Mark A. Kramer, Department of Mathematics and Statistics, Boston University,
Boston, MA, 02115, USA<EMAIL_ADDRESS>NIH, NIBIB, Grant/Award Number: R01EB026938;
NSF, Grant/Award Number: 1451384

# Golden rhythms as a theoretical framework for cross-frequency organization

Mark A. Kramer  Department of Mathematics and Statistics, Boston University

###### Abstract

While brain rhythms appear fundamental to brain function, why brain rhythms
consistently organize into the small set of discrete frequency bands observed
remains unknown. Here we propose that rhythms separated by factors of the
golden ratio ($\phi=(1+\sqrt{5})/2$) optimally support segregation and cross-
frequency integration of information transmission in the brain. Organized by
the golden ratio, pairs of transient rhythms support multiplexing by reducing
interference between separate communication channels, and triplets of
transient rhythms support integration of signals to establish a hierarchy of
cross-frequency interactions. We illustrate this framework in simulation and
apply this framework to propose four hypotheses.

###### keywords:

oscillations, cross-frequency coupling, multiplexing, neural communication
system

In memory of Prof. Miles A. Whittington

##  1 Introduction

The brain is organized into a hierarchy of functionally specialized regions,
which selectively coordinate during behavior [1, 2, 3, 4] and rest [5, 6, 7].
Effective function relies on dynamic coordination between brain regions, in
response to a changing environment, on an essentially fixed and limited
anatomical substrate [8, 9, 10, 11]. Through these anatomical connections
multiplexing occurs: multiple signals that combine for transmission through a
single communication channel must then be differentiated at a downstream
target location [12, 13]. How information – communicated via coordinated
transmission of spiking activity [14] – dynamically routes through the brain’s
complex, distributed, hierarchical network remains unknown [15].

Brain rhythms – approximately periodic fluctuations in neural population
activity – have been proposed to control the flow of information within the
brain network [12, 16, 17, 18, 19, 20] and proposed as the core of cognition
[21, 22, 23, 24]. Through periodic modulations in neuronal excitability,
rhythms may support flexible and selective communication, allowing exchange of
information through coordination of phase at rhythms of the same frequency
(e.g., coherence [16, 19, 25, 26, 27]) and different frequencies (e.g., phase-
amplitude coupling [18, 28, 29] or n:m phase locking [30, 31, 32]). Recent
evidence shows that neural oscillations appear as transient, isolated events
[33, 34]; how such transient oscillations route information through neural
networks remains unclear [35].

Significant evidence supports the organization of brain rhythms into a small
set of discrete frequency bands (e.g., theta [4-8 Hz], alpha [8-12 Hz], beta
[12-30 Hz], gamma [30-80 Hz]) [36, 37]. Consistent frequency bands appear
across mammalian species (mouse, rat, cat, macaque, and humans [38]) and in
some cases the biological mechanisms that pace a rhythm are well-established
(e.g., the decay time of inhibitory post-synaptic potentials sets the
timescale for the gamma rhythm [39]). Why brain rhythms organize into discrete
bands, and whether these rhythms are fixed by the brain’s biology or organized
to optimally support brain communication, remains unclear. For example, an
alternative organization of the brain’s rhythms (e.g., into a larger set of
different frequency bands) may better support communication but remain
inaccessible given the biological mechanisms available to pace brain rhythms.

While much evidence supports the existence of brain rhythms and their
importance to brain function, few theories explain their arrangement.
Different factors have been proposed for the spacing between the center
frequencies of neighboring bands: Euler’s number ($e\approx 2.718$) [40], the
integer 2 [41], or the golden ratio ($\phi\approx 1.618$) [42]. Existing
theory shows that irrational factors (e.g., $e$ and $\phi$) minimize
interference between frequency bands, in support of separate rhythmic
communication channels for multiplexing information in the brain [43, 44, 45].
However, if separate rhythmic channels communicate different information, and
the organization of brain rhythms prevents interference, how a target location
coordinates information across these rhythms is unclear. For example, how in
theory a neural population integrates top-down and bottom-up input
communicated in separate rhythmic channels (lower [$<$40 Hz] and higher [$>$40
Hz] frequency ranges, respectively [25, 26, 46, 47, 48]) remains unclear. We
propose a solution to this problem: addition of a third rhythm. Motivated by
an existing mathematical theory [43, 44, 45], we show that effective
communication among three rhythms is optimal for rhythms arranged according to
the golden ratio.

In what follows, we show that golden rhythms – rhythms organized by the golden
ratio – are the optimal choice to integrate information among separate
rhythmic communication channels. We propose that brain rhythms organize in the
discrete frequency bands observed, with the specific spacing observed, to
optimize segregation and integration of information transmission in the brain.

##  2 Methods

All simulations and analysis methods to reproduce the manuscript results and
figures are available at https://github.com/Mark-Kramer/Golden-Framework.

###  2.1 Damped harmonic oscillator model

As a simple model of rhythmic neural population activity (e.g., observed in
the local field potential (LFP) or magneto/electroencephalogram (M/EEG)) we
implement a network of coupled damped harmonic oscillators [49]. We choose the
damped harmonic oscillator for three reasons. First, a harmonic oscillator
(e.g., a spring) mimics the restorative mechanisms governing displacements
about a stable equilibrium in neural dynamics (e.g., excitation followed by
inhibition in the gamma rhythm [39, 50], depolarization followed by
hyperpolarization – and vice versa – in bursting rhythms [51]). Second, brain
rhythms are transient [33, 34]. In the model, damping (e.g., friction)
produces transient oscillations that decay to a stable equilibrium. Third, the
damped harmonic oscillator driven by noise is equivalent to an autoregressive
model of order two (AR(2), see Appendix A). The AR(2) model simulates
stochastic brain oscillations [52], consistent with the concept of a neural
population with resonant frequency driven by random inputs.

