Effects of extremely low-frequency magnetic fields on the response of a conductance-based neuron model.

Page Proof December 10, 2013 16:27 1450007

International Journal of Neural Systems, Vol. 24, No. 1 (2014) 1450007 (16 pages) c World Scientific Publishing Company DOI: 10.1142/S0129065714500075

EFFECTS OF EXTREMELY LOW-FREQUENCY MAGNETIC FIELDS ON THE RESPONSE OF A CONDUCTANCE-BASED NEURON MODEL

Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/02/15. For personal use only.

GUOSHENG YI∗ , JIANG WANG† , XILE WEI and BIN DENG School of Electrical Engineering and Automation Tianjin University Tianjin 300072, P. R. China ∗ [email protected] † [email protected] KAI-MING TSANG and WAI-LOK CHAN Department of Electrical Engineering The Hong Kong Polytechnic University Kowloon, Hong Kong CHUNXIAO HAN School of Automation and Electrical Engineering Tianjin University of Technology and Education Tianjin 300222, P. R. China

Accepted 31 October 2013 Published Online 12 December 2013 To provide insights into the modulation of neuronal activity by extremely low-frequency (ELF) magnetic field (MF), we present a conductance-based neuron model and introduce ELF sinusoidal MF as an additive voltage input. By analyzing spike times and spiking frequency, it is observed that neuron with distinct spiking patterns exhibits different response properties in the presence of MF exposure. For tonic spiking neuron, the perturbations of MF exposure on spike times is maximized at the harmonics of neuronal intrinsic spiking frequency, while it is maximized at the harmonics of bursting frequency for burst spiking neuron. As MF intensity increases, the perturbations also increase. Compared with tonic spiking, bursting dynamics are less sensitive to the perturbations of ELF MF exposure. Further, ELF MF exposure is more prone to perturb neuronal spike times relative to spiking frequency. Our finding suggests that the resonance may be one of the neural mechanisms underlying the modulatory effects of the low-intensity ELF MFs on neuronal activities. The results highlight the impacts of ELF MFs exposure on neuronal activity from the single cell level, and demonstrate various factors including ELF MF properties and neuronal spiking characteristics could determine the outcome of exposure. These insights into the mechanism of MF exposure may be relevant for the design of multi-intensity magnetic stimulus protocols, and may even contribute to the interpretation of MF effects on the central nervous systems. Keywords: Extremely low-frequency magnetic field; neuron; resonance; spike times; spiking frequency.

†

Corresponding author.

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1.

Introduction

Nowadays, using extremely low-frequency (ELF, below 300 Hz) magnetic field (MF) to actively stimulate central nervous system is a painless and noninvasive therapeutic option to alleviate the symptoms of a number of neuropsychiatric diseases.1 The most well-known technique is Transcranial Magnetic Stimulation (TMS),1,2 which places a TMS coil above the brain skull over a region of interest. Until now, TMS has been proved useful in the treatment of depression and is actively investigated as a therapeutic option in a number of other psychiatric and neurological diseases.2–4 It has also been successfully applied to probe changes in cortical excitability and effective connectivity associated with these diseases by combing with EEG,5–7 since the latter is a powerful tool for measuring and detecting abnormal brain rhythms in physiological and pathological conditions.8–15 Moreover, there are also other kinds of magnetic stimulations of growing interest, such as Pulsed Magnetic Fields (PMF) therapy16 and Low-field Magnetic stimulation (LFMS).17 Compared with TMS, the intensity levels of these two techniques are significantly lower, usually on the order of 1mT for PMF and LFMS versus 1T for TMS.1,16–18 All of the above techniques based on magnetic stimulation are noninvasive, which have become increasingly important investigative tools in cognitive neuroscience for exploring brain physiology and function.1,2,19 Despite the widespread and successful application of magnetic stimulation, the mechanism of neuronal excitation by magnetic stimulation is still largely unknown. To address this problem requires knowledge of the effects of ELF MF exposure on neuronal activity and the mechanism of the interactions between the applied MF and neural tissue. Previous investigations with animals and humans have indicated that ELF MF exposure can induce various effects on the activity of central and peripheral nervous systems, such as altering circadian rhythms,20 aborting epileptiform discharges,3,4 regulating synaptic plasticity,21 activating neurotransmitter release,21 increasing EEG power spectra,22,23 altering neuronal receptor activity,21,24 modulating learning and memory,21 and so on. Further, it has been indicated the effects of ELF magnetic stimulation are characterized by many physical properties of the

