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Edited by: Jun Ma, Lanzhou University of Technology, China

Reviewed by: Kang K. L. Liu, Brandeis University, United States; Otti D'Huys, Aston University, United Kingdom

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We investigate the dynamics of a non-linear network with noise, periodic forcing and delayed feedback. Our model reveals that there exist forcing regimes—called persistent entrainment regimes—in which the system displays oscillatory responses that outlast the termination of the forcing. Our analysis shows that in presence of delays, periodic forcing can selectively excite components of an infinite reservoir of intrinsic modes and hence display a wide range of damped frequencies. Mean-field and linear stability analysis allows a characterization of the magnitude and duration of these persistent oscillations, as well as their dependence on noise intensity and time delay. These results provide new perspectives on the control of non-linear delayed system using periodic forcing.

Brain stimulation has become increasingly popular in neuroscience to support a wide variety of experimental and clinical interventions [

To better understand this phenomenon, we consider a network of neural populations with delayed feedback implementing a particular type of dynamic memory. Such delayed feedback systems appear not only in neural system but also in optics [

In the present work, we analyze the dynamics of a network of globally (i.e., all-to-all) interacting inhibitory neural populations whose membrane potentials _{i}(

where _{ij} are such that < _{ij} >_{NxN} = _{ij} < 0 for all _{i} of intensity

Driven nonlinear network model.

Whenever noise in the system is sufficiently small (i.e.,

where the local fluctuations ν_{i} obey the Langevin equation

In this regime, we may easily derive the mean-field representation of the network dynamics in presence of global periodic forcing and independent noise sources. The mean field is given by the scalar non-linear delay-differential equation [

with the noise-corrected response function

In the presence of periodic forcing, Equation (1) exhibits frequency-selective and persistent entrainment: oscillatory forcing-induced responses, whose duration exceeds the delay τ, can be observed after the forcing offset and for which the dynamics differ significantly from the one seen in the autonomous regime (i.e., before the forcing onset). This form of forcing buffering allows fluctuations due to the forcing to outlast the stimulation application; in fact, the forcing is being transiently memorized by the system. An example of this persistent entrainment effect is shown in Figure

Example of persistent entrainment and sensitivity to stimulation frequency. ^{*}). In all cases, the model parameters are the same except for the stimulation frequency with τ = 100 ms:

To better understand the phenomenon observed in Figure

Stability of the equilibrium state _{o} is determined by considering small fluctuations around the fixed point to obtain the linearized dynamics. The linearized dynamics for Equation (4) with

where ^{−λτ}, from which we obtain the typical transcendental characteristic equation that defines the eigenvalues λ of Equation (6) in presence of delay [

where _{k} = α_{k} + _{k}| α_{k} = _{k}] ∈ ℝ, ω_{k} = _{k}] ∈ ℝ of the characteristic equation above form the spectrum Λ = {λ_{k}} of the mean-field in Eq. (4) and determine its linear stability. These eigenvalues correspond to branches of the Lambert function _{k}, where _{k} ∈ ℂ, λ_{k} ∈ Λ span the stable (

In this framework, the solution _{k} and eigenfrequencies ω_{k}. Stable limit cycle solutions emerge whenever a supercritical Hopf bifurcation occurs for which a pair of critical eigenvalues cross the imaginary axis; that is, for

for some critical values of _{c}. Here _{o} is the zeroth-order Lambert function. The set of critical values _{k} = 0,

Effect of noise on system's stability and eigenmodes in the mean-field equation. _{k} > 0. Increasing

The stability analysis above shows that in addition to the critical frequency ω_{c} observed near the HB in Equation (6), the set of eigenmodes in Equation (8) provide a reservoir Ω = {ω_{k}} of intrinsic resonances whose elements correspond to the imaginary parts of the eigenvalues λ_{k} of Equation (6). The analysis also shows that this set depends on noise intensity through the linear gain

The effect of noise on the network eigenmodes and resonances can be seen in Figure

For large delays τ, the purely oscillatory eigenvalues λ = iω at the HB can be approximated as follows. The ansatz

Using the fact that as τ → ∞, ω_{k} → 0 [

To understand the interaction between the forcing and the system's oscillatory modes, we can examine the particular solution in the linearized case in Equation (6) to reveal susceptibilities to persistent entrainment. The susceptibility to forcing of various frequencies can be characterized by computing the resonance curves of the periodically forced linear delayed system

where

Substituting this into Equation (13) and solving for the amplitudes

Where

and

Inserting Equations (16, 17) into Equation (14) yields the desired resonance curve

where

which is just the characteristic equation of Equation (6) evaluated at the forcing frequency ω. The amplitudes of the solutions thus diverge here whenever a pair of imaginary eigenvalues cross the imaginary axis. Resonance curves are shown in Figure

Resonance curves with variable delays and noise values for the linearized system in Equation (11). The system possesses a reservoir of resonances, corresponding to peaks in the amplitude of forced solutions. As noise in increased, the linear gain

The linear analysis above tells us that in Equation (15), forced solutions possess a resonance for all eigenfrequencies ω_{k}. However, it remains unclear how these resonances relate to one another with respect to persistent activity. Figure _{o} as the delay increases [