We simulate an 8-node network of damped, driven harmonic oscillators. We model
the activity $x_{k}$ at node $k$ as,

$\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=\left(\bar{g}_{C}+\bar{g}_{S}\cos{\omega_{S}t}\right)\sum_{j\neq
k}x_{j}\ ,$ (1)

where $\beta$ is the damping constant, and $\omega_{k}=2\pi f_{k}$ is the
natural frequency of node $k$. The activity $x_{j}$ summed from all other
nodes ($j\neq k$) drives node $k$. We modulate this drive by a gain function
with two terms: a constant gain $\bar{g}_{C}$ and a sinusoidal gain with
amplitude $\bar{g}_{S}$ and frequency $\omega_{S}=2\pi f_{S}$. To include
noise in the dynamics, we represent the second order differential equation in
Equation (1) as two first order differential equations for the position and
velocity of the oscillator. We add to the position dynamics a noise term,
normally distributed with mean zero and standard deviation equal to the
average standard deviation of the evoked response at all oscillators simulated
without noise, excluding the perturbed oscillator from the average. In this
way, we add meaningful noise of the same magnitude to all oscillators. We
numerically simulate the model with noise using the Euler-Maruyama method. To
examine the impact of different noise levels, we multiply the noise term by
factors $\\{0,0.5,1.0,1.5,2.0\\}$. For each noise level, we repeat the
simulation 100 times with random noise instantiations.

##  3 Results

In what follows, we propose that brain rhythms organized according to the
golden ratio produce triplets of rhythms that establish a hierarchy of cross-
frequency coupling. We conclude with four hypotheses deduced from this
framework and testable in experiments.

###  3.1 Rhythms organized by the golden ratio support selective cross-
frequency coupling

In the case of weakly-connected oscillatory populations, whether the
populations interact or not depends on their frequency ratios [43, 44, 45];
rational frequency ratios support interactions, while irrational frequency
ratios do not. Motivated by this theory, we consider a network of interacting,
rhythmic neural populations (Figure 1). We model each population as a damped
harmonic oscillator, with each oscillator assigned a natural frequency
$f_{k}$. To couple the populations, we drive each oscillator with the summed
activity of all other oscillators (i.e., the connectivity is all-to-all). We
modulate this drive by a gain function ($g$) with constant ($\bar{g}_{C}$) and
sinusoidal (amplitude $\bar{g}_{S}$, frequency $f_{S}$) terms:
$g=\bar{g}_{C}+\bar{g}_{S}\cos{\left(2\pi f_{S}t\right)}$; see Methods.
Analysis of this coupled oscillator system reveals resonance (i.e., a large
amplitude response) at a target oscillator in two cases. To describe these
cases, we denote the frequency of a target oscillator as $f_{T}$ and the
frequency of a driver oscillator as $f_{D}$. A large amplitude (resonant)
response occurs at the target oscillator in the following cases,

constant gain modulation:

$\displaystyle 0=f_{T}-f_{D}$ (2)

sinusoidal gain modulation:

$\displaystyle f_{S}=f_{T}-f_{D}$ (3a) $\displaystyle f_{S}=f_{D}-f_{T}$ (3b)
$\displaystyle f_{S}=f_{D}+f_{T}$ (3c)

The first case (Equation 2) corresponds to the standard result for a damped
target oscillator driven by sinusoidal input; when the sinusoidal driver
frequency $f_{D}$ matches the natural frequency of the target $f_{T}$, the
response amplitude at the target is largest (e.g., see Chapter 5 of [53]). The
next three cases (Equation 3) correspond to a damped target oscillator driven
by sinusoidal input modulated by sinusoidal gain. If the gain frequency
$f_{S}$ equals the sum or difference of the target and driver frequencies,
then the response amplitude at the target is largest (see Appendix B). We note
that the first case corresponds to within-frequency coupling (i.e., the driver
and target have the same frequency) while the next three cases correspond to
cross-frequency coupling (i.e., the driver and target have different
frequencies). We also note that, in this model, we assume an oscillator
responds to an input by exhibiting a large amplitude response; in this way, we
consider the oscillation amplitude as encoding information, consistent with
notion of information encoded in firing rate modulations [54].

Figure 1: Illustration of the coupled oscillator network. Oscillators with
frequency $f_{k}$ receive input from all other oscillators. Input from one
oscillator (frequency $f_{1}$) to all other oscillators
$(f_{2},f_{3},\ldots,f_{8})$ is shown (black curves); similar connectivities
exist from all other oscillators (not shown). Constant ($\bar{g}_{C}$) and
sinusoidal ($\bar{g}_{S}$) gain modulates each input (gray lines).

The results in Equations (2, 3) hold for any choice of driver, target, and
gain frequencies without additional restrictions. We now apply an additional
restriction, and consider the damped harmonic oscillator network with
oscillator and gain frequencies $f_{k}$ satisfying,

$\displaystyle f_{k}=f_{0}\,c^{k}\,,$ (4)

where $f_{0}>0$ determines the frequency at $k=0$. As discussed above,
candidate values for $c$ deduced from in vivo observations include Euler’s
number ($e\approx 2.718$) [40], the integer 2 [41], or the golden ratio
($\phi\approx 1.618$) [42]. Then, given the set of three neighboring
frequencies $\\{f_{k},f_{k+1},f_{k+2}\\}$, what choice of $c$ supports cross-
frequency coupling in the network? To answer this, we choose $f_{S}=f_{k+2}$,
$f_{D}=f_{k+1}$, and $f_{T}=f_{k}$ so that Equation (3c) becomes

$f_{k+2}=f_{k+1}+f_{k}\,.$

Substituting Equation (4) into this expression and solving for $c$, we find

$c^{2}-c-1=0$

with solution

$c=\dfrac{1+\sqrt{5}}{2}=\phi\,,$

the golden ratio. The same solution holds for all Equations (3) with
appropriate selection of $\\{f_{S},f_{D},f_{T}\\}$ from
$\\{f_{k},f_{k+1},f_{k+2}\\}$. We conclude that, for a system of damped
coupled oscillators with oscillator and gain frequencies spaced by the
multiplicative factor $c$, cross-frequency coupling between three neighboring
rhythms requires $c=\phi$, the golden ratio. In other words, we propose that
frequencies organized according to the golden ratio are particularly suited to
support these cross-frequency interactions.

To illustrate this result, we consider a network of 8 damped, coupled
oscillators each with a different natural frequency determined by the golden
ratio ($f_{k}=\phi^{k}$, where $\phi=\frac{(1+\sqrt{5})}{2}$; Figure 2); we
label these rhythms – scaled by factors of the golden ratio – as golden
rhythms. Starting all nodes in a resting state, we perturb one oscillator
($f_{D}=\phi^{6}\approx 17.9$ Hz) to produce a transient oscillation at that
node. With only a constant gain $(\bar{g}_{C}=50,\bar{g}_{S}=0)$, the impact
of the perturbation on the other oscillators is small (Figure 2B); because
$f_{T}\neq f_{D}$ for any oscillator pair, the network impact of the
perturbation is small, despite the constant coupling.