stimulus, including stimulus intensity and frequency, pulse waveform, pulse duration, and the diameter and shape of the coil.25,26 For instance, it is found that low-frequency rTMS (5 Hz) increases cortical excitability27 ; further, extracellular single cell recordings show that high-intensity TMS often lead to early inhibition followed by excitation in cat visual cortex, while low-intensity TMS typically results in early excitation followed by long lasting inhibition.26,28 It is difficult to integrate these findings into a unified framework, since the experimental results are not always reproducible among studies. Meanwhile, the mechanisms of the interactions between ELF MFs and neuronal activity still remain unclear. Apart from above experimental studies, there are also some theoretical and numerical investigations of the interactions between MFs and neural tissue. Most of them used multi-compartment model of neocortical neurons to explore MF effects on the isolated single neuron level. For example, Roth and Basser29 demonstrated that a brief magnetic pulse can generate an action potential in a long straight axon model; Nagarajan et al.30 explored the effects of magnetic stimulation on finite length neuronal structures using computer simulations; Kamitani et al.31 developed a theory to calculate the effects of magnetic stimulation on arbitrary neuronal structure. Recently, biophysical approaches have been used to examine cortical stimulation by MF exposure. Miyawaki and Okada32 studied the mechanisms of spike inhibition in a cortical network induced by TMS; Lazutkin and Husar17 developed a 3D model of TMS and LFMS based upon a layered sphere head model by using Maxwell’s classical electromagnetic theory and Faraday’s law; Pashut et al.25 developed a numerical scheme to combine realistic magnetic stimulation with compartmental modeling of neurons with arbitrary morphology; Modolo et al.18 used the Izhikevich model to study the effects of ELF magnetic stimulation on neuronal activity. However, all above results do not address the specific dynamical behaviors of neuron exposed to MF, and most of them focus on exploring induced electric field distribution inside the brain by MF exposure, lacking basic biophysical research and mechanism analysis. Therefore, it is necessary to take into account the specific effects of MF exposure on neuronal

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Effects of ELF MF on the Response of a Conductance-based Neuron Model

rhythms, and analyze impacts of MF exposure on neuronal responses through the dynamical interactions between neuronal activity and external ELF MF. In this study, we address the question of how ELF magnetic stimulation affects neuronal activity from the biophysical modeling point of view. We first describe our conductance-based neuron model during exposure to ELF sinusoidal MF based on a modified Morris–Lecar (ML) model.33 Then, we study how ELF MFs with different intensity and frequency modulate neuronal spike times and spiking frequency in the case of two spiking patterns, i.e. tonic spiking and burst spiking. Finally, we discuss our results by comparing with previous ones and give the conclusions. 2. Material and Method 2.1. Modified ML model To model the response of single cortical neuron during MF exposure, we used a modified ML model proposed by Prescott et al.33 The original ML model is a conductance-based model to describe the voltage oscillations in the barnacle giant muscle fiber, which is also used as a qualitatively accurate model of neuronal behaviors including excitability and spiking. In the original ML model, there are two ionic currents (i.e. Ca2+ and K+ ) and leakage current. Prescott et al. modified it by replacing Ca2+ with Na+ , to investigate how shunting and adaptation modulate neuronal repetitive spiking. In our study, we only focus on how MF modulates neuron activity and do not consider adaptation effects. Then, the modified ML model can be described by the following equations33 : C

dV = Is − g¯Na m∞ (V )(V − ENa ) dt − g¯K w(V − EK ) − gleak (V − Eleak ),

w∞ (V ) − w dw =ϕ , dt τw (V ) V − V1 m∞ (V ) = 0.5 1 + tanh , V2 V − V3 w∞ (V ) = 0.5 1 + tanh , V4 V − V3 τw (V ) = 1 cosh 2V4

which has a similar form with the two-dimensional Hodgkin-Huxley motor neuron model.34 There are two variables in this model: membrane potential V expressed in mV and a slower recovery variable w. The variable w is a function of membrane potential and represents the probability of an activation gate being in the open state for K + channels. ENa , EK and Eleak are the reversal potentials of Na+ , K+ and leakage channels, respectively. The corresponding maximum conductances are g¯Na , g¯K and gleak . C is the capacitance of the membrane, Is is the synaptic input current. V1 , V2 , V3 and V4 are system parameters. Despite its simplicity, this model is able to reproduce the main dynamical properties of neuron electric activity, such as resting state, the mechanisms for type I and II excitability. The numerical values of the parameters in model (1) are given in Table 1, which can give rise to type I excitability with a continuous f –Is curve. All the values for these parameters are taken from Prescott et al.33 2.2. Effects of ELF MF exposure on neuronal membrane potential activity According to Faraday’s law, applying a time-varying MF B(t) to brain can induce a time-varying induced EF E(t) in brain tissue, which is shown as18 : r dB(t) , (2) 2 dt where r stands for the radius of MF exposure. This induced EF E(t) can cause the displacement of the charges in brain tissue, which could further alter the concentrations of different ions in extra- and intra-cellular medium. It has been shown that the differential of ionic concentrations between the extra- and intra-cellular medium originates the membrane potential V (t) of neuron cell,35 which could be described by the Glodman–Hodgkin–Katz equation.36 Further, the electric activity in neurons E(t) =

Table 1.

