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Edited by: Olivier David, Institut National de la Santé et de la Recherche Médicale (INSERM), France

Reviewed by: Thomas R. Knösche, Max-Planck-Institut für Kognitions- und Neurowissenschaften, Germany; Tamer Demiralp, Istanbul University, Turkey

This article was submitted to Neural Technology, a section of the journal Frontiers in Neuroscience

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 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.

In the past decade, there has been a surge of interest in using patterned brain stimulation to manipulate cortical oscillations, in both experimental and clinical settings. But the relationship between stimulation waveform and its impact on ongoing oscillations remains poorly understood and severely restrains the development of new paradigms. To address some aspects of this intricate problem, we combine computational and mathematical approaches, providing new insights into the influence of waveform of both low and high-frequency stimuli on synchronous neural activity. Using a cellular-based cortical microcircuit network model, we performed numerical simulations to test the influence of different waveforms on ongoing alpha oscillations, and derived a mean-field description of stimulation-driven dynamics to better understand the observed responses. Our analysis shows that high-frequency periodic stimulation translates into an effective transformation of the neurons' response function, leading to waveform-dependent changes in oscillatory dynamics and resting state activity. Moreover, we found that randomly fluctuating stimulation linearizes the neuron response function while constant input moves its activation threshold. Taken together, our findings establish a new theoretical framework in which stimulation waveforms impact neural systems at the population-scale through non-linear interactions.

Oscillatory brain activity results from the collective and synchronous discharge of large populations of neurons, and is thought to play an important role in homeostasis, neural communication and information processing (Singer and Gray,

This study sets out to answer some of these questions by harnessing computational and mathematical techniques and study the effect of stimulation waveform on cortical alpha oscillations. Alpha oscillations have been implicated in a wide variety of physiological and cognitive functions (Başar,

To provide new insight into the effects of brain stimulation on neural populations, we use two computational models in parallel and explore the impact of stimulation waveform and polarity on alpha oscillations. The first model, which we study numerically, is a cortical microcircuit network model which has been used before by the authors to investigate alpha resonance and entrainment in the cortex (Herrmann et al.,

To study the influence of different stimulation waveforms on oscillatory dynamics in cortical microcircuits, we here consider a model of interacting cortical populations and investigate changes in limit cycle solutions when subjected to stimulation. This model has been thoroughly discussed and analyzed in previous work (Herrmann et al.,

This cortical network consists of spatially extended excitatory (

where

with _{n}. The linear operator

where ^{jk} = |^{−1}, with

with synaptic time constant _{m}.

Excitatory and inhibitory populations are subjected to endogenous sources of noise _{n}. Synaptic weights within (

Neuron ensembles in the network are distributed randomly within a one-dimensional spatial domain Ω. The constants

where

Cortical microcircuit network model parameters.

Ω | Network spatial size | 10 mm |

_{e} |
Number of excitatory neurons | 800 |

_{i} |
Number of inhibitory neurons | 200 |

β | Response function gain | 300 a.u. |

Response function threshold | −0.1 a.u. | |

τ_{m} |
synaptic time constant | 10 ms |

α_{e} |
Dendritic rate constant – excitatory | 1.0 |

α_{i} |
Dendritic rate constant – inhibitory | 1.5 |

Conduction velocity | 0.128 m/s | |

c | Connection probability | 0.6 |

60 | ||

70 | ||

−70 | ||

−70 | ||

Excitatory synaptic spatial decay rate | 1.0 a.u. | |

Inhibitory synaptic spatial decay rate | 0.5 a.u. | |

Intrinsic noise level | 0.0001 | |

Integration time step | 1 ms |

Spectral analysis was performed using a fast Fourier transform routine using freely available C++ scripts (Press et al.,

To better understand the mechanism involved in shaping oscillations in the cortical microcircuit model, we use a scalar and reduced non-linear network as a prototype to rigorously analyze the role of delayed and non-linear interactions in shaping emergent oscillations, and specifically how those are impacted by stimulation waveform. This simplified model sacrifices many physiological details in comparison to the cortical microcircuit model but preserves key components underlying the rhythmic activity seen in the cortical microcircuit model while remaining analytically tractable. Our goal here is to obtain a qualitative assessment of the different phenomena observed in our results.

