Exact computation of the Maximum Entropy Potential of spiking neural networks models

Understanding how stimuli and synaptic connectivity in uence the statistics of spike patterns in neural networks is a central question in computational neuroscience. Maximum Entropy approach has been successfully used to characterize the statistical response of simultaneously recorded spiking neurons responding to stimuli. But, in spite of good performance in terms of prediction, the fitting parameters do not explain the underlying mechanistic causes of the observed correlations. On the other hand, mathematical models of spiking neurons (neuro-mimetic models) provide a probabilistic mapping between stimulus, network architecture and spike patterns in terms of conditional proba- bilities. In this paper we build an exact analytical mapping between neuro-mimetic and Maximum Entropy models.Comment: arXiv admin note: text overlap with arXiv:1309.587

Linear response for spiking neuronal networks with unbounded memory

We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allows us to predict the influence of a weak amplitude time-dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how linear response is explicitly related to neuronal dynamics with an example, the gIF model, introduced by M. Rudolph and A. Destexhe. This example illustrates the collective effect of the stimuli, intrinsic neuronal dynamics, and network connectivity on spike statistics. We illustrate our results with numerical simulations.Comment: 60 pages, 8 figure

Entropy production of Multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain

Consciousness is supported by complex patterns of brain activity which are indicative of irreversible non-equilibrium dynamics. While the framework of stochastic thermodynamics has facilitated the understanding of physical systems of this kind, its application to infer the level of consciousness from empirical data remains elusive. We faced this challenge by calculating entropy production in a multivariate Ornstein-Uhlenbeck process fitted to fMRI brain activity recordings. To test this approach, we focused on the transition from wakefulness to deep sleep, revealing a monotonous relationship between entropy production and the level of consciousness. Our results constitute robust signatures of consciousness while also advancing our understanding of the link between consciousness and complexity from the fundamental perspective of statistical physics

Spike train statistics and Gibbs distributions

International audienceThis paper is based on a lecture given in the LACONEU summer school, Valparaiso, January 2012. We introduce Gibbs distribution in a general setting, including non stationary dynamics, and present then three examples of such Gibbs distributions, in the context of neural networks spike train statistics: (i) Maximum entropy model with spatio-temporal constraints; (ii) Generalized Linear Models; (iii) Conductance based Inte- grate and Fire model with chemical synapses and gap junctions

Can We Hear the Shape of a Maximum Entropy Potential From Spike Trains?

International audienceWe consider a spike-generating stationary Markov process whose transition probabilities are known. We show that there is a canonicalpotential whose Gibbs distribution, obtained from the MaximumEntropy Principle (MaxEnt), is the equilibrium distribution of thisprocess. We provide a method to compute explicitly and exactly thispotential as a linear combination of spatio-temporal interactions.The method is based on the Hammersley Clifford decomposition andon periodic orbits sampling. An explicit correspondence between theparameters of MaxEnt model and the parameters of Markovianmodels like the Generalized-Linear Model can be established. Wealso provide numerical results

Space-time correlations in spike trains and the neural code

International audiencePresentation given in the Workshop MATHEMATICS AND NEUROSCIENCE A DIALOGU

Estimating maximum entropy distributions from periodic orbits in spike trains

We present a method allowing to compute the shape of a Maximum Entropy potential with spatio-temporal constraints, from the periodic orbits appearing in the spike train

Spike train analysis and Gibbs distributions

International audienceSpikes in sensory neurons are conveyed collectively to the cortex using correlated binary patterns (in space and time) whichconstitute “the neural code”. Since patterns occur irregularly it is appropriate to characterize them using probabilistic descriptions or statistical models. Two major approaches attempt to characterize the spike train statistics: The Maximum Entropy Principle (MaxEnt) and NeuronalNetwork modeling (N.N). Remarkably, both approaches are related via the concept of Gibbs distributions. MaxEnt models arerestricted to time-invariant Gibbs distributions , vi the underlying assumption of stationarity, but this concept extends to non-stationary statistics (not defined via entropy), allowing to handle as well statistics of N.N models and GLM with non-stationary dynamics. We show inthis poster that, stationary N.N, GLMmodels and MaxEnt models are equivalent via an explicit mapping. This allows us, inparticular, to interpret the so-called "effective interactions" of MaxEnt models in terms of “real connections” models

Spatio-Temporal Linear Response of Spiking Neuronal Network Models

International audienceWe study the impact of a weak time-dependent external stimulus on the collective statistics of spiking responses in neuronal networks. We extend the current knowledge, assessing the impact over firing rates and cross correlations, to any higher order spatio-temporal correlation [1]. Our approach is based on Gibbs distributions (in a general setting considering non stationary dynamics and infinite memory) [2] and linear response theory. The linear response is written in terms of a correlation matrix, computed with respect to the spiking dynamics without stimulus. We give an example of application in a conductance based integrate-and fire model