Including the sinusoidal gain modulation ($\bar{g}_{C}=50,\bar{g}_{S}=50$)
results in selective communication between the oscillators. For example,
choosing $f_{S}=\phi^{7}\approx 29.0$ Hz, we observe an evoked response at two
oscillators (Figure 2C): $f_{T}=\phi^{8}\approx 47.0$ Hz (consistent with
Equation (3a)) and $f_{T}=\phi^{5}\approx 11.1$ Hz (consistent with Equation
(3c)). We note that the frequency of evoked responses matches the natural
frequency of each oscillator. We also note that no solution exists for
Equation (3b) because $f_{T}>0$. Different choices of gain frequency $f_{S}$
result in different pairs of cross-frequency coupling between the driver
($f_{D}$) and response oscillators (Figure 2D). Cross-frequency coupling
occurs when Equations (3) are satisfied with $f_{D}\approx 17.9$ Hz. The
coupling is selective; for example, choosing a gain modulation of $f_{S}=11.1$
Hz results in cross-frequency coupling between the driver ($f_{D}=17.9$ Hz)
and faster ($29$ Hz) and slower ($6.9$ Hz) golden rhythms. In this case,
sinusoidal gain frequencies $f_{S}$ exist that support cross-frequency
coupling and occur at factors of the golden ratio: i.e., $f_{S}=\phi^{k}$
(Figure 2D, circles). We note that evoked responses also occur when
$f_{S}\neq\phi^{k}$ (Figure 2D, X’s); in these cases, frequencies outside the
original rhythm sequence $f_{k}=\phi^{k}$ must exist to support cross-
frequency coupling. We conclude that if brain rhythmic activity – both
oscillator and gain frequencies – organizes according to the golden ratio,
then cross-frequency coupling is possible between a subset of separate
rhythmic communication channels.

Figure 2: Rhythms organized by the golden ratio support selective cross-
frequency coupling. (A) We perturb one oscillator (natural frequency 17.9 Hz,
red), with connectivity to all other oscillators; $\phi$ is the golden ratio.
(B) With only constant gain modulation, the perturbation ($t=0$, red) has
little impact on other nodes. (C) With sinusoidal gain modulation at 29 Hz,
two oscillators (natural frequencies 11.1 Hz and 47.0 Hz) selectively respond
to the perturbation. (D) Average response amplitude (logarithm base 10) from
$t=0$ to $t=1.5$ s at each oscillator versus gain frequency $f_{S}$. Different
choices of gain frequency support selective coupling between the perturbed
oscillator (natural frequency 17.9 Hz) and other oscillators. Peaks in
response amplitude occur at golden rhythms (yellow circles, vertical lines) or
other frequencies (red X’s). Minimum response set to 0 for each curve, and
vertical scale bar indicates 1. (E) Average amplitude (black curve) and range
($2.5\%$ to $97.5\%$ from 100 simulations, shaded region) of evoked responses
versus noise level. The oscillator with frequency $17.9$ Hz is directly
perturbed, and sinusoidal gain modulation occurs with $f_{S}\approx 29.0$ Hz.
Oscillators at golden rhythms exhibit different behavior with perturbation
(red) versus without perturbation (gray). Code to simulate this network and
create this figure is available here.

We now consider the impact of noise on this cross-frequency communication.
With sinusoidal gain modulation ($\bar{g}_{C}=50,\bar{g}_{S}=50$, and
$f_{S}=\phi^{7}\approx 29.0$ Hz) and including noise in the oscillator
dynamics (see Methods), we show the results for two cases: with perturbation
and without perturbation to one oscillator ($f_{D}\approx 17.9$ Hz, as above).
Without perturbation (gray in Figure 2E), we find no evidence of an evoked
response at any node, as expected; the amplitude remains small at all nodes,
with a small gradual increase as the noise increases. With the perturbation
(red in Figure 2E), we find an evoked response at the perturbed oscillator
($f_{D}\approx 17.9$ Hz) and two other oscillators: $f_{T}\approx 47.0$ Hz
(consistent with Equation (3a)) and $f_{T}\approx 11.1$ Hz (consistent with
Equation (3c)). As the noise increases, so does the variability in the evoked
response. For the lower frequency $f_{T}\approx 11.1$ Hz oscillator, the
evoked response remains evident as the noise increases; in Figure 2E, the
perturbed (red) and unperturbed (gray) responses remain separate. For the
higher frequency $f_{T}\approx 47.0$ Hz oscillator, the evoked response
becomes more difficult to distinguish from the unperturbed case as the noise
increases; in Figure 2E, the perturbed (red) and unperturbed (gray) responses
begin to overlap with increasing noise. We note that the amplitude of evoked
responses decreases with frequency. Therefore, the same amount of noise
impacts the higher frequency ($f_{T}\approx 47.0$ Hz) oscillator more than the
lower frequency ($f_{T}\approx 11.1$ Hz) oscillator, making an evoked response
more difficult to distinguish from background noise in the higher frequency
case. We also note that oscillators not satisfying Equation (3) (i.e.,
$f_{T}\approx\\{6.9,29.0,76.0\\}$ Hz when $f_{D}\approx 17.9$ Hz and
$f_{S}\approx 29.0$ Hz) exhibit little evidence of an evoked response at any
noise level.

To illustrate the utility of the golden ratio, we consider an alternative
network of oscillators with frequencies organized by a factor of 2 (Figure
3A); such integer relationships have been proposed as important to neural
communication [30, 55, 41]. As expected, with only constant gain
($\bar{g}_{C}=50$) a perturbation to one node ($f_{D}=16$ Hz) does not impact
the rest of the network (Figure 3B). Including sinusoidal gain with frequency
$f_{S}$ can produce cross-frequency coupling. For example, choosing $f_{S}=8$
Hz results in cross-frequency coupling between the $f_{D}=16$ Hz and $f_{T}=8$
Hz rhythms (Figure 3C). Similarly, choosing $f_{S}=16$ Hz results in cross-
frequency coupling between the $f_{D}=16$ Hz and $f_{T}=32$ Hz rhythms;
however, this choice of $f_{S}$ also results in strong cross-frequency
coupling between $f_{D}=16$ Hz and lower frequency rhythms ($f_{T}=8,4,2,1$
Hz; Figure 3D). Importantly, we note that cross-frequency coupling typically
occurs at sinusoidal gain frequencies that differ from the set of oscillator
frequencies at $2^{k}$ Hz (vertical lines in Figure 3D); a new set of rhythms
(and rhythm generators) must exist to support cross-frequency coupling in this
network.

