Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

Abstract

The Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of scienThe Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.ce, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

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

“…They proved, e.g., that if components of the series system are iid and belong to the IFR family, then the generalized cumulative residual entropy of the conditional residual system lifetime , , is decreasing. Other connections between Shannon entropy (see Wang et al [ 16 ], Jäntschi and Bolboacă [ 17 ], Tănăsescu and Popescu [ 18 ]) and order statistics were presented, e.g., by Koné [ 19 ], Jäntschi [ 20 ], Courtier et al [ 21 ] and Cofré et al [ 22 ].…”