Macroscopic equations governing noisy spiking neuronal populations

At functional scales, cortical behavior results from the complex interplay of a large number of excitable cells operating in noisy environments. Such systems resist to mathematical analysis, and computational neurosciences have largely relied on heuristic partial (and partially justified) macroscopic models, which successfully reproduced a number of relevant phenomena. The relationship between these macroscopic models and the spiking noisy dynamics of the underlying cells has since then been a great endeavor. Based on recent mean-field reductions for such spiking neurons, we present here {a principled reduction of large biologically plausible neuronal networks to firing-rate models, providing a rigorous} relationship between the macroscopic activity of populations of spiking neurons and popular macroscopic models, under a few assumptions (mainly linearity of the synapses). {The reduced model we derive consists of simple, low-dimensional ordinary differential equations with} parameters and {nonlinearities derived from} the underlying properties of the cells, and in particular the noise level. {These simple reduced models are shown to reproduce accurately the dynamics of large networks in numerical simulations}. Appropriate parameters and functions are made available {online} for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models

Channel Density Regulation of Firing Patterns in a Cortical Neuron Model

AbstractModifying the density and distribution of ion channels in a neuron (by natural up- and downregulation or by pharmacological intervention or by spontaneous mutations) changes its activity pattern. In this investigation we analyzed how the impulse patterns are regulated by the density of voltage-gated channels in a neuron model based on voltage-clamp measurements of hippocampal interneurons. At least three distinct oscillatory patterns, associated with three distinct regions in the Na-K channel density plane, were found. A stability analysis showed that the different regions are characterized by saddle-node, double-orbit, and Hopf-bifurcation threshold dynamics, respectively. Single, strongly graded action potentials occur in an area outside the oscillatory regions, but less graded action potentials occur together with repetitive firing over a considerable range of channel densities. The relationship found here between channel densities and oscillatory behavior may partly explain the difference between the principal spiking patterns previously described for crab axons (class 1 and 2) and cortical neurons (regular firing and fast spiking)

Travelling waves in a model of quasi-active dendrites with active spines

Dendrites, the major components of neurons, have many different types of branching structures and are involved in receiving and integrating thousands of synaptic inputs from other neurons. Dendritic spines with excitable channels can be present in large densities on the dendrites of many cells. The recently proposed Spike-Diffuse-Spike (SDS) model that is described by a system of point hot-spots (with an integrate-and-fire process) embedded throughout a passive tree has been shown to provide a reasonable caricature of a dendritic tree with supra-threshold dynamics. Interestingly, real dendrites equipped with voltage-gated ion channels can exhibit not only supra-threshold responses, but also sub-threshold dynamics. This sub-threshold resonant-like oscillatory behaviour has already been shown to be adequately described by a quasi-active membrane. In this paper we introduce a mathematical model of a branched dendritic tree based upon a generalisation of the SDS model where the active spines are assumed to be distributed along a quasi-active dendritic structure. We demonstrate how solitary and periodic travelling wave solutions can be constructed for both continuous and discrete spine distributions. In both cases the speed of such waves is calculated as a function of system parameters. We also illustrate that the model can be naturally generalised to an arbitrary branched dendritic geometry whilst remaining computationally simple. The spatio-temporal patterns of neuronal activity are shown to be significantly influenced by the properties of the quasi-active membrane. Active (sub- and supra-threshold) properties of dendrites are known to vary considerably among cell types and animal species, and this theoretical framework can be used in studying the combined role of complex dendritic morphologies and active conductances in rich neuronal dynamics

