Perfect Integrate and fire with noisy adaptation or fractional noise (Richard et al 2018)

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"Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having Long-range dependency (LRD). However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models."
1 . Richard A, Orio P, Tanré E (2018) An integrate-and-fire model to generate spike trains with long-range dependence Journal of Computational Neuroscience
Model Information (Click on a link to find other models with that property)
Model Type:
Brain Region(s)/Organism:
Cell Type(s): Abstract integrate-and-fire leaky neuron;
Gap Junctions:
Simulation Environment: Python;
Model Concept(s):
Implementer(s): Orio, Patricio [patricio.orio at]; Richard, Alexandre ;
Perfect Integrate and Fire (IF) model with noisy adaptation or fractional noise

This code is companion to the manuscript "An integrate-and-fire model to generate
spike trains with long-range dependence", 
by authors Alexandre Richard, Patricio Orio and Etienne Tanré

When run, this script will generate 5 plots with different simulation conditions:
    1) Noise in voltage (V)
    2) Noise in the adaptation (Z)
    3) Noise in both
    4) Noise in adaptation with larger time constant
    5) Fractional Noise with alpha=0.7
Plots will contain 9 panels (from top to bottom, then left to right):
    1) Short time course of both variables (V,Z)
    2) Long (500s) inter-spike interval (ISI) plot
    3) Windowed Kolmogorov-Smirnoff test, to visually show stationarity of ISIs
    4) R/S (rescaled range) plot for the whole ISI sequence. Red line is the 
       fit of all points to a power law (straight line in loglog). The slope (H),
       r value and p-value of the fit are indicated. Colored lines are fits of
       small sets of points, in a moving window fashion.
    5) Plot of the moving slopes (colored lines in above plot) against the length
       of the sequence. Red line and blue shadow indicate the mean and 2*SD
       obtained with 100x surrogate data
    6) Autocorrelation plot of the ISI sequence, in log-linear scale
    7) Detrended fluctuation analysis (DFA) plot. Red and colored lines are the
       same as in plot 4
    8) Similar to plot 5, but with the DFA
    9) Autocorrelation plot of the ISI sequence, in log-log scale

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