Gain control with A-type potassium current: IA as a switch between divisive and subtractive inhibition

Neurons process information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron's output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two "modes" of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. We use simulations and mathematical analysis of a neuron model to find the specific conditions for which inhibitory inputs have subtractive or divisive effects. We identify a novel role for the A-type Potassium current (IA). In our model, this fast-activating, slowly- inactivating outward current acts as a switch between subtractive and divisive inhibition. If IA is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if IA is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods to define how a spike threshold condition depends on synaptic inputs and IA. Our findings suggest that neurons can "self-regulate" the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for IA would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract.Comment: 20 pages, 11 figure

Gain Control With A-Type Potassium Current: IA As A Switch Between Divisive And Subtractive Inhibition

Neurons process and convey information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition typically suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron’s output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two “modes” of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. In this study, we use simulations and mathematical analysis of a neuron model to find the specific conditions (parameter sets) for which inhibitory inputs have subtractive or divisive effects. Significantly, we identify a novel role for the A-type Potassium current (IA). In our model, this fast-activating, slowly-inactivating outward current acts as a switch between subtractive and divisive inhibition. In particular, if IA is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if IA is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods (plane analysis and fast-slow dissection) to define how a spike threshold condition depends on synaptic inputs and IA. Our findings suggest that neurons can “self-regulate” the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for IA would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract

Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States

The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled activation subunits, while the DA was modeled using uncoupled activation subunits. Implementations of DA with coupled subunits, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable - allowing an easy and efficient DA implementation. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods.Comment: 32 text pages, 10 figures, 1 supplementary text + figur

The what and where of adding channel noise to the Hodgkin-Huxley equations

One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, synthesizes the impact of ionic currents on a cell's voltage -- and the highly nonlinear impact of that voltage back on the currents themselves -- into the rapid push and pull of the action potential. Latter studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic Hodgkin-Huxley equations. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly Matlab simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.Comment: 14 pages, 3 figures, review articl

Citizenship Education and Liberalism: A State of the Debate Analysis 1990–2010

What kind of citizenship education, if any, should schools in liberal societies promote? And what ends is such education supposed to serve? Over the last decades a respectable body of literature has emerged to address these and related issues. In this state of the debate analysis we examine a sample of journal articles dealing with these very issues spanning a twenty-year period with the aim to analyse debate patterns and developments in the research field. We first carry out a qualitative analysis where we design a two-dimensional theoretical framework in order to systematise the various liberal debate positions, and make us able to study their justifications, internal tensions and engagements with other positions. In the ensuing quantitative leg of the study we carry out a quantitative bibliometric analysis where we weigh the importance of specific scholars. We finally discuss possible merits and flaws in the research field, as evidenced in and by the analysis

The Resource Curse and Rentier States in the Caspian Region : A Need for Context Analysis

Although much attention is paid to the Caspian region with regard to energy issues, the domestic consequences of the region’s resource production have so far constituted a neglected field of research. A systematic survey of the latest research trends in the economic and political causalities of the resource curse and of rentier states reveals that there is a need for context analysis. In reference to this, the paper traces any shortcomings and promising approaches in the existent body of literature on the Caspian region. Following on from this, the paper then proposes a new approach; specifically, one in which any differences and similarities in the context conditions are captured. This enables a more precise exploration of the exact ways in which they form contemporary post-Soviet Caspian rentier states.Obwohl der Region am Kaspischen Meer im Zuge von Energiediskursen große Aufmerksamkeit zuteil wird, stellen die innerstaatlichen Folgen der Ressourcenproduktion in der Region ein bislang vernachlässigtes Forschungsfeld dar. Ein systematischer Überblick über die jüngsten Forschungstrends zu wirtschaftlichen und politischen Kausalzusammenhängen des Ressourcenfluchs und zu Rentierstaaten offenbart die Notwendigkeit von Kontextanalysen. Hierauf Bezug nehmend, analysiert der Aufsatz sowohl die Mängel als auch viel versprechende Ansätze in der betreffenden Literatur zur Region am Kaspischen Meer. Der Aufsatz stellt letztendlich einen neuen Ansatz vor, der Unterschiede und Gemeinsamkeiten in den Kontextbedingungen erfasst, um zu erforschen, wie diese die gegenwärtigen post-sowjetischen Rentierstaaten in der Region am Kaspischen Meer tatsächlich prägen

A Model of Electrically Stimulated Auditory Nerve Fiber Responses with Peripheral and Central Sites of Spike Generation

A computational model of cat auditory nerve fiber (ANF) responses to electrical stimulation is presented. The model assumes that (1) there exist at least two sites of spike generation along the ANF and (2) both an anodic (positive) and a cathodic (negative) charge in isolation can evoke a spike. A single ANF is modeled as a network of two exponential integrateand-fire point-neuron models, referred to as peripheral and central axons of the ANF. The peripheral axon is excited by the cathodic charge, inhibited by the anodic charge, and exhibits longer spike latencies than the central axon; the central axon is excited by the anodic charge, inhibited by the cathodic charge, and exhibits shorter spike latencies than the peripheral axon. The model also includes subthreshold and suprathreshold adaptive feedback loops which continuously modify the membrane potential and can account for effects of facilitation, accommodation, refractoriness, and spike-rate adaptation in ANF. Although the model is parameterized using data for either single or paired pulse stimulation with monophasic rectangular pulses, it correctly predicts effects of various stimulus pulse shapes, stimulation pulse rates, and level on the neural response statistics. The model may serve as a framework to explore the effects of different stimulus parameters on psychophysical performance measured in cochlear implant listeners