Traveling waves in the Baer and Rinzel model of spine studded dendritic tissue

The Baer and Rinzel model of dendritic spines uniformly distributed along a dendritic cable is shown to admit a variety of regular traveling wave solutions including solitary pulses, multiple pulses and periodic waves. We investigate numerically the speed of these waves and their propagation failure as functions of the system parameters by numerical continuation. Multiple pulse waves are shown to occur close to the primary pulse, except in certain exceptional regions of parameter space, which we identify. The propagation failure of solitary and multiple pulse waves is shown to be associated with the destruction of a saddle-node bifurcation of periodic orbits. The system also supports many types of irregular wave trains. These include waves which may be regarded as connections to periodics and bursting patterns in which pulses can cluster together in well-defined packets. The behavior and properties of both these irregular spike-trains is explained within a kinematic framework that is based on the times of wave pulses. The dispersion curve for periodic waves is important for such a description and is obtained in a straightforward manner using the numerical scheme developed for the study of the speed of a periodic wave. Stability of periodic waves within the kinematic theory is given in terms of the derivative of the dispersion curve and provides a weak form of stability that may be applied to solutions of the traveling wave equations. The kinematic theory correctly predicts the conditions for period doubling bifurcations and the generation of bursting states. Moreover, it also accurately describes the shape and speed of the traveling front that connects waves with two different periods

Phase-responsiveness transmission in a network of quadratic integrate-and-fire neurons

In the study of the dynamics of neuronal networks, it is interesting to see how the interaction between neurons can elicit different behaviours in each individual one. Moreover, this can lead to the population exhibiting collective phenomena that is not intrinsic to a single cell, such as synchronization. In this project, we work with a large-scale network and a firing-rate model of quadratic integrate-and-fire (QIF) neurons. After studying the dynamics of the QIF model and computing its phase response curve (PRC), we propose an algorithm to describe the population through the PRCs. Our method is able to replicate the same dynamics we observe with the aforementioned models and it also serves us to gain more insight into the transmission of pulses and to explain how a network can maintain a state of synchronized firing

Assembling life : models, the cell, and the reformations of biological science, 1920-1960

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Voltage dependence of Hodgkin-Huxley rate functions for a multi-stage K channel voltage sensor within a membrane

The activation of a $K^+$ channel sensor in two sequential stages during a voltage clamp may be described as the translocation of a Brownian particle in an energy landscape with two large barriers between states. A solution of the Smoluchowski equation for a square-well approximation to the potential function of the S4 voltage sensor satisfies a master equation, and has two frequencies that may be determined from the forward and backward rate functions. When the higher frequency terms have small amplitude, the solution reduces to the relaxation of a rate equation, where the derived two-state rate functions are dependent on the relative magnitude of the forward rates ($\alpha$ and $\gamma$) and the backward rates ($\beta$ and $\delta$) for each stage. In particular, the voltage dependence of the Hodgkin-Huxley rate functions for a $K^+$ channel may be derived by assuming that the rate functions of the first stage are large relative to those of the second stage - $\alpha \gg \gamma$ and $\beta \gg \delta$. For a {\em Shaker} IR $K^+$ channel, the first forward and backward transitions are rate limiting ($\alpha < \gamma$ and $\delta \ll \beta$), and for an activation process with either two or three stages, the derived two-state rate functions also have a voltage dependence that is of a similar form to that determined for the squid axon. The potential variation generated by the interaction between a two-stage $K^+$ ion channel and a noninactivating $Na^+$ ion channel is determined by the master equation for $K^+$ ion channel activation and the ionic current equation when the $Na^+$ ion channel activation time is small, and if $\beta \ll \delta$ and $\alpha \ll \gamma$, the system may exhibit a small amplitude oscillation between spikes, or mixed-mode oscillation.Comment: 31 pages, 14 figure

Modeling the coupling of action potential and electrodes

The present monograph is a study of pulse propagation in nerves. The main project of this work is modeling and simulation of the action potential propagation in a neuron and its interaction with the electrodes in the context of neurochip application. In the first part, I work with an adapted model of FitzHugh-Nagumo derived from the Hodgkin-Huxley model. The second part was the result of turning the spotlight-on onto the drawbacks of Hodgkin-Huxley model and to bring forth, an alternative model: soliton model. The purpose is to comprehend the role of membrane state in the pulse propagation