(1)

Parameter C g¯Na g¯K gleak EK ENa

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Parameter values in models (1) and (7). Value

Parameter

Value

2 µF/cm2 20 mS/cm2 20 mS/cm2 2 mS/cm2 −100 mV 50 mV

Eleak V1 V2 V3 V4 ϕ

−70 mV −1.2 mV 23 mV 10 mV 21 mV 0.15 (unitless)


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is sustained and propagated via ionic movements through neuron membranes.35 Thus, the external ELF MF applied could modulate neuron membrane electric activities. Previous studies have shown that applied EF could induce a field-induced membrane depolarization ∆V on neuron membrane to modulate neuron electric activity.37 Recently, Modolo et al.18 proposed a differential equation which could describe the field-induced membrane polarization for a steady EF E(t): 1 λ d∆V + ∆V = E cos θ, (3) dt τ τ where τ is the Maxwell–Wagner time constant, indicating at which speed charges displace on cell membrane, whose value depends on the properties of the medium, such as resistivity and permittivity.38 λ is the polarization length, which could be determined from recordings of neuron membrane potential for different EF values.39 θ is the angle between the axis of the EF and the cell. To simplify, we assume that the cell is either affected by ELF MF exposure, or not at all, thereby cos θ equals to 1 in the following.18 The field-induced membrane depolarization ∆V is the solution to the differential equation (3). If the external MF is a sinusoidal MF with intensity B and frequency f , which is: B(t) = B sin(2πft )

To describe the interactions between neuronal activity and applied MF, a widely used approach we can follow is to represent applied MF action as a membrane voltage perturbation at cellular level.18,40–42 Therefore, MF exposure can be modeled as an additive term ∆V (i.e. field-induced membrane polarization) over physiological membrane potential.18,42 Then, the equations of the modified ML model describing the dynamics of a neuron exposed to ELF sinusoidal MF can be expressed by: C

r dB(t) = rπfB cos(2πft ). (5) 2 dt For the steady EF in Eq. (5), the field-induced membrane polarization ∆V which is the solution to differential equation (3) becomes: E(t) =

d(V + ∆V ) = Is − g¯Na m∞ (V ) dt × (V + ∆V − ENa ) − g¯K w(V + ∆V − EK ) − gleak (V + ∆V − Eleak ), w∞ (V ) − w dw =ϕ , dt τw (V ) V + ∆V − V1 m∞ (V ) = 0.5 1 + tanh , V2 V + ∆V − V3 w∞ (V ) = 0.5 1 + tanh , V4 V + ∆V − V3 τw (V ) = 1 cosh . 2V4 (7)

(4)

according to Eq. (2), the induced EF on the membrane can be expressed as:

∆V (t) = λrBπf cos θ

2.3. Introducing MF to the modified ML model

The equivalent electric circuit for model (7) is shown in Fig. 1, where ∆V is induced by MF exposure. From Fig. 1 and model (7), it can be observed that

cos(2πft ) + 2πf τ sin(2πft ) . 1 + (2πf τ )2 (6)

Therefore, we can compute the membrane depolarization ∆V at any time for a given sinusoidal MF by using differential equation (3). In our study, the polarization length is λ = 0.5 mm, which is in accordance with a cortical neuron during external electric field stimulation in vitro.39 According to the results of Bedard et al.,38 the Maxwell–Wagner time constant is τ = 10−4 s in present work. The radius of exposure is r = 10 cm.

Fig. 1. The equivalent electric circuit of the modified ML model exposed to ELF MF. Is is the synaptic input current, the field-induced membrane polarization ∆V is introduced as an additive perturbation over the membrane potential.

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sinusoidal MF exposure can induce an additive perturbation ∆V in the nonlinear membrane potential equations. It does not cause any change in the basic structure of the neuron model, but just adds an additive nonlinear potential to the membrane potential V and produces a simple input current with amplitude C(d∆V /dt). The perturbation ∆V can influence the activity of voltage-dependent ionic channels. Its effects on ionic currents can be temporally integrated and reflected again on membrane potential, which indicates a feedback interaction occurs.43 Through this interaction mechanism, the applied MF could modulate neuron electric activity and coding dynamics. The parameter values in model (7) are also shown in Table 1.

precise spike time by ELF MF exposure. SST can be calculated by the differences between the spike time for each membrane potential with MF exposure and that without MF exposure. Then, if SST is positive, the spike time is delayed, and negative means spike time is advanced. Depending on ELF MF frequency and intensity, we compute the average SST value σmean that is induced by ELF MF exposure. Moreover, our stimulations are performed using a fourth-order Runge–Kutta method with a time step of 0.01 ms. The simulation experiment lasts 8000 ms for every journey to get the steady state. 3.