To summarize, in a network of damped coupled oscillators (Equation 1),
sinusoidal gain modulation supports cross-frequency coupling (Equation 3). If
oscillator and gain frequencies organize according to a multiplicative factor
(Equation 4), then cross-frequency coupling between neighboring frequencies
requires a multiplicative factor of $\phi$, the golden ratio (e.g., Figure
2D). While oscillators organized with a different multiplicative factor can
still produce cross-frequency coupling, the frequencies of effective gain
modulation are not part of the original rhythmic sequence (e.g., Figure 3D),
thus requiring the brain devote more resources to implementing a larger set of
rhythms in support of cross-frequency interactions.

Figure 3: An integer scaling between oscillators limits cross-frequency
interactions. (A) In a network of oscillators with frequencies organized by a
factor of 2, we perturb one oscillator (natural frequency 16 Hz, red). (B)
With constant gain, the impact of the perturbation is limited. (C) With
sinusoidal modulation at $f_{S}=8$ Hz, a response appears at another
oscillator (natural frequency 8 Hz). (D) The average response amplitude at
each oscillator versus gain frequency $f_{S}$. Many oscillators respond when
the gain frequency is 8 Hz, and responses tend not to occur at the oscillator
frequencies; see Figure 2D for plot details. Code to simulate this network and
create this figure is available here.

###  3.2 Rhythms organized by the golden ratio support ensembles of cross-
frequency coupling

In the previous section, we considered a network of nodes oscillating at
different natural frequencies. As an alternative example, we now consider a
network with two ensembles of nodes oscillating at different frequencies. The
two ensembles consist of nodes oscillating at frequencies $\phi^{k}$ or
$\phi^{k+2}$, where $\phi$ is the golden ratio. With only constant gain, a
perturbation to any node impacts only nodes of the same ensemble (i.e., with
the same frequency). Including sinusoidal gain modulation with (intermediate)
frequency $f_{S}=\phi^{k+1}$, a perturbation to any node impacts nodes in both
ensembles. We illustrate this in the 8-node network with 4 nodes in each
ensemble oscillating at natural frequencies $\phi^{4}\approx 6.85$ Hz or
$\phi^{6}\approx 17.9$ Hz (Figure 4A). With only constant gain
($\bar{g}_{C}=50,\bar{g}_{S}=0$), a perturbation to one $\phi^{6}\approx 17.9$
Hz (driver) node impacts the amplitude of all other nodes in the same ensemble
(Figure 4B). Including sinusoidal gain modulation
($\bar{g}_{C}=50,\bar{g}_{S}=50$) with frequency $\phi^{5}\approx 11.1$ Hz,
the same perturbation now impacts all nodes in both ensembles (Figure 4C).
From Equation 3 we determine that two sinusoidal gain frequencies support
cross-frequency coupling between the driver ($f_{D}=\phi^{6}\approx 17.9$ Hz)
and target ($f_{T}=\phi^{4}\approx 6.85$ Hz) ensembles,

$\displaystyle f_{S}=f_{T}-f_{D}=6.85-17.9<0.00\,,$ $\displaystyle
f_{S}=f_{D}-f_{T}=17.9-6.85=11.1\,\mathrm{Hz}\,,$ $\displaystyle
f_{S}=f_{D}+f_{T}=17.9+6.85=24.6\,\mathrm{Hz}\,.$

However, of these two frequencies, only the former ($f_{S}=11.1$ Hz) is also a
golden rhythm (Figure 4D, box). In this case, cross-frequency coupling occurs
when ensemble and gain rhythms organize in a “golden triplet”
$(f_{T},f_{S},f_{D})=(\phi^{k},\phi^{k+1},\phi^{k+2})\approx(6.85,11.1,17.9)$
Hz, where $\phi^{k}+\phi^{k+1}=\phi^{k+2}$.

Figure 4: Golden rhythms support coupling among an ensemble of nodes. (A) The
network consists of two ensembles with frequencies: $\phi^{4}$ and $\phi^{6}$.
(B) With constant gain, perturbing a node in one ensemble impacts (red) other
nodes in the same ensemble (black). (C) With sinusoidal gain at frequency
$f_{S}=11.1$ Hz, the same perturbation impacts both ensembles. (D) Average
amplitude response versus gain frequency for all nodes in both ensembles. The
response at the unperturbed ensemble (blue) increases when the gain frequency
is a golden rhythm (yellow box); see Figure 2D for additional plot details.
Code to simulate this network and create this figure is available here.

An alternative choice of irrational frequency ratio between the brain’s
rhythms is Euler’s number ($e$) [40]. Repeating the simulation with two
ensembles of frequency $e^{k}$ or $e^{k+2}$ results in cross-frequency
coupling between ensembles only when $f_{S}=e^{k+2}\pm e^{k}$ (see Figure 5
for an example with $k=2$). We therefore find similar results for the “Euler
triplet” $(f_{D},f_{T},f_{S})=(e^{k+2},e^{k},e^{k+2}\pm e^{k})$ or
specifically for $k=2$, $(f_{D},f_{T},f_{S})=(e^{4},e^{2},e^{4}\pm e^{2})$.
However, this Euler triplet is not consistent with the ratio of $e$ observed
in vivo, where three neighboring frequency bands appear at multiplicative
factors of $e$ (e.g., $(f,ef,e^{2}f)$) and the two slower rhythms do not sum
to equal the faster rhythm (e.g., $f+ef\neq e^{2}f)$. Only for three
neighboring frequency bands related by the golden ratio $(f,\phi f,\phi^{2}f)$
do the frequencies of the slower rhythms sum to the faster rhythm (i.e.,
$f+\phi f=\phi^{2}f$).

Figure 5: Rhythms organized by Euler’s number do not support coupling between
ensembles of nodes. (A) The network consists of two ensembles with
frequencies: $e^{4}$ and $e^{2}$. (B) With constant gain, perturbing a node in
one ensemble impacts (red) other nodes in the same ensemble (black). (C) With
sinusoidal gain at frequency $f_{S}=47.2$ Hz, the same perturbation impacts
both ensembles. (D) Average amplitude response versus gain frequency for all
nodes in both ensembles. The response at the unperturbed ensemble (blue) does
not increase when the gain frequency is a factor of the Euler number (black
vertical lines); see Figure 2D for additional plot details. Code to simulate
this network and create this figure is available here.

