Customization of coherence analysis by
relaxing its iso-frequency constraint
Pooya Pakarian1 & Arash Hadipour Niktarash2
1
2
Program in Neural Computation, Center for Neural Basis of
Cognition (CNBC), Carnegie Mellon University.
Laboratory of computational motor control, Department of
biomedical engineering, Johns Hopkins University.
Magnitude Cross Correlation:
dot product of complex numbers
(sizes multiplied, angles subtracted
because the second one is conjugated)
Sum of the vectors, depends not only on the
size of the vectors, but also on the angles:
consistency of phase difference between the
two original waves
The size of this vector
represents the magnitude
cross correlation between
signals x and Y:
=
MCCxy
Magnitude Auto Correlation:
The magnitude of each frequency (f=3 cycles per
window) over n windows (n=2 in this example)
represents a vector in an n dimensional space.
The size of this vector
represents the magnitude
auto correlation of signal x:
MACx
The size of this vector
represents the magnitude
auto correlation of signal Y:
MACy
Magnitude Squared Coherence:
(MCCxy)2
(MACx)(MACy)
Coherence is calculated between identical frequencies (shown
as f ) of the two signals (x and y), therefore can not probe the
relation between different frequencies of the two signals; for
example those described in:
*Liu et al (2002): High frequency (100 Hz) electrical stimulation of the subthalamic nucleus (STN) induced tremor (4 Hz) in both
forearms, and also oscillation of the contralateral STN (4 Hz). In contrast, low frequency (5 Hz) stimulation induced contralateral arrhythmic
involuntary movement (3 Hz).
*Paré et al (1990): 100-400 Hz activity in thalamic neurons related to Parkinson’s tremor (3-8 Hz).
*Brown et al (2004) Exp Neurol: 10 Hz stimulation of Sunbthalamic Area, potentiates 31-40 Hz oscillation in GPi.
Relaxing the iso-frequency constraint:
Let’s calculate the coherence for all pairs of
frequencies of the two signals, not only for identical
frequencies, therefore the functional relation of them
can be shown even in different frequencies.
For example:
Hadipour-Niktarash A. (2006) J Comput Neurosci. The simulation results show that, by an interaction
between the TC and RE neurons, the TC-RE network transforms a low-frequency oscillatory activity of the
GPi neurons to a higher frequency of oscillatory activity of the TC neurons. (superharmonic transformation)
(TC= thalamic cell; RE= reticular nucleus; GPi= Globus Pallidus )
GPi
TC
TC oscillation frequency
4 Hz simulated GPi burst
can trigger 8 HzTC burst.
GPi oscillation frequency
Future plan: Wavelet-base coherence analysis
Harmonic frequencies that pertain to the
non-sinusoidal shape of a signal can
induce false-positive high non-linear
coherence. They can be removed by
calculating this customized coherence
based on the appropriate wavelet
analysis, instead of Fourier analysis
whose base is sinusoidal wave.
Acknowledgement: We thank Dr. William MacKay of the University
of Toronto for insightful comments.
References:
Liu X, Ford-Dunn HL, Hayward GN, Nandi D, Miall RC, Aziz TZ, Stein JF. (2002) The oscillatory activity in the Parkinsonian subthalamic nucleus
investigated using the macro-electrodes for deep brain stimulation. Clin Neurophysiol. 113(11):1667-72.
Paré D, Curro'Dossi R, Steriade M. (1990) Neuronal basis of the parkinsonian resting tremor: a hypothesis and its implications for treatment. Neuroscience.
35(2):217-26.
Brown P, Mazzone P, Oliviero A, Altibrandi MG, Pilato F, Tonali PA, Di Lazzaro V. (2004) Effects of stimulation of the subthalamic area on oscillatory
pallidal activity in Parkinson's disease. Exp Neurol. 188(2):480-90.
Hadipour-Niktarash A. (2006) A computational model of how an interaction between the thalamocortical and thalamic reticular neurons transforms the lowfrequency oscillations of the globus pallidus. J Comput Neurosci. 20(3):299-320.