Persistent entrainment mediated by the selective excitation of unstable eigenmodes. Here, the noise intensity is set such that the first five eigenvalues are located to the right hand side of the imaginary axis. Then, forcing with frequency ω aligned with the eigenfrequencies ω_{o−4} is successively applied. One can see that the amplitude of the response during forcing is not monotonically changing with forcing frequency, suggesting the presence of non-linear resonances that arise from the coupling of the linear modes by the system's nonlinearities. Persistent entrainment can be observed in every case, but its duration decreases with forcing frequency. The frequencies used were:

To better characterize how the amplitude of persistent responses change as a function of forcing frequency, we numerically computed the power spectral density of persistent responses (after forcing offset) for all pairs of values of noise intensity and forcing frequency in the mean-field model given by Equation (4). We then computed the peak power at the forcing frequency during that period. Results are shown in Figure _{k}.

Persistent entrainment power and frequency as a function of forcing parameters in the mean-field equation in Equation (4).

We note that despite their persistence, persistent responses are not stable orbits. Rather, the system's activity will always relax back to the oscillation defined by the dominant bifurcating eigenvalues (i.e., the poles, responsible for the autonomous dynamics before forcing onset). As seen in Figure _{k} > 0) and thus solutions diverge. We may nonetheless obtain an estimate of the entrainment duration using an approximation based on the superposition of damped oscillations. Without loss of generality, let us consider the case where the mean activity in Equation (13) under the absence of stimulation (i.e.,

The principal eigenvalue _{λo ≡ λc} = α_{c} + _{c} defines the dominant oscillatory mode of the system and also corresponds to the eigenmode for which α_{c} > α_{k} and ω_{c} < ω_{k} | ∀_{k} ∈ Λ. As such, we may rewrite the solution as

We may consider Equation (21) as an oscillator _{k} − α_{c} < 0, taking the limit as

We assume that a forcing _{k}

for some constant _{k} − α_{c} and thus at a characteristic damping time that we define as the

According to this approximation, buffering time decreases as forcing frequency increases. We also note that the critical eigenvalue _{c} is stable. The dependence of the buffering time on the time delay and linear gain is shown in Figure

Buffering time _{o} = ω_{c} is infinite and thus only the next four modes are shown i.e., _{1} = 7.2 Hz; _{2} = 12.0 Hz; _{3} = 16.9 Hz; _{4} = 21.7 Hz. The delay is the dominant control parameter of buffering time: longer delays are responsible for more persistent responses. Note that these panels are related to the excited modes in Figures

We have also numerically investigated the response of the mean-field in Equation (4) to forcing with multiple frequencies to see whether persistent entrainment could carry multiple resonances—not only one. We have here considered the case a dual-frequency stimulation i.e., _{1}cos(ω_{1}_{2} cos(ω_{2}_{1} and ω_{2} were chosen to be eigenfrequencies of Equation (4). As shown in Figure

Response to dual-frequency stimulation. _{2} = 12.4 Hz, ω_{5} = 27.4 Hz. _{1} = _{2} = 1.

The analytic and numerical examinations of the phenomenon considered thus far have been done using simplified mean-field approximations and subsequent linear analysis (Figures

Persistent entrainment in a network model of large-scale human brain dynamics with realistic wiring topology and distributed delays.

Dominant frequency responses of this system as a function of noise and forcing frequency are shown in Figure

The optimization of stimulation, in particular the control and stabilization of its aftereffects, are important aspects of both fundamental and clinical research on normal and abnormal brain dynamics. Recent studies have shown that after rhythmic entrainment, neural oscillations may remain locked to the stimulation frequency even after stimulation offset [

Our analysis further indicates that noise acts as a resonance regulator, which can tune the response of the system by displacing the eigenvalue spectrum in the imaginary plane and thus through an effective change in linear stability. Through this change in stability, different resonances can be amplified and the buffering time can be increased. From a neuroscience perspective, this is consistent with results reporting the dependence of rhythmic stimulation effects on brain states. Assuming that different brain states may be associated with different noise levels, this noise shapes the susceptibility of neural populations to entrainment, and consequently the persistence of oscillations beyond the end of stimulation.

The results above are contingent upon linear approximations, while the original system (i.e., Equation 1) is not. In particular, for values of

Our analysis further demonstrates that persistent entrainment is prevalent in systems with longer time delays. Indeed, the density of resonant frequencies (number of resonances per frequency unit) increases with τ; persistent frequencies represent an increasingly larger portion of parameter space. This can be seen in Figure

It is clear that a range of delays, arising from the variety of feedback loops that may exist in brain networks, can support reverberations across a range of periods for a certain time. In particular, when delays are sufficiently large compared to intrinsic neural response times (typically when the ratio of these time scales exceeds 1—see [

Our work relates to other recent efforts to examine the buffering ability of neural networks, i.e., their capacity to temporarily store a signal over a short time delay and then play it back through readout neurons [

Another more recent line of research at the intersection of delayed dynamics and neural computation involves reservoir computing designs that rely on a simple delay with a nonlinear element [

It is clear that our ability to control networks to respond to certain signals and not others, and to use such effects as biomarkers for e.g., mental disease (see e.g., [

JL, JG, and SP performed the research and analysis. JL, JG, SP, and AL wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.