The four-dimensional Hodgkin–Huxley equations are considered as the prototype for description of neural pulse propagation. Their mathematical complexity and sophistication prompted a simplified two-dimensional model, the FitzHugh–Nagumo equations, which display many of the former’s dynamical features. Numerical and mathematical analysis are employed to demonstrate that the FitzHugh–Nagumo equat...

We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of 1-pulse undetermined width and propagating speed. Next, compute associated for this from which conditions existence saddle-node bifurcation as zeros its derivatives. obtain same by using diff...

Minimal nonlinear dynamic neuron models of the generic bifurcation type may provide the middle way between the detailed models favored by experimentalists and the simplified threshold and rate model of computational neuroscientists. This thesis investigates to which extent generic bifurcation type models grasp the essential dynamical features that may turn out play a role in cooperative neural ...

Abstract: The FitzHugh-Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level. We study a stochastic version of the model from a probabilistic point of view. The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution. The bi-dimensional stochastic process is β-mixing. The stationary density can be estim...

We introduce a generalized thermostatistics based on Kolmogorov-Nagumo averages and appropriately selected information measures. The formalism includes Tsallis non-extensive thermostatistics, but also extensive thermostatistics based on Rényi entropy. The Curie-Weiss model is discussed as an example.

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved f...

In this paper we adopt a temporal coding approach to neuronal modeling of the visual cortex, using oscillations. We propose a hierarchy of three processing modules corresponding to di erent levels of representation. The rst layer encodes the input image (stimulus) into an array of units, while the second layer consists of a network of FitzHugh-Nagumo oscillators. The dynamical behaviour of the ...

A space clamped neural axon along which flows a constant current exhibits current dependent oscillations of the action potential. With increasing current a characteristic Hopf bifurcation scenario can be observed: approximately harmonic small amplitude oscillations arise and bifurcate subsequently into relaxation oscillations of relatively large amplitude. Amplitude and period of the oscillatio...

An optoelectronic implementation of a spiking neuron model based on the FitzHugh-Nagumo equations is presented. A tunable semiconductor laser source and a spectral filter provide a nonlinear mapping from driver voltage to detected signal. Linear electronic feedback completes the implementation, which allows either electronic or optical input signals. Experimental results for a single system and...