In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministicmarkov process in infinite dimensions. Adv. Appl. Probab. 44(2012) 749–773], the authors addressed the question of averaging for a slow-fastPiecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the presentpaper, we carry on and complete this work by the mathematical analysis of the fluctuationsof the slow-fast system around the averaged limit. A central limit theorem is derived andthe associated Langevin approximation is considered. The motivation for this work is thestudy of stochastic conductance based neuron models which describe the propagation of anaction potential along a nerve fiber.

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

Tactile sensing is advantageous for the acquisition of local, proximal information such as the contact condition between a finger and an object. This type of sensing, however, is not suited for recognizing an entire object that is easily recognized by vision. The objective of this paper is to ease the limitations experienced in tactile sensing by using both a neural model based on the human tactile sensation and a tactile-oriented associative memory model to enable a robot to recognize object contours. In the model, first the direction vectors belonging to segments of the object contour are obtained from a filtered tactile pattern of the simulated neurons' excitation. Second, the vectors are quantized by the chain-symbolizing method and stored for use in a memory matrix that accumulates matrix-products between the vector and its transposition. In the recalling process, complete vectors are remembered even if some input vector elements are missing. In the experiments, a robotic manipulator equipped with a tactile sensor traces five types of contours, these being a circle, a square, a triangle, a star, and a hexagon. After the robot recalls the complete contours, it is able to recognize a complete contour by just touching even a part of a contour.