Download Nonlinear Harmonic Generation and Wave

Nonlinear Harmonic Generation
and Wave-Wave Interactions in
the Brain
Mariya Ferdousi
a,b
, T. Babaie
a,b
, P. A. Robinson
a,b
a
School of Physics, University of Sydney, Australia.
b
Center for Integrative Brain Function, University of Sydney, Australia.
This work was supported by Australian Research Council Grants CE140100007 and FL140100025
3. Nonlinear Power Spectrum
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1. Neural Field Model
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A neural field corticothalamic model is first used
to calculate the linear power spectrum via the
system transfer function. The model includes
corticothalamic interactions of four populations:
e
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Cortical excitatory population (e)
Cortical inhibitory population (i)
i
Thalamic reticular population (r)
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Thalamic relay population (s)
r
The transfer function that relates perturbations
in cortical excitatory activity
for our
corticothalamic model is
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s
Normalization coefficient is found
in the sleep and wake state, to
estimate absolute levels of activity in our model.
sleep state
The spindle harmonic is found at
Hz in
the nonlinear and total power spectrum which
arises due to nonlinear effects.
The linear power spectrum is dominated by a
single spindle peak and the nonlinear power
spectrum is dominated by a single and
symmetric peak at double the frequency of the
original peak.
The contribution of nonlinear power spectrum
to the total spindle peak is negligible. The
linear
and
nonlinear
power
spectrum
contribute to the harmonic peak by 80% and
20%, respectively.
The beta peak arises near harmonics of alpha
peak in the model.
The contributions of the linear and nonlinear
power spectrum in the total alpha peak are
62% and 38%, respectively. The linear and
nonlinear contributions for the total beta peak
are 46% and 54%, respectively.
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linear
nonlinear
total
-3
P (normalized)
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The linear power spectrum is convolved with itself to obtain the nonlinear power
spectrum [2], which is
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Strong peaks in the power spectrum of brain activity are often accompanied by one or more
harmonics. Examples occur for the alpha and beta peaks in wake, sleep spindles, steady state
visual evoked potentials (SSVEPs), and seizures. Some of these are thought to be generated by
linear delay loops, while others are expected to be nonlinear in origin. Here neural field theory
of the corticothalamic system is applied to quantitatively understand harmonic generation in
wake and sleep states and the wider question of wave-wave interactions in SSVEP
experiments.
log
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0
10
20
30
40
50
f (Hz)
wake state
-1
log10P (normalized)
Abstract
linear
nonlinear
total
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0
10
20
30
40
50
f (Hz)
4. Gaussian approximations
We approximately fit a Gaussian function to the
strong peak in the linear power spectrum which is
are gains for the excitatory corticothalamic, inhibitory corticothalamic, and intrathalamic
loops, respectively. The parameters
quantify the connection to population a from
population b.
The fitted Gaussian function shows accurate peak
value and bandwidth of the spindle peak in the
linear power spectrum.
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Theoretically, the mean frequency of harmonic peak
is twice the frequency of the fundamental peak and
the bandwidth is
times the fundamental peaks
bandwidth. The total second harmonic power is
proportional to the square of the power in the
fundamental peak. These yield a prediction for the
nonlinear power spectrum as
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2. Linear Power Spectrum
The frequency power spectrum is calculated by integrating
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over k:
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sleep state
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The spindle peak is found at
Hz
for sleep parameters from [1] in our
model.
log10P (s-1)
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The large sleep spindle peak has a
frequency
related to the rate
constants
with
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2
1
0
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-1
0
No harmonic is found in the linear
power spectrum.
10
20
30
40
50
f (Hz)
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wake state
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The linear alpha and beta peaks that are
found in the relaxed wake state originate
in the corticothalamic delay loop and the
frequency depends on the loop delay.
The alpha peak occurs at
Hz and
the beta peak is at
Hz for wake
parameters from [1] in the wake state.
3.5
log10P (s -1)
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This theoretical prediction is compared to the
calculated nonlinear power spectrum. The fit
is accurate at the peak value, however, it is
less precise in the bandwidth.
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Fit Eq. (3)
Linear power
spectrum
8
6
4
0
10
20
30
40
50
f (Hz)
sleep state
Fit Eq. (3)
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Nonlinear
power
spectrum
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10
9
0
10
20
30
40
50
f (Hz)
5. Preliminary Application to SSVEPs
3
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is the mean
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log10 P (normalized)
where k is the wave vector,
is the angular frequency, L incorporates the low pass filter
characteristics of synaptodendritic dynamics, and
is the external signal. The gain
is
the response in neurons a due to unit input from neurons b. The quantities
where Pr is the Gaussian peak,
frequency and
is the bandwidth.
sleep state
log10P (normalized)
●
3
An experiment performed by Herrmann [3], in
which human subjects were presented
flickering light at a range of frequencies,
showed the presence of nonlinear features in
SSVEPs spectral response.
wake state
Applying similar normalization and calculations
for SSVEPs shows generation of harmonics and
also sum and difference frequencies due to
wave-wave coalescence and decay.
Figure shows the SSVEPs spectral response at
frequency f for drive frequency fD in wake state.
Label shows alpha and beta peaks plus strong
responses at the drive frequency, its second
harmonic, and sum and difference frequencies.
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Future work:
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1
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In future, the SSVEPs result will be compared to the Herrmann's experimental result [3].
Similar analysis will also be applied in the sleep state and will be used to study
bifurcations in the model.
40
f (Hz)
Selected references
Summary
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[1] R. G. Abeysuriya, C. J. Rennie, and P. A. Robinson, J. Theor. Biol. 344, 70 (2014).
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[2] P. A. Robinson and N. Roy, Phys. Rev. E 91, 062719 (2015).
[3] C. S. Herrmann, Exp. Brain Res. 137, 346 (2001).
Further Information: [email protected]
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This analysis gives details of nonlinear harmonic generation and wave-wave
interactions in the brain.
Analysis of linear, nonlinear and total power spectra shows that nonlinear effects are
strong in the sleep state, causing a nonlinear harmonic peak, while nonlinear
contribution to the waking state has much less effect on spectral peaks.
This study is applied to SSVEPs and the generation of harmonic in the wake state is
observed.