An Introduction to Modeling Neuronal Dynamics

Charged particles, namely ions, diffuse in water in the brain. There is a field of study called electro-diffusion theory concerned with the diffusion of charged particles. In this chapter, we study one electro-diffusion problem that is crucial for understanding nerve cells.Nernst equilibrium

In the 1940s, Alan Hodgkin and Andrew Huxley clarified the fundamental physical mechanism by which electrical impulses are generated by nerve cells, and travel along axons, in animals and humans. They experimented with isolated pieces of the giant axon of the squid. They summarized their conclusions in a series of publications in 1952; the last of these papers [76] is arguably the single most influential paper ever written in neuroscience, and forms the foundation of the field of mathematical and computational neuroscience. This chapter is an introduction to the Hodgkin-Huxley model.

In practice, complicated differential equations such as the Hodgkin-Huxley ODEs are almost always solved numerically, that is, approximate solutions are obtained on a computer. The study of methods for the numerical solution of differential equations is the subject of a large and highly sophisticated branch of mathematics. However, here we will study only the two simplest methods: Euler’s method and the midpoint method. Euler’s method is explained here merely as a stepping stone to the midpoint method. For all simulations of this book, we will use the midpoint method, since it is far more efficient, and not much more complicated than Euler’s method.

Hodgkin and Huxley modeled the giant axon of the squid. Since then, many similar models of neurons in mammalian brains have been proposed. In this chapter, we list three examples, which will be used throughout the book.

The model proposed by Hodgkin and Huxley in 1952 is not a set of ODEs, but a set of PDEs — the dependent variables are not only functions of time, but also of space. This dependence will be neglected everywhere in this book, except in the present chapter. You can therefore safely skip this chapter, unless you are curious what the PDE-version of the Hodgkin-Huxley model looks like, and how it arises.

Nearly half a century before Hodgkin and Huxley, in 1907, Louis Édouard Lapicque proposed a mathematical model of nerve cells. Lapicque died in 1952, the year when the famous series of papers by Hodgkin and Huxley appeared in print. Lapicque’s model is nowadays known as the integrate-and-fire neuron. We will refer to it as the LIF neuron. Most authors take the L in “LIF” to stand for “leaky,” for reasons that will become clear shortly. We take it to stand for “linear,” to distinguish it from the quadratic integrate-and-fire (QIF) neuron discussed in Chapter 8 The LIF neuron is useful because of its utter mathematical simplicity. It can lead to insight, but as we will demonstrate with examples in later chapters, reduced models such as the LIF neuron are also dangerous — they can lead to incorrect conclusions.

As noted at the end of the preceding chapter, subthreshold voltage traces of neurons often have an inflection point; see, for instance, Figs. 5.2 and 5.4. Voltage traces of the LIF neuron (Fig. 7.4) don’t have this feature. However, we can modify the LIF neuron to introduce an inflection point, as follows:

Many neurons, in particular the excitatory pyramidal cells, have spike frequency adaptation currents. These are hyperpolarizing currents, activated when the membrane potential is high, and de-activated, typically slowly, when the membrane potential is low. As a consequence, many neurons cannot sustain rapid firing over a long time; spike frequency adaptation currents act as “brakes,” preventing hyper-activity. In this chapter, we discuss two kinds of adaptation currents, called M-currents (Section 9.1) and calcium-dependent afterhyperpolarization (AHP) currents (Section 9.2). Both kinds are found in many neurons in the brain. They have different properties. In particular, M-currents are active even before the neuron fires, while calcium-dependent AHP currents are firing-activated. As a result, the two kinds of currents affect the dynamics of neurons and neuronal network in different ways; see, for instance, exercises 17.3 and 17.4.

In this chapter, we use phase plane pictures to understand the mechanism of neuronal firing more clearly. The key is the presence of two time scales, a slow one and a fast one: Neurons build up towards firing gradually, then fire all of the sudden. This is what is captured (and in fact this is all that is captured) by the integrate-and-fire model.

Model neurons, as well as real neurons, don’t fire when they receive little or no input current, but fire periodically when they receive strong input current. The transition from rest to firing, as the input current is raised, is called a bifurcation. In general, a bifurcation is a sudden qualitative change in the solutions to a differential equation, or a system of differential equations, occurring as a parameter, called the bifurcation parameter in this context, is moved past a threshold value, also called the critical value. (To bifurcate, in general, means to divide or fork into two branches. This suggests that in a bifurcation, one thing turns into two. This is indeed the case in some bifurcations, but not in all.) Because the variation of drive to a neuron is the primary example we have in mind, we will denote the bifurcation parameter by I, and its threshold value by I _c , in this chapter.

For a model neuron, there is typically a critical value I _c with the property that for I < I _c , there is a stable equilibrium with a low membrane potential, whereas periodic firing is the only stable behavior for I > I _c , as long as I is not too high.

In some neuronal models, the transition from I < I _c to I > I _c involves not a SNIC, as in Chapter 12, but a Hopf bifurcation. In this chapter, we give a very brief introduction to Hopf bifurcations.