###  3.3 Golden rhythms establish a hierarchy of cross-frequency interactions

We now consider results derived for weakly coupled oscillators, which
motivated the study of (strongly) coupled damped harmonic oscillators
presented above. In [43, 44], Hoppensteadt and Izhikevich consider the general
case of intrinsically oscillating neural populations with weak synaptic
connections. When uncoupled, each neural population exhibits periodic activity
(i.e., a stable limit cycle attractor) described by the phase of oscillation.
We note that, in our study of coupled damped harmonic oscillators, we instead
consider the amplitude of each oscillator. When Hoppensteadt and Izhikevich
include weak synaptic connections between the neural populations, the phases
of the neural populations interact only when a resonance relation exists
between frequencies, i.e.,

$\sum_{i}{k_{i}f_{i}}=0,$

where $k_{i}$ is an integer and not all 0, and $f_{i}$ is the frequency of
neural population $i$. The resonance order is then defined as the summed
magnitudes of the integers $k_{i}$,

$\mathrm{resonance\ order}=\sum_{i}||{k_{i}}||.$

For the case of two neural populations, if

$k_{1}f_{1}+k_{2}f_{2}=0$

for integers $k_{1}$ and $k_{2}$, then

$\frac{f_{2}}{f_{1}}=-\frac{k_{1}}{k_{2}}=\mathrm{rational}.$

In other words, if the frequency ratio $f_{2}/f_{1}$ of the two neural
populations is rational (i.e., the ratio of two integers), then the neural
populations may interact, with the strength of interaction decreasing as
either $k_{1}$ or $k_{2}$ increases (i.e., stronger interactions correspond to
smaller resonance orders)111See Proposition 9.14 of [43]. Alternatively, if
this frequency ratio is irrational,

$\frac{f_{2}}{f_{1}}=\mathrm{irrational},$

then the two neural populations behave as if uncoupled.

Consistent with the results presented here, Hoppensteadt and Izhikevich show
that golden triplets possess the lowest resonance order, and therefore the
strongest cross-frequency coupling [43, 44, 45]. However, other resonances
exist due to the recursive nature of rhythms organized by the golden ratio. To
illustrate these relationships, we consider a set of golden rhythms
$\\{f^{k}\\}$ – rhythms organized by the golden ratio so that,

$f_{k-1}+f_{k}=f_{k+1}\,,$ (6)

where $k$ is an integer. Because

$f_{k-1}+f_{k}-f_{k+1}=0$

the resonance order is $3$; this golden triplet supports strong cross-rhythm
communication. Replacing $k$ with $k-1$ in Equation 6, we find

$f_{k-2}+f_{k-1}=f_{k}\,.$ (7)

Then, replacing $f_{k}$ in Equation 6 with the expression in Equation 7, we
find

$f_{k-1}+\big{(}f_{k-2}+f_{k-1}\big{)}=f_{k+1}$

or

$f_{k-2}+2f_{k-1}-f_{k+1}=0\,,$ (8)

which has resonance order $4$. Continuing this procedure to replace $f_{k-2}$
in the equation above, we find

${-f}_{k-3}+{3\ f}_{k-1}{-\ f}_{k+1}=0\,,$ (9)

which has resonance order $5$. In this way, golden rhythms support specific
patterns of preferred coupling between rhythmic triplets, with the strongest
coupling (lowest resonance order) between golden triplets.

As a specific example, we fix $f_{k+1}=40$ Hz and list in Table 1 the sequence
of golden rhythms beginning with this generating frequency. We expect strong
coupling between $(f_{k-1},f_{k},f_{k+1})=(15.3,25,40)$ Hz, a golden triplet,
which has resonance order $3$. Using Equations 8, 9, and Table 1, we compute
additional triplets with higher resonance orders: $(9.4,15.3,40)$ Hz with
resonance order $4$, and $(5.8,15.3,40)$ Hz with resonance order $5$.
Continuing this procedure organizes golden rhythms into triplets with
different resonance orders (Figure 6). Triplets with low resonance order
appear near the target frequency of $f_{k+1}=40$ Hz (see gold, silver, and
bronze circles in Figure 6), and resonance orders tend to increase for
frequencies further from $f_{k+1}=40$ Hz, with exceptions (e.g.,
$(f_{k-1},f_{k},f_{k+1})=(2.2,9.4,40)$ Hz has resonance order $6$). We
conclude that - based on theory developed for weakly coupled oscillators -
golden rhythms support both separate communication channels and a hierarchy of
cross-frequency interactions between rhythmic triplets with varying coupling
strengths. While here we consider three interacting rhythms, we note that the
theory also applies to four (or more) interacting rhythms. The implications of
these results for networks of (strongly) coupled (damped) oscillators remains
unclear.

Table 1: Example sequence of golden rhythms. Beginning with $f_{k+1}=40$ Hz we compute the sequence of golden rhythms by multiplying or dividing by the golden ratio. $f_{k-5}$ |  $f_{k-4}$ |  $f_{k-3}$ |  $f_{k-2}$ |  $f_{k-1}$ |  $f_{k}$ |  $f_{k+1}$ |  $f_{k+2}$ |  $f_{k+3}$ |  $f_{k+4}$ |  $f_{k+5}$
---|---|---|---|---|---|---|---|---|---|---
2.2 | 3.6 | 5.8 | 9.4 | 15.3 | 24.7 | 40 | 64.7 | 104.7 | 169.4 | 274.2
Figure 6: Golden rhythms establish triplets with a discrete set of resonance
orders. The resonance order (numerical value, marker size) for triplets
generated from 40 Hz. Lower resonance orders (3,4,5) indicated in color (gold,
silver, bronze, respectively). Code to simulate this network and create this
figure is available here.

###  3.4 Four experimental hypotheses

We propose that golden rhythms optimally support separate and integrated
communication channels between oscillatory neural populations. We now describe
four hypotheses deduced from this theory. First, if the organization of brain
rhythms follows the golden ratio, then we expect a discrete sequence of three
frequency bands subdivides the existing gamma frequency band, broadly defined
from 30-100 Hz [37, 50], with peak frequencies separated by a factor of
$\phi$. For example, using the sequence of golden rhythms with generating
frequency 40 Hz (Table 1), we identify multiple distinct rhythms (at 40 Hz, 65
Hz, 105 Hz) corresponding to this gamma band. Consistent with this hypothesis,
multiple distinct rhythms have been identified within the gamma band (e.g.,
[56, 57, 58, 59, 60, 61, 62, 63, 64]). While different choices of generating
frequency produce quantitatively different results, the qualitative result is
the same: organized according to the golden ratio, multiple distinct rhythms
exist within the gamma frequency range, each capable of supporting a separate
communication channel.

