Let us say i have 100 time series, each time series represents one neuron's activity over 10k time points. How should one test if these are generated by a dynamical system and an attractor?
Some neuroscience papers just assume this is a dynamical system. I was wondering if one can use the available measurement to proof that.
Thanks ahead!
If there are reasons to assume that a time series (or a set of time series are) are related with a certain dynamical system, and in there is not an already available mathematical model of a dynamical system that could explain the series, it is possible to use data to try to reconstruct a state-space of an hypothetical dynamis system using embedding tecniques. A d-dimensional vector of scalar values of the measured dependent variable x, taken with a variable lag l in the independent variable (time) is constructed: (x(t), x(t+l), x(t+2l),....,x(t+(d-1)l). A suitable selection of dimensions and time lags must be done, and from the evolution of these vectors a state-space reconstruction (embedding) can be done. A short explanation of the procedure can be found in Neil Gershenfeld´s book "The Nature of Mathematical Modeling".
Thank you so much for your answer. @Roberto Suárez-Ántola.
Just want to make sure i understand correctly:
1) To use time-delay embedding to reconstruct the state space, one has to assume that the time series is relate to a dynamical system. Is it right? Can i blindly use time-delay embedding without knowing if the time series is from a dynamical system?
2) And if the state space reconstructed by time-delay embedding resembles an attract-like structure (say in a 3D plot), can we say this time series forms an attractor?
1) You can apply embedding procedures without knowing if the time series originates in a dynamical system.
2) If the state space reconstructed by time-delay embedding resembles an attract-like structure you can accept the possibility of an attractor, in provisional form. (The behavior under embedding of a time series obtained from the numerical solution of the reduced order mathematical model of the athmosphere due to Lorenz with the emergence of a pattern in the plot that suggests an attractor is a suitable example, although in this case the existance of the attractor is proved from the structure of the mathematical model.)
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