J. MODOLO, A. LEGROS
The investigation of electromagnetic fields (EMF) effects on central nervous system (CNS) activity, notably in the extremely low-frequency (ELF, 300 Hz) range, has motivated extensive experimental efforts from in vitro to humans, with the aim to characterize possible physiological and behavioral effects. Among those studies, effects ranging from changes in neural excitability in neuronal cultures to altered learning after repetition of a cognitive task in humans have been reported. However, due to technical and ethical considerations, it is not possible in most situations to perform experimental recordings that would enable unambiguous identification of the biological mechanisms underlying reported effects of EMF exposure. In order to overcome this roadblock, one effective approach is the joint use of experimental techniques with biomathematical models describing the electrical activity of neural circuits. A number of biomathematical models are now available, from the microscopic level (simulating single-cell activity) to the macroscopic level (describing locally averaged activity in brain tissue), which can be used to distinguish between several mechanistic hypotheses.
In this presentation, we will first review the main biomathematical modelling approaches, emphasizing their assumptions and possibilities, while remaining accessible for nonspecialists, in order to maximize interactions and foster discussions with the audience. Second, we will cover previous research studies using such biomathematical models to unravel the possible mechanisms that explain the response of brain tissue to ELF EMF, with an emphasis on reconciling acute effects with reported lasting effects (i.e., effects that outlast the cessation of exposure). Finally, we will present future prospects in the field, involving 1) modelling the effects of peripheral nerve stimulation (PNS) and associated CNS responses in humans, and 2) modelling experimentally reported changes in excitability in neuronal cultures in response to high-level sinusoidal magnetic fields.
KeywordsPower-line magnetic field - extremely low frequency - human, biophysics - computational
The effects of extremely low-frequency (ELF, < 300 Hz) electromagnetic fields (EMF) on human health in general, and on the nervous system in particular, are a topic of considerable interest in the fields of public health, but also in medical engineering (e.g., diagnostic and therapeutic applications). One recurring question in the field, that is still largely an open question, is: what are the mechanisms of interaction that underlie the physiological response of the tissue during EMF exposure? The interest of understanding is twofold: first, this would better constrain the uncertainty parameters that are applied when setting the guidelines for safe EMF exposure; and second, this would open numerous avenues in terms of medical applications, that would exploit to mechanisms in order to probe the integrity of the nervous system (our example of interest) and regulate or reverse the pathophysiological changes involved in several disorders. Therefore, there is a considerable interest in identifying the mechanisms involved in the interaction of ELF EMFs on the human nervous system. This has motivated, over the last decades, studies ranging from single cell-level experiments (in vitro) quantifying neuronal excitability or firing rates, to human studies on motor function or cognitive performance. However, for issues of feasibility (technological limitations) or invasiveness (ethical considerations), the unambiguous identification of those mechanisms is still not achievable.
In order to overcome those difficulties and achieve significant progress regarding this issue, there has been a growing interest in the development of mathematical models that simulate specific neurophysiological processes to predict the impact of ELF EMFs on the electric or metabolic activity of brain tissue. Since the pioneering works of Lapique  and Hodgkin and Huxley , which involved respectively the first mathematical model describing neuron membrane polarization, and the first detailed model involving the role of ionic channels in shaping action potentials; there has been a plethora of mathematical models developed describing the generation of electrophysiological activity at various spatial scales. Here, we review the main types of neuro-inspired models that have been developed, how they can contribute to improving our understanding of EMF interaction with brain tissue, and discuss how investigating as opposed to lasting effects involves new modelling and experimental challenges.
2. Methodological considerations in neuro-inspired models
First, at the microscopic level, single-cell models describe the activity of only one cell (hence their name), generally by describing the membrane potential dynamics as a function of synaptic inputs and external stimuli (e.g., electric or magnetic stimulation). The “integrate-and-fire” model developed by Lapique and the Hodgkin-Huxley (HH) model are examples of microscopic levels. Generally, those models are written under the form of systems of differential equations, that range from single-equation, linear systems (Lapique model) to systems of coupled, non-linear differential equations (HH model). The HH model, despite having been developed almost 70 years ago, still remains the gold standard to simulate the activity of single cells, since it integrates key process such as voltage-gated sodium and potassium channels and the integration of inputs by the membrane. It is even possible, using the socalled multi-compartmental models, to simulate a single cell by taking into account its intricate morphology by dividing it into small compartments, each of those being described by a HH model. While such models are extremely realistic and have significant predictive power, they are also computationally intensive and limit simulations to networks involving a relatively small number of cells (e.g., hundreds).
Second, at the mesoscopic level, models have been developed to overcome the significant computational needs involved when circuits of thousands or millions of neurons are considered. Using a popular approach in physics called mean-field theory, models describing the averaged activity of large populations of neurons have been developed. This approach was initiated by Wilson and Cowan  and Amari , and basically considered homogenous populations of excitatory or inhibitory neurons (theoretically, an infinite number of cells per population) coupled with each other with synaptic projections with defined time constants (i.e., kinetics of the connection) and gain (i.e., strength of the connection). Such mesoscopic models are called neural field or neural mass models (depending if they describe the surface of the cortex as a continuum or discrete cortical columns, respectively). After about 50 years of development, those models have largely demonstrated their predictive and explanatory power, notably in the field of epilepsy, by accurately reproducing all the stages of an epileptic seizure . As explained above, those models have also been used to simulate the effect of EMFs on brain tissue.