A neuron is said to be of bifurcation type 2bifurcation type 2 if the transition that occurs as I crosses I _c is a Hopf bifurcation [47, 75, 129]. Examples of model neurons of bifurcation type 2 include the classical Hodgkin-Huxley model, and the Erisir model described in Section 5.3 The Hopf bifurcation in the classical Hodgkin-Huxley model is analyzed in great detail in [67]. For numerical evidence that the transition from rest to firing in the Erisir model involves a subcritical Hopf bifurcation, see [17], and also Fig. 17.9.

The bifurcation diagrams of neuronal models of bifurcation type 2 usually do not at all look very similar to the idealized bifurcation diagrams in, for instance, Figs. 13.3 and 13.9. Those idealized pictures miss a feature called thecanard explosioncanard explosion, a sudden extremely rapid growth in the amplitude of periodic spiking, resulting from the spike-generating mechanism of the neuron. What this has to do with “canard” (French for “duck”) will be explained near the end of Section 15.1. There is a very large literature on the theory of canards and their role in neuronal dynamics; see, for example, [8, 40, 99, 120, 134, 177]. The canard phenomenon plays a role in numerous other areas of science and engineering as well; see, for instance, [64, 112, 124, 141]. However, in this chapter, we will merely demonstrate the phenomenon in two very simple two-dimensional examples.

In neuronal models of bifurcation type 2 (Chapter 14), the possibility of stable rest is abolished via a Hopf bifurcation as I rises above I _c , and the possibility of stable periodic firing is abolished via a collision of the stable limit cycle with an unstable periodic orbit as I falls below I_∗, with I_∗ < I _c . In neuronal models of bifurcation type 1 (Chapter 12), there is no distinction between I_∗ and I _c ; stable rest is abolished as I rises above I _c , and stable periodic firing is abolished as I falls below I _c .

In this chapter, we study the periodic firing frequency, f, of a model neuron as a function of input current density, I. The graph of f as a function of I is commonly called the frequency-current curve, or f-I curve, of the model neuron. The f-I curve summarizes important features of the dynamics of a neuron in a single picture.

rebound firingWe have seen that some model neurons have continuous, single-valued f-I curves. With the exception of the LIF neuron, all of these model neurons transition from rest to firing via a SNIC. Other model neurons have discontinuous f-I curves with an interval of bistability, in which both rest and periodic firing are possible. The examples we have seen transition from rest to firing either via a subcritical Hopf bifurcation, or via a saddle-node bifurcation off an invariant cycle. The distinction between continuous and discontinuous f-I curves closely resembles the distinction between “class 1” and “class 2” neurons made by Hodgkin in 1948 [75]. class 1 and class 2 (Hodgkin)

Bursting is a very common behavior of neurons in the brain. bursting A bursting neuron fires groups of action potentials in quick succession, separated by pauses which we will call the inter-burst intervals. inter-burst interval Figure 19.1 shows an experimentally recorded voltage trace illustrating bursting. Figure 19.2 shows a simulation discussed in detail later in this chapter. bursting

We now think about neuronal communication via (chemical) synapses; see Section 1.1 for a brief general explanation of what this means. In this chapter, we describe how to model chemical synapses in the context of differential equations models of neuronal networks.

In addition to chemical synapses involving neurotransmitters, there are also electrical synapses at sites known as gap junctions. A gap junction is a location where two neuronal membranes come very close to each other, with holes in both membranes allowing direct passage of charged particles from one cell into the other. electrical synapsegap junctionWe will always assume that there is no preferred direction for the flow of ions [29]; in this regard, most gap-junctional coupling is very different from chemical synaptic coupling, where communication is asymmetric. In invertebrates, however, there do exist gap junctions that allow the passage of charged particles in only one direction; they are called rectifying [111]. rectifying gap junction

There is a large literature on neuronal network models attempting to model neuronal population activity without modeling individual neurons.

Part IV of the book is about synchronization in populations of nerve cells.

In this chapter, we begin our study of synchronization via mutual synaptic interactions.

Phase response curves (PRCs) describe how neurons respond to brief, transient input pulses.

There is a vast literature on the question when firing-triggered pulsatile signaling among neurons will lead to synchrony. In this chapter, we analyze the simplest formalization of this question. Perhaps the main worthwhile conclusion from the analysis presented here is that there is no general statement of the form “excitatory interactions synchronize” or “inhibitory interactions synchronize.” Whether or not pulse-coupling synchronizes depends on the detailed nature of the responses of the neurons to signals they receive; see Proposition 26.2. Approximately speaking, excitatory pulse-coupling synchronizes if the phase response is of type 2, or if there is strong refractoriness; this will be made precise in inequality (26.12).

Action potentials in the brain travel at a finitepulse-coupled oscillators with delays speed. The conduction delays between different brain areas can be on the order of 10 ​ms or more [151, Table 1]. Nonetheless, precise synchronization between oscillations in different parts of the brain has been reported numerous times [166]. This may sound surprising at first. However, there is a large body of theoretical work proposing mechanisms that could lead to synchronization in the presence of conductance delays. In this chapter, we discuss what is arguably the simplest result of this sort. We consider two identical oscillators, called A and B, pulse-coupled as in Chapter 26, but with a conduction delay.