Modulation of Tonic Spiking by ELF MF Exposure

3.1. Spiking rhythms exposed to ELF MF 2.4. Stimulation The focus of our investigation is to study how ELF MFs with different intensity and frequency modulate neuron electric activity and coding process. To attain this goal, we perform our stimulation as the following course: first, trigger different spiking patterns in the neuron by changing synaptic input current Is in the case of no ELF MF exposure [i.e. ∆V = 0 mV in Eq. (7)]; then, introduce perturbation ∆V to study how ELF MF exposure perturbs neuronal spiking behaviors. We examine two spiking pattern, i.e. tonic spiking and burst spiking, in our work. The tonic spiking is elicited by superthreshold constant input current with different amplitude, and the bursting dynamic is stimulated by sinusoidal input current with low frequency. To assess the performance and reliability of neurons in a quantitative way, many influential coding schemes are proposed in computational neuroscience, such as rate coding (the average number of spikes per unit time), time coding (the precise time of single spike), and population coding (encoding information by the joint activities of a large number of neurons).44 At the single cell level, the output of a neuron can be described in terms of precise spike times or the average firing rate. Both of these coding strategies convey significant information, possibly distinct, about the stimulus.45–51 Thus, we use them to characterize neuron response under the ELF MF exposure. To investigate the impacts of ELF MF exposure on neuronal time coding, we study the shift in spike times (SST),18 i.e. the perturbation in each

Figure 2 presents the sample membrane potentials of the modified ML neuron during ELF MF exposure in the case of tonic spiking. In Fig. 2, a constant bias current of amplitude Is = 17 µA/cm2 is injected into neuron membrane to induce tonic spiking at a frequency of 43.5 Hz. We take this spiking frequency without ELF MF exposure as neuronal intrinsic spiking frequency.18 It can be found the ELF MF with different intensity and frequency could induce different changes on neuronal spiking rhythms. The spike times and interspike intervals (ISIs) of neuron are all perturbed by ELF MF exposure. The neuronal spiking rhythms may be minimally affected, or can be greatly affected by MF exposure. As MF intensity increases, the effects of MF exposure on neuronal spike times also increase. Figure 3 presents the ISI sequences of neuronal spiking trains induced by ELF MF exposure. The constant bias current in Fig. 3 is also Is = 17 µA/cm2 , and neuron presents tonic spiking at a frequency of 43.5 Hz without MF exposure. In this case, the ISI sequences distribute at a same value which is ISI = 22.98 ms. When ELF MF is applied to neuron, the periodicity of spike is perturbed and the spiking trains become irregular. Then, the ISIs will be no longer a fixed value, but show a large variation around a value, as shown in Fig. 3. This effect will increase as MF intensity increases. Besides, for some MF frequencies, the ISI sequences will change to a fixed value once more under the exposure of MF. Here, the spiking rhythms will be regular again. The MF frequency ranges for this phenomenon are

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Fig. 2. (Color online) Membrane potentials of a tonic spiking neuron during exposure to ELF MFs with different frequency and intensity. Black curves represent the membrane potentials without MF exposure, and bright blue curves represent membrane potentials exposed to ELF sinusoidal MF.

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Fig. 3.

ISI sequences of neuronal spiking trains exposed to ELF MF. 1450007-6



almost around the harmonics of the intrinsic spiking frequency of neuron (here, approximately 43.5 Hz), which will widen as MF intensity increases. Next, we will systematically analyze how ELF MF exposure affects neuronal coding and electric activity.


3.2. Impacts of ELF MF exposure on tonic spike times Figure 4 shows the average SST σmean as a function of ELF MF frequency and intensity. Neurons without ELF MF exposure in Figs. 4(a)–(c) are all in tonic spiking with intrinsic spiking frequency of 31.25, 43.5 and 78.25 Hz, respectively. The corresponding input currents are 15.7, 17 and 31 µA/cm2 . Figure 5 presents the evolutions of average SST σmean with MF frequency f . From the results presented in Figs. 4 and 5, one can see that the SST of neuron induced by ELF MF exposure can be a positive value or a negative one, which indicates MF exposure can delay or advance tonic spike times. When MF intensity is relatively smaller, the perturbation of average SST σmean induced by ELF MF is also very small. In this case, neuronal spiking rhythm with MF exposure is almost the same as that without MF exposure, as shown in Figs. 2(a)–(c). As ELF MF intensity increases, the fluctuation of average SST σmean at the harmonics of intrinsic spiking frequency will increase. Between every two adjacent harmonics, the average SST σmean decreases gradually along the negative half-axis as MF intensity increases, which indicates the effects of MF exposure are mainly to advance neuronal spike times in the frequencies between every two adjacent harmonics. In the high-frequency area (approximately exceeds second harmonic), the average SST values are almost negative, which will continue to decrease as MF frequency and intensity increase. Further, it should be noted that if neuronal intrinsic spiking frequency is different, the perturbations of neuronal spike times induced by ELF MF exposure are different. In addition, there is a significant fluctuation of the average SST value σmean when MF frequencies are at the harmonics of the intrinsic spiking frequency of the neuron. One can clearly see that there is an obvious abrupt fluctuation for σmean at the harmonics of the intrinsic spiking frequency. The specific fluctuation is: as MF frequency increases, σmean rapidly rises to a local maximum, then quickly falls

