Magnetic Stimulation Of Curved Nerves Assaf Rotem, Elisha Moses Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel. Abstract Magnetic stimulation of nerves is attracting increased attention recently, as it has been found to be useful in therapy of neural disorders in humans. In an effort to explain the mechanisms of magnetic stimulation we focus on the dependence of magnetic stimulation on neuronal morphology and in particular on the importance of curvature of axonal bundles. Using the theory of passive membrane dynamics, we predict the Threshold Power (the minimum stimulation power required to initiate an action potential) of specific axonal morphologies. In the experimental section we show that magnetic stimulation of the frog sciatic nerve follows our theoretical predictions. Furthermore, the voltage length constant of the nerve can be measured based on these results alone. Model - The axon as a passive cable Straight axon Non-uniform external field Passive cable equation Curved axon Uniform external field VmVm Membrane potential ε External field x Axon direction Length constant λ Time constant τ A physical equation (A) is used to model the sub-threshold (passive) membrane potential of an axon in the presence of an external electric field (this electric field can result from external electric stimulation or from external magnetic stimulation). B) Any gradients in this electric field will induce an accumulation of ionic charge which will in turn will build up membrane potential. C) Alternatively, if the axon is curved, ionic charge will accumulate at the curve hillock even if the external electric field is uniform. ABC The significance of curvature in magnetic stimulation The figure below demonstrates that the effect of curvature gradients (section C in the bottom left frame) is more than 100 times greater than the effect of field gradients (section B in the bottom left frame) when considering the magnetic stimulation of a single axon with a Magstim Rapid Double 70mm Coil. Modeling the field gradients of the coil described in the left frame A) The maximal absolute value of electric field gradients along a straight axon oriented parallel to the y axis. Each point in the figure indicates the electric field gradient that would have been induced if the axon was positioned at that point. The central circle indicates the boundaries of the dish and the black line illustrates a straight axon above the coil center. B) The maximal absolute value of electric field gradients along an axon which bends from the –y direction to +y direction in half a loop of radius 0.4mm. Each point in the figure indicates the electric field gradient that would have been induced if the axon bend was positioned at that point. The central circle indicates the boundaries of the dish and the black line illustrates an axon bend over the center of the coil (not to scale). All distances are in units of the length constant. Background – Transcranial Magnetic Stimulation Taken from Mark George’s Article on Brain Stimulation in Scientific American 09/03 The induced electric field According to the physical phenomena of magnetic induction, rapidly varying magnetic fields induce electric fields. A designated magnetic coil positioned over a skull can induce electric fields which are strong enough to stimulate cortical neurons. The mechanism of stimulation is described below. Modeling the fields created by a Magstim Rapid Double 70mm Coil. Two magnetic coils (9 windings in each coil) in a figure eight configuration carry opposite electric currents. (right coil - clockwise current). Blue vector map indicates the maximum electric field induced by an abrupt discharge of current through the coils which are positioned 5mm below the plane of the figure. The central circle indicates the boundaries of the dish which was used in the experiment. Model - curving nerves In the figure below we predict the membrane potential induced by magnetic stimulation of axons curved into half a loop and a complete loop. Our model predicts a lower excitation level for complete loops compared to half loops as a result of a canceling effect between the maxima and the minima of the complete loop. A) Zooming in on the center of the stimulating coil, the passive effect of the induced Electric Field (blue vector map) on an axon (thick curve) is shown (color coding represents maximal membrane potential). The axon was curved to form half a loop of radius r t around the origin (see top right frame). B) Same as (A) with an axon curved to form a complete loop of radius r t around the origin C) The maximal membrane potential along the two axons curved at either half (solid line) or complete (dashed line) loop at the time of maximum induced field. All distances are in units of the length constant λ. Notice the canceling effects in the complete loop curvature. Methods Frog sciatic nerves were removed with the Gastrocnemius muscle intact (approved by the Weizmann IACUC). We used the Magstim Rapid with 70mm Double Coil. The variable measured in the experiment was the Threshold Power, i.e. the minimum Magstim power setting required in order to initiate an action potential in the nerve, indicated by muscle twitching*. Results 14 nerves were curved from one half loop (N=1) through single loops up to 3.5 loops (N=7). Threshold Powers of these configuration agree with the proposed model. We can use these results to estimate the length constant, by plotting the absolute difference between Thresholds of consecutive N's. It can be shown that these differences decay exponentially with a decay constant which is proportional to the ratio between the length constant and the radius of curvature. Experimental Setup. The double coil is placed below the plate, to which the nerve and muscle are fixed by glass pins. Inset shows a half loop configuration (N=1) with the nerve wrapped around the center glass pin. The table presents parameters used in the experiment and model (length and time constants are taken from [3]) * To verify that muscle twitching is indeed a good indicator of nerve excitation, we also measured the nerve’s electrical response, and found that the threshold potential as determined visually coincides precisely with the one determined electrically. Electric recording was performed with an extra- cellular pipette filled with frog ringer, using an Axon AM 3000 (no filtering) at X100-X1000 amplification. Exponential fit of Absolute difference between consecutive N’s provides an estimate of λ/r t. Using the known value for r t we can derive the average length constant of the nerve. Summary Stimulating mechanism of TMS on nerves is dominated by nerve morphology (curvature, branching or endings), which induces effects that are more than 100 times greater than the stimulating effect created by the field gradient of a typical TMS coil. The Power Threshold is a simple and reliable method to explore the passive properties of nerves. The measured voltage length constant is consistent with the range of measured values in the literature [2]. In the future, applying the method we have presented here on increasingly complex neural networks (neural cultures, slices and living brains) will help us predict and understand better the interaction of the stimulation with curved neural substrate, and possibly improve our ability to affect activity in the cortex of humans. References (1)Rotem A, Moses E. Magnetic stimulation of curved nerves, IEEE Trans Biomed Eng. Accepted Jun (2)B. Katz. Nerve, muscle, and synapse. New York, McGraw-Hill, (3)P. J. Basser and B. J. Roth. “Stimulation of a myelinated nerve axon by electromagnetic induction,” Med Biol Eng Comput, 29(3):261–268, May A) Threshold Power of 14 nerves was measured for curving of N half loops. B) Average Threshold Power of (A). C) Average Threshold Power of (A) inverted and normalized. The red line is the model prediction.