When coupling is weak, the equations of pulse-coupled oscillators can be simplified, in a way that is mathematically attractive and facilitates analysis, by averaging the effects of several discrete pulses over time.

A simple way of synchronizing a population of neurons is to subject all neurons to the same inhibitory synaptic input pulse.

When populations of excitatory and inhibitory neurons are synaptically connected, oscillations often emerge. The reason is apparent: Activity of the excitatory neurons (which we will call E-cells from here on, as we did in Chapter 22) generates activity of the inhibitory neurons (I-cells).E-cellI-cell The activity of the I-cells causes the activity of the E-cells to cease transiently, and when it resumes, the E-cell population is closer to synchrony, as discussed in Chapter 29 The oscillations in Chapter 22 are of a similar nature, although there individual cells were not modeled.E-cellI-cell Figure 30.1 (nearly identical with Fig. 22.1) represents the interaction of E- and I-cells symbolically.

Gamma rhythms can also be generated by the interaction of I-cells alone, without any involvement of E-cells. For example, in brain slices, gamma rhythms can be evoked even in the presence of drugs blocking AMPA and NMDA receptors. See Fig. 31.1 for an example, recorded from rat CA1.

In the PING model of Chapter 30, each E-cell and each I-cell fires once on each cycle of the oscillation.

Oscillations at frequencies of approximately 12–30 ​Hz — roughly half the gamma frequency — are called beta oscillations or beta rhythms in neuroscience. Many experimental studies have linked beta oscillations to motor function. They are, in particular, more pronounced during holding periods, and attenuated during voluntary movement. Engel and Fries [44] have hypothesized that more generally, beta oscillations may signal the expectation or intent of maintaining a sensorimotor or cognitive status quo. (The sensorimotor areas of the brain are those that combine sensory and motor functions.) This fits with the observation that in patients suffering from Parkinson’s disease and the associated slowed movement (bradykinesia), power and coherence of beta oscillations in the basal ganglia are abnormally high, and are attenuated by levodopa, a drug commonly used to treat Parkinson’s disease [20].

In many brain structures, in particular in the hippocampus, gamma oscillations appear near the crests of much slower, 4–11 ​Hz oscillations, called theta oscillations or theta rhythms. For an example, see Fig. 34.1.

In experiments described in [143], fast-firing inhibitory interneurons in the barrel cortex of mice, the part of the mouse brain that processes input from the whiskers, were driven to synchronize at 40 ​Hz, using optogenetic techniques. (In general, optogenetic techniques involve genetically sensitizing neurons to light, then using light to control them.) Figure 35.1 is a reproduction of Fig. 3A of [143]. The figure shows LFP recordings from barrel cortex during 40 ​Hz optogenetic stimulation of the fast-firing inhibitory interneurons. The central finding of [143] was that making the activity of fast-firing inhibitory interneurons rhythmic at 40 ​Hz improved the ability of the mice to perceive certain whisker deflections.

In Chapter 35, we showed that inhibition can become less powerful when it is made rhythmic. More precisely, a weak signal can elicit a response more easily when the inhibition in the receiving network oscillates. The opposite effect is seen for a strong signal. Here we will argue that excitation can become more powerful when it is made rhythmic. More precisely, weak signals benefit from being made rhythmic, while strong ones become less effective when made rhythmic.

In Section 33.2, we mentioned Hebb’s idea of cell assemblies. The hypothesis is that information is carried by membership in neuronal ensembles that (temporarily) fire together. One attractive aspect of this idea is that it would give a brain with 1011 neurons an unfathomably large storage capacity, since the number of subsets of a large set is huge.

During a PING oscillation, the pyramidal cells are subject to inhibition of oscillating strength. Inputs to the pyramidal cells should be more effective at times when inhibition is weak than when it is strong. Pascal Fries [56] has suggested that the brain may make use of this fact, making neuronal communication more or less effective by shifting the phase differences between rhythmic senders and oscillating receivers. This hypothesis, called the communication through coherence (CTC) hypothesis,communication through coherence (CTC) has attracted considerable interest in the neuroscience literature.

At many synapses, repeated firing of the pre-synaptic neuron can lead to a transient decrease in the strengths of post-synaptic currents, for instance, because of neurotransmitter depletion [53]. This is called short-term depression. A phenomenological model of short-term depression was proposed by Tsodyks and Markram [165]. We describe it in Section 39.1.

We mentioned spike timing-dependent plasticity (STDP) in Section 33.2 already. Experimental evidence for STDP was presented in [68]: The connection from cell A to cell B was shown to be strengthened when A fired just before B (Hebbian learning), and weakened when B fired just before A (anti-Hebbian learning). To model STDP at a synapse, we make the maximum conductance g¯syn $$\overline{g}_{\mathrm{syn}}$$ associated with the synapse (see eq. (20.8)) time-dependent. In analogy with Sections 39.1 and 39.2, we first describe a model in which the value of g¯syn $$\overline{g}_{\mathrm{syn}}$$ jumps discontinuously, and then turn it into a differential equations model in which the change is rapid but continuous.