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Fig. 4. Average SST σmean of the tonic spiking neuron as a function of ELF MF intensity B and frequency f . The intrinsic spiking frequency of neuron is: (a) 31.25 Hz, (b) 43.5 Hz and (c) 78.25 Hz.

to a local minimum, and finally stabilizes to a relative stability of lower equilibrium value. The fluctuations of σmean at the intrinsic spiking frequency and its second harmonic are much larger than other harmonics, especially at second harmonic. Furthermore, no matter how large the MF intensity is, the perturbations of average SST σmean induced by MF exposure

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Fig. 5. Evolutions of average SST σmean with MF frequency f . The intensity of ELF MF is 10mT, 30mT, 50mT and 70mT, respectively. The intrinsic spiking frequency of neuron is: (a) 31.25 Hz, (b) 43.5 Hz and (c) 78.25 Hz.

always have a significant fluctuation at the harmonics of the intrinsic spiking frequency of neuron. 3.3. Impacts of ELF MF exposure on tonic spiking frequency Figure 6 presents the neuronal average spiking frequency fout as a function of ELF MF frequency

(c)

Fig. 6. Average spiking frequency fout of the tonic spiking neuron as a function of ELF MF intensity B and frequency f . Neuronal intrinsic spiking frequency is: (a) 31.25 Hz, (b) 43.5 Hz and (c) 78.25 Hz.

and intensity. Neurons without ELF MF exposure in Figs. 6(a)–(c) are also in tonic spiking with intrinsic spiking frequency of 31.25, 43.5 and 78.25 Hz, respectively. From the results presented in Fig. 6, it can be found that the average spiking frequency fout of neuron with MF exposure can be increased or

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decreased. Thus, neuronal tonic spiking activity can be enhanced or inhibited by ELF MF exposure. When MF intensity is small, the perturbation of average spiking frequency fout induced by MF exposure is almost close to zero. As MF intensity increases, the fluctuations of fout at the harmonics of neuronal intrinsic spiking frequency will increase, especially at the intrinsic spiking frequency and its second harmonic. In the high-frequency area (approximately exceeds neuronal second harmonic), the average spiking frequency fout with MF exposure is obviously higher than its intrinsic spiking frequency, which continues to increase as MF frequency and intensity increase. Between every two adjacent harmonics, the average spiking frequency fout increases starting from the intrinsic spiking frequency as MF intensity increases. Thus, MF exposure is mainly to enhance neuronal spiking activity in these frequency ranges. Further, it should be noted that if neuronal intrinsic spiking frequency is different, ELF MF exposure will induce different impacts on neuronal spiking frequency. Moreover, when MF frequencies are at the harmonics of the intrinsic spiking frequency of neuron, there is a significant fluctuation of average spiking frequency fout . The specific change is: as MF frequency increases, fout will first rapidly fall to a local minimum and then quickly rise to a local maximum, and finally stabilize to a relative stability of higher equilibrium value. The fluctuations of fout at the intrinsic spiking frequency and its second harmonic are much greater than other harmonics, especially at the second harmonic. As MF intensity changes, the frequency ranges in which there are apparent fluctuations of average spiking frequency fout will almost not change. That is to say, no matter how large the MF intensity is, the perturbation of spiking frequency induced by MF exposure will always have a significant fluctuation at the harmonics of the intrinsic spiking frequency of neuron. Furthermore, it has been proposed that neuronal spiking frequency can be calculated based on the reciprocal of ISI.33 Then, the result in Fig. 6(b) should be comparable with that shown in Fig. 3. From Fig. 3, it can be observed the ISI sequences in the presence of ELF MF distribute at a fixed value when MF frequencies are around the harmonics of neuronal intrinsic spiking frequency. In this case, although neuronal spiking rhythms are regular, the

