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The fractal random telegraph signal: Signal analysis and applications

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Abstract

A random telegraph signal is a time series whose value S(t) at time t is either one of only two possible values. Many processes including chemical reactions, cell membrane ion channels, and electronic noise generate such signals. Usually, Markov models have been used to model and analyze such data. Instead, we present a new fractal random telegraph signal that is statistically self-similar in time. We show how to analyze such signals and apply those techniques to study burst noise in a defective operational amplifier and ion currents recorded through individual ion channels in a cell membrane.

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Abbreviations

A :

proportionality constant in the power lawk=At 1−D

A c :

proportionality constant for entering the closed state

A o :

proportionality constant for entering the open state

c (subscript):

closed level of the two state (open-closed) signal

D :

fractal dimension

D c :

fractal dimension associated with entering the closed state

D o :

fractal dimension associated with entering the open state

f(t) :

probability density of the durationst in a state

k :

transition rate (probability per unit time) of exiting a state

k eff :

effective transition rate measured at time scalet eff

k c(eff) :

effective transition rate for entering the closed state

k o(eff) :

effective transition rate for entering the open state

o (subscript):

open level of the two state (open-closed) signal

P(t) :

cumulative probability distribution of the durationst in a state

S(t) :

signal level at timet

t :

time

t eff :

effective time scale at which a measurement is done

X(t) :

state at timet

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Authors and Affiliations

  1. Department of Ophthalmology College of Physicians & Surgeons, Columbia University, 630 West 168th Street, 10032, New York, New York

    Larry S. Liebovitch

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  1. Larry S. Liebovitch
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Liebovitch, L.S. The fractal random telegraph signal: Signal analysis and applications. Ann Biomed Eng 16, 483–494 (1988). https://doi.org/10.1007/BF02368011

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  • DOI: https://doi.org/10.1007/BF02368011

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