Low-dimensional models of single neurons: A review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as sub-threshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type

The investigation of variable nernst equilibria on isolated neurons and coupled neurons forming discrete and continuous networks

Since the introduction of the Hodgkin-Huxley equations, used to describe the excitation of neurons, the Nernst equilibria for individual ion channels have assumed to be constant in time. Recent biological recordings call into question the validity of this assumption. Very little theoretical work has been done to address the issue of accounting for these non-static Nernst equilibria using the Hodgkin-Huxley formalism. This body of work incorporates non-static Nernst equilibria into the generalized Hodgkin-Huxley formalism by considering the first-order effects of the Nernst equation. It is further demonstrated that these effects are likely dominate in neurons with diameters much smaller than that of the squid giant axon and permeate important information processing regions of the brain such as the hippocampus. Particular results of interest include single-cell bursting due to the interplay of spatially separated neurons, pattern formation via spiral waves within a soliton-like regime, and quantifiable shifts in the multifractality of hippocampal neurons under the administration of various drugs at varying dosages. This work provides a new perspective on the variability of Nernst equilibria and demonstrates its utility in areas such as pharmacology and information processing

Frequency preference and reliability of signal integration

Die Eigenschaften einzelner Nervenzellen sind von grundlegender Bedeutung für die Verarbeitung von Informationen im Nervensystem. Neuronen antworten auf Eingangsreize durch Veränderung der elektrischen Spannung über die Zellmembran. Die Spannungsantwort wird dabei durch die Dynamik der Ionenkanäle in der Zellmembran bestimmt. In dieser Arbeit untersuche ich anhand von leitfähigkeits-basierten Modellneuronen den Einfluss von Ionenkanälen auf zwei Aspekte der Signalverarbeitung: die Frequenz-Selektivität sowie die Zuverlässigkeit und zeitliche Präzision von Aktionspotentialen. Zunächst werden die zell-intrinsischen Mechanismen identifiziert, welche the Frequenz-Selektivität und die Zuverlässigkeit bestimmen. Weiterhin wird untersucht, wie Ionenkanäle diese Mechanismen modulieren können, um die Integration von Signalen zu optimieren. Im ersten Teil der Arbeit wird demonstriert, dass der Mechanismus der unterschwelligen Resonanz, so wie er bisher für periodische Signale beobachtet wurde, auch auf nicht-periodische Signale anwendbar ist und sich ebenfalls in den Feuerraten niederschlägt. Im zweiten Teil wird gezeigt, dass zeitliche Präzision und Zuverlässigkeit von Aktionspotentialen mit der Stimulusfrequenz variieren und dass, in Abhängigkeit davon, ob das Stimulusmittel über- oder unterhalb der Feuerschwelle liegt, zwei Stimulusregime unterschieden werden müssen. In beiden Regimen existiert eine bevorzugte Stimulusfrequenz, welche durch die Gesamtleitfähigkeit und die Dynamik spezifischer Ionenkanäle moduliert werden kann. Im dritten Teil wird belegt, dass Ionenkanäle die Zuverlässigkeit auch direkt über eine Veränderung der Sensitivität einer Zelle gegenüber neuronalem Rauschen bestimmen können. Die Ergebnisse der Arbeit lassen auf eine wichtige Rolle der dynamischen Regulierung der Ionenkanäle für die Frequenz-Selektivität und die zeitliche Präzision und Zuverlässigkeit der Spannungsantworten schließen.The properties of individual neurons are of fundamental importance for the processing of information in the nervous system. The generation of voltage responses to input signals, in particular, depends on the properties of ion channels in the cell membrane. Within this thesis, I employ conductance-based model neurons to investigate the effect of ionic conductances and their dynamics on two aspects of signal processing: frequency-selectivity and temporal precision and reliability of spikes. First, the cell-intrinsic mechanisms that determine frequency selectivity and spike timing reliability are identified on the basis of conductance-based model neurons. Second, it is analyzed how ionic conductances can serve to modulate these mechanisms in order to optimize signal integration. In the first part, the frequency selectivity of subthreshold response amplitudes previously observed for periodic stimuli is proven to extend to nonperiodic stimuli and to translate into firing rates. In the second part, it is demonstrated that spike timing reliability is frequency-selective and that two different stimulus regimes have to be distinguished, depending on whether the stimulus mean is below or above threshold. In both cases, resonance effects determine the most reliable stimulus frequency. It is shown that this frequency preference can be modulated by the peak conductance and dynamics of specific ion channels. In the third part, evidence is provided that ionic conductances determine spike timing reliability beyond changes in the preferred frequency. It is demonstrated that ionic conductances also exert a direct influence on the sensitivity of the timing of spikes to neuronal noise. The findings suggest an important role for dynamic neuromodulation of ion channels with regard to frequency selectivity and spike timing reliability

Identification of Molecular Pathologies Sufficient to Cause Neuropathic Excitability in Primary Somatosensory Afferents Using Dynamical Systems Theory

Pain caused by nerve injury (i.e. neuropathic pain) is associated with development of neuronal hyperexcitability at several points along the pain pathway. Within primary afferents, numerous injury-induced changes have been identified but it remains unclear which molecular changes are necessary and sufficient to explain cellular hyperexcitability. To investigate this, we built computational models that reproduce the switch from a normal spiking pattern characterized by a single spike at the onset of depolarization to a neuropathic one characterized by repetitive spiking throughout depolarization. Parameter changes that were sufficient to switch the spiking pattern also enabled membrane potential oscillations and bursting, suggesting that all three pathological changes are mechanistically linked. Dynamical analysis confirmed this prediction by showing that excitability changes co-develop when the nonlinear mechanism responsible for spike initiation switches from a quasi-separatrix-crossing to a subcritical Hopf bifurcation. This switch stems from biophysical changes that bias competition between oppositely directed fast- and slow-activating conductances operating at subthreshold potentials. Competition between activation and inactivation of a single conductance can be similarly biased with equivalent consequences for excitability. “Bias” can arise from a multitude of molecular changes occurring alone or in combination; in the latter case, changes can add or offset one another. Thus, our results identify pathological change in the nonlinear interaction between processes affecting spike initiation as the critical determinant of how simple injury-induced changes at the molecular level manifest complex excitability changes at the cellular level. We demonstrate that multiple distinct molecular changes are sufficient to produce neuropathic changes in excitability; however, given that nerve injury elicits numerous molecular changes that may be individually sufficient to alter spike initiation, our results argue that no single molecular change is necessary to produce neuropathic excitability. This deeper understanding of degenerate causal relationships has important implications for how we understand and treat neuropathic pain

A biologically plausible model of time-scale invariant interval timing

The temporal durations between events often exert a strong influence over behavior. The details of this influence have been extensively characterized in behavioral experiments in different animal species. A remarkable feature of the data collected in these experiments is that they are often time-scale invariant. This means that response measurements obtained under intervals of different durations coincide when plotted as functions of relative time. Here we describe a biologically plausible model of an interval timing device and show that it is consistent with time-scale invariant behavior over a substantial range of interval durations. The model consists of a set of bistable units that switch from one state to the other at random times. We first use an abstract formulation of the model to derive exact expressions for some key quantities and to demonstrate time-scale invariance for any range of interval durations. We then show how the model could be implemented in the nervous system through a generic and biologically plausible mechanism. In particular, we show that any system that can display noise-driven transitions from one stable state to another can be used to implement the timing device. Our work demonstrates that a biologically plausible model can qualitatively account for a large body of data and thus provides a link between the biology and behavior of interval timing