ISI value is very different from that without MF. Then, the average spiking frequency fout could show an abrupt fluctuation in these MF frequency ranges. Moreover, although the ISI sequences show a large variation around a value between every two adjacent harmonics, the average value of ISIs by MF is a little smaller than that without MF. Then, the average spiking frequency fout is a little bigger than that without MF, which could not show significant fluctuation in this MF frequency range. In addition, comparing the effects of MF exposure on average SST σmean with that on average spiking frequency fout , it can be found that the perturbation of σmean induced by MF exposure is much bigger than that of fout . In other words, the effect of MF exposure on neuronal spiking rhythms is mainly to perturb its spike times, while it has relatively smaller effects on spiking frequency, which is in agreement with the results in Fig. 2. Further, ELF MF exposure has almost opposite effects on average SST σmean and spiking frequency fout . Specifically, if ELF MF exposure mainly advances neuronal spike times, the average spiking frequency fout will increase; if ELF MF exposure mainly delays spike times, the average spiking frequency fout will decrease; when MF frequency and intensity are relatively larger, the effects of ELF MF exposure are mainly to advance neuronal spike times (shown in Fig. 4) and increase neuronal spiking frequency (shown in Fig. 6).

4.

Modulation of Burst Spiking by ELF MF Exposure

Figure 7 shows the sample membrane potentials of the modified ML neuron during ELF MF exposure in the case of burst spiking. To stimulate bursting dynamics, we use a sinusoidal input current Is = 60 sin(2πfs t) with low frequency. In Fig. 7, the input current frequency is fs = 12 Hz, and the neuron is triggered to generate a regular bursting at 12 Hz. It should be noted that 12 Hz is the frequency of the burst, while the frequency of spikes within each burst is significantly higher, which could be up to 100 Hz. Further, the intrinsic spiking frequency of the neuron is 36 Hz, which is the average spiking frequency without MF exposure. Compared with tonic spiking, the effects of MF exposure on spiking times are significantly smaller.

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Fig. 7. (Color online) Membrane potentials of a burst spiking neuron during exposure to ELF MFs with different frequency and intensity. Black curves represent the membrane potentials without MF exposure, and bright blue curves represent membrane potentials exposed to ELF sinusoidal MF.

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Fig. 8. (a) and (b) show the average SST σmean of the burst spiking neuron as a function of ELF MF intensity B and frequency f . The intrinsic spiking frequency of neuron is: (a) 36 Hz and (b) 48 Hz. (c) and (d) show the evolutions of the average SST σmean with MF frequency for several MF intensities. The intensity of ELF MF in (c) and (d) is 20mT, 60mT and 90mT.

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Fig. 9. Average spiking frequency fout of the burst spiking neuron as a function of ELF MF intensity B and frequency f . Neuronal intrinsic spiking frequency is: (a) 36 Hz and (b) 48 Hz.

There are almost no obvious differences between the membrane potentials with and without MF exposure by visual inspection. Despite this, we could observe that the MF with higher intensity could induce a bigger perturbation on neuronal spike times by subtle differences. To give a visual depiction of how ELF MF exposure modulates neuronal bursting dynamics, we compute the average SST σmean and average spiking frequency fout in the case of different MF intensity and frequency, which is respectively shown in Figs. 8 and 9. In Fig. 8(a), the input current frequency fs is 12 Hz, neuronal bursting frequency is also 12 Hz and its intrinsic spiking frequency is 36 Hz. In Fig. 8(b), fs is 24 Hz, and neuronal bursting frequency is 24 Hz while its intrinsic spiking frequency is 48 Hz. From Fig. 8, it can be seen that the SST of burst spiking induced by ELF MF exposure can also be a positive value or a negative one, which means MF exposure can delay or advance burst spike times. When MF frequency is at the harmonic frequencies of the bursting frequency, neuronal average SST σmean exhibits a significant fluctuation. It can reach a local maximum or a local minimum, which indicates the perturbations of MF exposure on spike times are maximized at these special MF frequencies. This is different from tonic spiking. As MF intensity increases, the perturbations of MF exposure on spike times also get larger. Further, the average SST σmean induced by the same level of ELF MF exposure is significantly smaller in the case of burst spiking compared with tonic spiking. For instance, the maximal perturbation of spike times for bursting dynamic is about −0.19 ms [Fig. 8(a)], while it could be up to

−98.77 ms for tonic spiking [Fig. 4(a)]. Therefore, the bursting spiking pattern appears less prone to the perturbation of spiking times induced by ELF MF exposure relative to tonic spiking. That is, the bursting spiking neuron is less sensitive to the weak ELF MF exposure. From Fig. 9, it can be found that the average spiking frequency fout remains unchanged in the observed range of f and B. It indicates that MF exposure is more prone to perturb the spike times of burst spiking, while has little or even no effects on neuronal spiking frequency.

5.

