgkin–Huxley model in its original form describes only three types of ion channel. However, as we shall see in Section 2.3 it can be extended to include many other ion channel types. The Hodgkin–Huxley equations are the basis for detailed neuron models which account for different types of synapse, and the spatial geometry of an individual neuron. Synaptic dynamics and the spatial structure of dendrites are the topics of Chapter 3. The Hodgkin–Huxley model is also the starting point for the derivation of simplified neuron models in Chapter 4 and will serve as a reference throughout the discussion of generalized integrate-and-fire models in Part II of the book.

Before we can turn to the Hodgkin–Huxley equations, we need to give some additional information on the equilibrium potential of ion channels.

Neurons in the brain receive input from thousands of other, presynaptic neurons, which emit action potentials and send their spikes to their postsynaptic targets. From the perspective of a postsynaptic neuron receiving a barrage of spikes, spike arrival times may look completely random, even under the assumption that presynaptic neurons generate their spikes by a deterministic process. Indeed, as we have seen in the preceding chapter, internal noise sources of a cell, such as spontaneous opening of ion channels, do not account for all the variability of spike-trains encountered in freely behaving animals in vivo. Rather, it is likely that a large fraction of the apparent variability is generated by the network. Modeling studies confirm that networks with fixed random connectivity can lead to chaos on the microscopic level, so that spike arrival times appear to be random even if generated by a deterministic network.

In this chapter, we discuss the consequences of stochastic spike arrivals for modeling. The “noise” generated by the network is often described by a noise term in the differential equation of the membrane voltage (Section 8.1). Such a noise term, typically modeled as white noise or colored noise, can be derived in a framework of stochastic spike arrival, as shown in Section 8.2. Stochastic spike arrival leads to fluctuations of the membrane potential which will be discussed in the case of a passive membrane (Section 8.2.1) – or, more generally, for neuron models in the subthreshold regime.

Humans remember important events in their lives. You might be able to recall every detail of your first exam at college, or of your first public speech, or of your first day in kindergarten, or of the first time you went to a new school after your family moved to a new city. Human memory works with associations. If you hear the voice of an old friend on the phone, you may spontaneously recall stories that you had not thought of for years. If you are hungry and see a picture of a banana, you might vividly recall the taste and smell of a banana … and thereby realize that you are indeed hungry.

In this chapter, we present models of neural networks that describe the recall of previously stored items from memory. In Section 17.1 we start with a few examples of associative recall to prepare the stage for the modeling work later on. In Section 17.2 we introduce an abstract network model of memory recall, known as the Hopfield model. We take this network as a starting point and add, in subsequent sections, some biological realism to the model.

Associations and memory

A well-known demonstration of the strong associations which are deeply embedded in the human brain is given by the following task. The aim is to respond as quickly as possible to three questions.

Detailed conductance-based neuron models can reproduce electrophysiological measurements to a high degree of accuracy, but because of their intrinsic complexity these models are difficult to analyze. For this reason, simple phenomenological spiking neuron models are highly popular for studies of neural coding, memory, and network dynamics. In this chapter we discuss formal threshold models of neuronal firing, also called integrate-andfire models.

The shape of the action potential of a given neuron is rather stereotyped with very little change between one spike and the next. Thus, the shape of the action potential which travels along the axon to a postsynaptic neuron cannot be used to transmit information; rather, from the point of view of the receiving neuron, action potentials are “events” which are fully characterized by the arrival time of the spike at the synapse. Note that spikes from different neuron types can have different shapes and the duration and shape of the spike does influence neurotransmitter release; but the spikes that arrive at a given synapse all come from the same presynaptic neuron and – if we neglect effects of fatigue of ionic channels in the axon – we can assume that its time course is always the same. Therefore we make no effort to model the exact shape of an action potential. Rather, spikes are treated as events characterized by their firing time – and the task consists in finding a model so as to reliably predict spike timings.

There are various ways to introduce noise in formal spiking neuron models. In the previous chapter we focused on input noise in the form of stochastic spike arrival. In this chapter we assume that the input is known or can be estimated. Stochasticity arises at the level of the neuronal spike generation, i.e., at the moment of the output. The noisy output can be interpreted as arising from a “soft” threshold that enables an “escape” of the membrane potential across the threshold even before the threshold is reached. Models with a noisy threshold or escape noise are the basis of Generalized Linear Models which will be used in Chapters 10 and 11 as a powerful statistical tool for modeling spike-train data.

In Section 9.1, the notion of escape noise is introduced. In Section 9.2 we determine the likelihood that a specific spike train is generated by a neuron model with escape noise. In Section 9.3 we apply the escape noise formalism to the Spike Response Model already encountered in Chapter 6 and show an interesting link to the renewal statistics encountered in Chapter 7. The escape rate formalism gives rise to an efficient description of noise processes, independently of their biophysical nature, be it channel noise or stochastic spike arrival. Indeed, as shown in Section 9.4, noisy input models and noisy output models can behave rather similarly.

When an experimenter injects a strong step current into the soma of a neuron, the response consists of a series of spikes separated by long or short intervals. The stereotypical arrangement of short, long or very long interspike intervals defines the neuronal firing pattern. In Chapter 2 we have already encountered firing patterns such as tonic, adapting, or delayed spike firing. In addition to these, several variants of burst firing have also been observed in real neurons (see Fig. 6.1). This diversity of firing patterns can be explained, to a large extent, by adaptation mechanisms which in turn depend on the zoo of available ion channels (Chapter 2) and neuronal anatomy (Chapter 3).

In order to describe firing patterns, and in particular adaptation, in a transparent mathematical framework, we start in this chapter with the simplified model of spike initiation from Chapter 5 and include a phenomenological equation for subthreshold and spiketriggered adaptation. The resulting model is called the adaptive exponential integrate-andfire (AdEx; Section 6.1). We then use this simple model to explain the main firing patterns (Section 6.2). In Section 6.3, we describe how the parameters of the subthreshold and spike-triggered adaptation reflect the contribution of various ion channels and of dendritic morphology. Finally, we introduce the Spike Response Model (SRM; Section 6.4) as a transparent framework to describe neuronal dynamics. The Spike Response Model will serve as a starting point for the Generalized Linear Models which we will discuss later, in Chapter 9.

The brain contains millions of neurons which are organized in different brain areas, within a brain area organized in different subregions, inside each small region into different layers, inside each layer into various cell types. The first two parts of this book focused on the mathematical description of an isolated neuron. Starting with this chapter, we shift our attention to the collective properties of groups of neurons, which we call “neuronal populations.” Instead of modeling the spike times of a single neuron which belongs, for example, to the cell class “pyramidal” in layer 5 of subregion C4 in brain region S1 (the numbers here are completely arbitrary), we can ask the question: Suppose a human subject or animal receives a visual, auditory, or somatosensory stimulus – what is the activity of all the cells in this layer of this subregion that are of type “pyramidal” in response to the stimulus? What is the response of this subregion as a whole? What is the response of a brain area? In other words, at any of the scales of spatial resolution (Fig. 12.1), we may be interested in the response of the neuronal population as a whole, rather than in the spikes of individual neurons.