Second, if rhythms organize according to the golden ratio, then evidence for
this relationship should exist in vivo. To that end, we consider examples of
two or more frequency bands reported in the literature (predominately in
rodent hippocampus; Figure 7). These preliminary observations suggest that, in
these cases, frequency bands separated by a factor of $\phi$ or $\phi^{2}$
commonly occur.

Figure 7: Empirical observations of rhythms organized by the golden ratio in
vivo. (A) Pairs of frequencies reported in the literature; see legend. When
only a frequency band is reported, we select the mean frequency of the band.
(B) Histogram of the frequency ratio for each point in (A). Lines (golden)
indicate frequency bands organized by $\phi\approx 1.6$ or $\phi^{2}\approx
2.6$. Code to create this figure is available here.

Third, if rhythms organize according to the golden ratio, then we propose that
rhythmic triplets support cross-frequency communication. Nearly all existing
research in cross-frequency coupling focuses on interactions between two
rhythms (e.g., theta-gamma [18, 28, 29]), and many measures exist to assess
and interpret bivariate coupling between rhythms [29, 23, 65, 66]. Yet, brain
rhythms coordinate beyond pairwise interactions; trivariate interactions
between three brain rhythms include coordination of beta, low gamma, and high
gamma activity by theta phase [31, 56, 57, 58, 62, 67]; coordination between
ripples (140-200 Hz), sleep spindles (12-16 Hz), and slow oscillations
(0.5-1.5 Hz) [20]; and coordination between (top-down) beta, (bottom-up)
gamma, and theta rhythms [26]. To assess trivariate coupling, an obvious
initial choice is the bicoherence, which assesses the phase relationship
between three rhythms: $f_{1}$, $f_{2}$, and $f_{1}+f_{2}$ [68, 69, 70].
However, the bicoherence may be too restrictive (requiring a constant phase
relationship between the three rhythms), and estimation of alternative
interactions (e.g., between amplitudes and phases) will require application
and development of alternative methods [71].

Fourth, why brain rhythms occur at the specific frequency bands observed, and
not different bands, remains unknown. To address this, we combine the golden
ratio scaling proposed here with a fundamental timescale for life on Earth:
the time required for Earth to complete one rotation (i.e., the sidereal
period) of 23 hr, 56 min. Beginning from this fundamental frequency ($1/86160$
Hz), we compute higher frequency bands by repeated multiplication of the
golden ratio (Table 2). Doing so, we identify frequencies consistent with the
canonical frequency bands (i.e., delta, theta, alpha, beta, low gamma, middle
gamma, high gamma, ripples, fast ripples; see last column of Table 2). We note
that broad frequency ranges define the canonical frequency bands, for example
the gamma band from (30, 100) Hz. Therefore, model predictions that identify
rhythms within a band is not surprising. We propose instead that the relevant
model prediction is the subdivision of the canonical frequency bands (e.g.,
the 10-30 Hz beta band into two sub-bands, the 30-100 Hz gamma band into three
sub-bands), not the specific frequency values identified. We note that the
lower frequencies may include "body oscillations", such as heart rate and
breathing frequency [72], as proposed for a harmonic frequency relationship
(factor of 2) in [41, 73]. We hypothesize that if intelligent life were to
evolve on a planet like Earth, in a star system like our own, with neural
physiology like our own, then rhythmic bands would exist with center
frequencies that depend on the planet’s circadian cycle. We acknowledge that
this hypothesis, and the proposed association between neural rhythms and the
sidereal period in Table 2, remain speculation, without robust supporting
evidence.

Table 2: Golden rhythms, beginning with the sidereal period, align with the brain’s rhythms. The period ($T$) and frequency ($f$) of rhythms beginning with the sidereal period ($T=86160$ s) and multiplying the frequency by the golden ratio (number of multiplications indicated by the value in column Power). Traditional frequency band labels (from [37, 38, 42]) indicated in the last column. Power |  $T$ [s] |  $f$ [Hz] |  Power |  $T$ [s] |  $f$ [Hz] |  Power |  $T$ [s] |  $f$ [Hz] | Label
---|---|---|---|---|---|---|---|---|---
0 | 86160 | 1.16E-05 | 12 | 268 | 0.004 | 24 | 0.83 | 1.20 | Slow 1
1 | 53250 | 1.88E-05 | 13 | 165 | 0.006 | 25 | 0.51 | 2 | Delta
2 | 32910 | 3.04E-05 | 14 | 102 | 0.010 | 26 | 0.32 | 3 | Delta
3 | 20340 | 4.92E-05 | 15 | 63.2 | 0.016 | 27 | 0.20 | 5 | Theta
4 | 12571 | 7.96E-05 | 16 | 39.0 | 0.026 | 28 | 0.12 | 8 | Alpha
5 | 7769 | 1.29E-04 | 17 | 24.1 | 0.041 | 29 | 0.07 | 13 | Beta1
6 | 4802 | 2.08E-04 | 18 | 14.9 | 0.067 | 30 | 0.05 | 22 | Beta2
7 | 2968 | 3.37E-04 | 19 | 9.2 | 0.11 | 31 | 0.03 | 35 | Low Gamma
8 | 1834 | 5.45E-04 | 20 | 5.7 | 0.18 | 32 | 0.02 | 57 | Mid Gamma
9 | 1133 | 8.82E-04 | 21 | 3.5 | 0.28 | 33 | 0.01 | 91 | High Gamma
10 | 701 | 0.001 | 22 | 2.2 | 0.46 | 34 | 0.01 | 148 | Ripple
11 | 433 | 0.002 | 23 | 1.3 | 0.74 | 35 | 0.004 | 239 | Fast Ripples

##  4 Discussion

Why do brain rhythms organize into the small subset of discrete frequencies
observed? Why does the alpha rhythm peak at 8-12 Hz and the (low) gamma rhythm
peak at 35-55 Hz, across species [38]? Why does the brain not instead exhibit
a continuum of rhythms, or a denser set of frequency bands, or different
frequency bands? Here we provide a theoretical explanation for the
organization of brain rhythms. Imposing a ratio of $\phi$ (the golden ratio)
between the peaks of neighboring frequency bands, we constrain activity to a
small subset of discrete brain rhythms, consistent with those observed in
vivo. Organized in this way, brain rhythms optimally support the separation
and integration of information in distinct rhythmic communication channels.