Background - Motivation SINCE the first Transcranial Magnetic Stimulation (TMS) was conducted by Barker et al. [4] in 1985, it has become a remarkable tool for neuroscience research. As a painless means to probe into human brains, TMS continuously gains diagnostic and therapeutic applications [5] - [7]. Despite the impressive progress, there is a wide agreement that the full clinical potential of TMS, mainly as an alternative to Electroconvulsive Therapy (ECT - electric shocks) is still unrealized [8] - [10], with one of the main drawbacks being the incomplete comprehension of the physical mechanisms that underlie TMS. It is particularly not clear why certain regions in the brain are excitable while others are not, but brain geometry and the anatomy of sulci and gyri seem to play a role in this mechanism [11] - [15]. This fact motivates us to further explore how curvature modulates the effect of magnetic stimulation.

References (1)Rotem A, Moses E. Magnetic stimulation of curved nerves, IEEE Trans Biomed Eng. Accepted Jun (2)B. Katz. Nerve, muscle, and synapse. New York, McGraw-Hill, (3)P. J. Basser and B. J. Roth. “Stimulation of a myelinated nerve axon by electromagnetic induction,” Med Biol Eng Comput, 29(3):261–268, May (4)A. T. Barker, R. Jalinous and I. L. Freeston. “Non-invasive magnetic stimulation of human motor cortex,” Lancet, 1(8437):1106–1107, May Letter. (5)M S George, E M Wassermann and R M Post. “Transcranial magnetic stimulation: a neuropsychiatric tool for the 21st century,” J Neuropsychiatry Clin Neurosci, 8(4):373–382, Fall (6)M Hallett, “Transcranial magnetic stimulation and the human brain,” Nature, 406: , July (7)A. Pascual-Leone et al. Handbook of transcranial magnetic stimulation. London, England, Arnold, (8)S. Pridmore, “Substitution of rapid transcranial magnetic stimulation treatments for electroconvulsive therapy treatments in a course of electroconvulsive therapy,” Depression and Anxiety, 12(3): , Nov (9)A. Post and M. E. Keck, “Transcranial magnetic stimulation as a therapeutic tool in psychiatry: what do we know about the neurobiological mechanisms?,” Journal of Psychiatric Research 35:193–215, (10)M. K.obayashi and A Pascual-Leone, “Transcranial magnetic stimulation in neurology,” Lancet Neurol, 2(3):145-56, Mar (11)J. P. Brasil-Neto, L. G. Cohen, M. Panizza, J. Nilsson, B. J. Roth and M. Hallett, “Optimal focal transcranial magnetic activation of the human motor cortex: effects of coil orientation, shape of the induced current pulse, and stimulus intensity,” J Clin Neurophysiol 9(1):132–136, Jan (12)K. R. Mills, S. J. Boniface and M. Schubert, “Magnetic brain stimulation with a double coil: the importance of coil orientation,” Electroenceph clin Neurophysiol, 85(1): 17–21, Feb 1992 (13)A. Pascual-Leone, L. G. Cohen, J. P. Brasil-Neto and M. Hallett, “Non-invasive differentiation of motor cortical representation of hand muscles by mapping of optimal current directions,” Electroenceph clin Neurophysiol 93(1):42–48, Feb (14)A.G. Guggisberg, P. Dubach, C.W. Hess, C. Wüthrich, and J. Mathis, “Motor evoked potentials from masseter muscle induced by transcranial magnetic stimulation of the pyramidal tract: the importance of coil orientation,” Clin Neurophysiol 112(12):2312–2319, Dec (15)P. Dubach, A. G. Guggisberg, K. M. Rösler, C. W. Hess and J. Mathis, “Significance of coil orientation for motor evoked potentials from nasalis muscle elicited by transcranial magnetic stimulation,” Clin Neurophysiol, 115(4): , Apr 2004.