Complexity in Electrophysiological Dynamics. Emergence and measures of organization

Action potentials play an important role in the dynamics of cell-cell communication and they are thus of key relevance in neural tissues. They evidence the intrinsic dynamics of intracellular and membrane exchanges, and transmit nerve impulses both at gap junctions (through direct ionic exchange) and synapses (by initiating neurotransmission). In practice, action potential impulse trains exhibit a complex dynamics. Each impulse is highly anharmonic, and the shapes and amplitudes of impulses evolve continuously with regulation-based modulation. Additionally, the signal is quasi-periodic, and period fluctuations are far from random: they are highly structured across scales and follow what appears as non-Markovian dynamics. We propose to characterize anharmonicity in the signal as smooth perturbations of harmonicity, by introducing multiple symmetry breaks of variable strength at given phase axes. P. Hanusse has shown that the resulting framework is analytically treatable with a generalization of pseudo-trigonometric functions that include a shape factor as anharmonicity parameter r. These generalizations are defined up to an order n of convolutions, pcosn = sum (on k) k^(-n) r^(k-1) cos(nx) . Order 1 can be solved and it corresponds to a phase equation defined as a fraction of trigonometric polynomials. We show that typical real-world electrophysiological signals, with smooth deviations from harmonicity, are typically well described with just the first few terms and result in a rather compact, sparse representation. In particular, we have done an analysis of FitzHugh-Nagumo impulse trains; we have found that 3 anharmonic terms reconstruct better than an equivalent 8-term Fourier representation, with less than half the PSNR and no artifacts from Gibbs phenomenon. Reference: P. Hanusse, Rencontres du Non-Lineaire (2010)