Finally, at the macroscopic (e.g., brain-scale) level, several models have been developed (e.g., in the context of the Human Brain Project  or the Virtual Brain ), and are generally built by combining neural mass models located at the level of anatomical regions of interest, and connected using the known “connectome”, i.e. the physical wiring between brain regions that can be quantified using neuroimaging techniques such as Diffusion Tensor Imaging (DTI, ). Let us mention that current dosimetric models quantifying the effect of exposure on the brain typically do so by neglecting the coupling with neuronal activity, and therefore only account for “static” effects in terms of electric fields induced in situ. Let us mention that, however, coupling large-scale, macroscopic models of brain activity with realistic dosimetric models of EMF exposure still remains, to date, an unmet challenge that could considerably increase our insights in the field of bioelectromagnetics. In the following section, we provide examples of contributions from those models in the field of ELF EMFs, and possible future directions.
3. Contributions from models: past and future
The effects of EMFs on biological systems can be classified in two categories: acute and lasting effects. Acute effects refer to biological effects that are induced (almost) immediately in tissue by exposure, and that disappear as soon as exposure is stopped. A good example of acute effects is the perception of magnetophosphenes [9,10,11] as a consequence of head exposure to sinusoidal magnetic fields. Conversely, lasting effects refer to effects that outlast the duration of exposure, and that persist for some time after exposure cessation before fading away. An example of lasting effects induced by exposure to ELF electric fields is the change in brain tissue excitability that has been consistently reported as a consequence of electrical stimulation delivered using electrodes placed on the scalp and a low-intensity (1-2 mA) current (a technology known as transcranial current stimulation, tCS, see  for a review).
Acute effects have received the most attention so far, and the reason is twofold: 1) lasting effects need acute effects to occur, and 2) lasting effects are mechanistically more complex, therefore understanding acute effects is a pre-requisite. For example, at the single cell level, the HH model has been used to quantify the effect of ELF on the amplitude of membrane polarization and the potential role of stochastic resonance . Radman et al.  used a cellular-level model (integrate-and-fire type) to quantify how low magnitude electric fields could potentially impact the timing of action potentials, which is key to neural coding but also to synaptic plasticity processes. This study highlighted the plausibility for electric fields in the range of 1 V/m to impact ongoing activity by shifting the timing of action potentials. Using experimental recordings in combination with a simple model consisting in two interacting populations of excitatory and inhibitory neurons receiving sinusoidal inputs, Reato et al.  suggested that a relevant acute effect due to low magnitude stimulation with electric fields was phase entrainment of neuronal oscillations. Phase entrainment consists in increasing the coherence of phase between ongoing neuronal oscillations in tissue and the stimulus, which implies that to obtain such an acute effect it is necessary that the stimulus frequency has to be close to the frequency of endogenous neuronal oscillations. Interestingly, this concept of phase entrainment has been recently confirmed experimentally in a non-human primate model . Therefore, in the case of acute effects, those examples highlighted that neuro-inspired mathematical models can provide novel insights into the underlying mechanisms of ELF EMFs interaction, and lead to experimentally-testable predictions.
However, regarding lasting effects, few modelling efforts have been initiated. In addition to the aforementioned difficulty that lasting effects involve more complex mechanisms than acute effects, there is also the challenge that lasting effects are less attainable experimentally due to the difficulty to deliver long-duration exposure and record neuronal activity using either invasive (local field potentials, LFP) or non-invasive (for example, electroencephalography, EEG). Hence, it should be emphasized that characterizing the lasting effects of ELF EMFs also involves significant experimental efforts. Nevertheless, a few studies have attempted to evaluate the plausibility of candidate mechanisms. Using a neural mass model, we investigated the possible effect of 60 Hz MF on synaptic plasticity processes . The standard neural mass model formalism was extended to account for the insertion and removal of post-synaptic AMPA receptors as a result of calcium-based mechanisms, an established synaptic plasticity process. Simulation results supported the plausibility of this mechanism, and estimated threshold values as a function of the impacted cellular types.
In terms of future prospects, we suggest a few possible directions for the field. First, recent experimental results have provided novel insights into the effect of high-intensity MF on neuronal cultures , notably regarding potential synaptic plastic effects. One possibility to test the mechanisms suggested by the authors would be to use a model consisting in coupled neural masses, for example using the flexible modelling approach implemented into the NMMgenerator software . In brief, NMMgenerator can be used to design networks of coupled (excitatory / inhibitory) neural mass subject to any input, including sinusoidal stimuli. Furthermore, all parameters from the model (synaptic time constants, gains regarding the synaptic projections from one population of neurons to another…) can be easily modified and their impact on the simulated activity quantified. Second, another direction of research that has remains relatively unexplored to date would be to study the impact of peripheral nerve stimulation using ELF EMFs and its associated cortical-level neurophysiological response. An appropriate neuro-inspired model could assist in interpreting the associated evoked response, and frequency-dependent effects and thresholds of perception. Other directions include combining cortical-level neural mass models to investigate the acute (phase entrainment) and lasting (modulation of excitability and associated synaptic gains) due to stimulation using ELF EMFs. Let us mention that the possibility, say, to decrease neuronal excitability in a controlled way using electrical stimulation holds great promise for the treatment of neurological disorders such as epilepsy, which is characterized by pathological hyperexcitability in specific brain regions .
4. Discussion and conclusions remarks
We argue that the joint use of experimental approaches with neuro-inspired mathematical models enables accelerating the identification and validation of candidate mechanisms of action, by providing elements that are not accessible for experimental investigation, while also being timeefficient. Neuro-inspired models are also an ideal framework to investigate the validity of potential mechanisms, and formulate predictions that can tested experimentally to support or infirm them. Overall, the identification of such mechanisms is key to consolidate existing guidelines on EMF exposure in the ELF range, and also for designing novel diagnostic and therapeutic applications.
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