Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

Optimal Population Coding for Dynamic Input by Nonequilibrium Networks

The efficient coding hypothesis states that neural response should maximize its information about the external input. Theoretical studies focus on optimal response in single neuron and population code in networks with weak pairwise interactions. However, more biological settings with asymmetric connectivity and the encoding for dynamical stimuli have not been well-characterized. Here, we study the collective response in a kinetic Ising model that encodes the dynamic input. We apply gradient-based method and mean-field approximation to reconstruct networks given the neural code that encodes dynamic input patterns. We measure network asymmetry, decoding performance, and entropy production from networks that generate optimal population code. We analyze how stimulus correlation, time scale, and reliability of the network affect optimal encoding networks. Specifically, we find network dynamics altered by statistics of the dynamic input, identify stimulus encoding strategies, and show optimal effective temperature in the asymmetric networks. We further discuss how this approach connects to the Bayesian framework and continuous recurrent neural networks. Together, these results bridge concepts of nonequilibrium physics with the analyses of dynamics and coding in networks.

A framework for preparing a stochastic nonlinear integrate-and-fire model for integrated information theory

This paper presents a framework for spiking neural networks to be prepared for the Integrated Information Theory (IIT) analysis, using a stochastic nonlinear integrate-and-fire model. The model includes the crucial dynamics of the all-or-none law and after-spike refractoriness. The noise is modelled as an additive term in the system's equations. By preparing the model for the IIT analysis, it is meant to determine the length of the analysis time-window and the transition probability distributions required for the IIT 3.0. To this end, a system of differential equations is proposed to estimate the time evolution of the system's mean and covariance. Assuming the binary Fired/Silent activity as the possible states of each neuron, an algorithm is proposed to calculate the required probability distributions. As long as the Fired/Silent probabilities are only concerned, the Gaussian density assumption with the estimated moments is a reasonable estimate. The synaptic inputs are treated as random variables with low variances to avoid the costs of conditioning on the system's past activities. The Monte-Carlo simulation is used to validate the estimation methods. To increase the reliability of the inductive inference behind the Monte-Carlo method, various stimulation protocols are applied to evoke the dynamics of the equations.

Retinal Processing: Insights from Mathematical Modelling

The retina is the entrance of the visual system. Although based on common biophysical principles, the dynamics of retinal neurons are quite different from their cortical counterparts, raising interesting problems for modellers. In this paper, I address some mathematically stated questions in this spirit, discussing, in particular: (1) How could lateral amacrine cell connectivity shape the spatio-temporal spike response of retinal ganglion cells? (2) How could spatio-temporal stimuli correlations and retinal network dynamics shape the spike train correlations at the output of the retina? These questions are addressed, first, introducing a mathematically tractable model of the layered retina, integrating amacrine cells' lateral connectivity and piecewise linear rectification, allowing for computing the retinal ganglion cells receptive field together with the voltage and spike correlations of retinal ganglion cells resulting from the amacrine cells networks. Then, I review some recent results showing how the concept of spatio-temporal Gibbs distributions and linear response theory can be used to characterize the collective spike response to a spatio-temporal stimulus of a set of retinal ganglion cells, coupled via effective interactions corresponding to the amacrine cells network. On these bases, I briefly discuss several potential consequences of these results at the cortical level.

Bounds on the Lifetime Expectations of Series Systems with IFR Component Lifetimes

We consider series systems built of components which have independent identically distributed (iid) lifetimes with an increasing failure rate (IFR). We determine sharp upper bounds for the expectations of the system lifetimes expressed in terms of the mean, and various scale units based on absolute central moments of component lifetimes. We further establish analogous bounds under a more stringent assumption that the component lifetimes have an increasing density (ID) function. We also indicate the relationship between the IFR property of the components and the generalized cumulative residual entropy of the series system lifetime.