Oscillations in the cortical microcircuit model arise due to delayed recurrent inhibition conveyed by inhibitory synapses. In this regime, inhibitory interactions dominate the dynamics, and the cortical microcircuit model can be significantly simplified, preserving the key components responsible of the oscillations. Specifically, we focus on parameters that result in an inhibition driven regime in which

Consequently, the dynamics of the cortical model obeys in good approximation

This approximation renders independent the dynamics of the inhibitory population from the activity of the excitatory cells. The excitatory membrane potential is thus, on average, driven by the activity of the inhibitory population, such that one may fully characterize the activity of the network by considering inhibitory ensemble dynamics. Assuming that the firing rate is high and that σ_{i} is small enough, i.e., broad spatial connectivity, we can write

where

where ^{−1}. This kind of approximation has been used frequently in the literature (e.g., Curtu and Ermentrout,

A common approach when trying to understand the essential dynamical characteristics of an otherwise high-dimensional system is to derive mean-field representations. What is different here is that we apply the mean field reduction by including stimulation in the calculations. As such, let us further assume that limit cycle solutions occur in a mean-driven regime in which the local dynamics can be seen as small independent fluctuations around a slowly varying mean Ū i.e.,

where Ū is given by

and < >_{N} is an average performed over the ^{j} from the mean obey the zero mean processes

where we have used the fact that

for

Now the network dynamics are governed by the effective neuron response function (Hutt et al.,

where ρ(

To better understand the role of stimulation waveforms on the entrainment of network oscillations, we integrated numerically Equation (2) for different functional forms of the input term

Variability of the cortical microcircuit network responses to stimulation with diverse waveforms. Stimulation has different impact on network resting state oscillations depending on the waveform applied.

To understand how these results depend on stimulation settings and waveforms, we measured the response of the network while stimulation parameters were changed. For pulse trains and sinusoidal inputs, frequencies were systematically varied between 0 and 100 Hz with fixed amplitude. In the case of Gaussian white noise, the intensity of the noise was gradually increased between 0 and 0.01. Results are shown in Figure

Diverse effects of stimulation waveform and frequency on the power spectrum in the cortical microcircuit network model. In each case, a waveform was chosen and used to stimulate the network. Stimulation frequency was increased while the power spectrum of the network responses was calculated.

Impact of stimulation waveform on endogenous oscillations for near-resonant stimulation frequencies. This represents a close-up of the data plotted in Figure

For stimulation frequencies close to but larger than the endogenous frequency, positive pulses and sinusoidal stimulation entrain the endogenous rhythm. Conversely, negative pulses entrain the endogenous rhythm for a more narrow range of frequencies.

To understand the mechanism behind the numerical observations made with the cortical microcircuit model (Figures

As a first step to understand how the stimulation waveform affects endogenous oscillations, we applied the reduced neural oscillator model to characterize the effect of different stimulation waveforms on the response function of the network in Equation (15). These computations show that different stimulation patterns—leading to different statistics of the fluctuations around the activity mean—shape the effective response function in a plurality of waveform-dependent ways. The cases analyzed below are sequentially illustrated in Figure

Effect of stimulation polarity and fluctuation distribution on the effective response function of the reduced model. Stimulation-driven fluctuations around the mean change the effective response function of the system in a waveform-dependent way. _{s} = 12 Hz.

Reduced model parameters.

N | Number of Interacting Units | 100 |

β | Response function gain | 300 a.u. |

Response function threshold | −0.1 a.u. | |

Effective mean delay | 25 ms | |

Mean synaptic coupling | −15 a.u. | |

Linear operator gain constant | −1 a.u. | |

Integration time step | 1 ms |

Let us first consider the stimulus waveform

where δ(0) = 1 and zero otherwise. This pulse train has a rate

with μ_{S} =

In the limit of high gain,

Then the network mean activity ū obeys the mean-field dynamics

Note that in these calculations, we have made no assumptions on the value of

Let us now consider the periodic stimulation

with angle frequency ω_{s}. Fluctuations around the mean obey

since μ_{S} = 0. This case was studied in detail in Hutt et al. (

and consequently the effective non-linearity is given by

As illustrated in Figure

Next, we study Lefebvre and Hutt (

where ξ_{j} are Gaussian white noise processes such that < ξ_{j} ξ _{k}> = δ_{jk} i.e., all neurons in the network experience independent stochastic input of intensity

where μ_{S} = 0. The associated probability density function ρ(

Convolving with the threshold non-linearity yields

This equation states that additive noise linearizes the effective neuron response function as

At last, let us consider the simple tonic stimulus

In this particular case, the input has zero variance and the effective response function is thus unchanged

The effect on the dynamics can be understood by introducing the change of variable

which leads to the mean–field dynamics

with

As the derivations above have pointed out, stimulation statistics are reflected by changes in the effective response function, leading to mean-field equations with variable non-linear structures. Moreover, linear stability of the equilibria will also depend on stimulation statistics, which will be reflected on the features of oscillatory solutions. Regardless of stimuli waveform, limit cycle solutions are deployed around the implicitly defined equilibrium

The linearized dynamics around the steady state ū_{o} is

where _{o}] < 0 for the cases in Figure _{o}] depends explicitly on the stimulation waveform through the convolved statistics in the effective function _{o}. To determine the frequency of emergent alpha oscillations, the iterative Galerkin method (He,

Here we may consider small input stimuli i.e.,

Using the ansatz Ū(_{o}, one obtains for the first iteration

Setting _{lin} = 0 and solving for ω yields an approximation of the linear frequency as a function of _{o}] and delay τ i.e.

