Relationships between short and fast brain timescales

Abstract

Brain electric activity exhibits two important features: oscillations with different timescales, characterized by diverse functional and psychological outcomes, and a temporal power law distribution. In order to further investigate the relationships between low- and high- frequency spikes in the brain, we used a variant of the Borsuk–Ulam theorem which states that, when we assess the nervous activity as embedded in a sphere equipped with a fractal dimension, we achieve two antipodal points with similar features (the slow and fast, scale-free oscillations). We demonstrate that slow and fast nervous oscillations mirror each other over time via a sinusoid relationship and provide, through the Bloch theorem from solid-state physics, the possible equation which links the two timescale activities. We show that, based on topological findings, nervous activities occurring in micro-levels are projected to single activities at meso- and macro-levels. This means that brain functions assessed at the higher scale of the whole brain necessarily display a counterpart in the lower ones, and vice versa. Our topological approach makes it possible to assess brain functions both based on entropy, and in the general terms of particle trajectories taking place on donut-like manifolds. Condensed brain activities might give rise to ideas and concepts by combination of different functional and anatomical levels. Furthermore, cognitive phenomena, as well as social activity can be described by the laws of quantum mechanics; memories and decisions exhibit holographic organization. In physics, the term duality refers to a case where two seemingly different systems turn out to be equivalent. This topological duality holds for all the types of spatio-temporal brain activities, independent of their inter- and intra-level relationships, strength, magnitude and boundaries, allowing us to connect the physiological manifestations of consciousness to the electric activities of the brain.

Appendix: Bloch waves and Floquet theorem

The Bloch theorem states that, if you multiply a plane wave by a periodic function, you obtain a Bloch wave, which expresses the energy eigenstates for a particle in a lattice, written as ψn _k , where n is a discrete index.

There are different Bloch waves with the same k, each one with a different periodic component u. Further, the same Bloch wave can be built in different ways, involving different vectors k and different periodic functions u. However, if we take into account just the first Brillouin zone of our lattice, we obtain that every Bloch state has a unique k.

The concept of Bloch theorem from solid-state physics—a second order differential equation-is about crystals in any number of spatial dimensions and deals in particular with the Schrödinger equation. However, it can be also applied in theory of ordinary differential equations, through the Floquet theorem (Floquet 1883). Indeed, the two theorems are almost equivalent (Floquet). The Floquet theorem, through a coordinate change in the lattice, transforms the original periodic system into a more manageable traditional linear system with constant, real coefficients. In other words, we map a fundamental matrix solution into a matrix function depending on the time, giving rise to a time-dependent change of coordinates. The Floquet theorem holds for any homogeneous, linear system of first order differential equations with a periodic coefficient matrix. Through a coordinate change in the lattice, the Floquet’s theorem transforms the periodic system into a traditional linear system with constant, real coefficients.

We start from a linear first order differential equation:

$$\frac{{d\left( {x(t)} \right)}}{dt} = A\left( t \right)x,$$

where x(t) is a column vector of length n, and A(t) is an n × n periodic matrix with period T.Then, for all t ∊ R:

$$\varPhi \left( {t + T} \right) = \varPhi \left( t \right)\varPhi^{ - 1} \left( 0 \right)\varPhi \left( T \right),$$

where Φ(t) is a fundamental matrix solution of the above differential equation ${{d\left( {x(t)} \right)}}\div{dt} = A\left( t \right)x$, and Φ ⁻¹(0) Φ(T) is the monodromy matrix. Now consider the n × n matrices: B, P, Q, R: For each matrix B such that:

$$E^{TB} = \varPhi^{ - 1} \left( 0 \right)\varPhi \left( T \right),$$

there is a periodic (period T) matrix function t → P(t) such that:Φ(t) = P(t)E ^tB for all t∊ R. This representation is the Floquet normal form for the fundamental matrix Φ(t).

There is also a real matrix R and a real periodic function (period −2T) matrix function t → Q(t),  which is continuous and periodic—such that:

$$\varPhi \left( t \right) = Q\left( t \right)E^{tR} \, \quad {\text{for all }}t \in R.$$

The latter mapping gives rise to a time-dependent change of coordinates:

$$y = Q^{ - 1} \left( t \right)x,$$

under which the original system becomes a linear system with real constant coefficients y = Ry. The mapping of a fundamental matrix solution for such a differential equation into a (time-dependent) matrix function gives rise to a time-dependent change of coordinates, under which the original periodic system becomes a linear system with real constant coefficients y = Ry.

The eigenvalues of e ^TB are called the characteristic multipliers of the system, while the characteristic exponent, called the Floquet exponent, is a complex μ such that e ^μT is a characteristic multiplier of the system. Floquet exponents are not unique and their real parts correspond to the Lyapunov exponents.

The linear differential equations with periodic coefficients have been widely used in many scientific fields. In particular, they provide a versatile tool for the stability analysis of physical systems equipped with a periodic steady-state and infinite memory, such as Brownian particles and circuit resonators (Traversa et al. 2013). The Floquet multipliers have been also used to assess the stability of periodic motion in natural rhythmic - movements in humans and machines-, not just in linear systems, but also in stochastic noise and in limit-cycle, nonlinear oscillators (Ahn and Hogan 2015).