Conclusion and Discussion

To explore how ELF MF modulates neuron activity, we present a conductance-based neuron model and introduce ELF MF as a perturbation over membrane voltage. With this model, we systematically investigate the effects of ELF sinusoidal MF exposure on neuronal time coding and rate coding in the case of tonic spiking and burst spiking. For both cases, it is observed that the fieldinduced membrane depolarization induced by ELF MF exposure is able to modulate neuronal spike times. This modulatory effect is not only dependent on neuronal spiking properties (i.e. intrinsic spiking frequency and spiking patterns), but also tightly related to stimulus parameters (i.e. MF frequency and intensity), which highlights the previous proposal that the frequency and intensity of magnetic stimulation are the main determinants of cortical excitability modulation with MFs.27 Compared with the perturbation of spike times, MF exposure has relatively smaller effects on neuronal spiking frequency,

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which indicates ELF MF exposure is more prone to change neuronal spike times and further modulate neuronal spiking rhythms. All of these results theoretically highlight that the ELF MFs with lower intensity than TMS have the capabilities of modulating neuron spiking activity, which is consistent with the prediction from the Izhikevich model.18 Further, previous experimental and modeling studies52,53 have proposed that the small polarizations by weak fields can modulate spike times when neurons are already active. Due to the constant active state of the brain, these modulations could be amplified enough by network activity to affect brain functions.52,54 Our modeling studies confirm the modulatory capability of weak fields on spike times of the active neuron from the single cell level, which could help in understanding and intercepting how weak fields can have an effect on the brain activity. For tonic spiking, the perturbations induced by ELF MF exposure have influence on both of neuronal spike times and spiking frequency, which depends on neuronal intrinsic spiking frequency and MF parameters. The ELF MF exposure effects in terms of perturbation of spike times and spiking frequency for membrane responses is maximized at some special MF frequencies which are harmonics of the neuronal intrinsic spiking frequency. It indicates that the responses of neuron tissue exposed to ELF MF could occur in some narrow specific MF frequency bands, which will not vary as MF intensity changes. This conclusion is consistent with the previous theoretical results of Modolo et al.18 It could be confirmed by identifying and recording well-known EEG rhythms, such as alpha rhythm, during MF exposure to a similar frequency and its harmonics.18,22,23 One of the underlying mechanisms of the above effects may be resonance,55,56 which in physics refers to the tendency of a system to oscillate with greater amplitude at some frequencies than at others. The frequencies at which the response amplitude is a relative maximum are known as the system’s resonant frequencies. As the neuron model in our study is a highly nonlinear dynamical system, there could be multiple distinct resonant frequencies.55 Thus, the fluctuations of average SST and spiking frequency induced by ELF MF exposure could appear abrupt changes when MF frequency is around the harmonics of the neuronal intrinsic spiking frequency, which are resonant frequencies of tonic spiking neuron by

MFs. That is, there is superharmonic resonance57,58 (it is simply the resonance with one of the higher harmonics of a nonlinear oscillation) appearing in the tonic spiking neuron exposed to ELF MFs. Since the intensity of the field-induced membrane depolarization [shown in Eq. (6)] is directly related to both of MF intensity and frequency, the resonance effect could become obvious as MF frequency increases. Thus, the global maximum of the average SST and spiking frequency may not appear at the basic harmonic. Instead, it appears at the second harmonic for tonic spiking neuron. All of these results provide further support for previous findings that the resonance may be one of the neural mechanisms underlying the modulatory effects of the low-intensity or weak MFs on neuronal processing.42,59,60 For burst spiking, ELF MF exposure could only perturb neuronal spike times, while has no effects on spiking frequency in the observed range of MF intensity and frequency. Compared with tonic spiking, the fluctuation of average SST induced by the same level of ELF MF exposure is significantly smaller in the case of burst spiking, which indicates bursting spiking neuron is less sensitive to the weak ELF MF exposure. This suggests that it would be possible to affect brain area featuring mostly tonic spiking neurons with relatively weak ELF MF exposure. Further, the perturbations on spike times by MF are maximized at the harmonics of the bursting frequency, which is different from tonic spiking. This may due to that there are two different frequencies of bursting dynamics. The interactions between slow and fast dynamics could cause burst spiking neuron to exhibit different response property to MF stimulus. All these results demonstrate that neuronal spiking pattern is a critical factor in determining ELF MF exposure effects. Moreover, the bursting dynamics in our study is stimulated by low-frequency sinusoidal input current. In this case, the neuron exhibit p:1(p > 1) phase-locking spiking in the absence of MF exposure, which means neuronal intrinsic spiking frequency always appears at the harmonic of the bursting frequency. Therefore, to further distinguish whether bursting frequency or intrinsic spiking frequency plays key role in determining the effects of MF exposure, we should modify our model to break the proportional relation between these two frequencies. For instance, one can introduce a slow voltageor calcium-dependent process that can modulate fast

