Junko Inoue

Sensitive dependence of the coefficient of variation of interspike intervals on the lower boundary of membrane potential for the leaky integrate-and-fire neuron model

After the report of Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350], leaky integrate-and-fire models have been investigated to explain high coefficient of variation (CV) of interspike intervals (ISIs) at high firing rates observed in the cortex. The purpose of this paper is to study the effect of the position of a lower boundary of membrane potential on the possible value of CV of ISIs based on the diffusional leaky integrate-and-fire models with and without reversal potentials. Our result shows that the irregularity of ISIs for the diffusional leaky integrate-and-fire neuron significantly changes by imposing a lower boundary of membrane potential, which suggests the importance of the position of the lower boundary as well as that of the firing threshold when we study the statistical properties of leaky integrate-and-fire neuron models. It is worth pointing out that the mean-CV plot of ISIs for the diffusional leaky integrate-and-fire neuron with reversal potentials shows a close similarity to the experimental result obtained in Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350].

Chaotic spiking in the Hodgkin-Huxley nerve model with slow inactivation of the sodium current.

The Hodgkin-Huxley (HH) equations with a modification in which the inactivation process (h variable) of sodium channels is slightly slowed down are investigated. It is shown that this slight modification changes the HH dynamics to a completely different one, with chaotic spiking and very long interspike intervals appearing in a generic manner, although the initiation mechanism of repetitive firing is a simple Hopf bifurcation.

Chaos and Variability of Inter‐Spike Intervals in Neuronal Models with Slow‐Fast Dynamics

A neuron generates action potentials or spikes in response to electric stimuli, and also produces a train of spikes (periodic oscillation) when a continuous stimulus current is injected. Using the extended Bonhoeffer‐van der Pol (BVP) or FitzHugh‐Nagumo (FHN) equations, which is a simplified version of the famous Hodgkin‐Huxley neuronal model, we show that very slow spiking can appear near the (singular) Hopf bifurcation point in a certain generic situation. The patterns of the extraordinary slow spiking are phenomenologically classified into two types: a regular slow spiking and chaotic slow spiking. The variability of inter‐spike intervals (ISI’s) and the possible mechanism of slow spiking are discussed under slow‐fast decomposition analysis. The noise effects on such variability of ISI’s are also examined.

Computational and Mathematical Models of Neurons

What are models? The HH equations (2.3) are often called a physiological model, whereas the models appeared in the following sections are simplified models or abstract models. However, there is no model in which any simplifications or abstractions have not been made. Of course, many features of real neurons are ignored even in the HH equations. All models have their applicability and limits to describe natural phenomena. Therefore, all types of models whatever simplified or physiological, have their own values to model real phenomena. Starting with the BVP or FHN model which is a simplification of the HH equations, this chapter proceeds to several neuronal models with higher abstractions which are useful to tract some essential features of neurons.

Whole System Analysis of Hodgkin–Huxley Systems

In Chap. 2, we have explored the dynamics of the original Hodgkin–Huxley equations of a squid giant axon where only the parameter Iext was changed. The HH equations, however, include various constants or parameters whose values were determined based on physiological experiments, and thus the values inherently possess a certain ambiguity. Also, the “constants” are not really constant but change temporally. Thus, in this chapter, we study the effects of the change of the constants or parameters on the dynamics of the HH equations and consider the robustness and sensitivity of the equations; we study the bifurcation structure of the HH equations by changing their various parameters. To do so, in this chapter, we consider a slight modification of the original HH equations since the modification is mathematically more tractable.

The Hodgkin–Huxley Theory of Neuronal Excitation

Hodgkin and Huxley (1952) proposed the famous Hodgkin–Huxley (hereinafter referred to as HH) equations which quantitatively describe the generation of action potential of squid giant axon, although there are still arguments against it (Connor et al. 1977; Strassberg and DeFelice 1993; Rush and Rinzel 1995; Clay 1998). The HH equations are important not only in that it is one of the most successful mathematical model in quantitatively describing biological phenomena but also in that the method (the HH formalism or the HH theory) used in deriving the model of a squid is directly applicable to many kinds of neurons and other excitable cells. The equations derived following this HH formalism are called the HH-type equations.

Hodgkin–Huxley-Type Models of Cardiac Muscle Cells

Following the HH formalism introduced in Chap. 2, various kinds of HH-type models of neurons and other excitable cells are proposed (Canavier et al. 1991; Chay and Keizer 1983; Cronin 1987; Gerber and Jakobsson 1993; Hayashi and Ishizuka 1992; Keener and Sneyd 1998; Noble 1995; Rinzel 1990; Traub et al. 1991), and are analyzed (Alexander and Cai 1991; Av-Ron 1994; Bertram 1994; Bertram et al. 1995; Butera 1998; Canavier et al. 1993; Chay and Rinzel 1985; Doi and Kumagai 2005; Guckenheimer et al. 1993; Maeda et al. 1998; Rush and Rinzel 1994; Schweighofer et al. 1999; Terman 1991; Tsumoto et al. 2003, 2006; Yoshinaga et al. 1999). The HH-type equations include many variables depending on the number of different ionic currents and their gating variables considered in the equations, whereas the original HH equations possess only four variables (a membrane voltage, activation and inactivation variables of Na+ current and an activation variable of K+ current). Among the diverse family of HH-type equations, this chapter explores the dynamics and the bifurcation structure of the HH-type equations of heart muscle cells (cardiac myocytes).