Equation (36) approximates well the dependence of the network peak frequency on stimulation statistics whenever the stimulation amplitude remains small and its frequency high. Figure _{o}] cannot always be computed analytically due to the implicit condition for the equilibrium (34), some specific cases such as Gaussian white noise for instance, remain tractable and accurate (Hutt et al.,

Peak frequency of the reduced model as a function of the linear gain. For a fixed delay, changes in the linear gain due to the stimulation will mediate the non-linear entrainment of the endogenous oscillations. Increases in |

According to the local approximations above, effects of stimulation waveform on equilibrium states and oscillations can be characterized by local, stimulus-induced changes in the linearized gain, evaluated at the fixed point Ū_{o}. Equation (39) above states that ω is inversely proportional to |_{o}], translates into an acceleration (resp. deceleration) of the network peak frequency. For the stimulation types studied and shown in Figure

According to this framework, mathematically the sensitivity of network oscillations to the stimulation waveform depends fully on how the probability density ρ(

Using the approach outlined above, we computed the peak frequency of oscillations for various stimulation waveforms and compared them to the values computed numerically in the reduced network model. Results are presented in Figure

Waveform-dependent non-linear entrainment of endogenous oscillations in the reduced model. The convolution approach allows us to compute the dependence of the system's peak frequency on the stimulation parameters. _{o} is also plotted (bottom).

The results above hold for high-frequency (or stochastic) stimulation; what happens if this condition is relaxed? We investigated this question numerically and results are plotted in Figure

Numerically computed Arnold Tongues of reduced model for different waveforms. Given the different impact of the stimulation on the network response function and oscillatory properties, different waveforms possess different entrainment properties (linear or not).

Electromagnetic brain stimulation has become increasingly popular to support a wide variety of clinical interventions. It is routinely used in the treatment of various types of neuropsychiatric disorders such as treatment-resistant depression (Ferrucci et al.,

To answer some of these questions, we have combined computational and mathematical methods to reconcile the effect of stimulation at the mesoscopic neural ensemble and macroscopic population scales using mean-field techniques. Using a detailed cortical microcircuit model, we have numerically explored the effect of stimulation pulse trains, sinusoidal inputs and Gaussian white noise stimulation on resting state alpha oscillations. First, our simulations have confirmed that, as expected, distinct waveforms have different entrainment properties. Notably, while positive pulse trains were found to accelerate ongoing oscillations, negative pulses did the opposite. To understand the source of this novel finding, we have developed a framework in which the mesoscopic and waveform-dependent effects of stimulation on oscillatory activity can be characterized by a change in the neuron response function, mathematically speaking through a convolution with the probability density function associated with stimulation-induced fluctuations around the mean. Using this approach, it was possible to relate the statistics of stimuli to the acceleration/slowing down of non-linear oscillations using a reduced neural oscillator model that preserved the non-linear structure of the more detailed cortical microcircuit model. Taken together, these results show that the core differences in entrainment properties between various stimulation waveforms can be explained by population-scale changes in the effective response function of the network—an emerging perspective in line with recent findings in the literature (Kar et al.,

The analysis we have conducted has revealed that in regimes of high frequency stimulation, distinct waveforms may have similar impact on non-linear neural oscillations. Indeed, positive pulses, sinusoidal inputs and Gaussian white noise were all found to have similar influence on ongoing oscillations when the stimulation frequency (or intensity

Our analysis has shown that, in first approximation, the local features of the effective response function _{o} determine primarily the impact of stimulation waveforms on limit cycle solutions. This means that the stimulation efficacy is highly dependent on the state: changes to the response function are commensurate with the proximity of the fixed point, implying that the same waveform will have different impact on endogenous oscillations if the equilibrium location in phase space differs. This is in complete agreement with numerous recent findings showing the state dependence of entrainment efficacy (Neuling et al.,

The results presented in this study are aimed at obtaining a better understanding of the properties of mesoscopic neural population activity in regimes where the dynamics are dominated by recurrent inhibition. In our model, this arises due to differential spatial profiles of excitatory vs. inhibitory connections. Specifically, excitatory connections dominate at smaller spatial scales, whereas inhibitory connections dominate at larger spatial scales. This is the 1-D analog of classic “Mexican Hat” profiles of lateral connectivity, as observed for example in visual cortex (Amari,

In order to fully characterize the effects of electromagnetic brain stimulation, it is necessary to consider both local effects at the stimulation site, and distal effects that result from propagation of stimulated response throughout the brain via long-range white matter pathways (Massimini et al.,

AH and JL conceived the principle idea of the work. JL performed the numerical analysis. AH, JG, and JL have worked out the analytical results. JL and CH structured the manuscript and all authors have written the manuscript.

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.

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (JL).