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spiking activity.35,61 In this way, our neuron can generate burst spiking to constant input current. Besides, we adopt a new approach to model ELF MF effects in present work, which is different from previous studies.29–31,62–64 Most of the previous researches about modeling electromagnetic field effects used multi-compartment neuron model and regarded electromagnetic stimulation as an additional current input produced by the induced electric field on each compartment.29–31 Unlike them, we model the ELF MF effect as an additive perturbation over neuronal membrane potential by introducing the field-induced membrane depolarization into a single compartment neuron model. Although some previous studies of our group also treated the electromagnetic fields as an additive perturbation to neuronal membrane potential, they did not include this perturbation in voltage-dependent ionic channels,62–64 which ignores the effects of electromagnetic fields on neuronal ionic channels. The main advantage of our model is we can analyze how ELF MF exposure perturbs each action potential at the single cell level. However, there are also some imperfections of our model, which could be improved in several ways to investigate how ELF MF interacts with brain tissue in a more physiologically plausible manner. First, the approach and model in this paper only involve the effects of ELF MF exposure on the response of a patch of axon, and do not take spatial dependency and action potential propagation into consideration. Indeed, magnetic stimulation is a distributed stimulus, thereby it would be needed to check whether our conclusions still hold when take the action potential propagation along axon into consideration. Second, it is useful to explore the influence of ELF MF exposure on the perturbation of neuronal network activity. By changing the size and properties of neuronal network, one can analyze the effects of ELF MF exposure on network activity as measured by its power spectrum18 or spike-timing dependent plasticity (STDP).65,66 One can also investigate how ELF MF exposure affects the synaptic connections of neuronal network, or further explore how ELF MF exposure modulates the level of synchronization among neurons in the network. The above mentioned problems will be implemented in our following work. To summarize, our results demonstrate that specific ELF MF stimulation could modulate neuronal

spiking activity from the biophysical modeling point of view. The principles in our paper are significant for the oncoming experimental studies investigating human brain activity in response to a variety of ELF MF stimulus. The conclusions could be used to refine biophysical models describing the interactions between the external ELF electromagnetic fields and brain tissue. We further provide a plausible mechanism of the interactions between low-intensity ELF MF and neural tissue. These insights into the mechanism of magnetic stimulus may be relevant for the design of multi-intensity magnetic stimulus protocols and may facilitate the construction of magnetic stimulators, which may even aid the interpretation of the results of magnetic stimulations of the central nervous system. Thereby, it will significantly contribute to advance the therapeutic applications and developments of noninvasive neuromodulation techniques based on time-varying MF stimulation, such as TMS, rTMS, PMF, LFMS and so on. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant Nos. 61072012, 61172009 and 61372010, the Young Scientists Fund of the National Natural Science Foundation of China under Grant No. 60901035, and Tianjin Municipal Natural Science Foundation under Grant Nos. 12JCZDJC21100 and 13JCZDJC27900. References

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Dosimetry of extremely-low-frequency magnetic fields.

Effects of extremely low frequency pulsed magnetic fields on diabetic nephropathy in streptozotocin-treated rats.

A dynamic model of the eye nystagmus response to high magnetic fields.

Bio-effects of high magnetic fields: a study using a simple animal model.

Effects of refractive errors on visual evoked magnetic fields.

Perspectives on health effects of electric and magnetic fields.

Effects of extremely low-frequency electric fields at different intensities and exposure durations on mismatch negativity.

Effects of extremely low frequency electromagnetic fields on intracellular calcium transients in cardiomyocytes.

Hazard zoning around electric substations of petrochemical industries by stimulation of extremely low-frequency magnetic fields.

Effects of extremely low-frequency electromagnetic fields (ELF-EMF) exposure on B6C3F1 mice.

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Extremely low frequency magnetic fields inhibit adipogenesis of human mesenchymal stem cells.

Dosimetric extrapolations of extremely-low-frequency electric and magnetic fields across biological systems.

Occupational exposure to extremely low-frequency magnetic fields and brain tumor risks in the INTEROCC study.

Characterization of extremely low frequency magnetic fields from diesel, gasoline and hybrid cars under controlled conditions.

Extremely low-frequency pulsed magnetic fields and multiple sclerosis: effects on neurotransmission alone or also on immunomodulation? Building a working hypothesis.

Magnetic response of zigzag nanoribbons under electric fields.

More on static magnetic fields.

Comments on static magnetic fields.

Influence of magnetic fields on magneto-aerotaxis.

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A comment on a retinal neuron model.

Signal integration on the dendrites of a pyramidal neuron model.

Patient exposure to extremely low-frequency magnetic fields during laparoscopic and robotic surgeries.