The framework proposed here combines insights developed in existing works.
Mathematical analysis of weakly coupled oscillators established the importance
of resonance order for effective communication between neural populations
oscillating at different frequencies [43, 44, 45, 74]. Experimental
observations and computational models have established the importance of brain
rhythms [36, 39, 75], their interactions [18, 26], and their organization
according to the golden ratio [42, 76, 77]. Here, we combine these previous
results with simulations and analysis of a network of damped, coupled
oscillators in support of the proposed theory.

The framework proposed here is consistent with existing theories for the role
of brain rhythms. Like the communication-through-coherence (CTC) hypothesis
[16, 26] and the frequency-division multiplexing hypothesis [12, 78], in the
framework proposed here neural populations communicate dynamically along
anatomical connections via coordinated rhythms. Organization by the golden
ratio complements these existing theories in two ways. First, by proposing
which rhythms participate – namely, rhythms spaced by factors of the golden
ratio. Second, by proposing the importance of three rhythms to establish
cross-frequency interactions and proposing a hierarchical organization to
these interactions.

We considered a network of damped, coupled oscillators with sinusoidal gain
modulation. In that network, cross-frequency coupling occurs when the gain
frequency equals the sum or difference of the oscillator frequencies (Equation
3). This result holds without additional restrictions on the oscillator or
gain frequencies. However, golden rhythms are unique in that oscillator and
gain frequencies chosen from this set support cross-frequency coupling; no
rhythms beyond this set are required. Alternative irrational scaling factors
(e.g., Euler’s number $e$) establish different sets of oscillator frequencies
(e.g., $\ldots e^{1},e^{2},e^{3},\ldots$) and separate communication channels,
but require gain frequencies beyond this set to support cross-frequency
coupling. In this alternative scenario, two distinct sets of rhythms exist:
one reflecting local population activity, and another the cross-frequency
coupling between populations. Rhythms organized by the golden ratio support a
simpler framework: one set of frequencies (oscillator and gain) that reflect
both local oscillations and their cross-frequency coupling. Golden rhythms are
the smallest set of rhythms that support both separate communication channels
and their cross-frequency interactions. Requiring fewer rhythms simplifies
implementation, reducing the number of mechanisms required to produce these
rhythms.

These results are consistent with existing proposals that the golden ratio
organizes brain rhythms and minimizes cross-frequency interference [77, 79,
80]. We extend these proposals by showing how triplets of golden rhythms
facilitate cross-frequency coupling. An integer ratio of $2$ between frequency
bands (with bandwidth determined by the golden ratio) provides an alternative
organization to support cross-frequency coupling [41, 73]. In this scenario,
cross-frequency interactions have been proposed to occur via a shift in
frequency. For example, two regions - with an irrational frequency ratio -
remain decoupled until the center frequencies shift to establish $1:2$ phase
coupling [81, 82]. We instead propose both regions maintain their original
frequencies and couple when an appropriate third rhythm appears (e.g., a
golden triplet). Our simulation results suggest more widespread coupling
between populations oscillating at a $1:2$ frequency ratio compared to a
golden ratio (Figure 3D). Interpreted another way, integer ratios between
frequency bands may facilitate a "coupling superhighway"; a target region
shifts frequency to enter the coupling superhighway and receive strong inputs
from all upstream regions oscillating at integer multiples (or factors) of the
target frequency. Rhythms organized by a golden ratio require coordination
with a third input to establish cross-frequency coupling. Investigating these
proposals requires analysis of larger networks with multiple rhythms, and
perhaps multiple organizing frequency ratios.

While we do not propose the specific mechanisms that support golden rhythms,
proposals do exist. A biologically motivated sequence exists to create golden
rhythms from the beta1 (15 Hz), beta2 (25 Hz), and gamma (40 Hz) bands.
Through in vitro experiments and computational models, a process of period
concatenation – in which the mechanisms producing the faster beta2 and gamma
rhythms concatenate to create the slower beta1 rhythm – was proposed [77, 42,
76]. Alternatively, golden rhythms may emerge when two input rhythms undergo a
nonlinear transformation [71, 83]. The framework proposed here suggests that
the emergent rhythms – appearing at the sum and difference of the two rhythms
– may support local coordination of the input rhythms.

The simplicity of the proposed framework (compared to the complexity of brain
dynamics) results in at least four limitations. First, brain rhythms appear as
broad spectral bands, not sharply defined spectral peaks. Therefore, the
meaning of a precise frequency ratio, or the practical difference between an
irrational frequency ratio ($\phi$) and a rational frequency ratio (e.g.,
$1.6$) is unclear. Second, rhythm frequencies may vary systematically and
continuously with respect to stimulus or behavioral parameters [84, 85].
Whether the brain maintains a constant frequency ratio between varying
frequency rhythms, and what mechanisms could support this coordination, is
unclear. Third, no evidence suggests the tuning of brain rhythms specifically
to support separate communication channels. Instead, brain rhythms may occur
at the frequencies observed due to the biological mechanisms available for
coordination of neural activity (e.g., due to the decay time of inhibitory
postsynaptic potentials that coordinate excitatory cell activity). Fourth,
identification of rhythms in noisy brain signals remains a practical
challenge, with numerous opportunities for confounds [64, 86, 87]. Therefore,
the best approach to compare this theory with data remains unclear.

However, the simplicity of the golden framework is also an advantage. The
framework consists of only one parameter (the golden ratio) compared to the
many – typically poorly constrained – parameters of biologically detailed
models of neural rhythms. In this way, the golden framework is broadly
applicable and requires no specific biological mechanisms or rhythm
frequencies; instead, only the relationship between frequencies is
constrained.

No theoretical framework exists to explain the discrete set of brain rhythms
observed in nature. Here, we propose a candidate framework, simply stated:
brain rhythms are spaced according to the golden ratio. This simple statement
implies brain rhythms establish communication channels optimal for separate
and integrated information flow. While the specific purpose of brain rhythms
remains unknown, perhaps the brain evolved to these rhythms in support of
efficient multiplexing on a limited anatomical network.

## Acknowledgements

The author would like to acknowledge Dr. Catherine Chu for writing assistance
and tolerating many conversations about the golden ratio.

