Towards Equations for Brain Dynamics and the Concept of Extended Connectome

Peters JF, Tozzi A, Deli E.  

The brain is a system at the edge of chaos equipped with nonlinear dynamics and functional energetic
landscapes. However, still doubts exist concerning the type of attractors or the trajectories followed by particles in
the nervous phase space. Starting from a system governed by differential equations in which a dissipative strange
attractor coexists with an invariant conservative torus, we developed a 3D model of brain phase space which has the
potential to be operationalized and assessed empirically. We achieved a system displaying both a torus and a strange
attractor, depending just on the initial conditions. Further, the system generates a funnel-like attractor equipped with a
fractal structure. Changes in three brain phase parameters lead to modifications in funnel’s breadth or in torus/attractor
superimposition. We have found that the higher frequencies of evoked activities are more deterministic due to the
greater funnel breadth with decreasing degrees of freedom. In contrast, the resting state is formed by lower frequencies
represents greater degrees of freedom. Thus, our model explains a large repertoire of brain functions and activities,
such as sensations/perceptions, memory and self-generated thoughts.
 
 

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The brain is a system at the edge of chaos equipped with nonlinear dynamics and functional energetic
landscapes. However, still doubts exist concerning the type of attractors or the trajectories followed by particles in
the nervous phase space. Starting from a system governed by differential equations in which a dissipative strange
attractor coexists with an invariant conservative torus, we developed a 3D model of brain phase space which has the
potential to be operationalized and assessed empirically. We achieved a system displaying both a torus and a strange
attractor, depending just on the initial conditions. Further, the system generates a funnel-like attractor equipped with a
fractal structure. Changes in three brain phase parameters lead to modifications in funnel’s breadth or in torus/attractor
superimposition. We have found that the higher frequencies of evoked activities are more deterministic due to the
greater funnel breadth with decreasing degrees of freedom. In contrast, the resting state is formed by lower frequencies
represents greater degrees of freedom. Thus, our model explains a large repertoire of brain functions and activities,
such as sensations/perceptions, memory and self-generated thoughts.