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##  Appendix A Relationship between a damped harmonic oscillator and
autoregressive model of order two

Consider the damped harmonic oscillator driven by noise,

$\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=\xi(t)\,,$

where $x$ is the position of the oscillator, $\beta$ is the damping constant,
$\omega_{0}$ is the natural frequency, and $\xi(t)$ is a noise term evaluated
at time $t$ (e.g., see Equation (5.28) of [53]). Replacing each derivative
with a discrete approximation we find,

$\frac{x_{t}-2x_{t-1}+x_{t-2}}{\Delta^{2}}+2\beta\frac{x_{t}-x_{t-1}}{\Delta}+\omega_{0}^{2}\
x_{t}=\xi_{t}\,,$

where $x_{t}$ is the oscillator position at discrete time $t$, and $\Delta$ is
the time between $t$ and $t+1$. Collecting terms at the same discrete time, we
find,

$\left(1+2\beta\Delta+\omega_{0}^{2}\Delta^{2}\right)x_{t}-2\left(1+\beta\Delta\right)x_{t-1}+x_{t-2}=\Delta^{2}\xi_{t}\,,$

or

$x_{t}=\alpha_{1}x_{t-1}+\alpha_{2}x_{t-2}+\epsilon_{t},$ (10)

where

$\begin{array}[]{rl}\alpha_{1}=&\dfrac{2(1+\beta\Delta)}{1+2\beta\Delta+\omega_{0}^{2}\Delta^{2}}\,,\\\
\alpha_{2}=&\dfrac{-1}{1+2\beta\Delta+\omega_{0}^{2}\Delta^{2}}\\\
\epsilon_{t}=&\Delta^{2}\xi_{t}\,.\end{array}$

Equation 10 defines an autoregressive model of order 2 (i.e., an AR(2)).

##  Appendix B Resonance response for a damped, driven oscillator with
sinusoidal gain

We begin with Equation 1,

$\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=\left(\bar{g}_{C}+\bar{g}_{S}\cos{\omega_{S}t}\right)\sum_{j\neq
k}x_{j}\ ,$ (11)

and simplify by replacing each $x_{j}$ with,

$x_{j}\approx A_{j}\cos{(\omega_{j}t)}\,;$

i.e., we assume each input oscillator $x_{j}$ oscillates with fixed amplitude
($A_{j}$) at its natural frequency ($\omega_{j}$). Then Equation 11 becomes,

$\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=\bar{g}_{C}\sum_{j\neq
k}A_{j}\cos{(\omega_{j}t)}+\bar{g}_{S}\cos{(\omega_{S}t)}\sum_{j\neq\
k}{A_{j}\cos{(\omega_{j}}t)}\,.$ (12)

Considering the first summation in Equation 12 for the $j^{th}$ oscillator,

$\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=\bar{g}_{C}A_{j}\cos{(\omega_{j}t)}\,.$

and applying the standard approach to solving a damped oscillator with
sinusoidal driving force (e.g., see Chapter 5 of [53]), we determine the
amplitude $A_{k}$ of the driven oscillator,

$A_{k}^{2}=\dfrac{\bar{g}_{C}^{2}A_{j}^{2}}{(\omega_{k}^{2}-\omega_{j}^{2})^{2}+4\beta^{2}\omega_{j}^{2}}\,.$
(13)

The amplitude of the driven oscillator is largest when
$\omega_{j}=\omega_{k}$, i.e., when the frequency of the driving oscillator
$\omega_{j}$ equals the natural frequency of the driven oscillator
$\omega_{k}$.

We now consider the second summation in Equation 12 for the $j^{th}$
oscillator,

$\begin{array}[]{rl}\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=&\bar{g}_{S}A_{j}\cos{\left(\omega_{j}t\right)}\cos{\left(\omega_{S}t\right)}\\\
=&\dfrac{\bar{g}_{S}A_{j}}{2}\cos{\left((\omega_{j}-\omega_{S})t\right)}+\cos{\left((\omega_{j}+\omega_{S})t\right)}\,.\end{array}$

Any solution to this equation must also satisfy,

$\begin{array}[]{rl}\ddot{x}_{k}+2\beta\dot{x}_{k}+\omega_{k}^{2}x_{k}=&\dfrac{\bar{g}_{S}A_{j}}{2}\sin{\left((\omega_{j}-\omega_{S})t\right)}+\sin{\left((\omega_{j}+\omega_{S})t\right)}\,.\end{array}$

We define $z_{k}=x_{k}+iy_{k}$ and combine the two previous equations to find,

$\displaystyle\ddot{z}_{k}+2\beta\dot{z}_{k}+\omega_{k}^{2}z_{k}=$
$\displaystyle\dfrac{\bar{g}_{S}A_{j}}{2}e^{i\left(\omega_{j}-\omega_{S}\right)t}+\dfrac{\bar{g}_{S}A_{j}}{2}e^{i\left(\omega_{j}+\omega_{S}\right)t}\,.$

We now apply the standard approach to solving a damped oscillator with
sinusoidal driving force (e.g., see Chapter 5 of [53]) to determine the
amplitude $B_{k}$ of the driven oscillator,

$B_{k}^{2}=\dfrac{\bar{g}_{C}^{2}A_{j}^{2}/4}{\left(\omega_{k}^{2}-(\omega_{j}\pm\omega_{S})^{2}\right)^{2}+4\beta^{2}\left(\omega_{j}\pm\omega_{S}\right)^{2}}\,.$

The amplitude $B_{k}$ is largest when,

$\omega_{k}^{2}=\left(\omega_{j}\pm\omega_{S}\right)^{2}\,,$

which is satisfied when,

$\begin{array}[]{ccc}\omega_{j}+\omega_{S}=\omega_{k}&\rightarrow&\omega_{S}=\omega_{k}-\omega_{j}\,,\\\
\omega_{j}-\omega_{S}=\omega_{k}&\rightarrow&\omega_{S}=\omega_{j}-\omega_{k}\,.\end{array}$
(14)

Considering the equivalent expression,

$\displaystyle\omega_{k}^{2}=$
$\displaystyle(-1)^{2}\left(\omega_{j}\pm\omega_{S}\right)^{2}=$
$\displaystyle\left(-\omega_{j}\mp\omega_{S}\right)^{2}\,,$

we find an additional solution,

$\begin{array}[]{ccc}-\omega_{j}+\omega_{S}=\omega_{k}&\rightarrow&\omega_{S}=\omega_{k}+\omega_{j}\,.\end{array}$
(15)

We conclude that the amplitude of the driven oscillator $B_{k}$ is largest
when the frequency of the sinusoidal gain modulation $\omega_{S}$ equals the
difference (Equation 14) or sum (Equation 15) of the natural frequencies of
the driven $\omega_{k}$ and driving $\omega_{j}$ oscillators.